Repository: claytonjwong/Algorithms-Illuminated Branch: main Commit: 7874d6752bef Files: 174 Total size: 85.3 MB Directory structure: gitextract_gtzmg6or/ ├── .gitignore ├── Algorithms-Illuminated/ │ ├── .gitignore │ ├── Algorithms-Illuminated.iml │ ├── Cargo.toml │ ├── README.md │ ├── src/ │ │ └── main.rs │ └── target/ │ ├── .rustc_info.json │ └── CACHEDIR.TAG ├── README.md ├── _challenge_problems/ │ ├── index_equal_element.py │ └── unimodal_array.py ├── bellman_ford/ │ ├── CMakeLists.txt │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ └── test.txt ├── closest_pair/ │ ├── naive.py │ └── recursive.py ├── dijkstra/ │ ├── CMakeLists.txt │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ ├── problem9.8.txt │ └── problem9.8test.txt ├── floyd_warshall/ │ ├── CMakeLists.txt │ ├── floyd_warshall │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ ├── problem18.8file1.txt │ ├── problem18.8file2.txt │ ├── problem18.8file3.txt │ ├── problem18.8file4.txt │ ├── problem18.8test1.txt │ └── problem18.8test2.txt ├── greedy_scheduling/ │ ├── CMakeLists.txt │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ ├── problem13.4.txt │ ├── problem13.4test1.txt │ └── problem13.4test2.txt ├── huffman/ │ ├── CMakeLists.txt │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ ├── problem14.6.txt │ ├── problem14.6test1.txt │ └── problem14.6test2.txt ├── karatsuba/ │ ├── main.jl │ └── main.py ├── knapsack/ │ ├── CMakeLists.txt │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ ├── problem16.7.txt │ └── problem16.7test.txt ├── kosaraju/ │ ├── CMakeLists.txt │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ ├── problem8.10.txt │ ├── problem8.10test1.txt │ ├── problem8.10test2.txt │ ├── problem8.10test3.txt │ ├── problem8.10test4.txt │ ├── problem8.10test5.txt │ └── section8.6.5page64.txt ├── kotlin/ │ ├── dijkstra.kt │ ├── greedy_scheduling.kt │ ├── huffman.kt │ ├── knapsack.kt │ ├── kosaraju.kt │ ├── kruskal.kt │ ├── merge_sort.kt │ ├── merge_sort_inversions.kt │ ├── prim.kt │ ├── quick_sort.kt │ ├── rselect.kt │ ├── topo_sort.kt │ └── weighted_independent_set.kt ├── kruskal/ │ ├── CMakeLists.txt │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ ├── problem15.9.txt │ └── problem15.9test.txt ├── matrix_multiplication/ │ ├── main.jl │ └── main.py ├── merge_sort/ │ ├── CMakeLists.txt │ ├── merge_sort.cpp │ ├── merge_sort.jl │ ├── merge_sort.js │ ├── merge_sort.kt │ └── merge_sort.py ├── merge_sort_inversions/ │ ├── CMakeLists.txt │ ├── merge_sort_inversions.cpp │ ├── merge_sort_inversions.jl │ ├── merge_sort_inversions.js │ ├── merge_sort_inversions.kt │ ├── merge_sort_inversions.py │ ├── problem3.5.txt │ └── problem3.5test.txt ├── prim/ │ ├── CMakeLists.txt │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ ├── problem15.9.txt │ └── problem15.9test.txt ├── quick_sort/ │ ├── CMakeLists.txt │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ ├── problem5.6.txt │ ├── problem5.6test1.txt │ └── problem5.6test2.txt ├── rec_int_mult/ │ └── main.jl ├── rec_mat_mult/ │ ├── main.py │ └── pyproject.toml ├── rselect/ │ ├── CMakeLists.txt │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ ├── problem6.5test1.txt │ └── problem6.5test2.txt ├── strassen/ │ ├── main.py │ ├── pyproject.toml │ └── variants.py ├── topo_sort/ │ ├── CMakeLists.txt │ ├── main.cpp │ ├── main.js │ ├── main.kt │ └── main.py ├── traveling_salesman/ │ ├── README.md │ ├── main.cpp │ ├── main.js │ ├── main.kt │ ├── main.py │ ├── package.json │ ├── quiz19.2.txt │ └── quiz20.7.txt ├── traveling_salesman_nearest_neighbor/ │ ├── README.md │ ├── input.txt │ └── main.py └── weighted_independent_set/ ├── CMakeLists.txt ├── main.cpp ├── main.js ├── main.kt ├── main.py ├── package.json ├── problem16.6.txt └── problem16.6test.txt ================================================ FILE CONTENTS ================================================ ================================================ FILE: .gitignore ================================================ # IDE .idea .vscode .python-version # Prerequisites *.d # Compiled Object files *.slo *.lo *.o *.obj # Precompiled Headers *.gch *.pch # Compiled Dynamic libraries *.so *.dylib *.dll # Fortran module files *.mod *.smod # Compiled Static libraries *.lai *.la *.a *.lib # Executables *.exe *.out *.app *.jar cmake-build-debug # Dependencies node_modules # rust target/debug target/package # C++ main main.dSYM ================================================ FILE: Algorithms-Illuminated/.gitignore ================================================ # IDE .idea .vscode .python-version # Prerequisites *.d # Compiled Object files *.slo *.lo *.o *.obj # Precompiled Headers *.gch *.pch # Compiled Dynamic libraries *.so *.dylib *.dll # Fortran module files *.mod *.smod # Compiled Static libraries *.lai *.la *.a *.lib # Executables *.exe *.out *.app *.jar cmake-build-debug # Dependencies node_modules # rust target/debug target/package ================================================ FILE: Algorithms-Illuminated/Algorithms-Illuminated.iml ================================================ ================================================ FILE: Algorithms-Illuminated/Cargo.toml ================================================ [package] name = "Algorithms-Illuminated" version = "0.1.1" edition = "2021" description = "https://github.com/claytonjwong/Algorithms-Illuminated" license = "MIT" # See more keys and their definitions at https://doc.rust-lang.org/cargo/reference/manifest.html [dependencies] ================================================ FILE: Algorithms-Illuminated/README.md ================================================ # Algorithms Illuminated * [algorithmsilluminated.org](https://www.algorithmsilluminated.org/) Coincidentally, my [algorithm learning journey](https://github.com/claytonjwong/Algorithms) which began in 2017 has occurred in parallel with the publication of Tim Roughgarden's (TR) 4-book series about algorithms and data structures. Over these years, I've purchased, studied, and provided feedback on TR's books. I was totally stoked when TR sent me a free copy of his 4th book for review before publication in 2020! I'm amazed by what can be done in near-linear time (ie. the amount of time to perform an algorithm is on the order of time to simply read the input), and it's awesome we can leverage these "for-free primitives" based upon computationally tractable problems as "building blocks" towards more complex solutions to computationally intractable (NP-Hard) problems via selective compromise on generality, correctness, and speed (ie. pick 2 of 3). [`💡` Can we do better?](https://en.wikipedia.org/wiki/Millennium_Prize_Problems#P_versus_NP)

--- # Part 1: The Basics
--- ### Merge Sort
📚 Lectures
* [MergeSort: Motivation and Example](https://www.youtube.com/watch?v=kiyRJ7GVWro&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=4) (Section 1.4, part 1) * [MergeSort: Pseudocode](https://www.youtube.com/watch?v=rBd5w0rQaFo&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=5) (Section 1.4, part 2) * [MergeSort: Analysis](https://www.youtube.com/watch?v=8ArtRiTkYEw&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=6) (Section 1.5)
🎯 Solutions
*Kotlin* ```java fun sort(A: IntArray): IntArray { fun merge(A: IntArray, B: IntArray): IntArray { var C = mutableListOf() var i = 0 var j = 0 while (i < A.size && j < B.size) if (A[i] < B[j]) C.add(A[i++]) else C.add(B[j++]) A.slice(i..A.lastIndex).forEach { C.add(it) } B.slice(j..B.lastIndex).forEach { C.add(it) } return C.toIntArray() } fun go(A: IntArray): IntArray { var N = A.size if (N < 2) return A var half = Math.floor(N / 2.0).toInt() var first = go(A.slice(0 until half).toIntArray()) var second = go(A.slice(half until N).toIntArray()) return merge(first, second) } return go(A) } fun main(args: Array) { sort(intArrayOf(5,3,8,9,1,7,0,2,6,4)).forEach { print("$it ") } // 0 1 2 3 4 5 6 7 8 9 println() } ``` *Javascript* ```javascript let sort = A => { let go = A => { let N = A.length; if (N < 2) return A; let half = Math.floor(N / 2); let first = go([...A.slice(0, half)]), second = go([...A.slice(half, N)]); return merge(first, second); }; let merge = (A, B, C = []) => { let M = A.length, N = B.length; let i = 0, j = 0; while (i < M && j < N) C.push(A[i] < B[j] ? A[i++] : B[j++]); C.push(...A.slice(i, M)); C.push(...B.slice(j, N)); return C; }; return go(A); }; console.log(sort([5,3,8,9,1,7,0,2,6,4])); // (10) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] ``` *Python3* ```python from math import floor def sort(A): def go(A): N = len(A) if N < 2: return A half = floor(N / 2) first = go(A[:half]) second = go(A[half:]) return merge(first, second) def merge(A, B): C = [] i = 0 j = 0 while i < len(A) and j < len(B): if A[i] < B[j]: C.append(A[i]); i += 1 else: C.append(B[j]); j += 1 C.extend(A[i:]) C.extend(B[j:]) return C return go(A) print(sort([5,3,8,9,1,7,0,2,6,4])) # [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] ``` *C++* ```cpp #include #include using namespace std; class Solution { public: using VI = vector; VI mergesort(VI& A) { return go(move(A)); } private: VI go(VI&& A) { auto N = A.size(); if( N < 2 ) return A; auto half = A.begin() + (N / 2); auto first = go({ A.begin(), half }), second = go({ half, A.end() }); return merge(first, second); } VI merge(VI& A, VI& B, VI C = {}) { auto i{ 0 }, j{ 0 }; while (i < A.size() && j < B.size()) C.push_back(A[i] < B[j] ? A[i++] : B[j++]); C.insert(C.end(), A.begin() + i, A.end()); C.insert(C.end(), B.begin() + j, B.end()); return C; } }; int main() { Solution::VI A{ 3,5,7,1,3,9,2,0 }; auto ans = Solution().mergesort(A); copy(ans.begin(), ans.end(), ostream_iterator(cout, " ")), cout << endl; // 0 1 2 3 4 5 6 7 8 9 return 0; } ```
--- ### Counting Inversions
📚 Lectures
* [The Divide-and-Conquer Paradigm](https://www.youtube.com/watch?v=7_AJfusC6UQ&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=13) (Section 3.1; part 1 of Section 3.2) * [Counting Inversions in O(n log n)](https://www.youtube.com/watch?v=I6ygiW8xN7Y&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=14) Time (Section 3.2, part 2)
🎯 Solutions
*Kotlin* ```java import java.io.File fun sort(A: IntArray): Pair { fun merge(A: IntArray, B: IntArray): Pair { var C = mutableListOf() var inv: Long = 0 var i = 0 var j = 0 while (i < A.size && j < B.size) if (A[i] < B[j]) { C.add(A[i++]) } else { inv += A.size - i // ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.add(B[j++]) } A.slice(i..A.lastIndex).forEach { C.add(it) } B.slice(j..B.lastIndex).forEach { C.add(it) } return Pair(C.toIntArray(), inv) } fun go(A: IntArray): Pair { var N = A.size if (N < 2) return Pair(A, 0) var half = Math.floor(N / 2.0).toInt() var (first, inv1) = go(A.slice(0 until half).toIntArray()) var (second, inv2) = go(A.slice(half until N).toIntArray()) var (third, inv3) = merge(first, second) return Pair(third, inv1 + inv2 + inv3) } return go(A) } fun run(filename: String): Long { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } var (_, inv) = sort(A.toIntArray()) return inv } fun main() { println("problem3.5test.txt: " + run("problem3.5test.txt")) // problem3.5test.txt: 28 println("problem3.5.txt: " + run("problem3.5.txt")) // problem3.5.txt: 2407905288 } ``` *Javascript* ```javascript let sort = A => { let go = A => { let N = A.length; if (N < 2) return [A, 0]; let half = Math.floor(N / 2); let [first, inv1] = go([...A.slice(0, half)]), [second, inv2] = go([...A.slice(half, N)]), [third, inv3] = merge(first, second); return [third, inv1 + inv2 + inv3]; }; let merge = (A, B, C = [], inv = 0) => { let M = A.length, N = B.length; let i = 0, j = 0; while (i < M && j < N) if (A[i] < B[j]) C.push(A[i++]); else inv += M - i, // ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.push(B[j++]); C.push(...A.slice(i, M)); C.push(...B.slice(j, N)); return [C, inv]; }; return go(A); }; let run = filename => { let A = []; require('fs').readFileSync(filename, 'utf-8').split(/\r?\n/).forEach(line => A.push(Number(line))); let [_, inv] = sort(A); return inv; } console.log(`problem3.5test.txt: ${run('problem3.5test.txt')}`); // problem3.5test.txt: 28 console.log(`problem3.5.txt: ${run('problem3.5.txt')}`); // problem3.5.txt: 2407905288 ``` *Python3* ```python from math import floor def sort(A): def go(A): N = len(A) if N < 2: return [A, 0] half = floor(N / 2) first, inv1 = go(A[:half]) second, inv2 = go(A[half:]) third, inv3 = merge(first, second) return [third, inv1 + inv2 + inv3] def merge(A, B, inv = 0): C = [] i = 0 j = 0 while i < len(A) and j < len(B): if A[i] < B[j]: C.append(A[i]); i += 1 else: inv += len(A) - i # ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.append(B[j]); j += 1 C.extend(A[i:]) C.extend(B[j:]) return [C, inv] return go(A) def run(filename): A = [] with open(filename) as fin: while True: line = fin.readline() if not line: break A.append(int(line)) _, inv = sort(A) return inv print(f"problem3.5test.txt: {run('problem3.5test.txt')}") # problem3.5test.txt: 28 print(f"problem3.5.txt: {run('problem3.5.txt')}") # problem3.5.txt: 2407905288 ``` *C++* ```cpp #include #include #include using namespace std; class Solution { public: using VL = vector; using Pair = pair; using fun = function; Pair merge(VL& A, VL& B, VL C = {}, long inv = 0) { auto i = 0, j = 0; while (i < A.size() && j < B.size()) { if (A[i] < B[j]) { C.push_back(A[i++]); } else { inv += A.size() - i; // ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.push_back(B[j++]); } } C.insert(C.end(), A.begin() + i, A.end()); C.insert(C.end(), B.begin() + j, B.end()); return { C, inv }; } Pair inversions(VL& A) { fun go = [&](VL&& A) -> Pair { int N = A.size(); if (N < 2) return { A, 0 }; int half = N / 2; auto [first, inv1] = go({ A.begin(), A.begin() + half }); auto [second, inv2] = go({ A.begin() + half, A.end() }); auto [third, inv3] = merge(first, second); return { third, inv1 + inv2 + inv3 }; }; return go(move(A)); } }; long run(string filename) { Solution solution; Solution::VL A; fstream fin{ filename }; for (string line; fin >> line; A.push_back(stol(line))); auto [_, inv] = solution.inversions(A); return inv; } int main() { cout << "problem3.5test.txt: " << run("problem3.5test.txt") << endl // problem3.5test.txt: 28 << "problem3.5.txt: " << run("problem3.5.txt") << endl; // problem3.5.txt: 2407905288 return 0; } ```
--- ### Quick Sort
📚 Lectures
* [QuickSort: Overview](https://www.youtube.com/watch?v=ETo1cpLN7kk&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=24) (Section 5.1) * [Partitioning Around a Pivot Element](https://www.youtube.com/watch?v=LYzdRN5iFdA&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=25) (Section 5.2) * [Choosing a Good Pivot](https://www.youtube.com/watch?v=kqO46FOUTbI&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=26) (Sections 5.3 and 5.4) * [QuickSort Analysis (Part 1)](https://www.youtube.com/watch?v=sToWtKSYlMw&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=27) (Section 5.5, part 1) * [QuickSort Analysis (Part 2)](https://www.youtube.com/watch?v=4t_Y-aGLkok&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=28) (Section 5.5, part 2) * [QuickSort Analysis (Part 3)](https://www.youtube.com/watch?v=IBTvneWhFsA&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=29) (Section 5.5, part 3) * [Sorting Requires Omega(n log n) Comparisons](https://www.youtube.com/watch?v=aFveIyII5D4&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=30) (Section 5.6) * [Proofs by Induction and the Correctness of QuickSort](https://www.youtube.com/watch?v=Colb_4jAy8A&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=36) (Appendix A) * [Quick Review of Discrete Probability](https://www.youtube.com/watch?v=uLeIMwMHX5U&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=37) (Appendix B)
🎯 Solutions
*Kotlin* ```java import java.io.File typealias PivotFunc = (A: MutableList, L: Int, R: Int) -> (Int) var pivotLeft: PivotFunc = { _: MutableList, L: Int, _: Int -> L } var pivotRight: PivotFunc = { _: MutableList, _: Int, R: Int -> R } fun _pivotMedian(A: MutableList, L: Int, R: Int): Int { var M = L + (R - L) / 2 var cand = intArrayOf(A[L], A[M], A[R]) cand.sort() var target = cand[1] if (target == A[L]) return L if (target == A[M]) return M if (target == A[R]) return R return -1 } var pivotMedian: PivotFunc = { A: MutableList, L: Int, R: Int -> _pivotMedian(A, L, R) } fun partition(A: MutableList, L: Int, R: Int, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { var i = L + 1 var j = L + 1 var k = choosePivot(A, L, R) A[k] = A[L].also { A[L] = A[k] } // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] A[i] = A[j].also { A[j] = A[i] } ++i } ++j } A[L] = A[i - 1].also { A[i - 1] = A[L] } // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 } fun quicksort(A: MutableList, L: Int, R: Int, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { if (R <= L) return 0 var k = partition(A, L, R, choosePivot) return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot) } fun run(filename: String, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } return quicksort(A, 0, A.size - 1, choosePivot) } fun main() { var filename = "problem5.6.txt" println(" left: ${run(filename, pivotLeft)}") // left: 162085 println(" right: ${run(filename, pivotRight)}") // right: 164123 println("median: ${run(filename, pivotMedian)}") // median: 138382 } ``` *Javascript* ```javascript let pivotLeft = (A, L, R) => L; let pivotRight = (A, L, R) => R; let pivotMedian = (A, L, R) => { let M = L + Math.floor((R - L) / 2); let cand = [A[L], A[M], A[R]].sort((a, b) => a - b), target = cand[1]; if (target == A[L]) return L; if (target == A[M]) return M; if (target == A[R]) return R; }; let partition = (A, L, R, choosePivot) => { let i = L + 1, j = L + 1, k = choosePivot(A, L, R); [A[L], A[k]] = [A[k], A[L]]; // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] [A[i], A[j]] = [A[j], A[i]]; ++i; } ++j; } [A[L], A[i - 1]] = [A[i - 1], A[L]]; // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; }; let quicksort = (A, L, R, choosePivot) => { if (R <= L) return 0; let k = partition(A, L, R, choosePivot); return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot); }; let run = (filename, choosePivot) => { let A = []; let LineByLine = require("n-readlines"); let input = new LineByLine(filename); for (let line; line = input.next(); A.push(Number(line))); return quicksort(A, 0, A.length - 1, choosePivot); } let filename = 'problem5.6.txt'; console.log(` left: ${run(filename, pivotLeft)}`); // left: 162085 console.log(` right: ${run(filename, pivotRight)}`); // right: 164123 console.log(`median: ${run(filename, pivotMedian)}`); // median: 138382 ``` *Python3* ```python def pivotLeft(A, L, R): return L def pivotRight(A, L, R): return R def pivotMedian(A, L, R): M = L + (R - L) // 2 cand = sorted([A[L], A[M], A[R]]) target = cand[1] if target == A[L]: return L if target == A[M]: return M if target == A[R]: return R def partition(A, L, R, choosePivot): i = L + 1 j = L + 1 k = choosePivot(A, L, R) A[L], A[k] = A[k], A[L] # swap pivot A[k] with first element of subarray A[L] while j <= R: if A[j] < A[L]: # maintain loop invariant A[i] < pivot < A[j] A[i], A[j] = A[j], A[i] i += 1 j += 1 A[L], A[i - 1] = A[i - 1], A[L] # swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 def quicksort(A, L, R, choosePivot): if R <= L: return 0 k = partition(A, L, R, choosePivot) return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot) def run(filename, choosePivot): A = [] with open(filename) as fin: while True: line = fin.readline() if not line: break A.append(int(line)) return quicksort(A, 0, len(A) - 1, choosePivot) filename = 'problem5.6.txt' print(f' left: {run(filename, pivotLeft)}') # left: 162085 print(f' right: {run(filename, pivotRight)}') # right: 164123 print(f'median: {run(filename, pivotMedian)}') # median: 138382 ``` *C++* ```cpp #include #include #include using namespace std; using VI = vector; using fun = function; fun pivotLeft = [](VI& A, int L, int R) { return L; }; fun pivotRight = [](VI& A, int L, int R) { return R; }; fun pivotMedian = [](VI& A, int L, int R) { auto M = L + (R - L) / 2; VI cand{ A[L], A[M], A[R] }; sort(cand.begin(), cand.end()); auto target = cand[1]; if (target == A[L]) return L; if (target == A[M]) return M; if (target == A[R]) return R; }; int partition(VI& A, int L, int R, fun choosePivot) { auto i = L + 1, j = L + 1, k = choosePivot(A, L, R); swap(A[L], A[k]); // swap pivot A[k] with first element of the subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] swap(A[i], A[j]); ++i; } ++j; } swap(A[L], A[i - 1]); // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; } int quicksort(VI& A, int L, int R, fun choosePivot) { if (R <= L) return 0; auto k = partition(A, L, R, choosePivot); return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot); } int run(string& filename, fun choosePivot) { VI A; fstream fin{ filename }; for (string line; fin >> line; A.push_back(stoi(line))); int N = A.size(); return quicksort(A, 0, N - 1, choosePivot); } int main() { string filename{ "problem5.6.txt" }; cout << " left: " << run(filename, pivotLeft) << endl // left: 162085 << " right: " << run(filename, pivotRight) << endl // right: 164123 << "median: " << run(filename, pivotMedian) << endl; // median: 138382 return 0; } ```
--- ### Randomized Linear-Time Selection
📚 Lectures
* [Randomized Linear-Time Selection](https://www.youtube.com/watch?v=nFw6x7DoYbs&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=31) (Section 6.1) * [Randomized Linear-Time Selection (Analysis)](https://www.youtube.com/watch?v=rX2u2CnpveQ&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=32) (Section 6.2)
🎯 Solutions
*Kotlin* ```java import java.io.File import kotlin.random.Random fun partition(A: MutableList, L: Int, R: Int): Int { var i = L + 1 var j = L + 1 var k = Random.nextInt(L, R + 1) // +1 for L..R inclusive A[L] = A[k].also { A[k] = A[L] } // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] A[i] = A[j].also { A[j] = A[i] } ++i } ++j } A[L] = A[i - 1].also { A[i - 1] = A[L] } // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 } fun rselect(A: MutableList, i: Int, L_: Int, R_: Int): Int { var L = L_ var R = R_ var k = partition(A, L, R) if (i == k) return A[k] // 🎯 lucky guess if (i < k) R = k - 1 else L = k + 1 return rselect(A, i, L, R) } fun run(filename: String, i: Int): Int { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } var N = A.size return rselect(A, i - 1, 0 , N - 1) // -1 for 0-based indexing } fun main() { println("problem6.5test1.txt: " + run("problem6.5test1.txt", 5)) // problem6.5test1.txt: 5469 println("problem6.5test2.txt: " + run("problem6.5test2.txt", 50)) // problem6.5test2.txt: 4715 } ``` *Javascript* ```javascript let random = (L, R) => Math.floor(Math.random() * (R + 1 - L) + L); // +1 for L..R inclusive let partition = (A, L, R) => { let i = L + 1, j = L + 1, k = random(L, R); [A[L], A[k]] = [A[k], A[L]]; // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] [A[i], A[j]] = [A[j], A[i]]; ++i; } ++j; } [A[L], A[i - 1]] = [A[i - 1], A[L]]; // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; }; let rselect = (A, i, L, R) => { let k = partition(A, L, R); if (i == k) return A[k]; // 🎯 lucky guess if (i < k) R = k - 1; else L = k + 1; return rselect(A, i, L, R); } let run = (filename, i) => { let A = []; let LineByLine = require("n-readlines"); let input = new LineByLine(filename); for (let line; line = input.next(); A.push(Number(line))); let N = A.length; return rselect(A, i - 1, 0, N - 1); // -1 for 0-based indexing }; console.log(`problem6.5test1.txt: ${run('problem6.5test1.txt', 5)}`); // problem6.5test1.txt: 5469 console.log(`problem6.5test2.txt: ${run('problem6.5test2.txt', 50)}`); // problem6.5test2.txt: 4715 ``` *Python3* ```python from random import uniform from math import floor def partition(A, L, R): i = L + 1 j = L + 1 k = floor(uniform(L, R)) A[L], A[k] = A[k], A[L] # swap pivot A[k] with first element of subarray A[L] while j <= R: if A[j] < A[L]: # maintain loop invariant A[i] < pivot < A[j] A[i], A[j] = A[j], A[i] i += 1 j += 1 A[L], A[i - 1] = A[i - 1], A[L] # swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 def rselect(A, i, L, R): k = partition(A, L, R) if i == k: return A[k] # 🎯 lucky guess if i < k: R = k - 1 else: L = k + 1 return rselect(A, i, L, R) def run(filename, i): A = [] with open(filename) as fin: while True: line = fin.readline() if not line: break A.append(int(line)) N = len(A) return rselect(A, i - 1, 0, N - 1) # -1 for 0-based indexing print('problem6.5test1.txt:', run('problem6.5test1.txt', 5)) # problem6.5test1.txt: 5469 print('problem6.5test2.txt:', run('problem6.5test2.txt', 50)) # problem6.5test2.txt: 4715 ``` *C++* ```cpp #include #include #include #include using namespace std; using VI = vector; int random(int L, int R) { random_device rd; mt19937 gen{ rd() }; uniform_int_distribution dist(L, R); return dist(gen); } int partition(VI& A, int L, int R) { auto i = L + 1, j = L + 1, k = random(L, R); swap(A[L], A[k]); // swap pivot A[k] with first element of the subarray A[L] while (j <= R) { if (A[j] < A[L]) // maintain loop invariant A[i] < pivot < A[j] swap(A[i++], A[j]); ++j; } swap(A[L], A[i - 1]); // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; } int rselect(VI& A, int i, int L, int R) { auto k = partition(A, L, R); if (i == k) return A[k]; // 🎯 lucky guess if (i < k) R = k - 1; else L = k + 1; return rselect(A, i, L, R); } int run(string filename, int i, VI A = {}) { fstream fin{ filename }; for (string line; fin >> line; A.push_back(stoi(line))); int N = A.size(); return rselect(A, i - 1, 0, N - 1); // -1 for 0-based indexing } int main() { cout << "problem6.5test1.txt: " << run("problem6.5test1.txt", 5) << endl; // problem6.5test1.txt: 5469 cout << "problem6.5test2.txt: " << run("problem6.5test2.txt", 50) << endl; // problem6.5test2.txt: 4715 return 0; } ```
--- # Part 2: Graph Algorithms and Data Structures
--- ### Topological Sort
📚 Lectures
* [Graphs: The Basics (from 2:06 to 6:39)](https://www.youtube.com/watch?v=4Ih3UhVuEtw&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=1) (Sections 7.1 and 7.2) * [Graph Representations](https://www.youtube.com/watch?v=b-Mfu8dPv9U&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=2) (Sections 7.3 and 7.4) * [Graph Search Overview](https://www.youtube.com/watch?v=SW6jwg7WS48&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=3) (Section 8.1) * [Breadth-First Search](https://www.youtube.com/watch?v=73qCvXsYkfk&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=4) (Section 8.2, Part 1) * [Depth-First Search](https://www.youtube.com/watch?v=73qCvXsYkfk&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=4) (Section 8.4) * [Topological Sort](https://www.youtube.com/watch?v=ozso3xxkVGU&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=8) (Section 8.5)
🎯 Solutions
*Kotlin* ```java import java.util.Queue import java.util.LinkedList class Solution(val adj: MutableMap>) { var N: Int var color: Int var m = mutableMapOf() var seen = mutableSetOf() init { N = adj.size color = 0 } fun init(start: Int) { color = start m.clear() seen.clear() } fun topoSortBFS(): String { init(1) // 👉 color forward from 1..N bfs() return toString() } fun topoSortDFS(): String { init(N) // 👈 color reverse from N..1 (as the recursive stack unwinds) adj.forEach{ (u, _) -> dfs(u) } return toString() } fun bfs() { var degree = mutableMapOf() adj.forEach{ (_, neighbors) -> neighbors.forEach{ v -> degree[v] = 1 + degree.getOrDefault(v, 0) } } var q: Queue = LinkedList(adj.map{ (u, _) -> u }.filter{ !degree.contains(it) }) while (0 < q.size) { var u = q.poll() m[u] = color++ adj[u]!!.forEach{ v -> degree[v] = degree[v]!!.minus(1) if (degree[v] == 0 && !seen.contains(v)) { q.add(v); seen.add(v) } } } } fun dfs(u: Char) { if (seen.contains(u)) return seen.add(u) adj[u]!!.forEach{ v -> dfs(v) } m[u] = color-- } override fun toString(): String { var s = mutableListOf() adj.forEach{ (u, _) -> s.add("$u: ${m[u]}") } return s.joinToString("\n") } } fun main() { var adj = mutableMapOf>( 's' to listOf('v', 'w'), 'v' to listOf('t'), 'w' to listOf('t'), 't' to listOf() ) var solution = Solution(adj) println("BFS:\n${solution.topoSortBFS()}\n\nDFS:\n${solution.topoSortDFS()}") // BFS: // s: 1 // v: 2 // w: 3 // t: 4 // DFS: // s: 1 // v: 3 // w: 2 // t: 4 } ``` *Javascript* ```javascript class Solution { constructor(adj) { this.adj = adj; this.N = this.adj.size; } init(start) { this.color = start; this.seen = new Set(); this.m = new Map(); } topo_sort_bfs() { this.init(1); // 👉 color forward from 1..N this.bfs(); return this.to_string(); } topo_sort_dfs() { this.init(this.N); // 👈 color reverse from N..1 (as the recursive stack unwinds) for (let [u, _] of [...this.adj]) this.dfs(u); return this.to_string(); } bfs() { let degree = new Map(); for (let [u, _] of [...this.adj]) { degree.set(u, (degree.get(u) || 0)); for (let v of this.adj.get(u)) degree.set(v, 1 + (degree.get(v) || 0)); } let q = [...this.adj].map(([u, _]) => u).filter(u => !degree.get(u)); let seen = new Set(q); while (q.length) { let u = q.shift(); this.m.set(u, this.color++); for (let v of this.adj.get(u)) { degree.set(v, -1 + degree.get(v)); if (!degree.get(v) && !seen.has(v)) q.push(v), seen.add(v); } } } dfs(u) { if (this.seen.has(u)) return; this.seen.add(u); for (let v of this.adj.get(u)) if (!this.seen.has(v)) this.dfs(v); this.m.set(u, this.color--); } to_string() { let s = []; for (let [u, color] of [...this.m]) s.push(`${u}: ${color}`); return s.join('\n'); } } let adj = new Map(); adj.set('s', ['v', 'w']); adj.set('v', ['t']); adj.set('w', ['t']); adj.set('t', []); let solution = new Solution(adj); console.log(`BFS:\n${solution.topo_sort_bfs()}\n\nDFS:\n${solution.topo_sort_dfs()}`); // BFS: // s: 1 // v: 2 // w: 3 // t: 4 // DFS: // t: 4 // v: 3 // w: 2 // s: 1 ``` *Python3* ```python from collections import deque class Solution: def __init__(self, adj): self.adj = adj self.N = len(adj) self.seen = set() self.m = {} def init(self, start): self.color = start self.seen.clear() self.m.clear() def topo_sort_bfs(self): self.init(1) # 👉 color forward from 1..N self.bfs() return self.to_string() def topo_sort_dfs(self): self.init(self.N) # 👈 color reverse from N..1 (as the recursive stack unwinds) for u, _ in self.adj.items(): self.dfs(u) return self.to_string() def bfs(self): degree = {} for _, neighbors in self.adj.items(): for v in neighbors: degree[v] = 1 + (degree[v] if v in degree else 0) q = deque(u for u, _ in self.adj.items() if u not in degree) self.seen.update(*q) while q: u = q.popleft() self.m[u] = self.color; self.color += 1 for v in adj[u]: degree[v] -= 1 if not degree[v] and v not in self.seen: q.append(v); self.seen.add(v) def dfs(self, u): if u in self.seen: return self.seen.add(u) for v in adj[u]: self.dfs(v) self.m[u] = self.color; self.color -= 1 def to_string(self): s = [] for u, color in self.m.items(): s.append(f'{u}: {color}') return '\n'.join(s) # # graph from Quiz 8.3 on page 45 of Algorithms Illuminated: Part 2 # adj = { 's': ['v', 'w'], 'v': ['t'], 'w': ['t'], 't': [] } solution = Solution(adj) print(f'BFS:\n{solution.topo_sort_bfs()}\n\nDFS:\n{solution.topo_sort_dfs()}') # BFS: # s: 1 # v: 2 # w: 3 # t: 4 # DFS: # t: 4 # v: 3 # w: 2 # s: 1 ``` *C++* ```cpp #include #include #include #include #include #include using namespace std; using VI = vector; using AdjList = unordered_map; using Set = unordered_set; using Map = unordered_map; using Queue = queue; using fun = function; class Solution { private: AdjList adj; const int N; Map m; Set seen; int color; public: Solution(AdjList& adj) : adj{ adj }, N{ int(adj.size()) } { } void init(int start) { m.clear(); seen.clear(); color = start; } string topo_sort_bfs() { init(1); // 👉 color forward from 1..N bfs(); return to_string(); } string topo_sort_dfs() { init(N); // 👈 color reverse from N..1 (as the recursive stack unwinds) for (auto [u, _]: adj) dfs(u); return to_string(); } void bfs() { Map degree; for (auto [_, neighbors]: adj) for (auto v: neighbors) ++degree[v]; Queue q; for (auto [u, _]: adj) if (!degree[u] && seen.insert(u).second) q.push(u); while (q.size()) { auto u = q.front(); q.pop(); m[u] = color++; for (auto v: adj[u]) if (!--degree[v] && seen.insert(v).second) q.push(v); } } void dfs(char start) { fun go = [&](auto u) { if (!seen.insert(u).second) return; for (auto v: adj[u]) go(v); m[u] = color--; }; go(start); } string to_string() { ostringstream os; for (auto [u, color]: m) os << u << ": " << color << endl; return os.str(); } }; int main() { // // graph from Quiz 8.3 on page 45 of Algorithms Illuminated: Part 2 // AdjList adj{ { 's', { 'v', 'w' } }, { 'v', { 't' } }, { 'w', { 't' } }, { 't', {} } }; Solution solution{ adj }; cout << "BFS:" << endl << solution.topo_sort_bfs() << endl << "DFS:" << endl << solution.topo_sort_dfs() << endl; // BFS: // t: 4 // w: 3 // v: 2 // s: 1 // // DFS: // s: 1 // w: 2 // v: 3 // t: 4 return 0; } ```
--- ### Kosaraju
📚 Lectures
* [Computing Strongly Connected Components (Part 1) (Section 8.6, Part 1)](https://www.youtube.com/watch?v=O98hLTYVN3c&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=9) * [Computing Strongly Connected Components (Part 2) (Section 8.6, Part 2)](https://www.youtube.com/watch?v=gbs3UNRJIYk&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=10) * [The Structure of the Web (Section 8.7)](https://www.youtube.com/watch?v=7YodysGShlo&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=11)
🎯 Solutions
*Kotlin* ```java import java.util.Stack import java.io.File class RecursiveSolution(var adj: MutableMap>, var rev: MutableMap>) { fun topo_sort(): MutableList { var list = mutableListOf() var seen = mutableSetOf() fun go(u: Int) { if (seen.contains(u)) return seen.add(u) for (v in rev[u]!!) go(v) list.add(0, u) } for ((u, _) in rev) go(u) return list } fun kosaraju(): MutableList> { var lists = mutableListOf>() var seen = mutableSetOf() fun go(u: Int, list: MutableList) { if (seen.contains(u)) return list.add(u); seen.add(u) for (v in adj[u]!!) go(v, list) } for (u in topo_sort()) { if (seen.contains(u)) continue var list = mutableListOf() go(u, list) lists.add(list.toList()) } return lists } } class IterativeSolution(var adj: MutableMap>, var rev: MutableMap>) { fun topo_sort(): MutableList { var list = mutableListOf() var seen = mutableSetOf() for ((u, _) in rev) { if (seen.contains(u)) continue var stack = Stack() stack.push(u); seen.add(u) while (!stack.empty()) { var u = stack.last() for (v in rev[u]!!) { if (!seen.contains(v)) { stack.push(v); seen.add(v) } } if (u == stack.last()) list.add(0, stack.pop()) } } return list } fun kosaraju(): MutableList> { var lists = mutableListOf>() var seen = mutableSetOf() for (u in topo_sort()) { if (seen.contains(u)) continue var list = mutableListOf() var stack = Stack() stack.push(u); seen.add(u) while (!stack.empty()) { var u = stack.last() for (v in adj[u]!!) { if (!seen.contains(v)) { stack.push(v); seen.add(v) } } if (u == stack.last()) list.add(stack.pop()) } lists.add(list.toList()) } return lists } } fun run(filename: String) { var adj = mutableMapOf>() var rev = mutableMapOf>() File(filename).forEachLine { var (u, v) = it.trim().split(" ").map{ it.toInt() } if (!adj.contains(u)) adj[u] = mutableListOf(); if (!adj.contains(v)) adj[v] = mutableListOf() if (!rev.contains(u)) rev[u] = mutableListOf(); if (!rev.contains(v)) rev[v] = mutableListOf() adj[u]!!.add(v) rev[v]!!.add(u) } // var solution = RecursiveSolution(adj, rev) var solution = IterativeSolution(adj, rev) var A = solution.kosaraju() A.sortWith(Comparator{ a: List, b: List -> b.size - a.size }) println(filename + ": " + A.map{ it.size }.slice(0 until Math.min(A.size, 5)).joinToString(" ")) } fun main() { run("section8.6.5page64.txt"); // Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run("problem8.10test1.txt"); // Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run("problem8.10test2.txt"); // Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run("problem8.10test3.txt"); // Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run("problem8.10test4.txt"); // Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run("problem8.10test5.txt"); // Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run("problem8.10.txt"); // Challenge data set: Vertices are labeled as positive integers from 1 to 875714 // section8.6.5page64.txt: 4 3 3 1 // problem8.10test1.txt: 3 3 3 // problem8.10test2.txt: 3 3 2 // problem8.10test3.txt: 3 3 1 1 // problem8.10test4.txt: 7 1 // problem8.10test5.txt: 6 3 2 1 // problem8.10.txt: 434821 968 459 313 211 } ``` *Javascript* ```javascript class BaseSolution { constructor(adj, rev) { this.adj = adj; this.rev = rev; } } class RecursiveSolution extends BaseSolution { constructor(adj, rev) { super(adj, rev); } topo_sort() { let list = []; let seen = new Set(); let go = u => { if (seen.has(u)) return; seen.add(u); for (let v of [...this.rev.get(u)]) go(v); list.unshift(u); }; for (let [u, _] of [...this.rev]) go(u); return list; } kosaraju() { let lists = []; let seen = new Set(); let go = (u, list) => { if (seen.has(u)) return; seen.add(u); list.push(u); for (let v of [...this.adj.get(u)]) go(v, list); }; for (let u of this.topo_sort()) { let list = []; go(u, list); lists.push([...list]); } lists.sort((a, b) => b.length - a.length); return lists; } } class IterativeSolution extends BaseSolution { constructor(adj, rev) { super(adj, rev); } topo_sort() { let list = []; let seen = new Set(); for (let [u, _] of [...this.rev]) { if (seen.has(u)) continue; let stack = [ u ]; seen.add(u); stack.back = () => stack[stack.length - 1]; while (stack.length) { let u = stack.back(); for (let v of [...this.rev.get(u)]) if (!seen.has(v)) stack.push(v), seen.add(v); if (u == stack.back()) list.unshift(stack.pop()); } } return list; } kosaraju() { let lists = []; let seen = new Set(); for (let u of this.topo_sort()) { if (seen.has(u)) continue; let list = []; let stack = [ u ]; seen.add(u); stack.back = () => stack[stack.length - 1]; while (stack.length) { let u = stack.back(); for (let v of [...this.adj.get(u)]) if (!seen.has(v)) stack.push(v), seen.add(v); if (u == stack.back()) list.push(stack.pop()); } lists.push([...list]); } lists.sort((a, b) => b.length - a.length); return lists; } } let run = filename => { let adj = new Map(), rev = new Map(); let LineByLine = require('n-readlines'); let input = new LineByLine(filename); let line; while (line = input.next()) { let [u, v] = String.fromCharCode(...line).split(' ').map(Number); if (!adj.has(u)) adj.set(u, []); if (!adj.has(v)) adj.set(v, []); if (!rev.has(u)) rev.set(u, []); if (!rev.has(v)) rev.set(v, []); adj.get(u).push(v); rev.get(v).push(u); } // let A = new RecursiveSolution(adj, rev).kosaraju(); let A = new IterativeSolution(adj, rev).kosaraju(); console.log(`${filename}: ${A.slice(0, Math.min(A.length, 5)).map(scc => scc.length).join(' ')}`); }; run('section8.6.5page64.txt') // Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run('problem8.10test1.txt') // Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run('problem8.10test2.txt') // Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run('problem8.10test3.txt') // Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run('problem8.10test4.txt') // Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run('problem8.10test5.txt') // Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run('problem8.10.txt') // Challenge data set: Vertices are labeled as positive integers from 1 to 875714 // section8.6.5page64.txt: 4 3 3 1 // problem8.10test1.txt: 3 3 3 // problem8.10test2.txt: 3 3 2 // problem8.10test3.txt: 3 3 1 1 // problem8.10test4.txt: 7 1 // problem8.10test5.txt: 6 3 2 1 // problem8.10.txt: 434821 968 459 313 211 ``` *Python3* ```python from collections import deque from functools import cmp_to_key class BaseSolution: def __init__(self, adj, rev): self.adj = adj self.rev = rev class RecursiveSolution(BaseSolution): def topo_sort(self): list = deque() seen = set() def go(u): if u in seen: return seen.add(u) for v in self.rev[u]: go(v) list.appendleft(u) for u in self.rev.keys(): go(u) return list def kosaraju(self): lists = [] seen = set() def go(u, list): if u in seen: return seen.add(u) list.append(u) for v in self.adj[u]: go(v, list) for u in self.topo_sort(): list = [] go(u, list) lists.append(list.copy()) lists.sort(key = cmp_to_key(lambda a, b: len(b) - len(a))) return lists class IterativeSolution(BaseSolution): def topo_sort(self): list = deque() seen = set() for u in self.rev.keys(): if u in seen: continue stack = [ u ]; seen.add(u) while len(stack): u = stack[-1] for v in self.rev[u]: if v not in seen: stack.append(v); seen.add(v) if u == stack[-1]: list.appendleft(stack.pop()) return list def kosaraju(self): lists = [] seen = set() for u in self.topo_sort(): if u in seen: continue list = deque() stack = [ u ]; seen.add(u) while len(stack): u = stack[-1] for v in self.adj[u]: if v not in seen: stack.append(v); seen.add(v) if u == stack[-1]: list.appendleft(stack.pop()) lists.append(list.copy()) lists.sort(key = cmp_to_key(lambda a, b: len(b) - len(a))) return lists def run(filename): adj, rev = {}, {} with open(filename) as fin: while True: line = fin.readline().strip() if not line: break u, v = [int(x) for x in line.split()] if u not in adj: adj[u] = [] if v not in adj: adj[v] = [] if u not in rev: rev[u] = [] if v not in rev: rev[v] = [] adj[u].append(v) rev[v].append(u) # solution = RecursiveSolution(adj, rev) solution = IterativeSolution(adj, rev) A = solution.kosaraju() print(filename + ': ' + ' '.join(str(len(scc)) for scc in A[:5])) run('section8.6.5page64.txt') # Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run('problem8.10test1.txt') # Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run('problem8.10test2.txt') # Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run('problem8.10test3.txt') # Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run('problem8.10test4.txt') # Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run('problem8.10test5.txt') # Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run('problem8.10.txt') # Challenge data set: Vertices are labeled as positive integers from 1 to 875714 # section8.6.5page64.txt: 4 3 3 1 # problem8.10test1.txt: 3 3 3 # problem8.10test2.txt: 3 3 2 # problem8.10test3.txt: 3 3 1 1 # problem8.10test4.txt: 7 1 # problem8.10test5.txt: 6 3 2 1 # problem8.10.txt: 434821 968 459 313 211 ``` *C++* ```cpp #include #include #include #include #include #include using namespace std; using List = deque; using Lists = deque; using AdjList = unordered_map; using Set = unordered_set; using Map = unordered_map; namespace Base { class Solution { protected: AdjList adj, rev; public: Solution(AdjList& adj, AdjList& rev) : adj{ adj }, rev{ rev } {} }; } namespace Recursive { struct Solution : public Base::Solution { Solution(AdjList& adj, AdjList& rev) : Base::Solution{ adj, rev } {} Lists kosaraju() { Lists lists; Set seen; using fun = function; fun go = [&](auto u, auto& list) { if (!seen.insert(u).second) return; list.push_back(u); for (auto v: adj[u]) go(v, list); }; for (auto u: topo_sort()) { List list; go(u, list); lists.emplace_back(list); } sort(lists.begin(), lists.end(), [](auto& a, auto& b) { return b.size() < a.size(); }); return lists; } List topo_sort() { List list; Set seen; using fun = function; fun go = [&](auto u) { if (!seen.insert(u).second) return; for (auto v: rev[u]) go(v); list.push_front(u); }; for (auto [u, _]: rev) go(u); return list; } }; } namespace Iterative { struct Solution : public Base::Solution { Solution(AdjList& adj, AdjList& rev) : Base::Solution{ adj, rev } {} Lists kosaraju() { Lists lists; Set seen; for (auto u: topo_sort()) { if (seen.find(u) != seen.end()) continue; List list; List stack{ u }; seen.insert(u); while (stack.size()) { auto u = stack.back(); for (auto v: adj[u]) if (seen.insert(v).second) stack.push_back(v); if (u == stack.back()) list.push_back(u), stack.pop_back(); } lists.emplace_back(list); } sort(lists.begin(), lists.end(), [](auto& a, auto& b) { return b.size() < a.size(); }); return lists; } List topo_sort() { List list; Set seen; for (auto [u, _]: rev) { if (seen.find(u) != seen.end()) continue; List stack{ u }; seen.insert(u); while (stack.size()) { auto u = stack.back(); for (auto v: rev[u]) if (seen.insert(v).second) stack.push_back(v); if (u == stack.back()) list.push_front(stack.back()), stack.pop_back(); } } return list; } }; } void run(string filename) { int u, v; AdjList adj, rev; fstream fin{ filename }; for (string line; fin >> u >> v;) { adj[u].push_back(v); rev[v].push_back(u); } auto A = Iterative::Solution{ adj, rev }.kosaraju(); A.resize(min(A.size(), size_t(5))); cout << filename << ": "; for (auto i{ 0 }; i < A.size(); cout << A[i++].size() << " "); cout << endl; } int main() { run("section8.6.5page64.txt"); // Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run("problem8.10test1.txt"); // Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run("problem8.10test2.txt"); // Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run("problem8.10test3.txt"); // Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run("problem8.10test4.txt"); // Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run("problem8.10test5.txt"); // Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run("problem8.10.txt"); // Challenge data set: Vertices are labeled as positive integers from 1 to 875714 // section8.6.5page64.txt: 4 3 3 1 // problem8.10test1.txt: 3 3 3 // problem8.10test2.txt: 3 3 2 // problem8.10test3.txt: 3 3 1 1 // problem8.10test4.txt: 7 1 // problem8.10test5.txt: 6 3 2 1 // problem8.10.txt: 434821 968 459 313 211 return 0; } ```
--- ### Dijkstra
📚 Lectures
* [Shortest Paths and Dijkstra's Algorithm (Sections 9.1 and 9.2, Part 1)](https://www.youtube.com/watch?v=jRlNVmRjdRk&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=12) * [Dijkstra's Algorithm: Examples (Section 9.2, Part 2)](https://www.youtube.com/watch?v=ahYhIzLklYo&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=13) * [Correctness of Dijkstra's Algorithm (Section 9.3)](https://www.youtube.com/watch?v=sb7j3EW055M&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=14) * [Implementation and Running Time of Dijkstra's Algorithm (0:00-4:30) (Section 9.4)](https://www.youtube.com/watch?v=00LtSn_PQjc&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=15) * [Data Structures Overview (Section 10.1)](https://www.youtube.com/watch?v=cMrQxxrKg8I&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=16) * [Heaps: Operations and Applications (Sections 10.2 and 10.3)](https://www.youtube.com/watch?v=mNYHDv7SbDI&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=17) * [Speeding Up Dijkstra's Algorithm With Heaps (4:30-26:27) (Section 10.4)](https://www.youtube.com/watch?v=00LtSn_PQjc&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=15) * [Heaps: Implementation Details (Section 10.5)](https://www.youtube.com/watch?v=6VI5kJu8Mv4&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=19&t=0s)
🎯 Solutions
*Kotlin* ```java import java.io.File import java.util.PriorityQueue var INF = (1e9 + 7).toInt() interface BaseSolution { fun run(filename: String, queries: Array): String } class NaiveSolution : BaseSolution { fun dijkstra(E: List>): MutableMap { var dist = mutableMapOf() var seen = mutableSetOf() var start = 1 dist[start] = 0; seen.add(start) var found: Boolean; do { found = false var best_v = INF var best_w = INF for ((u, v, w) in E) { if (!seen.contains(u) || seen.contains(v)) continue found = true if (best_w > dist[u]!! + w) { best_v = v best_w = dist[u]!! + w } } var v = best_v var w = best_w dist[v] = w; seen.add(v) } while (found) return dist } override fun run(filename: String, queries: Array): String { var E = mutableListOf>() File(filename).forEachLine { var words = it.trim().split("\t") var u = words[0].toInt() for (i in 1 until words.size) { var (v, w) = words[i].split(",").map{ it.toInt() } E.add(Triple(u, v, w)) } } var dist = dijkstra(E.toList()) return queries.map{ dist[it] }.joinToString(" ") } } class HeapSolution : BaseSolution { fun dijkstra(adj: MutableMap>>): MutableMap { var dist = mutableMapOf() var seen = mutableSetOf() var start = 1 dist[start] = 0 var q = PriorityQueue>(Comparator{ a: Pair, b: Pair -> a.first.compareTo(b.first) }) q.add(Pair(0, start)) while (0 < q.size) { var (cost, u) = q.poll() if (seen.contains(u)) continue dist[u] = cost; seen.add(u) for ((w, v) in adj[u]!!) { if (seen.contains(v)) continue q.add(Pair(cost + w, v)) } } return dist } override fun run(filename: String, queries: Array): String { var adj = mutableMapOf>>() File(filename).forEachLine { var words = it.trim().split("\t") var u = words[0].toInt() if (!adj.contains(u)) adj[u] = mutableListOf() for (i in 1 until words.size) { var (v, w) = words[i].split(",").map{ it.toInt() } adj[u]!!.add(Pair(w, v)) } } var dist = dijkstra(adj) return queries.map{ dist[it] }.joinToString(" ") } } fun run(solution: BaseSolution) { println(solution.run("problem9.8test.txt", arrayOf(1, 2, 3, 4, 5, 6, 7, 8))) println(solution.run("problem9.8.txt", arrayOf(7, 37, 59, 82, 99, 115, 133, 165, 188, 197))) } fun main() { run(NaiveSolution()) // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(HeapSolution()) // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 } ``` *Javascript* ```javascript let LineByLine = require('n-readlines'); let INF = Number(1e9 + 7); class NaiveSolution { dijkstra(E) { let dist = new Map(); let seen = new Set(); let start = 1; dist[start] = 0; seen.add(start); for (;;) { let found = false; let best_v = INF, best_w = INF; for (let [u, v, w] of E) { if (!seen.has(u) || seen.has(v)) continue; found = true; if (best_w > dist[u] + w) best_v = v, best_w = dist[u] + w; } if (!found) break; let [v, w] = [best_v, best_w]; dist[v] = w; seen.add(v); } return dist; } run(filename, queries) { let E = []; let input = new LineByLine(filename); let line; while (line = input.next()) { let words = String.fromCharCode(...line).trim().split(/\s+/); let u = Number(words[0]); for (let i = 1; i < words.length; ++i) { let [v, w] = words[i].split(',').map(Number); E.push([ u, v, w ]); } } let dist = this.dijkstra(E); return queries.map(x => dist[x]).join(' '); } } let heapkey = x => Array.isArray(x) ? x[0] : x; let heappush = (A, x, f = Math.min) => { let P = i => Math.floor((i - 1) / 2); // parent A.push(x); let N = A.length, i = N - 1; while (0 < i && heapkey(A[i]) == f(heapkey(A[i]), heapkey(A[P(i)]))) { [A[i], A[P(i)]] = [A[P(i)], A[i]]; i = P(i); } }; let heappop = (A, f = Math.min) => { let L = i => 2 * i + 1, // children R = i => 2 * i + 2; let N = A.length, i = 0; let top = A[0]; [A[0], A[N - 1]] = [A[N - 1], A[0]], A.pop(), --N; let ok = true; do { ok = true; let left = f == Math.min ? Infinity : -Infinity, right = left; if (L(i) < N && heapkey(A[i]) != f(heapkey(A[i]), heapkey(A[L(i)]))) ok = false, left = heapkey(A[L(i)]); if (R(i) < N && heapkey(A[i]) != f(heapkey(A[i]), heapkey(A[R(i)]))) ok = false, right = heapkey(A[R(i)]); if (!ok) { let j = left == f(left, right) ? L(i) : R(i); [A[i], A[j]] = [A[j], A[i]]; i = j; } } while (!ok); return top; }; class HeapSolution { dijkstra(adj) { let dist = {}; let seen = new Set(); let start = 1; let q = [[ 0, start ]]; while (q.length) { let [cost, u] = heappop(q); if (seen.has(u)) continue; dist[u] = cost, seen.add(u); for (let [w, v] of (adj[u] || [])) heappush(q, [ dist[u] + w, v ]); } return dist; } run(filename, queries) { let adj = {}; let input = new LineByLine(filename); let line; while (line = input.next()) { let words = String.fromCharCode(...line).trim().split('\t'); let u = Number(words[0]); if (!(u in adj)) adj[u] = []; for (let i = 1; i < words.length; ++i) { let [v, w] = words[i].split(',').map(Number); adj[u].push([ w, v ]); } } let dist = this.dijkstra(adj); return queries.map(x => dist[x]).join(' '); } } let run = solution => { console.log(solution.run('problem9.8test.txt', [1, 2, 3, 4, 5, 6, 7, 8])); console.log(solution.run('problem9.8.txt', [7, 37, 59, 82, 99, 115, 133, 165, 188, 197])); }; run(new NaiveSolution()); // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(new HeapSolution()); // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 ``` *Python3* ```python from abc import ABC, abstractmethod from heapq import heappush, heappop INF = int(1e9 + 7) class BaseSolution(ABC): @abstractmethod def run(self, filename, queries): raise NotImplementedError class NaiveSolution(BaseSolution): def dijkstra(self, E): dist = {} seen = set() start = 1 dist[start] = 0; seen.add(start) while True: found = False best_v = INF best_w = INF for u, v, w in E: if u not in seen or v in seen: continue found = True if best_w > dist[u] + w: best_v = v best_w = dist[u] + w if not found: break v, w = best_v, best_w dist[v] = w; seen.add(v) return dist def run(self, filename, queries): E = [] with open(filename) as fin: while True: line = fin.readline() if not line: break words = line.split() u = int(words[0]) for i in range(1, len(words)): v, w = map(int, words[i].split(',')) E.append([ u, v, w ]) dist = self.dijkstra(E) return ' '.join(str(dist[x]) for x in queries) class HeapSolution(BaseSolution): def dijkstra(self, adj, start = 1): dist = {} seen = set() q = [[ 0, start ]] while len(q): cost, u = heappop(q) if u in seen: continue dist[u] = cost; seen.add(u) for w, v in adj[u]: if v not in seen: heappush(q, [ dist[u] + w, v ]) return dist def run(self, filename, queries): adj = {} with open(filename) as fin: while True: line = fin.readline() if not line: break words = line.split() u = int(words[0]) if u not in adj: adj[u] = [] for i in range(1, len(words)): v, w = map(int, words[i].split(',')) adj[u].append([ w, v ]) dist = self.dijkstra(adj) return ' '.join(str(dist[x]) for x in queries) def run(solution): print(solution.run('problem9.8test.txt', [1, 2, 3, 4, 5, 6, 7, 8])) print(solution.run('problem9.8.txt', [7, 37, 59, 82, 99, 115, 133, 165, 188, 197])) run(NaiveSolution()) # 0 1 2 3 4 4 3 2 # 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(HeapSolution()) # 0 1 2 3 4 4 3 2 # 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 ``` *C++* ```cpp #include #include #include #include #include #include #include using namespace std; using Queries = vector; using Distance = unordered_map; using Set = unordered_set; class BaseSolution { protected: static constexpr auto INF = int(1e9 + 7); public: virtual string run(string filename, Queries&& queries) = 0; }; class NaiveSolution : public BaseSolution { using Edge = tuple; using Edges = vector; public: Distance dijkstra(Edges& E) { Distance dist; Set seen; auto start{ 1 }; dist[start] = 0, seen.insert(start); for (;;) { auto found = false; auto best_v = INF, best_w = INF; for (auto [u, v, w]: E) { if (seen.find(u) == seen.end() || seen.find(v) != seen.end()) continue; found = true; if (best_w > dist[u] + w) best_v = v, best_w = dist[u] + w; } if (!found) break; auto [v, w] = tie(best_v, best_w); dist[v] = w, seen.insert(v); } return dist; } string run(string filename, Queries&& queries) { Edges E; fstream fin{ filename }; string line; int u, v, w; char _; while (getline(fin, line)) { istringstream is{ line }; for (is >> u; is >> v >> _ >> w; E.push_back({ u, v, w })); } auto dist = dijkstra(E); ostringstream os; transform(queries.begin(), queries.end(), ostream_iterator(os, " "), [&](auto x) { return dist[x]; }); return os.str(); } }; class HeapSolution : public BaseSolution { using Pair = pair; using Pairs = vector; using AdjList = unordered_map; priority_queue> q; public: Distance dijkstra(AdjList& adj) { Distance dist; Set seen; for (auto [u, _]: adj) dist[u] = INF; auto start{ 1 }; q.push({ 0, start }); while (q.size()) { auto [cost, u] = q.top(); q.pop(); if (!seen.insert(u).second) continue; dist[u] = cost; for (auto [w, v]: adj[u]) if (seen.find(v) == seen.end()) q.push({ dist[u] + w, v }); } return dist; } string run(string filename, Queries&& queries) { AdjList adj; fstream fin{ filename }; string line; int u, v, w; char _; while (getline(fin, line)) { istringstream is{ line }; for (is >> u; is >> v >> _ >> w; adj[u].push_back({ w, v })); } auto dist = dijkstra(adj); ostringstream os; transform(queries.begin(), queries.end(), ostream_iterator(os, " "), [&](auto x) { return dist[x]; }); return os.str(); } }; void run(BaseSolution&& solution) { cout << "problem9.8test.txt: " << solution.run("problem9.8test.txt", Queries{1, 2, 3, 4, 5, 6, 7, 8 }) << endl << "problem9.8.txt " << solution.run("problem9.8.txt", Queries{7, 37, 59, 82, 99, 115, 133, 165, 188, 197 }) << endl; } int main() { run(NaiveSolution()); // problem9.8test.txt: 0 1 2 3 4 4 3 2 // problem9.8.txt 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(HeapSolution()); // problem9.8test.txt: 0 1 2 3 4 4 3 2 // problem9.8.txt 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 return 0; } ```
--- # Part 3: Greedy Algorithms and Dynamic Programming
--- ### Greedy Scheduling
📚 Lectures
* [Introduction to Greedy Algorithms (Section 13.1)](https://www.youtube.com/watch?v=NTFmxA3qgoo&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=4&t=0s) * [A Scheduling Problem (Section 13.2)](https://www.youtube.com/watch?v=jK3d_GLzKQA&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=5) * [Developing a Greedy Algorithm (Section 13.3)](https://www.youtube.com/watch?v=AJOTgHYjL6A&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=6) * [Scheduling: Correctness Proof (Part 1) (Section 13.4, Part 1)](https://www.youtube.com/watch?v=oyLxydVkQwo&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=7) * [Scheduling: Correctness Proof (Part 2) (Section 13.4, Part 2)](https://www.youtube.com/watch?v=fmXVPYxmFsU&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=8) * [Scheduling: Correctness Proof (Part 3) (Section 13.4, Part 3)](https://www.youtube.com/watch?v=aBFG5PghX68&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=9)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File data class Job(val weight: Long, val length: Long) class Solution { fun minSum(jobs: Array): Pair { class Diff: Comparator { override fun compare(a: Job?, b: Job?): Int { if (a == null || b == null) return 0 var first = a.weight - a.length var second = b.weight - b.length return if (first == second) b.weight.compareTo(a.weight) else second.compareTo(first) // sort by descending difference, break ties in favor of jobs with larger weights } } class Ratio: Comparator { override fun compare(a: Job?, b: Job?): Int { if (a == null || b == null) return 0 var first = a.weight.toDouble() / a.length var second = b.weight.toDouble() / b.length return if (first == second) b.weight.compareTo(a.weight) else second.compareTo(first) // sort by descending difference, break ties in favor of jobs with larger weights } } return Pair(calcSum(jobs, Diff()), calcSum(jobs, Ratio())) } private fun calcSum(jobs: Array, comp: Comparator): Long { jobs.sortWith(comp) var time: Long = 0 var total: Long = 0 jobs.forEach { job -> time += job.length total += job.weight * time } return total } } fun run(filename: String) { var jobs = mutableListOf() var first = true File(filename).forEachLine { if (!first) { var words = it.trim().split(" ").map{ it.toLong() } var (weight, length) = words jobs.add(Job(weight, length)) } else { first = false } } var (diff, ratio) = Solution().minSum(jobs.toTypedArray()) println("$diff, $ratio") // sub-optimal, optimal } fun main() { run("problem13.4test1.txt") // 23, 22 run("problem13.4test2.txt") // 68615, 67247 run("problem13.4.txt") // 69119377652, 67311454237 } ``` *Javascript* ```javascript let LineByLine = require('n-readlines'); class Job { constructor(weight, length) { this.weight = weight; this.length = length; } } class Solution { minSum(jobs) { let diff = (a, b) => { let first = a.weight - a.length, second = b.weight - b.length; return first == second ? b.weight - a.weight : second - first; // sort by descending difference, break ties in favor of jobs with larger weights }; let ratio = (a, b) => { let first = a.weight / a.length, second = b.weight / b.length; return first == second ? b.weight - a.weight : second - first; // sort by descending ratio, break ties in favor of jobs with larger weights }; return [ this._calcSum(jobs, diff), this._calcSum(jobs, ratio) ]; } _calcSum(jobs, comp, time = 0) { jobs.sort((a, b) => comp(a, b)); return jobs.reduce((total, job) => total + job.weight * (time += job.length), 0); } } let run = filename => { let jobs = []; let input = new LineByLine(filename); let line = input.next(); // N while (line = input.next()) { let words = String.fromCharCode(...line).trim().split(' '); let [weight, length] = words.map(Number); jobs.push(new Job(weight, length)); } let [diff, ratio] = new Solution().minSum(jobs); console.log(`${diff}, ${ratio}`); // sub-optimal, optimal }; run('problem13.4test1.txt'); // 23, 22 run('problem13.4test2.txt'); // 68615, 67247 run('problem13.4.txt'); // 69119377652, 67311454237 ``` *Python3* ```python from functools import cmp_to_key class Job: def __init__(self, weight, length): self.weight = weight self.length = length class Solution: def minSum(self, jobs): def diff(a, b): first = a.weight - a.length second = b.weight - b.length return b.weight - a.weight if first == second else second - first # sort by descending difference, break ties in favor of jobs with larger weights def ratio(a, b): first = a.weight / a.length second = b.weight / b.length return b.weight - a.weight if first == second else second - first # sort by descending difference, break ties in favor of jobs with larger weights return [ self._calcSum(jobs, diff), self._calcSum(jobs, ratio) ] def _calcSum(self, jobs, comp, time = 0, total = 0): jobs.sort(key = cmp_to_key(lambda a, b: comp(a, b))) for job in jobs: time += job.length total += job.weight * time return total def run(filename): jobs = [] with open(filename) as fin: line = fin.readline() # N while True: line = fin.readline().strip() if not line: break words = line.split() weight, length = [int(x) for x in words] jobs.append(Job(weight, length)) diff, ratio = Solution().minSum(jobs) print(f'{diff}, {ratio}') # sub-optimal, optimal run('problem13.4test1.txt') # 23, 22 run('problem13.4test2.txt') # 68615, 67247 run('problem13.4.txt') # 69119377652, 67311454237 ``` *C++* ```cpp #include #include #include #include using namespace std; using LL = long long; struct Job { LL weight, length; Job(LL weight, LL length) : weight{ weight }, length{ length } {} }; using Jobs = vector; class Solution { public: using Pair = pair; // sub-optimal, optimal Pair minSum(Jobs& jobs) { auto diff = [](auto& a, auto& b) { auto first = a.weight - a.length, second = b.weight - b.length; return first == second ? b.weight < a.weight : second < first; // sort by descending difference, break ties in favor of jobs with larger weights }; auto ratio = [](auto& a, auto& b) { auto first = double(a.weight) / a.length, second = double(b.weight) / b.length; return first == second ? b.weight < a.weight : second < first; // sort by descending ratio, break ties in favor of jobs with larger weights }; return { calcSum(jobs, diff), calcSum(jobs, ratio) }; } private: template LL calcSum(Jobs& jobs, Comp comp, LL time = 0LL) { sort(jobs.begin(), jobs.end(), comp); return accumulate(jobs.begin(), jobs.end(), 0LL, [&](LL total, auto& job) { return total += job.weight * (time += job.length); }); } }; void run(const string& filename) { Jobs jobs; LL N, weight, length; fstream fin{ filename }; for (fin >> N; fin >> weight >> length; jobs.emplace_back(Job{ weight, length })); auto [diff, ratio] = Solution().minSum(jobs); cout << diff << ", " << ratio << endl; } int main() { run("problem13.4test1.txt"); // 23, 22 run("problem13.4test2.txt"); // 68615, 67247 run("problem13.4.txt"); // 69119377652, 67311454237 return 0; } ```
--- ### Huffman Codes
📚 Lectures
* [Codes (Section 14.1)](https://www.youtube.com/watch?v=K3WZhFZT6Y0&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=33) * [Codes as Trees (Section 14.2)](https://www.youtube.com/watch?v=HESrV5VDu8c&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=34) * [Huffman's Greedy Algorithm (Part 1) (Section 14.3, Part 1)](https://www.youtube.com/watch?v=NM6FZB7IfS8&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=35) * [Huffman's Greedy Algorithm (Part 2) (Section 14.3, Part 2)](https://www.youtube.com/watch?v=PPmn9osMDyI&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=36) * [Huffman's Algorithm: Correctness Proof (Part 1) (Section 14.4, Part 1)](https://www.youtube.com/watch?v=jibgSDjWxdI&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=37) * [Huffman's Algorithm: Correctness Proof (Part 2) (Section 14.4, Part 2)](https://www.youtube.com/watch?v=dAjCcqZKYf4&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=38)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File import java.util.PriorityQueue import java.util.Queue import java.util.LinkedList var INF = (1e9 + 7).toInt() data class Tree(val weight: Int, val left: Tree? = null, val right: Tree? = null) /* fun encode(A: List): Tree { var q = PriorityQueue(Comparator{ a: Tree, b: Tree -> a.weight.compareTo(b.weight) }) for (weight in A) q.add(Tree(weight)) while (1 < q.size) { var a = q.poll() var b = q.poll() var c = Tree(a.weight + b.weight, a, b) q.add(c) } return q.poll() } */ /* * Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation * of a sorting subroutine, followed by a linear amount of additional work. */ fun encode(A: MutableList): Tree { A.sort() var first: Queue = LinkedList(A.map{ weight -> Tree(weight) }.toList()) var second: Queue = LinkedList() var next = mutableListOf() while (1 < first.size + second.size) { next.clear() do { if (0 < first.size && 0 < second.size) { if (first.peek().weight < second.peek().weight) next.add(first.poll()) else next.add(second.poll()) } else if (0 < first.size) next.add(first.poll()) else if (0 < second.size) next.add(second.poll()) } while (next.size < 2) var (a, b) = next var c = Tree(a.weight + b.weight, a, b) second.add(c) } return second.poll() } fun run(filename: String): Pair { var A = mutableListOf() var first = true File(filename).forEachLine { if (!first) { var weight = it.trim().toInt() A.add(weight) } else { first = false } } var tree = encode(A.toMutableList()) var lo = INF var hi = -INF fun go(root: Tree? = tree, depth: Int = 0) { if (root == null) return var isLeaf = { node: Tree? -> node?.left == null && node?.right == null } if (isLeaf(root)) { lo = Math.min(lo, depth) hi = Math.max(hi, depth) } else { go(root.left, depth + 1) go(root.right, depth + 1) } } go() return Pair(lo, hi) } fun main() { for (filename in listOf("problem14.6test1.txt", "problem14.6test2.txt", "problem14.6.txt")) { var (lo, hi) = run(filename) println("$filename: $lo, $hi") // min, max encoding length in the corresponding optimal prefix-free tree } } // problem14.6test1.txt: 2, 5 // problem14.6test2.txt: 3, 6 // problem14.6.txt: 9, 19 ``` *Javascript* ```javascript let LineByLine = require('n-readlines'); class Tree { constructor(weight, left = null, right = null) { this.weight = weight; this.left = left; this.right = right; } } /* let key = x => Array.isArray(x) ? x[0] : x; let heappush = (A, x, f = Math.min) => { let P = i => Math.floor((i - 1) / 2); // parent A.push(x); let N = A.length, i = N - 1; while (0 < i && key(A[i]) == f(key(A[i]), key(A[P(i)]))) { [A[i], A[P(i)]] = [A[P(i)], A[i]]; i = P(i); } }; let heappop = (A, f = Math.min) => { let L = i => 2 * i + 1, // children R = i => 2 * i + 2; let N = A.length, i = 0; let top = A[0]; [A[0], A[N - 1]] = [A[N - 1], A[0]], A.pop(), --N; let ok; do { ok = true; let left = f == Math.min ? Infinity : -Infinity, right = left; if (L(i) < N && key(A[i]) != f(key(A[i]), key(A[L(i)]))) ok = false, left = key(A[L(i)]); if (R(i) < N && key(A[i]) != f(key(A[i]), key(A[R(i)]))) ok = false, right = key(A[R(i)]); if (!ok) { let j = left == f(left, right) ? L(i) : R(i); [A[i], A[j]] = [A[j], A[i]]; i = j; } } while (!ok); return top; }; let encode = A => { let T = []; for (let weight of A) heappush(T, [ weight, new Tree(weight) ]); while (1 < T.length) { let [ a, b ] = [ heappop(T), heappop(T) ]; let c = [ a[0] + b[0], new Tree(a[0] + b[0], a[1], b[1]) ]; heappush(T, c); } return T[0][1]; }; */ /* * Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation * of a sorting subroutine, followed by a linear amount of additional work. */ let encode = A => { A.sort((a, b) => a - b) let first = A.map(weight => new Tree(weight)), second = []; while (1 < first.length + second.length) { let next = []; while (next.length < 2) { if (first.length && second.length) { next.push(first[0].weight < second[0].weight ? first.shift() : second.shift()); } else if (first.length) next.push(first.shift()); else if (second.length) next.push(second.shift()); } let [a, b] = next; let c = new Tree(a.weight + b.weight, a, b); second.push(c); } return second.shift(); }; let run = filename => { let A = []; let input = new LineByLine(filename); let line; line = input.next(); // N while (line = input.next()) { let weight = Number(String.fromCharCode(...line).trim()); A.push(weight); } let tree = encode(A); let [lo, hi] = [Infinity, -Infinity]; let go = (root = tree, depth = 0) => { if (!root) return; let isLeaf = root => !root.left && !root.right; if (isLeaf(root)) lo = Math.min(lo, depth), hi = Math.max(hi, depth); else go(root.left, depth + 1), go(root.right, depth + 1); }; go(); return [ lo, hi ]; } for (let filename of [ 'problem14.6test1.txt', 'problem14.6test2.txt', 'problem14.6.txt' ]) { let [lo, hi] = run(filename); console.log(`${filename}: ${lo}, ${hi}`); // min, max encoding length in the corresponding optimal prefix-free tree } // problem14.6test1.txt: 2, 5 // problem14.6test2.txt: 3, 6 // problem14.6.txt: 9, 19 ``` *Python3* ```python class Tree: def __init__(self, weight, left = None, right = None): self.weight = weight self.left = left self.right = right def __lt__(self, other): return self.weight < other.weight # # priority queue # # from heapq import heappush # from heapq import heappop # def encode(A): # T = [] # for weight in A: # heappush(T, Tree(weight)) # while 1 < len(T): # a, b = heappop(T), heappop(T) # c = Tree(a.weight + b.weight, a, b) # heappush(T, c) # return heappop(T) # # Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation # of a sorting subroutine, followed by a linear amount of additional work. # from collections import deque def encode(A): A.sort() first, second = deque([Tree(weight) for weight in A]), deque() while 1 < len(first) + len(second): next = [] while len(next) < 2: if len(first) and len(second): next.append(first.popleft() if first[0].weight < second[0].weight else second.popleft()) elif len(first): next.append(first.popleft()) elif len(second): next.append(second.popleft()) a, b = next c = Tree(a.weight + b.weight, a, b) second.append(c) return second.popleft() def run(filename): A = [] with open(filename) as fin: N = int(fin.readline()) while True: line = fin.readline() if not line: break weight = int(line.strip()) A.append(weight) tree = encode(A) lo, hi = float('inf'), float('-inf') def go(root = tree, depth = 0): nonlocal lo, hi if not root: return isLeaf = lambda root: not root.left and not root.right if isLeaf(root): lo = min(lo, depth) hi = max(hi, depth) else: go(root.left, depth + 1) go(root.right, depth + 1) go() return [ lo, hi ] for filename in [ 'problem14.6test1.txt', 'problem14.6test2.txt', 'problem14.6.txt' ]: lo, hi = run(filename) print(f'{filename}: {lo}, {hi}') # min, max encoding length in the corresponding optimal prefix-free tree # problem14.6test1.txt: 2, 5 # problem14.6test2.txt: 3, 6 # problem14.6.txt: 9, 19 ``` *C++* ```cpp #include #include #include #include #include #define PRIORITY_QUEUE // O(N * logN) #ifndef PRIORITY_QUEUE #define TWO_QUEUES // O(N) #endif using namespace std; using LL = long long; using Weight = LL; using Weights = vector; struct Tree; using TreePtr = shared_ptr; struct Tree { Weight weight; TreePtr left, right; Tree(Weight weight, TreePtr left = nullptr, TreePtr right = nullptr) : weight{ weight }, left{ left }, right{ right } {} }; using TreePtrs = vector; #ifdef PRIORITY_QUEUE struct Comp { size_t operator()(const TreePtr& a, const TreePtr& b) const { return b->weight < a->weight; } }; using Queue = priority_queue; TreePtr encode(const Weights& A, Queue q = {}) { for (auto weight: A) q.emplace(make_shared(weight)); while (1 < q.size()) { auto a = q.top(); q.pop(); auto b = q.top(); q.pop(); auto c = make_shared(a->weight + b->weight, a, b); q.emplace(c); } return q.top(); } #else // TWO_QUEUES /* * Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation * of a sorting subroutine, followed by a linear amount of additional work. */ using Queue = queue; TreePtr encode(Weights& A, Queue first = {}, Queue second = {}) { sort(A.begin(), A.end()); for (auto weight: A) first.push(make_shared(weight)); TreePtrs next; auto takeFirst = [&]() { next.push_back(first.front()), first.pop(); }; auto takeSecond = [&]() { next.push_back(second.front()), second.pop(); }; while (1 < first.size() + second.size()) { next.clear(); do { if (first.size() && second.size()) { if (first.front()->weight < second.front()->weight) takeFirst(); else takeSecond(); } else if (first.size()) takeFirst(); else if (second.size()) takeSecond(); } while (next.size() < 2); auto [a, b] = tie(next[0], next[1]); auto c = make_shared(a->weight + b->weight, a, b); second.emplace(c); } return second.front(); } #endif using MinMax = pair; constexpr auto Min = numeric_limits::min(); constexpr auto Max = numeric_limits::max(); MinMax run(const string& filename) { Weights A; // weight of each symbol fstream fin{ filename }; LL N, weight; for (fin >> N; fin >> weight; A.push_back(weight)); auto tree = encode(A); LL lo = Max, hi = Min; using fun = function; fun go = [&](auto root, LL depth) { if (!root) return; auto isLeaf = [](auto root) { return !root->left && !root->right; }; if (isLeaf(root)) lo = min(lo, depth), hi = max(hi, depth); else go(root->left, depth + 1), go(root->right, depth + 1); }; go(tree, 0); return make_pair(lo, hi); } int main() { for (auto& filename: { "problem14.6test1.txt", "problem14.6test2.txt", "problem14.6.txt" }) { auto [lo, hi] = run(filename); cout << filename << ": " << lo << ", " << hi << endl; // min, max encoding length in the corresponding optimal prefix-free tree } // problem14.6test1.txt: 2, 5 // problem14.6test2.txt: 3, 6 // problem14.6.txt: 9, 19 return 0; } ```
--- ### Prim's MST
📚 Lectures
* [Minimum Spanning Trees: Problem Definition (Section 15.1)](https://www.youtube.com/watch?v=tDj9BkaQDO8&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=10) * [Prim's MST Algorithm (Section 15.2)](https://www.youtube.com/watch?v=jsvOPssDVJA&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=11) * [Speeding Up Prim's Algorithm via Heaps (Part 1) (Section 15.3, Part 1)](https://www.youtube.com/watch?v=cDtQnXMZGtg&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=15) * [Speeding Up Prim's Algorithm via Heaps (Part 2) (Section 15.3, Part 2)](https://www.youtube.com/watch?v=jGR_LAwGLGk&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=16) * [Prim's Algorithm: Correctness Proof (Part 1) (Section 15.4, Part 1) [Note: this video provides an alternative treatment to that in the book.]](https://www.youtube.com/watch?v=pGUzn3S7bp4&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=12) * [Prim's Algorithm: Correctness Proof (Part 2) (Section 15.4, Part 2) [Note: this video provides an alternative treatment to that in the book.]](https://www.youtube.com/watch?v=199ItGt4mE8&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=13)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File import java.util.PriorityQueue import java.util.Random fun prim(N: Int, adj: MutableMap>>): Int { var total: Int = 0 var start = Random().nextInt(N) + 1 var q = PriorityQueue>(Comparator{ a: Pair, b: Pair -> a.first.compareTo(b.first) }) var seen = mutableSetOf(start) for ((w, v) in adj[start]!!) q.add(Pair(w, v)) while (0 < q.size) { var (cost, u) = q.poll() if (seen.contains(u)) continue total += cost; seen.add(u) for ((w, v) in adj[u]!!) if (!seen.contains(v)) q.add(Pair(w, v)) } return total } fun run(filename: String) { var N: Int = 0 var adj = mutableMapOf>>() var first = true File(filename).forEachLine { line -> if (!first) { var (u, v, w) = line.split(" ").map{ it.toInt() } if (!adj.contains(u)) adj[u] = mutableListOf>() if (!adj.contains(v)) adj[v] = mutableListOf>() adj[u]!!.add(Pair(w, v)) adj[v]!!.add(Pair(w, u)) } else { var (numVertex, _) = line.split(" ").map{ it.toInt() } N = numVertex first = false } } var cost = prim(N, adj) println("$filename: $cost") } fun main() { run("problem15.9test.txt") // problem15.9test.txt: 14 run("problem15.9.txt") // problem15.9.txt: -3612829 } ``` *Javascript* ```javascript let LineByLine = require('n-readlines'); let key = x => Array.isArray(x) ? x[0] : x; let heappush = (A, x, f = Math.min) => { let P = i => Math.floor((i - 1) / 2); // parent A.push(x); let N = A.length, i = N - 1; while (0 < i && key(A[i]) == f(key(A[i]), key(A[P(i)]))) { [A[i], A[P(i)]] = [A[P(i)], A[i]]; i = P(i); } }; let heappop = (A, f = Math.min) => { let L = i => 2 * i + 1, // children R = i => 2 * i + 2; let N = A.length, i = 0; let top = A[0]; [A[0], A[N - 1]] = [A[N - 1], A[0]], A.pop(), --N; let ok; do { ok = true; let left = f == Math.min ? Infinity : -Infinity, right = left; if (L(i) < N && key(A[i]) != f(key(A[i]), key(A[L(i)]))) ok = false, left = key(A[L(i)]); if (R(i) < N && key(A[i]) != f(key(A[i]), key(A[R(i)]))) ok = false, right = key(A[R(i)]); if (!ok) { let j = left == f(left, right) ? L(i) : R(i); [A[i], A[j]] = [A[j], A[i]]; i = j; } } while (!ok); return top; }; let prim = (N, adj, q = [], seen = new Set(), total = 0) => { let start = Math.ceil(N * Math.random()); seen.add(start); for (let [w, v] of adj.get(start)) heappush(q, [w, v]); while (q.length) { let [cost, u] = heappop(q); if (seen.has(u)) continue; total += cost; seen.add(u); for (let [w, v] of adj.get(u)) if (!seen.has(v)) heappush(q, [w, v]); } return total; }; let run = filename => { let adj = new Map(); let input = new LineByLine(filename); let line = input.next(); let [N, M] = String.fromCharCode(...line).trim().split(' ').map(Number); while (line = input.next()) { let [u, v, w] = String.fromCharCode(...line).trim().split(' ').map(Number); if (!adj.has(u)) adj.set(u, []); if (!adj.has(v)) adj.set(v, []); adj.get(u).push([w, v]); adj.get(v).push([w, u]); } let cost = prim(N, adj); console.log(`${filename}: ${cost}`); } run('problem15.9test.txt') // problem15.9test.txt: 14 run('problem15.9.txt') // problem15.9.txt: -3612829 ``` *Python3* ```python from random import randint from heapq import heappush, heappop def prim(N, adj, total = 0): q = [] seen = set() start = randint(1, N); seen.add(start) for w, v in adj[start]: heappush(q, [w, v]) while q: cost, u = heappop(q) if u in seen: continue total += cost; seen.add(u) for w, v in adj[u]: if v not in seen: heappush(q, [w, v]) return total def run(filename): adj = {} first = True with open(filename) as fin: while True: line = fin.readline() if not line: break words = line.split() if not first: u, v, w = [int(x) for x in words] if u not in adj: adj[u] = [] if v not in adj: adj[v] = [] adj[u].append([w, v]) adj[v].append([w, u]) else: N, M = [int(x) for x in words] first = False cost = prim(N, adj) print(f'{filename}: {cost}') run('problem15.9test.txt') # problem15.9test.txt: 14 run('problem15.9.txt') # problem15.9.txt: -3612829 ``` *C++* ```cpp #include #include #include #include #include #include using namespace std; using Pair = pair; using Pairs = vector; using AdjList = unordered_map; using Queue = priority_queue>; using Set = unordered_set; constexpr auto INF = int(1e9 + 7); int getRandom(int N) { default_random_engine generator; uniform_int_distribution distribution(1, N); return distribution(generator); } int prim(int N, AdjList& adj, Queue q = {}, Set seen = {}, int total = 0) { auto start = getRandom(N); seen.insert(start); for (auto [w, v]: adj[start]) q.push({ w, v }); while (q.size()) { auto [cost, u] = q.top(); q.pop(); if (!seen.insert(u).second) continue; total += cost; for (auto [w, v]: adj[u]) if (seen.find(v) == seen.end()) q.push({ w, v }); } return total; } void run(const string& filename) { AdjList adj; fstream fin{ filename }; int N, M; fin >> N >> M; // N vertices and M edges int u, v, w; // edge u -> v of weight w while (fin >> u >> v >> w) { adj[u].emplace_back(w, v); adj[v].emplace_back(w, u); } auto cost = prim(N, adj); cout << filename << ": " << cost << endl; } int main() { run("problem15.9test.txt"); // problem15.9test.txt: 14 run("problem15.9.txt"); // problem15.9.txt: -3612829 return 0; } ```
--- ### Kruskal's MST
📚 Lectures
* [Minimum Spanning Trees: Problem Definition (Section 15.1)](https://www.youtube.com/watch?v=tDj9BkaQDO8&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=10) * [Kruskal's MST Algorithm (Section 15.5)](https://www.youtube.com/watch?v=SZuCspj5AJc&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=17) * [Speeding Up Kruskal's Algorithm via Union-Find (Part 1) (Section 15.6, Part 1)](https://www.youtube.com/watch?v=fItEZEVyJKE&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=19) * [Speeding Up Kruskal's Algorithm via Union-Find (Part 2) (Section 15.6, Part 2) [Note: this video provides an alternative treatment to that in the book.]](https://www.youtube.com/watch?v=jY-vY6d18W4&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=20) * [Lazy Unions (Section 15.6, Part 3) [Note: this video is closer to the union-find implementation in the book.]](https://www.youtube.com/watch?v=bRwTSPIEI9k&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=24) * [Kruskal's Algorithm: Correctness Proof (Section 15.7) [Note: this video provides an alternative treatment to that in the book.]](https://www.youtube.com/watch?v=AjLjL0Rp10g&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=18) * [Application: Single-Link Clustering (Section 15.8)](https://www.youtube.com/watch?v=MSSzOs1X4K8&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=22)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File fun kruskal(E: MutableList>): Int { var total: Int = 0 var M = E.size var P = IntArray(M) { it } // 🙂 parent representatives of 1..M disjoint sets fun find(x: Int): Int { P[x] = if (P[x] == x) x else find(P[x]) return P[x] } fun union(a: Int, b: Int): Boolean { var x = find(a) var y = find(b) if (x == y) return false P[x] = y // 🎲 arbitrary choice return true } E.sortWith(Comparator{ a, b -> a.third.compareTo(b.third) }) // sort edges by nondecreasing weight for ((u, v, w) in E) if (union(u, v)) total += w return total } fun run(filename: String) { var E = mutableListOf>() var first = true File(filename).forEachLine { line -> if (!first) { var (u, v, w) = line.trim().split(" ").map{ it.toInt() } E.add(Triple(u, v, w)) } else { first = false // ignore first line with N vertices and M edges } } var cost = kruskal(E) println("$filename: $cost") } fun main() { run("problem15.9test.txt") // problem15.9test.txt: 14 run("problem15.9.txt") // problem15.9.txt: -3612829 } ``` *Javascipt* ```javascript let LineByLine = require('n-readlines'); let kruskal = E => { let total = 0; let M = E.length; let P = [...Array(M).keys()]; // 🙂 parent representatives of 1..M disjoint sets let find = x => P[x] = P[x] == x ? x : find(P[x]); let union = (a, b) => { a = find(a); b = find(b); if (a == b) return false; P[a] = b; // 🎲 arbitrary choice return true; }; E.sort((first, second) => { // sort edges by nondecreasing weight let [u1, v1, w1] = first, [u2, v2, w2] = second; return w1 - w2; }); for (let [u, v, w] of E) if (union(u, v)) total += w; return total; }; let run = filename => { let E = []; let input = new LineByLine(filename); let line = input.next(); // ignore first line with N vertices and M edges while (line = input.next()) { let [u, v, w] = String.fromCharCode(...line).trim().split(' ').map(Number); E.push([ u, v, w ]); } let cost = kruskal(E); console.log(`${filename}: ${cost}`); }; run('problem15.9test.txt'); // problem15.9test.txt: 14 run('problem15.9.txt'); // problem15.9.txt: -3612829 ``` *Python3* ```python from functools import cmp_to_key def kruskal(E, total = 0): M = len(E) P = [i for i in range(M)] # 🙂 parent representatives of 1..M disjoint sets def find(x): P[x] = P[x] if P[x] == x else find(P[x]) return P[x] def union(a, b): a = find(a) b = find(b) if a == b: return False P[a] = b # 🎲 arbitary choice return True E.sort(key = cmp_to_key(lambda first, second: first[2] - second[2])) # sort edges by nondecreasing weight for u, v, w in E: if union(u, v): total += w return total def run(filename): E = [] first = True with open(filename) as fin: while True: line = fin.readline() if not line: break values = [int(x) for x in line.strip().split()] if not first: u, v, w = values # edge u -> v of weight w E.append([ u, v, w ]) else: first = False # ignore first line with N vertices and M edges cost = kruskal(E) print(f'{filename}: {cost}') run('problem15.9test.txt') # problem15.9test.txt: 14 run('problem15.9.txt') # problem15.9.txt: -3612829 ``` *C++* ```cpp #include #include #include #include #include using namespace std; using Edge = tuple; using Edges = vector; using Parents = vector; using fun = function; int kruskal(Edges& E, int total = 0) { auto M = E.size(); Parents P(M); iota(P.begin(), P.end(), 0); // 🙂 parent representatives of 1..M disjoint sets fun find = [&](auto x) { return P[x] = P[x] == x ? x : find(P[x]); }; auto join = [&](auto a, auto b) { a = find(a); b = find(b); if (a == b) return false; P[a] = b; // 🎲 arbitrary choice return true; }; sort(E.begin(), E.end(), [](auto& first, auto& second) { // sort edges by nondecreasing weight auto [u1, v1, w1] = first; auto [u2, v2, w2] = second; return w1 < w2; }); for (auto [u, v, w]: E) if (join(u, v)) total += w; return total; } void run(const string& filename, Edges E = {}) { fstream fin{ filename }; int N, M; fin >> N >> M; // ignore first line with N vertices and M edges int u, v, w; // edge u -> v of weight w while (fin >> u >> v >> w) E.emplace_back(u, v, w); auto cost = kruskal(E); cout << filename << ": " << cost << endl; } int main() { run("problem15.9test.txt"); // problem15.9test.txt: 14 run("problem15.9.txt"); // problem15.9.txt: -3612829 return 0; } ```
--- ### Weighted Independent Set
📚 Lectures
* [The Weighted Independent Set Problem (Section 16.1)](https://www.youtube.com/watch?v=0awkct8SkxA&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=39) * [A Linear-Time Algorithm for WIS in Path Graphs (Part 1) (Section 16.2, Part 1)](https://www.youtube.com/watch?v=pLOkbHGRsv0&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=40) * [A Linear-Time Algorithm for WIS in Path Graphs (Part 2) (Section 16.2, Part 2)](https://www.youtube.com/watch?v=Im_zjFkZDCY&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=41) * [A Reconstruction Algorithm (Section 16.3)](https://www.youtube.com/watch?v=W2ncNfksRqo&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=42)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File fun topDown(A: MutableList): Long { var N = A.size var m = mutableMapOf() fun go(i: Int = N - 1): Long { if (m.contains(i)) // 🤔 memo return m[i]!! if (i < 0) { // 🛑 empty set m[i] = 0 return 0 } if (i == 0) { // 🛑 single set m[i] = A[0] return A[0] } var include = go(i - 2) + A[i] // ✅ include A[i] var exclude = go(i - 1) // 🚫 exclude A[i] m[i] = Math.max(include, exclude) // 🎯 best return m[i]!! } return go() } fun bottomUp(A: MutableList): Long { var N = A.size var dp = LongArray(N + 1) // 🤔 memo dp[0] = 0 // 🛑 empty set dp[1] = A[0] // 🛑 single set for (i in 2..N) { var include = dp[i - 2] + A[i - 1] // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) var exclude = dp[i - 1] // 🚫 exclude A[i] dp[i] = Math.max(include, exclude) // 🎯 best } return dp[N] } fun bottomUpMemOpt(A: MutableList): Long { var N = A.size var a: Long = 0 // 🤔 memo + 🛑 empty set var b: Long = A[0] // 🤔 memo + 🛑 single set var c: Long = -1 for (i in 2..N) { var include = a + A[i - 1] // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) var exclude = b // 🚫 exclude A[i] c = Math.max(include, exclude) // 🎯 best a = b; b = c // 👈 slide window } return c } fun run(filename: String) { var A = mutableListOf() var first = true File(filename).forEachLine { line -> if (!first) { A.add(line.toLong()) } else { first = false } } var a = topDown(A) var b = bottomUp(A) var c = bottomUpMemOpt(A) assert(a == b && b == c) // 💩 sanity check println("$filename: $a") } fun main() { run("problem16.6test.txt") // problem16.6test.txt: 2617 run("problem16.6.txt") // problem16.6.txt: 2955353732 } ``` *Javascript* ```javascript const assert = require('assert'); const LineByLine = require('n-readlines'); let top_down = (A, m = {}) => { let N = A.length; let go = (i = N - 1) => { if (m[i]) // 🤔 memo return m[i]; if (i < 0) return m[i] = 0; // 🛑 empty set if (!i) return m[i] = A[0]; // 🛑 single set let include = go(i - 2) + A[i], // ✅ include A[i] exclude = go(i - 1); // 🚫 exclude A[i] return m[i] = Math.max(include, exclude); // 🎯 best }; return go(); }; let bottom_up = A => { let N = A.length; let dp = Array(N + 1); // 🤔 memo dp[0] = 0; // 🛑 empty set dp[1] = A[0]; // 🛑 single set for (let i = 2; i <= N; ++i) { let include = dp[i - 2] + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = dp[i - 1]; // 🚫 exclude A[i] dp[i] = Math.max(include, exclude); // 🎯 best } return dp[N]; }; let bottom_up_memopt = A => { let N = A.length; let a = 0, // 🤔 memo + 🛑 empty set b = A[0], // 🤔 memo + 🛑 single set c = -1; for (let i = 2; i <= N; ++i) { let include = a + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = b; // 🚫 exclude A[i] c = Math.max(include, exclude); // 🎯 best a = b, b = c; // 👈 slide window } return c; }; let run = filename => { let A = []; let input = new LineByLine(filename); let line; let first = true; while (line = input.next()) { if (!first) { A.push(Number(line.toString('ascii'))); } else { first = false; } } let a = top_down(A), b = bottom_up(A), c = bottom_up_memopt(A); assert(a == b && b == c); // 💩 sanity check console.log(`${filename}: ${a}`); }; run('problem16.6test.txt'); // problem16.6test.txt: 2617 run('problem16.6.txt'); // problem16.6.txt: 2955353732 ``` *Python3* ```python from functools import lru_cache def top_down(A): N = len(A) @lru_cache # 🤔 memo def go(i = N - 1): if i < 0: return 0 # 🛑 empty set if i == 0: return A[0] # 🛑 single set include = go(i - 2) + A[i] # ✅ include A[i] exclude = go(i - 1) # 🚫 exclude A[i] return max(include, exclude) # 🎯 best return go() def bottom_up(A): N = len(A) dp = [0] * (N + 1) # 🤔 memo dp[0] = 0 # 🛑 empty set dp[1] = A[0] # 🛑 single set for i in range(2, N + 1): include = dp[i - 2] + A[i - 1] # ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = dp[i - 1] # 🚫 exclude A[i] dp[i] = max(include, exclude) # 🎯 best return dp[N] def bottom_up_memopt(A): N = len(A) a = 0 # 🤔 memo + 🛑 empty set b = A[0] # 🤔 memo + 🛑 single set c = -1 for i in range(2, N + 1): include = a + A[i - 1] # ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = b # 🚫 exclude A[i] c = max(include, exclude) # 🎯 best a = b; b = c # 👈 slide window return c def run(filename): A = [] with open(filename) as fin: first = True while True: line = fin.readline() if not line: break x = int(line) if not first: A.append(x) else: first = False N = x a = top_down(A) b = bottom_up(A) c = bottom_up_memopt(A) assert(a == b and b == c) # 💩 sanity check print(f'{filename}: {a}') run('problem16.6test.txt') # problem16.6test.txt: 2617 run('problem16.6.txt') # problem16.6.txt: 2955353732 ``` *C++* ```cpp #include #include #include #include #include using namespace std; using LL = long long; using List = vector; using Map = unordered_map; namespace TopDown { LL best(List& A, Map m = {}) { int N = A.size(); using fun = function; fun go = [&](auto i) { if (m[i]) return m[i]; // 🤔 memo if (i < 0) return m[i] = 0LL; // 🛑 empty set if (!i) return m[i] = A[0]; // 🛑 single set auto include = go(i - 2) + A[i], // ✅ include A[i] exclude = go(i - 1); // 🚫 exclude A[i] return m[i] = max(include, exclude); // 🎯 best }; return go(N - 1); } } namespace BottomUp { LL best(List& A, Map m = {}) { int N = A.size(); List dp(N + 1); // 🤔 memo dp[0] = 0LL; // 🛑 empty set dp[1] = A[0]; // 🛑 single set for (auto i{ 2 }; i <= N; ++i) { auto include = dp[i - 2] + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = dp[i - 1]; // 🚫 exclude A[i] dp[i] = max(include, exclude); // 🎯 best } return dp[N]; } } namespace BottomUpMemOpt { LL best(List& A) { int N = A.size(); LL a = 0LL, // 🤔 memo + 🛑 empty set b = A[0], // 🤔 memo + 🛑 single set c = -1; for (auto i{ 2 }; i <= N; ++i) { auto include = a + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = b; // 🚫 exclude A[i] c = max(include, exclude); // 🎯 best a = b, b = c; // 👈 slide window } return c; } } void run(const string& filename) { List A; fstream fin{ filename }; int N; fin >> N; copy_n(istream_iterator(fin), N, back_inserter(A)); auto a = TopDown::best(A), b = BottomUp::best(A), c = BottomUpMemOpt::best(A); assert(a == b && b == c); // 💩 sanity check cout << filename << ": " << a << endl; } int main() { run("problem16.6test.txt"); // problem16.6test.txt: 2617 run("problem16.6.txt"); // problem16.6.txt: 2955353732 return 0; } ```
--- ### Knapsack
📚 Lectures
* [Principles of Dynamic Programming (Section 16.4)](https://www.youtube.com/watch?v=27nK8C-GCPM&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=43) * [The Knapsack Problem (Part 1) (Section 16.5, Part 1)](https://www.youtube.com/watch?v=jlHIBaUizOU&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=44) * [The Knapsack Problem (Part 2) (Section 16.5, Part 2)](https://www.youtube.com/watch?v=KX_6OF8X6HQ&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=45) * [The Knapsack Problem (Part 3) (Section 16.5, Part 3)](https://www.youtube.com/watch?v=1dRUOZKcvYs&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=46)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File var INF = (1e9 + 7).toInt() fun top_down(A: List>, K: Int): Int { var N = A.size var m = mutableMapOf() fun go(i: Int = 0, k: Int = K): Int { if (i == N) // 🛑 empty set return 0 var key = "$i,$k" if (m.contains(key)) // 🤔 memo return m[key]!! var (value, weight) = A[i] var include = if (0 <= k - weight) go(i + 1, k - weight) + value else -INF // ✅ include A[i] var exclude = go(i + 1, k) // 🚫 exclude A[i] m[key] = Math.max(include, exclude) // 🎯 best return m[key]!! } return go() } fun bottom_up(A: List>, K: Int): Int { var N = A.size var dp = Array(N + 1){ Array(K + 1){ -INF } } // 🤔 memo for (k in 0..K) // 🛑 empty set dp[0][k] = 0 for (i in 1..N) { for (k in 0..K) { var (value, weight) = A[i - 1] var include = if (0 <= k - weight) dp[i - 1][k - weight] + value else -INF // ✅ include A[i] var exclude = dp[i - 1][k] // 🚫 exclude A[i] dp[i][k] = Math.max(include, exclude) // 🎯 best } } return dp[N][K] } fun bottom_up_memopt(A: List>, K: Int): Int { var N = A.size var pre = Array(K + 1) { 0 } // 🤔 memo + 🛑 empty set for (i in 1..N) { var cur = Array(K + 1) { -INF } for (k in 0..K) { var (value, weight) = A[i - 1] var include = if (0 <= k - weight) pre[k - weight] + value else -INF // ✅ include A[i] var exclude = pre[k] // 🚫 exclude A[i] cur[k] = Math.max(include, exclude) // 🎯 best } pre = cur.also { cur = pre } } return pre[K] } fun run(filename: String) { var A = mutableListOf>() var K = 0 var N = 0 var first = true File(filename).forEachLine { line -> if (!first) { var (value, weight) = line.trim().split(" ").map{ it -> it.toInt() } A.add(Pair(value, weight)) } else { var (a, b) = line.trim().split(" ").map{ it -> it.toInt() } K = a N = b first = false } } var a = top_down(A, K) var b = bottom_up(A, K) var c = bottom_up_memopt(A, K) assert(a == b && b == c) // 💩 sanity check println("$filename: $a") } fun main() { run("problem16.7test.txt") // problem16.7test.txt: 2493893 } ``` *Javascript* ```javascript const assert = require('assert'); const LineByLine = require('n-readlines'); let top_down = (A, K, m = new Map()) => { let N = A.length; let go = (i = 0, k = K) => { if (i == N) // 🛑 empty set return 0; let key = `${i},${k}`; if (m.has(key)) // 🤔 memo return m.get(key); let [value, weight] = A[i]; let include = 0 <= k - weight ? go(i + 1, k - weight) + value : -Infinity, // ✅ include A[i] exclude = go(i + 1, k); // 🚫 exclude A[i] return m.set(key, Math.max(include, exclude)) // 🎯 best .get(key); }; return go(); }; let bottom_up = (A, K) => { let N = A.length; let dp = [...Array(N + 1)].map(_ => Array(K + 1).fill(-Infinity)); // 🤔 memo for (let k = 0; k < K; dp[0][k++] = 0); // 🛑 empty set for (let i = 1; i <= N; ++i) { for (let k = 0; k <= K; ++k) { let [value, weight] = A[i - 1]; let include = 0 <= k - weight ? dp[i - 1][k - weight] + value : -Infinity, // ✅ include A[i] exclude = dp[i - 1][k]; // 🚫 exclude A[i] dp[i][k] = Math.max(include, exclude); // 🎯 best } } return dp[N][K]; }; let bottom_up_memopt = (A, K) => { let N = A.length; let pre = Array(K + 1).fill(0); // 🤔 memo + 🛑 empty set for (let i = 1; i <= N; ++i) { let cur = Array(K + 1).fill(-Infinity); for (let k = 0; k <= K; ++k) { let [value, weight] = A[i - 1]; let include = 0 <= k - weight ? pre[k - weight] + value : -Infinity, // ✅ include A[i] exclude = pre[k]; // 🚫 exclude A[i] cur[k] = Math.max(include, exclude); // 🎯 best } [pre, cur] = [cur, pre]; } return pre[K]; }; let run = filename => { let A = []; const input = new LineByLine(filename) let [K, N] = input.next().toString().split(' ').map(Number); // K capacity, N items let line; while (line = input.next()) { let [value, weight] = line.toString().split(' ').map(Number); A.push([value, weight]); } let a = top_down(A, K), b = bottom_up(A, K), c = bottom_up_memopt(A, K); assert(a == b && b == c); // 💩 sanity check console.log(`${filename}: ${a}`); }; run('problem16.7test.txt') // problem16.7test.txt: 2493893 ``` *Python3* ```python from functools import lru_cache def top_down(A, K): N = len(A) total = [0] * N @lru_cache(maxsize = None) # 🤔 memo def go(i = 0, k = K): if i == N: # 🛑 empty set return 0 value, weight = A[i] include = go(i + 1, k - weight) + value if 0 <= k - weight else float('-inf') # ✅ include A[i] exclude = go(i + 1, k) # 🚫 exclude A[i] return max(include, exclude) # 🎯 best return go() def bottom_up(A, K): N = len(A) dp = [[float('-inf')] * (K + 1) for _ in range(N + 1)] # 🤔 memo for j in range(K): # 🛑 empty set dp[0][j] = 0 for i in range(1, N + 1): for k in range(1, K + 1): value, weight = A[i - 1] include = dp[i - 1][k - weight] + value if 0 <= k - weight else float('-inf') # ✅ include A[i] exclude = dp[i - 1][k] # 🚫 exclude A[i] dp[i][k] = max(include, exclude) # 🎯 best return dp[N][K] def bottom_up_memopt(A, K): N = len(A) pre = [0] * (K + 1) # 🤔 memo + 🛑 empty set for i in range(1, N + 1): cur = [float('-inf')] * (K + 1) for k in range(1, K + 1): value, weight = A[i - 1] include = pre[k - weight] + value if 0 <= k - weight else float('-inf') # ✅ include A[i] exclude = pre[k] # 🚫 exclude A[i] cur[k] = max(include, exclude) # 🎯 best pre, cur = cur, pre return pre[K] def run(filename): A = [] with open(filename) as fin: line = fin.readline() [K, N] = [int(word) for word in line.split()] # K capacity, N items while True: line = fin.readline() if not line: break value, weight = [int(word) for word in line.split()] A.append([value, weight]) a = top_down(A, K) b = bottom_up(A, K) c = bottom_up_memopt(A, K) assert(a == b and b == c) # 💩 sanity check print(f'{filename}: {a}') run('problem16.7test.txt') # problem16.7test.txt: 2493893 ``` *C++* ```cpp #include #include #include #include #include using namespace std; using Pair = pair; // value, weight using Pairs = vector; using fun = function; using Map = unordered_map; int INF = 1e9 + 7; int top_down(Pairs& A, int K, Map m = {}) { auto N = A.size(); fun go = [&](auto i, auto k) { if (i == N) // 🛑 empty set return 0; stringstream key; key << i << "," << k; if (m.find(key.str()) != m.end()) // 🤔 memo return m[key.str()]; auto [value, weight] = A[i]; auto include = 0 <= k - weight ? go(i + 1, k - weight) + value : -INF, // ✅ include A[i] exclude = go(i + 1, k); // 🚫 exclude A[i] return m[key.str()] = max(include, exclude); // 🎯 best }; return go(0, K); } int bottom_up(Pairs& A, int K) { auto N = A.size(); using VI = vector; using VVI = vector; VVI dp(N + 1, VI(K + 1, -INF)); // 🤔 memo for (auto k{ 0 }; k < K; dp[0][k++] = 0); // 🛑 empty set for (auto i{ 1 }; i <= N; ++i) { for (auto k{ 0 }; k <= K; ++k) { auto [value, weight] = A[i - 1]; auto include = 0 <= k - weight ? dp[i - 1][k - weight] +value : -INF, // ✅ include A[i] exclude = dp[i - 1][k]; // 🚫 exclude A[i] dp[i][k] = max(include, exclude); // 🎯 best } } return dp[N][K]; } int bottom_up_memopt(Pairs& A, int K) { auto N = A.size(); using VI = vector; VI pre(K + 1, 0); // 🤔 memo + 🛑 empty set for (auto i{ 1 }; i <= N; ++i) { VI cur(K + 1, -INF); for (auto k{ 0 }; k <= K; ++k) { auto [value, weight] = A[i - 1]; auto include = 0 <= k - weight ? pre[k - weight] +value : -INF, // ✅ include A[i] exclude = pre[k]; // 🚫 exclude A[i] cur[k] = max(include, exclude); // 🎯 best } swap(pre, cur); } return pre[K]; } void run(const string& filename) { Pairs A; fstream fin{ filename }; int K, N; // K capacity, N items fin >> K >> N; for (int value, weight; fin >> value >> weight; A.emplace_back(value, weight)); auto a = top_down(A, K), b = bottom_up(A, K), c = bottom_up_memopt(A, K); assert(a == b && b == c); // 💩 sanity check cout << filename << ": " << a << endl; } int main() { run("problem16.7test.txt"); // problem16.7test.txt: 2493893 return 0; } ```
--- ### Bellman-Ford
📚 Lectures
* [Shortest Paths with Negative Edge Lengths (Section 18.1)](https://www.youtube.com/watch?v=oyHoqtxHiOs&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=54) * [The Bellman-Ford Algorithm (Part 1) (Section 18.2, Part 1)](https://www.youtube.com/watch?v=06OK99Aak60&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=55) * [The Bellman-Ford Algorithm (Part 2) (Section 18.2, Part 2)](https://www.youtube.com/watch?v=d1TWZa20Mkw&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=56) * [The Bellman-Ford Algorithm (Part 3) (Section 18.2, Part 3)](https://www.youtube.com/watch?v=HaXTsC8eOfo&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=57) * [The Bellman-Ford Algorithm (Part 4) (Section 18.2, Part 4)](https://www.youtube.com/watch?v=Otn0kbliRUQ&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=58)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File import java.util.LinkedList import java.util.Queue // bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices fun bell(E: Array>, N: Int, start: Int = 1, INF: Int = (1e6).toInt()): IntArray { var dist = IntArray(N) { INF } dist[start] = 0 var K = N - 1 while (0 < K--) E.forEach{ (u, v, w) -> dist[v] = Math.min(dist[v], dist[u] + w)} return dist } // shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) fun spfa(E: Array>, N: Int, start: Int = 1, INF: Int = (1e6).toInt()): IntArray { var dist = IntArray(N) { INF } dist[start] = 0 var adj = Array>>(N) { mutableListOf>() } for ((u, v, w) in E) adj[u].add(Pair(v, w)) var q: Queue = LinkedList(listOf(start)) while (0 < q.size) { var u = q.poll() for ((v, w) in adj[u]) { if (dist[v] > dist[u] + w) { dist[v] = dist[u] + w; q.add(v) } } } return dist } fun run(filename: String): IntArray { var N = 0 var E = mutableListOf>() File(filename).forEachLine { var A = ArrayDeque(it.trim().split("\t")) var u = A.removeFirst().toInt() for ((v, w) in A.map{ it.split(",").map{ it.toInt() } }) E.add(Triple(u, v, w)) ++N; } var A = E.toTypedArray() var a = bell(A, N + 1) // +1 for 1-based indexing var b = spfa(A, N + 1) assert(a == b) // 💩 sanity check: single source shortest paths are the same return b } fun main() { var dist = run("test.txt") println(listOf(7, 37, 59, 82, 99, 115, 133, 165, 188, 197).map{ dist[it] }.joinToString(",")) // 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068 } ``` *Javascript* ```javascript const assert = require('assert'); const zip = require('lodash/zip'); const LineByLine = require('n-readlines'); // bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices let bell = (E, N, start = 1, INF = Number(1e6)) => { let dist = Array(N).fill(INF); dist[start] = 0; let K = N - 1; while (K--) E.forEach(([u, v, w]) => dist[v] = Math.min(dist[v], dist[u] + w)); return dist; }; // shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) let spfa = (E, N, start = 1, INF = Number(1e6)) => { let dist = Array(N).fill(INF); dist[start] = 0; let adj = [...Array(N)].map(_ => []); E.forEach(([u, v, w]) => adj[u].push([v, w])); let q = [start]; while (q.length) { let u = q.shift(); for (let [v, w] of adj[u]) if (dist[v] > dist[u] + w) dist[v] = dist[u] + w, q.push(v); } return dist; }; let run = filename => { let N = 0; let E = []; let input = new LineByLine(filename); let line; while (line = input.next()) { let A = line.toString('ascii').split('\t').filter(it => it.length); let u = Number(A.shift()); A.map(pair => pair.split(',').map(Number)).forEach(([v, w]) => E.push([u, v, w])); ++N; } let a = bell(E, N + 1), // +1 for 1-based indexing b = spfa(E, N + 1); zip(a, b).forEach(([x, y]) => assert(x == y)); // 💩 sanity check: single source shortest paths are the same return a; }; let dist = run('test.txt'); console.log([7, 37, 59, 82, 99, 115, 133, 165, 188, 197].map(x => dist[x]).join(',')); // 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068 ``` *Python3* ```python from collections import deque # bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices def bell(E, N, start = 1, INF = int(1e6)): dist = [INF] * N dist[start] = 0 k = N - 1 while k: for u, v, w in E: dist[v] = min(dist[v], dist[u] + w) k -= 1 return dist # shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) def spfa(E, N, start = 1, INF = int(1e6)): dist = [INF] * N dist[start] = 0 adj = {i: [] for i in range(N)} for u, v, w in E: adj[u].append([v, w]) q = deque([start]) while q: u = q.popleft() for v, w in adj[u]: if dist[v] > dist[u] + w: dist[v] = dist[u] + w; q.append(v) return dist def run(filename): E = [] N = 0 with open(filename) as fin: while True: line = fin.readline() if not line: break line = line.strip() A = [word for word in line.split('\t') if len(word)] u = int(A[0]) for i in range(1, len(A)): v, w = [int(x) for x in A[i].split(',')] E.append([u, v, w]) N += 1 a = bell(E, N + 1) # +1 for 1-based indexing b = spfa(E, N + 1) assert(a == b) return b dist = run('test.txt') print(','.join(str(dist[x]) for x in [7, 37, 59, 82, 99, 115, 133, 165, 188, 197])) # 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068 ``` *C++* ```cpp #include #include #include #include #include #include using namespace std; using VI = vector; using VS = vector; using Edge = tuple; using Edges = vector; using Pair = pair; using Pairs = vector; using AdjList = unordered_map; using Queue = queue; // bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices VI bell(Edges& E, int N, int start = 1, int INF = 1e6) { VI dist(N, INF); dist[start] = 0; auto K = N - 1; while (K--) for (auto [u, v, w]: E) dist[v] = min(dist[v], dist[u] + w); return dist; } // shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) VI spfa(Edges& E, int N, int start = 1, int INF = 1e6, AdjList adj = {}) { VI dist(N, INF); dist[start] = 0; for (auto [u, v, w]: E) adj[u].emplace_back(v, w); Queue q{{ start }}; while (q.size()) { auto u = q.front(); q.pop(); for (auto [v, w]: adj[u]) if (dist[v] > dist[u] + w) dist[v] = dist[u] + w, q.push(v); } return dist; } VI run(const string& filename) { auto N = 0; Edges E; fstream fin{ filename }; VS A; for (string line; getline(fin, line); A.emplace_back(line)); for (auto& s: A) { transform(s.begin(), s.end(), s.begin(), [](auto c) { return c == ',' ? ' ' : c; }); stringstream ss{ s }; auto [u, v, w] = make_tuple(0, 0, 0); ss >> u; while (ss >> v >> w) E.emplace_back(u, v, w); ++N; } auto a = bell(E, N + 1), // +1 for 1-based indexing b = spfa(E, N + 1); assert(a == b); return b; } int main() { auto dist = run("test.txt"); VI A{ 7, 37, 59, 82, 99, 115, 133, 165, 188, 197 }; transform(A.begin(), A.end(), A.begin(), [&](auto x) { return dist[x]; }); copy(A.begin(), A.end(), ostream_iterator(cout, ",")); // 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068, return 0; } ```
--- ### Floyd-Warshall
📚 Lectures
* [The All-Pairs Shortest Path Problem (Section 18.3)](https://www.youtube.com/watch?v=TENbWZPz3Ho&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=62) * [The Floyd-Warshall Algorithm (Part 1) (Section 18.4, Part 1)](https://www.youtube.com/watch?v=ogcvCr02gqM&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=63) * [The Floyd-Warshall Algorithm (Part 2) (Section 18.4, Part 2)](https://www.youtube.com/watch?v=3cBHwPjDZxg&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=64)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File var key = { i: Int, j: Int -> "$i,$j" } var INF = (1e9 + 7).toInt() fun floyd_warshall(E: MutableMap, N: Int): Array { var dp = Array(N + 1) { Array(N + 1) { IntArray(N + 1) { INF } } } for (i in 0..N) for (j in 0..N) if (i == j) dp[0][i][j] = 0 else if (E.contains(key(i, j))) dp[0][i][j] = E[key(i, j)]!! for (k in 1..N) for (i in 1..N) for (j in 1..N) dp[k][i][j] = Math.min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]) return dp[N] } fun floyd_warshall_memopt(E: MutableMap, N: Int): Array { var pre = Array(N + 1) { IntArray(N + 1) { INF } } for (i in 0..N) for (j in 0..N) if (i == j) pre[i][j] = 0 else if (E.contains(key(i, j))) pre[i][j] = E[key(i, j)]!! for (k in 1..N) { var cur = Array(N + 1) { IntArray(N + 1) { INF } } for (i in 1..N) for (j in 1..N) cur[i][j] = Math.min(pre[i][j], pre[i][k] + pre[k][j]) pre = cur.also{ cur = pre } } return pre } fun run(filename: String) { var N = 0 var E = mutableMapOf() var first = true File(filename).forEachLine { if (!first) { var (u, v, w) = it.trim().split(" ").map{ it.toInt() } E[key(u, v)] = w } else { N = it.trim().split(" ").map{ it.toInt() }[0] first = false } } var a = floyd_warshall_memopt(E, N) var b = floyd_warshall(E, N) for (i in 1..N) for (j in 1..N) assert(a[i][j] == b[i][j]) // 💩 sanity check var cycle = false for (i in 1..N) if (a[i][i] < 0) cycle = true if (cycle) { println("$filename: contains a negative cycle") return } var best = INF for (i in 1..N) for (j in 1..N) best = Math.min(best, a[i][j]) println("$filename: $best") } fun main() { run("problem18.8test1.txt"); // problem18.8test1.txt: -2 run("problem18.8test2.txt"); // problem18.8test2.txt: contains a negative cycle run("problem18.8file1.txt"); // problem18.8file1.txt: contains a negative cycle run("problem18.8file2.txt"); // problem18.8file2.txt: contains a negative cycle run("problem18.8file3.txt"); // problem18.8file3.txt: -19 // run("problem18.8file4.txt"); } ``` *Javascript* ```javascript const LineByLine = require('n-readlines'); const assert = require('assert'); let key = (i, j) => `${i},${j}`; let floyd_warshall = (E, N) => { let dp = [...Array(N + 1)].map(_ => [...Array(N + 1)].map(_ => Array(N + 1).fill(Infinity))); for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) if (i == j) dp[0][i][j] = 0; else if (E.has(key(i, j))) dp[0][i][j] = E.get(key(i, j)); for (let k = 1; k <= N; ++k) for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) dp[k][i][j] = Math.min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]); return dp[N]; }; let floyd_warshall_memopt = (E, N) => { let pre = [...Array(N + 1)].map(_ => Array(N + 1).fill(Infinity)); for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) if (i == j) pre[i][j] = 0; else if (E.has(key(i, j))) pre[i][j] = E.get(key(i, j)); for (let k = 1; k <= N; ++k) { let cur = [...Array(N + 1)].map(_ => Array(N + 1).fill(Infinity)); for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) cur[i][j] = Math.min(pre[i][j], pre[i][k] + pre[k][j]); [pre, cur] = [cur, pre]; } return pre; }; let run = filename => { let E = new Map(); let input = new LineByLine(filename); let [N, _] = input.next().toString('ascii').split(' ').map(Number); let line; while (line = input.next()) { let [u, v, w] = line.toString('ascii').split(' ').map(Number); E.set(key(u, v), w); } let a = floyd_warshall_memopt(E, N), b = floyd_warshall(E, N); for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) assert(a[i][j] == b[i][j]); let cycle = false; for (let i = 1; i <= N; ++i) if (a[i][i] < 0) cycle = true; if (cycle) { console.log(`${filename}: contains a negative cycle`); return; } var best = Infinity; for (row of a) best = Math.min(best, ...row); console.log(`${filename}: ${best}`); }; run('problem18.8test1.txt'); // problem18.8test1.txt: -2 run('problem18.8test2.txt'); // problem18.8test2.txt: contains a negative cycle run('problem18.8file1.txt'); // problem18.8file1.txt: contains a negative cycle run('problem18.8file2.txt'); // problem18.8file2.txt: contains a negative cycle run('problem18.8file3.txt'); // problem18.8file3.txt: -19 // run('problem18.8file4.txt'); ``` *Python3* ```python key = lambda i, j: f'{i},{j}' def floyd_warshall(E, N): dp = [[[float('inf')] * (N + 1) for _ in range(N + 1)] for _ in range(N + 1)] for i in range(1, N + 1): for j in range(1, N + 1): if i == j: dp[0][i][j] = 0 elif key(i, j) in E: dp[0][i][j] = E[key(i, j)] for k in range(1, N + 1): for i in range(1, N + 1): for j in range(1, N + 1): dp[k][i][j] = min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]) return dp[N] def floyd_warshall_memopt(E, N): pre = [[float('inf')] * (N + 1) for _ in range(N + 1)] for i in range(1, N + 1): for j in range(1, N + 1): if i == j: pre[i][j] = 0 elif key(i, j) in E: pre[i][j] = E[key(i, j)] for k in range(1, N + 1): cur = [[float('inf')] * (N + 1) for _ in range(N + 1)] for i in range(1, N + 1): for j in range(1, N + 1): cur[i][j] = min(pre[i][j], pre[i][k] + pre[k][j]) pre, cur = cur, pre return pre def run(filename): E = {} N = 0 first = True with open(filename) as fin: while True: line = fin.readline() if not line: break if not first: u, v, w = [int(x) for x in line.strip().split(' ')] E[key(u, v)] = w else: N = [int(x) for x in line.strip().split(' ')][0] first = False a = floyd_warshall_memopt(E, N) b = floyd_warshall(E, N) for i in range(1, N + 1): for j in range(1, N + 1): assert(a[i][j] == b[i][j]) # 💩 sanity check cycle = False for i in range(1, N + 1): if a[i][i] < 0: cycle = True if cycle: print(f'{filename}: contains a negative cycle') return best = float('inf') for row in a: best = min(best, *row) print(f'{filename}: {best}') run('problem18.8test1.txt') # problem18.8test1.txt: -2 run('problem18.8test2.txt') # problem18.8test2.txt: contains a negative cycle run('problem18.8file1.txt') # problem18.8file1.txt: contains a negative cycle run('problem18.8file2.txt') # problem18.8file2.txt: contains a negative cycle run('problem18.8file3.txt') # problem18.8file3.txt: -19 # run('problem18.8file4.txt') ``` *C++* ```cpp #include #include #include #include #include #define PERF_TEST using namespace std; using LL = long long; using VL = vector; using VVL = vector; using VVVL = vector; using Edges = unordered_map; LL INF = 1e9 + 7; string key(int i, int j) { stringstream ss; ss << i << "," << j; return ss.str(); } VVL floyd_warshall(Edges& E, int N) { VVVL dp(N + 1, VVL(N + 1, VL(N + 1, INF))); for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) if (i == j) dp[0][i][j] = 0; else if (E.find(key(i, j)) != E.end()) dp[0][i][j] = E[key(i, j)]; for (auto k{ 1 }; k <= N; ++k) for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) dp[k][i][j] = min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]); return dp[N]; } VVL floyd_warshall_memopt(Edges& E, int N) { VVL pre(N + 1, VL(N + 1, INF)); for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) if (i == j) pre[i][j] = 0; else if (E.find(key(i, j)) != E.end()) pre[i][j] = E[key(i, j)]; for (auto k{ 1 }; k <= N; ++k) { VVL cur(N + 1, VL(N + 1, INF)); for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) cur[i][j] = min(pre[i][j], pre[i][k] + pre[k][j]); swap(pre, cur); } return pre; } void run(const string& filename) { Edges E; fstream fin{ filename }; int N, M; fin >> N >> M; for (int u, v, w; fin >> u >> v >> w; E[key(u, v)] = w); #ifdef PERF_TEST auto a = floyd_warshall_memopt(E, N); #else auto a = floyd_warshall_memopt(E, N), b = floyd_warshall(E, N); assert(a == b); // 💩 sanity check #endif auto cycle = false; for (auto i{ 1 }; i <= N && !cycle; ++i) cycle = a[i][i] < 0; if (cycle) { cout << filename << ": contains a negative cycle" << endl; return; } auto best = INF; for (auto& row: a) best = min(best, *min_element(row.begin(), row.end())); cout << filename << ": " << best << endl; } int main() { run("problem18.8test1.txt"); // problem18.8test1.txt: -2 run("problem18.8test2.txt"); // problem18.8test2.txt: contains a negative cycle run("problem18.8file1.txt"); // problem18.8file1.txt: contains a negative cycle run("problem18.8file2.txt"); // problem18.8file2.txt: contains a negative cycle run("problem18.8file3.txt"); // problem18.8file3.txt: -19 // run("problem18.8file4.txt"); return 0; } ```
--- # Part 4: Algorithms for NP-Hard Problems
--- ================================================ FILE: Algorithms-Illuminated/src/main.rs ================================================ fn main() { println!("Hello, world!"); } ================================================ FILE: Algorithms-Illuminated/target/.rustc_info.json ================================================ {"rustc_fingerprint":10491184667718451318,"outputs":{"17598535894874457435":{"success":true,"status":"","code":0,"stdout":"rustc 1.57.0 (f1edd0429 2021-11-29)\nbinary: rustc\ncommit-hash: f1edd0429582dd29cccacaf50fd134b05593bd9c\ncommit-date: 2021-11-29\nhost: x86_64-apple-darwin\nrelease: 1.57.0\nLLVM version: 13.0.0\n","stderr":""},"2797684049618456168":{"success":true,"status":"","code":0,"stdout":"___\nlib___.rlib\nlib___.dylib\nlib___.dylib\nlib___.a\nlib___.dylib\n","stderr":""},"931469667778813386":{"success":true,"status":"","code":0,"stdout":"___\nlib___.rlib\nlib___.dylib\nlib___.dylib\nlib___.a\nlib___.dylib\n/Users/clayton.wonng/.rustup/toolchains/stable-x86_64-apple-darwin\ndebug_assertions\nproc_macro\ntarget_arch=\"x86_64\"\ntarget_endian=\"little\"\ntarget_env=\"\"\ntarget_family=\"unix\"\ntarget_feature=\"fxsr\"\ntarget_feature=\"sse\"\ntarget_feature=\"sse2\"\ntarget_feature=\"sse3\"\ntarget_feature=\"ssse3\"\ntarget_os=\"macos\"\ntarget_pointer_width=\"64\"\ntarget_vendor=\"apple\"\nunix\n","stderr":""},"15537503139010883884":{"success":true,"status":"","code":0,"stdout":"___\nlib___.rlib\nlib___.dylib\nlib___.dylib\nlib___.a\nlib___.dylib\n","stderr":""}},"successes":{}} ================================================ FILE: Algorithms-Illuminated/target/CACHEDIR.TAG ================================================ Signature: 8a477f597d28d172789f06886806bc55 # This file is a cache directory tag created by cargo. # For information about cache directory tags see https://bford.info/cachedir/ ================================================ FILE: README.md ================================================ # Algorithms Illuminated * [algorithmsilluminated.org](https://www.algorithmsilluminated.org/) Coincidentally, my [algorithm learning journey](https://github.com/claytonjwong/Algorithms) which began in 2017 has occurred in parallel with the publication of Tim Roughgarden's (TR) 4-book series about algorithms and data structures. Over these years, I've purchased, studied, and provided feedback on TR's books. I was totally stoked when TR sent me a free copy of his 4th book for review before publication in 2020! I'm amazed by what can be done in near-linear time (ie. the amount of time to perform an algorithm is on the order of time to simply read the input), and it's awesome we can leverage these "for-free primitives" based upon computationally tractable problems as "building blocks" towards more complex solutions to computationally intractable (NP-Hard) problems via selective compromise on generality, correctness, and speed (ie. pick 2 of 3). [`💡` Can we do better?](https://en.wikipedia.org/wiki/Millennium_Prize_Problems#P_versus_NP)

--- # Part 1: The Basics
⚡️ TL;DR ⚡️
> A "fast algorithm" is an algorithm whose worst-case runtime grows slowly with the input size. * **Recursion Tree:** each child node corresponds to recusive invocations by its parent node * helpful visualization of work done by recursive algorithms to quantify aymptotic bounds * **Algorithm Analysis Principles** 1. Worst-Case * no input assumptions for general-purpose analysis 2. Big-Picture * drop constant factors and lower-order terms for runtime upper bound * mathematical tractability * primitive operation runtime depends on environmental factors * retains accurate predictions for slow/fast algorithms 3. Asymptotic Analysis * focus on the runtime growth for large inputs * only large inputs require algorithmic ingenuity * **For-Free Primitives** * Linear-time algorithms (the "holy grail") * Runtime is proportional to the input size * "Free" since the time to perform the algorithm is on the order of time to simply read the input * **Asymptotic Notation**: suppress constant factors and lower-order terms to focus on how an algorithm's runtime scales with input size * constant factors are too system dependent * lower-order terms are irrelevant for large inputs * **Big-O Notation:** `T(n) = O(f(n))` if-and-only-if `T(n)` is eventually bounded above by a constant multiple of `f(n)` * `T(n) = O(f(n))` if-and-only-if there exists positive constants *c* and *n*0 such that `T(n) <= c * O(f(n))` for all *n* greater-than-or-equal-to *n*0
--- ### Integer Multiplication
📚 Lectures
* [Introduction: Why Study Algorithms?](https://www.youtube.com/watch?v=yRM3sc57q0c&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=1) * [Introduction: Integer Multiplication](https://www.youtube.com/watch?v=6u0Vaj4nn54&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=2)
🎯 Solutions
*Python3* ```python def go(x, y): if x < 10 or y < 10: return x * y n = max(len(str(x)), len(str(y))) if n & 1: n += 1 # +1 is n is odd m = 10 ** (n // 2) a, b = x // m, x % m c, d = y // m, y % m p, q = a + b, c + d ac = go(a, c) bd = go(b, d) pq = go(p, q) adbc = pq - ac - bd return 10**n * ac + 10**(n//2) * adbc + bd x = 3141592653589793238462643383279502884197169399375105820974944592 y = 2718281828459045235360287471352662497757247093699959574966967627 assert(x * y == go(x, y)) ``` *Julia* ```julia using Test, Random function go(x, y) if x < 10 || y < 10 return x * y end n = max(length(string(x)), length(string(y))) if isodd(n) n += 1 end m = 10 ^ div(n, 2) # middle decimal value a, b = div(x, m), mod(x, m) c, d = div(y, m), mod(y, m) ac = go(a, c) ad = go(a, d) bc = go(b, c) bd = go(b, d) return 10^n * ac + 10^div(n, 2) * (ad + bc) + bd end Random.seed!(123456789) @testset "Recursive Integer Multiplication tests" begin for _ in 1:100 n = rand((1, 2, 4, 8, 16)) lo = 10^(n-1) hi = 10^n - 1 x = rand(lo:hi) y = rand(lo:hi) expect, actual = x * y, go(x, y) @test expect == actual println("($x x $y)\nexpect: $expect\nactual: $actual\n") end end ```
--- ### Karatsuba Multiplication
📚 Lectures
* [Introduction: Karatsuba Multiplication](https://www.youtube.com/watch?v=JCbZayFr9RE&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=3)
🎯 Solutions
*Julia* ```julia using Test, Random function go(x, y) if x < 10 || y < 10 return x * y end n = max(length(string(x)), length(string(y))) if isodd(n) n += 1 end m = 10 ^ div(n, 2) # middle decimal value a, b = div(x, m), mod(x, m) c, d = div(y, m), mod(y, m) p, q = a + b, c + d ac = go(a, c) bd = go(b, d) pq = go(p, q) adbc = pq - ac - bd return 10^n * ac + 10^div(n, 2) * adbc + bd end Random.seed!(123456789) @testset "Karatsuba Multiplication tests" begin for _ in 1:100 n = rand((1, 2, 4, 8, 16)) lo = 10^(n-1) hi = 10^n - 1 x = rand(lo:hi) y = rand(lo:hi) expect, actual = x * y, go(x, y) @test expect == actual println("($x x $y)\nexpect: $expect\nactual: $actual\n") end end ```
--- ### Merge Sort
📚 Lectures
* [MergeSort: Motivation and Example](https://www.youtube.com/watch?v=kiyRJ7GVWro&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=4) (Section 1.4, part 1) * [MergeSort: Pseudocode](https://www.youtube.com/watch?v=rBd5w0rQaFo&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=5) (Section 1.4, part 2) * [MergeSort: Analysis](https://www.youtube.com/watch?v=8ArtRiTkYEw&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=6) (Section 1.5)
🎯 Solutions
*Kotlin* ```java fun sort(A: IntArray): IntArray { fun merge(A: IntArray, B: IntArray): IntArray { var C = mutableListOf() var i = 0 var j = 0 while (i < A.size && j < B.size) if (A[i] < B[j]) C.add(A[i++]) else C.add(B[j++]) A.slice(i..A.lastIndex).forEach { C.add(it) } B.slice(j..B.lastIndex).forEach { C.add(it) } return C.toIntArray() } fun go(A: IntArray): IntArray { var N = A.size if (N < 2) return A var half = Math.floor(N / 2.0).toInt() var first = go(A.slice(0 until half).toIntArray()) var second = go(A.slice(half until N).toIntArray()) return merge(first, second) } return go(A) } fun main(args: Array) { sort(intArrayOf(5,3,8,9,1,7,0,2,6,4)).forEach { print("$it ") } // 0 1 2 3 4 5 6 7 8 9 println() } ``` *Javascript* ```javascript let sort = A => { let go = A => { let N = A.length; if (N < 2) return A; let half = Math.floor(N / 2); let first = go([...A.slice(0, half)]), second = go([...A.slice(half, N)]); return merge(first, second); }; let merge = (A, B, C = []) => { let M = A.length, N = B.length; let i = 0, j = 0; while (i < M && j < N) C.push(A[i] < B[j] ? A[i++] : B[j++]); C.push(...A.slice(i, M)); C.push(...B.slice(j, N)); return C; }; return go(A); }; console.log(sort([5,3,8,9,1,7,0,2,6,4])); // (10) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] ``` ```julia function go(A) N = length(A) if N < 2 return A end half = div(N, 2) first = go(A[begin:half]) second = go(A[half+1:end]) return merge(first, second) end function merge(A, B) C = [] i, j, k = 1, 1, 1 while i <= length(A) && j <= length(B) if A[i] < B[j] push!(C, A[i]) i += 1 else push!(C, B[j]) j += 1 end k += 1 end append!(C, @view A[i:end]) append!(C, @view B[j:end]) return C end println(go([5, 3, 8, 9, 1, 7, 0, 2, 6, 4])) # Any[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] ``` *Python3* ```python from math import floor def sort(A): def go(A): N = len(A) if N < 2: return A half = floor(N / 2) first = go(A[:half]) second = go(A[half:]) return merge(first, second) def merge(A, B): C = [] i = 0 j = 0 while i < len(A) and j < len(B): if A[i] < B[j]: C.append(A[i]); i += 1 else: C.append(B[j]); j += 1 C.extend(A[i:]) C.extend(B[j:]) return C return go(A) print(sort([5,3,8,9,1,7,0,2,6,4])) # [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] ``` *C++* ```cpp #include #include using namespace std; class Solution { public: using VI = vector; VI mergesort(VI& A) { return go(move(A)); } private: VI go(VI&& A) { auto N = A.size(); if( N < 2 ) return A; auto half = A.begin() + (N / 2); auto first = go({ A.begin(), half }), second = go({ half, A.end() }); return merge(first, second); } VI merge(VI& A, VI& B, VI C = {}) { auto i{ 0 }, j{ 0 }; while (i < A.size() && j < B.size()) C.push_back(A[i] < B[j] ? A[i++] : B[j++]); C.insert(C.end(), A.begin() + i, A.end()); C.insert(C.end(), B.begin() + j, B.end()); return C; } }; int main() { Solution::VI A{ 3,5,7,1,3,9,2,0 }; auto ans = Solution().mergesort(A); copy(ans.begin(), ans.end(), ostream_iterator(cout, " ")), cout << endl; // 0 1 2 3 4 5 6 7 8 9 return 0; } ```
--- ### Counting Inversions
📚 Lectures
* [The Divide-and-Conquer Paradigm](https://www.youtube.com/watch?v=7_AJfusC6UQ&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=13) (Section 3.1; part 1 of Section 3.2) * [Counting Inversions in O(n log n)](https://www.youtube.com/watch?v=I6ygiW8xN7Y&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=14) Time (Section 3.2, part 2)
🎯 Solutions
*Kotlin* ```java import java.io.File fun sort(A: IntArray): Pair { fun merge(A: IntArray, B: IntArray): Pair { var C = mutableListOf() var inv: Long = 0 var i = 0 var j = 0 while (i < A.size && j < B.size) if (A[i] < B[j]) { C.add(A[i++]) } else { inv += A.size - i // ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.add(B[j++]) } A.slice(i..A.lastIndex).forEach { C.add(it) } B.slice(j..B.lastIndex).forEach { C.add(it) } return Pair(C.toIntArray(), inv) } fun go(A: IntArray): Pair { var N = A.size if (N < 2) return Pair(A, 0) var half = Math.floor(N / 2.0).toInt() var (first, inv1) = go(A.slice(0 until half).toIntArray()) var (second, inv2) = go(A.slice(half until N).toIntArray()) var (third, inv3) = merge(first, second) return Pair(third, inv1 + inv2 + inv3) } return go(A) } fun run(filename: String): Long { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } var (_, inv) = sort(A.toIntArray()) return inv } fun main() { println("problem3.5test.txt: " + run("problem3.5test.txt")) // problem3.5test.txt: 28 println("problem3.5.txt: " + run("problem3.5.txt")) // problem3.5.txt: 2407905288 } ``` *Javascript* ```javascript let sort = A => { let go = A => { let N = A.length; if (N < 2) return [A, 0]; let half = Math.floor(N / 2); let [first, inv1] = go([...A.slice(0, half)]), [second, inv2] = go([...A.slice(half, N)]), [third, inv3] = merge(first, second); return [third, inv1 + inv2 + inv3]; }; let merge = (A, B, C = [], inv = 0) => { let M = A.length, N = B.length; let i = 0, j = 0; while (i < M && j < N) if (A[i] < B[j]) C.push(A[i++]); else inv += M - i, // ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.push(B[j++]); C.push(...A.slice(i, M)); C.push(...B.slice(j, N)); return [C, inv]; }; return go(A); }; let run = filename => { let A = []; require('fs').readFileSync(filename, 'utf-8').split(/\r?\n/).forEach(line => A.push(Number(line))); let [_, inv] = sort(A); return inv; } console.log(`problem3.5test.txt: ${run('problem3.5test.txt')}`); // problem3.5test.txt: 28 console.log(`problem3.5.txt: ${run('problem3.5.txt')}`); // problem3.5.txt: 2407905288 ``` *Julia* ```julia function go(A) N = length(A) if N < 2 return A, 0 end half = div(N, 2) first, inv1 = go(A[begin:half]) second, inv2 = go(A[half+1:end]) third, inv3 = merge(first, second) return third, inv1 + inv2 + inv3 end function merge(A, B) C, inv = [], 0 i, j, k = 1, 1, 1 while i <= length(A) && j <= length(B) if A[i] < B[j] push!(C, A[i]) i += 1 else push!(C, B[j]); inv += length(A) - i + 1 # ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions j += 1 end k += 1 end append!(C, @view A[i:end]) append!(C, @view B[j:end]) return C, inv end function run(filename) input = Int[] open(filename, "r") do file for line in eachline(file) push!(input, parse(Int, strip(line))) end end _, inv = go(input) return inv end println(run("problem3.5test.txt")) # 28 println(run("problem3.5.txt")) # 2407905288 ``` *Python3* ```python from math import floor def sort(A): def go(A): N = len(A) if N < 2: return [A, 0] half = floor(N / 2) first, inv1 = go(A[:half]) second, inv2 = go(A[half:]) third, inv3 = merge(first, second) return [third, inv1 + inv2 + inv3] def merge(A, B, inv = 0): C = [] i = 0 j = 0 while i < len(A) and j < len(B): if A[i] < B[j]: C.append(A[i]); i += 1 else: inv += len(A) - i # ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.append(B[j]); j += 1 C.extend(A[i:]) C.extend(B[j:]) return [C, inv] return go(A) def run(filename): A = [] with open(filename) as fin: while True: line = fin.readline() if not line: break A.append(int(line)) _, inv = sort(A) return inv print(f"problem3.5test.txt: {run('problem3.5test.txt')}") # problem3.5test.txt: 28 print(f"problem3.5.txt: {run('problem3.5.txt')}") # problem3.5.txt: 2407905288 ``` *C++* ```cpp #include #include #include using namespace std; class Solution { public: using VL = vector; using Pair = pair; using fun = function; Pair merge(VL& A, VL& B, VL C = {}, long inv = 0) { auto i = 0, j = 0; while (i < A.size() && j < B.size()) { if (A[i] < B[j]) { C.push_back(A[i++]); } else { inv += A.size() - i; // ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.push_back(B[j++]); } } C.insert(C.end(), A.begin() + i, A.end()); C.insert(C.end(), B.begin() + j, B.end()); return { C, inv }; } Pair inversions(VL& A) { fun go = [&](VL&& A) -> Pair { int N = A.size(); if (N < 2) return { A, 0 }; int half = N / 2; auto [first, inv1] = go({ A.begin(), A.begin() + half }); auto [second, inv2] = go({ A.begin() + half, A.end() }); auto [third, inv3] = merge(first, second); return { third, inv1 + inv2 + inv3 }; }; return go(move(A)); } }; long run(string filename) { Solution solution; Solution::VL A; fstream fin{ filename }; for (string line; fin >> line; A.push_back(stol(line))); auto [_, inv] = solution.inversions(A); return inv; } int main() { cout << "problem3.5test.txt: " << run("problem3.5test.txt") << endl // problem3.5test.txt: 28 << "problem3.5.txt: " << run("problem3.5.txt") << endl; // problem3.5.txt: 2407905288 return 0; } ```
--- ### Recursive Matrix Multiplication
📚 Lectures
* [Strassen 's Subcubic Matrix Multiplication Algorithm](https://www.youtube.com/watch?v=ORrM-aSNZUs&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=15)
🎯 Solutions
* Assume N is a power of 2 for simplicity #### Iterative Matrix Multiplication * **Cubic runtime:** O(N3) *Python3* ```python N = 2 X = [[1, 2], [3, 4]] Y = [[5, 6], [7, 8]] Z = [[0] * N for _ in range(N)] for i in range(N): for j in range(N): for k in range(N): Z[i][j] += X[i][k] * Y[k][j] ``` #### Standard Recursive Matrix Multiplication * **Cubic runtime:** O(N3) *Python3* ```python import numpy as np def go(X, Y): n = X.shape[0] # Base case: 1x1 matrix if n == 1: return X * Y # Divide: Partition A and B into n/2 x n/2 submatrices k = n // 2 A, B = X[:k, :k], X[:k, k:] C, D = X[k:, :k], X[k:, k:] E, F = Y[:k, :k], Y[:k, k:] G, H = Y[k:, :k], Y[k:, k:] # Combine: Reconstruct the full matrix from quadrants Z = np.zeros((n, n), dtype=X.dtype) Z[:k, :k], Z[:k, k:] = go(A, E) + go(B, G), go(A, F) + go(B, H) Z[k:, :k], Z[k:, k:] = go(C, E) + go(D, G), go(C, F) + go(D, H) return Z ``` #### Strassen's Matrix Multiplication * **Sub-cubic runtime:** O(Nlog2(7)) *Python3* ```python import numpy as np def go(X, Y): n = X.shape[0] # Base case: 1x1 matrix if n == 1: return X * Y # Divide: Partition A and B into n/2 x n/2 submatrices k = n // 2 A, B = X[:k, :k], X[:k, k:] C, D = X[k:, :k], X[k:, k:] E, F = Y[:k, :k], Y[:k, k:] G, H = Y[k:, :k], Y[k:, k:] # Combine: Reconstruct the full matrix from quadrants Z = np.zeros((n, n), dtype=X.dtype) P1 = go(A, F - H) P2 = go(A + B, H) P3 = go(C + D, E) P4 = go(D, G - E) P5 = go(A + D, E + H) P6 = go(B - D, G + H) P7 = go(A - C, E + F) Z[:k, :k], Z[:k, k:] = (P5 + P4 - P2 + P6), (P1 + P2) Z[k:, :k], Z[k:, k:] = (P3 + P4), (P1 + P5 - P3 - P7) return Z ``` #### Strassen's Matrix Multiplication (Variants) * **Sub-cubic runtime:** O(Nlog2(7)) *Python3* ```python # # Strassen-family Recursive Matrix Multiplication (2x2 block form) # # Methods: # - "strassen" : classical Strassen scheme (7 recursive multiplies) # - "winograd" : Winograd rearrangement of Strassen (7 multiplies, fewer additions) # - "basis2017" : alternative-basis 7-multiply scheme (basis change + recombination) # # All methods: # - Recursively split X and Y into 2x2 blocks # - Perform 7 recursive multiplications # - Recombine into result blocks # # Assumption: n is a power of 2 # def go(X: np.ndarray, Y: np.ndarray, method: str = "strassen") -> np.ndarray: n = X.shape[0] # Base case: 1x1 matrix multiply if n == 1: return X * Y k = n // 2 # Partition X into 2x2 block matrix # X = [[A11 A12] # [A21 A22]] A11, A12 = X[:k, :k], X[:k, k:] A21, A22 = X[k:, :k], X[k:, k:] # Partition Y into 2x2 block matrix # Y = [[B11 B12] # [B21 B22]] B11, B12 = Y[:k, :k], Y[:k, k:] B21, B22 = Y[k:, :k], Y[k:, k:] # Allocate result matrix Z = np.zeros((n, n), dtype=X.dtype) # ============================================================ # Classical Strassen scheme # ============================================================ if method == "strassen": # 7 recursive multiplications M1 = go(A11 + A22, B11 + B22, method) M2 = go(A21 + A22, B11, method) M3 = go(A11, B12 - B22, method) M4 = go(A22, B21 - B11, method) M5 = go(A11 + A12, B22, method) M6 = go(A21 - A11, B11 + B12, method) M7 = go(A12 - A22, B21 + B22, method) # Recombine into C blocks C11 = M1 + M4 - M5 + M7 C12 = M3 + M5 C21 = M2 + M4 C22 = M1 - M2 + M3 + M6 Z[:k, :k] = C11 Z[:k, k:] = C12 Z[k:, :k] = C21 Z[k:, k:] = C22 return Z # ============================================================ # Winograd rearrangement of Strassen # Same 7 multiplies, different additive structure # ============================================================ if method == "winograd": # Core recursive products t = go(A11, B11, method) u = go(A21 - A11, B12 - B22, method) v = go(A21 + A22, B12 - B11, method) # w reuses t to reduce total additions w = t + go(A21 + A22 - A11, B11 + B22 - B12, method) # Recombine result blocks C11 = t + go(A12, B21, method) C12 = w + v + go(A11 + A12 - A21 - A22, B22, method) C21 = w + u + go(A22, B21 + B12 - B11 - B22, method) C22 = w + u + v Z[:k, :k] = C11 Z[:k, k:] = C12 Z[k:, :k] = C21 Z[k:, k:] = C22 return Z # ============================================================ # 2017 alternative-basis 7-multiply scheme # Performs a basis change on A22 and B22 blocks # ============================================================ if method == "basis2017": # Basis transformation of lower-right blocks A22p = A12 - A21 + A22 B22p = B12 - B21 + B22 # Linear combinations used by recursive multiplies t1 = A21 + A22p t2 = A22p - A12 t3 = A22p - A11 t4 = B22p - B11 t5 = B21 + B22p t6 = B22p - B12 # 7 recursive multiplications M1 = go(A11, B11, method) M2 = go(A12, B21, method) M3 = go(A21, t4, method) M4 = go(A22p, B22p, method) M5 = go(t1, t5, method) M6 = go(t2, t6, method) M7 = go(t3, B12, method) # Initial recombination C11 = M1 + M2 C12 = M5 - M7 C21 = M3 + M6 C22 = M5 + M6 - M2 - M4 # Final correction step required by basis change C12 = C12 - C22 C21 = C22 - C21 Z[:k, :k] = C11 Z[:k, k:] = C12 Z[k:, :k] = C21 Z[k:, k:] = C22 return Z ```
--- ### Closest Pair
📚 Lectures
* [O(nlogn) Algorithm for Closest Pair: Part One](https://www.youtube.com/watch?v=3pUOv_ocJyA&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=16) * [O(nlogn) Algorithm for Closest Pair: Part Two](https://www.youtube.com/watch?v=7tiafUFrlBw&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=17)
🎯 Solutions
* **Input:** `n >= 2` points p1=(x1,y1), p2=(x2,y2), ... , pn=(xn,yn) * **Output:** The pair pi,pj of distinct points with smallest Euclidean distance d(pi,pj) #### Brute-Force * **Quadratic runtime:** O(N2) *Python3* ```python P = [(1, 8), (2, 5), (4, 7), (6, 3)] N = len(P) INF = 1234567890 # arbitary choice for infinity def dist(i, j): x1, y1 = P[i] x2, y2 = P[j] return (x1 - x2) ** 2 \ + (y1 - y2) ** 2 best_dist = 123456789 best_pair = None D = [[INF] * N for _ in range(N)] for i in range(N): for j in range(N): D[i][j] = dist(i, j) if i != j: # candidates for best pair cannot be the same point if best_dist >= D[i][j]: best_dist = D[i][j] best_pair = (i, j) i, j = best_pair print(f'best pair: {P[i]}, {P[j]}') ``` #### Recursive * **Super-linear runtime:** O(NlogN) ```python INF = 1234567890 # arbitary choice for infinity def distance(a, b): x1, y1 = a x2, y2 = b return (x1 - x2) ** 2 \ + (y1 - y2) ** 2 def split(Px, Py, d): # median x-coordinate median = Px[len(Px) // 2][0] # identify points near left/right boundary Sy = [(x, y) for x, y in Py if median - d <= x <= median + d] # return the best split pair (if it exists) best_dist, best_pairs = INF, set() for i in range(len(Sy) - 1): for j in range(i + 1, min(i + 1 + 7, len(Sy))): cand = distance(Sy[i], Sy[j]) if best_dist > cand: best_dist = cand best_pairs = set() if best_dist == cand: best_pairs.add((Sy[i], Sy[j])) return best_dist, best_pairs def best(P): best_dist, best_pairs = INF, set() for i in range(len(P)): for j in range(i + 1, len(P)): cand = distance(P[i], P[j]) if P[i] != P[j] else INF if best_dist > cand: best_dist = cand best_pairs = set() if best_dist == cand: best_pairs.add((P[i], P[j])) return best_dist, best_pairs def go(Px, Py): if len(Px) <= 3: # Base case return best(Px) Lx, Rx = Px[:len(Px) // 2], Px[len(Px) // 2:] Ly, Ry = [], [] for x, y in Py: if x <= Lx[-1][0]: Ly.append((x, y)) else: Ry.append((x, y)) dist_left, best_left = go(Lx, Ly) dist_right, best_right = go(Rx, Ry) dist_split, best_split = split(Px, Py, min(dist_left, dist_right)) cands = sorted([(dist_left, best_left), (dist_right, best_right), (dist_split, best_split)], key=lambda it: it[0]) best_dist, best_pairs = cands[0][0], set() for dist, pairs in cands: if dist == best_dist: for a, b in pairs: best_pairs.add((a, b) if a < b else (b, a)) return best_dist, best_pairs def run(points, expected_best_pair): Px = sorted(points, key=lambda it: it[0]) Py = sorted(points, key=lambda it: it[1]) _, best_pair = go(Px, Py) print(f'points: {points}') print(f'actual: {sorted(best_pair)}') print(f'expect: {sorted(expected_best_pair)}') print() assert(sorted(best_pair) == sorted(expected_best_pair)) ```
--- ### Quick Sort
📚 Lectures
* [QuickSort: Overview](https://www.youtube.com/watch?v=ETo1cpLN7kk&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=24) (Section 5.1) * [Partitioning Around a Pivot Element](https://www.youtube.com/watch?v=LYzdRN5iFdA&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=25) (Section 5.2) * [Choosing a Good Pivot](https://www.youtube.com/watch?v=kqO46FOUTbI&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=26) (Sections 5.3 and 5.4) * [QuickSort Analysis (Part 1)](https://www.youtube.com/watch?v=sToWtKSYlMw&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=27) (Section 5.5, part 1) * [QuickSort Analysis (Part 2)](https://www.youtube.com/watch?v=4t_Y-aGLkok&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=28) (Section 5.5, part 2) * [QuickSort Analysis (Part 3)](https://www.youtube.com/watch?v=IBTvneWhFsA&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=29) (Section 5.5, part 3) * [Sorting Requires Omega(n log n) Comparisons](https://www.youtube.com/watch?v=aFveIyII5D4&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=30) (Section 5.6) * [Proofs by Induction and the Correctness of QuickSort](https://www.youtube.com/watch?v=Colb_4jAy8A&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=36) (Appendix A) * [Quick Review of Discrete Probability](https://www.youtube.com/watch?v=uLeIMwMHX5U&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=37) (Appendix B)
🎯 Solutions
*Kotlin* ```java import java.io.File typealias PivotFunc = (A: MutableList, L: Int, R: Int) -> (Int) var pivotLeft: PivotFunc = { _: MutableList, L: Int, _: Int -> L } var pivotRight: PivotFunc = { _: MutableList, _: Int, R: Int -> R } fun _pivotMedian(A: MutableList, L: Int, R: Int): Int { var M = L + (R - L) / 2 var cand = intArrayOf(A[L], A[M], A[R]) cand.sort() var target = cand[1] if (target == A[L]) return L if (target == A[M]) return M if (target == A[R]) return R return -1 } var pivotMedian: PivotFunc = { A: MutableList, L: Int, R: Int -> _pivotMedian(A, L, R) } fun partition(A: MutableList, L: Int, R: Int, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { var i = L + 1 var j = L + 1 var k = choosePivot(A, L, R) A[k] = A[L].also { A[L] = A[k] } // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] A[i] = A[j].also { A[j] = A[i] } ++i } ++j } A[L] = A[i - 1].also { A[i - 1] = A[L] } // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 } fun quicksort(A: MutableList, L: Int, R: Int, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { if (R <= L) return 0 var k = partition(A, L, R, choosePivot) return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot) } fun run(filename: String, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } return quicksort(A, 0, A.size - 1, choosePivot) } fun main() { var filename = "problem5.6.txt" println(" left: ${run(filename, pivotLeft)}") // left: 162085 println(" right: ${run(filename, pivotRight)}") // right: 164123 println("median: ${run(filename, pivotMedian)}") // median: 138382 } ``` *Javascript* ```javascript let pivotLeft = (A, L, R) => L; let pivotRight = (A, L, R) => R; let pivotMedian = (A, L, R) => { let M = L + Math.floor((R - L) / 2); let cand = [A[L], A[M], A[R]].sort((a, b) => a - b), target = cand[1]; if (target == A[L]) return L; if (target == A[M]) return M; if (target == A[R]) return R; }; let partition = (A, L, R, choosePivot) => { let i = L + 1, j = L + 1, k = choosePivot(A, L, R); [A[L], A[k]] = [A[k], A[L]]; // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] [A[i], A[j]] = [A[j], A[i]]; ++i; } ++j; } [A[L], A[i - 1]] = [A[i - 1], A[L]]; // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; }; let quicksort = (A, L, R, choosePivot) => { if (R <= L) return 0; let k = partition(A, L, R, choosePivot); return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot); }; let run = (filename, choosePivot) => { let A = []; let LineByLine = require("n-readlines"); let input = new LineByLine(filename); for (let line; line = input.next(); A.push(Number(line))); return quicksort(A, 0, A.length - 1, choosePivot); } let filename = 'problem5.6.txt'; console.log(` left: ${run(filename, pivotLeft)}`); // left: 162085 console.log(` right: ${run(filename, pivotRight)}`); // right: 164123 console.log(`median: ${run(filename, pivotMedian)}`); // median: 138382 ``` *Python3* ```python def pivotLeft(A, L, R): return L def pivotRight(A, L, R): return R def pivotMedian(A, L, R): M = L + (R - L) // 2 cand = sorted([A[L], A[M], A[R]]) target = cand[1] if target == A[L]: return L if target == A[M]: return M if target == A[R]: return R def partition(A, L, R, choosePivot): i = L + 1 j = L + 1 k = choosePivot(A, L, R) A[L], A[k] = A[k], A[L] # swap pivot A[k] with first element of subarray A[L] while j <= R: if A[j] < A[L]: # maintain loop invariant A[i] < pivot < A[j] A[i], A[j] = A[j], A[i] i += 1 j += 1 A[L], A[i - 1] = A[i - 1], A[L] # swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 def quicksort(A, L, R, choosePivot): if R <= L: return 0 k = partition(A, L, R, choosePivot) return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot) def run(filename, choosePivot): A = [] with open(filename) as fin: while True: line = fin.readline() if not line: break A.append(int(line)) return quicksort(A, 0, len(A) - 1, choosePivot) filename = 'problem5.6.txt' print(f' left: {run(filename, pivotLeft)}') # left: 162085 print(f' right: {run(filename, pivotRight)}') # right: 164123 print(f'median: {run(filename, pivotMedian)}') # median: 138382 ``` *C++* ```cpp #include #include #include using namespace std; using VI = vector; using fun = function; fun pivotLeft = [](VI& A, int L, int R) { return L; }; fun pivotRight = [](VI& A, int L, int R) { return R; }; fun pivotMedian = [](VI& A, int L, int R) { auto M = L + (R - L) / 2; VI cand{ A[L], A[M], A[R] }; sort(cand.begin(), cand.end()); auto target = cand[1]; if (target == A[L]) return L; if (target == A[M]) return M; if (target == A[R]) return R; }; int partition(VI& A, int L, int R, fun choosePivot) { auto i = L + 1, j = L + 1, k = choosePivot(A, L, R); swap(A[L], A[k]); // swap pivot A[k] with first element of the subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] swap(A[i], A[j]); ++i; } ++j; } swap(A[L], A[i - 1]); // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; } int quicksort(VI& A, int L, int R, fun choosePivot) { if (R <= L) return 0; auto k = partition(A, L, R, choosePivot); return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot); } int run(string& filename, fun choosePivot) { VI A; fstream fin{ filename }; for (string line; fin >> line; A.push_back(stoi(line))); int N = A.size(); return quicksort(A, 0, N - 1, choosePivot); } int main() { string filename{ "problem5.6.txt" }; cout << " left: " << run(filename, pivotLeft) << endl // left: 162085 << " right: " << run(filename, pivotRight) << endl // right: 164123 << "median: " << run(filename, pivotMedian) << endl; // median: 138382 return 0; } ```
--- ### Randomized Linear-Time Selection
📚 Lectures
* [Randomized Linear-Time Selection](https://www.youtube.com/watch?v=nFw6x7DoYbs&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=31) (Section 6.1) * [Randomized Linear-Time Selection (Analysis)](https://www.youtube.com/watch?v=rX2u2CnpveQ&list=PLEGCF-WLh2RLHqXx6-GZr_w7LgqKDXxN_&index=32) (Section 6.2)
🎯 Solutions
*Kotlin* ```java import java.io.File import kotlin.random.Random fun partition(A: MutableList, L: Int, R: Int): Int { var i = L + 1 var j = L + 1 var k = Random.nextInt(L, R + 1) // +1 for L..R inclusive A[L] = A[k].also { A[k] = A[L] } // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] A[i] = A[j].also { A[j] = A[i] } ++i } ++j } A[L] = A[i - 1].also { A[i - 1] = A[L] } // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 } fun rselect(A: MutableList, i: Int, L_: Int, R_: Int): Int { var L = L_ var R = R_ var k = partition(A, L, R) if (i == k) return A[k] // 🎯 lucky guess if (i < k) R = k - 1 else L = k + 1 return rselect(A, i, L, R) } fun run(filename: String, i: Int): Int { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } var N = A.size return rselect(A, i - 1, 0 , N - 1) // -1 for 0-based indexing } fun main() { println("problem6.5test1.txt: " + run("problem6.5test1.txt", 5)) // problem6.5test1.txt: 5469 println("problem6.5test2.txt: " + run("problem6.5test2.txt", 50)) // problem6.5test2.txt: 4715 } ``` *Javascript* ```javascript let random = (L, R) => Math.floor(Math.random() * (R + 1 - L) + L); // +1 for L..R inclusive let partition = (A, L, R) => { let i = L + 1, j = L + 1, k = random(L, R); [A[L], A[k]] = [A[k], A[L]]; // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] [A[i], A[j]] = [A[j], A[i]]; ++i; } ++j; } [A[L], A[i - 1]] = [A[i - 1], A[L]]; // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; }; let rselect = (A, i, L, R) => { let k = partition(A, L, R); if (i == k) return A[k]; // 🎯 lucky guess if (i < k) R = k - 1; else L = k + 1; return rselect(A, i, L, R); } let run = (filename, i) => { let A = []; let LineByLine = require("n-readlines"); let input = new LineByLine(filename); for (let line; line = input.next(); A.push(Number(line))); let N = A.length; return rselect(A, i - 1, 0, N - 1); // -1 for 0-based indexing }; console.log(`problem6.5test1.txt: ${run('problem6.5test1.txt', 5)}`); // problem6.5test1.txt: 5469 console.log(`problem6.5test2.txt: ${run('problem6.5test2.txt', 50)}`); // problem6.5test2.txt: 4715 ``` *Python3* ```python from random import uniform from math import floor def partition(A, L, R): i = L + 1 j = L + 1 k = floor(uniform(L, R)) A[L], A[k] = A[k], A[L] # swap pivot A[k] with first element of subarray A[L] while j <= R: if A[j] < A[L]: # maintain loop invariant A[i] < pivot < A[j] A[i], A[j] = A[j], A[i] i += 1 j += 1 A[L], A[i - 1] = A[i - 1], A[L] # swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 def rselect(A, i, L, R): k = partition(A, L, R) if i == k: return A[k] # 🎯 lucky guess if i < k: R = k - 1 else: L = k + 1 return rselect(A, i, L, R) def run(filename, i): A = [] with open(filename) as fin: while True: line = fin.readline() if not line: break A.append(int(line)) N = len(A) return rselect(A, i - 1, 0, N - 1) # -1 for 0-based indexing print('problem6.5test1.txt:', run('problem6.5test1.txt', 5)) # problem6.5test1.txt: 5469 print('problem6.5test2.txt:', run('problem6.5test2.txt', 50)) # problem6.5test2.txt: 4715 ``` *C++* ```cpp #include #include #include #include using namespace std; using VI = vector; int random(int L, int R) { random_device rd; mt19937 gen{ rd() }; uniform_int_distribution dist(L, R); return dist(gen); } int partition(VI& A, int L, int R) { auto i = L + 1, j = L + 1, k = random(L, R); swap(A[L], A[k]); // swap pivot A[k] with first element of the subarray A[L] while (j <= R) { if (A[j] < A[L]) // maintain loop invariant A[i] < pivot < A[j] swap(A[i++], A[j]); ++j; } swap(A[L], A[i - 1]); // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; } int rselect(VI& A, int i, int L, int R) { auto k = partition(A, L, R); if (i == k) return A[k]; // 🎯 lucky guess if (i < k) R = k - 1; else L = k + 1; return rselect(A, i, L, R); } int run(string filename, int i, VI A = {}) { fstream fin{ filename }; for (string line; fin >> line; A.push_back(stoi(line))); int N = A.size(); return rselect(A, i - 1, 0, N - 1); // -1 for 0-based indexing } int main() { cout << "problem6.5test1.txt: " << run("problem6.5test1.txt", 5) << endl; // problem6.5test1.txt: 5469 cout << "problem6.5test2.txt: " << run("problem6.5test2.txt", 50) << endl; // problem6.5test2.txt: 4715 return 0; } ```
--- # Part 2: Graph Algorithms and Data Structures
--- ### Topological Sort
📚 Lectures
* [Graphs: The Basics (from 2:06 to 6:39)](https://www.youtube.com/watch?v=4Ih3UhVuEtw&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=1) (Sections 7.1 and 7.2) * [Graph Representations](https://www.youtube.com/watch?v=b-Mfu8dPv9U&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=2) (Sections 7.3 and 7.4) * [Graph Search Overview](https://www.youtube.com/watch?v=SW6jwg7WS48&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=3) (Section 8.1) * [Breadth-First Search](https://www.youtube.com/watch?v=73qCvXsYkfk&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=4) (Section 8.2, Part 1) * [Depth-First Search](https://www.youtube.com/watch?v=73qCvXsYkfk&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=4) (Section 8.4) * [Topological Sort](https://www.youtube.com/watch?v=ozso3xxkVGU&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=8) (Section 8.5)
🎯 Solutions
*Kotlin* ```java import java.util.Queue import java.util.LinkedList class Solution(val adj: MutableMap>) { var N: Int var color: Int var m = mutableMapOf() var seen = mutableSetOf() init { N = adj.size color = 0 } fun init(start: Int) { color = start m.clear() seen.clear() } fun topoSortBFS(): String { init(1) // 👉 color forward from 1..N bfs() return toString() } fun topoSortDFS(): String { init(N) // 👈 color reverse from N..1 (as the recursive stack unwinds) adj.forEach{ (u, _) -> dfs(u) } return toString() } fun bfs() { var degree = mutableMapOf() adj.forEach{ (_, neighbors) -> neighbors.forEach{ v -> degree[v] = 1 + degree.getOrDefault(v, 0) } } var q: Queue = LinkedList(adj.map{ (u, _) -> u }.filter{ !degree.contains(it) }) while (0 < q.size) { var u = q.poll() m[u] = color++ adj[u]!!.forEach{ v -> degree[v] = degree[v]!!.minus(1) if (degree[v] == 0 && !seen.contains(v)) { q.add(v); seen.add(v) } } } } fun dfs(u: Char) { if (seen.contains(u)) return seen.add(u) adj[u]!!.forEach{ v -> dfs(v) } m[u] = color-- } override fun toString(): String { var s = mutableListOf() adj.forEach{ (u, _) -> s.add("$u: ${m[u]}") } return s.joinToString("\n") } } fun main() { var adj = mutableMapOf>( 's' to listOf('v', 'w'), 'v' to listOf('t'), 'w' to listOf('t'), 't' to listOf() ) var solution = Solution(adj) println("BFS:\n${solution.topoSortBFS()}\n\nDFS:\n${solution.topoSortDFS()}") // BFS: // s: 1 // v: 2 // w: 3 // t: 4 // DFS: // s: 1 // v: 3 // w: 2 // t: 4 } ``` *Javascript* ```javascript class Solution { constructor(adj) { this.adj = adj; this.N = this.adj.size; } init(start) { this.color = start; this.seen = new Set(); this.m = new Map(); } topo_sort_bfs() { this.init(1); // 👉 color forward from 1..N this.bfs(); return this.to_string(); } topo_sort_dfs() { this.init(this.N); // 👈 color reverse from N..1 (as the recursive stack unwinds) for (let [u, _] of [...this.adj]) this.dfs(u); return this.to_string(); } bfs() { let degree = new Map(); for (let [u, _] of [...this.adj]) { degree.set(u, (degree.get(u) || 0)); for (let v of this.adj.get(u)) degree.set(v, 1 + (degree.get(v) || 0)); } let q = [...this.adj].map(([u, _]) => u).filter(u => !degree.get(u)); let seen = new Set(q); while (q.length) { let u = q.shift(); this.m.set(u, this.color++); for (let v of this.adj.get(u)) { degree.set(v, -1 + degree.get(v)); if (!degree.get(v) && !seen.has(v)) q.push(v), seen.add(v); } } } dfs(u) { if (this.seen.has(u)) return; this.seen.add(u); for (let v of this.adj.get(u)) if (!this.seen.has(v)) this.dfs(v); this.m.set(u, this.color--); } to_string() { let s = []; for (let [u, color] of [...this.m]) s.push(`${u}: ${color}`); return s.join('\n'); } } let adj = new Map(); adj.set('s', ['v', 'w']); adj.set('v', ['t']); adj.set('w', ['t']); adj.set('t', []); let solution = new Solution(adj); console.log(`BFS:\n${solution.topo_sort_bfs()}\n\nDFS:\n${solution.topo_sort_dfs()}`); // BFS: // s: 1 // v: 2 // w: 3 // t: 4 // DFS: // t: 4 // v: 3 // w: 2 // s: 1 ``` *Python3* ```python from collections import deque class Solution: def __init__(self, adj): self.adj = adj self.N = len(adj) self.seen = set() self.m = {} def init(self, start): self.color = start self.seen.clear() self.m.clear() def topo_sort_bfs(self): self.init(1) # 👉 color forward from 1..N self.bfs() return self.to_string() def topo_sort_dfs(self): self.init(self.N) # 👈 color reverse from N..1 (as the recursive stack unwinds) for u, _ in self.adj.items(): self.dfs(u) return self.to_string() def bfs(self): degree = {} for _, neighbors in self.adj.items(): for v in neighbors: degree[v] = 1 + (degree[v] if v in degree else 0) q = deque(u for u, _ in self.adj.items() if u not in degree) self.seen.update(*q) while q: u = q.popleft() self.m[u] = self.color; self.color += 1 for v in adj[u]: degree[v] -= 1 if not degree[v] and v not in self.seen: q.append(v); self.seen.add(v) def dfs(self, u): if u in self.seen: return self.seen.add(u) for v in adj[u]: self.dfs(v) self.m[u] = self.color; self.color -= 1 def to_string(self): s = [] for u, color in self.m.items(): s.append(f'{u}: {color}') return '\n'.join(s) # # graph from Quiz 8.3 on page 45 of Algorithms Illuminated: Part 2 # adj = { 's': ['v', 'w'], 'v': ['t'], 'w': ['t'], 't': [] } solution = Solution(adj) print(f'BFS:\n{solution.topo_sort_bfs()}\n\nDFS:\n{solution.topo_sort_dfs()}') # BFS: # s: 1 # v: 2 # w: 3 # t: 4 # DFS: # t: 4 # v: 3 # w: 2 # s: 1 ``` *C++* ```cpp #include #include #include #include #include #include using namespace std; using VI = vector; using AdjList = unordered_map; using Set = unordered_set; using Map = unordered_map; using Queue = queue; using fun = function; class Solution { private: AdjList adj; const int N; Map m; Set seen; int color; public: Solution(AdjList& adj) : adj{ adj }, N{ int(adj.size()) } { } void init(int start) { m.clear(); seen.clear(); color = start; } string topo_sort_bfs() { init(1); // 👉 color forward from 1..N bfs(); return to_string(); } string topo_sort_dfs() { init(N); // 👈 color reverse from N..1 (as the recursive stack unwinds) for (auto [u, _]: adj) dfs(u); return to_string(); } void bfs() { Map degree; for (auto [_, neighbors]: adj) for (auto v: neighbors) ++degree[v]; Queue q; for (auto [u, _]: adj) if (!degree[u] && seen.insert(u).second) q.push(u); while (q.size()) { auto u = q.front(); q.pop(); m[u] = color++; for (auto v: adj[u]) if (!--degree[v] && seen.insert(v).second) q.push(v); } } void dfs(char start) { fun go = [&](auto u) { if (!seen.insert(u).second) return; for (auto v: adj[u]) go(v); m[u] = color--; }; go(start); } string to_string() { ostringstream os; for (auto [u, color]: m) os << u << ": " << color << endl; return os.str(); } }; int main() { // // graph from Quiz 8.3 on page 45 of Algorithms Illuminated: Part 2 // AdjList adj{ { 's', { 'v', 'w' } }, { 'v', { 't' } }, { 'w', { 't' } }, { 't', {} } }; Solution solution{ adj }; cout << "BFS:" << endl << solution.topo_sort_bfs() << endl << "DFS:" << endl << solution.topo_sort_dfs() << endl; // BFS: // t: 4 // w: 3 // v: 2 // s: 1 // // DFS: // s: 1 // w: 2 // v: 3 // t: 4 return 0; } ```
--- ### Kosaraju
📚 Lectures
* [Computing Strongly Connected Components (Part 1) (Section 8.6, Part 1)](https://www.youtube.com/watch?v=O98hLTYVN3c&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=9) * [Computing Strongly Connected Components (Part 2) (Section 8.6, Part 2)](https://www.youtube.com/watch?v=gbs3UNRJIYk&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=10) * [The Structure of the Web (Section 8.7)](https://www.youtube.com/watch?v=7YodysGShlo&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=11)
🎯 Solutions
*Kotlin* ```java import java.util.Stack import java.io.File class RecursiveSolution(var adj: MutableMap>, var rev: MutableMap>) { fun topo_sort(): MutableList { var list = mutableListOf() var seen = mutableSetOf() fun go(u: Int) { if (seen.contains(u)) return seen.add(u) for (v in rev[u]!!) go(v) list.add(0, u) } for ((u, _) in rev) go(u) return list } fun kosaraju(): MutableList> { var lists = mutableListOf>() var seen = mutableSetOf() fun go(u: Int, list: MutableList) { if (seen.contains(u)) return list.add(u); seen.add(u) for (v in adj[u]!!) go(v, list) } for (u in topo_sort()) { if (seen.contains(u)) continue var list = mutableListOf() go(u, list) lists.add(list.toList()) } return lists } } class IterativeSolution(var adj: MutableMap>, var rev: MutableMap>) { fun topo_sort(): MutableList { var list = mutableListOf() var seen = mutableSetOf() for ((u, _) in rev) { if (seen.contains(u)) continue var stack = Stack() stack.push(u); seen.add(u) while (!stack.empty()) { var u = stack.last() for (v in rev[u]!!) { if (!seen.contains(v)) { stack.push(v); seen.add(v) } } if (u == stack.last()) list.add(0, stack.pop()) } } return list } fun kosaraju(): MutableList> { var lists = mutableListOf>() var seen = mutableSetOf() for (u in topo_sort()) { if (seen.contains(u)) continue var list = mutableListOf() var stack = Stack() stack.push(u); seen.add(u) while (!stack.empty()) { var u = stack.last() for (v in adj[u]!!) { if (!seen.contains(v)) { stack.push(v); seen.add(v) } } if (u == stack.last()) list.add(stack.pop()) } lists.add(list.toList()) } return lists } } fun run(filename: String) { var adj = mutableMapOf>() var rev = mutableMapOf>() File(filename).forEachLine { var (u, v) = it.trim().split(" ").map{ it.toInt() } if (!adj.contains(u)) adj[u] = mutableListOf(); if (!adj.contains(v)) adj[v] = mutableListOf() if (!rev.contains(u)) rev[u] = mutableListOf(); if (!rev.contains(v)) rev[v] = mutableListOf() adj[u]!!.add(v) rev[v]!!.add(u) } // var solution = RecursiveSolution(adj, rev) var solution = IterativeSolution(adj, rev) var A = solution.kosaraju() A.sortWith(Comparator{ a: List, b: List -> b.size - a.size }) println(filename + ": " + A.map{ it.size }.slice(0 until Math.min(A.size, 5)).joinToString(" ")) } fun main() { run("section8.6.5page64.txt"); // Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run("problem8.10test1.txt"); // Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run("problem8.10test2.txt"); // Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run("problem8.10test3.txt"); // Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run("problem8.10test4.txt"); // Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run("problem8.10test5.txt"); // Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run("problem8.10.txt"); // Challenge data set: Vertices are labeled as positive integers from 1 to 875714 // section8.6.5page64.txt: 4 3 3 1 // problem8.10test1.txt: 3 3 3 // problem8.10test2.txt: 3 3 2 // problem8.10test3.txt: 3 3 1 1 // problem8.10test4.txt: 7 1 // problem8.10test5.txt: 6 3 2 1 // problem8.10.txt: 434821 968 459 313 211 } ``` *Javascript* ```javascript class BaseSolution { constructor(adj, rev) { this.adj = adj; this.rev = rev; } } class RecursiveSolution extends BaseSolution { constructor(adj, rev) { super(adj, rev); } topo_sort() { let list = []; let seen = new Set(); let go = u => { if (seen.has(u)) return; seen.add(u); for (let v of [...this.rev.get(u)]) go(v); list.unshift(u); }; for (let [u, _] of [...this.rev]) go(u); return list; } kosaraju() { let lists = []; let seen = new Set(); let go = (u, list) => { if (seen.has(u)) return; seen.add(u); list.push(u); for (let v of [...this.adj.get(u)]) go(v, list); }; for (let u of this.topo_sort()) { let list = []; go(u, list); lists.push([...list]); } lists.sort((a, b) => b.length - a.length); return lists; } } class IterativeSolution extends BaseSolution { constructor(adj, rev) { super(adj, rev); } topo_sort() { let list = []; let seen = new Set(); for (let [u, _] of [...this.rev]) { if (seen.has(u)) continue; let stack = [ u ]; seen.add(u); stack.back = () => stack[stack.length - 1]; while (stack.length) { let u = stack.back(); for (let v of [...this.rev.get(u)]) if (!seen.has(v)) stack.push(v), seen.add(v); if (u == stack.back()) list.unshift(stack.pop()); } } return list; } kosaraju() { let lists = []; let seen = new Set(); for (let u of this.topo_sort()) { if (seen.has(u)) continue; let list = []; let stack = [ u ]; seen.add(u); stack.back = () => stack[stack.length - 1]; while (stack.length) { let u = stack.back(); for (let v of [...this.adj.get(u)]) if (!seen.has(v)) stack.push(v), seen.add(v); if (u == stack.back()) list.push(stack.pop()); } lists.push([...list]); } lists.sort((a, b) => b.length - a.length); return lists; } } let run = filename => { let adj = new Map(), rev = new Map(); let LineByLine = require('n-readlines'); let input = new LineByLine(filename); let line; while (line = input.next()) { let [u, v] = String.fromCharCode(...line).split(' ').map(Number); if (!adj.has(u)) adj.set(u, []); if (!adj.has(v)) adj.set(v, []); if (!rev.has(u)) rev.set(u, []); if (!rev.has(v)) rev.set(v, []); adj.get(u).push(v); rev.get(v).push(u); } // let A = new RecursiveSolution(adj, rev).kosaraju(); let A = new IterativeSolution(adj, rev).kosaraju(); console.log(`${filename}: ${A.slice(0, Math.min(A.length, 5)).map(scc => scc.length).join(' ')}`); }; run('section8.6.5page64.txt') // Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run('problem8.10test1.txt') // Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run('problem8.10test2.txt') // Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run('problem8.10test3.txt') // Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run('problem8.10test4.txt') // Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run('problem8.10test5.txt') // Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run('problem8.10.txt') // Challenge data set: Vertices are labeled as positive integers from 1 to 875714 // section8.6.5page64.txt: 4 3 3 1 // problem8.10test1.txt: 3 3 3 // problem8.10test2.txt: 3 3 2 // problem8.10test3.txt: 3 3 1 1 // problem8.10test4.txt: 7 1 // problem8.10test5.txt: 6 3 2 1 // problem8.10.txt: 434821 968 459 313 211 ``` *Python3* ```python from collections import deque from functools import cmp_to_key class BaseSolution: def __init__(self, adj, rev): self.adj = adj self.rev = rev class RecursiveSolution(BaseSolution): def topo_sort(self): list = deque() seen = set() def go(u): if u in seen: return seen.add(u) for v in self.rev[u]: go(v) list.appendleft(u) for u in self.rev.keys(): go(u) return list def kosaraju(self): lists = [] seen = set() def go(u, list): if u in seen: return seen.add(u) list.append(u) for v in self.adj[u]: go(v, list) for u in self.topo_sort(): list = [] go(u, list) lists.append(list.copy()) lists.sort(key = cmp_to_key(lambda a, b: len(b) - len(a))) return lists class IterativeSolution(BaseSolution): def topo_sort(self): list = deque() seen = set() for u in self.rev.keys(): if u in seen: continue stack = [ u ]; seen.add(u) while len(stack): u = stack[-1] for v in self.rev[u]: if v not in seen: stack.append(v); seen.add(v) if u == stack[-1]: list.appendleft(stack.pop()) return list def kosaraju(self): lists = [] seen = set() for u in self.topo_sort(): if u in seen: continue list = deque() stack = [ u ]; seen.add(u) while len(stack): u = stack[-1] for v in self.adj[u]: if v not in seen: stack.append(v); seen.add(v) if u == stack[-1]: list.appendleft(stack.pop()) lists.append(list.copy()) lists.sort(key = cmp_to_key(lambda a, b: len(b) - len(a))) return lists def run(filename): adj, rev = {}, {} with open(filename) as fin: while True: line = fin.readline().strip() if not line: break u, v = [int(x) for x in line.split()] if u not in adj: adj[u] = [] if v not in adj: adj[v] = [] if u not in rev: rev[u] = [] if v not in rev: rev[v] = [] adj[u].append(v) rev[v].append(u) # solution = RecursiveSolution(adj, rev) solution = IterativeSolution(adj, rev) A = solution.kosaraju() print(filename + ': ' + ' '.join(str(len(scc)) for scc in A[:5])) run('section8.6.5page64.txt') # Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run('problem8.10test1.txt') # Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run('problem8.10test2.txt') # Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run('problem8.10test3.txt') # Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run('problem8.10test4.txt') # Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run('problem8.10test5.txt') # Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run('problem8.10.txt') # Challenge data set: Vertices are labeled as positive integers from 1 to 875714 # section8.6.5page64.txt: 4 3 3 1 # problem8.10test1.txt: 3 3 3 # problem8.10test2.txt: 3 3 2 # problem8.10test3.txt: 3 3 1 1 # problem8.10test4.txt: 7 1 # problem8.10test5.txt: 6 3 2 1 # problem8.10.txt: 434821 968 459 313 211 ``` *C++* ```cpp #include #include #include #include #include #include using namespace std; using List = deque; using Lists = deque; using AdjList = unordered_map; using Set = unordered_set; using Map = unordered_map; namespace Base { class Solution { protected: AdjList adj, rev; public: Solution(AdjList& adj, AdjList& rev) : adj{ adj }, rev{ rev } {} }; } namespace Recursive { struct Solution : public Base::Solution { Solution(AdjList& adj, AdjList& rev) : Base::Solution{ adj, rev } {} Lists kosaraju() { Lists lists; Set seen; using fun = function; fun go = [&](auto u, auto& list) { if (!seen.insert(u).second) return; list.push_back(u); for (auto v: adj[u]) go(v, list); }; for (auto u: topo_sort()) { List list; go(u, list); lists.emplace_back(list); } sort(lists.begin(), lists.end(), [](auto& a, auto& b) { return b.size() < a.size(); }); return lists; } List topo_sort() { List list; Set seen; using fun = function; fun go = [&](auto u) { if (!seen.insert(u).second) return; for (auto v: rev[u]) go(v); list.push_front(u); }; for (auto [u, _]: rev) go(u); return list; } }; } namespace Iterative { struct Solution : public Base::Solution { Solution(AdjList& adj, AdjList& rev) : Base::Solution{ adj, rev } {} Lists kosaraju() { Lists lists; Set seen; for (auto u: topo_sort()) { if (seen.find(u) != seen.end()) continue; List list; List stack{ u }; seen.insert(u); while (stack.size()) { auto u = stack.back(); for (auto v: adj[u]) if (seen.insert(v).second) stack.push_back(v); if (u == stack.back()) list.push_back(u), stack.pop_back(); } lists.emplace_back(list); } sort(lists.begin(), lists.end(), [](auto& a, auto& b) { return b.size() < a.size(); }); return lists; } List topo_sort() { List list; Set seen; for (auto [u, _]: rev) { if (seen.find(u) != seen.end()) continue; List stack{ u }; seen.insert(u); while (stack.size()) { auto u = stack.back(); for (auto v: rev[u]) if (seen.insert(v).second) stack.push_back(v); if (u == stack.back()) list.push_front(stack.back()), stack.pop_back(); } } return list; } }; } void run(string filename) { int u, v; AdjList adj, rev; fstream fin{ filename }; for (string line; fin >> u >> v;) { adj[u].push_back(v); rev[v].push_back(u); } auto A = Iterative::Solution{ adj, rev }.kosaraju(); A.resize(min(A.size(), size_t(5))); cout << filename << ": "; for (auto i{ 0 }; i < A.size(); cout << A[i++].size() << " "); cout << endl; } int main() { run("section8.6.5page64.txt"); // Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run("problem8.10test1.txt"); // Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run("problem8.10test2.txt"); // Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run("problem8.10test3.txt"); // Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run("problem8.10test4.txt"); // Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run("problem8.10test5.txt"); // Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run("problem8.10.txt"); // Challenge data set: Vertices are labeled as positive integers from 1 to 875714 // section8.6.5page64.txt: 4 3 3 1 // problem8.10test1.txt: 3 3 3 // problem8.10test2.txt: 3 3 2 // problem8.10test3.txt: 3 3 1 1 // problem8.10test4.txt: 7 1 // problem8.10test5.txt: 6 3 2 1 // problem8.10.txt: 434821 968 459 313 211 return 0; } ```
--- ### Dijkstra
📚 Lectures
* [Shortest Paths and Dijkstra's Algorithm (Sections 9.1 and 9.2, Part 1)](https://www.youtube.com/watch?v=jRlNVmRjdRk&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=12) * [Dijkstra's Algorithm: Examples (Section 9.2, Part 2)](https://www.youtube.com/watch?v=ahYhIzLklYo&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=13) * [Correctness of Dijkstra's Algorithm (Section 9.3)](https://www.youtube.com/watch?v=sb7j3EW055M&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=14) * [Implementation and Running Time of Dijkstra's Algorithm (0:00-4:30) (Section 9.4)](https://www.youtube.com/watch?v=00LtSn_PQjc&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=15) * [Data Structures Overview (Section 10.1)](https://www.youtube.com/watch?v=cMrQxxrKg8I&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=16) * [Heaps: Operations and Applications (Sections 10.2 and 10.3)](https://www.youtube.com/watch?v=mNYHDv7SbDI&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=17) * [Speeding Up Dijkstra's Algorithm With Heaps (4:30-26:27) (Section 10.4)](https://www.youtube.com/watch?v=00LtSn_PQjc&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=15) * [Heaps: Implementation Details (Section 10.5)](https://www.youtube.com/watch?v=6VI5kJu8Mv4&list=PLEGCF-WLh2RJ5W-pt-KE9GUArTDzVwL1P&index=19&t=0s)
🎯 Solutions
*Kotlin* ```java import java.io.File import java.util.PriorityQueue var INF = (1e9 + 7).toInt() interface BaseSolution { fun run(filename: String, queries: Array): String } class NaiveSolution : BaseSolution { fun dijkstra(E: List>): MutableMap { var dist = mutableMapOf() var seen = mutableSetOf() var start = 1 dist[start] = 0; seen.add(start) var found: Boolean; do { found = false var best_v = INF var best_w = INF for ((u, v, w) in E) { if (!seen.contains(u) || seen.contains(v)) continue found = true if (best_w > dist[u]!! + w) { best_v = v best_w = dist[u]!! + w } } var v = best_v var w = best_w dist[v] = w; seen.add(v) } while (found) return dist } override fun run(filename: String, queries: Array): String { var E = mutableListOf>() File(filename).forEachLine { var words = it.trim().split("\t") var u = words[0].toInt() for (i in 1 until words.size) { var (v, w) = words[i].split(",").map{ it.toInt() } E.add(Triple(u, v, w)) } } var dist = dijkstra(E.toList()) return queries.map{ dist[it] }.joinToString(" ") } } class HeapSolution : BaseSolution { fun dijkstra(adj: MutableMap>>): MutableMap { var dist = mutableMapOf() var seen = mutableSetOf() var start = 1 dist[start] = 0 var q = PriorityQueue>(Comparator{ a: Pair, b: Pair -> a.first.compareTo(b.first) }) q.add(Pair(0, start)) while (0 < q.size) { var (cost, u) = q.poll() if (seen.contains(u)) continue dist[u] = cost; seen.add(u) for ((w, v) in adj[u]!!) { if (seen.contains(v)) continue q.add(Pair(cost + w, v)) } } return dist } override fun run(filename: String, queries: Array): String { var adj = mutableMapOf>>() File(filename).forEachLine { var words = it.trim().split("\t") var u = words[0].toInt() if (!adj.contains(u)) adj[u] = mutableListOf() for (i in 1 until words.size) { var (v, w) = words[i].split(",").map{ it.toInt() } adj[u]!!.add(Pair(w, v)) } } var dist = dijkstra(adj) return queries.map{ dist[it] }.joinToString(" ") } } fun run(solution: BaseSolution) { println(solution.run("problem9.8test.txt", arrayOf(1, 2, 3, 4, 5, 6, 7, 8))) println(solution.run("problem9.8.txt", arrayOf(7, 37, 59, 82, 99, 115, 133, 165, 188, 197))) } fun main() { run(NaiveSolution()) // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(HeapSolution()) // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 } ``` *Javascript* ```javascript let LineByLine = require('n-readlines'); let INF = Number(1e9 + 7); class NaiveSolution { dijkstra(E) { let dist = new Map(); let seen = new Set(); let start = 1; dist[start] = 0; seen.add(start); for (;;) { let found = false; let best_v = INF, best_w = INF; for (let [u, v, w] of E) { if (!seen.has(u) || seen.has(v)) continue; found = true; if (best_w > dist[u] + w) best_v = v, best_w = dist[u] + w; } if (!found) break; let [v, w] = [best_v, best_w]; dist[v] = w; seen.add(v); } return dist; } run(filename, queries) { let E = []; let input = new LineByLine(filename); let line; while (line = input.next()) { let words = String.fromCharCode(...line).trim().split(/\s+/); let u = Number(words[0]); for (let i = 1; i < words.length; ++i) { let [v, w] = words[i].split(',').map(Number); E.push([ u, v, w ]); } } let dist = this.dijkstra(E); return queries.map(x => dist[x]).join(' '); } } let heapkey = x => Array.isArray(x) ? x[0] : x; let heappush = (A, x, f = Math.min) => { let P = i => Math.floor((i - 1) / 2); // parent A.push(x); let N = A.length, i = N - 1; while (0 < i && heapkey(A[i]) == f(heapkey(A[i]), heapkey(A[P(i)]))) { [A[i], A[P(i)]] = [A[P(i)], A[i]]; i = P(i); } }; let heappop = (A, f = Math.min) => { let L = i => 2 * i + 1, // children R = i => 2 * i + 2; let N = A.length, i = 0; let top = A[0]; [A[0], A[N - 1]] = [A[N - 1], A[0]], A.pop(), --N; let ok = true; do { ok = true; let left = f == Math.min ? Infinity : -Infinity, right = left; if (L(i) < N && heapkey(A[i]) != f(heapkey(A[i]), heapkey(A[L(i)]))) ok = false, left = heapkey(A[L(i)]); if (R(i) < N && heapkey(A[i]) != f(heapkey(A[i]), heapkey(A[R(i)]))) ok = false, right = heapkey(A[R(i)]); if (!ok) { let j = left == f(left, right) ? L(i) : R(i); [A[i], A[j]] = [A[j], A[i]]; i = j; } } while (!ok); return top; }; class HeapSolution { dijkstra(adj) { let dist = {}; let seen = new Set(); let start = 1; let q = [[ 0, start ]]; while (q.length) { let [cost, u] = heappop(q); if (seen.has(u)) continue; dist[u] = cost, seen.add(u); for (let [w, v] of (adj[u] || [])) heappush(q, [ dist[u] + w, v ]); } return dist; } run(filename, queries) { let adj = {}; let input = new LineByLine(filename); let line; while (line = input.next()) { let words = String.fromCharCode(...line).trim().split('\t'); let u = Number(words[0]); if (!(u in adj)) adj[u] = []; for (let i = 1; i < words.length; ++i) { let [v, w] = words[i].split(',').map(Number); adj[u].push([ w, v ]); } } let dist = this.dijkstra(adj); return queries.map(x => dist[x]).join(' '); } } let run = solution => { console.log(solution.run('problem9.8test.txt', [1, 2, 3, 4, 5, 6, 7, 8])); console.log(solution.run('problem9.8.txt', [7, 37, 59, 82, 99, 115, 133, 165, 188, 197])); }; run(new NaiveSolution()); // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(new HeapSolution()); // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 ``` *Python3* ```python from abc import ABC, abstractmethod from heapq import heappush, heappop INF = int(1e9 + 7) class BaseSolution(ABC): @abstractmethod def run(self, filename, queries): raise NotImplementedError class NaiveSolution(BaseSolution): def dijkstra(self, E): dist = {} seen = set() start = 1 dist[start] = 0; seen.add(start) while True: found = False best_v = INF best_w = INF for u, v, w in E: if u not in seen or v in seen: continue found = True if best_w > dist[u] + w: best_v = v best_w = dist[u] + w if not found: break v, w = best_v, best_w dist[v] = w; seen.add(v) return dist def run(self, filename, queries): E = [] with open(filename) as fin: while True: line = fin.readline() if not line: break words = line.split() u = int(words[0]) for i in range(1, len(words)): v, w = map(int, words[i].split(',')) E.append([ u, v, w ]) dist = self.dijkstra(E) return ' '.join(str(dist[x]) for x in queries) class HeapSolution(BaseSolution): def dijkstra(self, adj, start = 1): dist = {} seen = set() q = [[ 0, start ]] while len(q): cost, u = heappop(q) if u in seen: continue dist[u] = cost; seen.add(u) for w, v in adj[u]: if v not in seen: heappush(q, [ dist[u] + w, v ]) return dist def run(self, filename, queries): adj = {} with open(filename) as fin: while True: line = fin.readline() if not line: break words = line.split() u = int(words[0]) if u not in adj: adj[u] = [] for i in range(1, len(words)): v, w = map(int, words[i].split(',')) adj[u].append([ w, v ]) dist = self.dijkstra(adj) return ' '.join(str(dist[x]) for x in queries) def run(solution): print(solution.run('problem9.8test.txt', [1, 2, 3, 4, 5, 6, 7, 8])) print(solution.run('problem9.8.txt', [7, 37, 59, 82, 99, 115, 133, 165, 188, 197])) run(NaiveSolution()) # 0 1 2 3 4 4 3 2 # 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(HeapSolution()) # 0 1 2 3 4 4 3 2 # 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 ``` *C++* ```cpp #include #include #include #include #include #include #include using namespace std; using Queries = vector; using Distance = unordered_map; using Set = unordered_set; class BaseSolution { protected: static constexpr auto INF = int(1e9 + 7); public: virtual string run(string filename, Queries&& queries) = 0; }; class NaiveSolution : public BaseSolution { using Edge = tuple; using Edges = vector; public: Distance dijkstra(Edges& E) { Distance dist; Set seen; auto start{ 1 }; dist[start] = 0, seen.insert(start); for (;;) { auto found = false; auto best_v = INF, best_w = INF; for (auto [u, v, w]: E) { if (seen.find(u) == seen.end() || seen.find(v) != seen.end()) continue; found = true; if (best_w > dist[u] + w) best_v = v, best_w = dist[u] + w; } if (!found) break; auto [v, w] = tie(best_v, best_w); dist[v] = w, seen.insert(v); } return dist; } string run(string filename, Queries&& queries) { Edges E; fstream fin{ filename }; string line; int u, v, w; char _; while (getline(fin, line)) { istringstream is{ line }; for (is >> u; is >> v >> _ >> w; E.push_back({ u, v, w })); } auto dist = dijkstra(E); ostringstream os; transform(queries.begin(), queries.end(), ostream_iterator(os, " "), [&](auto x) { return dist[x]; }); return os.str(); } }; class HeapSolution : public BaseSolution { using Pair = pair; using Pairs = vector; using AdjList = unordered_map; priority_queue> q; public: Distance dijkstra(AdjList& adj) { Distance dist; Set seen; for (auto [u, _]: adj) dist[u] = INF; auto start{ 1 }; q.push({ 0, start }); while (q.size()) { auto [cost, u] = q.top(); q.pop(); if (!seen.insert(u).second) continue; dist[u] = cost; for (auto [w, v]: adj[u]) if (seen.find(v) == seen.end()) q.push({ dist[u] + w, v }); } return dist; } string run(string filename, Queries&& queries) { AdjList adj; fstream fin{ filename }; string line; int u, v, w; char _; while (getline(fin, line)) { istringstream is{ line }; for (is >> u; is >> v >> _ >> w; adj[u].push_back({ w, v })); } auto dist = dijkstra(adj); ostringstream os; transform(queries.begin(), queries.end(), ostream_iterator(os, " "), [&](auto x) { return dist[x]; }); return os.str(); } }; void run(BaseSolution&& solution) { cout << "problem9.8test.txt: " << solution.run("problem9.8test.txt", Queries{1, 2, 3, 4, 5, 6, 7, 8 }) << endl << "problem9.8.txt " << solution.run("problem9.8.txt", Queries{7, 37, 59, 82, 99, 115, 133, 165, 188, 197 }) << endl; } int main() { run(NaiveSolution()); // problem9.8test.txt: 0 1 2 3 4 4 3 2 // problem9.8.txt 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(HeapSolution()); // problem9.8test.txt: 0 1 2 3 4 4 3 2 // problem9.8.txt 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 return 0; } ```
--- # Part 3: Greedy Algorithms and Dynamic Programming
--- ### Greedy Scheduling
📚 Lectures
* [Introduction to Greedy Algorithms (Section 13.1)](https://www.youtube.com/watch?v=NTFmxA3qgoo&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=4&t=0s) * [A Scheduling Problem (Section 13.2)](https://www.youtube.com/watch?v=jK3d_GLzKQA&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=5) * [Developing a Greedy Algorithm (Section 13.3)](https://www.youtube.com/watch?v=AJOTgHYjL6A&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=6) * [Scheduling: Correctness Proof (Part 1) (Section 13.4, Part 1)](https://www.youtube.com/watch?v=oyLxydVkQwo&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=7) * [Scheduling: Correctness Proof (Part 2) (Section 13.4, Part 2)](https://www.youtube.com/watch?v=fmXVPYxmFsU&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=8) * [Scheduling: Correctness Proof (Part 3) (Section 13.4, Part 3)](https://www.youtube.com/watch?v=aBFG5PghX68&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=9)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File data class Job(val weight: Long, val length: Long) class Solution { fun minSum(jobs: Array): Pair { class Diff: Comparator { override fun compare(a: Job?, b: Job?): Int { if (a == null || b == null) return 0 var first = a.weight - a.length var second = b.weight - b.length return if (first == second) b.weight.compareTo(a.weight) else second.compareTo(first) // sort by descending difference, break ties in favor of jobs with larger weights } } class Ratio: Comparator { override fun compare(a: Job?, b: Job?): Int { if (a == null || b == null) return 0 var first = a.weight.toDouble() / a.length var second = b.weight.toDouble() / b.length return if (first == second) b.weight.compareTo(a.weight) else second.compareTo(first) // sort by descending difference, break ties in favor of jobs with larger weights } } return Pair(calcSum(jobs, Diff()), calcSum(jobs, Ratio())) } private fun calcSum(jobs: Array, comp: Comparator): Long { jobs.sortWith(comp) var time: Long = 0 var total: Long = 0 jobs.forEach { job -> time += job.length total += job.weight * time } return total } } fun run(filename: String) { var jobs = mutableListOf() var first = true File(filename).forEachLine { if (!first) { var words = it.trim().split(" ").map{ it.toLong() } var (weight, length) = words jobs.add(Job(weight, length)) } else { first = false } } var (diff, ratio) = Solution().minSum(jobs.toTypedArray()) println("$diff, $ratio") // sub-optimal, optimal } fun main() { run("problem13.4test1.txt") // 23, 22 run("problem13.4test2.txt") // 68615, 67247 run("problem13.4.txt") // 69119377652, 67311454237 } ``` *Javascript* ```javascript let LineByLine = require('n-readlines'); class Job { constructor(weight, length) { this.weight = weight; this.length = length; } } class Solution { minSum(jobs) { let diff = (a, b) => { let first = a.weight - a.length, second = b.weight - b.length; return first == second ? b.weight - a.weight : second - first; // sort by descending difference, break ties in favor of jobs with larger weights }; let ratio = (a, b) => { let first = a.weight / a.length, second = b.weight / b.length; return first == second ? b.weight - a.weight : second - first; // sort by descending ratio, break ties in favor of jobs with larger weights }; return [ this._calcSum(jobs, diff), this._calcSum(jobs, ratio) ]; } _calcSum(jobs, comp, time = 0) { jobs.sort((a, b) => comp(a, b)); return jobs.reduce((total, job) => total + job.weight * (time += job.length), 0); } } let run = filename => { let jobs = []; let input = new LineByLine(filename); let line = input.next(); // N while (line = input.next()) { let words = String.fromCharCode(...line).trim().split(' '); let [weight, length] = words.map(Number); jobs.push(new Job(weight, length)); } let [diff, ratio] = new Solution().minSum(jobs); console.log(`${diff}, ${ratio}`); // sub-optimal, optimal }; run('problem13.4test1.txt'); // 23, 22 run('problem13.4test2.txt'); // 68615, 67247 run('problem13.4.txt'); // 69119377652, 67311454237 ``` *Python3* ```python from functools import cmp_to_key class Job: def __init__(self, weight, length): self.weight = weight self.length = length class Solution: def minSum(self, jobs): def diff(a, b): first = a.weight - a.length second = b.weight - b.length return b.weight - a.weight if first == second else second - first # sort by descending difference, break ties in favor of jobs with larger weights def ratio(a, b): first = a.weight / a.length second = b.weight / b.length return b.weight - a.weight if first == second else second - first # sort by descending difference, break ties in favor of jobs with larger weights return [ self._calcSum(jobs, diff), self._calcSum(jobs, ratio) ] def _calcSum(self, jobs, comp, time = 0, total = 0): jobs.sort(key = cmp_to_key(lambda a, b: comp(a, b))) for job in jobs: time += job.length total += job.weight * time return total def run(filename): jobs = [] with open(filename) as fin: line = fin.readline() # N while True: line = fin.readline().strip() if not line: break words = line.split() weight, length = [int(x) for x in words] jobs.append(Job(weight, length)) diff, ratio = Solution().minSum(jobs) print(f'{diff}, {ratio}') # sub-optimal, optimal run('problem13.4test1.txt') # 23, 22 run('problem13.4test2.txt') # 68615, 67247 run('problem13.4.txt') # 69119377652, 67311454237 ``` *C++* ```cpp #include #include #include #include using namespace std; using LL = long long; struct Job { LL weight, length; Job(LL weight, LL length) : weight{ weight }, length{ length } {} }; using Jobs = vector; class Solution { public: using Pair = pair; // sub-optimal, optimal Pair minSum(Jobs& jobs) { auto diff = [](auto& a, auto& b) { auto first = a.weight - a.length, second = b.weight - b.length; return first == second ? b.weight < a.weight : second < first; // sort by descending difference, break ties in favor of jobs with larger weights }; auto ratio = [](auto& a, auto& b) { auto first = double(a.weight) / a.length, second = double(b.weight) / b.length; return first == second ? b.weight < a.weight : second < first; // sort by descending ratio, break ties in favor of jobs with larger weights }; return { calcSum(jobs, diff), calcSum(jobs, ratio) }; } private: template LL calcSum(Jobs& jobs, Comp comp, LL time = 0LL) { sort(jobs.begin(), jobs.end(), comp); return accumulate(jobs.begin(), jobs.end(), 0LL, [&](LL total, auto& job) { return total += job.weight * (time += job.length); }); } }; void run(const string& filename) { Jobs jobs; LL N, weight, length; fstream fin{ filename }; for (fin >> N; fin >> weight >> length; jobs.emplace_back(Job{ weight, length })); auto [diff, ratio] = Solution().minSum(jobs); cout << diff << ", " << ratio << endl; } int main() { run("problem13.4test1.txt"); // 23, 22 run("problem13.4test2.txt"); // 68615, 67247 run("problem13.4.txt"); // 69119377652, 67311454237 return 0; } ```
--- ### Huffman Codes
📚 Lectures
* [Codes (Section 14.1)](https://www.youtube.com/watch?v=K3WZhFZT6Y0&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=33) * [Codes as Trees (Section 14.2)](https://www.youtube.com/watch?v=HESrV5VDu8c&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=34) * [Huffman's Greedy Algorithm (Part 1) (Section 14.3, Part 1)](https://www.youtube.com/watch?v=NM6FZB7IfS8&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=35) * [Huffman's Greedy Algorithm (Part 2) (Section 14.3, Part 2)](https://www.youtube.com/watch?v=PPmn9osMDyI&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=36) * [Huffman's Algorithm: Correctness Proof (Part 1) (Section 14.4, Part 1)](https://www.youtube.com/watch?v=jibgSDjWxdI&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=37) * [Huffman's Algorithm: Correctness Proof (Part 2) (Section 14.4, Part 2)](https://www.youtube.com/watch?v=dAjCcqZKYf4&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=38)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File import java.util.PriorityQueue import java.util.Queue import java.util.LinkedList var INF = (1e9 + 7).toInt() data class Tree(val weight: Int, val left: Tree? = null, val right: Tree? = null) /* fun encode(A: List): Tree { var q = PriorityQueue(Comparator{ a: Tree, b: Tree -> a.weight.compareTo(b.weight) }) for (weight in A) q.add(Tree(weight)) while (1 < q.size) { var a = q.poll() var b = q.poll() var c = Tree(a.weight + b.weight, a, b) q.add(c) } return q.poll() } */ /* * Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation * of a sorting subroutine, followed by a linear amount of additional work. */ fun encode(A: MutableList): Tree { A.sort() var first: Queue = LinkedList(A.map{ weight -> Tree(weight) }.toList()) var second: Queue = LinkedList() var next = mutableListOf() while (1 < first.size + second.size) { next.clear() do { if (0 < first.size && 0 < second.size) { if (first.peek().weight < second.peek().weight) next.add(first.poll()) else next.add(second.poll()) } else if (0 < first.size) next.add(first.poll()) else if (0 < second.size) next.add(second.poll()) } while (next.size < 2) var (a, b) = next var c = Tree(a.weight + b.weight, a, b) second.add(c) } return second.poll() } fun run(filename: String): Pair { var A = mutableListOf() var first = true File(filename).forEachLine { if (!first) { var weight = it.trim().toInt() A.add(weight) } else { first = false } } var tree = encode(A.toMutableList()) var lo = INF var hi = -INF fun go(root: Tree? = tree, depth: Int = 0) { if (root == null) return var isLeaf = { node: Tree? -> node?.left == null && node?.right == null } if (isLeaf(root)) { lo = Math.min(lo, depth) hi = Math.max(hi, depth) } else { go(root.left, depth + 1) go(root.right, depth + 1) } } go() return Pair(lo, hi) } fun main() { for (filename in listOf("problem14.6test1.txt", "problem14.6test2.txt", "problem14.6.txt")) { var (lo, hi) = run(filename) println("$filename: $lo, $hi") // min, max encoding length in the corresponding optimal prefix-free tree } } // problem14.6test1.txt: 2, 5 // problem14.6test2.txt: 3, 6 // problem14.6.txt: 9, 19 ``` *Javascript* ```javascript let LineByLine = require('n-readlines'); class Tree { constructor(weight, left = null, right = null) { this.weight = weight; this.left = left; this.right = right; } } /* let key = x => Array.isArray(x) ? x[0] : x; let heappush = (A, x, f = Math.min) => { let P = i => Math.floor((i - 1) / 2); // parent A.push(x); let N = A.length, i = N - 1; while (0 < i && key(A[i]) == f(key(A[i]), key(A[P(i)]))) { [A[i], A[P(i)]] = [A[P(i)], A[i]]; i = P(i); } }; let heappop = (A, f = Math.min) => { let L = i => 2 * i + 1, // children R = i => 2 * i + 2; let N = A.length, i = 0; let top = A[0]; [A[0], A[N - 1]] = [A[N - 1], A[0]], A.pop(), --N; let ok; do { ok = true; let left = f == Math.min ? Infinity : -Infinity, right = left; if (L(i) < N && key(A[i]) != f(key(A[i]), key(A[L(i)]))) ok = false, left = key(A[L(i)]); if (R(i) < N && key(A[i]) != f(key(A[i]), key(A[R(i)]))) ok = false, right = key(A[R(i)]); if (!ok) { let j = left == f(left, right) ? L(i) : R(i); [A[i], A[j]] = [A[j], A[i]]; i = j; } } while (!ok); return top; }; let encode = A => { let T = []; for (let weight of A) heappush(T, [ weight, new Tree(weight) ]); while (1 < T.length) { let [ a, b ] = [ heappop(T), heappop(T) ]; let c = [ a[0] + b[0], new Tree(a[0] + b[0], a[1], b[1]) ]; heappush(T, c); } return T[0][1]; }; */ /* * Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation * of a sorting subroutine, followed by a linear amount of additional work. */ let encode = A => { A.sort((a, b) => a - b) let first = A.map(weight => new Tree(weight)), second = []; while (1 < first.length + second.length) { let next = []; while (next.length < 2) { if (first.length && second.length) { next.push(first[0].weight < second[0].weight ? first.shift() : second.shift()); } else if (first.length) next.push(first.shift()); else if (second.length) next.push(second.shift()); } let [a, b] = next; let c = new Tree(a.weight + b.weight, a, b); second.push(c); } return second.shift(); }; let run = filename => { let A = []; let input = new LineByLine(filename); let line; line = input.next(); // N while (line = input.next()) { let weight = Number(String.fromCharCode(...line).trim()); A.push(weight); } let tree = encode(A); let [lo, hi] = [Infinity, -Infinity]; let go = (root = tree, depth = 0) => { if (!root) return; let isLeaf = root => !root.left && !root.right; if (isLeaf(root)) lo = Math.min(lo, depth), hi = Math.max(hi, depth); else go(root.left, depth + 1), go(root.right, depth + 1); }; go(); return [ lo, hi ]; } for (let filename of [ 'problem14.6test1.txt', 'problem14.6test2.txt', 'problem14.6.txt' ]) { let [lo, hi] = run(filename); console.log(`${filename}: ${lo}, ${hi}`); // min, max encoding length in the corresponding optimal prefix-free tree } // problem14.6test1.txt: 2, 5 // problem14.6test2.txt: 3, 6 // problem14.6.txt: 9, 19 ``` *Python3* ```python class Tree: def __init__(self, weight, left = None, right = None): self.weight = weight self.left = left self.right = right def __lt__(self, other): return self.weight < other.weight # # priority queue # # from heapq import heappush # from heapq import heappop # def encode(A): # T = [] # for weight in A: # heappush(T, Tree(weight)) # while 1 < len(T): # a, b = heappop(T), heappop(T) # c = Tree(a.weight + b.weight, a, b) # heappush(T, c) # return heappop(T) # # Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation # of a sorting subroutine, followed by a linear amount of additional work. # from collections import deque def encode(A): A.sort() first, second = deque([Tree(weight) for weight in A]), deque() while 1 < len(first) + len(second): next = [] while len(next) < 2: if len(first) and len(second): next.append(first.popleft() if first[0].weight < second[0].weight else second.popleft()) elif len(first): next.append(first.popleft()) elif len(second): next.append(second.popleft()) a, b = next c = Tree(a.weight + b.weight, a, b) second.append(c) return second.popleft() def run(filename): A = [] with open(filename) as fin: N = int(fin.readline()) while True: line = fin.readline() if not line: break weight = int(line.strip()) A.append(weight) tree = encode(A) lo, hi = float('inf'), float('-inf') def go(root = tree, depth = 0): nonlocal lo, hi if not root: return isLeaf = lambda root: not root.left and not root.right if isLeaf(root): lo = min(lo, depth) hi = max(hi, depth) else: go(root.left, depth + 1) go(root.right, depth + 1) go() return [ lo, hi ] for filename in [ 'problem14.6test1.txt', 'problem14.6test2.txt', 'problem14.6.txt' ]: lo, hi = run(filename) print(f'{filename}: {lo}, {hi}') # min, max encoding length in the corresponding optimal prefix-free tree # problem14.6test1.txt: 2, 5 # problem14.6test2.txt: 3, 6 # problem14.6.txt: 9, 19 ``` *C++* ```cpp #include #include #include #include #include #define PRIORITY_QUEUE // O(N * logN) #ifndef PRIORITY_QUEUE #define TWO_QUEUES // O(N) #endif using namespace std; using LL = long long; using Weight = LL; using Weights = vector; struct Tree; using TreePtr = shared_ptr; struct Tree { Weight weight; TreePtr left, right; Tree(Weight weight, TreePtr left = nullptr, TreePtr right = nullptr) : weight{ weight }, left{ left }, right{ right } {} }; using TreePtrs = vector; #ifdef PRIORITY_QUEUE struct Comp { size_t operator()(const TreePtr& a, const TreePtr& b) const { return b->weight < a->weight; } }; using Queue = priority_queue; TreePtr encode(const Weights& A, Queue q = {}) { for (auto weight: A) q.emplace(make_shared(weight)); while (1 < q.size()) { auto a = q.top(); q.pop(); auto b = q.top(); q.pop(); auto c = make_shared(a->weight + b->weight, a, b); q.emplace(c); } return q.top(); } #else // TWO_QUEUES /* * Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation * of a sorting subroutine, followed by a linear amount of additional work. */ using Queue = queue; TreePtr encode(Weights& A, Queue first = {}, Queue second = {}) { sort(A.begin(), A.end()); for (auto weight: A) first.push(make_shared(weight)); TreePtrs next; auto takeFirst = [&]() { next.push_back(first.front()), first.pop(); }; auto takeSecond = [&]() { next.push_back(second.front()), second.pop(); }; while (1 < first.size() + second.size()) { next.clear(); do { if (first.size() && second.size()) { if (first.front()->weight < second.front()->weight) takeFirst(); else takeSecond(); } else if (first.size()) takeFirst(); else if (second.size()) takeSecond(); } while (next.size() < 2); auto [a, b] = tie(next[0], next[1]); auto c = make_shared(a->weight + b->weight, a, b); second.emplace(c); } return second.front(); } #endif using MinMax = pair; constexpr auto Min = numeric_limits::min(); constexpr auto Max = numeric_limits::max(); MinMax run(const string& filename) { Weights A; // weight of each symbol fstream fin{ filename }; LL N, weight; for (fin >> N; fin >> weight; A.push_back(weight)); auto tree = encode(A); LL lo = Max, hi = Min; using fun = function; fun go = [&](auto root, LL depth) { if (!root) return; auto isLeaf = [](auto root) { return !root->left && !root->right; }; if (isLeaf(root)) lo = min(lo, depth), hi = max(hi, depth); else go(root->left, depth + 1), go(root->right, depth + 1); }; go(tree, 0); return make_pair(lo, hi); } int main() { for (auto& filename: { "problem14.6test1.txt", "problem14.6test2.txt", "problem14.6.txt" }) { auto [lo, hi] = run(filename); cout << filename << ": " << lo << ", " << hi << endl; // min, max encoding length in the corresponding optimal prefix-free tree } // problem14.6test1.txt: 2, 5 // problem14.6test2.txt: 3, 6 // problem14.6.txt: 9, 19 return 0; } ```
--- ### Prim's MST
📚 Lectures
* [Minimum Spanning Trees: Problem Definition (Section 15.1)](https://www.youtube.com/watch?v=tDj9BkaQDO8&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=10) * [Prim's MST Algorithm (Section 15.2)](https://www.youtube.com/watch?v=jsvOPssDVJA&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=11) * [Speeding Up Prim's Algorithm via Heaps (Part 1) (Section 15.3, Part 1)](https://www.youtube.com/watch?v=cDtQnXMZGtg&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=15) * [Speeding Up Prim's Algorithm via Heaps (Part 2) (Section 15.3, Part 2)](https://www.youtube.com/watch?v=jGR_LAwGLGk&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=16) * [Prim's Algorithm: Correctness Proof (Part 1) (Section 15.4, Part 1) [Note: this video provides an alternative treatment to that in the book.]](https://www.youtube.com/watch?v=pGUzn3S7bp4&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=12) * [Prim's Algorithm: Correctness Proof (Part 2) (Section 15.4, Part 2) [Note: this video provides an alternative treatment to that in the book.]](https://www.youtube.com/watch?v=199ItGt4mE8&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=13)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File import java.util.PriorityQueue import java.util.Random fun prim(N: Int, adj: MutableMap>>): Int { var total: Int = 0 var start = Random().nextInt(N) + 1 var q = PriorityQueue>(Comparator{ a: Pair, b: Pair -> a.first.compareTo(b.first) }) var seen = mutableSetOf(start) for ((w, v) in adj[start]!!) q.add(Pair(w, v)) while (0 < q.size) { var (cost, u) = q.poll() if (seen.contains(u)) continue total += cost; seen.add(u) for ((w, v) in adj[u]!!) if (!seen.contains(v)) q.add(Pair(w, v)) } return total } fun run(filename: String) { var N: Int = 0 var adj = mutableMapOf>>() var first = true File(filename).forEachLine { line -> if (!first) { var (u, v, w) = line.split(" ").map{ it.toInt() } if (!adj.contains(u)) adj[u] = mutableListOf>() if (!adj.contains(v)) adj[v] = mutableListOf>() adj[u]!!.add(Pair(w, v)) adj[v]!!.add(Pair(w, u)) } else { var (numVertex, _) = line.split(" ").map{ it.toInt() } N = numVertex first = false } } var cost = prim(N, adj) println("$filename: $cost") } fun main() { run("problem15.9test.txt") // problem15.9test.txt: 14 run("problem15.9.txt") // problem15.9.txt: -3612829 } ``` *Javascript* ```javascript let LineByLine = require('n-readlines'); let key = x => Array.isArray(x) ? x[0] : x; let heappush = (A, x, f = Math.min) => { let P = i => Math.floor((i - 1) / 2); // parent A.push(x); let N = A.length, i = N - 1; while (0 < i && key(A[i]) == f(key(A[i]), key(A[P(i)]))) { [A[i], A[P(i)]] = [A[P(i)], A[i]]; i = P(i); } }; let heappop = (A, f = Math.min) => { let L = i => 2 * i + 1, // children R = i => 2 * i + 2; let N = A.length, i = 0; let top = A[0]; [A[0], A[N - 1]] = [A[N - 1], A[0]], A.pop(), --N; let ok; do { ok = true; let left = f == Math.min ? Infinity : -Infinity, right = left; if (L(i) < N && key(A[i]) != f(key(A[i]), key(A[L(i)]))) ok = false, left = key(A[L(i)]); if (R(i) < N && key(A[i]) != f(key(A[i]), key(A[R(i)]))) ok = false, right = key(A[R(i)]); if (!ok) { let j = left == f(left, right) ? L(i) : R(i); [A[i], A[j]] = [A[j], A[i]]; i = j; } } while (!ok); return top; }; let prim = (N, adj, q = [], seen = new Set(), total = 0) => { let start = Math.ceil(N * Math.random()); seen.add(start); for (let [w, v] of adj.get(start)) heappush(q, [w, v]); while (q.length) { let [cost, u] = heappop(q); if (seen.has(u)) continue; total += cost; seen.add(u); for (let [w, v] of adj.get(u)) if (!seen.has(v)) heappush(q, [w, v]); } return total; }; let run = filename => { let adj = new Map(); let input = new LineByLine(filename); let line = input.next(); let [N, M] = String.fromCharCode(...line).trim().split(' ').map(Number); while (line = input.next()) { let [u, v, w] = String.fromCharCode(...line).trim().split(' ').map(Number); if (!adj.has(u)) adj.set(u, []); if (!adj.has(v)) adj.set(v, []); adj.get(u).push([w, v]); adj.get(v).push([w, u]); } let cost = prim(N, adj); console.log(`${filename}: ${cost}`); } run('problem15.9test.txt') // problem15.9test.txt: 14 run('problem15.9.txt') // problem15.9.txt: -3612829 ``` *Python3* ```python from random import randint from heapq import heappush, heappop def prim(N, adj, total = 0): q = [] seen = set() start = randint(1, N); seen.add(start) for w, v in adj[start]: heappush(q, [w, v]) while q: cost, u = heappop(q) if u in seen: continue total += cost; seen.add(u) for w, v in adj[u]: if v not in seen: heappush(q, [w, v]) return total def run(filename): adj = {} first = True with open(filename) as fin: while True: line = fin.readline() if not line: break words = line.split() if not first: u, v, w = [int(x) for x in words] if u not in adj: adj[u] = [] if v not in adj: adj[v] = [] adj[u].append([w, v]) adj[v].append([w, u]) else: N, M = [int(x) for x in words] first = False cost = prim(N, adj) print(f'{filename}: {cost}') run('problem15.9test.txt') # problem15.9test.txt: 14 run('problem15.9.txt') # problem15.9.txt: -3612829 ``` *C++* ```cpp #include #include #include #include #include #include using namespace std; using Pair = pair; using Pairs = vector; using AdjList = unordered_map; using Queue = priority_queue>; using Set = unordered_set; constexpr auto INF = int(1e9 + 7); int getRandom(int N) { default_random_engine generator; uniform_int_distribution distribution(1, N); return distribution(generator); } int prim(int N, AdjList& adj, Queue q = {}, Set seen = {}, int total = 0) { auto start = getRandom(N); seen.insert(start); for (auto [w, v]: adj[start]) q.push({ w, v }); while (q.size()) { auto [cost, u] = q.top(); q.pop(); if (!seen.insert(u).second) continue; total += cost; for (auto [w, v]: adj[u]) if (seen.find(v) == seen.end()) q.push({ w, v }); } return total; } void run(const string& filename) { AdjList adj; fstream fin{ filename }; int N, M; fin >> N >> M; // N vertices and M edges int u, v, w; // edge u -> v of weight w while (fin >> u >> v >> w) { adj[u].emplace_back(w, v); adj[v].emplace_back(w, u); } auto cost = prim(N, adj); cout << filename << ": " << cost << endl; } int main() { run("problem15.9test.txt"); // problem15.9test.txt: 14 run("problem15.9.txt"); // problem15.9.txt: -3612829 return 0; } ```
--- ### Kruskal's MST
📚 Lectures
* [Minimum Spanning Trees: Problem Definition (Section 15.1)](https://www.youtube.com/watch?v=tDj9BkaQDO8&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=10) * [Kruskal's MST Algorithm (Section 15.5)](https://www.youtube.com/watch?v=SZuCspj5AJc&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=17) * [Speeding Up Kruskal's Algorithm via Union-Find (Part 1) (Section 15.6, Part 1)](https://www.youtube.com/watch?v=fItEZEVyJKE&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=19) * [Speeding Up Kruskal's Algorithm via Union-Find (Part 2) (Section 15.6, Part 2) [Note: this video provides an alternative treatment to that in the book.]](https://www.youtube.com/watch?v=jY-vY6d18W4&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=20) * [Lazy Unions (Section 15.6, Part 3) [Note: this video is closer to the union-find implementation in the book.]](https://www.youtube.com/watch?v=bRwTSPIEI9k&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=24) * [Kruskal's Algorithm: Correctness Proof (Section 15.7) [Note: this video provides an alternative treatment to that in the book.]](https://www.youtube.com/watch?v=AjLjL0Rp10g&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=18) * [Application: Single-Link Clustering (Section 15.8)](https://www.youtube.com/watch?v=MSSzOs1X4K8&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=22)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File fun kruskal(E: MutableList>): Int { var total: Int = 0 var M = E.size var P = IntArray(M) { it } // 🙂 parent representatives of 1..M disjoint sets fun find(x: Int): Int { P[x] = if (P[x] == x) x else find(P[x]) return P[x] } fun union(a: Int, b: Int): Boolean { var x = find(a) var y = find(b) if (x == y) return false P[x] = y // 🎲 arbitrary choice return true } E.sortWith(Comparator{ a, b -> a.third.compareTo(b.third) }) // sort edges by nondecreasing weight for ((u, v, w) in E) if (union(u, v)) total += w return total } fun run(filename: String) { var E = mutableListOf>() var first = true File(filename).forEachLine { line -> if (!first) { var (u, v, w) = line.trim().split(" ").map{ it.toInt() } E.add(Triple(u, v, w)) } else { first = false // ignore first line with N vertices and M edges } } var cost = kruskal(E) println("$filename: $cost") } fun main() { run("problem15.9test.txt") // problem15.9test.txt: 14 run("problem15.9.txt") // problem15.9.txt: -3612829 } ``` *Javascipt* ```javascript let LineByLine = require('n-readlines'); let kruskal = E => { let total = 0; let M = E.length; let P = [...Array(M).keys()]; // 🙂 parent representatives of 1..M disjoint sets let find = x => P[x] = P[x] == x ? x : find(P[x]); let union = (a, b) => { a = find(a); b = find(b); if (a == b) return false; P[a] = b; // 🎲 arbitrary choice return true; }; E.sort((first, second) => { // sort edges by nondecreasing weight let [u1, v1, w1] = first, [u2, v2, w2] = second; return w1 - w2; }); for (let [u, v, w] of E) if (union(u, v)) total += w; return total; }; let run = filename => { let E = []; let input = new LineByLine(filename); let line = input.next(); // ignore first line with N vertices and M edges while (line = input.next()) { let [u, v, w] = String.fromCharCode(...line).trim().split(' ').map(Number); E.push([ u, v, w ]); } let cost = kruskal(E); console.log(`${filename}: ${cost}`); }; run('problem15.9test.txt'); // problem15.9test.txt: 14 run('problem15.9.txt'); // problem15.9.txt: -3612829 ``` *Python3* ```python from functools import cmp_to_key def kruskal(E, total = 0): M = len(E) P = [i for i in range(M)] # 🙂 parent representatives of 1..M disjoint sets def find(x): P[x] = P[x] if P[x] == x else find(P[x]) return P[x] def union(a, b): a = find(a) b = find(b) if a == b: return False P[a] = b # 🎲 arbitary choice return True E.sort(key = cmp_to_key(lambda first, second: first[2] - second[2])) # sort edges by nondecreasing weight for u, v, w in E: if union(u, v): total += w return total def run(filename): E = [] first = True with open(filename) as fin: while True: line = fin.readline() if not line: break values = [int(x) for x in line.strip().split()] if not first: u, v, w = values # edge u -> v of weight w E.append([ u, v, w ]) else: first = False # ignore first line with N vertices and M edges cost = kruskal(E) print(f'{filename}: {cost}') run('problem15.9test.txt') # problem15.9test.txt: 14 run('problem15.9.txt') # problem15.9.txt: -3612829 ``` *C++* ```cpp #include #include #include #include #include using namespace std; using Edge = tuple; using Edges = vector; using Parents = vector; using fun = function; int kruskal(Edges& E, int total = 0) { auto M = E.size(); Parents P(M); iota(P.begin(), P.end(), 0); // 🙂 parent representatives of 1..M disjoint sets fun find = [&](auto x) { return P[x] = P[x] == x ? x : find(P[x]); }; auto join = [&](auto a, auto b) { a = find(a); b = find(b); if (a == b) return false; P[a] = b; // 🎲 arbitrary choice return true; }; sort(E.begin(), E.end(), [](auto& first, auto& second) { // sort edges by nondecreasing weight auto [u1, v1, w1] = first; auto [u2, v2, w2] = second; return w1 < w2; }); for (auto [u, v, w]: E) if (join(u, v)) total += w; return total; } void run(const string& filename, Edges E = {}) { fstream fin{ filename }; int N, M; fin >> N >> M; // ignore first line with N vertices and M edges int u, v, w; // edge u -> v of weight w while (fin >> u >> v >> w) E.emplace_back(u, v, w); auto cost = kruskal(E); cout << filename << ": " << cost << endl; } int main() { run("problem15.9test.txt"); // problem15.9test.txt: 14 run("problem15.9.txt"); // problem15.9.txt: -3612829 return 0; } ```
--- ### Weighted Independent Set
📚 Lectures
* [The Weighted Independent Set Problem (Section 16.1)](https://www.youtube.com/watch?v=0awkct8SkxA&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=39) * [A Linear-Time Algorithm for WIS in Path Graphs (Part 1) (Section 16.2, Part 1)](https://www.youtube.com/watch?v=pLOkbHGRsv0&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=40) * [A Linear-Time Algorithm for WIS in Path Graphs (Part 2) (Section 16.2, Part 2)](https://www.youtube.com/watch?v=Im_zjFkZDCY&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=41) * [A Reconstruction Algorithm (Section 16.3)](https://www.youtube.com/watch?v=W2ncNfksRqo&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=42)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File fun topDown(A: MutableList): Long { var N = A.size var m = mutableMapOf() fun go(i: Int = N - 1): Long { if (m.contains(i)) // 🤔 memo return m[i]!! if (i < 0) { // 🛑 empty set m[i] = 0 return 0 } if (i == 0) { // 🛑 single set m[i] = A[0] return A[0] } var include = go(i - 2) + A[i] // ✅ include A[i] var exclude = go(i - 1) // 🚫 exclude A[i] m[i] = Math.max(include, exclude) // 🎯 best return m[i]!! } return go() } fun bottomUp(A: MutableList): Long { var N = A.size var dp = LongArray(N + 1) // 🤔 memo dp[0] = 0 // 🛑 empty set dp[1] = A[0] // 🛑 single set for (i in 2..N) { var include = dp[i - 2] + A[i - 1] // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) var exclude = dp[i - 1] // 🚫 exclude A[i] dp[i] = Math.max(include, exclude) // 🎯 best } return dp[N] } fun bottomUpMemOpt(A: MutableList): Long { var N = A.size var a: Long = 0 // 🤔 memo + 🛑 empty set var b: Long = A[0] // 🤔 memo + 🛑 single set var c: Long = -1 for (i in 2..N) { var include = a + A[i - 1] // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) var exclude = b // 🚫 exclude A[i] c = Math.max(include, exclude) // 🎯 best a = b; b = c // 👈 slide window } return c } fun run(filename: String) { var A = mutableListOf() var first = true File(filename).forEachLine { line -> if (!first) { A.add(line.toLong()) } else { first = false } } var a = topDown(A) var b = bottomUp(A) var c = bottomUpMemOpt(A) assert(a == b && b == c) // 💩 sanity check println("$filename: $a") } fun main() { run("problem16.6test.txt") // problem16.6test.txt: 2617 run("problem16.6.txt") // problem16.6.txt: 2955353732 } ``` *Javascript* ```javascript const assert = require('assert'); const LineByLine = require('n-readlines'); let top_down = (A, m = {}) => { let N = A.length; let go = (i = N - 1) => { if (m[i]) // 🤔 memo return m[i]; if (i < 0) return m[i] = 0; // 🛑 empty set if (!i) return m[i] = A[0]; // 🛑 single set let include = go(i - 2) + A[i], // ✅ include A[i] exclude = go(i - 1); // 🚫 exclude A[i] return m[i] = Math.max(include, exclude); // 🎯 best }; return go(); }; let bottom_up = A => { let N = A.length; let dp = Array(N + 1); // 🤔 memo dp[0] = 0; // 🛑 empty set dp[1] = A[0]; // 🛑 single set for (let i = 2; i <= N; ++i) { let include = dp[i - 2] + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = dp[i - 1]; // 🚫 exclude A[i] dp[i] = Math.max(include, exclude); // 🎯 best } return dp[N]; }; let bottom_up_memopt = A => { let N = A.length; let a = 0, // 🤔 memo + 🛑 empty set b = A[0], // 🤔 memo + 🛑 single set c = -1; for (let i = 2; i <= N; ++i) { let include = a + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = b; // 🚫 exclude A[i] c = Math.max(include, exclude); // 🎯 best a = b, b = c; // 👈 slide window } return c; }; let run = filename => { let A = []; let input = new LineByLine(filename); let line; let first = true; while (line = input.next()) { if (!first) { A.push(Number(line.toString('ascii'))); } else { first = false; } } let a = top_down(A), b = bottom_up(A), c = bottom_up_memopt(A); assert(a == b && b == c); // 💩 sanity check console.log(`${filename}: ${a}`); }; run('problem16.6test.txt'); // problem16.6test.txt: 2617 run('problem16.6.txt'); // problem16.6.txt: 2955353732 ``` *Python3* ```python from functools import lru_cache def top_down(A): N = len(A) @lru_cache # 🤔 memo def go(i = N - 1): if i < 0: return 0 # 🛑 empty set if i == 0: return A[0] # 🛑 single set include = go(i - 2) + A[i] # ✅ include A[i] exclude = go(i - 1) # 🚫 exclude A[i] return max(include, exclude) # 🎯 best return go() def bottom_up(A): N = len(A) dp = [0] * (N + 1) # 🤔 memo dp[0] = 0 # 🛑 empty set dp[1] = A[0] # 🛑 single set for i in range(2, N + 1): include = dp[i - 2] + A[i - 1] # ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = dp[i - 1] # 🚫 exclude A[i] dp[i] = max(include, exclude) # 🎯 best return dp[N] def bottom_up_memopt(A): N = len(A) a = 0 # 🤔 memo + 🛑 empty set b = A[0] # 🤔 memo + 🛑 single set c = -1 for i in range(2, N + 1): include = a + A[i - 1] # ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = b # 🚫 exclude A[i] c = max(include, exclude) # 🎯 best a = b; b = c # 👈 slide window return c def run(filename): A = [] with open(filename) as fin: first = True while True: line = fin.readline() if not line: break x = int(line) if not first: A.append(x) else: first = False N = x a = top_down(A) b = bottom_up(A) c = bottom_up_memopt(A) assert(a == b and b == c) # 💩 sanity check print(f'{filename}: {a}') run('problem16.6test.txt') # problem16.6test.txt: 2617 run('problem16.6.txt') # problem16.6.txt: 2955353732 ``` *C++* ```cpp #include #include #include #include #include using namespace std; using LL = long long; using List = vector; using Map = unordered_map; namespace TopDown { LL best(List& A, Map m = {}) { int N = A.size(); using fun = function; fun go = [&](auto i) { if (m[i]) return m[i]; // 🤔 memo if (i < 0) return m[i] = 0LL; // 🛑 empty set if (!i) return m[i] = A[0]; // 🛑 single set auto include = go(i - 2) + A[i], // ✅ include A[i] exclude = go(i - 1); // 🚫 exclude A[i] return m[i] = max(include, exclude); // 🎯 best }; return go(N - 1); } } namespace BottomUp { LL best(List& A, Map m = {}) { int N = A.size(); List dp(N + 1); // 🤔 memo dp[0] = 0LL; // 🛑 empty set dp[1] = A[0]; // 🛑 single set for (auto i{ 2 }; i <= N; ++i) { auto include = dp[i - 2] + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = dp[i - 1]; // 🚫 exclude A[i] dp[i] = max(include, exclude); // 🎯 best } return dp[N]; } } namespace BottomUpMemOpt { LL best(List& A) { int N = A.size(); LL a = 0LL, // 🤔 memo + 🛑 empty set b = A[0], // 🤔 memo + 🛑 single set c = -1; for (auto i{ 2 }; i <= N; ++i) { auto include = a + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = b; // 🚫 exclude A[i] c = max(include, exclude); // 🎯 best a = b, b = c; // 👈 slide window } return c; } } void run(const string& filename) { List A; fstream fin{ filename }; int N; fin >> N; copy_n(istream_iterator(fin), N, back_inserter(A)); auto a = TopDown::best(A), b = BottomUp::best(A), c = BottomUpMemOpt::best(A); assert(a == b && b == c); // 💩 sanity check cout << filename << ": " << a << endl; } int main() { run("problem16.6test.txt"); // problem16.6test.txt: 2617 run("problem16.6.txt"); // problem16.6.txt: 2955353732 return 0; } ```
--- ### Knapsack
📚 Lectures
* [Principles of Dynamic Programming (Section 16.4)](https://www.youtube.com/watch?v=27nK8C-GCPM&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=43) * [The Knapsack Problem (Part 1) (Section 16.5, Part 1)](https://www.youtube.com/watch?v=jlHIBaUizOU&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=44) * [The Knapsack Problem (Part 2) (Section 16.5, Part 2)](https://www.youtube.com/watch?v=KX_6OF8X6HQ&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=45) * [The Knapsack Problem (Part 3) (Section 16.5, Part 3)](https://www.youtube.com/watch?v=1dRUOZKcvYs&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=46)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File var INF = (1e9 + 7).toInt() fun top_down(A: List>, K: Int): Int { var N = A.size var m = mutableMapOf() fun go(i: Int = 0, k: Int = K): Int { if (i == N) // 🛑 empty set return 0 var key = "$i,$k" if (m.contains(key)) // 🤔 memo return m[key]!! var (value, weight) = A[i] var include = if (0 <= k - weight) go(i + 1, k - weight) + value else -INF // ✅ include A[i] var exclude = go(i + 1, k) // 🚫 exclude A[i] m[key] = Math.max(include, exclude) // 🎯 best return m[key]!! } return go() } fun bottom_up(A: List>, K: Int): Int { var N = A.size var dp = Array(N + 1){ Array(K + 1){ -INF } } // 🤔 memo for (k in 0..K) // 🛑 empty set dp[0][k] = 0 for (i in 1..N) { for (k in 0..K) { var (value, weight) = A[i - 1] var include = if (0 <= k - weight) dp[i - 1][k - weight] + value else -INF // ✅ include A[i] var exclude = dp[i - 1][k] // 🚫 exclude A[i] dp[i][k] = Math.max(include, exclude) // 🎯 best } } return dp[N][K] } fun bottom_up_memopt(A: List>, K: Int): Int { var N = A.size var pre = Array(K + 1) { 0 } // 🤔 memo + 🛑 empty set for (i in 1..N) { var cur = Array(K + 1) { -INF } for (k in 0..K) { var (value, weight) = A[i - 1] var include = if (0 <= k - weight) pre[k - weight] + value else -INF // ✅ include A[i] var exclude = pre[k] // 🚫 exclude A[i] cur[k] = Math.max(include, exclude) // 🎯 best } pre = cur.also { cur = pre } } return pre[K] } fun run(filename: String) { var A = mutableListOf>() var K = 0 var N = 0 var first = true File(filename).forEachLine { line -> if (!first) { var (value, weight) = line.trim().split(" ").map{ it -> it.toInt() } A.add(Pair(value, weight)) } else { var (a, b) = line.trim().split(" ").map{ it -> it.toInt() } K = a N = b first = false } } var a = top_down(A, K) var b = bottom_up(A, K) var c = bottom_up_memopt(A, K) assert(a == b && b == c) // 💩 sanity check println("$filename: $a") } fun main() { run("problem16.7test.txt") // problem16.7test.txt: 2493893 } ``` *Javascript* ```javascript const assert = require('assert'); const LineByLine = require('n-readlines'); let top_down = (A, K, m = new Map()) => { let N = A.length; let go = (i = 0, k = K) => { if (i == N) // 🛑 empty set return 0; let key = `${i},${k}`; if (m.has(key)) // 🤔 memo return m.get(key); let [value, weight] = A[i]; let include = 0 <= k - weight ? go(i + 1, k - weight) + value : -Infinity, // ✅ include A[i] exclude = go(i + 1, k); // 🚫 exclude A[i] return m.set(key, Math.max(include, exclude)) // 🎯 best .get(key); }; return go(); }; let bottom_up = (A, K) => { let N = A.length; let dp = [...Array(N + 1)].map(_ => Array(K + 1).fill(-Infinity)); // 🤔 memo for (let k = 0; k < K; dp[0][k++] = 0); // 🛑 empty set for (let i = 1; i <= N; ++i) { for (let k = 0; k <= K; ++k) { let [value, weight] = A[i - 1]; let include = 0 <= k - weight ? dp[i - 1][k - weight] + value : -Infinity, // ✅ include A[i] exclude = dp[i - 1][k]; // 🚫 exclude A[i] dp[i][k] = Math.max(include, exclude); // 🎯 best } } return dp[N][K]; }; let bottom_up_memopt = (A, K) => { let N = A.length; let pre = Array(K + 1).fill(0); // 🤔 memo + 🛑 empty set for (let i = 1; i <= N; ++i) { let cur = Array(K + 1).fill(-Infinity); for (let k = 0; k <= K; ++k) { let [value, weight] = A[i - 1]; let include = 0 <= k - weight ? pre[k - weight] + value : -Infinity, // ✅ include A[i] exclude = pre[k]; // 🚫 exclude A[i] cur[k] = Math.max(include, exclude); // 🎯 best } [pre, cur] = [cur, pre]; } return pre[K]; }; let run = filename => { let A = []; const input = new LineByLine(filename) let [K, N] = input.next().toString().split(' ').map(Number); // K capacity, N items let line; while (line = input.next()) { let [value, weight] = line.toString().split(' ').map(Number); A.push([value, weight]); } let a = top_down(A, K), b = bottom_up(A, K), c = bottom_up_memopt(A, K); assert(a == b && b == c); // 💩 sanity check console.log(`${filename}: ${a}`); }; run('problem16.7test.txt') // problem16.7test.txt: 2493893 ``` *Python3* ```python from functools import lru_cache def top_down(A, K): N = len(A) total = [0] * N @lru_cache(maxsize = None) # 🤔 memo def go(i = 0, k = K): if i == N: # 🛑 empty set return 0 value, weight = A[i] include = go(i + 1, k - weight) + value if 0 <= k - weight else float('-inf') # ✅ include A[i] exclude = go(i + 1, k) # 🚫 exclude A[i] return max(include, exclude) # 🎯 best return go() def bottom_up(A, K): N = len(A) dp = [[float('-inf')] * (K + 1) for _ in range(N + 1)] # 🤔 memo for j in range(K): # 🛑 empty set dp[0][j] = 0 for i in range(1, N + 1): for k in range(1, K + 1): value, weight = A[i - 1] include = dp[i - 1][k - weight] + value if 0 <= k - weight else float('-inf') # ✅ include A[i] exclude = dp[i - 1][k] # 🚫 exclude A[i] dp[i][k] = max(include, exclude) # 🎯 best return dp[N][K] def bottom_up_memopt(A, K): N = len(A) pre = [0] * (K + 1) # 🤔 memo + 🛑 empty set for i in range(1, N + 1): cur = [float('-inf')] * (K + 1) for k in range(1, K + 1): value, weight = A[i - 1] include = pre[k - weight] + value if 0 <= k - weight else float('-inf') # ✅ include A[i] exclude = pre[k] # 🚫 exclude A[i] cur[k] = max(include, exclude) # 🎯 best pre, cur = cur, pre return pre[K] def run(filename): A = [] with open(filename) as fin: line = fin.readline() [K, N] = [int(word) for word in line.split()] # K capacity, N items while True: line = fin.readline() if not line: break value, weight = [int(word) for word in line.split()] A.append([value, weight]) a = top_down(A, K) b = bottom_up(A, K) c = bottom_up_memopt(A, K) assert(a == b and b == c) # 💩 sanity check print(f'{filename}: {a}') run('problem16.7test.txt') # problem16.7test.txt: 2493893 ``` *C++* ```cpp #include #include #include #include #include using namespace std; using Pair = pair; // value, weight using Pairs = vector; using fun = function; using Map = unordered_map; int INF = 1e9 + 7; int top_down(Pairs& A, int K, Map m = {}) { auto N = A.size(); fun go = [&](auto i, auto k) { if (i == N) // 🛑 empty set return 0; stringstream key; key << i << "," << k; if (m.find(key.str()) != m.end()) // 🤔 memo return m[key.str()]; auto [value, weight] = A[i]; auto include = 0 <= k - weight ? go(i + 1, k - weight) + value : -INF, // ✅ include A[i] exclude = go(i + 1, k); // 🚫 exclude A[i] return m[key.str()] = max(include, exclude); // 🎯 best }; return go(0, K); } int bottom_up(Pairs& A, int K) { auto N = A.size(); using VI = vector; using VVI = vector; VVI dp(N + 1, VI(K + 1, -INF)); // 🤔 memo for (auto k{ 0 }; k < K; dp[0][k++] = 0); // 🛑 empty set for (auto i{ 1 }; i <= N; ++i) { for (auto k{ 0 }; k <= K; ++k) { auto [value, weight] = A[i - 1]; auto include = 0 <= k - weight ? dp[i - 1][k - weight] +value : -INF, // ✅ include A[i] exclude = dp[i - 1][k]; // 🚫 exclude A[i] dp[i][k] = max(include, exclude); // 🎯 best } } return dp[N][K]; } int bottom_up_memopt(Pairs& A, int K) { auto N = A.size(); using VI = vector; VI pre(K + 1, 0); // 🤔 memo + 🛑 empty set for (auto i{ 1 }; i <= N; ++i) { VI cur(K + 1, -INF); for (auto k{ 0 }; k <= K; ++k) { auto [value, weight] = A[i - 1]; auto include = 0 <= k - weight ? pre[k - weight] +value : -INF, // ✅ include A[i] exclude = pre[k]; // 🚫 exclude A[i] cur[k] = max(include, exclude); // 🎯 best } swap(pre, cur); } return pre[K]; } void run(const string& filename) { Pairs A; fstream fin{ filename }; int K, N; // K capacity, N items fin >> K >> N; for (int value, weight; fin >> value >> weight; A.emplace_back(value, weight)); auto a = top_down(A, K), b = bottom_up(A, K), c = bottom_up_memopt(A, K); assert(a == b && b == c); // 💩 sanity check cout << filename << ": " << a << endl; } int main() { run("problem16.7test.txt"); // problem16.7test.txt: 2493893 return 0; } ```
--- ### Bellman-Ford
📚 Lectures
* [Shortest Paths with Negative Edge Lengths (Section 18.1)](https://www.youtube.com/watch?v=oyHoqtxHiOs&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=54) * [The Bellman-Ford Algorithm (Part 1) (Section 18.2, Part 1)](https://www.youtube.com/watch?v=06OK99Aak60&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=55) * [The Bellman-Ford Algorithm (Part 2) (Section 18.2, Part 2)](https://www.youtube.com/watch?v=d1TWZa20Mkw&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=56) * [The Bellman-Ford Algorithm (Part 3) (Section 18.2, Part 3)](https://www.youtube.com/watch?v=HaXTsC8eOfo&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=57) * [The Bellman-Ford Algorithm (Part 4) (Section 18.2, Part 4)](https://www.youtube.com/watch?v=Otn0kbliRUQ&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=58)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File import java.util.LinkedList import java.util.Queue // bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices fun bell(E: Array>, N: Int, start: Int = 1, INF: Int = (1e6).toInt()): IntArray { var dist = IntArray(N) { INF } dist[start] = 0 var K = N - 1 while (0 < K--) E.forEach{ (u, v, w) -> dist[v] = Math.min(dist[v], dist[u] + w)} return dist } // shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) fun spfa(E: Array>, N: Int, start: Int = 1, INF: Int = (1e6).toInt()): IntArray { var dist = IntArray(N) { INF } dist[start] = 0 var adj = Array>>(N) { mutableListOf>() } for ((u, v, w) in E) adj[u].add(Pair(v, w)) var q: Queue = LinkedList(listOf(start)) while (0 < q.size) { var u = q.poll() for ((v, w) in adj[u]) { if (dist[v] > dist[u] + w) { dist[v] = dist[u] + w; q.add(v) } } } return dist } fun run(filename: String): IntArray { var N = 0 var E = mutableListOf>() File(filename).forEachLine { var A = ArrayDeque(it.trim().split("\t")) var u = A.removeFirst().toInt() for ((v, w) in A.map{ it.split(",").map{ it.toInt() } }) E.add(Triple(u, v, w)) ++N; } var A = E.toTypedArray() var a = bell(A, N + 1) // +1 for 1-based indexing var b = spfa(A, N + 1) assert(a == b) // 💩 sanity check: single source shortest paths are the same return b } fun main() { var dist = run("test.txt") println(listOf(7, 37, 59, 82, 99, 115, 133, 165, 188, 197).map{ dist[it] }.joinToString(",")) // 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068 } ``` *Javascript* ```javascript const assert = require('assert'); const zip = require('lodash/zip'); const LineByLine = require('n-readlines'); // bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices let bell = (E, N, start = 1, INF = Number(1e6)) => { let dist = Array(N).fill(INF); dist[start] = 0; let K = N - 1; while (K--) E.forEach(([u, v, w]) => dist[v] = Math.min(dist[v], dist[u] + w)); return dist; }; // shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) let spfa = (E, N, start = 1, INF = Number(1e6)) => { let dist = Array(N).fill(INF); dist[start] = 0; let adj = [...Array(N)].map(_ => []); E.forEach(([u, v, w]) => adj[u].push([v, w])); let q = [start]; while (q.length) { let u = q.shift(); for (let [v, w] of adj[u]) if (dist[v] > dist[u] + w) dist[v] = dist[u] + w, q.push(v); } return dist; }; let run = filename => { let N = 0; let E = []; let input = new LineByLine(filename); let line; while (line = input.next()) { let A = line.toString('ascii').split('\t').filter(it => it.length); let u = Number(A.shift()); A.map(pair => pair.split(',').map(Number)).forEach(([v, w]) => E.push([u, v, w])); ++N; } let a = bell(E, N + 1), // +1 for 1-based indexing b = spfa(E, N + 1); zip(a, b).forEach(([x, y]) => assert(x == y)); // 💩 sanity check: single source shortest paths are the same return a; }; let dist = run('test.txt'); console.log([7, 37, 59, 82, 99, 115, 133, 165, 188, 197].map(x => dist[x]).join(',')); // 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068 ``` *Python3* ```python from collections import deque # bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices def bell(E, N, start = 1, INF = int(1e6)): dist = [INF] * N dist[start] = 0 k = N - 1 while k: for u, v, w in E: dist[v] = min(dist[v], dist[u] + w) k -= 1 return dist # shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) def spfa(E, N, start = 1, INF = int(1e6)): dist = [INF] * N dist[start] = 0 adj = {i: [] for i in range(N)} for u, v, w in E: adj[u].append([v, w]) q = deque([start]) while q: u = q.popleft() for v, w in adj[u]: if dist[v] > dist[u] + w: dist[v] = dist[u] + w; q.append(v) return dist def run(filename): E = [] N = 0 with open(filename) as fin: while True: line = fin.readline() if not line: break line = line.strip() A = [word for word in line.split('\t') if len(word)] u = int(A[0]) for i in range(1, len(A)): v, w = [int(x) for x in A[i].split(',')] E.append([u, v, w]) N += 1 a = bell(E, N + 1) # +1 for 1-based indexing b = spfa(E, N + 1) assert(a == b) return b dist = run('test.txt') print(','.join(str(dist[x]) for x in [7, 37, 59, 82, 99, 115, 133, 165, 188, 197])) # 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068 ``` *C++* ```cpp #include #include #include #include #include #include using namespace std; using VI = vector; using VS = vector; using Edge = tuple; using Edges = vector; using Pair = pair; using Pairs = vector; using AdjList = unordered_map; using Queue = queue; // bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices VI bell(Edges& E, int N, int start = 1, int INF = 1e6) { VI dist(N, INF); dist[start] = 0; auto K = N - 1; while (K--) for (auto [u, v, w]: E) dist[v] = min(dist[v], dist[u] + w); return dist; } // shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) VI spfa(Edges& E, int N, int start = 1, int INF = 1e6, AdjList adj = {}) { VI dist(N, INF); dist[start] = 0; for (auto [u, v, w]: E) adj[u].emplace_back(v, w); Queue q{{ start }}; while (q.size()) { auto u = q.front(); q.pop(); for (auto [v, w]: adj[u]) if (dist[v] > dist[u] + w) dist[v] = dist[u] + w, q.push(v); } return dist; } VI run(const string& filename) { auto N = 0; Edges E; fstream fin{ filename }; VS A; for (string line; getline(fin, line); A.emplace_back(line)); for (auto& s: A) { transform(s.begin(), s.end(), s.begin(), [](auto c) { return c == ',' ? ' ' : c; }); stringstream ss{ s }; auto [u, v, w] = make_tuple(0, 0, 0); ss >> u; while (ss >> v >> w) E.emplace_back(u, v, w); ++N; } auto a = bell(E, N + 1), // +1 for 1-based indexing b = spfa(E, N + 1); assert(a == b); return b; } int main() { auto dist = run("test.txt"); VI A{ 7, 37, 59, 82, 99, 115, 133, 165, 188, 197 }; transform(A.begin(), A.end(), A.begin(), [&](auto x) { return dist[x]; }); copy(A.begin(), A.end(), ostream_iterator(cout, ",")); // 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068, return 0; } ```
--- ### Floyd-Warshall
📚 Lectures
* [The All-Pairs Shortest Path Problem (Section 18.3)](https://www.youtube.com/watch?v=TENbWZPz3Ho&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=62) * [The Floyd-Warshall Algorithm (Part 1) (Section 18.4, Part 1)](https://www.youtube.com/watch?v=ogcvCr02gqM&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=63) * [The Floyd-Warshall Algorithm (Part 2) (Section 18.4, Part 2)](https://www.youtube.com/watch?v=3cBHwPjDZxg&list=PLXFMmlk03Dt5EMI2s2WQBsLsZl7A5HEK6&index=64)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File var key = { i: Int, j: Int -> "$i,$j" } var INF = (1e9 + 7).toInt() fun floyd_warshall(E: MutableMap, N: Int): Array { var dp = Array(N + 1) { Array(N + 1) { IntArray(N + 1) { INF } } } for (i in 0..N) for (j in 0..N) if (i == j) dp[0][i][j] = 0 else if (E.contains(key(i, j))) dp[0][i][j] = E[key(i, j)]!! for (k in 1..N) for (i in 1..N) for (j in 1..N) dp[k][i][j] = Math.min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]) return dp[N] } fun floyd_warshall_memopt(E: MutableMap, N: Int): Array { var pre = Array(N + 1) { IntArray(N + 1) { INF } } for (i in 0..N) for (j in 0..N) if (i == j) pre[i][j] = 0 else if (E.contains(key(i, j))) pre[i][j] = E[key(i, j)]!! for (k in 1..N) { var cur = Array(N + 1) { IntArray(N + 1) { INF } } for (i in 1..N) for (j in 1..N) cur[i][j] = Math.min(pre[i][j], pre[i][k] + pre[k][j]) pre = cur.also{ cur = pre } } return pre } fun run(filename: String) { var N = 0 var E = mutableMapOf() var first = true File(filename).forEachLine { if (!first) { var (u, v, w) = it.trim().split(" ").map{ it.toInt() } E[key(u, v)] = w } else { N = it.trim().split(" ").map{ it.toInt() }[0] first = false } } var a = floyd_warshall_memopt(E, N) var b = floyd_warshall(E, N) for (i in 1..N) for (j in 1..N) assert(a[i][j] == b[i][j]) // 💩 sanity check var cycle = false for (i in 1..N) if (a[i][i] < 0) cycle = true if (cycle) { println("$filename: contains a negative cycle") return } var best = INF for (i in 1..N) for (j in 1..N) best = Math.min(best, a[i][j]) println("$filename: $best") } fun main() { run("problem18.8test1.txt"); // problem18.8test1.txt: -2 run("problem18.8test2.txt"); // problem18.8test2.txt: contains a negative cycle run("problem18.8file1.txt"); // problem18.8file1.txt: contains a negative cycle run("problem18.8file2.txt"); // problem18.8file2.txt: contains a negative cycle run("problem18.8file3.txt"); // problem18.8file3.txt: -19 // run("problem18.8file4.txt"); } ``` *Javascript* ```javascript const LineByLine = require('n-readlines'); const assert = require('assert'); let key = (i, j) => `${i},${j}`; let floyd_warshall = (E, N) => { let dp = [...Array(N + 1)].map(_ => [...Array(N + 1)].map(_ => Array(N + 1).fill(Infinity))); for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) if (i == j) dp[0][i][j] = 0; else if (E.has(key(i, j))) dp[0][i][j] = E.get(key(i, j)); for (let k = 1; k <= N; ++k) for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) dp[k][i][j] = Math.min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]); return dp[N]; }; let floyd_warshall_memopt = (E, N) => { let pre = [...Array(N + 1)].map(_ => Array(N + 1).fill(Infinity)); for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) if (i == j) pre[i][j] = 0; else if (E.has(key(i, j))) pre[i][j] = E.get(key(i, j)); for (let k = 1; k <= N; ++k) { let cur = [...Array(N + 1)].map(_ => Array(N + 1).fill(Infinity)); for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) cur[i][j] = Math.min(pre[i][j], pre[i][k] + pre[k][j]); [pre, cur] = [cur, pre]; } return pre; }; let run = filename => { let E = new Map(); let input = new LineByLine(filename); let [N, _] = input.next().toString('ascii').split(' ').map(Number); let line; while (line = input.next()) { let [u, v, w] = line.toString('ascii').split(' ').map(Number); E.set(key(u, v), w); } let a = floyd_warshall_memopt(E, N), b = floyd_warshall(E, N); for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) assert(a[i][j] == b[i][j]); let cycle = false; for (let i = 1; i <= N; ++i) if (a[i][i] < 0) cycle = true; if (cycle) { console.log(`${filename}: contains a negative cycle`); return; } var best = Infinity; for (row of a) best = Math.min(best, ...row); console.log(`${filename}: ${best}`); }; run('problem18.8test1.txt'); // problem18.8test1.txt: -2 run('problem18.8test2.txt'); // problem18.8test2.txt: contains a negative cycle run('problem18.8file1.txt'); // problem18.8file1.txt: contains a negative cycle run('problem18.8file2.txt'); // problem18.8file2.txt: contains a negative cycle run('problem18.8file3.txt'); // problem18.8file3.txt: -19 // run('problem18.8file4.txt'); ``` *Python3* ```python key = lambda i, j: f'{i},{j}' def floyd_warshall(E, N): dp = [[[float('inf')] * (N + 1) for _ in range(N + 1)] for _ in range(N + 1)] for i in range(1, N + 1): for j in range(1, N + 1): if i == j: dp[0][i][j] = 0 elif key(i, j) in E: dp[0][i][j] = E[key(i, j)] for k in range(1, N + 1): for i in range(1, N + 1): for j in range(1, N + 1): dp[k][i][j] = min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]) return dp[N] def floyd_warshall_memopt(E, N): pre = [[float('inf')] * (N + 1) for _ in range(N + 1)] for i in range(1, N + 1): for j in range(1, N + 1): if i == j: pre[i][j] = 0 elif key(i, j) in E: pre[i][j] = E[key(i, j)] for k in range(1, N + 1): cur = [[float('inf')] * (N + 1) for _ in range(N + 1)] for i in range(1, N + 1): for j in range(1, N + 1): cur[i][j] = min(pre[i][j], pre[i][k] + pre[k][j]) pre, cur = cur, pre return pre def run(filename): E = {} N = 0 first = True with open(filename) as fin: while True: line = fin.readline() if not line: break if not first: u, v, w = [int(x) for x in line.strip().split(' ')] E[key(u, v)] = w else: N = [int(x) for x in line.strip().split(' ')][0] first = False a = floyd_warshall_memopt(E, N) b = floyd_warshall(E, N) for i in range(1, N + 1): for j in range(1, N + 1): assert(a[i][j] == b[i][j]) # 💩 sanity check cycle = False for i in range(1, N + 1): if a[i][i] < 0: cycle = True if cycle: print(f'{filename}: contains a negative cycle') return best = float('inf') for row in a: best = min(best, *row) print(f'{filename}: {best}') run('problem18.8test1.txt') # problem18.8test1.txt: -2 run('problem18.8test2.txt') # problem18.8test2.txt: contains a negative cycle run('problem18.8file1.txt') # problem18.8file1.txt: contains a negative cycle run('problem18.8file2.txt') # problem18.8file2.txt: contains a negative cycle run('problem18.8file3.txt') # problem18.8file3.txt: -19 # run('problem18.8file4.txt') ``` *C++* ```cpp #include #include #include #include #include #define PERF_TEST using namespace std; using LL = long long; using VL = vector; using VVL = vector; using VVVL = vector; using Edges = unordered_map; LL INF = 1e9 + 7; string key(int i, int j) { stringstream ss; ss << i << "," << j; return ss.str(); } VVL floyd_warshall(Edges& E, int N) { VVVL dp(N + 1, VVL(N + 1, VL(N + 1, INF))); for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) if (i == j) dp[0][i][j] = 0; else if (E.find(key(i, j)) != E.end()) dp[0][i][j] = E[key(i, j)]; for (auto k{ 1 }; k <= N; ++k) for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) dp[k][i][j] = min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]); return dp[N]; } VVL floyd_warshall_memopt(Edges& E, int N) { VVL pre(N + 1, VL(N + 1, INF)); for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) if (i == j) pre[i][j] = 0; else if (E.find(key(i, j)) != E.end()) pre[i][j] = E[key(i, j)]; for (auto k{ 1 }; k <= N; ++k) { VVL cur(N + 1, VL(N + 1, INF)); for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) cur[i][j] = min(pre[i][j], pre[i][k] + pre[k][j]); swap(pre, cur); } return pre; } void run(const string& filename) { Edges E; fstream fin{ filename }; int N, M; fin >> N >> M; for (int u, v, w; fin >> u >> v >> w; E[key(u, v)] = w); #ifdef PERF_TEST auto a = floyd_warshall_memopt(E, N); #else auto a = floyd_warshall_memopt(E, N), b = floyd_warshall(E, N); assert(a == b); // 💩 sanity check #endif auto cycle = false; for (auto i{ 1 }; i <= N && !cycle; ++i) cycle = a[i][i] < 0; if (cycle) { cout << filename << ": contains a negative cycle" << endl; return; } auto best = INF; for (auto& row: a) best = min(best, *min_element(row.begin(), row.end())); cout << filename << ": " << best << endl; } int main() { run("problem18.8test1.txt"); // problem18.8test1.txt: -2 run("problem18.8test2.txt"); // problem18.8test2.txt: contains a negative cycle run("problem18.8file1.txt"); // problem18.8file1.txt: contains a negative cycle run("problem18.8file2.txt"); // problem18.8file2.txt: contains a negative cycle run("problem18.8file3.txt"); // problem18.8file3.txt: -19 // run("problem18.8file4.txt"); return 0; } ```
--- # Part 4: Algorithms for NP-Hard Problems
--- ### Traveling Salesman
📚 Lectures
* [Overview and Prerequisites (Section 19.0)](https://www.youtube.com/watch?v=qOFLAjxqoK0&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=1) * [MST vs. TSP: An Algorithmic Mystery (Section 19.1)](https://www.youtube.com/watch?v=9uImnCfKMlA&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=2) * [Possible Levels of Expertise (Section 19.2)](https://www.youtube.com/watch?v=TQ_f5PqmtYo&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=3) * [Easy and Hard Problems (Section 19.3)](https://www.youtube.com/watch?v=PEyRWZA3ESE&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=4) * [Algorithmic Strategies for NP-Hard Problems (Section 19.4)](https://www.youtube.com/watch?v=K4HZKRZ3v-M&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=5) * [Proving NP-Hardness: A Simple Recipe (Section 19.5)](https://www.youtube.com/watch?v=ZqDT-tlIAyY&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=6) * [Rookie Mistakes (Section 19.6)](https://www.youtube.com/watch?v=SBsCjEeSoeQ&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=7)
🎯 Solutions
*Kotlin* ```kotlin import java.io.File class Solution() { private var best = (1e9 + 7).toInt() private var best_path = listOf() private var start = 0 private var M = 0 private var N = 0 private var adj = mutableMapOf>() private var cost = mutableMapOf() private var key = { u: Int, v: Int -> "$u,$v" } private fun init(input_file: String) { best = (1e9 + 7).toInt() best_path = mutableListOf() adj = mutableMapOf>() cost = mutableMapOf() var i = 0 File(input_file).forEachLine { line -> var A = line.trim().split(" ").map{ it.toInt() } if (0 < i++) { var (u, v, w) = A if (!adj.contains(u)) adj[u] = mutableSetOf() if (!adj.contains(v)) adj[v] = mutableSetOf() adj[u]!!.add(v); cost[key(u, v)] = w adj[v]!!.add(u); cost[key(v, u)] = w } else { N = A[0] M = A[1] } } } fun run(input_file: String): Pair> { init(input_file) start = 1 go(start, mutableListOf(start), mutableSetOf(start)) return Pair>(best, best_path) } fun go(u: Int, path: MutableList, seen: MutableSet, t_: Int = 0) { if (seen.size == N) { var t = t_ + (cost[key(u, start)] ?: 0) if (adj[u]!!.contains(start) && t < best) { best = t; best_path = path.toList() } return } for (v in adj[u]!!) { if (seen.contains(v)) continue path.add(v); seen.add(v) go(v, path, seen, t_ + cost[key(u, v)]!!) path.removeLast(); seen.remove(v) } } } fun main() { var s = Solution() for (input_file in listOf("quiz19.2.txt", "quiz20.7.txt")) { var (best, path) = s.run(input_file) println("$input_file best: $best path: ${path.joinToString()}") } } // quiz19.2.txt best: 13 path: 1, 2, 4, 3 // quiz20.7.txt best: 23 path: 1, 3, 2, 5, 4 ``` *Javascript* ```javascript let LineByLine = require('n-readlines'); class Solution { key = (u, v) => `${u},${v}`; init(input_file) { this.best = Number(1e9 + 7); this.best_path = []; this.M = 0; this.N = 0; this.adj = new Map(); this.cost = new Map(); let [A, line, i] = [[], '', 0]; let input = new LineByLine(input_file); while (line = input.next()) { let A = String.fromCharCode(...line).trim().split(' ').map(Number); if (0 < i++) { let [u, v, w] = A; if (!this.adj.has(u)) this.adj.set(u, new Set()); if (!this.adj.has(v)) this.adj.set(v, new Set()); this.adj.get(u).add(v), this.cost.set(this.key(u, v), w); this.adj.get(v).add(u), this.cost.set(this.key(v, u), w); } else { [this.N, this.M] = A; } } } run(input_file) { this.init(input_file); this.start = 1; this.go(this.start, [this.start], new Set([this.start])); return [this.best, this.best_path]; } go(u, path, seen, t = 0) { if (seen.size == this.N) { t += this.cost.get(this.key(u, this.start)) || 0; if (this.adj.get(u).has(this.start) && t < this.best) this.best = t, this.best_path = [...path]; return; } for (let v of this.adj.get(u)) { if (seen.has(v)) continue; path.push(v), seen.add(v); this.go(v, path, seen, t + this.cost.get(this.key(u, v))); path.pop(), seen.delete(v); } } } let s = new Solution(); for (let input_file of ['quiz19.2.txt', 'quiz20.7.txt']) { let [best, path] = s.run(input_file); console.log(`${input_file} best: ${best} path: ${path}`); } // quiz19.2.txt best: 13 path: 1,2,4,3 // quiz20.7.txt best: 23 path: 1,3,2,5,4 ``` *Python3* ```python from random import randint from collections import defaultdict class Solution(): def init(self, input_file): self.best = int(1e9 + 7) self.best_path = [] self.M = 0 self.N = 0 self.adj = defaultdict(set) self.cost = defaultdict(int) with open(input_file) as input: for i, line in enumerate(input): A = [int(x) for x in line.strip().split(' ')] if 0 < i: u, v, w = A self.adj[u].add(v); self.cost[(u, v)] = w self.adj[v].add(u); self.cost[(v, u)] = w else: self.N, self.M = A def run(self, input_file): self.init(input_file) self.start = 1 self.go(self.start, [self.start], set([self.start])) return self.best, self.best_path def go(self, u, path, seen, t = 0): if len(seen) == self.N: t += self.cost[(u, self.start)] # connect ultimate edge of tour if self.start in self.adj[u] and t < self.best: self.best = t self.best_path = path[:] return for v in self.adj[u]: if v not in seen: path.append(v); seen.add(v) self.go(v, path, seen, t + self.cost[(u, v)]) path.pop(); seen.remove(v) s = Solution() for input_file in ['quiz19.2.txt', 'quiz20.7.txt']: best, path = s.run(input_file) print(f'{input_file} best: {best} path: {path}') # quiz19.2.txt best: 13 path: [1, 2, 4, 3] # quiz20.7.txt best: 23 path: [1, 3, 2, 5, 4] ``` *C++* ```cpp #include #include #include #include #include #include #include using namespace std; using VI = vector; using Pair = pair; using Set = unordered_set; using Adj = unordered_map; using Cost = unordered_map; class Solution { int best, start, M, N; VI best_path; Adj adj; Cost cost; string key(int u, int v) { stringstream ss; ss << u << "," << v; return ss.str(); } void init(const string& input_file, string line = {}) { best = 1e9 + 7; start = M = N = 0; best_path.clear(); adj.clear(); cost.clear(); auto i = 0; fstream fin{ input_file }; while (getline(fin, line)) { VI A; istringstream is{ line }; copy(istream_iterator(is), istream_iterator(), back_inserter(A)); if (0 < i++) { auto [u, v, w] = tie(A[0], A[1], A[2]); adj[u].insert(v); cost[key(u, v)] = w; adj[v].insert(u); cost[key(v, u)] = w; } else { N = A[0]; M = A[1]; } } } void go(int u, VI&& path, Set&& seen, int t = 0) { if (seen.size() == N) { t += cost[key(u, start)]; // connect ultimate edge of tour if (adj[u].find(start) != adj[u].end() && t < best) best = t, best_path = path; return; } for (auto v: adj[u]) { if (seen.find(v) != seen.end()) continue; path.push_back(v), seen.insert(v); go(v, move(path), move(seen), t + cost[key(u, v)]); path.pop_back(), seen.erase(v); } } public: Pair run(const string& input_file) { init(input_file); start = 1; go(start, {start}, {start}); return {best, best_path}; } }; int main() { auto s = Solution(); for (auto& input_file: {"quiz19.2.txt", "quiz20.7.txt"}) { auto [best, path] = s.run(input_file); cout << input_file << " best: " << best << " path: "; copy(path.begin(), path.end(), ostream_iterator(cout, " ")); cout << endl; } // quiz19.2.txt best: 13 path: 1 3 4 2 // quiz20.7.txt best: 23 path: 1 4 5 2 3 return 0; } ```
### Traveling Salesman (Heuristic)
📚 Lectures
* [Makespan Minimization (Part 1) (Section 20.1, Part 1)](https://www.youtube.com/watch?v=fn5HNoiMwHM&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=8) * [Makespan Minimization (Part 2) (Section 20.1, Part 2)](https://www.youtube.com/watch?v=CcdYLcLymV4&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=9) * [Maximum Coverage (Part 1) (Section 20.2, Part 1)](https://www.youtube.com/watch?v=GcJy8uzSUbA&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=10) * [Maximum Coverage (Part 2) (Section 20.2, Part 2)](https://www.youtube.com/watch?v=fHldsoBMFKI&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=11) * [Influence Maximization (Part 1) (Section 20.3, Part 1)](https://www.youtube.com/watch?v=GqKbx3LCQqE&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=12) * [Influence Maximization (Part 2) (Section 20.3, Part 2)](https://www.youtube.com/watch?v=_1-Vv5Vid9Q&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=13) * [2-OPT Heuristic for the TSP (Part 1) (Section 20.4, Part 1)](https://www.youtube.com/watch?v=dYEWqrp-mho&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=14) * [2-OPT Heuristic for the TSP (Part 2) (Section 20.4, Part 2)](https://www.youtube.com/watch?v=8vbKIfpDPJI&list=PLEGCF-WLh2RK6lq3iSsiU84rWVee3A-hz&index=15)
🎯 Solutions
*TSP Nearest Neighbor Heuristic (Python3)* ```python from collections import defaultdict N, M = -1, -1 adj = defaultdict(list) cost, key = {}, lambda u, v: f'{u},{v}' with open('input.txt') as input: for line in input: A = [int(x) for x in line.strip().split(' ')] if len(A) == 3: u, v, w = A adj[u].append(v); cost[key(u, v)] = w adj[v].append(u); cost[key(v, u)] = w elif len(A) == 2: N, M = A start = 1 u, seen, path = start, set([start]), [start] while len(seen) < N: best, best_v = float('inf'), -1 for v in adj[u]: if v not in seen: cand = cost[key(u, v)] if best > cand: best = cand; best_v = v u = best_v; seen.add(best_v); path.append(best_v) path.append(start) # add last edge to complete the tour t = sum(cost[key(path[i - 1], path[i])] for i in range(1, len(path))) print(f'total: {t} path: {path}') # total: 29 path: [1, 2, 3, 4, 5, 1] ```
================================================ FILE: _challenge_problems/index_equal_element.py ================================================ # You are given a sorted (from smallest to largest) array A of N distinct integers which can be positive, negative, or zero. Design the fastest algorithm you can for deciding whether there is an index i such that A[i] = i. # use case: if the value is less-than the index, search right half # -3 -2 -1 0 4 # 0 1 2 3 4 # ^ # use case: if the value is greater than the index, search left half # 0 2 3 4 5 # 0 1 2 3 4 # ^ def same(A): i, j = 0, len(A) - 1 while i < j: k = (i + j) // 2 if A[k] < k: i = k + 1 else: j = k return A[i] == i def run(A, ok=False): assert(same(A) == ok) run([0, 2, 3, 4, 5], True) run([-3, -2, -1, 0, 4], True) ================================================ FILE: _challenge_problems/unimodal_array.py ================================================ # Problem: You are given a unimodal array of n distinct elemnts, meaning that its entries are in increasing order up until its maximum element, after which its elemnets are in decreasing order. Give an algorithm to compute the maximum element of a unimodal array that runs in O(logN) time. def best(A): i, j = 0, len(A) - 1 while i < j: k = (i + j) // 2 if A[k] < A[k + 1]: i = k + 1 else: j = k return A[i] def run(A): error = f'NOT ({best(A)} == {max(A)})' assert best(A) == max(A), error # small inputs to test edge cases run([0, 1, 0]) run([1, 3, 1]) run([2, 5, 4]) # mediuim inputs run([0, 2, 4, 6, 8, 6, 4, 2]) run([1, 4, 7, 9, 8, 3]) run([3, 5, 9, 12, 10, 6, 2]) # peak near beginning/end run([1, 5, 4, 3, 2]) run([0, 2, 4, 6, 5]) # larger inputs run([0, 1, 3, 6, 10, 15, 21, 18, 14, 9, 5, 2]) run([2, 4, 6, 8, 11, 14, 17, 16, 13, 9, 5, 1]) run([1, 5, 10, 20, 35, 50, 49, 30, 15, 7, 2]) run([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]) run([0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 34, 21, 13, 8, 5, 3, 2, 1, 0]) import random def random_unimodal(seed=0): A = [] random.seed(seed) while len(A) < int(1e4): A.append(random.randint(0, int(1e9))) A.sort() L = A[:len(A) // 2] R = A[len(A) // 2:] return L + list(reversed(R)) for _ in range(10): run(random_unimodal()) ================================================ FILE: bellman_ford/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(bellman_ford) set(CMAKE_CXX_STANDARD 17) add_executable(bellman_ford main.cpp) ================================================ FILE: bellman_ford/main.cpp ================================================ #include #include #include #include #include #include using namespace std; using VI = vector; using VS = vector; using Edge = tuple; using Edges = vector; using Pair = pair; using Pairs = vector; using AdjList = unordered_map; using Queue = queue; // bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices VI bell(Edges& E, int N, int start = 1, int INF = 1e6) { VI dist(N, INF); dist[start] = 0; auto K = N - 1; while (K--) for (auto [u, v, w]: E) dist[v] = min(dist[v], dist[u] + w); return dist; } // shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) VI spfa(Edges& E, int N, int start = 1, int INF = 1e6, AdjList adj = {}) { VI dist(N, INF); dist[start] = 0; for (auto [u, v, w]: E) adj[u].emplace_back(v, w); Queue q{{ start }}; while (q.size()) { auto u = q.front(); q.pop(); for (auto [v, w]: adj[u]) if (dist[v] > dist[u] + w) dist[v] = dist[u] + w, q.push(v); } return dist; } VI run(const string& filename) { auto N = 0; Edges E; fstream fin{ filename }; VS A; for (string line; getline(fin, line); A.emplace_back(line)); for (auto& s: A) { transform(s.begin(), s.end(), s.begin(), [](auto c) { return c == ',' ? ' ' : c; }); stringstream ss{ s }; auto [u, v, w] = make_tuple(0, 0, 0); ss >> u; while (ss >> v >> w) E.emplace_back(u, v, w); ++N; } auto a = bell(E, N + 1), // +1 for 1-based indexing b = spfa(E, N + 1); assert(a == b); return b; } int main() { auto dist = run("test.txt"); VI A{ 7, 37, 59, 82, 99, 115, 133, 165, 188, 197 }; transform(A.begin(), A.end(), A.begin(), [&](auto x) { return dist[x]; }); copy(A.begin(), A.end(), ostream_iterator(cout, ",")); // 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068, return 0; } ================================================ FILE: bellman_ford/main.js ================================================ const assert = require('assert'); const zip = require('lodash/zip'); const LineByLine = require('n-readlines'); // bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices let bell = (E, N, start = 1, INF = Number(1e6)) => { let dist = Array(N).fill(INF); dist[start] = 0; let K = N - 1; while (K--) E.forEach(([u, v, w]) => dist[v] = Math.min(dist[v], dist[u] + w)); return dist; }; // shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) let spfa = (E, N, start = 1, INF = Number(1e6)) => { let dist = Array(N).fill(INF); dist[start] = 0; let adj = [...Array(N)].map(_ => []); E.forEach(([u, v, w]) => adj[u].push([v, w])); let q = [start]; while (q.length) { let u = q.shift(); for (let [v, w] of adj[u]) if (dist[v] > dist[u] + w) dist[v] = dist[u] + w, q.push(v); } return dist; }; let run = filename => { let N = 0; let E = []; let input = new LineByLine(filename); let line; while (line = input.next()) { let A = line.toString('ascii').split('\t').filter(it => it.length); let u = Number(A.shift()); A.map(pair => pair.split(',').map(Number)).forEach(([v, w]) => E.push([u, v, w])); ++N; } let a = bell(E, N + 1), // +1 for 1-based indexing b = spfa(E, N + 1); zip(a, b).forEach(([x, y]) => assert(x == y)); // 💩 sanity check: single source shortest paths are the same return a; }; let dist = run('test.txt'); console.log([7, 37, 59, 82, 99, 115, 133, 165, 188, 197].map(x => dist[x]).join(',')); // 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068 ================================================ FILE: bellman_ford/main.kt ================================================ import java.io.File import java.util.LinkedList import java.util.Queue // bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices fun bell(E: Array>, N: Int, start: Int = 1, INF: Int = (1e6).toInt()): IntArray { var dist = IntArray(N) { INF } dist[start] = 0 var K = N - 1 while (0 < K--) E.forEach{ (u, v, w) -> dist[v] = Math.min(dist[v], dist[u] + w)} return dist } // shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) fun spfa(E: Array>, N: Int, start: Int = 1, INF: Int = (1e6).toInt()): IntArray { var dist = IntArray(N) { INF } dist[start] = 0 var adj = Array>>(N) { mutableListOf>() } for ((u, v, w) in E) adj[u].add(Pair(v, w)) var q: Queue = LinkedList(listOf(start)) while (0 < q.size) { var u = q.poll() for ((v, w) in adj[u]) { if (dist[v] > dist[u] + w) { dist[v] = dist[u] + w; q.add(v) } } } return dist } fun run(filename: String): IntArray { var N = 0 var E = mutableListOf>() File(filename).forEachLine { var A = ArrayDeque(it.trim().split("\t")) var u = A.removeFirst().toInt() for ((v, w) in A.map{ it.split(",").map{ it.toInt() } }) E.add(Triple(u, v, w)) ++N; } var A = E.toTypedArray() var a = bell(A, N + 1) // +1 for 1-based indexing var b = spfa(A, N + 1) assert(a == b) // 💩 sanity check: single source shortest paths are the same return b } fun main() { var dist = run("test.txt") println(listOf(7, 37, 59, 82, 99, 115, 133, 165, 188, 197).map{ dist[it] }.joinToString(",")) // 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068 } ================================================ FILE: bellman_ford/main.py ================================================ from collections import deque # bellman-ford: N - 1 edge relaxations (u -> v of cost w) [ie. attempting to relax M edges N - 1 times] given N vertices def bell(E, N, start = 1, INF = int(1e6)): dist = [INF] * N dist[start] = 0 k = N - 1 while k: for u, v, w in E: dist[v] = min(dist[v], dist[u] + w) k -= 1 return dist # shortest-paths faster algorithm: only attempt to relax candidate edges (note: adjacency list needed) def spfa(E, N, start = 1, INF = int(1e6)): dist = [INF] * N dist[start] = 0 adj = {i: [] for i in range(N)} for u, v, w in E: adj[u].append([v, w]) q = deque([start]) while q: u = q.popleft() for v, w in adj[u]: if dist[v] > dist[u] + w: dist[v] = dist[u] + w; q.append(v) return dist def run(filename): E = [] N = 0 with open(filename) as fin: while True: line = fin.readline() if not line: break line = line.strip() A = [word for word in line.split('\t') if len(word)] u = int(A[0]) for i in range(1, len(A)): v, w = [int(x) for x in A[i].split(',')] E.append([u, v, w]) N += 1 a = bell(E, N + 1) # +1 for 1-based indexing b = spfa(E, N + 1) assert(a == b) return b dist = run('test.txt') print(','.join(str(dist[x]) for x in [7, 37, 59, 82, 99, 115, 133, 165, 188, 197])) # 2599,2610,2947,2052,2367,2399,2029,2442,2505,3068 ================================================ FILE: bellman_ford/package.json ================================================ { "name": "bellman_ford", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "clayton wong", "license": "ISC", "dependencies": { "assert": "^2.0.0", "lodash": "^4.17.21", "n-readlines": "^1.0.1" } } ================================================ FILE: bellman_ford/test.txt ================================================ 1 80,982 163,8164 170,2620 145,648 200,8021 173,2069 92,647 26,4122 140,546 11,1913 160,6461 27,7905 40,9047 150,2183 61,9146 159,7420 198,1724 114,508 104,6647 30,4612 99,2367 138,7896 169,8700 49,2437 125,2909 117,2597 55,6399 2 42,1689 127,9365 5,8026 170,9342 131,7005 172,1438 34,315 30,2455 26,2328 6,8847 11,1873 17,5409 157,8643 159,1397 142,7731 182,7908 93,8177 3 57,1239 101,3381 43,7313 41,7212 91,2483 31,3031 167,3877 106,6521 76,7729 122,9640 144,285 44,2165 6,9006 177,7097 119,7711 4 162,3924 70,5285 195,2490 72,6508 126,2625 121,7639 31,399 118,3626 90,9446 127,6808 135,7582 159,6133 106,4769 52,9267 190,7536 78,8058 75,7044 116,6771 49,619 107,4383 89,6363 54,313 5 200,4009 112,1522 25,3496 23,9432 64,7836 56,8262 120,1862 2,8026 90,8919 142,1195 81,2469 182,8806 17,2514 83,8407 146,5308 147,1087 51,22 6 141,8200 98,5594 66,6627 159,9500 143,3110 129,8525 118,8547 88,2039 83,4949 165,6473 162,6897 184,8021 123,13 176,3512 195,2233 42,7265 47,274 132,1514 2,8847 171,3722 3,9006 7 156,7027 187,9522 87,4976 121,8739 56,6616 10,2904 71,8206 53,179 146,4823 165,6019 125,5670 27,4888 63,9920 150,9031 84,4061 8 152,1257 189,2780 58,4708 26,8342 199,1918 31,3987 35,3160 71,5829 27,3483 69,8815 130,55 168,2076 122,5338 73,4528 28,9996 17,3535 40,3193 72,7308 24,8434 87,2833 25,3949 175,1022 177,8508 9 152,1087 115,7827 17,7002 72,794 150,4539 190,3613 95,9480 36,5284 166,8702 63,1753 199,70 131,700 76,9340 70,2 139,8701 140,4163 180,5995 10 57,9988 78,3771 62,4816 137,5273 7,2904 187,4786 184,3207 96,807 31,1184 88,2539 135,4650 168,9495 164,3866 11,8988 116,1493 51,5578 171,2029 11 1,1913 185,2045 77,815 22,8425 181,8448 47,8727 81,7299 150,4802 178,1696 28,2275 183,594 131,833 157,8497 25,5057 59,3203 10,8988 2,1873 134,294 83,4211 124,6180 12 78,5753 17,4602 62,5676 16,8068 60,5933 67,371 71,6734 53,7001 72,3626 34,6690 59,761 18,1520 128,7542 38,6699 57,9416 13 144,9987 59,9801 97,7026 50,758 43,5400 163,3870 178,4194 151,9629 45,1794 105,6821 29,2784 172,2070 57,6850 77,8638 135,861 14 149,4352 187,4874 26,3841 128,9662 155,4446 118,373 123,2733 106,7912 169,4333 53,9197 161,4275 126,9602 73,4106 160,7860 131,358 141,4477 119,960 43,3199 47,7898 175,6718 177,6741 60,2464 127,5682 31,1945 143,5848 94,3551 82,3283 15 42,1789 22,3571 25,7019 163,818 56,2334 100,809 143,1041 107,4589 190,6854 169,7485 94,9606 34,7961 54,8983 157,2136 24,8040 16 200,2848 198,2223 92,2896 18,8663 27,8673 75,4116 150,1680 36,1555 41,2747 90,4558 68,5894 12,8068 42,2596 185,6280 171,3482 109,1469 127,9807 178,1714 35,839 56,9828 134,5203 55,6680 110,4252 17 26,1275 45,5114 142,8016 83,4615 140,6440 8,3535 69,3610 153,8545 9,7002 12,4602 173,7312 114,8915 108,1942 54,3115 66,6176 190,7000 70,3899 5,2514 178,7464 166,4762 2,5409 146,5362 117,6266 18 57,4216 80,5252 86,7517 62,1926 120,44 173,7256 133,2702 148,589 167,7625 16,8663 170,4989 118,6388 142,332 95,6122 99,5717 154,453 150,5150 149,2664 146,9000 171,4403 111,785 12,1520 19 33,6938 77,7013 187,107 109,8397 88,2002 95,8691 132,3157 195,5038 154,4320 23,8560 152,9751 185,5896 119,7406 160,3997 80,62 20 66,2667 173,2676 43,8105 135,6434 33,6387 74,6183 106,8785 75,2484 130,9048 56,7194 50,9507 88,3014 124,392 61,2580 90,7372 92,1704 87,2639 154,2398 41,4203 85,1435 169,5990 166,6086 28,2234 145,8099 21 23,5183 40,2199 31,2556 71,4986 165,2151 193,494 154,1845 111,3060 85,2880 101,2775 182,2447 80,9884 87,2681 102,6643 131,3748 22 92,5592 64,4257 11,8425 24,594 15,3571 42,3783 41,1374 114,9960 144,9362 146,3620 71,3243 143,8603 131,6075 192,4606 108,9656 168,4356 177,8713 132,1560 23 143,7543 161,6863 45,8074 165,208 21,5183 118,5079 40,8336 27,9054 112,3201 135,4560 167,2133 188,4236 166,8077 195,3179 48,4485 137,7591 99,6485 5,9432 71,3316 96,2431 125,922 19,8560 24 141,6862 197,9337 66,5879 59,6941 70,4670 55,4106 103,8083 61,7906 48,7959 151,784 177,393 102,8731 199,2838 73,3509 8,8434 187,9327 22,594 150,5669 164,7312 157,9540 15,8040 25 115,9233 197,3875 185,3573 72,2332 104,4899 137,5378 8,3949 5,3496 77,2729 136,9251 143,108 83,9569 15,7019 48,3214 155,3242 153,2477 129,3005 132,219 11,5057 37,1591 68,4188 26 14,3841 8,8342 1,4122 147,5759 113,5553 157,7 65,9434 116,4221 66,2747 138,7027 145,6697 130,5706 60,701 127,9896 136,7200 17,1275 120,5788 175,6165 70,9252 95,36 106,6940 2,2328 96,425 51,9329 183,4842 196,6754 27 23,9054 78,3066 8,3483 1,7905 152,2124 108,9929 63,3896 151,5915 111,3101 34,8912 182,6234 133,7749 16,8673 192,5344 114,714 168,1578 175,210 138,5918 7,4888 122,84 28 8,9996 188,3816 116,2638 132,5604 20,2234 178,3642 76,3705 122,9165 184,4164 198,366 161,9217 160,9059 56,5375 120,8874 11,2275 111,4495 193,9441 157,6880 48,2803 29 78,8190 144,6452 114,9478 156,5083 62,9692 121,4537 184,9797 109,6873 153,5446 67,3449 172,5830 111,1005 100,1642 148,3252 13,2784 30 78,5469 119,7372 144,1616 130,1356 59,4458 40,9818 79,503 43,6233 148,4760 42,263 1,4612 57,5668 185,3846 101,6979 94,6976 106,7819 2,2455 71,9294 31 4,399 8,3987 50,2598 75,7688 47,7840 99,8583 190,5055 112,5231 114,7617 118,6949 180,3598 21,2556 199,5564 14,1945 3,3031 35,9855 10,1184 146,2837 51,3739 83,6588 46,5964 32 136,3823 77,1689 92,3395 121,1615 85,7494 173,9631 177,6902 88,8129 36,7329 116,6065 61,3332 68,7352 119,1914 82,8571 70,9909 33 144,4841 173,5949 170,3648 113,652 110,1986 82,3577 61,1837 97,5671 55,1252 19,6938 48,914 74,3642 125,67 89,3089 176,3258 20,6387 138,6960 153,6574 171,3913 34 86,6435 156,8641 72,2540 181,5267 27,8912 58,8824 179,8528 62,9864 70,2348 57,5471 53,236 168,3923 101,3383 142,7791 55,7174 2,315 147,9758 15,7961 199,8196 12,6690 35 57,3693 8,3160 144,3087 114,490 65,8910 178,5774 172,992 16,839 118,8640 41,6749 31,9855 39,853 64,6071 166,2816 184,7437 49,3098 182,7369 110,4985 93,8775 36 80,2032 130,7589 123,6226 16,1555 150,116 88,7759 100,8612 9,5284 198,6280 49,953 143,5111 42,4917 134,979 159,6043 32,7329 67,2380 148,9550 48,7266 37 197,9188 119,9313 187,4105 191,3573 109,2135 75,751 200,7541 139,8208 155,609 142,6433 25,1591 132,821 156,7714 107,1144 99,7757 38 91,7087 88,502 132,6092 126,5441 147,8391 12,6699 130,5227 146,4400 108,8712 100,1369 134,4730 87,2975 99,6169 183,5213 109,4945 39 200,4319 98,3993 130,2414 40,2489 196,9267 133,8145 82,3528 44,9175 42,5464 127,6103 93,6132 180,9506 192,7454 119,1376 115,983 81,7400 35,853 40 23,8336 1,9047 120,7760 101,2885 21,2199 144,7772 96,5739 136,4658 184,4306 189,4263 30,9818 39,2489 108,8883 8,3193 80,9657 181,2338 162,3056 71,2826 68,5800 41 200,2622 78,63 66,4654 198,7215 59,284 75,7333 22,1374 181,5235 16,2747 154,901 150,7278 3,7212 103,7917 163,5256 20,4203 91,7776 35,6749 147,1858 165,3741 107,8116 42 160,2382 156,6539 6,7265 15,1789 61,8096 164,347 194,6498 172,5383 104,2726 124,3496 161,4792 159,5951 117,7074 2,1689 186,9391 62,3249 79,9404 39,5464 187,3075 22,3783 30,263 16,2596 137,4572 163,1278 60,6663 70,9396 36,4917 73,9154 43 200,8943 159,9621 97,3906 20,8105 164,6849 13,5400 3,7313 133,8488 108,8964 30,6233 79,5052 131,8231 167,8120 14,3199 130,2685 138,7965 177,9544 143,1171 65,5805 118,8008 140,4482 93,8479 44 197,4900 144,2276 198,2619 39,9175 87,7875 191,8130 166,6953 170,6940 163,18 79,9988 145,2888 173,5518 57,9979 82,3134 54,4113 3,2165 45 57,4630 23,8074 112,9496 130,4994 86,8207 17,5114 120,5279 169,662 162,3436 170,8060 118,5918 124,3290 110,8317 13,1794 167,1163 46 57,2413 152,9550 86,7512 123,132 138,2860 195,8206 176,9923 119,2687 54,9328 196,9632 73,5109 31,5964 173,2969 193,199 80,7968 194,2429 47 57,9584 114,9480 145,9483 190,5892 182,8382 31,7840 129,9533 142,5297 58,1229 146,2959 6,274 14,7898 189,5939 11,8727 76,2138 70,2236 48 152,5835 23,4485 33,914 24,7959 25,3214 135,8869 53,3578 162,201 28,2803 141,7941 36,7266 85,2792 86,3588 124,2593 130,7921 49 160,8648 154,2962 109,7520 36,953 178,9747 192,3113 112,2935 35,3098 71,3441 4,619 96,9901 171,9736 163,4688 1,2437 133,5167 117,2896 105,9278 50 152,5767 112,6454 185,3968 77,5220 20,9507 165,2667 98,990 187,2485 198,3798 13,758 128,2987 189,7031 52,9931 127,3622 31,2598 179,2502 191,5026 153,4905 51 80,7589 72,4882 137,1096 138,8755 109,662 67,4225 181,158 132,6107 189,8899 159,3017 5,22 10,5578 31,3739 120,5675 26,9329 176,1625 52 4,9267 115,4973 159,7816 185,8925 188,7805 97,9063 50,9931 137,9846 91,424 150,634 56,2416 107,3647 68,7601 168,1134 179,3504 53 14,9197 114,7352 156,4662 62,153 85,1227 177,9852 34,236 7,179 12,7001 48,3578 71,9285 86,7353 150,662 183,5304 125,8054 54,8361 54 197,2223 66,2906 136,1794 188,4883 17,3115 109,7832 44,4113 182,438 15,8983 200,4899 112,2279 169,2296 4,313 53,8361 138,6261 46,9328 55 33,1252 188,5181 101,6050 24,4106 169,7795 149,3088 34,7174 193,8583 1,6399 145,3342 105,8477 166,3686 121,44 16,6680 82,3547 56 101,3516 20,7194 179,5284 127,3031 5,8262 161,9811 16,9828 15,2334 52,2416 7,6616 77,7923 182,7267 88,3375 61,1315 117,1934 28,5375 124,552 100,361 57 18,4216 94,558 186,8815 3,1239 85,6678 45,4630 46,2413 35,3693 84,6563 185,9772 67,8012 47,9584 155,893 64,810 10,9988 80,8722 160,2058 59,2689 79,2330 30,5668 184,7592 44,9979 162,6483 116,656 34,5471 106,4868 131,6342 183,9093 13,6850 12,9416 58 152,5877 98,3677 8,4708 130,7020 59,5735 121,8818 47,1229 102,6906 150,4857 90,7141 86,5989 175,3675 79,2365 34,8824 186,8993 125,1050 74,7934 147,2267 193,6166 59 86,1293 147,2651 149,2405 141,9126 112,4585 58,5735 74,4470 24,6941 199,8958 57,2689 13,9801 162,391 30,4458 180,2435 41,284 72,7154 101,1804 87,4628 168,4170 99,671 70,8055 11,3203 12,761 60 200,3269 98,2073 26,701 185,6670 120,2231 14,2464 127,1402 12,5933 42,6663 189,4415 107,52 146,2317 112,2570 154,6667 177,5345 172,2781 61 1,9146 159,49 33,1837 42,8096 20,2580 24,7906 87,9053 163,448 190,9775 155,5301 173,4803 115,3324 196,5577 171,6888 32,3332 56,1315 131,6924 195,8928 62 97,9163 53,153 120,3851 18,1926 154,3238 12,5676 88,9007 152,7404 29,9692 161,4144 10,4816 105,2736 42,3249 107,5324 115,1913 121,4145 116,7419 34,9864 193,6610 103,8383 63 141,5607 77,5873 27,3896 169,5160 95,5264 69,2323 125,1315 158,5709 102,5806 9,1753 103,9314 71,3007 131,5257 92,9006 96,5638 7,9920 64 57,810 98,3909 97,2201 22,4257 120,2385 177,7660 83,2716 81,9744 111,2663 145,2685 130,2493 148,6419 106,256 141,158 86,414 87,9403 121,771 102,4635 5,7836 67,2090 35,6071 131,4631 182,4701 110,6711 65 152,3595 66,6930 26,9434 97,6170 123,9599 175,7920 155,5533 102,1652 77,4069 198,3575 81,3054 199,11 95,6605 35,8910 43,5805 71,439 134,9956 74,6617 165,3705 140,5376 66 80,2902 68,8312 142,777 156,2965 41,4654 6,6627 84,7710 102,3328 65,6930 54,2906 24,5879 112,2271 93,5873 94,3424 20,2667 26,2747 130,5826 17,6176 69,824 89,3012 67 57,8012 102,5417 175,5048 153,6204 12,371 137,1414 133,3802 64,2090 98,980 200,475 171,1394 36,2380 29,3449 124,1880 51,4225 195,5737 100,6216 103,1468 68 141,3540 197,8223 78,7924 66,8312 144,2277 174,7082 16,5894 163,4920 146,3895 52,7601 140,9624 40,5800 25,4188 32,7352 186,2528 69 8,8815 198,6284 17,3610 156,9959 75,3354 168,2357 102,1172 190,8022 139,9030 161,6171 96,4815 189,5215 66,824 94,1427 63,2323 70 4,5285 24,4670 148,7231 26,9252 17,3899 59,8055 47,2236 42,9396 175,3256 149,2366 92,96 153,6532 178,3394 168,1295 156,4830 34,2348 9,2 124,9089 32,9909 183,5332 71 8,5829 22,3243 138,1229 81,1711 170,1539 49,3441 23,3316 134,7485 12,6734 30,9294 21,4986 142,6038 65,439 7,8206 40,2826 145,6127 53,9285 63,3007 186,7143 171,6702 72 4,6508 78,5839 119,6215 114,8350 9,794 8,7308 113,8782 102,3377 34,2540 25,2332 59,7154 172,3153 89,4836 178,5128 51,4882 120,2287 174,2019 153,541 96,859 146,4264 171,8573 157,604 12,3626 73 14,4106 8,4528 159,4969 97,6534 77,2438 24,3509 174,2581 150,8061 139,4428 149,5233 42,9154 90,5133 78,212 194,8521 172,2239 46,5109 74 159,8960 33,3642 59,4470 20,6183 99,7031 179,1223 93,5576 164,8627 58,7934 65,6617 110,6731 108,8251 165,2602 121,1468 182,1873 176,8129 75 115,9140 141,9237 80,2187 86,259 20,2484 92,6095 97,1883 41,7333 87,3244 69,3354 120,6892 131,5902 31,7688 108,5943 4,7044 16,4116 191,1403 81,2609 37,751 76 115,7291 185,3674 181,3275 47,2138 143,1079 28,3705 125,1865 178,8433 3,7729 114,9690 100,1793 200,4623 199,6878 138,5683 141,1969 126,9595 9,9340 83,4424 89,6942 77 112,3500 160,105 189,5702 191,5135 124,8896 198,5081 19,7013 73,2438 63,5873 129,2337 11,815 133,2481 192,561 32,1689 50,5220 87,7040 25,2729 65,4069 106,9161 153,4483 56,7923 172,4771 13,8638 78 10,3771 68,7924 12,5753 30,5469 158,6367 122,6207 27,3066 116,2732 41,63 72,5839 161,6310 4,8058 104,1377 83,3955 29,8190 98,6603 154,8423 137,1910 135,6919 73,212 145,7244 79 141,7918 101,3205 165,3768 96,3059 119,4117 152,6519 57,2330 42,9404 166,8726 161,8395 30,503 89,5169 134,5792 117,9043 129,7314 43,5052 109,9677 58,2365 44,9988 167,820 193,7737 194,5784 80 36,2032 84,4645 1,982 115,1417 151,6728 112,5208 51,7589 152,9606 113,917 18,5252 121,2257 75,2187 57,8722 133,7217 179,7729 119,108 66,2902 40,9657 97,7213 172,7715 89,7224 19,62 46,7968 21,9884 81 115,2608 197,5540 97,8866 101,4493 64,9744 11,7299 71,1711 109,2519 136,1409 39,7400 75,2609 142,424 141,4032 183,3061 184,4485 95,7627 5,2469 143,9810 65,3054 89,6124 82 33,3577 130,3349 156,4691 39,3528 173,591 177,7882 44,3134 116,8491 132,4162 135,519 131,3457 128,6834 32,8571 55,3547 14,3283 83 78,3955 6,4949 185,9306 17,4615 64,2716 25,9569 149,6823 5,8407 167,8200 117,8516 165,1555 151,162 31,6588 76,4424 11,4211 84 57,6563 80,4645 119,6417 66,7710 198,5999 136,4270 86,195 104,5330 154,5421 137,4367 95,3812 159,8763 170,2436 107,2954 85,9888 134,9312 7,4061 85 57,6678 160,3613 156,6669 168,6193 136,6221 180,5525 32,7494 118,1102 192,544 129,517 93,2349 87,7478 189,1147 53,1227 20,1435 167,8110 133,836 84,9888 132,3873 128,4644 110,6060 21,2880 48,2792 86 115,4291 197,9714 144,8808 59,1293 126,937 189,1115 18,7517 45,8207 46,7512 177,7010 180,4604 75,259 157,4447 84,195 34,6435 120,5230 64,414 184,801 58,5989 142,1663 53,7353 117,4220 48,3588 87 160,8712 119,518 75,3244 94,9647 59,4628 61,9053 44,7875 168,9716 64,9403 164,3629 20,2639 8,2833 77,7040 7,4976 159,19 85,7478 191,6921 88,8011 167,1022 158,4081 110,1219 21,2681 38,2975 88 6,2039 62,9007 20,3014 113,7322 136,9026 32,8129 38,502 151,2295 150,6770 183,5547 36,7759 87,8011 94,4629 115,6611 19,2002 161,1726 56,3375 10,2539 125,5012 89,6267 89 33,3089 72,4836 123,1723 79,5169 174,858 76,6942 4,6363 199,2446 105,2736 66,3012 180,6612 80,7224 163,4055 88,6267 81,6124 90 152,4427 4,9446 115,1117 119,928 185,7284 20,7372 16,4558 108,9076 179,3149 139,7846 58,7141 5,8919 73,5133 144,6223 174,6914 91 160,383 181,5060 174,3418 113,4626 95,1806 3,2483 192,6625 52,424 115,1105 137,4129 142,9164 41,7776 158,5553 38,7087 200,1988 92 1,647 130,4320 108,1844 134,610 194,426 177,3182 75,6095 20,1704 94,6085 128,556 22,5592 16,2896 186,7980 32,3395 139,6763 121,3819 138,8080 70,96 63,9006 93 66,5873 39,6132 181,4071 154,4073 85,2349 106,7477 74,5576 150,9213 98,6617 147,7807 43,8479 152,6543 35,8775 167,5670 2,8177 94 57,558 66,3424 92,6085 120,2733 87,9647 30,6976 191,8318 139,7116 109,1299 88,4629 170,9318 69,1427 14,3551 115,3350 171,9959 15,9606 95 119,8126 112,555 120,1104 18,6122 91,1806 173,4092 196,231 26,36 147,1278 19,8691 125,2917 9,9480 63,5264 81,7627 84,3812 65,6605 105,6026 96 40,5739 79,3059 104,9639 113,712 162,3737 155,1251 10,807 49,9901 151,5643 23,2431 72,859 26,425 69,4815 143,9274 183,5939 63,5638 147,6736 193,8831 97 33,5671 185,7065 52,9063 64,2201 188,4695 192,6411 43,3906 73,6534 13,7026 112,7969 81,8866 80,7213 62,9163 65,6170 140,6527 75,1883 137,4667 98 115,5825 131,4614 64,3909 155,5515 139,1235 39,3993 102,8330 60,2073 200,2690 166,2364 78,6603 162,6139 58,3677 117,9545 6,5594 144,7198 50,990 150,2093 143,4300 67,980 93,6617 99 104,9140 18,5717 174,5675 157,6818 132,6234 182,2897 151,4990 183,3577 59,671 133,2090 23,6485 153,4560 31,8583 74,7031 1,2367 127,1408 37,7757 193,4566 194,5832 38,6169 100 159,6567 137,7178 163,9709 190,6674 36,8612 142,2994 76,1793 67,6216 29,1642 56,361 144,6605 128,2584 153,9522 145,5512 15,809 38,1369 101 188,8531 40,2885 157,1393 171,4083 55,6050 144,3619 3,3381 113,5024 81,4493 163,6033 56,3516 129,8821 184,9591 59,1804 79,3205 30,6979 138,2902 143,4042 34,3383 21,2775 102 98,8330 66,3328 144,7884 72,3377 24,8731 181,3585 137,6814 172,6572 58,6906 64,4635 117,2689 177,4462 67,5417 183,9634 69,1172 65,1652 178,1334 161,8230 63,5806 140,6370 21,6643 103 200,6851 123,8756 24,8083 41,7917 191,9683 63,9314 112,7409 110,491 131,2920 196,696 186,9654 62,8383 113,5248 67,1468 114,1318 104 141,8162 78,1377 42,2726 123,9213 1,6647 126,8615 200,7083 197,4174 84,5330 192,4219 142,6236 99,9140 96,9639 25,4899 172,561 179,8827 169,3712 105 185,1033 62,2736 113,3388 116,7899 89,2736 164,4661 183,7722 55,8477 190,2518 180,341 95,6026 119,9930 120,9333 13,6821 49,9278 106 14,7912 4,4769 115,9598 141,674 112,4854 20,8785 64,256 181,5332 190,3305 3,6521 30,7819 93,7477 26,6940 77,9161 57,4868 111,6460 107 114,2032 123,4581 62,5324 187,2610 60,52 116,9864 84,2954 182,8313 37,1144 169,668 52,3647 4,4383 41,8116 146,7862 112,7448 15,4589 176,6806 108 200,9976 185,8699 17,1942 40,8883 156,7039 92,1844 75,5943 22,9656 43,8964 27,9929 174,5669 90,9076 145,521 143,972 113,4342 74,8251 126,525 38,8712 109 152,2743 136,8658 81,2519 169,382 51,662 49,7520 129,2464 79,9677 54,7832 37,2135 94,1299 185,6644 29,6873 19,8397 16,1469 38,4945 110 197,751 33,1986 145,6099 118,9403 74,6731 126,1073 103,491 35,4985 137,8848 165,4097 85,6060 87,1219 45,8317 64,6711 16,4252 111 197,4083 144,5456 114,2027 64,2663 27,3101 191,5723 162,8771 152,1940 28,4495 106,6460 29,1005 130,9137 133,6767 18,785 160,9702 21,3060 112 23,3201 141,130 80,5208 181,2524 95,555 77,3500 183,9037 164,1492 155,7915 106,4854 50,6454 133,9083 5,1522 45,9496 173,7338 66,2271 59,4585 97,7969 60,2570 31,5231 149,8736 49,2935 158,383 128,7645 107,7448 103,7409 54,2279 196,5663 113 80,917 33,652 26,5553 72,8782 101,5024 108,4342 132,5383 116,8036 184,4999 88,7322 105,3388 187,6332 190,697 136,1984 96,712 91,4626 103,5248 114 29,9478 180,8490 189,3102 111,2027 192,6813 141,6388 72,8350 115,3112 152,9627 53,7352 129,168 107,2032 1,508 47,9480 35,490 17,8915 22,9960 27,714 31,7617 76,9690 117,7876 193,4000 103,1318 194,949 115 152,4383 176,8236 75,9140 25,9233 9,7827 98,5825 52,4973 146,9828 81,2608 128,9072 86,4291 76,7291 90,1117 106,9598 144,3825 80,1417 114,3112 62,1913 39,983 91,1105 88,6611 129,7465 166,3001 61,3324 117,1399 94,3350 116 78,2732 26,4221 113,8036 179,8099 32,6065 105,7899 62,7419 107,9864 82,8491 186,8639 176,4512 192,5906 57,656 4,6771 28,2638 10,1493 117 98,9545 198,3936 42,7074 79,9043 102,2689 56,1934 114,7876 83,8516 86,4220 196,3366 1,2597 49,2896 138,7762 17,6266 115,1399 118 14,373 23,5079 4,3626 6,8547 123,2088 181,8129 18,6388 85,1102 31,6949 166,3979 35,8640 43,8008 45,5918 177,8279 110,9403 128,6289 119 80,108 154,741 130,730 84,6417 72,6215 30,7372 170,275 168,1890 157,9158 90,928 121,6261 37,9313 95,8126 14,960 87,518 79,4117 39,1376 163,5189 169,3511 195,617 3,7711 32,1914 19,7406 46,2687 196,2824 105,9930 120 40,7760 62,3851 72,2287 94,2733 86,5230 95,1104 164,5926 26,5788 18,44 155,6822 60,2231 185,556 45,5279 179,3327 159,5811 75,6892 64,2385 5,1862 178,8906 28,8874 51,5675 105,9333 121 4,7639 80,2257 197,6502 119,6261 136,1320 156,195 29,4537 7,8739 58,8818 32,1615 168,7186 62,4145 92,3819 173,2976 64,771 175,8821 191,8606 162,1977 132,3867 74,1468 165,7147 148,8115 55,44 122 78,6207 8,5338 174,8205 168,1574 162,3518 166,6712 135,6345 28,9165 192,1494 128,7247 189,2017 3,9640 148,3230 27,84 179,7377 123 14,2733 198,3493 6,13 104,9213 107,4581 89,1723 118,2088 128,1602 155,3251 46,132 36,6226 184,2832 103,8756 65,9599 124,6490 145,8804 193,7460 124 141,168 159,6580 42,3496 123,6490 77,8896 20,392 67,1880 158,1870 147,9014 165,9797 136,7388 56,552 11,6180 70,9089 45,3290 48,2593 125 33,67 174,9048 95,2917 53,8054 1,2909 63,1315 76,1865 88,5012 58,1050 23,922 173,2615 188,230 172,8515 196,4519 138,9932 183,9920 7,5670 126 14,9602 4,2625 159,2980 86,937 181,8920 104,8615 179,7436 191,3989 161,4512 108,525 155,307 110,1073 134,5002 38,5441 76,9595 180,7176 127 4,6808 160,7534 26,9896 39,6103 50,3622 137,4069 60,1402 156,6372 2,9365 16,9807 56,3031 139,2526 14,5682 99,1408 167,3756 135,1752 161,6643 146,8151 128 14,9662 115,9072 123,1602 92,556 50,2987 190,2968 118,6289 157,6815 132,2789 184,6339 198,1860 112,7645 122,7247 85,4644 82,6834 100,2584 196,1760 12,7542 129 6,8525 114,168 101,8821 77,2337 79,7314 47,9533 85,517 175,7121 184,5623 109,2464 143,8021 167,5370 25,3005 159,6895 115,7465 130 119,730 8,55 26,5706 45,4994 36,7589 30,1356 184,8488 178,615 92,4320 58,7020 82,3349 174,4481 66,5826 39,2414 194,9429 156,2264 20,9048 64,2493 43,2685 137,5926 190,3429 147,9251 111,9137 48,7921 38,5227 131 14,358 98,4614 159,4355 75,5902 22,6075 43,8231 163,8625 11,833 57,6342 61,6924 82,3457 64,4631 134,6293 167,6269 2,7005 63,5257 9,700 103,2920 21,3748 132 198,2393 113,5383 99,6234 138,5667 28,5604 19,3157 38,6092 85,3873 82,4162 25,219 182,6433 22,1560 147,2847 6,1514 121,3867 128,2789 51,6107 37,821 133 80,7217 112,9083 136,3739 77,2481 39,8145 43,8488 18,2702 27,7749 168,899 99,2090 190,9958 139,4719 182,8241 191,4296 85,836 153,8437 67,3802 49,5167 111,6767 134 197,6225 198,4071 92,610 79,5792 175,4489 36,979 131,6293 71,7485 146,9556 158,119 11,294 65,9956 135,276 16,5203 84,9312 126,5002 38,4730 135 23,4560 4,7582 20,6434 174,6977 150,9732 190,1431 173,5664 144,6396 127,1752 122,6345 48,8869 82,519 158,8348 184,7629 78,6919 10,4650 134,276 194,5726 13,861 136 161,3866 195,4279 32,3823 84,4270 168,9519 54,1794 170,1529 197,9068 121,1320 194,2496 109,8658 199,7783 133,3739 145,1769 179,6711 26,7200 40,4658 174,5711 85,6221 113,1984 25,9251 184,6682 81,1409 88,9026 178,2752 124,7388 137 188,9117 100,7178 42,4572 51,1096 52,9846 97,4667 25,5378 102,6814 130,5926 84,4367 141,7872 23,7591 127,4069 157,5286 78,1910 91,4129 155,5736 67,1414 10,5273 110,8848 138 200,678 160,9902 26,7027 101,2902 51,8755 132,5667 43,7965 92,8080 1,7896 173,8555 33,6960 71,1229 46,2860 27,5918 188,892 169,7498 178,6589 125,9932 76,5683 117,7762 54,6261 140,5446 139 197,1386 98,1235 92,6763 181,5456 176,8186 182,2354 133,4719 158,3451 196,3988 73,4428 90,7846 155,2100 194,6966 69,9030 94,7116 127,2526 162,4510 9,8701 37,8208 140 1,546 17,6440 97,6527 158,4029 151,5289 68,9624 138,5446 43,4482 169,2230 9,4163 155,5001 188,6223 102,6370 166,9829 65,5376 141 154,3188 68,3540 106,674 79,7918 104,8162 24,6862 63,5607 6,8200 75,9237 150,2634 124,168 14,4477 112,130 164,6499 198,2621 114,6388 185,6820 59,9126 64,158 137,7872 81,4032 76,1969 48,7941 142 66,777 17,8016 104,6236 18,332 47,5297 81,424 91,9164 150,2598 5,1195 34,7791 155,6613 169,7506 86,1663 100,2994 190,8070 2,7731 71,6038 145,5378 37,6433 143 23,7543 6,3110 22,8603 108,972 25,108 81,9810 43,1171 186,8430 14,5848 129,8021 101,4042 76,1079 98,4300 155,2442 177,3884 36,5111 153,9940 96,9274 15,1041 176,5062 144 115,3825 173,8693 29,6452 98,7198 68,2277 195,652 102,7884 30,1616 111,5456 33,4841 13,9987 44,2276 86,8808 148,3617 35,3087 40,7772 101,3619 22,9362 184,5222 135,6396 90,6223 3,285 100,6605 145 200,2753 1,648 136,1769 26,6697 64,2685 174,7252 47,9483 108,521 44,2888 123,8804 71,6127 142,5378 20,8099 78,7244 110,6099 100,5512 55,3342 146 115,9828 22,3620 47,2959 60,2317 5,5308 127,8151 7,4823 134,9556 18,9000 31,2837 17,5362 68,3895 72,4264 188,5715 167,52 107,7862 38,4400 147 26,5759 59,2651 163,7221 95,1278 132,2847 124,9014 5,1087 34,9758 96,6736 38,8391 41,1858 167,5546 130,9251 149,6379 58,2267 93,7807 148 160,3995 144,3617 185,2814 64,6419 18,589 30,4760 173,6725 179,6374 70,7231 155,6603 122,3230 192,4834 36,9550 121,8115 29,3252 149 14,4352 59,2405 188,1003 163,9897 70,2366 83,6823 187,9580 174,2824 73,5233 112,8736 55,3088 18,2664 176,2255 190,3755 180,3407 147,6379 150 141,2634 1,2183 16,1680 41,7278 36,116 135,9732 11,4802 98,2093 58,4857 142,2598 73,8061 9,4539 18,5150 24,5669 52,634 190,5243 88,6770 53,662 7,9031 93,9213 151 80,6728 156,6169 24,784 27,5915 174,1846 168,9216 99,4990 88,2295 178,5979 96,5643 83,162 13,9629 140,5289 189,9732 163,3624 152 161,6939 58,5877 48,5835 9,1087 8,1257 46,9550 189,8140 199,2697 109,2743 65,3595 186,2075 115,4383 50,5767 193,5444 90,4427 80,9606 114,9627 62,7404 79,6519 27,2124 111,1940 19,9751 93,6543 153 17,8545 99,4560 70,6532 164,8748 29,5446 25,2477 72,541 143,9940 173,7613 77,4483 50,4905 165,897 133,8437 33,6574 67,6204 100,9522 154 141,3188 119,741 198,5078 156,8062 62,3238 20,2398 18,453 49,2962 187,1808 168,6317 200,229 185,1448 93,4073 78,8423 84,5421 41,901 60,6667 170,5971 19,4320 21,1845 199,5786 155 57,893 14,4446 98,5515 112,7915 123,3251 120,6822 25,3242 139,2100 143,2442 65,5533 142,6613 96,1251 137,5736 148,6603 61,5301 126,307 37,609 140,5001 194,6768 156 66,2965 42,6539 82,4691 53,4662 151,6169 7,7027 85,6669 29,5083 154,8062 130,2264 108,7039 34,8641 121,195 175,2205 69,9959 70,4830 127,6372 37,7714 157 119,9158 26,7 86,4447 101,1393 187,7184 137,5286 99,6818 11,8497 2,8643 158,1881 128,6815 175,1597 72,604 24,9540 28,6880 15,2136 158 78,6367 187,6714 91,5553 139,3451 169,1709 135,8348 63,5709 195,4912 140,4029 157,1881 124,1870 181,969 112,383 87,4081 134,119 159 4,6133 160,4243 61,49 180,5757 6,9500 194,9391 43,9621 100,6567 126,2980 73,4969 131,4355 124,6580 1,7420 52,7816 74,8960 198,8330 42,5951 120,5811 87,19 36,6043 84,8763 129,6895 2,1397 51,3017 193,3786 160 14,7860 138,9902 127,7534 49,8648 85,3613 189,1530 57,2058 159,4243 183,9293 148,3995 87,8712 180,624 179,2542 91,383 42,2382 1,6461 77,105 28,9059 111,9702 19,3997 161 152,6939 14,4275 23,6863 78,6310 136,3866 42,4792 62,4144 79,8395 127,6643 69,6171 126,4512 56,9811 196,9729 102,8230 88,1726 28,9217 162 4,3924 98,6139 6,6897 59,391 194,1483 122,3518 57,6483 121,1977 179,9718 139,4510 199,7749 40,3056 111,8771 96,3737 45,3436 186,6114 48,201 163 1,8164 101,6033 41,5256 173,5799 15,818 13,3870 149,9897 131,8625 181,5593 119,5189 61,448 42,1278 68,4920 100,9709 147,7221 44,18 49,4688 192,2762 151,3624 89,4055 164 141,6499 112,1492 42,347 120,5926 43,6849 87,3629 184,9774 170,328 153,8748 10,3866 172,9550 74,8627 195,9730 105,4661 24,7312 183,615 176,8681 165 23,208 6,6473 79,3768 187,1291 50,2667 153,897 166,4221 74,2602 21,2151 121,7147 124,9797 83,1555 65,3705 7,6019 41,3741 110,4097 166 23,8077 98,2364 79,8726 173,6917 44,6953 167,7080 118,3979 165,4221 168,4399 17,4762 9,8702 122,6712 20,6086 169,5530 115,3001 35,2816 55,3686 140,9829 167 23,2133 188,6549 187,9538 43,8120 18,7625 127,3756 180,6183 87,1022 79,820 83,8200 3,3877 129,5370 85,8110 193,8116 166,7080 131,6269 146,52 147,5546 45,1163 199,2001 93,5670 168 119,1890 8,2076 136,9519 87,9716 154,6317 121,7186 133,899 122,1574 85,6193 173,3859 59,4170 27,1578 69,2357 151,9216 22,4356 70,1295 34,3923 166,4399 10,9495 52,1134 169 14,4333 175,9164 177,2274 45,662 20,5990 63,5160 119,3511 104,3712 187,9225 192,8603 109,382 158,1709 55,7795 1,8700 138,7498 142,7506 166,5530 107,668 15,7485 54,2296 140,2230 170 119,275 1,2620 198,8089 136,1529 33,3648 18,4989 177,2774 44,6940 84,2436 2,9342 154,5971 164,328 45,8060 71,1539 94,9318 171 188,6981 101,4083 16,3482 67,1394 61,6888 49,9736 182,4704 94,9959 10,2029 33,3913 185,5113 71,6702 6,3722 18,4403 72,8573 172 200,6888 42,5383 72,3153 104,561 174,3433 102,6572 175,5353 35,992 73,2239 164,9550 29,5830 80,7715 77,4771 2,1438 60,2781 125,8515 13,2070 173 112,7338 1,2069 144,8693 33,5949 17,7312 20,2676 18,7256 121,2976 168,3859 32,9631 148,6725 82,591 163,5799 192,2550 166,6917 179,4234 138,8555 44,5518 95,4092 153,7613 135,5664 61,4803 125,2615 193,4808 46,2969 174 130,4481 89,858 73,2581 135,6977 68,7082 136,5711 91,3418 125,9048 122,8205 145,7252 72,2019 151,1846 172,3433 108,5669 99,5675 179,1886 149,2824 90,6914 175 156,2205 187,2864 27,210 199,3779 67,5048 121,8821 169,9164 134,4489 65,7920 26,6165 172,5353 197,5909 8,1022 129,7121 14,6718 184,9107 70,3256 58,3675 157,1597 176 200,1709 115,8236 6,3512 33,3258 187,1128 191,3352 139,8186 149,2255 116,4512 46,9923 143,5062 51,1625 164,8681 107,6806 74,8129 177 86,7010 92,3182 24,393 64,7660 102,4462 43,9544 22,8713 190,1332 14,6741 8,8508 170,2774 82,7882 169,2274 32,6902 53,9852 60,5345 143,3884 178,7547 118,8279 3,7097 178 130,615 72,5128 70,3394 13,4194 120,8906 11,1696 151,5979 16,1714 138,6589 102,1334 17,7464 49,9747 177,7547 35,5774 136,2752 28,3642 76,8433 179 80,7729 160,2542 136,6711 120,3327 56,5284 50,2502 186,9993 180,664 104,8827 90,3149 74,1223 148,6374 174,1886 126,7436 173,4234 162,9718 116,8099 34,8528 52,3504 122,7377 180 160,624 159,5757 114,8490 59,2435 86,4604 39,9506 85,5525 179,664 191,6312 31,3598 149,3407 167,6183 89,6612 9,5995 126,7176 105,341 181 112,2524 187,7956 41,5235 118,8129 185,9712 139,5456 76,3275 40,2338 11,8448 34,5267 102,3585 126,8920 106,5332 93,4071 91,5060 191,4844 163,5593 158,969 51,158 182 27,6234 47,8382 99,2897 139,2354 5,8806 133,8241 132,6433 56,7267 74,1873 35,7369 64,4701 107,8313 54,438 171,4704 2,7908 21,2447 183 160,9293 112,9037 102,9634 99,3577 81,3061 88,5547 11,594 96,5939 53,5304 105,7722 26,4842 57,9093 70,5332 125,9920 164,615 38,5213 184 6,8021 130,8488 188,198 40,4306 123,2832 101,9591 113,4999 129,5623 144,5222 175,9107 136,6682 164,9774 86,801 57,7592 29,9797 81,4485 35,7437 135,7629 28,4164 10,3207 128,6339 185 57,9772 105,1033 90,7284 52,8925 83,9306 97,7065 148,2814 25,3573 197,3140 50,3968 11,2045 141,6820 76,3674 108,8699 60,6670 188,700 120,556 181,9712 154,1448 30,3846 16,6280 109,6644 171,5113 19,5896 196,1339 186 57,8815 152,2075 197,1324 42,9391 188,5027 92,7980 179,9993 191,8255 58,8993 143,8430 116,8639 162,6114 71,7143 189,8128 195,2099 68,2528 103,9654 187 14,4874 200,6767 176,1128 37,4105 7,9522 175,2864 19,107 107,2610 167,9538 157,7184 24,9327 181,7956 42,3075 158,6714 165,1291 50,2485 154,1808 113,6332 169,9225 149,9580 10,4786 188 23,4236 198,9114 137,9117 184,198 101,8531 167,6549 54,4883 149,1003 28,3816 196,88 52,7805 55,5181 185,700 186,5027 171,6981 97,4695 138,892 125,230 146,5715 140,6223 189 152,8140 160,1530 8,2780 114,3102 86,1115 40,4263 77,5702 50,7031 47,5939 85,1147 60,4415 69,5215 186,8128 198,9430 122,2017 51,8899 151,9732 193,2375 190 4,7536 200,5999 47,5892 113,697 130,3429 135,1431 150,5243 61,9775 69,8022 133,9958 128,2968 100,6674 17,7000 106,3305 9,3613 177,1332 31,5055 149,3755 142,8070 15,6854 105,2518 191 77,5135 126,3989 176,3352 94,8318 186,8255 121,8606 87,6921 44,8130 103,9683 75,1403 181,4844 50,5026 180,6312 111,5723 37,3573 133,4296 192 114,6813 97,6411 77,561 39,7454 22,4606 104,4219 27,5344 85,544 173,2550 91,6625 169,8603 116,5906 49,3113 163,2762 122,1494 148,4834 193 152,5444 167,8116 114,4000 55,8583 58,6166 189,2375 159,3786 96,8831 79,7737 21,494 28,9441 62,6610 123,7460 99,4566 173,4808 46,199 194 200,2915 159,9391 136,2496 130,9429 42,6498 92,426 139,6966 162,1483 73,8521 99,5832 155,6768 114,949 46,2429 79,5784 135,5726 195 23,3179 4,2490 197,5337 144,652 136,4279 6,2233 158,4912 186,2099 200,8114 61,8928 46,8206 164,9730 119,617 19,5038 67,5737 196 188,88 39,9267 139,3988 95,231 61,5577 161,9729 125,4519 117,3366 103,696 46,9632 26,6754 112,5663 128,1760 119,2824 185,1339 197 54,2223 186,1324 121,6502 134,6225 68,8223 24,9337 25,3875 139,1386 81,5540 44,4900 195,5337 37,9188 111,4083 86,9714 110,751 136,9068 185,3140 104,4174 175,5909 198 141,2621 170,8089 44,2619 188,9114 41,7215 1,1724 154,5078 84,5999 16,2223 117,3936 123,3493 159,8330 69,6284 132,2393 134,4071 77,5081 50,3798 65,3575 36,6280 28,366 128,1860 189,9430 199 152,2697 8,1918 136,7783 59,8958 24,2838 175,3779 31,5564 162,7749 65,11 76,6878 9,70 89,2446 34,8196 167,2001 154,5786 200 108,9976 103,6851 145,2753 41,2622 187,6767 190,5999 16,2848 194,2915 5,4009 172,6888 39,4319 176,1709 60,3269 138,678 43,8943 98,2690 1,8021 104,7083 154,229 91,1988 67,475 76,4623 195,8114 37,7541 54,4899 ================================================ FILE: closest_pair/naive.py ================================================ P = [(1, 8), (2, 5), (4, 7), (6, 3)] N = len(P) INF = 1234567890 # arbitary choice for infinity def dist(i, j): x1, y1 = P[i] x2, y2 = P[j] return (x1 - x2) ** 2 \ + (y1 - y2) ** 2 best_dist = INF best_pair = None D = [[INF] * N for _ in range(N)] for i in range(N): for j in range(N): D[i][j] = dist(i, j) if i != j: # candidates for best pair cannot be the same point if best_dist >= D[i][j]: best_dist = D[i][j] best_pair = (i, j) i, j = best_pair print(f'best pair: {P[i]}, {P[j]}') # ➜ closest_pair git:(main) ✗ python3 ./naive.py # best pair: (4, 7), (2, 5) ================================================ FILE: closest_pair/recursive.py ================================================ INF = 1234567890 # arbitary choice for infinity def distance(a, b): x1, y1 = a x2, y2 = b return (x1 - x2) ** 2 \ + (y1 - y2) ** 2 def split(Px, Py, d): # median x-coordinate median = Px[len(Px) // 2][0] # identify points near left/right boundary Sy = [(x, y) for x, y in Py if median - d <= x <= median + d] # return the best split pair (if it exists) best_dist, best_pairs = INF, set() for i in range(len(Sy) - 1): for j in range(i + 1, min(i + 1 + 7, len(Sy))): cand = distance(Sy[i], Sy[j]) if best_dist > cand: best_dist = cand best_pairs = set() if best_dist == cand: best_pairs.add((Sy[i], Sy[j])) return best_dist, best_pairs def best(P): best_dist, best_pairs = INF, set() for i in range(len(P)): for j in range(i + 1, len(P)): cand = distance(P[i], P[j]) if P[i] != P[j] else INF if best_dist > cand: best_dist = cand best_pairs = set() if best_dist == cand: best_pairs.add((P[i], P[j])) return best_dist, best_pairs def go(Px, Py): if len(Px) <= 3: # Base case return best(Px) Lx, Rx = Px[:len(Px) // 2], Px[len(Px) // 2:] Ly, Ry = [], [] for x, y in Py: if x <= Lx[-1][0]: Ly.append((x, y)) else: Ry.append((x, y)) dist_left, best_left = go(Lx, Ly) dist_right, best_right = go(Rx, Ry) dist_split, best_split = split(Px, Py, min(dist_left, dist_right)) cands = sorted([(dist_left, best_left), (dist_right, best_right), (dist_split, best_split)], key=lambda it: it[0]) best_dist, best_pairs = cands[0][0], set() for dist, pairs in cands: if dist == best_dist: for a, b in pairs: best_pairs.add((a, b) if a < b else (b, a)) return best_dist, best_pairs def run(points, expected_best_pair): Px = sorted(points, key=lambda it: it[0]) Py = sorted(points, key=lambda it: it[1]) _, best_pair = go(Px, Py) print(f'points: {points}') print(f'actual: {sorted(best_pair)}') print(f'expect: {sorted(expected_best_pair)}') print() assert(sorted(best_pair) == sorted(expected_best_pair)) # initial example run(points=[(1, 8), (2, 5), (4, 7), (6, 3)], expected_best_pair=set([((2, 5), (4, 7))])) # simple, obvious nearest neighbors run(points=[(0, 0), (3, 4), (7, 1), (10, 6)], expected_best_pair=set([((0, 0), (3, 4)), ((3, 4), (7, 1))])) # closest pair not adjacent in sorted-by-x run(points=[(1, 1), (4, 10), (7, 4), (9, 8)], expected_best_pair=set([((7, 4), (9, 8))])) # tight cluster inside larger spread run(points=[(2, 9), (5, 3), (6, 5), (11, 1)], expected_best_pair=set([((5, 3), (6, 5))])) # stress: diagonal-ish run(points=[(1, 10), (4, 7), (8, 3), (13, 6)], expected_best_pair=set([((1, 10), (4, 7))])) # larger input: 10 points run(points=[(1, 17), (3, 4), (6, 14), (8, 9), (11, 2), (14, 7), (17, 12), (19, 5), (22, 15), (25, 1)], expected_best_pair=set([((6, 14), (8, 9)), ((14, 7), (19, 5))])) # larger input: 12 points more spread out run(points=[(2, 21), (4, 6), (7, 15), (9, 2), (12, 18), (14, 9), (16, 4), (19, 13), (21, 7), (24, 16), (27, 1), (30, 11)], expected_best_pair=set([((14, 9), (16, 4))])) # same x-coordinate (vertical stack) run(points=[(5, 1), (5, 4), (5, 9), (8, 20), (12, 30)], expected_best_pair=set([((5, 1), (5, 4))])) # same y-coordinate (horizontal line) run(points=[(1, 7), (4, 7), (9, 7), (20, 2), (25, 11)], expected_best_pair=set([((1, 7), (4, 7))])) # redundant x-coordinate to test split pairs run(points=[(10, 1), (10, 4), (10, 8), (10, 13), (11, 7), (20, 50)], expected_best_pair=set([((10, 8), (11, 7))])) # redundant x-coordinate to test split pairs run(points=[(0, 0), (0, 1000), (0, 2000), (0, 3000), (0, 4000), (0, 5000), (0, 6000), (0, 7000), (5, 1), (5, 7002)], expected_best_pair=set([((0, 0), (5, 1))])) # tie for best run(points=[(0, 0), (0, 1), (1, 0)], expected_best_pair=set([((0, 0), (0, 1)), ((0, 0), (1, 0))])) # ➜ closest_pair git:(main) ✗ python3 recursive.py # points: [(1, 8), (2, 5), (4, 7), (6, 3)] # actual: [((2, 5), (4, 7))] # expect: [((2, 5), (4, 7))] # points: [(0, 0), (3, 4), (7, 1), (10, 6)] # actual: [((0, 0), (3, 4)), ((3, 4), (7, 1))] # expect: [((0, 0), (3, 4)), ((3, 4), (7, 1))] # points: [(1, 1), (4, 10), (7, 4), (9, 8)] # actual: [((7, 4), (9, 8))] # expect: [((7, 4), (9, 8))] # points: [(2, 9), (5, 3), (6, 5), (11, 1)] # actual: [((5, 3), (6, 5))] # expect: [((5, 3), (6, 5))] # points: [(1, 10), (4, 7), (8, 3), (13, 6)] # actual: [((1, 10), (4, 7))] # expect: [((1, 10), (4, 7))] # points: [(1, 17), (3, 4), (6, 14), (8, 9), (11, 2), (14, 7), (17, 12), (19, 5), (22, 15), (25, 1)] # actual: [((6, 14), (8, 9)), ((14, 7), (19, 5))] # expect: [((6, 14), (8, 9)), ((14, 7), (19, 5))] # points: [(2, 21), (4, 6), (7, 15), (9, 2), (12, 18), (14, 9), (16, 4), (19, 13), (21, 7), (24, 16), (27, 1), (30, 11)] # actual: [((14, 9), (16, 4))] # expect: [((14, 9), (16, 4))] # points: [(5, 1), (5, 4), (5, 9), (8, 20), (12, 30)] # actual: [((5, 1), (5, 4))] # expect: [((5, 1), (5, 4))] # points: [(1, 7), (4, 7), (9, 7), (20, 2), (25, 11)] # actual: [((1, 7), (4, 7))] # expect: [((1, 7), (4, 7))] # points: [(10, 1), (10, 4), (10, 8), (10, 13), (11, 7), (20, 50)] # actual: [((10, 8), (11, 7))] # expect: [((10, 8), (11, 7))] # points: [(0, 0), (0, 1000), (0, 2000), (0, 3000), (0, 4000), (0, 5000), (0, 6000), (0, 7000), (5, 1), (5, 7002)] # actual: [((0, 0), (5, 1))] # expect: [((0, 0), (5, 1))] # points: [(0, 0), (0, 1), (1, 0)] # actual: [((0, 0), (0, 1)), ((0, 0), (1, 0))] # expect: [((0, 0), (0, 1)), ((0, 0), (1, 0))] ================================================ FILE: dijkstra/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(dijkstra) set(CMAKE_CXX_STANDARD 17) add_executable(dijkstra main.cpp) ================================================ FILE: dijkstra/main.cpp ================================================ #include #include #include #include #include #include #include using namespace std; using Queries = vector; using Distance = unordered_map; using Set = unordered_set; class BaseSolution { protected: static constexpr auto INF = int(1e9 + 7); public: virtual string run(string filename, Queries&& queries) = 0; }; class NaiveSolution : public BaseSolution { using Edge = tuple; using Edges = vector; public: Distance dijkstra(Edges& E) { Distance dist; Set seen; auto start{ 1 }; dist[start] = 0, seen.insert(start); for (;;) { auto found = false; auto best_v = INF, best_w = INF; for (auto [u, v, w]: E) { if (seen.find(u) == seen.end() || seen.find(v) != seen.end()) continue; found = true; if (best_w > dist[u] + w) best_v = v, best_w = dist[u] + w; } if (!found) break; auto [v, w] = tie(best_v, best_w); dist[v] = w, seen.insert(v); } return dist; } string run(string filename, Queries&& queries) { Edges E; fstream fin{ filename }; string line; int u, v, w; char _; while (getline(fin, line)) { istringstream is{ line }; for (is >> u; is >> v >> _ >> w; E.push_back({ u, v, w })); } auto dist = dijkstra(E); ostringstream os; transform(queries.begin(), queries.end(), ostream_iterator(os, " "), [&](auto x) { return dist[x]; }); return os.str(); } }; class HeapSolution : public BaseSolution { using Pair = pair; using Pairs = vector; using AdjList = unordered_map; priority_queue> q; public: Distance dijkstra(AdjList& adj) { Distance dist; Set seen; for (auto [u, _]: adj) dist[u] = INF; auto start{ 1 }; q.push({ 0, start }); while (q.size()) { auto [cost, u] = q.top(); q.pop(); if (!seen.insert(u).second) continue; dist[u] = cost; for (auto [w, v]: adj[u]) if (seen.find(v) == seen.end()) q.push({ dist[u] + w, v }); } return dist; } string run(string filename, Queries&& queries) { AdjList adj; fstream fin{ filename }; string line; int u, v, w; char _; while (getline(fin, line)) { istringstream is{ line }; for (is >> u; is >> v >> _ >> w; adj[u].push_back({ w, v })); } auto dist = dijkstra(adj); ostringstream os; transform(queries.begin(), queries.end(), ostream_iterator(os, " "), [&](auto x) { return dist[x]; }); return os.str(); } }; void run(BaseSolution&& solution) { cout << "problem9.8test.txt: " << solution.run("problem9.8test.txt", Queries{1, 2, 3, 4, 5, 6, 7, 8 }) << endl << "problem9.8.txt " << solution.run("problem9.8.txt", Queries{7, 37, 59, 82, 99, 115, 133, 165, 188, 197 }) << endl; } int main() { run(NaiveSolution()); // problem9.8test.txt: 0 1 2 3 4 4 3 2 // problem9.8.txt 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(HeapSolution()); // problem9.8test.txt: 0 1 2 3 4 4 3 2 // problem9.8.txt 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 return 0; } ================================================ FILE: dijkstra/main.js ================================================ let LineByLine = require('n-readlines'); let INF = Number(1e9 + 7); class NaiveSolution { dijkstra(E) { let dist = new Map(); let seen = new Set(); let start = 1; dist[start] = 0; seen.add(start); for (;;) { let found = false; let best_v = INF, best_w = INF; for (let [u, v, w] of E) { if (!seen.has(u) || seen.has(v)) continue; found = true; if (best_w > dist[u] + w) best_v = v, best_w = dist[u] + w; } if (!found) break; let [v, w] = [best_v, best_w]; dist[v] = w; seen.add(v); } return dist; } run(filename, queries) { let E = []; let input = new LineByLine(filename); let line; while (line = input.next()) { let words = String.fromCharCode(...line).trim().split(/\s+/); let u = Number(words[0]); for (let i = 1; i < words.length; ++i) { let [v, w] = words[i].split(',').map(Number); E.push([ u, v, w ]); } } let dist = this.dijkstra(E); return queries.map(x => dist[x]).join(' '); } } let heapkey = x => Array.isArray(x) ? x[0] : x; let heappush = (A, x, f = Math.min) => { let P = i => Math.floor((i - 1) / 2); // parent A.push(x); let N = A.length, i = N - 1; while (0 < i && heapkey(A[i]) == f(heapkey(A[i]), heapkey(A[P(i)]))) { [A[i], A[P(i)]] = [A[P(i)], A[i]]; i = P(i); } }; let heappop = (A, f = Math.min) => { let L = i => 2 * i + 1, // children R = i => 2 * i + 2; let N = A.length, i = 0; let top = A[0]; [A[0], A[N - 1]] = [A[N - 1], A[0]], A.pop(), --N; let ok; do { ok = true; let left = f == Math.min ? Infinity : -Infinity, right = left; if (L(i) < N && heapkey(A[i]) != f(heapkey(A[i]), heapkey(A[L(i)]))) ok = false, left = heapkey(A[L(i)]); if (R(i) < N && heapkey(A[i]) != f(heapkey(A[i]), heapkey(A[R(i)]))) ok = false, right = heapkey(A[R(i)]); if (!ok) { let j = left == f(left, right) ? L(i) : R(i); [A[i], A[j]] = [A[j], A[i]]; i = j; } } while (!ok); return top; }; class HeapSolution { dijkstra(adj) { let dist = {}; let seen = new Set(); let start = 1; let q = [[ 0, start ]]; while (q.length) { let [cost, u] = heappop(q); if (seen.has(u)) continue; dist[u] = cost, seen.add(u); for (let [w, v] of (adj[u] || [])) heappush(q, [ dist[u] + w, v ]); } return dist; } run(filename, queries) { let adj = {}; let input = new LineByLine(filename); let line; while (line = input.next()) { let words = String.fromCharCode(...line).trim().split('\t'); let u = Number(words[0]); if (!(u in adj)) adj[u] = []; for (let i = 1; i < words.length; ++i) { let [v, w] = words[i].split(',').map(Number); adj[u].push([ w, v ]); } } let dist = this.dijkstra(adj); return queries.map(x => dist[x]).join(' '); } } let run = solution => { console.log(solution.run('problem9.8test.txt', [1, 2, 3, 4, 5, 6, 7, 8])); console.log(solution.run('problem9.8.txt', [7, 37, 59, 82, 99, 115, 133, 165, 188, 197])); }; run(new NaiveSolution()); // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(new HeapSolution()); // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 ================================================ FILE: dijkstra/main.kt ================================================ import java.io.File import java.util.PriorityQueue var INF = (1e9 + 7).toInt() interface BaseSolution { fun run(filename: String, queries: Array): String } class NaiveSolution : BaseSolution { fun dijkstra(E: List>): MutableMap { var dist = mutableMapOf() var seen = mutableSetOf() var start = 1 dist[start] = 0; seen.add(start) var found: Boolean; do { found = false var best_v = INF var best_w = INF for ((u, v, w) in E) { if (!seen.contains(u) || seen.contains(v)) continue found = true if (best_w > dist[u]!! + w) { best_v = v best_w = dist[u]!! + w } } var v = best_v var w = best_w dist[v] = w; seen.add(v) } while (found) return dist } override fun run(filename: String, queries: Array): String { var E = mutableListOf>() File(filename).forEachLine { var words = it.trim().split("\t") var u = words[0].toInt() for (i in 1 until words.size) { var (v, w) = words[i].split(",").map{ it.toInt() } E.add(Triple(u, v, w)) } } var dist = dijkstra(E.toList()) return queries.map{ dist[it] }.joinToString(" ") } } class HeapSolution : BaseSolution { fun dijkstra(adj: MutableMap>>): MutableMap { var dist = mutableMapOf() var seen = mutableSetOf() var start = 1 dist[start] = 0 var q = PriorityQueue>(Comparator{ a: Pair, b: Pair -> a.first.compareTo(b.first) }) q.add(Pair(0, start)) while (0 < q.size) { var (cost, u) = q.poll() if (seen.contains(u)) continue dist[u] = cost; seen.add(u) for ((w, v) in adj[u]!!) { if (seen.contains(v)) continue q.add(Pair(cost + w, v)) } } return dist } override fun run(filename: String, queries: Array): String { var adj = mutableMapOf>>() File(filename).forEachLine { var words = it.trim().split("\t") var u = words[0].toInt() if (!adj.contains(u)) adj[u] = mutableListOf() for (i in 1 until words.size) { var (v, w) = words[i].split(",").map{ it.toInt() } adj[u]!!.add(Pair(w, v)) } } var dist = dijkstra(adj) return queries.map{ dist[it] }.joinToString(" ") } } fun run(solution: BaseSolution) { println(solution.run("problem9.8test.txt", arrayOf(1, 2, 3, 4, 5, 6, 7, 8))) println(solution.run("problem9.8.txt", arrayOf(7, 37, 59, 82, 99, 115, 133, 165, 188, 197))) } fun main() { run(NaiveSolution()) // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(HeapSolution()) // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 } ================================================ FILE: dijkstra/main.py ================================================ from abc import ABC, abstractmethod from heapq import heappush, heappop INF = int(1e9 + 7) class BaseSolution(ABC): @abstractmethod def run(self, filename, queries): raise NotImplementedError class NaiveSolution(BaseSolution): def dijkstra(self, E): dist = {} seen = set() start = 1 dist[start] = 0; seen.add(start) while True: found = False best_v = INF best_w = INF for u, v, w in E: if u not in seen or v in seen: continue found = True if best_w > dist[u] + w: best_v = v best_w = dist[u] + w if not found: break v, w = best_v, best_w dist[v] = w; seen.add(v) return dist def run(self, filename, queries): E = [] with open(filename) as fin: while True: line = fin.readline() if not line: break words = line.split() u = int(words[0]) for i in range(1, len(words)): v, w = map(int, words[i].split(',')) E.append([ u, v, w ]) dist = self.dijkstra(E) return ' '.join(str(dist[x]) for x in queries) class HeapSolution(BaseSolution): def dijkstra(self, adj, start = 1): dist = {} seen = set() q = [[ 0, start ]] while len(q): cost, u = heappop(q) if u in seen: continue dist[u] = cost; seen.add(u) for w, v in adj[u]: if v not in seen: heappush(q, [ dist[u] + w, v ]) return dist def run(self, filename, queries): adj = {} with open(filename) as fin: while True: line = fin.readline() if not line: break words = line.split() u = int(words[0]) if u not in adj: adj[u] = [] for i in range(1, len(words)): v, w = map(int, words[i].split(',')) adj[u].append([ w, v ]) dist = self.dijkstra(adj) return ' '.join(str(dist[x]) for x in queries) def run(solution): print(solution.run('problem9.8test.txt', [1, 2, 3, 4, 5, 6, 7, 8])) print(solution.run('problem9.8.txt', [7, 37, 59, 82, 99, 115, 133, 165, 188, 197])) run(NaiveSolution()) # 0 1 2 3 4 4 3 2 # 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(HeapSolution()) # 0 1 2 3 4 4 3 2 # 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 ================================================ FILE: dijkstra/package.json ================================================ { "name": "dijkstra", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "", "license": "ISC", "dependencies": { "n-readlines": "^1.0.1" } } ================================================ FILE: dijkstra/problem9.8.txt ================================================ 1 80,982 163,8164 170,2620 145,648 200,8021 173,2069 92,647 26,4122 140,546 11,1913 160,6461 27,7905 40,9047 150,2183 61,9146 159,7420 198,1724 114,508 104,6647 30,4612 99,2367 138,7896 169,8700 49,2437 125,2909 117,2597 55,6399 2 42,1689 127,9365 5,8026 170,9342 131,7005 172,1438 34,315 30,2455 26,2328 6,8847 11,1873 17,5409 157,8643 159,1397 142,7731 182,7908 93,8177 3 57,1239 101,3381 43,7313 41,7212 91,2483 31,3031 167,3877 106,6521 76,7729 122,9640 144,285 44,2165 6,9006 177,7097 119,7711 4 162,3924 70,5285 195,2490 72,6508 126,2625 121,7639 31,399 118,3626 90,9446 127,6808 135,7582 159,6133 106,4769 52,9267 190,7536 78,8058 75,7044 116,6771 49,619 107,4383 89,6363 54,313 5 200,4009 112,1522 25,3496 23,9432 64,7836 56,8262 120,1862 2,8026 90,8919 142,1195 81,2469 182,8806 17,2514 83,8407 146,5308 147,1087 51,22 6 141,8200 98,5594 66,6627 159,9500 143,3110 129,8525 118,8547 88,2039 83,4949 165,6473 162,6897 184,8021 123,13 176,3512 195,2233 42,7265 47,274 132,1514 2,8847 171,3722 3,9006 7 156,7027 187,9522 87,4976 121,8739 56,6616 10,2904 71,8206 53,179 146,4823 165,6019 125,5670 27,4888 63,9920 150,9031 84,4061 8 152,1257 189,2780 58,4708 26,8342 199,1918 31,3987 35,3160 71,5829 27,3483 69,8815 130,55 168,2076 122,5338 73,4528 28,9996 17,3535 40,3193 72,7308 24,8434 87,2833 25,3949 175,1022 177,8508 9 152,1087 115,7827 17,7002 72,794 150,4539 190,3613 95,9480 36,5284 166,8702 63,1753 199,70 131,700 76,9340 70,2 139,8701 140,4163 180,5995 10 57,9988 78,3771 62,4816 137,5273 7,2904 187,4786 184,3207 96,807 31,1184 88,2539 135,4650 168,9495 164,3866 11,8988 116,1493 51,5578 171,2029 11 1,1913 185,2045 77,815 22,8425 181,8448 47,8727 81,7299 150,4802 178,1696 28,2275 183,594 131,833 157,8497 25,5057 59,3203 10,8988 2,1873 134,294 83,4211 124,6180 12 78,5753 17,4602 62,5676 16,8068 60,5933 67,371 71,6734 53,7001 72,3626 34,6690 59,761 18,1520 128,7542 38,6699 57,9416 13 144,9987 59,9801 97,7026 50,758 43,5400 163,3870 178,4194 151,9629 45,1794 105,6821 29,2784 172,2070 57,6850 77,8638 135,861 14 149,4352 187,4874 26,3841 128,9662 155,4446 118,373 123,2733 106,7912 169,4333 53,9197 161,4275 126,9602 73,4106 160,7860 131,358 141,4477 119,960 43,3199 47,7898 175,6718 177,6741 60,2464 127,5682 31,1945 143,5848 94,3551 82,3283 15 42,1789 22,3571 25,7019 163,818 56,2334 100,809 143,1041 107,4589 190,6854 169,7485 94,9606 34,7961 54,8983 157,2136 24,8040 16 200,2848 198,2223 92,2896 18,8663 27,8673 75,4116 150,1680 36,1555 41,2747 90,4558 68,5894 12,8068 42,2596 185,6280 171,3482 109,1469 127,9807 178,1714 35,839 56,9828 134,5203 55,6680 110,4252 17 26,1275 45,5114 142,8016 83,4615 140,6440 8,3535 69,3610 153,8545 9,7002 12,4602 173,7312 114,8915 108,1942 54,3115 66,6176 190,7000 70,3899 5,2514 178,7464 166,4762 2,5409 146,5362 117,6266 18 57,4216 80,5252 86,7517 62,1926 120,44 173,7256 133,2702 148,589 167,7625 16,8663 170,4989 118,6388 142,332 95,6122 99,5717 154,453 150,5150 149,2664 146,9000 171,4403 111,785 12,1520 19 33,6938 77,7013 187,107 109,8397 88,2002 95,8691 132,3157 195,5038 154,4320 23,8560 152,9751 185,5896 119,7406 160,3997 80,62 20 66,2667 173,2676 43,8105 135,6434 33,6387 74,6183 106,8785 75,2484 130,9048 56,7194 50,9507 88,3014 124,392 61,2580 90,7372 92,1704 87,2639 154,2398 41,4203 85,1435 169,5990 166,6086 28,2234 145,8099 21 23,5183 40,2199 31,2556 71,4986 165,2151 193,494 154,1845 111,3060 85,2880 101,2775 182,2447 80,9884 87,2681 102,6643 131,3748 22 92,5592 64,4257 11,8425 24,594 15,3571 42,3783 41,1374 114,9960 144,9362 146,3620 71,3243 143,8603 131,6075 192,4606 108,9656 168,4356 177,8713 132,1560 23 143,7543 161,6863 45,8074 165,208 21,5183 118,5079 40,8336 27,9054 112,3201 135,4560 167,2133 188,4236 166,8077 195,3179 48,4485 137,7591 99,6485 5,9432 71,3316 96,2431 125,922 19,8560 24 141,6862 197,9337 66,5879 59,6941 70,4670 55,4106 103,8083 61,7906 48,7959 151,784 177,393 102,8731 199,2838 73,3509 8,8434 187,9327 22,594 150,5669 164,7312 157,9540 15,8040 25 115,9233 197,3875 185,3573 72,2332 104,4899 137,5378 8,3949 5,3496 77,2729 136,9251 143,108 83,9569 15,7019 48,3214 155,3242 153,2477 129,3005 132,219 11,5057 37,1591 68,4188 26 14,3841 8,8342 1,4122 147,5759 113,5553 157,7 65,9434 116,4221 66,2747 138,7027 145,6697 130,5706 60,701 127,9896 136,7200 17,1275 120,5788 175,6165 70,9252 95,36 106,6940 2,2328 96,425 51,9329 183,4842 196,6754 27 23,9054 78,3066 8,3483 1,7905 152,2124 108,9929 63,3896 151,5915 111,3101 34,8912 182,6234 133,7749 16,8673 192,5344 114,714 168,1578 175,210 138,5918 7,4888 122,84 28 8,9996 188,3816 116,2638 132,5604 20,2234 178,3642 76,3705 122,9165 184,4164 198,366 161,9217 160,9059 56,5375 120,8874 11,2275 111,4495 193,9441 157,6880 48,2803 29 78,8190 144,6452 114,9478 156,5083 62,9692 121,4537 184,9797 109,6873 153,5446 67,3449 172,5830 111,1005 100,1642 148,3252 13,2784 30 78,5469 119,7372 144,1616 130,1356 59,4458 40,9818 79,503 43,6233 148,4760 42,263 1,4612 57,5668 185,3846 101,6979 94,6976 106,7819 2,2455 71,9294 31 4,399 8,3987 50,2598 75,7688 47,7840 99,8583 190,5055 112,5231 114,7617 118,6949 180,3598 21,2556 199,5564 14,1945 3,3031 35,9855 10,1184 146,2837 51,3739 83,6588 46,5964 32 136,3823 77,1689 92,3395 121,1615 85,7494 173,9631 177,6902 88,8129 36,7329 116,6065 61,3332 68,7352 119,1914 82,8571 70,9909 33 144,4841 173,5949 170,3648 113,652 110,1986 82,3577 61,1837 97,5671 55,1252 19,6938 48,914 74,3642 125,67 89,3089 176,3258 20,6387 138,6960 153,6574 171,3913 34 86,6435 156,8641 72,2540 181,5267 27,8912 58,8824 179,8528 62,9864 70,2348 57,5471 53,236 168,3923 101,3383 142,7791 55,7174 2,315 147,9758 15,7961 199,8196 12,6690 35 57,3693 8,3160 144,3087 114,490 65,8910 178,5774 172,992 16,839 118,8640 41,6749 31,9855 39,853 64,6071 166,2816 184,7437 49,3098 182,7369 110,4985 93,8775 36 80,2032 130,7589 123,6226 16,1555 150,116 88,7759 100,8612 9,5284 198,6280 49,953 143,5111 42,4917 134,979 159,6043 32,7329 67,2380 148,9550 48,7266 37 197,9188 119,9313 187,4105 191,3573 109,2135 75,751 200,7541 139,8208 155,609 142,6433 25,1591 132,821 156,7714 107,1144 99,7757 38 91,7087 88,502 132,6092 126,5441 147,8391 12,6699 130,5227 146,4400 108,8712 100,1369 134,4730 87,2975 99,6169 183,5213 109,4945 39 200,4319 98,3993 130,2414 40,2489 196,9267 133,8145 82,3528 44,9175 42,5464 127,6103 93,6132 180,9506 192,7454 119,1376 115,983 81,7400 35,853 40 23,8336 1,9047 120,7760 101,2885 21,2199 144,7772 96,5739 136,4658 184,4306 189,4263 30,9818 39,2489 108,8883 8,3193 80,9657 181,2338 162,3056 71,2826 68,5800 41 200,2622 78,63 66,4654 198,7215 59,284 75,7333 22,1374 181,5235 16,2747 154,901 150,7278 3,7212 103,7917 163,5256 20,4203 91,7776 35,6749 147,1858 165,3741 107,8116 42 160,2382 156,6539 6,7265 15,1789 61,8096 164,347 194,6498 172,5383 104,2726 124,3496 161,4792 159,5951 117,7074 2,1689 186,9391 62,3249 79,9404 39,5464 187,3075 22,3783 30,263 16,2596 137,4572 163,1278 60,6663 70,9396 36,4917 73,9154 43 200,8943 159,9621 97,3906 20,8105 164,6849 13,5400 3,7313 133,8488 108,8964 30,6233 79,5052 131,8231 167,8120 14,3199 130,2685 138,7965 177,9544 143,1171 65,5805 118,8008 140,4482 93,8479 44 197,4900 144,2276 198,2619 39,9175 87,7875 191,8130 166,6953 170,6940 163,18 79,9988 145,2888 173,5518 57,9979 82,3134 54,4113 3,2165 45 57,4630 23,8074 112,9496 130,4994 86,8207 17,5114 120,5279 169,662 162,3436 170,8060 118,5918 124,3290 110,8317 13,1794 167,1163 46 57,2413 152,9550 86,7512 123,132 138,2860 195,8206 176,9923 119,2687 54,9328 196,9632 73,5109 31,5964 173,2969 193,199 80,7968 194,2429 47 57,9584 114,9480 145,9483 190,5892 182,8382 31,7840 129,9533 142,5297 58,1229 146,2959 6,274 14,7898 189,5939 11,8727 76,2138 70,2236 48 152,5835 23,4485 33,914 24,7959 25,3214 135,8869 53,3578 162,201 28,2803 141,7941 36,7266 85,2792 86,3588 124,2593 130,7921 49 160,8648 154,2962 109,7520 36,953 178,9747 192,3113 112,2935 35,3098 71,3441 4,619 96,9901 171,9736 163,4688 1,2437 133,5167 117,2896 105,9278 50 152,5767 112,6454 185,3968 77,5220 20,9507 165,2667 98,990 187,2485 198,3798 13,758 128,2987 189,7031 52,9931 127,3622 31,2598 179,2502 191,5026 153,4905 51 80,7589 72,4882 137,1096 138,8755 109,662 67,4225 181,158 132,6107 189,8899 159,3017 5,22 10,5578 31,3739 120,5675 26,9329 176,1625 52 4,9267 115,4973 159,7816 185,8925 188,7805 97,9063 50,9931 137,9846 91,424 150,634 56,2416 107,3647 68,7601 168,1134 179,3504 53 14,9197 114,7352 156,4662 62,153 85,1227 177,9852 34,236 7,179 12,7001 48,3578 71,9285 86,7353 150,662 183,5304 125,8054 54,8361 54 197,2223 66,2906 136,1794 188,4883 17,3115 109,7832 44,4113 182,438 15,8983 200,4899 112,2279 169,2296 4,313 53,8361 138,6261 46,9328 55 33,1252 188,5181 101,6050 24,4106 169,7795 149,3088 34,7174 193,8583 1,6399 145,3342 105,8477 166,3686 121,44 16,6680 82,3547 56 101,3516 20,7194 179,5284 127,3031 5,8262 161,9811 16,9828 15,2334 52,2416 7,6616 77,7923 182,7267 88,3375 61,1315 117,1934 28,5375 124,552 100,361 57 18,4216 94,558 186,8815 3,1239 85,6678 45,4630 46,2413 35,3693 84,6563 185,9772 67,8012 47,9584 155,893 64,810 10,9988 80,8722 160,2058 59,2689 79,2330 30,5668 184,7592 44,9979 162,6483 116,656 34,5471 106,4868 131,6342 183,9093 13,6850 12,9416 58 152,5877 98,3677 8,4708 130,7020 59,5735 121,8818 47,1229 102,6906 150,4857 90,7141 86,5989 175,3675 79,2365 34,8824 186,8993 125,1050 74,7934 147,2267 193,6166 59 86,1293 147,2651 149,2405 141,9126 112,4585 58,5735 74,4470 24,6941 199,8958 57,2689 13,9801 162,391 30,4458 180,2435 41,284 72,7154 101,1804 87,4628 168,4170 99,671 70,8055 11,3203 12,761 60 200,3269 98,2073 26,701 185,6670 120,2231 14,2464 127,1402 12,5933 42,6663 189,4415 107,52 146,2317 112,2570 154,6667 177,5345 172,2781 61 1,9146 159,49 33,1837 42,8096 20,2580 24,7906 87,9053 163,448 190,9775 155,5301 173,4803 115,3324 196,5577 171,6888 32,3332 56,1315 131,6924 195,8928 62 97,9163 53,153 120,3851 18,1926 154,3238 12,5676 88,9007 152,7404 29,9692 161,4144 10,4816 105,2736 42,3249 107,5324 115,1913 121,4145 116,7419 34,9864 193,6610 103,8383 63 141,5607 77,5873 27,3896 169,5160 95,5264 69,2323 125,1315 158,5709 102,5806 9,1753 103,9314 71,3007 131,5257 92,9006 96,5638 7,9920 64 57,810 98,3909 97,2201 22,4257 120,2385 177,7660 83,2716 81,9744 111,2663 145,2685 130,2493 148,6419 106,256 141,158 86,414 87,9403 121,771 102,4635 5,7836 67,2090 35,6071 131,4631 182,4701 110,6711 65 152,3595 66,6930 26,9434 97,6170 123,9599 175,7920 155,5533 102,1652 77,4069 198,3575 81,3054 199,11 95,6605 35,8910 43,5805 71,439 134,9956 74,6617 165,3705 140,5376 66 80,2902 68,8312 142,777 156,2965 41,4654 6,6627 84,7710 102,3328 65,6930 54,2906 24,5879 112,2271 93,5873 94,3424 20,2667 26,2747 130,5826 17,6176 69,824 89,3012 67 57,8012 102,5417 175,5048 153,6204 12,371 137,1414 133,3802 64,2090 98,980 200,475 171,1394 36,2380 29,3449 124,1880 51,4225 195,5737 100,6216 103,1468 68 141,3540 197,8223 78,7924 66,8312 144,2277 174,7082 16,5894 163,4920 146,3895 52,7601 140,9624 40,5800 25,4188 32,7352 186,2528 69 8,8815 198,6284 17,3610 156,9959 75,3354 168,2357 102,1172 190,8022 139,9030 161,6171 96,4815 189,5215 66,824 94,1427 63,2323 70 4,5285 24,4670 148,7231 26,9252 17,3899 59,8055 47,2236 42,9396 175,3256 149,2366 92,96 153,6532 178,3394 168,1295 156,4830 34,2348 9,2 124,9089 32,9909 183,5332 71 8,5829 22,3243 138,1229 81,1711 170,1539 49,3441 23,3316 134,7485 12,6734 30,9294 21,4986 142,6038 65,439 7,8206 40,2826 145,6127 53,9285 63,3007 186,7143 171,6702 72 4,6508 78,5839 119,6215 114,8350 9,794 8,7308 113,8782 102,3377 34,2540 25,2332 59,7154 172,3153 89,4836 178,5128 51,4882 120,2287 174,2019 153,541 96,859 146,4264 171,8573 157,604 12,3626 73 14,4106 8,4528 159,4969 97,6534 77,2438 24,3509 174,2581 150,8061 139,4428 149,5233 42,9154 90,5133 78,212 194,8521 172,2239 46,5109 74 159,8960 33,3642 59,4470 20,6183 99,7031 179,1223 93,5576 164,8627 58,7934 65,6617 110,6731 108,8251 165,2602 121,1468 182,1873 176,8129 75 115,9140 141,9237 80,2187 86,259 20,2484 92,6095 97,1883 41,7333 87,3244 69,3354 120,6892 131,5902 31,7688 108,5943 4,7044 16,4116 191,1403 81,2609 37,751 76 115,7291 185,3674 181,3275 47,2138 143,1079 28,3705 125,1865 178,8433 3,7729 114,9690 100,1793 200,4623 199,6878 138,5683 141,1969 126,9595 9,9340 83,4424 89,6942 77 112,3500 160,105 189,5702 191,5135 124,8896 198,5081 19,7013 73,2438 63,5873 129,2337 11,815 133,2481 192,561 32,1689 50,5220 87,7040 25,2729 65,4069 106,9161 153,4483 56,7923 172,4771 13,8638 78 10,3771 68,7924 12,5753 30,5469 158,6367 122,6207 27,3066 116,2732 41,63 72,5839 161,6310 4,8058 104,1377 83,3955 29,8190 98,6603 154,8423 137,1910 135,6919 73,212 145,7244 79 141,7918 101,3205 165,3768 96,3059 119,4117 152,6519 57,2330 42,9404 166,8726 161,8395 30,503 89,5169 134,5792 117,9043 129,7314 43,5052 109,9677 58,2365 44,9988 167,820 193,7737 194,5784 80 36,2032 84,4645 1,982 115,1417 151,6728 112,5208 51,7589 152,9606 113,917 18,5252 121,2257 75,2187 57,8722 133,7217 179,7729 119,108 66,2902 40,9657 97,7213 172,7715 89,7224 19,62 46,7968 21,9884 81 115,2608 197,5540 97,8866 101,4493 64,9744 11,7299 71,1711 109,2519 136,1409 39,7400 75,2609 142,424 141,4032 183,3061 184,4485 95,7627 5,2469 143,9810 65,3054 89,6124 82 33,3577 130,3349 156,4691 39,3528 173,591 177,7882 44,3134 116,8491 132,4162 135,519 131,3457 128,6834 32,8571 55,3547 14,3283 83 78,3955 6,4949 185,9306 17,4615 64,2716 25,9569 149,6823 5,8407 167,8200 117,8516 165,1555 151,162 31,6588 76,4424 11,4211 84 57,6563 80,4645 119,6417 66,7710 198,5999 136,4270 86,195 104,5330 154,5421 137,4367 95,3812 159,8763 170,2436 107,2954 85,9888 134,9312 7,4061 85 57,6678 160,3613 156,6669 168,6193 136,6221 180,5525 32,7494 118,1102 192,544 129,517 93,2349 87,7478 189,1147 53,1227 20,1435 167,8110 133,836 84,9888 132,3873 128,4644 110,6060 21,2880 48,2792 86 115,4291 197,9714 144,8808 59,1293 126,937 189,1115 18,7517 45,8207 46,7512 177,7010 180,4604 75,259 157,4447 84,195 34,6435 120,5230 64,414 184,801 58,5989 142,1663 53,7353 117,4220 48,3588 87 160,8712 119,518 75,3244 94,9647 59,4628 61,9053 44,7875 168,9716 64,9403 164,3629 20,2639 8,2833 77,7040 7,4976 159,19 85,7478 191,6921 88,8011 167,1022 158,4081 110,1219 21,2681 38,2975 88 6,2039 62,9007 20,3014 113,7322 136,9026 32,8129 38,502 151,2295 150,6770 183,5547 36,7759 87,8011 94,4629 115,6611 19,2002 161,1726 56,3375 10,2539 125,5012 89,6267 89 33,3089 72,4836 123,1723 79,5169 174,858 76,6942 4,6363 199,2446 105,2736 66,3012 180,6612 80,7224 163,4055 88,6267 81,6124 90 152,4427 4,9446 115,1117 119,928 185,7284 20,7372 16,4558 108,9076 179,3149 139,7846 58,7141 5,8919 73,5133 144,6223 174,6914 91 160,383 181,5060 174,3418 113,4626 95,1806 3,2483 192,6625 52,424 115,1105 137,4129 142,9164 41,7776 158,5553 38,7087 200,1988 92 1,647 130,4320 108,1844 134,610 194,426 177,3182 75,6095 20,1704 94,6085 128,556 22,5592 16,2896 186,7980 32,3395 139,6763 121,3819 138,8080 70,96 63,9006 93 66,5873 39,6132 181,4071 154,4073 85,2349 106,7477 74,5576 150,9213 98,6617 147,7807 43,8479 152,6543 35,8775 167,5670 2,8177 94 57,558 66,3424 92,6085 120,2733 87,9647 30,6976 191,8318 139,7116 109,1299 88,4629 170,9318 69,1427 14,3551 115,3350 171,9959 15,9606 95 119,8126 112,555 120,1104 18,6122 91,1806 173,4092 196,231 26,36 147,1278 19,8691 125,2917 9,9480 63,5264 81,7627 84,3812 65,6605 105,6026 96 40,5739 79,3059 104,9639 113,712 162,3737 155,1251 10,807 49,9901 151,5643 23,2431 72,859 26,425 69,4815 143,9274 183,5939 63,5638 147,6736 193,8831 97 33,5671 185,7065 52,9063 64,2201 188,4695 192,6411 43,3906 73,6534 13,7026 112,7969 81,8866 80,7213 62,9163 65,6170 140,6527 75,1883 137,4667 98 115,5825 131,4614 64,3909 155,5515 139,1235 39,3993 102,8330 60,2073 200,2690 166,2364 78,6603 162,6139 58,3677 117,9545 6,5594 144,7198 50,990 150,2093 143,4300 67,980 93,6617 99 104,9140 18,5717 174,5675 157,6818 132,6234 182,2897 151,4990 183,3577 59,671 133,2090 23,6485 153,4560 31,8583 74,7031 1,2367 127,1408 37,7757 193,4566 194,5832 38,6169 100 159,6567 137,7178 163,9709 190,6674 36,8612 142,2994 76,1793 67,6216 29,1642 56,361 144,6605 128,2584 153,9522 145,5512 15,809 38,1369 101 188,8531 40,2885 157,1393 171,4083 55,6050 144,3619 3,3381 113,5024 81,4493 163,6033 56,3516 129,8821 184,9591 59,1804 79,3205 30,6979 138,2902 143,4042 34,3383 21,2775 102 98,8330 66,3328 144,7884 72,3377 24,8731 181,3585 137,6814 172,6572 58,6906 64,4635 117,2689 177,4462 67,5417 183,9634 69,1172 65,1652 178,1334 161,8230 63,5806 140,6370 21,6643 103 200,6851 123,8756 24,8083 41,7917 191,9683 63,9314 112,7409 110,491 131,2920 196,696 186,9654 62,8383 113,5248 67,1468 114,1318 104 141,8162 78,1377 42,2726 123,9213 1,6647 126,8615 200,7083 197,4174 84,5330 192,4219 142,6236 99,9140 96,9639 25,4899 172,561 179,8827 169,3712 105 185,1033 62,2736 113,3388 116,7899 89,2736 164,4661 183,7722 55,8477 190,2518 180,341 95,6026 119,9930 120,9333 13,6821 49,9278 106 14,7912 4,4769 115,9598 141,674 112,4854 20,8785 64,256 181,5332 190,3305 3,6521 30,7819 93,7477 26,6940 77,9161 57,4868 111,6460 107 114,2032 123,4581 62,5324 187,2610 60,52 116,9864 84,2954 182,8313 37,1144 169,668 52,3647 4,4383 41,8116 146,7862 112,7448 15,4589 176,6806 108 200,9976 185,8699 17,1942 40,8883 156,7039 92,1844 75,5943 22,9656 43,8964 27,9929 174,5669 90,9076 145,521 143,972 113,4342 74,8251 126,525 38,8712 109 152,2743 136,8658 81,2519 169,382 51,662 49,7520 129,2464 79,9677 54,7832 37,2135 94,1299 185,6644 29,6873 19,8397 16,1469 38,4945 110 197,751 33,1986 145,6099 118,9403 74,6731 126,1073 103,491 35,4985 137,8848 165,4097 85,6060 87,1219 45,8317 64,6711 16,4252 111 197,4083 144,5456 114,2027 64,2663 27,3101 191,5723 162,8771 152,1940 28,4495 106,6460 29,1005 130,9137 133,6767 18,785 160,9702 21,3060 112 23,3201 141,130 80,5208 181,2524 95,555 77,3500 183,9037 164,1492 155,7915 106,4854 50,6454 133,9083 5,1522 45,9496 173,7338 66,2271 59,4585 97,7969 60,2570 31,5231 149,8736 49,2935 158,383 128,7645 107,7448 103,7409 54,2279 196,5663 113 80,917 33,652 26,5553 72,8782 101,5024 108,4342 132,5383 116,8036 184,4999 88,7322 105,3388 187,6332 190,697 136,1984 96,712 91,4626 103,5248 114 29,9478 180,8490 189,3102 111,2027 192,6813 141,6388 72,8350 115,3112 152,9627 53,7352 129,168 107,2032 1,508 47,9480 35,490 17,8915 22,9960 27,714 31,7617 76,9690 117,7876 193,4000 103,1318 194,949 115 152,4383 176,8236 75,9140 25,9233 9,7827 98,5825 52,4973 146,9828 81,2608 128,9072 86,4291 76,7291 90,1117 106,9598 144,3825 80,1417 114,3112 62,1913 39,983 91,1105 88,6611 129,7465 166,3001 61,3324 117,1399 94,3350 116 78,2732 26,4221 113,8036 179,8099 32,6065 105,7899 62,7419 107,9864 82,8491 186,8639 176,4512 192,5906 57,656 4,6771 28,2638 10,1493 117 98,9545 198,3936 42,7074 79,9043 102,2689 56,1934 114,7876 83,8516 86,4220 196,3366 1,2597 49,2896 138,7762 17,6266 115,1399 118 14,373 23,5079 4,3626 6,8547 123,2088 181,8129 18,6388 85,1102 31,6949 166,3979 35,8640 43,8008 45,5918 177,8279 110,9403 128,6289 119 80,108 154,741 130,730 84,6417 72,6215 30,7372 170,275 168,1890 157,9158 90,928 121,6261 37,9313 95,8126 14,960 87,518 79,4117 39,1376 163,5189 169,3511 195,617 3,7711 32,1914 19,7406 46,2687 196,2824 105,9930 120 40,7760 62,3851 72,2287 94,2733 86,5230 95,1104 164,5926 26,5788 18,44 155,6822 60,2231 185,556 45,5279 179,3327 159,5811 75,6892 64,2385 5,1862 178,8906 28,8874 51,5675 105,9333 121 4,7639 80,2257 197,6502 119,6261 136,1320 156,195 29,4537 7,8739 58,8818 32,1615 168,7186 62,4145 92,3819 173,2976 64,771 175,8821 191,8606 162,1977 132,3867 74,1468 165,7147 148,8115 55,44 122 78,6207 8,5338 174,8205 168,1574 162,3518 166,6712 135,6345 28,9165 192,1494 128,7247 189,2017 3,9640 148,3230 27,84 179,7377 123 14,2733 198,3493 6,13 104,9213 107,4581 89,1723 118,2088 128,1602 155,3251 46,132 36,6226 184,2832 103,8756 65,9599 124,6490 145,8804 193,7460 124 141,168 159,6580 42,3496 123,6490 77,8896 20,392 67,1880 158,1870 147,9014 165,9797 136,7388 56,552 11,6180 70,9089 45,3290 48,2593 125 33,67 174,9048 95,2917 53,8054 1,2909 63,1315 76,1865 88,5012 58,1050 23,922 173,2615 188,230 172,8515 196,4519 138,9932 183,9920 7,5670 126 14,9602 4,2625 159,2980 86,937 181,8920 104,8615 179,7436 191,3989 161,4512 108,525 155,307 110,1073 134,5002 38,5441 76,9595 180,7176 127 4,6808 160,7534 26,9896 39,6103 50,3622 137,4069 60,1402 156,6372 2,9365 16,9807 56,3031 139,2526 14,5682 99,1408 167,3756 135,1752 161,6643 146,8151 128 14,9662 115,9072 123,1602 92,556 50,2987 190,2968 118,6289 157,6815 132,2789 184,6339 198,1860 112,7645 122,7247 85,4644 82,6834 100,2584 196,1760 12,7542 129 6,8525 114,168 101,8821 77,2337 79,7314 47,9533 85,517 175,7121 184,5623 109,2464 143,8021 167,5370 25,3005 159,6895 115,7465 130 119,730 8,55 26,5706 45,4994 36,7589 30,1356 184,8488 178,615 92,4320 58,7020 82,3349 174,4481 66,5826 39,2414 194,9429 156,2264 20,9048 64,2493 43,2685 137,5926 190,3429 147,9251 111,9137 48,7921 38,5227 131 14,358 98,4614 159,4355 75,5902 22,6075 43,8231 163,8625 11,833 57,6342 61,6924 82,3457 64,4631 134,6293 167,6269 2,7005 63,5257 9,700 103,2920 21,3748 132 198,2393 113,5383 99,6234 138,5667 28,5604 19,3157 38,6092 85,3873 82,4162 25,219 182,6433 22,1560 147,2847 6,1514 121,3867 128,2789 51,6107 37,821 133 80,7217 112,9083 136,3739 77,2481 39,8145 43,8488 18,2702 27,7749 168,899 99,2090 190,9958 139,4719 182,8241 191,4296 85,836 153,8437 67,3802 49,5167 111,6767 134 197,6225 198,4071 92,610 79,5792 175,4489 36,979 131,6293 71,7485 146,9556 158,119 11,294 65,9956 135,276 16,5203 84,9312 126,5002 38,4730 135 23,4560 4,7582 20,6434 174,6977 150,9732 190,1431 173,5664 144,6396 127,1752 122,6345 48,8869 82,519 158,8348 184,7629 78,6919 10,4650 134,276 194,5726 13,861 136 161,3866 195,4279 32,3823 84,4270 168,9519 54,1794 170,1529 197,9068 121,1320 194,2496 109,8658 199,7783 133,3739 145,1769 179,6711 26,7200 40,4658 174,5711 85,6221 113,1984 25,9251 184,6682 81,1409 88,9026 178,2752 124,7388 137 188,9117 100,7178 42,4572 51,1096 52,9846 97,4667 25,5378 102,6814 130,5926 84,4367 141,7872 23,7591 127,4069 157,5286 78,1910 91,4129 155,5736 67,1414 10,5273 110,8848 138 200,678 160,9902 26,7027 101,2902 51,8755 132,5667 43,7965 92,8080 1,7896 173,8555 33,6960 71,1229 46,2860 27,5918 188,892 169,7498 178,6589 125,9932 76,5683 117,7762 54,6261 140,5446 139 197,1386 98,1235 92,6763 181,5456 176,8186 182,2354 133,4719 158,3451 196,3988 73,4428 90,7846 155,2100 194,6966 69,9030 94,7116 127,2526 162,4510 9,8701 37,8208 140 1,546 17,6440 97,6527 158,4029 151,5289 68,9624 138,5446 43,4482 169,2230 9,4163 155,5001 188,6223 102,6370 166,9829 65,5376 141 154,3188 68,3540 106,674 79,7918 104,8162 24,6862 63,5607 6,8200 75,9237 150,2634 124,168 14,4477 112,130 164,6499 198,2621 114,6388 185,6820 59,9126 64,158 137,7872 81,4032 76,1969 48,7941 142 66,777 17,8016 104,6236 18,332 47,5297 81,424 91,9164 150,2598 5,1195 34,7791 155,6613 169,7506 86,1663 100,2994 190,8070 2,7731 71,6038 145,5378 37,6433 143 23,7543 6,3110 22,8603 108,972 25,108 81,9810 43,1171 186,8430 14,5848 129,8021 101,4042 76,1079 98,4300 155,2442 177,3884 36,5111 153,9940 96,9274 15,1041 176,5062 144 115,3825 173,8693 29,6452 98,7198 68,2277 195,652 102,7884 30,1616 111,5456 33,4841 13,9987 44,2276 86,8808 148,3617 35,3087 40,7772 101,3619 22,9362 184,5222 135,6396 90,6223 3,285 100,6605 145 200,2753 1,648 136,1769 26,6697 64,2685 174,7252 47,9483 108,521 44,2888 123,8804 71,6127 142,5378 20,8099 78,7244 110,6099 100,5512 55,3342 146 115,9828 22,3620 47,2959 60,2317 5,5308 127,8151 7,4823 134,9556 18,9000 31,2837 17,5362 68,3895 72,4264 188,5715 167,52 107,7862 38,4400 147 26,5759 59,2651 163,7221 95,1278 132,2847 124,9014 5,1087 34,9758 96,6736 38,8391 41,1858 167,5546 130,9251 149,6379 58,2267 93,7807 148 160,3995 144,3617 185,2814 64,6419 18,589 30,4760 173,6725 179,6374 70,7231 155,6603 122,3230 192,4834 36,9550 121,8115 29,3252 149 14,4352 59,2405 188,1003 163,9897 70,2366 83,6823 187,9580 174,2824 73,5233 112,8736 55,3088 18,2664 176,2255 190,3755 180,3407 147,6379 150 141,2634 1,2183 16,1680 41,7278 36,116 135,9732 11,4802 98,2093 58,4857 142,2598 73,8061 9,4539 18,5150 24,5669 52,634 190,5243 88,6770 53,662 7,9031 93,9213 151 80,6728 156,6169 24,784 27,5915 174,1846 168,9216 99,4990 88,2295 178,5979 96,5643 83,162 13,9629 140,5289 189,9732 163,3624 152 161,6939 58,5877 48,5835 9,1087 8,1257 46,9550 189,8140 199,2697 109,2743 65,3595 186,2075 115,4383 50,5767 193,5444 90,4427 80,9606 114,9627 62,7404 79,6519 27,2124 111,1940 19,9751 93,6543 153 17,8545 99,4560 70,6532 164,8748 29,5446 25,2477 72,541 143,9940 173,7613 77,4483 50,4905 165,897 133,8437 33,6574 67,6204 100,9522 154 141,3188 119,741 198,5078 156,8062 62,3238 20,2398 18,453 49,2962 187,1808 168,6317 200,229 185,1448 93,4073 78,8423 84,5421 41,901 60,6667 170,5971 19,4320 21,1845 199,5786 155 57,893 14,4446 98,5515 112,7915 123,3251 120,6822 25,3242 139,2100 143,2442 65,5533 142,6613 96,1251 137,5736 148,6603 61,5301 126,307 37,609 140,5001 194,6768 156 66,2965 42,6539 82,4691 53,4662 151,6169 7,7027 85,6669 29,5083 154,8062 130,2264 108,7039 34,8641 121,195 175,2205 69,9959 70,4830 127,6372 37,7714 157 119,9158 26,7 86,4447 101,1393 187,7184 137,5286 99,6818 11,8497 2,8643 158,1881 128,6815 175,1597 72,604 24,9540 28,6880 15,2136 158 78,6367 187,6714 91,5553 139,3451 169,1709 135,8348 63,5709 195,4912 140,4029 157,1881 124,1870 181,969 112,383 87,4081 134,119 159 4,6133 160,4243 61,49 180,5757 6,9500 194,9391 43,9621 100,6567 126,2980 73,4969 131,4355 124,6580 1,7420 52,7816 74,8960 198,8330 42,5951 120,5811 87,19 36,6043 84,8763 129,6895 2,1397 51,3017 193,3786 160 14,7860 138,9902 127,7534 49,8648 85,3613 189,1530 57,2058 159,4243 183,9293 148,3995 87,8712 180,624 179,2542 91,383 42,2382 1,6461 77,105 28,9059 111,9702 19,3997 161 152,6939 14,4275 23,6863 78,6310 136,3866 42,4792 62,4144 79,8395 127,6643 69,6171 126,4512 56,9811 196,9729 102,8230 88,1726 28,9217 162 4,3924 98,6139 6,6897 59,391 194,1483 122,3518 57,6483 121,1977 179,9718 139,4510 199,7749 40,3056 111,8771 96,3737 45,3436 186,6114 48,201 163 1,8164 101,6033 41,5256 173,5799 15,818 13,3870 149,9897 131,8625 181,5593 119,5189 61,448 42,1278 68,4920 100,9709 147,7221 44,18 49,4688 192,2762 151,3624 89,4055 164 141,6499 112,1492 42,347 120,5926 43,6849 87,3629 184,9774 170,328 153,8748 10,3866 172,9550 74,8627 195,9730 105,4661 24,7312 183,615 176,8681 165 23,208 6,6473 79,3768 187,1291 50,2667 153,897 166,4221 74,2602 21,2151 121,7147 124,9797 83,1555 65,3705 7,6019 41,3741 110,4097 166 23,8077 98,2364 79,8726 173,6917 44,6953 167,7080 118,3979 165,4221 168,4399 17,4762 9,8702 122,6712 20,6086 169,5530 115,3001 35,2816 55,3686 140,9829 167 23,2133 188,6549 187,9538 43,8120 18,7625 127,3756 180,6183 87,1022 79,820 83,8200 3,3877 129,5370 85,8110 193,8116 166,7080 131,6269 146,52 147,5546 45,1163 199,2001 93,5670 168 119,1890 8,2076 136,9519 87,9716 154,6317 121,7186 133,899 122,1574 85,6193 173,3859 59,4170 27,1578 69,2357 151,9216 22,4356 70,1295 34,3923 166,4399 10,9495 52,1134 169 14,4333 175,9164 177,2274 45,662 20,5990 63,5160 119,3511 104,3712 187,9225 192,8603 109,382 158,1709 55,7795 1,8700 138,7498 142,7506 166,5530 107,668 15,7485 54,2296 140,2230 170 119,275 1,2620 198,8089 136,1529 33,3648 18,4989 177,2774 44,6940 84,2436 2,9342 154,5971 164,328 45,8060 71,1539 94,9318 171 188,6981 101,4083 16,3482 67,1394 61,6888 49,9736 182,4704 94,9959 10,2029 33,3913 185,5113 71,6702 6,3722 18,4403 72,8573 172 200,6888 42,5383 72,3153 104,561 174,3433 102,6572 175,5353 35,992 73,2239 164,9550 29,5830 80,7715 77,4771 2,1438 60,2781 125,8515 13,2070 173 112,7338 1,2069 144,8693 33,5949 17,7312 20,2676 18,7256 121,2976 168,3859 32,9631 148,6725 82,591 163,5799 192,2550 166,6917 179,4234 138,8555 44,5518 95,4092 153,7613 135,5664 61,4803 125,2615 193,4808 46,2969 174 130,4481 89,858 73,2581 135,6977 68,7082 136,5711 91,3418 125,9048 122,8205 145,7252 72,2019 151,1846 172,3433 108,5669 99,5675 179,1886 149,2824 90,6914 175 156,2205 187,2864 27,210 199,3779 67,5048 121,8821 169,9164 134,4489 65,7920 26,6165 172,5353 197,5909 8,1022 129,7121 14,6718 184,9107 70,3256 58,3675 157,1597 176 200,1709 115,8236 6,3512 33,3258 187,1128 191,3352 139,8186 149,2255 116,4512 46,9923 143,5062 51,1625 164,8681 107,6806 74,8129 177 86,7010 92,3182 24,393 64,7660 102,4462 43,9544 22,8713 190,1332 14,6741 8,8508 170,2774 82,7882 169,2274 32,6902 53,9852 60,5345 143,3884 178,7547 118,8279 3,7097 178 130,615 72,5128 70,3394 13,4194 120,8906 11,1696 151,5979 16,1714 138,6589 102,1334 17,7464 49,9747 177,7547 35,5774 136,2752 28,3642 76,8433 179 80,7729 160,2542 136,6711 120,3327 56,5284 50,2502 186,9993 180,664 104,8827 90,3149 74,1223 148,6374 174,1886 126,7436 173,4234 162,9718 116,8099 34,8528 52,3504 122,7377 180 160,624 159,5757 114,8490 59,2435 86,4604 39,9506 85,5525 179,664 191,6312 31,3598 149,3407 167,6183 89,6612 9,5995 126,7176 105,341 181 112,2524 187,7956 41,5235 118,8129 185,9712 139,5456 76,3275 40,2338 11,8448 34,5267 102,3585 126,8920 106,5332 93,4071 91,5060 191,4844 163,5593 158,969 51,158 182 27,6234 47,8382 99,2897 139,2354 5,8806 133,8241 132,6433 56,7267 74,1873 35,7369 64,4701 107,8313 54,438 171,4704 2,7908 21,2447 183 160,9293 112,9037 102,9634 99,3577 81,3061 88,5547 11,594 96,5939 53,5304 105,7722 26,4842 57,9093 70,5332 125,9920 164,615 38,5213 184 6,8021 130,8488 188,198 40,4306 123,2832 101,9591 113,4999 129,5623 144,5222 175,9107 136,6682 164,9774 86,801 57,7592 29,9797 81,4485 35,7437 135,7629 28,4164 10,3207 128,6339 185 57,9772 105,1033 90,7284 52,8925 83,9306 97,7065 148,2814 25,3573 197,3140 50,3968 11,2045 141,6820 76,3674 108,8699 60,6670 188,700 120,556 181,9712 154,1448 30,3846 16,6280 109,6644 171,5113 19,5896 196,1339 186 57,8815 152,2075 197,1324 42,9391 188,5027 92,7980 179,9993 191,8255 58,8993 143,8430 116,8639 162,6114 71,7143 189,8128 195,2099 68,2528 103,9654 187 14,4874 200,6767 176,1128 37,4105 7,9522 175,2864 19,107 107,2610 167,9538 157,7184 24,9327 181,7956 42,3075 158,6714 165,1291 50,2485 154,1808 113,6332 169,9225 149,9580 10,4786 188 23,4236 198,9114 137,9117 184,198 101,8531 167,6549 54,4883 149,1003 28,3816 196,88 52,7805 55,5181 185,700 186,5027 171,6981 97,4695 138,892 125,230 146,5715 140,6223 189 152,8140 160,1530 8,2780 114,3102 86,1115 40,4263 77,5702 50,7031 47,5939 85,1147 60,4415 69,5215 186,8128 198,9430 122,2017 51,8899 151,9732 193,2375 190 4,7536 200,5999 47,5892 113,697 130,3429 135,1431 150,5243 61,9775 69,8022 133,9958 128,2968 100,6674 17,7000 106,3305 9,3613 177,1332 31,5055 149,3755 142,8070 15,6854 105,2518 191 77,5135 126,3989 176,3352 94,8318 186,8255 121,8606 87,6921 44,8130 103,9683 75,1403 181,4844 50,5026 180,6312 111,5723 37,3573 133,4296 192 114,6813 97,6411 77,561 39,7454 22,4606 104,4219 27,5344 85,544 173,2550 91,6625 169,8603 116,5906 49,3113 163,2762 122,1494 148,4834 193 152,5444 167,8116 114,4000 55,8583 58,6166 189,2375 159,3786 96,8831 79,7737 21,494 28,9441 62,6610 123,7460 99,4566 173,4808 46,199 194 200,2915 159,9391 136,2496 130,9429 42,6498 92,426 139,6966 162,1483 73,8521 99,5832 155,6768 114,949 46,2429 79,5784 135,5726 195 23,3179 4,2490 197,5337 144,652 136,4279 6,2233 158,4912 186,2099 200,8114 61,8928 46,8206 164,9730 119,617 19,5038 67,5737 196 188,88 39,9267 139,3988 95,231 61,5577 161,9729 125,4519 117,3366 103,696 46,9632 26,6754 112,5663 128,1760 119,2824 185,1339 197 54,2223 186,1324 121,6502 134,6225 68,8223 24,9337 25,3875 139,1386 81,5540 44,4900 195,5337 37,9188 111,4083 86,9714 110,751 136,9068 185,3140 104,4174 175,5909 198 141,2621 170,8089 44,2619 188,9114 41,7215 1,1724 154,5078 84,5999 16,2223 117,3936 123,3493 159,8330 69,6284 132,2393 134,4071 77,5081 50,3798 65,3575 36,6280 28,366 128,1860 189,9430 199 152,2697 8,1918 136,7783 59,8958 24,2838 175,3779 31,5564 162,7749 65,11 76,6878 9,70 89,2446 34,8196 167,2001 154,5786 200 108,9976 103,6851 145,2753 41,2622 187,6767 190,5999 16,2848 194,2915 5,4009 172,6888 39,4319 176,1709 60,3269 138,678 43,8943 98,2690 1,8021 104,7083 154,229 91,1988 67,475 76,4623 195,8114 37,7541 54,4899 ================================================ FILE: dijkstra/problem9.8test.txt ================================================ 1 2,1 8,2 2 1,1 3,1 3 2,1 4,1 4 3,1 5,1 5 4,1 6,1 6 5,1 7,1 7 6,1 8,1 8 7,1 1,2 ================================================ FILE: floyd_warshall/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(floyd_warshall) set(CMAKE_CXX_STANDARD 17) add_executable(floyd_warshall main.cpp) ================================================ FILE: floyd_warshall/main.cpp ================================================ /* * Programming Problem 18.8: All-Pairs Shortest Paths * * In this problem, each file describes a directed graph. The first line of the file indicates the number * of vertices and edges, respectively. Each subsequent line describes an edge (the first two numbers are its tail * and head, respectively) and its length (the third number). NOTE: edge lengths might be negative, and the graphs * may or may not be negative cycles. * * Test cases: (contributed by Matt Davis) What is the shortest shortest path in the graphs described in this file * and in this file? (I.e., the minimum, over choices of an origin s and a destination t, of a shortest s-t path in the graph.) * (If the graph has a negative cycle, your algorithm should detect that fact.) (Answer: -2 and "contains a negative cycle," respectively.) * * Challenge data set: Repeat the previous problem for the graphs described in the following files: graph #1, graph #2, graph #3, and graph #4. */ #include #include #include #include #include #define PERF_TEST using namespace std; using LL = long long; using VL = vector; using VVL = vector; using VVVL = vector; using Edges = unordered_map; LL INF = 1e9 + 7; string key(int i, int j) { stringstream ss; ss << i << "," << j; return ss.str(); } VVL floyd_warshall(Edges& E, int N) { VVVL dp(N + 1, VVL(N + 1, VL(N + 1, INF))); for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) if (i == j) dp[0][i][j] = 0; else if (E.find(key(i, j)) != E.end()) dp[0][i][j] = E[key(i, j)]; for (auto k{ 1 }; k <= N; ++k) for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) dp[k][i][j] = min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]); return dp[N]; } VVL floyd_warshall_memopt(Edges& E, int N) { VVL pre(N + 1, VL(N + 1, INF)); for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) if (i == j) pre[i][j] = 0; else if (E.find(key(i, j)) != E.end()) pre[i][j] = E[key(i, j)]; for (auto k{ 1 }; k <= N; ++k) { VVL cur(N + 1, VL(N + 1, INF)); for (auto i{ 1 }; i <= N; ++i) for (auto j{ 1 }; j <= N; ++j) cur[i][j] = min(pre[i][j], pre[i][k] + pre[k][j]); swap(pre, cur); } return pre; } void run(const string& filename) { Edges E; fstream fin{ filename }; int N, M; fin >> N >> M; for (int u, v, w; fin >> u >> v >> w; E[key(u, v)] = w); #ifdef PERF_TEST auto a = floyd_warshall_memopt(E, N); #else auto a = floyd_warshall_memopt(E, N), b = floyd_warshall(E, N); assert(a == b); // 💩 sanity check #endif auto cycle = false; for (auto i{ 1 }; i <= N && !cycle; ++i) cycle = a[i][i] < 0; if (cycle) { cout << filename << ": contains a negative cycle" << endl; return; } auto best = INF; for (auto& row: a) best = min(best, *min_element(row.begin(), row.end())); cout << filename << ": " << best << endl; } int main() { run("problem18.8test1.txt"); // problem18.8test1.txt: -2 run("problem18.8test2.txt"); // problem18.8test2.txt: contains a negative cycle run("problem18.8file1.txt"); // problem18.8file1.txt: contains a negative cycle run("problem18.8file2.txt"); // problem18.8file2.txt: contains a negative cycle run("problem18.8file3.txt"); // problem18.8file3.txt: -19 // run("problem18.8file4.txt"); return 0; } ================================================ FILE: floyd_warshall/main.js ================================================ /* * Programming Problem 18.8: All-Pairs Shortest Paths * * In this problem, each file describes a directed graph. The first line of the file indicates the number * of vertices and edges, respectively. Each subsequent line describes an edge (the first two numbers are its tail * and head, respectively) and its length (the third number). NOTE: edge lengths might be negative, and the graphs * may or may not be negative cycles. * * Test cases: (contributed by Matt Davis) What is the shortest shortest path in the graphs described in this file * and in this file? (I.e., the minimum, over choices of an origin s and a destination t, of a shortest s-t path in the graph.) * (If the graph has a negative cycle, your algorithm should detect that fact.) (Answer: -2 and "contains a negative cycle," respectively.) * * Challenge data set: Repeat the previous problem for the graphs described in the following files: graph #1, graph #2, graph #3, and graph #4. */ const LineByLine = require('n-readlines'); const assert = require('assert'); let key = (i, j) => `${i},${j}`; let floyd_warshall = (E, N) => { let dp = [...Array(N + 1)].map(_ => [...Array(N + 1)].map(_ => Array(N + 1).fill(Infinity))); for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) if (i == j) dp[0][i][j] = 0; else if (E.has(key(i, j))) dp[0][i][j] = E.get(key(i, j)); for (let k = 1; k <= N; ++k) for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) dp[k][i][j] = Math.min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]); return dp[N]; }; let floyd_warshall_memopt = (E, N) => { let pre = [...Array(N + 1)].map(_ => Array(N + 1).fill(Infinity)); for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) if (i == j) pre[i][j] = 0; else if (E.has(key(i, j))) pre[i][j] = E.get(key(i, j)); for (let k = 1; k <= N; ++k) { let cur = [...Array(N + 1)].map(_ => Array(N + 1).fill(Infinity)); for (let i = 1; i <= N; ++i) for (let j = 1; j <= N; ++j) cur[i][j] = Math.min(pre[i][j], pre[i][k] + pre[k][j]); [pre, cur] = [cur, pre]; } return pre; }; let run = filename => { let E = new Map(); let input = new LineByLine(filename); let [N, _] = input.next().toString('ascii').split(' ').map(Number); let line; while (line = input.next()) { let [u, v, w] = line.toString('ascii').split(' ').map(Number); E.set(key(u, v), w); } let a = floyd_warshall_memopt(E, N); // let b = floyd_warshall(E, N); // for (let i = 1; i <= N; ++i) // for (let j = 1; j <= N; ++j) // assert(a[i][j] == b[i][j]); let cycle = false; for (let i = 1; i <= N; ++i) if (a[i][i] < 0) cycle = true; if (cycle) { console.log(`${filename}: contains a negative cycle`); return; } var best = Infinity; for (row of a) best = Math.min(best, ...row); console.log(`${filename}: ${best}`); }; run('problem18.8test1.txt'); // problem18.8test1.txt: -2 run('problem18.8test2.txt'); // problem18.8test2.txt: contains a negative cycle run('problem18.8file1.txt'); // problem18.8file1.txt: contains a negative cycle run('problem18.8file2.txt'); // problem18.8file2.txt: contains a negative cycle run('problem18.8file3.txt'); // problem18.8file3.txt: -19 // run('problem18.8file4.txt'); ================================================ FILE: floyd_warshall/main.kt ================================================ /* * Programming Problem 18.8: All-Pairs Shortest Paths * * In this problem, each file describes a directed graph. The first line of the file indicates the number * of vertices and edges, respectively. Each subsequent line describes an edge (the first two numbers are its tail * and head, respectively) and its length (the third number). NOTE: edge lengths might be negative, and the graphs * may or may not be negative cycles. * * Test cases: (contributed by Matt Davis) What is the shortest shortest path in the graphs described in this file * and in this file? (I.e., the minimum, over choices of an origin s and a destination t, of a shortest s-t path in the graph.) * (If the graph has a negative cycle, your algorithm should detect that fact.) (Answer: -2 and "contains a negative cycle," respectively.) * * Challenge data set: Repeat the previous problem for the graphs described in the following files: graph #1, graph #2, graph #3, and graph #4. */ import java.io.File var key = { i: Int, j: Int -> "$i,$j" } var INF = (1e9 + 7).toInt() fun floyd_warshall(E: MutableMap, N: Int): Array { var dp = Array(N + 1) { Array(N + 1) { IntArray(N + 1) { INF } } } for (i in 0..N) for (j in 0..N) if (i == j) dp[0][i][j] = 0 else if (E.contains(key(i, j))) dp[0][i][j] = E[key(i, j)]!! for (k in 1..N) for (i in 1..N) for (j in 1..N) dp[k][i][j] = Math.min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]) return dp[N] } fun floyd_warshall_memopt(E: MutableMap, N: Int): Array { var pre = Array(N + 1) { IntArray(N + 1) { INF } } for (i in 0..N) for (j in 0..N) if (i == j) pre[i][j] = 0 else if (E.contains(key(i, j))) pre[i][j] = E[key(i, j)]!! for (k in 1..N) { var cur = Array(N + 1) { IntArray(N + 1) { INF } } for (i in 1..N) for (j in 1..N) cur[i][j] = Math.min(pre[i][j], pre[i][k] + pre[k][j]) pre = cur.also{ cur = pre } } return pre } fun run(filename: String) { var N = 0 var E = mutableMapOf() var first = true File(filename).forEachLine { if (!first) { var (u, v, w) = it.trim().split(" ").map{ it.toInt() } E[key(u, v)] = w } else { N = it.trim().split(" ").map{ it.toInt() }[0] first = false } } var a = floyd_warshall_memopt(E, N) // var b = floyd_warshall(E, N) // for (i in 1..N) // for (j in 1..N) // assert(a[i][j] == b[i][j]) // 💩 sanity check var cycle = false for (i in 1..N) if (a[i][i] < 0) cycle = true if (cycle) { println("$filename: contains a negative cycle") return } var best = INF for (i in 1..N) for (j in 1..N) best = Math.min(best, a[i][j]) println("$filename: $best") } fun main() { run("problem18.8test1.txt"); // problem18.8test1.txt: -2 run("problem18.8test2.txt"); // problem18.8test2.txt: contains a negative cycle run("problem18.8file1.txt"); // problem18.8file1.txt: contains a negative cycle run("problem18.8file2.txt"); // problem18.8file2.txt: contains a negative cycle run("problem18.8file3.txt"); // problem18.8file3.txt: -19 // run("problem18.8file4.txt"); } ================================================ FILE: floyd_warshall/main.py ================================================ # # Programming Problem 18.8: All-Pairs Shortest Paths # # In this problem, each file describes a directed graph. The first line of the file indicates the number # of vertices and edges, respectively. Each subsequent line describes an edge (the first two numbers are its tail # and head, respectively) and its length (the third number). NOTE: edge lengths might be negative, and the graphs # may or may not be negative cycles. # # Test cases: (contributed by Matt Davis) What is the shortest shortest path in the graphs described in this file # and in this file? (I.e., the minimum, over choices of an origin s and a destination t, of a shortest s-t path in the graph.) # (If the graph has a negative cycle, your algorithm should detect that fact.) (Answer: -2 and "contains a negative cycle," respectively.) # # Challenge data set: Repeat the previous problem for the graphs described in the following files: graph #1, graph #2, graph #3, and graph #4. # key = lambda i, j: f'{i},{j}' def floyd_warshall(E, N): dp = [[[float('inf')] * (N + 1) for _ in range(N + 1)] for _ in range(N + 1)] for i in range(1, N + 1): for j in range(1, N + 1): if i == j: dp[0][i][j] = 0 elif key(i, j) in E: dp[0][i][j] = E[key(i, j)] for k in range(1, N + 1): for i in range(1, N + 1): for j in range(1, N + 1): dp[k][i][j] = min(dp[k - 1][i][j], dp[k - 1][i][k] + dp[k - 1][k][j]) return dp[N] def floyd_warshall_memopt(E, N): pre = [[float('inf')] * (N + 1) for _ in range(N + 1)] for i in range(1, N + 1): for j in range(1, N + 1): if i == j: pre[i][j] = 0 elif key(i, j) in E: pre[i][j] = E[key(i, j)] for k in range(1, N + 1): cur = [[float('inf')] * (N + 1) for _ in range(N + 1)] for i in range(1, N + 1): for j in range(1, N + 1): cur[i][j] = min(pre[i][j], pre[i][k] + pre[k][j]) pre, cur = cur, pre return pre def run(filename): E = {} N = 0 first = True with open(filename) as fin: while True: line = fin.readline() if not line: break if not first: u, v, w = [int(x) for x in line.strip().split(' ')] E[key(u, v)] = w else: N = [int(x) for x in line.strip().split(' ')][0] first = False a = floyd_warshall_memopt(E, N) # b = floyd_warshall(E, N) # for i in range(1, N + 1): # for j in range(1, N + 1): # assert(a[i][j] == b[i][j]) # 💩 sanity check cycle = False for i in range(1, N + 1): if a[i][i] < 0: cycle = True if cycle: print(f'{filename}: contains a negative cycle') return best = float('inf') for row in a: best = min(best, *row) print(f'{filename}: {best}') run('problem18.8test1.txt') # problem18.8test1.txt: -2 run('problem18.8test2.txt') # problem18.8test2.txt: contains a negative cycle run('problem18.8file1.txt') # problem18.8file1.txt: contains a negative cycle run('problem18.8file2.txt') # problem18.8file2.txt: contains a negative cycle run('problem18.8file3.txt') # problem18.8file3.txt: -19 # run('problem18.8file4.txt') ================================================ FILE: floyd_warshall/package.json ================================================ { "name": "floyd_warshall", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "clayton wong", "license": "ISC", "dependencies": { "assert": "^2.0.0", "n-readlines": "^1.0.1" } } ================================================ FILE: floyd_warshall/problem18.8file1.txt ================================================ 1000 47978 1 14 6 1 25 47 1 70 22 1 82 31 1 98 17 1 134 39 1 146 7 1 189 44 1 192 20 1 261 38 1 283 30 1 340 21 1 380 37 1 381 24 1 403 3 1 422 40 1 423 40 1 425 13 1 518 23 1 520 38 1 547 5 1 558 24 1 578 32 1 601 42 1 604 37 1 640 38 1 648 18 1 699 48 1 707 10 1 712 9 1 723 47 1 729 26 1 730 18 1 769 49 1 793 32 1 830 15 1 841 50 1 852 16 1 874 39 1 880 9 1 890 6 1 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41 998 755 30 998 768 32 998 790 21 998 804 37 998 807 50 998 811 27 998 836 -2 998 888 35 998 890 9 998 930 3 998 949 13 998 972 10 998 975 41 998 982 32 998 988 19 999 51 9 999 107 45 999 122 15 999 151 33 999 155 34 999 166 50 999 209 41 999 221 11 999 239 14 999 255 25 999 259 16 999 267 12 999 269 30 999 294 29 999 301 32 999 316 9 999 370 37 999 373 21 999 379 9 999 386 15 999 391 10 999 454 10 999 471 20 999 490 17 999 498 49 999 512 19 999 553 41 999 559 3 999 584 48 999 585 17 999 625 48 999 629 18 999 631 9 999 639 43 999 667 46 999 702 43 999 703 33 999 731 45 999 732 50 999 787 50 999 792 39 999 822 39 999 917 18 999 926 26 999 937 31 999 995 12 1000 3 42 1000 9 45 1000 37 39 1000 40 17 1000 183 24 1000 217 31 1000 231 49 1000 234 10 1000 252 20 1000 321 25 1000 329 19 1000 339 49 1000 398 42 1000 401 27 1000 451 40 1000 496 35 1000 506 24 1000 527 21 1000 541 12 1000 559 22 1000 560 25 1000 642 49 1000 718 14 1000 772 1 1000 786 43 1000 847 46 1000 852 48 1000 864 44 1000 874 37 1000 926 48 1000 946 39 1000 951 16 1000 958 23 1000 961 18 1000 977 33 1000 979 1 ================================================ FILE: floyd_warshall/problem18.8file4.txt ================================================ [File too large to display: 13.2 MB] ================================================ FILE: floyd_warshall/problem18.8test1.txt ================================================ 5 8 1 2 2 1 5 3 2 4 -2 3 1 1 4 1 4 4 3 1 4 5 2 5 3 -1 ================================================ FILE: floyd_warshall/problem18.8test2.txt ================================================ 5 8 1 2 2 1 5 3 2 4 -2 3 1 1 4 1 4 4 3 1 4 5 -1 5 3 -1 ================================================ FILE: greedy_scheduling/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(greedy_scheduling) set(CMAKE_CXX_STANDARD 17) add_executable(greedy_scheduling main.cpp) ================================================ FILE: greedy_scheduling/main.cpp ================================================ #include #include #include #include using namespace std; using LL = long long; struct Job { LL weight, length; Job(LL weight, LL length) : weight{ weight }, length{ length } {} }; using Jobs = vector; class Solution { public: using Pair = pair; // sub-optimal, optimal Pair minSum(Jobs& jobs) { auto diff = [](auto& a, auto& b) { auto first = a.weight - a.length, second = b.weight - b.length; return first == second ? b.weight < a.weight : second < first; // sort by descending difference, break ties in favor of jobs with larger weights }; auto ratio = [](auto& a, auto& b) { auto first = double(a.weight) / a.length, second = double(b.weight) / b.length; return first == second ? b.weight < a.weight : second < first; // sort by descending ratio, break ties in favor of jobs with larger weights }; return { calcSum(jobs, diff), calcSum(jobs, ratio) }; } private: template LL calcSum(Jobs& jobs, Comp comp, LL time = 0LL) { sort(jobs.begin(), jobs.end(), comp); return accumulate(jobs.begin(), jobs.end(), 0LL, [&](LL total, auto& job) { return total += job.weight * (time += job.length); }); } }; void run(const string& filename) { Jobs jobs; LL N, weight, length; fstream fin{ filename }; for (fin >> N; fin >> weight >> length; jobs.emplace_back(Job{ weight, length })); auto [diff, ratio] = Solution().minSum(jobs); cout << diff << ", " << ratio << endl; } int main() { run("problem13.4test1.txt"); // 23, 22 run("problem13.4test2.txt"); // 68615, 67247 run("problem13.4.txt"); // 69119377652, 67311454237 return 0; } ================================================ FILE: greedy_scheduling/main.js ================================================ let LineByLine = require('n-readlines'); class Job { constructor(weight, length) { this.weight = weight; this.length = length; } } class Solution { minSum(jobs) { let diff = (a, b) => { let first = a.weight - a.length, second = b.weight - b.length; return first == second ? b.weight - a.weight : second - first; // sort by descending difference, break ties in favor of jobs with larger weights }; let ratio = (a, b) => { let first = a.weight / a.length, second = b.weight / b.length; return first == second ? b.weight - a.weight : second - first; // sort by descending ratio, break ties in favor of jobs with larger weights }; return [ this._calcSum(jobs, diff), this._calcSum(jobs, ratio) ]; } _calcSum(jobs, comp, time = 0) { jobs.sort((a, b) => comp(a, b)); return jobs.reduce((total, job) => total + job.weight * (time += job.length), 0); } } let run = filename => { let jobs = []; let input = new LineByLine(filename); let line = input.next(); // N while (line = input.next()) { let words = String.fromCharCode(...line).trim().split(' '); let [weight, length] = words.map(Number); jobs.push(new Job(weight, length)); } let [diff, ratio] = new Solution().minSum(jobs); console.log(`${diff}, ${ratio}`); // sub-optimal, optimal }; run('problem13.4test1.txt'); // 23, 22 run('problem13.4test2.txt'); // 68615, 67247 run('problem13.4.txt'); // 69119377652, 67311454237 ================================================ FILE: greedy_scheduling/main.kt ================================================ import java.io.File data class Job(val weight: Long, val length: Long) class Solution { fun minSum(jobs: Array): Pair { class Diff: Comparator { override fun compare(a: Job?, b: Job?): Int { if (a == null || b == null) return 0 var first = a.weight - a.length var second = b.weight - b.length return if (first == second) b.weight.compareTo(a.weight) else second.compareTo(first) // sort by descending difference, break ties in favor of jobs with larger weights } } class Ratio: Comparator { override fun compare(a: Job?, b: Job?): Int { if (a == null || b == null) return 0 var first = a.weight.toDouble() / a.length var second = b.weight.toDouble() / b.length return if (first == second) b.weight.compareTo(a.weight) else second.compareTo(first) // sort by descending difference, break ties in favor of jobs with larger weights } } return Pair(calcSum(jobs, Diff()), calcSum(jobs, Ratio())) } private fun calcSum(jobs: Array, comp: Comparator): Long { jobs.sortWith(comp) var time: Long = 0 var total: Long = 0 jobs.forEach { job -> time += job.length total += job.weight * time } return total } } fun run(filename: String) { var jobs = mutableListOf() var first = true File(filename).forEachLine { if (!first) { var words = it.trim().split(" ").map{ it.toLong() } var (weight, length) = words jobs.add(Job(weight, length)) } else { first = false } } var (diff, ratio) = Solution().minSum(jobs.toTypedArray()) println("$diff, $ratio") // sub-optimal, optimal } fun main() { run("problem13.4test1.txt") // 23, 22 run("problem13.4test2.txt") // 68615, 67247 run("problem13.4.txt") // 69119377652, 67311454237 } ================================================ FILE: greedy_scheduling/main.py ================================================ from functools import cmp_to_key class Job: def __init__(self, weight, length): self.weight = weight self.length = length class Solution: def minSum(self, jobs): def diff(a, b): first = a.weight - a.length second = b.weight - b.length return b.weight - a.weight if first == second else second - first # sort by descending difference, break ties in favor of jobs with larger weights def ratio(a, b): first = a.weight / a.length second = b.weight / b.length return b.weight - a.weight if first == second else second - first # sort by descending difference, break ties in favor of jobs with larger weights return [ self._calcSum(jobs, diff), self._calcSum(jobs, ratio) ] def _calcSum(self, jobs, comp, time = 0, total = 0): jobs.sort(key = cmp_to_key(lambda a, b: comp(a, b))) for job in jobs: time += job.length total += job.weight * time return total def run(filename): jobs = [] with open(filename) as fin: line = fin.readline() # N while True: line = fin.readline().strip() if not line: break words = line.split() weight, length = [int(x) for x in words] jobs.append(Job(weight, length)) diff, ratio = Solution().minSum(jobs) print(f'{diff}, {ratio}') # sub-optimal, optimal run('problem13.4test1.txt') # 23, 22 run('problem13.4test2.txt') # 68615, 67247 run('problem13.4.txt') # 69119377652, 67311454237 ================================================ FILE: greedy_scheduling/package.json ================================================ { "name": "greedy_scheduling", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "Clayton Wong", "license": "ISC", "dependencies": { "n-readlines": "^1.0.1" } } ================================================ FILE: greedy_scheduling/problem13.4.txt ================================================ 10000 8 50 74 59 31 73 45 79 24 10 41 66 93 43 88 4 28 30 41 13 4 70 10 58 61 34 100 79 17 36 98 27 13 68 11 34 80 50 80 22 68 73 94 37 86 46 29 92 95 58 2 54 9 45 69 91 25 97 31 4 23 67 50 25 2 54 78 9 29 34 99 82 36 14 66 15 64 37 26 70 16 95 30 2 18 96 6 5 52 99 89 24 6 83 53 67 17 38 39 45 2 98 72 29 38 59 78 98 95 5 10 32 46 76 36 99 43 100 69 13 61 58 95 9 96 69 14 31 7 63 43 66 83 53 68 22 96 13 72 2 91 32 39 58 17 91 41 80 36 7 73 99 96 20 55 24 90 61 6 27 24 7 14 71 39 95 21 45 67 35 27 95 64 39 45 91 51 60 24 48 86 18 73 40 48 86 97 86 24 21 45 69 36 16 26 35 43 12 80 53 45 96 19 97 93 16 92 34 70 98 54 48 26 11 63 12 9 78 62 26 36 69 96 77 68 40 75 32 57 2 73 61 95 85 56 90 8 16 94 70 81 56 56 7 64 3 77 9 50 32 45 39 9 98 52 50 4 32 32 15 20 76 10 81 30 24 55 61 38 46 18 26 1 57 65 98 49 5 51 51 77 13 55 98 5 82 49 66 79 100 10 30 54 84 48 8 74 23 6 77 54 25 31 67 1 45 71 86 17 99 56 34 58 77 79 67 58 12 79 15 20 38 34 92 21 63 34 98 32 89 90 74 78 5 59 1 55 61 16 48 81 31 56 47 8 39 1 27 36 58 14 15 94 61 55 19 58 86 30 16 64 3 18 44 20 68 48 70 96 4 74 4 49 86 59 60 7 31 25 33 74 4 98 21 51 32 81 4 1 21 34 59 4 77 51 35 81 80 33 75 50 43 50 72 11 80 84 71 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61 13 92 50 29 21 49 25 12 70 95 98 22 8 53 52 25 76 20 6 3 76 64 77 61 64 86 28 87 62 86 85 57 27 55 99 77 95 60 89 60 99 21 26 15 58 9 27 69 16 86 2 49 81 53 2 91 45 61 40 89 60 56 75 61 50 40 94 74 31 1 7 72 90 62 55 13 24 54 80 89 65 33 22 81 34 93 39 22 74 54 99 55 90 94 11 43 74 86 49 50 58 58 59 89 57 74 74 7 94 56 47 9 68 59 49 4 59 44 81 50 89 97 50 43 23 10 54 24 27 89 33 7 34 88 75 70 67 62 88 57 10 74 7 16 15 43 94 45 93 99 87 79 66 38 38 19 22 49 63 21 8 98 19 50 38 47 86 29 45 92 64 62 75 1 91 4 35 43 15 71 10 19 27 59 9 100 90 12 43 33 75 41 73 47 29 48 57 17 82 55 65 79 12 80 100 23 96 11 30 94 32 60 24 73 69 71 66 76 68 84 5 29 56 84 21 66 85 47 34 78 98 4 99 59 100 55 62 47 39 15 22 17 18 78 60 70 80 73 11 75 76 68 56 90 57 92 5 36 19 8 61 48 49 69 51 8 43 48 79 99 54 54 75 86 58 52 78 67 58 100 44 46 69 38 93 16 95 69 34 63 47 71 48 68 8 19 62 87 74 27 88 54 56 85 93 71 63 31 38 21 7 27 15 29 53 31 80 28 39 91 67 5 35 67 26 24 32 5 10 99 45 76 75 60 57 89 30 31 85 86 95 59 90 89 42 83 77 69 87 38 1 69 50 30 60 46 8 2 96 34 94 40 99 66 99 78 65 81 8 37 81 22 32 74 31 13 15 74 7 8 18 1 97 17 83 54 100 63 86 78 17 36 28 64 51 100 9 14 92 8 23 78 26 32 63 10 84 41 55 4 23 43 72 38 92 83 25 56 99 100 47 100 26 24 17 24 34 32 10 22 99 71 98 96 59 99 74 2 75 70 54 24 36 75 15 75 93 98 82 52 9 42 82 38 3 5 20 70 19 56 92 96 8 48 66 77 66 30 13 96 24 75 74 8 70 99 63 99 16 72 93 24 3 15 86 55 4 90 55 69 52 22 37 79 75 51 60 42 33 47 9 50 84 5 32 35 62 53 97 70 51 88 3 33 61 60 65 40 68 100 96 7 44 54 100 79 2 79 90 11 99 51 66 10 91 44 16 2 46 44 8 28 9 70 56 20 25 13 53 70 38 43 23 37 70 97 59 50 48 14 75 79 59 75 65 38 73 18 37 19 30 94 52 61 20 57 76 100 37 11 39 50 28 34 78 28 14 98 55 70 39 96 76 47 59 40 86 21 14 36 98 86 6 59 45 80 99 39 61 2 23 30 51 93 96 84 71 36 36 72 4 89 64 8 89 8 58 79 53 80 47 10 42 63 14 49 18 55 39 96 47 42 26 90 6 60 52 42 85 28 17 68 9 10 33 45 11 2 53 74 33 36 16 21 27 57 89 55 40 9 96 61 2 44 97 91 90 7 69 84 4 9 62 46 50 6 49 93 33 66 23 30 75 5 27 83 11 19 18 14 89 87 45 56 48 34 54 11 82 6 77 95 44 16 27 73 53 71 83 92 97 34 63 25 38 57 88 51 40 69 25 85 17 15 35 91 23 50 49 99 64 17 74 74 9 95 9 37 53 60 54 56 63 4 38 68 15 ================================================ FILE: greedy_scheduling/problem13.4test1.txt ================================================ 2 3 5 1 2 ================================================ FILE: greedy_scheduling/problem13.4test2.txt ================================================ 12 8 50 74 59 31 73 45 79 24 10 41 66 93 43 88 4 28 30 41 13 4 70 10 58 ================================================ FILE: huffman/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(huffman) set(CMAKE_CXX_STANDARD 17) add_executable(huffman main.cpp) ================================================ FILE: huffman/main.cpp ================================================ /* * Programming Problem 14.6: Huffman Codes * * In this problem the file format is: * [number_of_symbols] * [weight of symbol #1] * [weight of symbol #2] * ... */ #include #include #include #include #include #define PRIORITY_QUEUE // O(N * logN) #ifndef PRIORITY_QUEUE #define TWO_QUEUES // O(N) #endif using namespace std; using LL = long long; using Weight = LL; using Weights = vector; struct Tree; using TreePtr = shared_ptr; struct Tree { Weight weight; TreePtr left, right; Tree(Weight weight, TreePtr left = nullptr, TreePtr right = nullptr) : weight{ weight }, left{ left }, right{ right } {} }; using TreePtrs = vector; #ifdef PRIORITY_QUEUE struct Comp { size_t operator()(const TreePtr& a, const TreePtr& b) const { return b->weight < a->weight; } }; using Queue = priority_queue; TreePtr encode(const Weights& A, Queue q = {}) { for (auto weight: A) q.emplace(make_shared(weight)); while (1 < q.size()) { auto a = q.top(); q.pop(); auto b = q.top(); q.pop(); auto c = make_shared(a->weight + b->weight, a, b); q.emplace(c); } return q.top(); } #else // TWO_QUEUES /* * Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation * of a sorting subroutine, followed by a linear amount of additional work. */ using Queue = queue; TreePtr encode(Weights& A, Queue first = {}, Queue second = {}) { sort(A.begin(), A.end()); for (auto weight: A) first.push(make_shared(weight)); TreePtrs next; auto takeFirst = [&]() { next.push_back(first.front()), first.pop(); }; auto takeSecond = [&]() { next.push_back(second.front()), second.pop(); }; while (1 < first.size() + second.size()) { next.clear(); do { if (first.size() && second.size()) { if (first.front()->weight < second.front()->weight) takeFirst(); else takeSecond(); } else if (first.size()) takeFirst(); else if (second.size()) takeSecond(); } while (next.size() < 2); auto [a, b] = tie(next[0], next[1]); auto c = make_shared(a->weight + b->weight, a, b); second.emplace(c); } return second.front(); } #endif using MinMax = pair; constexpr auto Min = numeric_limits::min(); constexpr auto Max = numeric_limits::max(); MinMax run(const string& filename) { Weights A; // weight of each symbol fstream fin{ filename }; LL N, weight; for (fin >> N; fin >> weight; A.push_back(weight)); auto tree = encode(A); LL lo = Max, hi = Min; using fun = function; fun go = [&](auto root, LL depth) { if (!root) return; auto isLeaf = [](auto root) { return !root->left && !root->right; }; if (isLeaf(root)) lo = min(lo, depth), hi = max(hi, depth); else go(root->left, depth + 1), go(root->right, depth + 1); }; go(tree, 0); return make_pair(lo, hi); } int main() { for (auto& filename: { "problem14.6test1.txt", "problem14.6test2.txt", "problem14.6.txt" }) { auto [lo, hi] = run(filename); cout << filename << ": " << lo << ", " << hi << endl; // min, max encoding length in the corresponding optimal prefix-free tree } // problem14.6test1.txt: 2, 5 // problem14.6test2.txt: 3, 6 // problem14.6.txt: 9, 19 return 0; } ================================================ FILE: huffman/main.js ================================================ /* * Programming Problem 14.6: Huffman Codes * * In this problem the file format is: * [number_of_symbols] * [weight of symbol #1] * [weight of symbol #2] * ... */ let LineByLine = require('n-readlines'); class Tree { constructor(weight, left = null, right = null) { this.weight = weight; this.left = left; this.right = right; } } /* let key = x => Array.isArray(x) ? x[0] : x; let heappush = (A, x, f = Math.min) => { let P = i => Math.floor((i - 1) / 2); // parent A.push(x); let N = A.length, i = N - 1; while (0 < i && key(A[i]) == f(key(A[i]), key(A[P(i)]))) { [A[i], A[P(i)]] = [A[P(i)], A[i]]; i = P(i); } }; let heappop = (A, f = Math.min) => { let L = i => 2 * i + 1, // children R = i => 2 * i + 2; let N = A.length, i = 0; let top = A[0]; [A[0], A[N - 1]] = [A[N - 1], A[0]], A.pop(), --N; let ok; do { ok = true; let left = f == Math.min ? Infinity : -Infinity, right = left; if (L(i) < N && key(A[i]) != f(key(A[i]), key(A[L(i)]))) ok = false, left = key(A[L(i)]); if (R(i) < N && key(A[i]) != f(key(A[i]), key(A[R(i)]))) ok = false, right = key(A[R(i)]); if (!ok) { let j = left == f(left, right) ? L(i) : R(i); [A[i], A[j]] = [A[j], A[i]]; i = j; } } while (!ok); return top; }; let encode = A => { let T = []; for (let weight of A) heappush(T, [ weight, new Tree(weight) ]); while (1 < T.length) { let [ a, b ] = [ heappop(T), heappop(T) ]; let c = [ a[0] + b[0], new Tree(a[0] + b[0], a[1], b[1]) ]; heappush(T, c); } return T[0][1]; }; */ /* * Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation * of a sorting subroutine, followed by a linear amount of additional work. */ let encode = A => { A.sort((a, b) => a - b) let first = A.map(weight => new Tree(weight)), second = []; while (1 < first.length + second.length) { let next = []; while (next.length < 2) { if (first.length && second.length) { next.push(first[0].weight < second[0].weight ? first.shift() : second.shift()); } else if (first.length) next.push(first.shift()); else if (second.length) next.push(second.shift()); } let [a, b] = next; let c = new Tree(a.weight + b.weight, a, b); second.push(c); } return second.shift(); }; let run = filename => { let A = []; let input = new LineByLine(filename); let line; line = input.next(); // N while (line = input.next()) { let weight = Number(String.fromCharCode(...line).trim()); A.push(weight); } let tree = encode(A); let [lo, hi] = [Infinity, -Infinity]; let go = (root = tree, depth = 0) => { if (!root) return; let isLeaf = root => !root.left && !root.right; if (isLeaf(root)) lo = Math.min(lo, depth), hi = Math.max(hi, depth); else go(root.left, depth + 1), go(root.right, depth + 1); }; go(); return [ lo, hi ]; } for (let filename of [ 'problem14.6test1.txt', 'problem14.6test2.txt', 'problem14.6.txt' ]) { let [lo, hi] = run(filename); console.log(`${filename}: ${lo}, ${hi}`); // min, max encoding length in the corresponding optimal prefix-free tree } // problem14.6test1.txt: 2, 5 // problem14.6test2.txt: 3, 6 // problem14.6.txt: 9, 19 ================================================ FILE: huffman/main.kt ================================================ /* * Programming Problem 14.6: Huffman Codes * * In this problem the file format is: * [number_of_symbols] * [weight of symbol #1] * [weight of symbol #2] * ... */ import java.io.File import java.util.PriorityQueue import java.util.Queue import java.util.LinkedList var INF = (1e9 + 7).toInt() data class Tree(val weight: Int, val left: Tree? = null, val right: Tree? = null) /* fun encode(A: List): Tree { var q = PriorityQueue(Comparator{ a: Tree, b: Tree -> a.weight.compareTo(b.weight) }) for (weight in A) q.add(Tree(weight)) while (1 < q.size) { var a = q.poll() var b = q.poll() var c = Tree(a.weight + b.weight, a, b) q.add(c) } return q.poll() } */ /* * Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation * of a sorting subroutine, followed by a linear amount of additional work. */ fun encode(A: MutableList): Tree { A.sort() var first: Queue = LinkedList(A.map{ weight -> Tree(weight) }.toList()) var second: Queue = LinkedList() var next = mutableListOf() while (1 < first.size + second.size) { next.clear() do { if (0 < first.size && 0 < second.size) { if (first.peek().weight < second.peek().weight) next.add(first.poll()) else next.add(second.poll()) } else if (0 < first.size) next.add(first.poll()) else if (0 < second.size) next.add(second.poll()) } while (next.size < 2) var (a, b) = next var c = Tree(a.weight + b.weight, a, b) second.add(c) } return second.poll() } fun run(filename: String): Pair { var A = mutableListOf() var first = true File(filename).forEachLine { if (!first) { var weight = it.trim().toInt() A.add(weight) } else { first = false } } var tree = encode(A.toMutableList()) var lo = INF var hi = -INF fun go(root: Tree? = tree, depth: Int = 0) { if (root == null) return var isLeaf = { node: Tree? -> node?.left == null && node?.right == null } if (isLeaf(root)) { lo = Math.min(lo, depth) hi = Math.max(hi, depth) } else { go(root.left, depth + 1) go(root.right, depth + 1) } } go() return Pair(lo, hi) } fun main() { for (filename in listOf("problem14.6test1.txt", "problem14.6test2.txt", "problem14.6.txt")) { var (lo, hi) = run(filename) println("$filename: $lo, $hi") // min, max encoding length in the corresponding optimal prefix-free tree } } // problem14.6test1.txt: 2, 5 // problem14.6test2.txt: 3, 6 // problem14.6.txt: 9, 19 ================================================ FILE: huffman/main.py ================================================ # # Programming Problem 14.6: Huffman Codes # # In this problem the file format is: # [number_of_symbols] # [weight of symbol #1] # [weight of symbol #2] # ... # class Tree: def __init__(self, weight, left = None, right = None): self.weight = weight self.left = left self.right = right def __lt__(self, other): return self.weight < other.weight # # priority queue # # from heapq import heappush # from heapq import heappop # def encode(A): # T = [] # for weight in A: # heappush(T, Tree(weight)) # while 1 < len(T): # a, b = heappop(T), heappop(T) # c = Tree(a.weight + b.weight, a, b) # heappush(T, c) # return heappop(T) # # Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation # of a sorting subroutine, followed by a linear amount of additional work. # from collections import deque def encode(A): A.sort() first, second = deque([Tree(weight) for weight in A]), deque() while 1 < len(first) + len(second): next = [] while len(next) < 2: if len(first) and len(second): next.append(first.popleft() if first[0].weight < second[0].weight else second.popleft()) elif len(first): next.append(first.popleft()) elif len(second): next.append(second.popleft()) a, b = next c = Tree(a.weight + b.weight, a, b) second.append(c) return second.popleft() def run(filename): A = [] with open(filename) as fin: N = int(fin.readline()) while True: line = fin.readline() if not line: break weight = int(line.strip()) A.append(weight) tree = encode(A) lo, hi = float('inf'), float('-inf') def go(root = tree, depth = 0): nonlocal lo, hi if not root: return isLeaf = lambda root: not root.left and not root.right if isLeaf(root): lo = min(lo, depth) hi = max(hi, depth) else: go(root.left, depth + 1) go(root.right, depth + 1) go() return [ lo, hi ] for filename in [ 'problem14.6test1.txt', 'problem14.6test2.txt', 'problem14.6.txt' ]: lo, hi = run(filename) print(f'{filename}: {lo}, {hi}') # min, max encoding length in the corresponding optimal prefix-free tree # problem14.6test1.txt: 2, 5 # problem14.6test2.txt: 3, 6 # problem14.6.txt: 9, 19 ================================================ FILE: huffman/package.json ================================================ { "name": "huffman", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "clayton wong", "license": "ISC", "dependencies": { "n-readlines": "^1.0.1" } } ================================================ FILE: huffman/problem14.6.txt ================================================ 1000 7540662 6852892 3235725 8045172 2667794 2595511 7030103 5882478 2731795 8630567 7224817 9147454 9180184 4194220 801991 8773930 7498447 5429618 1948993 4161112 7231836 3512965 6037327 8518300 2917342 547194 5015100 837907 341970 6249282 9353243 5546257 7031847 9959436 1082955 7132656 9863608 2548250 2209647 8069760 8572628 9344483 8874074 8638786 7812182 4849731 2492922 6698031 7404507 745731 8379593 4119022 4123053 4401250 5421516 8134188 8319394 9611963 3780734 6096612 6971484 7377399 6529211 6097359 6332343 5781826 279273 5153724 838231 4437902 3398506 3432892 2868587 3079930 6449351 7063764 9110714 2325311 5883213 911693 1779925 8305333 3636477 5712230 1858307 9079534 8865961 616791 2362911 2935267 3366532 1151019 7813585 6552397 3805014 7693039 795351 5346031 5699199 5297037 8236308 2817584 4715257 4633814 9478541 9933557 789849 3943007 2440768 615656 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3273528 379629 4483866 6416998 7966718 7807042 6305964 4966440 3210575 5630654 8499854 2207795 4094756 7364964 899516 8149244 5225044 7409718 8157209 7662259 7760348 9106972 2648302 1833054 1359669 9978358 9752431 7835150 7039906 4874388 7566994 6755902 9602458 7621071 4888538 7809809 2417608 7939434 7630425 2471915 1991528 2867487 4910075 3667457 2041172 6836791 7482135 9034598 5676907 107416 9568704 6633667 6493718 8509618 2403363 4521616 4142717 3825501 5734943 6781157 7043308 2714778 2562210 4467454 3294809 57802 6843026 616277 3807379 9353508 3101183 1360510 8348712 9455063 4007404 2990314 4679226 3439397 2400452 3794129 2600105 1850113 2659662 3127710 849244 1161427 1119946 8386556 8696999 5684375 4978411 6673706 9201552 8456014 7509368 3452778 8664366 8113437 8983823 652742 2057013 5920285 8530242 4010767 3315346 2524081 651730 341040 9448969 83429 2642101 1433786 3649759 555199 5474772 8325637 2263107 9288002 2097012 4012152 9496180 7599357 3119784 6849327 7368417 3918571 5019143 7816600 3698908 9386292 506213 1687743 169883 4473394 5880820 3370978 8525526 7720062 6094764 6975371 9034588 1049472 7472233 8616597 1331899 6725820 505108 5111362 1803917 3482963 3605378 5407258 2625367 1465096 2464278 758393 1121696 1968038 6320843 2917881 7807840 2199230 2691670 7583934 5029628 3055273 5523150 2603228 5666836 4305906 7922171 9787624 3164901 8298150 3720363 8155910 5145168 545323 2847654 3121259 7952235 829684 1456967 6839437 7253416 535893 3020761 3333465 3851724 7438085 9679393 1155576 5070284 6891277 8752837 8278436 8723147 5467399 1256676 794517 7395024 6524438 6111922 4388265 8132181 3389703 8089365 9807139 7057278 8063236 7392465 304460 9879636 6993002 2860509 810943 8727902 6823970 5760205 8982619 1189333 996638 1990671 394075 6710898 2970241 4769175 705755 8960359 7919966 9844247 4457249 6305295 3384562 9311730 3696071 569367 8026529 9018656 8786437 2822849 2612451 683042 1573997 1246625 4810066 4034299 6398176 8102279 6927406 4128359 3912623 4172411 1280821 4502892 8938413 3101492 4503602 1783438 6081576 7492446 6859568 6592440 37164 8126217 4756682 7950114 159892 4227759 8409663 7049520 9553411 7991122 5404616 9273380 5784690 8079351 8558865 5545395 9572785 4647771 2388923 2925254 3882138 9879739 2654229 7803982 1062936 4082368 2165077 3042587 9968891 7262102 1385141 2525180 8501245 8452218 1851594 3527804 8873317 3087832 9582323 4555928 5802025 4306883 9407054 1531412 6514736 1116902 12710 3310966 8556302 715468 6235973 4604649 133947 405875 5439070 4206464 5313145 9218187 356432 5883301 9827097 6358863 7705902 1774454 3446979 247376 7127672 771293 1017408 3591569 4932798 5856066 3079629 7487083 6682658 5442652 5734430 4286579 1614167 9061552 7636648 6215627 4503170 9236790 8521382 7749368 5247205 1358976 6368177 2272567 8007799 8902975 6620440 8001611 934209 2180122 192218 8075767 5278246 3030695 9357701 1528193 6081372 4032029 7152249 8534425 6093548 1219060 9407116 931792 3580190 8788289 5694987 7424854 3671960 61282 3572962 1747121 2258158 4214951 9018342 7765611 9706446 2655517 9916957 5112225 141897 1339459 440233 3011010 9772019 4905242 1109382 1309337 2587005 4536352 7702653 7802187 8006005 6497708 1386887 1064131 1536730 2564491 9911792 7131759 6905830 1659776 3987281 2419919 3882653 6103143 5164890 3665750 7567713 2024175 1350643 7870654 3959682 2346297 715503 8053278 3717635 7003767 3556415 4226994 5833572 3426133 8660808 8566596 6389614 1289868 8655532 192227 201512 1618977 5113351 7114006 515470 7283473 3171828 4819908 1378107 7276455 368071 7156144 3802117 3189072 3208861 160666 8888567 3500210 5135748 8769878 218073 4917463 5767214 5972689 6261982 4140557 732979 9614188 3186821 3257410 4564278 70116 1771529 9454158 3259365 258498 3601052 636731 6993050 6485811 7986006 5499670 1379613 7351266 1797443 149350 3009608 3382161 3371230 497856 6812480 8250599 4939811 4602102 1462877 8447562 1170616 5121988 795887 804116 1281873 191692 7386698 5461863 3896473 5080401 8044456 4844078 4563360 8568320 780907 8850726 8798444 9179767 ================================================ FILE: huffman/problem14.6test1.txt ================================================ 10 37 59 43 27 30 96 96 71 8 76 ================================================ FILE: huffman/problem14.6test2.txt ================================================ 15 895 121 188 953 378 849 153 579 144 727 589 301 442 327 930 ================================================ FILE: karatsuba/main.jl ================================================ # # Karatsuba Multiplication # # Input: two n-digit positive integers x and y # Output: the product x * y # Assumption: n is a power of 2 # using Test, Random function go(x, y) if x < 10 || y < 10 return x * y end n = max(length(string(x)), length(string(y))) if isodd(n) n += 1 end m = 10 ^ div(n, 2) # middle decimal value a, b = div(x, m), mod(x, m) c, d = div(y, m), mod(y, m) p, q = a + b, c + d ac = go(a, c) bd = go(b, d) pq = go(p, q) adbc = pq - ac - bd return 10^n * ac + 10^div(n, 2) * adbc + bd end Random.seed!(123456789) @testset "Karatsuba Multiplication tests" begin for _ in 1:100 n = rand((1, 2, 4, 8, 16)) lo = 10^(n-1) hi = 10^n - 1 x = rand(lo:hi) y = rand(lo:hi) expect, actual = x * y, go(x, y) @test expect == actual println("($x x $y)\nexpect: $expect\nactual: $actual\n") end end @testset "Challenge Problem" begin x = 3141592653589793238462643383279502884197169399375105820974944592 y = 2718281828459045235360287471352662497757247093699959574966967627 expect, actual = x * y, go(x, y) @test expect == actual println("($x x $y)\nexpect: $expect\nactual: $actual\n") end ================================================ FILE: karatsuba/main.py ================================================ def go(x, y): if x < 10 or y < 10: return x * y n = max(len(str(x)), len(str(y))) if n & 1: n += 1 # +1 is n is odd m = 10 ** (n // 2) a, b = x // m, x % m c, d = y // m, y % m p, q = a + b, c + d ac = go(a, c) bd = go(b, d) pq = go(p, q) adbc = pq - ac - bd return 10**n * ac + 10**(n//2) * adbc + bd x = 3141592653589793238462643383279502884197169399375105820974944592 y = 2718281828459045235360287471352662497757247093699959574966967627 assert(x * y == go(x, y)) print(x * y) print(go(x, y)) # 8539734222673567065463550869546574495034888535765114961879601127067743044893204848617875072216249073013374895871952806582723184 # 8539734222673567065463550869546574495034888535765114961879601127067743044893204848617875072216249073013374895871952806582723184 ================================================ FILE: knapsack/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(knapsack) set(CMAKE_CXX_STANDARD 17) add_executable(knapsack main.cpp) ================================================ FILE: knapsack/main.cpp ================================================ #include #include #include #include #include using namespace std; using Pair = pair; // value, weight using Pairs = vector; using fun = function; using Map = unordered_map; int INF = 1e9 + 7; int top_down(Pairs& A, int K, Map m = {}) { auto N = A.size(); fun go = [&](auto i, auto k) { if (i == N) // 🛑 empty set return 0; stringstream key; key << i << "," << k; if (m.find(key.str()) != m.end()) // 🤔 memo return m[key.str()]; auto [value, weight] = A[i]; auto include = 0 <= k - weight ? go(i + 1, k - weight) + value : -INF, // ✅ include A[i] exclude = go(i + 1, k); // 🚫 exclude A[i] return m[key.str()] = max(include, exclude); // 🎯 best }; return go(0, K); } int bottom_up(Pairs& A, int K) { auto N = A.size(); using VI = vector; using VVI = vector; VVI dp(N + 1, VI(K + 1, -INF)); // 🤔 memo for (auto k{ 0 }; k < K; dp[0][k++] = 0); // 🛑 empty set for (auto i{ 1 }; i <= N; ++i) { for (auto k{ 0 }; k <= K; ++k) { auto [value, weight] = A[i - 1]; auto include = 0 <= k - weight ? dp[i - 1][k - weight] +value : -INF, // ✅ include A[i] exclude = dp[i - 1][k]; // 🚫 exclude A[i] dp[i][k] = max(include, exclude); // 🎯 best } } return dp[N][K]; } int bottom_up_memopt(Pairs& A, int K) { auto N = A.size(); using VI = vector; VI pre(K + 1, 0); // 🤔 memo + 🛑 empty set for (auto i{ 1 }; i <= N; ++i) { VI cur(K + 1, -INF); for (auto k{ 0 }; k <= K; ++k) { auto [value, weight] = A[i - 1]; auto include = 0 <= k - weight ? pre[k - weight] +value : -INF, // ✅ include A[i] exclude = pre[k]; // 🚫 exclude A[i] cur[k] = max(include, exclude); // 🎯 best } swap(pre, cur); } return pre[K]; } void run(const string& filename) { Pairs A; fstream fin{ filename }; int K, N; // K capacity, N items fin >> K >> N; for (int value, weight; fin >> value >> weight; A.emplace_back(value, weight)); auto a = top_down(A, K), b = bottom_up(A, K), c = bottom_up_memopt(A, K); assert(a == b && b == c); // 💩 sanity check cout << filename << ": " << a << endl; } int main() { run("problem16.7test.txt"); // problem16.7test.txt: 2493893 return 0; } ================================================ FILE: knapsack/main.js ================================================ const assert = require('assert'); const LineByLine = require('n-readlines'); let top_down = (A, K, m = new Map()) => { let N = A.length; let go = (i = 0, k = K) => { if (i == N) // 🛑 empty set return 0; let key = `${i},${k}`; if (m.has(key)) // 🤔 memo return m.get(key); let [value, weight] = A[i]; let include = 0 <= k - weight ? go(i + 1, k - weight) + value : -Infinity, // ✅ include A[i] exclude = go(i + 1, k); // 🚫 exclude A[i] return m.set(key, Math.max(include, exclude)) // 🎯 best .get(key); }; return go(); }; let bottom_up = (A, K) => { let N = A.length; let dp = [...Array(N + 1)].map(_ => Array(K + 1).fill(-Infinity)); // 🤔 memo for (let k = 0; k < K; dp[0][k++] = 0); // 🛑 empty set for (let i = 1; i <= N; ++i) { for (let k = 0; k <= K; ++k) { let [value, weight] = A[i - 1]; let include = 0 <= k - weight ? dp[i - 1][k - weight] + value : -Infinity, // ✅ include A[i] exclude = dp[i - 1][k]; // 🚫 exclude A[i] dp[i][k] = Math.max(include, exclude); // 🎯 best } } return dp[N][K]; }; let bottom_up_memopt = (A, K) => { let N = A.length; let pre = Array(K + 1).fill(0); // 🤔 memo + 🛑 empty set for (let i = 1; i <= N; ++i) { let cur = Array(K + 1).fill(-Infinity); for (let k = 0; k <= K; ++k) { let [value, weight] = A[i - 1]; let include = 0 <= k - weight ? pre[k - weight] + value : -Infinity, // ✅ include A[i] exclude = pre[k]; // 🚫 exclude A[i] cur[k] = Math.max(include, exclude); // 🎯 best } [pre, cur] = [cur, pre]; } return pre[K]; }; let run = filename => { let A = []; const input = new LineByLine(filename) let [K, N] = input.next().toString().split(' ').map(Number); // K capacity, N items let line; while (line = input.next()) { let [value, weight] = line.toString().split(' ').map(Number); A.push([value, weight]); } let a = top_down(A, K), b = bottom_up(A, K), c = bottom_up_memopt(A, K); assert(a == b && b == c); // 💩 sanity check console.log(`${filename}: ${a}`); }; run('problem16.7test.txt') // problem16.7test.txt: 2493893 ================================================ FILE: knapsack/main.kt ================================================ import java.io.File var INF = (1e9 + 7).toInt() fun top_down(A: List>, K: Int): Int { var N = A.size var m = mutableMapOf() fun go(i: Int = 0, k: Int = K): Int { if (i == N) // 🛑 empty set return 0 var key = "$i,$k" if (m.contains(key)) // 🤔 memo return m[key]!! var (value, weight) = A[i] var include = if (0 <= k - weight) go(i + 1, k - weight) + value else -INF // ✅ include A[i] var exclude = go(i + 1, k) // 🚫 exclude A[i] m[key] = Math.max(include, exclude) // 🎯 best return m[key]!! } return go() } fun bottom_up(A: List>, K: Int): Int { var N = A.size var dp = Array(N + 1){ Array(K + 1){ -INF } } // 🤔 memo for (k in 0..K) // 🛑 empty set dp[0][k] = 0 for (i in 1..N) { for (k in 0..K) { var (value, weight) = A[i - 1] var include = if (0 <= k - weight) dp[i - 1][k - weight] + value else -INF // ✅ include A[i] var exclude = dp[i - 1][k] // 🚫 exclude A[i] dp[i][k] = Math.max(include, exclude) // 🎯 best } } return dp[N][K] } fun bottom_up_memopt(A: List>, K: Int): Int { var N = A.size var pre = Array(K + 1) { 0 } // 🤔 memo + 🛑 empty set for (i in 1..N) { var cur = Array(K + 1) { -INF } for (k in 0..K) { var (value, weight) = A[i - 1] var include = if (0 <= k - weight) pre[k - weight] + value else -INF // ✅ include A[i] var exclude = pre[k] // 🚫 exclude A[i] cur[k] = Math.max(include, exclude) // 🎯 best } pre = cur.also { cur = pre } } return pre[K] } fun run(filename: String) { var A = mutableListOf>() var K = 0 var N = 0 var first = true File(filename).forEachLine { line -> if (!first) { var (value, weight) = line.trim().split(" ").map{ it -> it.toInt() } A.add(Pair(value, weight)) } else { var (a, b) = line.trim().split(" ").map{ it -> it.toInt() } K = a N = b first = false } } var a = top_down(A, K) var b = bottom_up(A, K) var c = bottom_up_memopt(A, K) assert(a == b && b == c) // 💩 sanity check println("$filename: $a") } fun main() { run("problem16.7test.txt") // problem16.7test.txt: 2493893 } ================================================ FILE: knapsack/main.py ================================================ from functools import lru_cache def top_down(A, K): N = len(A) total = [0] * N @lru_cache(maxsize = None) # 🤔 memo def go(i = 0, k = K): if i == N: # 🛑 empty set return 0 value, weight = A[i] include = go(i + 1, k - weight) + value if 0 <= k - weight else float('-inf') # ✅ include A[i] exclude = go(i + 1, k) # 🚫 exclude A[i] return max(include, exclude) # 🎯 best return go() def bottom_up(A, K): N = len(A) dp = [[float('-inf')] * (K + 1) for _ in range(N + 1)] # 🤔 memo for j in range(K): # 🛑 empty set dp[0][j] = 0 for i in range(1, N + 1): for k in range(1, K + 1): value, weight = A[i - 1] include = dp[i - 1][k - weight] + value if 0 <= k - weight else float('-inf') # ✅ include A[i] exclude = dp[i - 1][k] # 🚫 exclude A[i] dp[i][k] = max(include, exclude) # 🎯 best return dp[N][K] def bottom_up_memopt(A, K): N = len(A) pre = [0] * (K + 1) # 🤔 memo + 🛑 empty set for i in range(1, N + 1): cur = [float('-inf')] * (K + 1) for k in range(1, K + 1): value, weight = A[i - 1] include = pre[k - weight] + value if 0 <= k - weight else float('-inf') # ✅ include A[i] exclude = pre[k] # 🚫 exclude A[i] cur[k] = max(include, exclude) # 🎯 best pre, cur = cur, pre return pre[K] def run(filename): A = [] with open(filename) as fin: line = fin.readline() [K, N] = [int(word) for word in line.split()] # K capacity, N items while True: line = fin.readline() if not line: break value, weight = [int(word) for word in line.split()] A.append([value, weight]) a = top_down(A, K) b = bottom_up(A, K) c = bottom_up_memopt(A, K) assert(a == b and b == c) # 💩 sanity check print(f'{filename}: {a}') run('problem16.7test.txt') # problem16.7test.txt: 2493893 ================================================ FILE: knapsack/package.json ================================================ { "name": "knapsack", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "clayton wong", "license": "ISC", "dependencies": { "assert": "^2.0.0", "n-readlines": "^1.0.1" } } ================================================ FILE: knapsack/problem16.7.txt ================================================ 2000000 2000 16808 241486 50074 834558 8931 738037 27545 212860 77924 494349 64441 815107 84493 723724 7988 421316 82328 652893 78841 402599 44304 631607 17710 318556 29561 608119 93100 429390 51817 431959 99098 365899 13513 90282 23811 467558 80980 743542 36580 896948 11968 883369 1394 604449 25486 562244 25229 333236 40195 157443 35002 696933 16709 539123 15669 202584 88125 759690 9531 69730 27723 110467 28550 651058 97802 231944 40978 186803 8229 572887 60299 797491 28636 692529 23866 503157 39064 243688 39426 182032 24116 262772 75630 246257 46518 605917 30106 685556 19452 522241 82189 419114 99506 622799 6753 512332 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================================================ FILE: knapsack/problem16.7test.txt ================================================ 10000 100 16808 250 50074 659 8931 273 27545 879 77924 710 64441 166 84493 43 7988 504 82328 730 78841 613 44304 170 17710 158 29561 934 93100 279 51817 336 99098 827 13513 268 23811 634 80980 150 36580 822 11968 673 1394 337 25486 746 25229 92 40195 358 35002 154 16709 945 15669 491 88125 197 9531 904 27723 667 28550 25 97802 854 40978 409 8229 934 60299 982 28636 14 23866 815 39064 537 39426 670 24116 95 75630 502 46518 196 30106 405 19452 299 82189 124 99506 883 6753 567 36717 338 54439 145 51502 898 83872 829 11138 359 53178 398 22295 905 21610 232 59746 176 53636 299 98143 400 27969 413 261 558 41595 9 16396 969 19114 531 71007 963 97943 366 42083 853 30768 822 85696 713 73672 902 48591 832 14739 58 31617 791 55641 680 37336 7 97973 99 49096 320 83455 224 12290 761 48906 127 36124 507 45814 771 35239 95 96221 845 12367 535 25227 395 41364 739 7845 591 36551 160 8624 948 97386 218 95273 540 99248 386 13497 886 40624 421 28145 969 35736 916 61626 535 46043 12 54680 153 ================================================ FILE: kosaraju/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(kosaraju) set(CMAKE_CXX_STANDARD 17) add_executable(kosaraju main.cpp) ================================================ FILE: kosaraju/main.cpp ================================================ #include #include #include #include #include #include using namespace std; using List = deque; using Lists = deque; using AdjList = unordered_map; using Set = unordered_set; using Map = unordered_map; namespace Base { class Solution { protected: AdjList adj, rev; public: Solution(AdjList& adj, AdjList& rev) : adj{ adj }, rev{ rev } {} }; } namespace Recursive { struct Solution : public Base::Solution { Solution(AdjList& adj, AdjList& rev) : Base::Solution{ adj, rev } {} Lists kosaraju() { Lists lists; Set seen; using fun = function; fun go = [&](auto u, auto& list) { if (!seen.insert(u).second) return; list.push_back(u); for (auto v: adj[u]) go(v, list); }; for (auto u: topo_sort()) { List list; go(u, list); lists.emplace_back(list); } sort(lists.begin(), lists.end(), [](auto& a, auto& b) { return b.size() < a.size(); }); return lists; } List topo_sort() { List list; Set seen; using fun = function; fun go = [&](auto u) { if (!seen.insert(u).second) return; for (auto v: rev[u]) go(v); list.push_front(u); }; for (auto [u, _]: rev) go(u); return list; } }; } namespace Iterative { struct Solution : public Base::Solution { Solution(AdjList& adj, AdjList& rev) : Base::Solution{ adj, rev } {} Lists kosaraju() { Lists lists; Set seen; for (auto u: topo_sort()) { if (seen.find(u) != seen.end()) continue; List list; List stack{ u }; seen.insert(u); while (stack.size()) { auto u = stack.back(); for (auto v: adj[u]) if (seen.insert(v).second) stack.push_back(v); if (u == stack.back()) list.push_back(u), stack.pop_back(); } lists.emplace_back(list); } sort(lists.begin(), lists.end(), [](auto& a, auto& b) { return b.size() < a.size(); }); return lists; } List topo_sort() { List list; Set seen; for (auto [u, _]: rev) { if (seen.find(u) != seen.end()) continue; List stack{ u }; seen.insert(u); while (stack.size()) { auto u = stack.back(); for (auto v: rev[u]) if (seen.insert(v).second) stack.push_back(v); if (u == stack.back()) list.push_front(stack.back()), stack.pop_back(); } } return list; } }; } void run(string filename) { int u, v; AdjList adj, rev; fstream fin{ filename }; for (string line; fin >> u >> v;) { adj[u].push_back(v); rev[v].push_back(u); } auto A = Iterative::Solution{ adj, rev }.kosaraju(); A.resize(min(A.size(), size_t(5))); cout << filename << ": "; for (auto i{ 0 }; i < A.size(); cout << A[i++].size() << " "); cout << endl; } int main() { run("section8.6.5page64.txt"); // Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run("problem8.10test1.txt"); // Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run("problem8.10test2.txt"); // Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run("problem8.10test3.txt"); // Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run("problem8.10test4.txt"); // Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run("problem8.10test5.txt"); // Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run("problem8.10.txt"); // Challenge data set: Vertices are labeled as positive integers from 1 to 875714 // section8.6.5page64.txt: 4 3 3 1 // problem8.10test1.txt: 3 3 3 // problem8.10test2.txt: 3 3 2 // problem8.10test3.txt: 3 3 1 1 // problem8.10test4.txt: 7 1 // problem8.10test5.txt: 6 3 2 1 // problem8.10.txt: 434821 968 459 313 211 return 0; } ================================================ FILE: kosaraju/main.js ================================================ class BaseSolution { constructor(adj, rev) { this.adj = adj; this.rev = rev; } } class RecursiveSolution extends BaseSolution { constructor(adj, rev) { super(adj, rev); } topo_sort() { let list = []; let seen = new Set(); let go = u => { if (seen.has(u)) return; seen.add(u); for (let v of [...this.rev.get(u)]) go(v); list.unshift(u); }; for (let [u, _] of [...this.rev]) go(u); return list; } kosaraju() { let lists = []; let seen = new Set(); let go = (u, list) => { if (seen.has(u)) return; seen.add(u); list.push(u); for (let v of [...this.adj.get(u)]) go(v, list); }; for (let u of this.topo_sort()) { let list = []; go(u, list); lists.push([...list]); } lists.sort((a, b) => b.length - a.length); return lists; } } class IterativeSolution extends BaseSolution { constructor(adj, rev) { super(adj, rev); } topo_sort() { let list = []; let seen = new Set(); for (let [u, _] of [...this.rev]) { if (seen.has(u)) continue; let stack = [ u ]; seen.add(u); stack.back = () => stack[stack.length - 1]; while (stack.length) { let u = stack.back(); for (let v of [...this.rev.get(u)]) if (!seen.has(v)) stack.push(v), seen.add(v); if (u == stack.back()) list.unshift(stack.pop()); } } return list; } kosaraju() { let lists = []; let seen = new Set(); for (let u of this.topo_sort()) { if (seen.has(u)) continue; let list = []; let stack = [ u ]; seen.add(u); stack.back = () => stack[stack.length - 1]; while (stack.length) { let u = stack.back(); for (let v of [...this.adj.get(u)]) if (!seen.has(v)) stack.push(v), seen.add(v); if (u == stack.back()) list.push(stack.pop()); } lists.push([...list]); } lists.sort((a, b) => b.length - a.length); return lists; } } let run = filename => { let adj = new Map(), rev = new Map(); let LineByLine = require('n-readlines'); let input = new LineByLine(filename); let line; while (line = input.next()) { let [u, v] = String.fromCharCode(...line).split(' ').map(Number); if (!adj.has(u)) adj.set(u, []); if (!adj.has(v)) adj.set(v, []); if (!rev.has(u)) rev.set(u, []); if (!rev.has(v)) rev.set(v, []); adj.get(u).push(v); rev.get(v).push(u); } // let A = new RecursiveSolution(adj, rev).kosaraju(); let A = new IterativeSolution(adj, rev).kosaraju(); console.log(`${filename}: ${A.slice(0, Math.min(A.length, 5)).map(scc => scc.length).join(' ')}`); }; run('section8.6.5page64.txt') // Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run('problem8.10test1.txt') // Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run('problem8.10test2.txt') // Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run('problem8.10test3.txt') // Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run('problem8.10test4.txt') // Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run('problem8.10test5.txt') // Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run('problem8.10.txt') // Challenge data set: Vertices are labeled as positive integers from 1 to 875714 // section8.6.5page64.txt: 4 3 3 1 // problem8.10test1.txt: 3 3 3 // problem8.10test2.txt: 3 3 2 // problem8.10test3.txt: 3 3 1 1 // problem8.10test4.txt: 7 1 // problem8.10test5.txt: 6 3 2 1 // problem8.10.txt: 434821 968 459 313 211 ================================================ FILE: kosaraju/main.kt ================================================ import java.util.Stack import java.io.File class RecursiveSolution(var adj: MutableMap>, var rev: MutableMap>) { fun topo_sort(): MutableList { var list = mutableListOf() var seen = mutableSetOf() fun go(u: Int) { if (seen.contains(u)) return seen.add(u) for (v in rev[u]!!) go(v) list.add(0, u) } for ((u, _) in rev) go(u) return list } fun kosaraju(): MutableList> { var lists = mutableListOf>() var seen = mutableSetOf() fun go(u: Int, list: MutableList) { if (seen.contains(u)) return list.add(u); seen.add(u) for (v in adj[u]!!) go(v, list) } for (u in topo_sort()) { if (seen.contains(u)) continue var list = mutableListOf() go(u, list) lists.add(list.toList()) } return lists } } class IterativeSolution(var adj: MutableMap>, var rev: MutableMap>) { fun topo_sort(): MutableList { var list = mutableListOf() var seen = mutableSetOf() for ((u, _) in rev) { if (seen.contains(u)) continue var stack = Stack() stack.push(u); seen.add(u) while (!stack.empty()) { var u = stack.last() for (v in rev[u]!!) { if (!seen.contains(v)) { stack.push(v); seen.add(v) } } if (u == stack.last()) list.add(0, stack.pop()) } } return list } fun kosaraju(): MutableList> { var lists = mutableListOf>() var seen = mutableSetOf() for (u in topo_sort()) { if (seen.contains(u)) continue var list = mutableListOf() var stack = Stack() stack.push(u); seen.add(u) while (!stack.empty()) { var u = stack.last() for (v in adj[u]!!) { if (!seen.contains(v)) { stack.push(v); seen.add(v) } } if (u == stack.last()) list.add(stack.pop()) } lists.add(list.toList()) } return lists } } fun run(filename: String) { var adj = mutableMapOf>() var rev = mutableMapOf>() File(filename).forEachLine { var (u, v) = it.trim().split(" ").map{ it.toInt() } if (!adj.contains(u)) adj[u] = mutableListOf(); if (!adj.contains(v)) adj[v] = mutableListOf() if (!rev.contains(u)) rev[u] = mutableListOf(); if (!rev.contains(v)) rev[v] = mutableListOf() adj[u]!!.add(v) rev[v]!!.add(u) } // var solution = RecursiveSolution(adj, rev) var solution = IterativeSolution(adj, rev) var A = solution.kosaraju() A.sortWith(Comparator{ a: List, b: List -> b.size - a.size }) println(filename + ": " + A.map{ it.size }.slice(0 until Math.min(A.size, 5)).joinToString(" ")) } fun main() { run("section8.6.5page64.txt"); // Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run("problem8.10test1.txt"); // Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run("problem8.10test2.txt"); // Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run("problem8.10test3.txt"); // Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run("problem8.10test4.txt"); // Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run("problem8.10test5.txt"); // Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run("problem8.10.txt"); // Challenge data set: Vertices are labeled as positive integers from 1 to 875714 // section8.6.5page64.txt: 4 3 3 1 // problem8.10test1.txt: 3 3 3 // problem8.10test2.txt: 3 3 2 // problem8.10test3.txt: 3 3 1 1 // problem8.10test4.txt: 7 1 // problem8.10test5.txt: 6 3 2 1 // problem8.10.txt: 434821 968 459 313 211 } ================================================ FILE: kosaraju/main.py ================================================ from collections import deque from functools import cmp_to_key class BaseSolution: def __init__(self, adj, rev): self.adj = adj self.rev = rev class RecursiveSolution(BaseSolution): def topo_sort(self): list = deque() seen = set() def go(u): if u in seen: return seen.add(u) for v in self.rev[u]: go(v) list.appendleft(u) for u in self.rev.keys(): go(u) return list def kosaraju(self): lists = [] seen = set() def go(u, list): if u in seen: return seen.add(u) list.append(u) for v in self.adj[u]: go(v, list) for u in self.topo_sort(): list = [] go(u, list) lists.append(list.copy()) lists.sort(key = cmp_to_key(lambda a, b: len(b) - len(a))) return lists class IterativeSolution(BaseSolution): def topo_sort(self): list = deque() seen = set() for u in self.rev.keys(): if u in seen: continue stack = [ u ]; seen.add(u) while len(stack): u = stack[-1] for v in self.rev[u]: if v not in seen: stack.append(v); seen.add(v) if u == stack[-1]: list.appendleft(stack.pop()) return list def kosaraju(self): lists = [] seen = set() for u in self.topo_sort(): if u in seen: continue list = deque() stack = [ u ]; seen.add(u) while len(stack): u = stack[-1] for v in self.adj[u]: if v not in seen: stack.append(v); seen.add(v) if u == stack[-1]: list.appendleft(stack.pop()) lists.append(list.copy()) lists.sort(key = cmp_to_key(lambda a, b: len(b) - len(a))) return lists def run(filename): adj, rev = {}, {} with open(filename) as fin: while True: line = fin.readline().strip() if not line: break u, v = [int(x) for x in line.split()] if u not in adj: adj[u] = [] if v not in adj: adj[v] = [] if u not in rev: rev[u] = [] if v not in rev: rev[v] = [] adj[u].append(v) rev[v].append(u) # solution = RecursiveSolution(adj, rev) solution = IterativeSolution(adj, rev) A = solution.kosaraju() print(filename + ': ' + ' '.join(str(len(scc)) for scc in A[:5])) run('section8.6.5page64.txt') # Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run('problem8.10test1.txt') # Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run('problem8.10test2.txt') # Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run('problem8.10test3.txt') # Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run('problem8.10test4.txt') # Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run('problem8.10test5.txt') # Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run('problem8.10.txt') # Challenge data set: Vertices are labeled as positive integers from 1 to 875714 # section8.6.5page64.txt: 4 3 3 1 # problem8.10test1.txt: 3 3 3 # problem8.10test2.txt: 3 3 2 # problem8.10test3.txt: 3 3 1 1 # problem8.10test4.txt: 7 1 # problem8.10test5.txt: 6 3 2 1 # problem8.10.txt: 434821 968 459 313 211 ================================================ FILE: kosaraju/package.json ================================================ { "name": "kosaraju", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "clayton wong", "license": "ISC", "dependencies": { "n-readlines": "^1.0.1" } } ================================================ FILE: kosaraju/problem8.10.txt ================================================ [File too large to display: 69.3 MB] ================================================ FILE: kosaraju/problem8.10test1.txt ================================================ 1 4 2 8 3 6 4 7 5 2 6 9 7 1 8 5 8 6 9 7 9 3 ================================================ FILE: kosaraju/problem8.10test2.txt ================================================ 1 2 2 6 2 3 2 4 3 1 3 4 4 5 5 4 6 5 6 7 7 6 7 8 8 5 8 7 ================================================ FILE: kosaraju/problem8.10test3.txt ================================================ 1 2 2 3 3 1 3 4 5 4 6 4 8 6 6 7 7 8 ================================================ FILE: kosaraju/problem8.10test4.txt ================================================ 1 2 2 3 3 1 3 4 5 4 6 4 8 6 6 7 7 8 4 3 4 6 ================================================ FILE: kosaraju/problem8.10test5.txt ================================================ 1 2 2 3 2 4 2 5 3 6 4 5 4 7 5 2 5 6 5 7 6 3 6 8 7 8 7 10 8 7 9 7 10 9 10 11 11 12 12 10 ================================================ FILE: kosaraju/section8.6.5page64.txt ================================================ 1 5 2 9 3 1 4 2 4 9 5 3 6 8 6 11 7 4 7 5 8 9 8 10 8 11 9 5 9 7 10 2 10 6 11 3 ================================================ FILE: kotlin/dijkstra.kt ================================================ import java.io.File import java.util.PriorityQueue var INF = (1e9 + 7).toInt() interface BaseSolution { fun run(filename: String, queries: Array): String } class NaiveSolution : BaseSolution { fun dijkstra(E: List>): MutableMap { var dist = mutableMapOf() var seen = mutableSetOf() var start = 1 dist[start] = 0; seen.add(start) var found: Boolean; do { found = false var best_v = INF var best_w = INF for ((u, v, w) in E) { if (!seen.contains(u) || seen.contains(v)) continue found = true if (best_w > dist[u]!! + w) { best_v = v best_w = dist[u]!! + w } } var v = best_v var w = best_w dist[v] = w; seen.add(v) } while (found) return dist } override fun run(filename: String, queries: Array): String { var E = mutableListOf>() File(filename).forEachLine { var words = it.trim().split("\t") var u = words[0].toInt() for (i in 1 until words.size) { var (v, w) = words[i].split(",").map{ it.toInt() } E.add(Triple(u, v, w)) } } var dist = dijkstra(E.toList()) return queries.map{ dist[it] }.joinToString(" ") } } class HeapSolution : BaseSolution { fun dijkstra(adj: MutableMap>>): MutableMap { var dist = mutableMapOf() var seen = mutableSetOf() var start = 1 dist[start] = 0 var q = PriorityQueue>(Comparator{ a: Pair, b: Pair -> a.first.compareTo(b.first) }) q.add(Pair(0, start)) while (0 < q.size) { var (cost, u) = q.poll() if (seen.contains(u)) continue dist[u] = cost; seen.add(u) for ((w, v) in adj[u]!!) { if (seen.contains(v)) continue q.add(Pair(cost + w, v)) } } return dist } override fun run(filename: String, queries: Array): String { var adj = mutableMapOf>>() File(filename).forEachLine { var words = it.trim().split("\t") var u = words[0].toInt() if (!adj.contains(u)) adj[u] = mutableListOf() for (i in 1 until words.size) { var (v, w) = words[i].split(",").map{ it.toInt() } adj[u]!!.add(Pair(w, v)) } } var dist = dijkstra(adj) return queries.map{ dist[it] }.joinToString(" ") } } fun run(solution: BaseSolution) { println(solution.run("problem9.8test.txt", arrayOf(1, 2, 3, 4, 5, 6, 7, 8))) println(solution.run("problem9.8.txt", arrayOf(7, 37, 59, 82, 99, 115, 133, 165, 188, 197))) } fun main() { run(NaiveSolution()) // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 run(HeapSolution()) // 0 1 2 3 4 4 3 2 // 2599 2610 2947 2052 2367 2399 2029 2442 2505 3068 } ================================================ FILE: kotlin/greedy_scheduling.kt ================================================ import java.io.File data class Job(val weight: Long, val length: Long) class Solution { fun minSum(jobs: Array): Pair { class Diff: Comparator { override fun compare(a: Job?, b: Job?): Int { if (a == null || b == null) return 0 var first = a.weight - a.length var second = b.weight - b.length return if (first == second) b.weight.compareTo(a.weight) else second.compareTo(first) // sort by descending difference, break ties in favor of jobs with larger weights } } class Ratio: Comparator { override fun compare(a: Job?, b: Job?): Int { if (a == null || b == null) return 0 var first = a.weight.toDouble() / a.length var second = b.weight.toDouble() / b.length return if (first == second) b.weight.compareTo(a.weight) else second.compareTo(first) // sort by descending difference, break ties in favor of jobs with larger weights } } return Pair(calcSum(jobs, Diff()), calcSum(jobs, Ratio())) } private fun calcSum(jobs: Array, comp: Comparator): Long { jobs.sortWith(comp) var time: Long = 0 var total: Long = 0 jobs.forEach { job -> time += job.length total += job.weight * time } return total } } fun run(filename: String) { var jobs = mutableListOf() var first = true File(filename).forEachLine { if (!first) { var words = it.trim().split(" ").map{ it.toLong() } var (weight, length) = words jobs.add(Job(weight, length)) } else { first = false } } var (diff, ratio) = Solution().minSum(jobs.toTypedArray()) println("$diff, $ratio") // sub-optimal, optimal } fun main() { run("problem13.4test1.txt") // 23, 22 run("problem13.4test2.txt") // 68615, 67247 run("problem13.4.txt") // 69119377652, 67311454237 } ================================================ FILE: kotlin/huffman.kt ================================================ /* * Programming Problem 14.6: Huffman Codes * * In this problem the file format is: * [number_of_symbols] * [weight of symbol #1] * [weight of symbol #2] * ... */ import java.io.File import java.util.PriorityQueue import java.util.Queue import java.util.LinkedList var INF = (1e9 + 7).toInt() data class Tree(val weight: Int, val left: Tree? = null, val right: Tree? = null) /* fun encode(A: List): Tree { var q = PriorityQueue(Comparator{ a: Tree, b: Tree -> a.weight.compareTo(b.weight) }) for (weight in A) q.add(Tree(weight)) while (1 < q.size) { var a = q.poll() var b = q.poll() var c = Tree(a.weight + b.weight, a, b) q.add(c) } return q.poll() } */ /* * Problem 14.5: Give an implementation of Huffman's greedy algorithm that uses a single invocation * of a sorting subroutine, followed by a linear amount of additional work. */ fun encode(A: MutableList): Tree { A.sort() var first: Queue = LinkedList(A.map{ weight -> Tree(weight) }.toList()) var second: Queue = LinkedList() var next = mutableListOf() while (1 < first.size + second.size) { next.clear() do { if (0 < first.size && 0 < second.size) { if (first.peek().weight < second.peek().weight) next.add(first.poll()) else next.add(second.poll()) } else if (0 < first.size) next.add(first.poll()) else if (0 < second.size) next.add(second.poll()) } while (next.size < 2) var (a, b) = next var c = Tree(a.weight + b.weight, a, b) second.add(c) } return second.poll() } fun run(filename: String): Pair { var A = mutableListOf() var first = true File(filename).forEachLine { if (!first) { var weight = it.trim().toInt() A.add(weight) } else { first = false } } var tree = encode(A.toMutableList()) var lo = INF var hi = -INF fun go(root: Tree? = tree, depth: Int = 0) { if (root == null) return var isLeaf = { node: Tree? -> node?.left == null && node?.right == null } if (isLeaf(root)) { lo = Math.min(lo, depth) hi = Math.max(hi, depth) } else { go(root.left, depth + 1) go(root.right, depth + 1) } } go() return Pair(lo, hi) } fun main() { for (filename in listOf("problem14.6test1.txt", "problem14.6test2.txt", "problem14.6.txt")) { var (lo, hi) = run(filename) println("$filename: $lo, $hi") // min, max encoding length in the corresponding optimal prefix-free tree } } // problem14.6test1.txt: 2, 5 // problem14.6test2.txt: 3, 6 // problem14.6.txt: 9, 19 ================================================ FILE: kotlin/knapsack.kt ================================================ import java.io.File var INF = (1e9 + 7).toInt() fun top_down(A: List>, K: Int): Int { var N = A.size var m = mutableMapOf() fun go(i: Int = 0, k: Int = K): Int { if (i == N) // 🛑 empty set return 0 var key = "$i,$k" if (m.contains(key)) // 🤔 memo return m[key]!! var (value, weight) = A[i] var include = if (0 <= k - weight) go(i + 1, k - weight) + value else -INF // ✅ include A[i] var exclude = go(i + 1, k) // 🚫 exclude A[i] m[key] = Math.max(include, exclude) // 🎯 best return m[key]!! } return go() } fun bottom_up(A: List>, K: Int): Int { var N = A.size var dp = Array(N + 1){ Array(K + 1){ -INF } } // 🤔 memo for (k in 0..K) // 🛑 empty set dp[0][k] = 0 for (i in 1..N) { for (k in 0..K) { var (value, weight) = A[i - 1] var include = if (0 <= k - weight) dp[i - 1][k - weight] + value else -INF // ✅ include A[i] var exclude = dp[i - 1][k] // 🚫 exclude A[i] dp[i][k] = Math.max(include, exclude) // 🎯 best } } return dp[N][K] } fun run(filename: String) { var A = mutableListOf>() var K = 0 var N = 0 var first = true File(filename).forEachLine { line -> if (!first) { var (value, weight) = line.trim().split(" ").map{ it -> it.toInt() } A.add(Pair(value, weight)) } else { var (a, b) = line.trim().split(" ").map{ it -> it.toInt() } K = a N = b first = false } } var a = top_down(A, K) var b = bottom_up(A, K) assert(a == b) // 💩 sanity check println("$filename: $a") } fun main() { run("problem16.7test.txt") // problem16.7test.txt: 2493893 } ================================================ FILE: kotlin/kosaraju.kt ================================================ import java.util.Stack import java.io.File class RecursiveSolution(var adj: MutableMap>, var rev: MutableMap>) { fun topo_sort(): MutableList { var list = mutableListOf() var seen = mutableSetOf() fun go(u: Int) { if (seen.contains(u)) return seen.add(u) for (v in rev[u]!!) go(v) list.add(0, u) } for ((u, _) in rev) go(u) return list } fun kosaraju(): MutableList> { var lists = mutableListOf>() var seen = mutableSetOf() fun go(u: Int, list: MutableList) { if (seen.contains(u)) return list.add(u); seen.add(u) for (v in adj[u]!!) go(v, list) } for (u in topo_sort()) { if (seen.contains(u)) continue var list = mutableListOf() go(u, list) lists.add(list.toList()) } return lists } } class IterativeSolution(var adj: MutableMap>, var rev: MutableMap>) { fun topo_sort(): MutableList { var list = mutableListOf() var seen = mutableSetOf() for ((u, _) in rev) { if (seen.contains(u)) continue var stack = Stack() stack.push(u); seen.add(u) while (!stack.empty()) { var u = stack.last() for (v in rev[u]!!) { if (!seen.contains(v)) { stack.push(v); seen.add(v) } } if (u == stack.last()) list.add(0, stack.pop()) } } return list } fun kosaraju(): MutableList> { var lists = mutableListOf>() var seen = mutableSetOf() for (u in topo_sort()) { if (seen.contains(u)) continue var list = mutableListOf() var stack = Stack() stack.push(u); seen.add(u) while (!stack.empty()) { var u = stack.last() for (v in adj[u]!!) { if (!seen.contains(v)) { stack.push(v); seen.add(v) } } if (u == stack.last()) list.add(stack.pop()) } lists.add(list.toList()) } return lists } } fun run(filename: String) { var adj = mutableMapOf>() var rev = mutableMapOf>() File(filename).forEachLine { var (u, v) = it.trim().split(" ").map{ it.toInt() } if (!adj.contains(u)) adj[u] = mutableListOf(); if (!adj.contains(v)) adj[v] = mutableListOf() if (!rev.contains(u)) rev[u] = mutableListOf(); if (!rev.contains(v)) rev[v] = mutableListOf() adj[u]!!.add(v) rev[v]!!.add(u) } // var solution = RecursiveSolution(adj, rev) var solution = IterativeSolution(adj, rev) var A = solution.kosaraju() A.sortWith(Comparator{ a: List, b: List -> b.size - a.size }) println(filename + ": " + A.map{ it.size }.slice(0 until Math.min(A.size, 5)).joinToString(" ")) } fun main() { run("section8.6.5page64.txt"); // Graph from section 8.6.5 on page 64 of Algorithms Illuminated: Part 2 run("problem8.10test1.txt"); // Test case #1: A 9-vertex 11-edge graph. Top 5 SCC sizes: 3,3,3,0,0 run("problem8.10test2.txt"); // Test case #2: An 8-vertex 14-edge graph. Top 5 SCC sizes: 3,3,2,0,0 run("problem8.10test3.txt"); // Test case #3: An 8-vertex 9-edge graph. Top 5 SCC sizes: 3,3,1,1,0 run("problem8.10test4.txt"); // Test case #4: An 8-vertex 11-edge graph. Top 5 SCC sizes: 7,1,0,0,0 run("problem8.10test5.txt"); // Test case #5: A 12-vertex 20-edge graph. Top 5 SCC sizes: 6,3,2,1,0 run("problem8.10.txt"); // Challenge data set: Vertices are labeled as positive integers from 1 to 875714 // section8.6.5page64.txt: 4 3 3 1 // problem8.10test1.txt: 3 3 3 // problem8.10test2.txt: 3 3 2 // problem8.10test3.txt: 3 3 1 1 // problem8.10test4.txt: 7 1 // problem8.10test5.txt: 6 3 2 1 // problem8.10.txt: 434821 968 459 313 211 } ================================================ FILE: kotlin/kruskal.kt ================================================ import java.io.File fun kruskal(E: MutableList>): Int { var total: Int = 0 var M = E.size var P = IntArray(M) { it } // 🙂 parent representatives of 1..M disjoint sets fun find(x: Int): Int { P[x] = if (P[x] == x) x else find(P[x]) return P[x] } fun union(a: Int, b: Int): Boolean { var x = find(a) var y = find(b) if (x == y) return false P[x] = y // 🎲 arbitrary choice return true } E.sortWith(Comparator{ a, b -> a.third.compareTo(b.third) }) // sort edges by nondecreasing weight for ((u, v, w) in E) if (union(u, v)) total += w return total } fun run(filename: String) { var E = mutableListOf>() var first = true File(filename).forEachLine { line -> if (!first) { var (u, v, w) = line.trim().split(" ").map{ it.toInt() } E.add(Triple(u, v, w)) } else { first = false // ignore first line with N vertices and M edges } } var cost = kruskal(E) println("$filename: $cost") } fun main() { run("problem15.9test.txt") // problem15.9test.txt: 14 run("problem15.9.txt") // problem15.9.txt: -3612829 } ================================================ FILE: kotlin/merge_sort.kt ================================================ fun sort(A: IntArray): IntArray { fun merge(A: IntArray, B: IntArray): IntArray { var C = mutableListOf() var i = 0 var j = 0 while (i < A.size && j < B.size) if (A[i] < B[j]) C.add(A[i++]) else C.add(B[j++]) A.slice(i..A.lastIndex).forEach { C.add(it) } B.slice(j..B.lastIndex).forEach { C.add(it) } return C.toIntArray() } fun go(A: IntArray): IntArray { var N = A.size if (N < 2) return A var half = Math.floor(N / 2.0).toInt() var first = go(A.slice(0 until half).toIntArray()) var second = go(A.slice(half until N).toIntArray()) return merge(first, second) } return go(A) } fun main(args: Array) { sort(intArrayOf(5,3,8,9,1,7,0,2,6,4)).forEach { print("$it ") } // 0 1 2 3 4 5 6 7 8 9 println() } ================================================ FILE: kotlin/merge_sort_inversions.kt ================================================ import java.io.File fun sort(A: IntArray): Pair { fun merge(A: IntArray, B: IntArray): Pair { var C = mutableListOf() var inv: Long = 0 var i = 0 var j = 0 while (i < A.size && j < B.size) if (A[i] < B[j]) { C.add(A[i++]) } else { inv += A.size - i // ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.add(B[j++]) } A.slice(i..A.lastIndex).forEach { C.add(it) } B.slice(j..B.lastIndex).forEach { C.add(it) } return Pair(C.toIntArray(), inv) } fun go(A: IntArray): Pair { var N = A.size if (N < 2) return Pair(A, 0) var half = Math.floor(N / 2.0).toInt() var (first, inv1) = go(A.slice(0 until half).toIntArray()) var (second, inv2) = go(A.slice(half until N).toIntArray()) var (third, inv3) = merge(first, second) return Pair(third, inv1 + inv2 + inv3) } return go(A) } fun run(filename: String): Long { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } var (_, inv) = sort(A.toIntArray()) return inv } fun main() { println("problem3.5test.txt: " + run("problem3.5test.txt")) // problem3.5test.txt: 28 println("problem3.5.txt: " + run("problem3.5.txt")) // problem3.5.txt: 2407905288 } ================================================ FILE: kotlin/prim.kt ================================================ import java.io.File import java.util.PriorityQueue import java.util.Random fun prim(N: Int, adj: MutableMap>>): Int { var total: Int = 0 var start = Random().nextInt(N) + 1 var q = PriorityQueue>(Comparator{ a: Pair, b: Pair -> a.first.compareTo(b.first) }) var seen = mutableSetOf(start) for ((w, v) in adj[start]!!) q.add(Pair(w, v)) while (0 < q.size) { var (cost, u) = q.poll() if (seen.contains(u)) continue total += cost; seen.add(u) for ((w, v) in adj[u]!!) if (!seen.contains(v)) q.add(Pair(w, v)) } return total } fun run(filename: String) { var N: Int = 0 var adj = mutableMapOf>>() var first = true File(filename).forEachLine { line -> if (!first) { var (u, v, w) = line.split(" ").map{ it.toInt() } if (!adj.contains(u)) adj[u] = mutableListOf>() if (!adj.contains(v)) adj[v] = mutableListOf>() adj[u]!!.add(Pair(w, v)) adj[v]!!.add(Pair(w, u)) } else { var (numVertex, _) = line.split(" ").map{ it.toInt() } N = numVertex first = false } } var cost = prim(N, adj) println("$filename: $cost") } fun main() { run("problem15.9test.txt") // problem15.9test.txt: 14 run("problem15.9.txt") // problem15.9.txt: -3612829 } ================================================ FILE: kotlin/quick_sort.kt ================================================ import java.io.File typealias PivotFunc = (A: MutableList, L: Int, R: Int) -> (Int) var pivotLeft: PivotFunc = { _: MutableList, L: Int, _: Int -> L } var pivotRight: PivotFunc = { _: MutableList, _: Int, R: Int -> R } fun _pivotMedian(A: MutableList, L: Int, R: Int): Int { var M = L + (R - L) / 2 var cand = intArrayOf(A[L], A[M], A[R]) cand.sort() var target = cand[1] if (target == A[L]) return L if (target == A[M]) return M if (target == A[R]) return R return -1 } var pivotMedian: PivotFunc = { A: MutableList, L: Int, R: Int -> _pivotMedian(A, L, R) } fun partition(A: MutableList, L: Int, R: Int, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { var i = L + 1 var j = L + 1 var k = choosePivot(A, L, R) A[k] = A[L].also { A[L] = A[k] } // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] A[i] = A[j].also { A[j] = A[i] } ++i } ++j } A[L] = A[i - 1].also { A[i - 1] = A[L] } // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 } fun quicksort(A: MutableList, L: Int, R: Int, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { if (R <= L) return 0 var k = partition(A, L, R, choosePivot) return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot) } fun run(filename: String, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } return quicksort(A, 0, A.size - 1, choosePivot) } fun main() { var filename = "problem5.6.txt" println(" left: ${run(filename, pivotLeft)}") // left: 162085 println(" right: ${run(filename, pivotRight)}") // right: 164123 println("median: ${run(filename, pivotMedian)}") // median: 138382 } ================================================ FILE: kotlin/rselect.kt ================================================ import java.io.File import kotlin.random.Random fun partition(A: MutableList, L: Int, R: Int): Int { var i = L + 1 var j = L + 1 var k = Random.nextInt(L, R + 1) // +1 for L..R inclusive A[L] = A[k].also { A[k] = A[L] } // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] A[i] = A[j].also { A[j] = A[i] } ++i } ++j } A[L] = A[i - 1].also { A[i - 1] = A[L] } // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 } fun rselect(A: MutableList, i: Int, L_: Int, R_: Int): Int { var L = L_ var R = R_ var k = partition(A, L, R) if (i == k) return A[k] // 🎯 lucky guess if (i < k) R = k - 1 else L = k + 1 return rselect(A, i, L, R) } fun run(filename: String, i: Int): Int { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } var N = A.size return rselect(A, i - 1, 0 , N - 1) // -1 for 0-based indexing } fun main() { println("problem6.5test1.txt: " + run("problem6.5test1.txt", 5)) // problem6.5test1.txt: 5469 println("problem6.5test2.txt: " + run("problem6.5test2.txt", 50)) // problem6.5test2.txt: 4715 } ================================================ FILE: kotlin/topo_sort.kt ================================================ import java.util.Queue import java.util.LinkedList class Solution(val adj: MutableMap>) { var N: Int var color: Int var m = mutableMapOf() var seen = mutableSetOf() init { N = adj.size color = 0 } fun init(start: Int) { color = start m.clear() seen.clear() } fun topoSortBFS(): String { init(1) // 👉 color forward from 1..N bfs() return toString() } fun topoSortDFS(): String { init(N) // 👈 color reverse from N..1 (as the recursive stack unwinds) adj.forEach{ (u, _) -> dfs(u) } return toString() } fun bfs() { var degree = mutableMapOf() adj.forEach{ (_, neighbors) -> neighbors.forEach{ v -> degree[v] = 1 + degree.getOrDefault(v, 0) } } var q: Queue = LinkedList(adj.map{ (u, _) -> u }.filter{ !degree.contains(it) }) while (0 < q.size) { var u = q.poll() m[u] = color++ adj[u]!!.forEach{ v -> degree[v] = degree[v]!!.minus(1) if (degree[v] == 0 && !seen.contains(v)) { q.add(v); seen.add(v) } } } } fun dfs(u: Char) { if (seen.contains(u)) return seen.add(u) adj[u]!!.forEach{ v -> dfs(v) } m[u] = color-- } override fun toString(): String { var s = mutableListOf() adj.forEach{ (u, _) -> s.add("$u: ${m[u]}") } return s.joinToString("\n") } } fun main() { var adj = mutableMapOf>( 's' to listOf('v', 'w'), 'v' to listOf('t'), 'w' to listOf('t'), 't' to listOf() ) var solution = Solution(adj) println("BFS:\n${solution.topoSortBFS()}\n\nDFS:\n${solution.topoSortDFS()}") // BFS: // s: 1 // v: 2 // w: 3 // t: 4 // DFS: // s: 1 // v: 3 // w: 2 // t: 4 } ================================================ FILE: kotlin/weighted_independent_set.kt ================================================ /* * In this problem, each file describes the weights of vertices in a path graph and has the format: * [number_of_vertices_in_path_graph] * [weight of first vertex] * [weight of second vertex] * ... * Test case: (contributed by Logan Travis) What is the value of a maximum-weight independent set of the 10-vertex path graph described in this file, and which vertices belong to the MWIS? (Answer: 2617, and the vertices 2, 4, 7, and 10). * Challenge data set: Repeat the previous problem for the 1000-vertex path graph described in this file. */ import java.io.File fun topDown(A: MutableList): Long { var N = A.size var m = mutableMapOf() fun go(i: Int = N - 1): Long { if (m.contains(i)) // 🤔 memo return m[i]!! if (i < 0) { // 🛑 empty set m[i] = 0 return 0 } if (i == 0) { // 🛑 single set m[i] = A[0] return A[0] } var include = go(i - 2) + A[i] // ✅ include A[i] var exclude = go(i - 1) // 🚫 exclude A[i] m[i] = Math.max(include, exclude) // 🎯 best return m[i]!! } return go() } fun bottomUp(A: MutableList): Long { var N = A.size var dp = LongArray(N + 1) // 🤔 memo dp[0] = 0 // 🛑 empty set dp[1] = A[0] // 🛑 single set for (i in 2..N) { var include = dp[i - 2] + A[i - 1] // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) var exclude = dp[i - 1] // 🚫 exclude A[i] dp[i] = Math.max(include, exclude) // 🎯 best } return dp[N] } fun run(filename: String) { var A = mutableListOf() var first = true File(filename).forEachLine { line -> if (!first) { A.add(line.toLong()) } else { first = false } } var a = topDown(A) var b = bottomUp(A) assert(a == b) // 💩 sanity check println("$filename: $a") } fun main() { run("problem16.6test.txt") // problem16.6test.txt: 2617 run("problem16.6.txt") // problem16.6.txt: 2955353732 } ================================================ FILE: kruskal/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(prim) set(CMAKE_CXX_STANDARD 17) add_executable(prim main.cpp) ================================================ FILE: kruskal/main.cpp ================================================ #include #include #include #include #include using namespace std; using Edge = tuple; using Edges = vector; using Parents = vector; using fun = function; int kruskal(Edges& E, int total = 0) { auto M = E.size(); Parents P(M); iota(P.begin(), P.end(), 0); // 🙂 parent representatives of 1..M disjoint sets fun find = [&](auto x) { return P[x] = P[x] == x ? x : find(P[x]); }; auto join = [&](auto a, auto b) { a = find(a); b = find(b); if (a == b) return false; P[a] = b; // 🎲 arbitrary choice return true; }; sort(E.begin(), E.end(), [](auto& first, auto& second) { // sort edges by nondecreasing weight auto [u1, v1, w1] = first; auto [u2, v2, w2] = second; return w1 < w2; }); for (auto [u, v, w]: E) if (join(u, v)) total += w; return total; } void run(const string& filename, Edges E = {}) { fstream fin{ filename }; int N, M; fin >> N >> M; // ignore first line with N vertices and M edges int u, v, w; // edge u -> v of weight w while (fin >> u >> v >> w) E.emplace_back(u, v, w); auto cost = kruskal(E); cout << filename << ": " << cost << endl; } int main() { run("problem15.9test.txt"); // problem15.9test.txt: 14 run("problem15.9.txt"); // problem15.9.txt: -3612829 return 0; } ================================================ FILE: kruskal/main.js ================================================ let LineByLine = require('n-readlines'); let kruskal = E => { let total = 0; let M = E.length; let P = [...Array(M).keys()]; // 🙂 parent representatives of 1..M disjoint sets let find = x => P[x] = P[x] == x ? x : find(P[x]); let union = (a, b) => { a = find(a); b = find(b); if (a == b) return false; P[a] = b; // 🎲 arbitrary choice return true; }; E.sort((first, second) => { // sort edges by nondecreasing weight let [u1, v1, w1] = first, [u2, v2, w2] = second; return w1 - w2; }); for (let [u, v, w] of E) if (union(u, v)) total += w; return total; }; let run = filename => { let E = []; let input = new LineByLine(filename); let line = input.next(); // ignore first line with N vertices and M edges while (line = input.next()) { let [u, v, w] = String.fromCharCode(...line).trim().split(' ').map(Number); E.push([ u, v, w ]); } let cost = kruskal(E); console.log(`${filename}: ${cost}`); }; run('problem15.9test.txt'); // problem15.9test.txt: 14 run('problem15.9.txt'); // problem15.9.txt: -3612829 ================================================ FILE: kruskal/main.kt ================================================ import java.io.File fun kruskal(E: MutableList>): Int { var total: Int = 0 var M = E.size var P = IntArray(M) { it } // 🙂 parent representatives of 1..M disjoint sets fun find(x: Int): Int { P[x] = if (P[x] == x) x else find(P[x]) return P[x] } fun union(a: Int, b: Int): Boolean { var x = find(a) var y = find(b) if (x == y) return false P[x] = y // 🎲 arbitrary choice return true } E.sortWith(Comparator{ a, b -> a.third.compareTo(b.third) }) // sort edges by nondecreasing weight for ((u, v, w) in E) if (union(u, v)) total += w return total } fun run(filename: String) { var E = mutableListOf>() var first = true File(filename).forEachLine { line -> if (!first) { var (u, v, w) = line.trim().split(" ").map{ it.toInt() } E.add(Triple(u, v, w)) } else { first = false // ignore first line with N vertices and M edges } } var cost = kruskal(E) println("$filename: $cost") } fun main() { run("problem15.9test.txt") // problem15.9test.txt: 14 run("problem15.9.txt") // problem15.9.txt: -3612829 } ================================================ FILE: kruskal/main.py ================================================ from functools import cmp_to_key def kruskal(E, total = 0): M = len(E) P = [i for i in range(M)] # 🙂 parent representatives of 1..M disjoint sets def find(x): P[x] = P[x] if P[x] == x else find(P[x]) return P[x] def union(a, b): a = find(a) b = find(b) if a == b: return False P[a] = b # 🎲 arbitary choice return True E.sort(key = cmp_to_key(lambda first, second: first[2] - second[2])) # sort edges by nondecreasing weight for u, v, w in E: if union(u, v): total += w return total def run(filename): E = [] first = True with open(filename) as fin: while True: line = fin.readline() if not line: break values = [int(x) for x in line.strip().split()] if not first: u, v, w = values # edge u -> v of weight w E.append([ u, v, w ]) else: first = False # ignore first line with N vertices and M edges cost = kruskal(E) print(f'{filename}: {cost}') run('problem15.9test.txt') # problem15.9test.txt: 14 run('problem15.9.txt') # problem15.9.txt: -3612829 ================================================ FILE: kruskal/package.json ================================================ { "name": "kruskal", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "clayton wong", "license": "ISC", "dependencies": { "n-readlines": "^1.0.1" } } ================================================ FILE: kruskal/problem15.9.txt ================================================ 500 2184 1 2 6807 2 3 -8874 3 4 -1055 4 5 4414 5 6 1728 6 7 -2237 7 8 -7507 8 9 7990 9 10 -5012 10 11 7353 11 12 -6736 12 13 -7604 13 14 5273 14 15 9331 15 16 -7753 16 17 -4370 17 18 267 18 19 903 19 20 7674 20 21 5436 21 22 9479 22 23 9432 23 24 4472 24 25 7258 25 26 8709 26 27 4358 27 28 -6775 28 29 68 29 30 -2849 30 31 -9951 31 32 -8643 32 33 -4010 33 34 -4667 34 35 -5829 35 36 -5222 36 37 -7609 37 38 3054 38 39 7935 39 40 173 40 41 9596 41 42 -4557 42 43 -4823 43 44 -6198 44 45 8450 45 46 5533 46 47 -6059 47 48 7411 48 49 9102 49 50 914 50 51 9501 51 52 2704 52 53 -140 53 54 1179 54 55 -1572 55 56 7095 56 57 3613 57 58 5888 58 59 -4184 59 60 1910 60 61 4401 61 62 591 62 63 -9504 63 64 3116 64 65 -6230 65 66 1867 66 67 3816 67 68 2402 68 69 7667 69 70 -3532 70 71 -2985 71 72 -3462 72 73 7737 73 74 8886 74 75 -102 75 76 -3013 76 77 9948 77 78 8032 78 79 -841 79 80 7663 80 81 8974 81 82 5234 82 83 -3366 83 84 3926 84 85 2583 85 86 3295 86 87 -3398 87 88 1058 88 89 2518 89 90 2141 90 91 8894 91 92 7151 92 93 -1766 93 94 -8183 94 95 1773 95 96 -6715 96 97 -9266 97 98 2472 98 99 9986 99 100 -4478 100 101 -1456 101 102 3812 102 103 4658 103 104 3034 104 105 6682 105 106 4633 106 107 7981 107 108 -2976 108 109 8849 109 110 -5089 110 111 -5617 111 112 9034 112 113 -4112 113 114 -9800 114 115 -1462 115 116 -3454 116 117 5890 117 118 -9012 118 119 7557 119 120 1231 120 121 6409 121 122 491 122 123 -5925 123 124 -8728 124 125 -6474 125 126 3510 126 127 -244 127 128 8413 128 129 8772 129 130 1584 130 131 6869 131 132 -7293 132 133 5355 133 134 -4059 134 135 -3303 135 136 -657 136 137 2192 137 138 2039 138 139 -1128 139 140 626 140 141 -887 141 142 -4769 142 143 -7750 143 144 5026 144 145 7151 145 146 -3960 146 147 5902 147 148 -9587 148 149 27 149 150 265 150 151 -5445 151 152 -1850 152 153 2907 153 154 5564 154 155 3872 155 156 -823 156 157 7660 157 158 8685 158 159 -3164 159 160 -7839 160 161 9902 161 162 -487 162 163 8353 163 164 -5987 164 165 4948 165 166 2192 166 167 2283 167 168 428 168 169 2596 169 170 636 170 171 6480 171 172 1141 172 173 -9697 173 174 1467 174 175 -8677 175 176 -9786 176 177 -9636 177 178 -620 178 179 653 179 180 -5463 180 181 6062 181 182 9858 182 183 8655 183 184 -1583 184 185 5507 185 186 -719 186 187 9275 187 188 -9471 188 189 7296 189 190 3768 190 191 -9 191 192 7546 192 193 7645 193 194 1272 194 195 2135 195 196 -9529 196 197 3904 197 198 -6105 198 199 -1925 199 200 -2491 200 201 5606 201 202 -8731 202 203 -8100 203 204 2830 204 205 -2078 205 206 6293 206 207 752 207 208 5289 208 209 -5201 209 210 1564 210 211 4944 211 212 -2368 212 213 6432 213 214 -1421 214 215 -7142 215 216 6331 216 217 3374 217 218 -2164 218 219 7518 219 220 -6876 220 221 8576 221 222 -7797 222 223 -4499 223 224 -1751 224 225 5777 225 226 -3832 226 227 -5058 227 228 8187 228 229 6101 229 230 9131 230 231 7884 231 232 5904 232 233 -5000 233 234 -8309 234 235 -5961 235 236 -4598 236 237 3563 237 238 -1734 238 239 -9874 239 240 6294 240 241 -2489 241 242 -8966 242 243 6243 243 244 -3343 244 245 -9103 245 246 1757 246 247 792 247 248 9403 248 249 -3384 249 250 5555 250 251 6155 251 252 7781 252 253 -2456 253 254 7775 254 255 -8096 255 256 4382 256 257 -1 257 258 1925 258 259 -7340 259 260 4493 260 261 -1791 261 262 9304 262 263 -2202 263 264 1858 264 265 -8596 265 266 -4101 266 267 -2373 267 268 -8907 268 269 -4668 269 270 -1966 270 271 -202 271 272 -4915 272 273 5789 273 274 -7658 274 275 -4649 275 276 -2753 276 277 101 277 278 -6763 278 279 3542 279 280 -9809 280 281 4856 281 282 -8850 282 283 -9803 283 284 7959 284 285 7229 285 286 6301 286 287 -9632 287 288 -9903 288 289 -5683 289 290 4849 290 291 -1292 291 292 -730 292 293 -1806 293 294 2480 294 295 7872 295 296 -4634 296 297 3695 297 298 7216 298 299 7174 299 300 -424 300 301 -157 301 302 5299 302 303 -3618 303 304 2745 304 305 2356 305 306 9341 306 307 6215 307 308 997 308 309 6059 309 310 -2124 310 311 -8711 311 312 -3721 312 313 -5356 313 314 6085 314 315 3445 315 316 7783 316 317 -7213 317 318 1999 318 319 -5649 319 320 5770 320 321 -1747 321 322 -2500 322 323 5951 323 324 -6212 324 325 -3338 325 326 -810 326 327 2013 327 328 3270 328 329 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9321 462 471 -2894 463 473 -5410 463 477 1205 464 468 -2595 464 475 -9291 467 473 9837 473 500 1052 474 491 7245 478 480 -4790 496 500 -1519 ================================================ FILE: kruskal/problem15.9test.txt ================================================ 6 10 1 2 6 1 4 5 1 5 4 2 4 1 2 5 2 2 3 5 2 6 3 3 6 4 4 5 2 5 6 4 ================================================ FILE: matrix_multiplication/main.jl ================================================ # # Straightforward Matrix Multiplication # # Input: n x n integer matrices X and Y # # Output: Z = X * Y # N = 2 X = [1 2 3 4] Y = [5 6 7 8] Z = zeros(Int64, N, N) for i in 1:N for j in 1:N for k in 1:N Z[i, j] += X[i, k] * Y[k, j] end end end function pretty_print(A) for row in eachrow(A) println(row) end println() end pretty_print(X) pretty_print(Y) pretty_print(Z) # ➜ matrix_multiplication git:(main) ✗ julia ./main.jl # [1, 2] # [3, 4] # [5, 6] # [7, 8] # [19, 22] # [43, 50] ================================================ FILE: matrix_multiplication/main.py ================================================ # # Straightforward Matrix Multiplication # # Input: n x n integer matrices X and Y # # Output: Z = X * Y # N = 2 X = [[1, 2], [3, 4]] Y = [[5, 6], [7, 8]] Z = [[0] * N for _ in range(N)] for i in range(N): for j in range(N): for k in range(N): Z[i][j] += X[i][k] * Y[k][j] pretty_print = lambda A: [print(row) for row in A] and print() pretty_print(X) pretty_print(Y) pretty_print(Z) # ➜ matrix_multiplication git:(main) ✗ python3 ./main.py # [1, 2] # [3, 4] # [5, 6] # [7, 8] # [19, 22] # [43, 50] ================================================ FILE: merge_sort/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.14) project(mergesort) set(CMAKE_CXX_STANDARD 17) add_executable(mergesort merge_sort.cpp) ================================================ FILE: merge_sort/merge_sort.cpp ================================================ #include #include using namespace std; class Solution { public: using VI = vector; VI mergesort(VI& A) { return go(move(A)); } private: VI go(VI&& A) { auto N = A.size(); if( N < 2 ) return A; auto half = A.begin() + (N / 2); auto first = go({ A.begin(), half }), second = go({ half, A.end() }); return merge(first, second); } VI merge(VI& A, VI& B, VI C = {}) { auto i{ 0 }, j{ 0 }; while (i < A.size() && j < B.size()) C.push_back(A[i] < B[j] ? A[i++] : B[j++]); C.insert(C.end(), A.begin() + i, A.end()); C.insert(C.end(), B.begin() + j, B.end()); return C; } }; int main() { Solution::VI A{ 3,5,7,1,3,9,2,0 }; auto ans = Solution().mergesort(A); copy(ans.begin(), ans.end(), ostream_iterator(cout, " ")), cout << endl; // 0 1 2 3 4 5 6 7 8 9 return 0; } ================================================ FILE: merge_sort/merge_sort.jl ================================================ # # Merge Sort # # Input: array A of n distinct integers # Output: array with the same integers, sorted from smallest to largest # function go(A) N = length(A) if N < 2 return A end half = div(N, 2) first = go(A[begin:half]) second = go(A[half+1:end]) return merge(first, second) end function merge(A, B) C = [] i, j, k = 1, 1, 1 while i <= length(A) && j <= length(B) if A[i] < B[j] push!(C, A[i]) i += 1 else push!(C, B[j]) j += 1 end k += 1 end append!(C, @view A[i:end]) append!(C, @view B[j:end]) return C end println(go([5, 3, 8, 9, 1, 7, 0, 2, 6, 4])) # Any[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] # ➜ merge_sort git:(main) ✗ julia ./merge_sort.jl # Any[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] ================================================ FILE: merge_sort/merge_sort.js ================================================ let sort = A => { let go = A => { let N = A.length; if (N < 2) return A; let half = Math.floor(N / 2); let first = go([...A.slice(0, half)]), second = go([...A.slice(half, N)]); return merge(first, second); }; let merge = (A, B, C = []) => { let M = A.length, N = B.length; let i = 0, j = 0; while (i < M && j < N) C.push(A[i] < B[j] ? A[i++] : B[j++]); C.push(...A.slice(i, M)); C.push(...B.slice(j, N)); return C; }; return go(A); }; console.log(sort([5,3,8,9,1,7,0,2,6,4])); // (10) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] ================================================ FILE: merge_sort/merge_sort.kt ================================================ fun sort(A: IntArray): IntArray { fun merge(A: IntArray, B: IntArray): IntArray { var C = mutableListOf() var i = 0 var j = 0 while (i < A.size && j < B.size) if (A[i] < B[j]) C.add(A[i++]) else C.add(B[j++]) A.slice(i..A.lastIndex).forEach { C.add(it) } B.slice(j..B.lastIndex).forEach { C.add(it) } return C.toIntArray() } fun go(A: IntArray): IntArray { var N = A.size if (N < 2) return A var half = Math.floor(N / 2.0).toInt() var first = go(A.slice(0 until half).toIntArray()) var second = go(A.slice(half until N).toIntArray()) return merge(first, second) } return go(A) } fun main(args: Array) { sort(intArrayOf(5,3,8,9,1,7,0,2,6,4)).forEach { print("$it ") } // 0 1 2 3 4 5 6 7 8 9 println() } ================================================ FILE: merge_sort/merge_sort.py ================================================ from math import floor def sort(A): def go(A): N = len(A) if N < 2: return A half = floor(N / 2) first = go(A[:half]) second = go(A[half:]) return merge(first, second) def merge(A, B): C = [] i = 0 j = 0 while i < len(A) and j < len(B): if A[i] < B[j]: C.append(A[i]); i += 1 else: C.append(B[j]); j += 1 C.extend(A[i:]) C.extend(B[j:]) return C return go(A) print(sort([5,3,8,9,1,7,0,2,6,4])) # [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] ================================================ FILE: merge_sort_inversions/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(merge_sort_inversions) set(CMAKE_CXX_STANDARD 17) add_executable(merge_sort_inversions merge_sort_inversions.cpp) ================================================ FILE: merge_sort_inversions/merge_sort_inversions.cpp ================================================ #include #include #include using namespace std; class Solution { public: using VL = vector; using Pair = pair; using fun = function; Pair merge(VL& A, VL& B, VL C = {}, long inv = 0) { auto i = 0, j = 0; while (i < A.size() && j < B.size()) { if (A[i] < B[j]) { C.push_back(A[i++]); } else { inv += A.size() - i; // ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.push_back(B[j++]); } } C.insert(C.end(), A.begin() + i, A.end()); C.insert(C.end(), B.begin() + j, B.end()); return { C, inv }; } Pair inversions(VL& A) { fun go = [&](VL&& A) -> Pair { int N = A.size(); if (N < 2) return { A, 0 }; int half = N / 2; auto [first, inv1] = go({ A.begin(), A.begin() + half }); auto [second, inv2] = go({ A.begin() + half, A.end() }); auto [third, inv3] = merge(first, second); return { third, inv1 + inv2 + inv3 }; }; return go(move(A)); } }; long run(string filename) { Solution solution; Solution::VL A; fstream fin{ filename }; for (string line; fin >> line; A.push_back(stol(line))); auto [_, inv] = solution.inversions(A); return inv; } int main() { cout << "problem3.5test.txt: " << run("problem3.5test.txt") << endl // problem3.5test.txt: 28 << "problem3.5.txt: " << run("problem3.5.txt") << endl; // problem3.5.txt: 2407905288 return 0; } ================================================ FILE: merge_sort_inversions/merge_sort_inversions.jl ================================================ # # Merge Sort Inversions # # Input: array A of n distinct integers # Output: the number of inversions of A # function go(A) N = length(A) if N < 2 return A, 0 end half = div(N, 2) first, inv1 = go(A[begin:half]) second, inv2 = go(A[half+1:end]) third, inv3 = merge(first, second) return third, inv1 + inv2 + inv3 end function merge(A, B) C, inv = [], 0 i, j, k = 1, 1, 1 while i <= length(A) && j <= length(B) if A[i] < B[j] push!(C, A[i]) i += 1 else push!(C, B[j]); inv += length(A) - i + 1 # ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions j += 1 end k += 1 end append!(C, @view A[i:end]) append!(C, @view B[j:end]) return C, inv end function run(filename) input = Int[] open(filename, "r") do file for line in eachline(file) push!(input, parse(Int, strip(line))) end end _, inv = go(input) return inv end println(run("problem3.5test.txt")) # 28 println(run("problem3.5.txt")) # 2407905288 ================================================ FILE: merge_sort_inversions/merge_sort_inversions.js ================================================ let sort = A => { let go = A => { let N = A.length; if (N < 2) return [A, 0]; let half = Math.floor(N / 2); let [first, inv1] = go([...A.slice(0, half)]), [second, inv2] = go([...A.slice(half, N)]), [third, inv3] = merge(first, second); return [third, inv1 + inv2 + inv3]; }; let merge = (A, B, C = [], inv = 0) => { let M = A.length, N = B.length; let i = 0, j = 0; while (i < M && j < N) if (A[i] < B[j]) C.push(A[i++]); else inv += M - i, // ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.push(B[j++]); C.push(...A.slice(i, M)); C.push(...B.slice(j, N)); return [C, inv]; }; return go(A); }; let run = filename => { let A = []; require('fs').readFileSync(filename, 'utf-8').split(/\r?\n/).forEach(line => A.push(Number(line))); let [_, inv] = sort(A); return inv; } console.log(`problem3.5test.txt: ${run('problem3.5test.txt')}`); // problem3.5test.txt: 28 console.log(`problem3.5.txt: ${run('problem3.5.txt')}`); // problem3.5.txt: 2407905288 ================================================ FILE: merge_sort_inversions/merge_sort_inversions.kt ================================================ import java.io.File fun sort(A: IntArray): Pair { fun merge(A: IntArray, B: IntArray): Pair { var C = mutableListOf() var inv: Long = 0 var i = 0 var j = 0 while (i < A.size && j < B.size) if (A[i] < B[j]) { C.add(A[i++]) } else { inv += A.size - i // ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.add(B[j++]) } A.slice(i..A.lastIndex).forEach { C.add(it) } B.slice(j..B.lastIndex).forEach { C.add(it) } return Pair(C.toIntArray(), inv) } fun go(A: IntArray): Pair { var N = A.size if (N < 2) return Pair(A, 0) var half = Math.floor(N / 2.0).toInt() var (first, inv1) = go(A.slice(0 until half).toIntArray()) var (second, inv2) = go(A.slice(half until N).toIntArray()) var (third, inv3) = merge(first, second) return Pair(third, inv1 + inv2 + inv3) } return go(A) } fun run(filename: String): Long { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } var (_, inv) = sort(A.toIntArray()) return inv } fun main() { println("problem3.5test.txt: " + run("problem3.5test.txt")) // problem3.5test.txt: 28 println("problem3.5.txt: " + run("problem3.5.txt")) // problem3.5.txt: 2407905288 } ================================================ FILE: merge_sort_inversions/merge_sort_inversions.py ================================================ from math import floor def sort(A): def go(A): N = len(A) if N < 2: return [A, 0] half = floor(N / 2) first, inv1 = go(A[:half]) second, inv2 = go(A[half:]) third, inv3 = merge(first, second) return [third, inv1 + inv2 + inv3] def merge(A, B, inv = 0): C = [] i = 0 j = 0 while i < len(A) and j < len(B): if A[i] < B[j]: C.append(A[i]); i += 1 else: inv += len(A) - i # ⭐️ B[j] comes before all remaining A[i...], thus all remaining A[i...] are inversions C.append(B[j]); j += 1 C.extend(A[i:]) C.extend(B[j:]) return [C, inv] return go(A) def run(filename): A = [] with open(filename) as fin: while True: line = fin.readline() if not line: break A.append(int(line)) _, inv = sort(A) return inv print(f"problem3.5test.txt: {run('problem3.5test.txt')}") # problem3.5test.txt: 28 print(f"problem3.5.txt: {run('problem3.5.txt')}") # problem3.5.txt: 2407905288 ================================================ FILE: merge_sort_inversions/problem3.5.txt ================================================ 54044 14108 79294 29649 25260 60660 2995 53777 49689 9083 16122 90436 4615 40660 25675 58943 92904 9900 95588 46120 29390 91323 85363 45738 80717 57415 7637 8540 6336 45434 65895 61811 8959 9139 31027 87662 2484 65550 23260 15616 3490 49568 5979 44737 52808 72122 37957 34826 21419 73531 94323 52910 84496 71799 50162 1692 1565 59279 56864 20141 13893 63942 6055 33424 44771 25678 33639 27793 41268 1857 63388 32976 46195 61291 61740 61680 45264 76361 49243 73250 67432 29124 15198 30626 18950 13857 32569 45179 47696 21283 77169 26357 97885 54741 57246 61929 25997 45859 3353 32204 97451 60550 49516 38558 57674 36443 666 32486 54910 24666 45119 40779 3441 83779 60645 63003 59616 7364 69058 41651 12201 13234 19717 11536 70350 15611 74484 96153 4127 60035 19418 58613 28590 32917 97780 17644 15836 88046 26489 38410 54618 16967 81604 60064 5833 80689 43137 57274 8002 4445 42933 75378 15130 10689 98579 42780 38379 76394 66337 15126 59535 53639 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92741 28504 9196 36420 79291 10401 83019 16123 4849 29672 55766 92155 95266 19643 71578 15264 16291 22935 93486 84428 35132 39530 24601 24387 44016 96051 84783 81830 86168 99507 21009 65059 55314 49665 1555 10893 15948 57460 94851 90366 5762 54950 86021 50048 62140 45364 32295 17859 95774 39670 54988 30531 27868 89380 22239 49887 43070 95590 77665 9741 88062 30223 1030 44136 21824 75906 90983 39618 52445 34312 44653 9545 59249 95189 35320 57725 83926 53499 79337 83121 53216 44605 87909 57389 75628 15921 93418 29440 40021 16023 77489 86815 67489 32131 67231 92487 48221 72361 49582 95596 16996 76965 21200 39847 82268 64948 2913 30480 31996 6494 14744 40647 79396 55844 29701 41150 1281 24169 76769 15240 16040 50930 66139 81059 65990 27756 93168 86078 8143 62161 76000 93838 20281 72269 56417 16398 95739 83526 38417 22213 60409 51016 91652 90945 16635 74235 37793 19243 9687 94272 16837 70033 80780 4938 12563 38605 85583 63400 50217 23434 12214 73625 15085 39375 97528 3898 94356 32208 89273 77821 15833 21557 63614 55517 16188 35130 71486 44163 32464 44239 86378 31081 99488 25883 95079 83797 63441 2008 24448 84996 82221 63937 14915 77773 50395 54389 4567 14174 64922 86930 41745 82210 4571 77558 7919 51570 29231 37032 99687 64450 99335 45854 57476 26093 86227 75606 9240 66229 30509 29618 47123 36971 15733 42961 79706 56098 92362 86625 80310 35775 7140 34988 93966 50132 11474 41040 99373 91117 97274 31433 8402 7850 85710 83163 74831 91751 83658 75671 7348 61785 34135 5122 61552 27934 74575 90822 99963 9600 11665 63327 52624 19580 82213 7698 48135 32579 94715 50867 4865 97416 93361 50560 91820 47697 32015 27044 90189 23893 50341 30860 68525 45306 84749 89346 19381 32075 72690 39462 36326 26401 80383 40265 29576 33026 94296 10253 20436 82586 836 79862 37223 46982 64399 16774 74018 71187 91901 ================================================ FILE: merge_sort_inversions/problem3.5test.txt ================================================ 54044 14108 79294 29649 25260 60660 2995 53777 49689 9083 ================================================ FILE: prim/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(prim) set(CMAKE_CXX_STANDARD 17) add_executable(prim main.cpp) ================================================ FILE: prim/main.cpp ================================================ #include #include #include #include #include #include using namespace std; using Pair = pair; using Pairs = vector; using AdjList = unordered_map; using Queue = priority_queue>; using Set = unordered_set; constexpr auto INF = int(1e9 + 7); int getRandom(int N) { default_random_engine generator; uniform_int_distribution distribution(1, N); return distribution(generator); } int prim(int N, AdjList& adj, Queue q = {}, Set seen = {}, int total = 0) { auto start = getRandom(N); seen.insert(start); for (auto [w, v]: adj[start]) q.push({ w, v }); while (q.size()) { auto [cost, u] = q.top(); q.pop(); if (!seen.insert(u).second) continue; total += cost; for (auto [w, v]: adj[u]) if (seen.find(v) == seen.end()) q.push({ w, v }); } return total; } void run(const string& filename) { AdjList adj; fstream fin{ filename }; int N, M; fin >> N >> M; // N vertices and M edges int u, v, w; // edge u -> v of weight w while (fin >> u >> v >> w) { adj[u].emplace_back(w, v); adj[v].emplace_back(w, u); } auto cost = prim(N, adj); cout << filename << ": " << cost << endl; } int main() { run("problem15.9test.txt"); // problem15.9test.txt: 14 run("problem15.9.txt"); // problem15.9.txt: -3612829 return 0; } ================================================ FILE: prim/main.js ================================================ let LineByLine = require('n-readlines'); let key = x => Array.isArray(x) ? x[0] : x; let heappush = (A, x, f = Math.min) => { let P = i => Math.floor((i - 1) / 2); // parent A.push(x); let N = A.length, i = N - 1; while (0 < i && key(A[i]) == f(key(A[i]), key(A[P(i)]))) { [A[i], A[P(i)]] = [A[P(i)], A[i]]; i = P(i); } }; let heappop = (A, f = Math.min) => { let L = i => 2 * i + 1, // children R = i => 2 * i + 2; let N = A.length, i = 0; let top = A[0]; [A[0], A[N - 1]] = [A[N - 1], A[0]], A.pop(), --N; let ok; do { ok = true; let left = f == Math.min ? Infinity : -Infinity, right = left; if (L(i) < N && key(A[i]) != f(key(A[i]), key(A[L(i)]))) ok = false, left = key(A[L(i)]); if (R(i) < N && key(A[i]) != f(key(A[i]), key(A[R(i)]))) ok = false, right = key(A[R(i)]); if (!ok) { let j = left == f(left, right) ? L(i) : R(i); [A[i], A[j]] = [A[j], A[i]]; i = j; } } while (!ok); return top; }; let prim = (N, adj, q = [], seen = new Set(), total = 0) => { let start = Math.ceil(N * Math.random()); seen.add(start); for (let [w, v] of adj.get(start)) heappush(q, [w, v]); while (q.length) { let [cost, u] = heappop(q); if (seen.has(u)) continue; total += cost; seen.add(u); for (let [w, v] of adj.get(u)) if (!seen.has(v)) heappush(q, [w, v]); } return total; }; let run = filename => { let adj = new Map(); let input = new LineByLine(filename); let line = input.next(); let [N, M] = String.fromCharCode(...line).trim().split(' ').map(Number); while (line = input.next()) { let [u, v, w] = String.fromCharCode(...line).trim().split(' ').map(Number); if (!adj.has(u)) adj.set(u, []); if (!adj.has(v)) adj.set(v, []); adj.get(u).push([w, v]); adj.get(v).push([w, u]); } let cost = prim(N, adj); console.log(`${filename}: ${cost}`); } run('problem15.9test.txt') // problem15.9test.txt: 14 run('problem15.9.txt') // problem15.9.txt: -3612829 ================================================ FILE: prim/main.kt ================================================ import java.io.File import java.util.PriorityQueue import java.util.Random fun prim(N: Int, adj: MutableMap>>): Int { var total: Int = 0 var start = Random().nextInt(N) + 1 var q = PriorityQueue>(Comparator{ a: Pair, b: Pair -> a.first.compareTo(b.first) }) var seen = mutableSetOf(start) for ((w, v) in adj[start]!!) q.add(Pair(w, v)) while (0 < q.size) { var (cost, u) = q.poll() if (seen.contains(u)) continue total += cost; seen.add(u) for ((w, v) in adj[u]!!) if (!seen.contains(v)) q.add(Pair(w, v)) } return total } fun run(filename: String) { var N: Int = 0 var adj = mutableMapOf>>() var first = true File(filename).forEachLine { line -> if (!first) { var (u, v, w) = line.split(" ").map{ it.toInt() } if (!adj.contains(u)) adj[u] = mutableListOf>() if (!adj.contains(v)) adj[v] = mutableListOf>() adj[u]!!.add(Pair(w, v)) adj[v]!!.add(Pair(w, u)) } else { var (numVertex, _) = line.split(" ").map{ it.toInt() } N = numVertex first = false } } var cost = prim(N, adj) println("$filename: $cost") } fun main() { run("problem15.9test.txt") // problem15.9test.txt: 14 run("problem15.9.txt") // problem15.9.txt: -3612829 } ================================================ FILE: prim/main.py ================================================ from random import randint from heapq import heappush, heappop def prim(N, adj, total = 0): q = [] seen = set() start = randint(1, N); seen.add(start) for w, v in adj[start]: heappush(q, [w, v]) while q: cost, u = heappop(q) if u in seen: continue total += cost; seen.add(u) for w, v in adj[u]: if v not in seen: heappush(q, [w, v]) return total def run(filename): adj = {} first = True with open(filename) as fin: while True: line = fin.readline() if not line: break words = line.split() if not first: u, v, w = [int(x) for x in words] if u not in adj: adj[u] = [] if v not in adj: adj[v] = [] adj[u].append([w, v]) adj[v].append([w, u]) else: N, M = [int(x) for x in words] first = False cost = prim(N, adj) print(f'{filename}: {cost}') run('problem15.9test.txt') # problem15.9test.txt: 14 run('problem15.9.txt') # problem15.9.txt: -3612829 ================================================ FILE: prim/package.json ================================================ { "name": "prim", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "clayton wong", "license": "ISC", "dependencies": { "n-readlines": "^1.0.1" } } ================================================ FILE: prim/problem15.9.txt ================================================ 500 2184 1 2 6807 2 3 -8874 3 4 -1055 4 5 4414 5 6 1728 6 7 -2237 7 8 -7507 8 9 7990 9 10 -5012 10 11 7353 11 12 -6736 12 13 -7604 13 14 5273 14 15 9331 15 16 -7753 16 17 -4370 17 18 267 18 19 903 19 20 7674 20 21 5436 21 22 9479 22 23 9432 23 24 4472 24 25 7258 25 26 8709 26 27 4358 27 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488 4881 488 489 7498 489 490 -9312 490 491 7282 491 492 2219 492 493 -4854 493 494 -421 494 495 2927 495 496 327 496 497 4963 497 498 -5404 498 499 -6861 499 500 8293 1 132 -151 1 171 8358 1 244 6723 1 310 9791 1 316 569 1 324 -1612 1 397 -5942 1 414 3655 2 25 6267 2 39 907 2 104 -8744 2 157 1036 2 173 -7751 2 309 -7230 3 91 -8754 3 110 504 3 136 -4218 3 144 -9056 3 231 -494 3 303 5310 3 353 9242 3 482 -1048 4 43 566 4 77 -3033 4 82 -6069 4 93 -2804 4 247 2557 4 266 1686 4 284 8282 4 364 -8897 4 388 5176 4 423 6365 5 122 722 5 153 8816 6 15 -1593 6 73 7693 6 297 -3253 6 343 154 7 103 8257 7 314 -1756 7 416 -5246 7 451 7495 8 124 -5969 8 162 7179 8 194 -507 8 294 -4333 8 318 -4261 8 359 2771 8 432 2206 9 17 -3864 9 250 -1984 9 307 -4003 9 495 5260 10 114 223 10 168 4271 10 330 7863 10 390 -8598 11 43 4323 11 69 171 11 176 -9470 11 469 6195 12 20 -686 12 277 6273 13 24 -9600 13 61 5960 13 152 4272 13 164 -1175 13 486 7432 14 64 -1515 14 209 9744 14 219 -9493 14 276 7321 14 359 9452 14 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-4790 496 500 -1519 ================================================ FILE: prim/problem15.9test.txt ================================================ 6 10 1 2 6 1 4 5 1 5 4 2 4 1 2 5 2 2 3 5 2 6 3 3 6 4 4 5 2 5 6 4 ================================================ FILE: quick_sort/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(quick_sort) set(CMAKE_CXX_STANDARD 17) add_executable(quick_sort main.cpp) ================================================ FILE: quick_sort/main.cpp ================================================ #include #include #include using namespace std; using VI = vector; using fun = function; fun pivotLeft = [](VI& A, int L, int R) { return L; }; fun pivotRight = [](VI& A, int L, int R) { return R; }; fun pivotMedian = [](VI& A, int L, int R) { auto M = L + (R - L) / 2; VI cand{ A[L], A[M], A[R] }; sort(cand.begin(), cand.end()); auto target = cand[1]; if (target == A[L]) return L; if (target == A[M]) return M; if (target == A[R]) return R; }; int partition(VI& A, int L, int R, fun choosePivot) { auto i = L + 1, j = L + 1, k = choosePivot(A, L, R); swap(A[L], A[k]); // swap pivot A[k] with first element of the subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] swap(A[i], A[j]); ++i; } ++j; } swap(A[L], A[i - 1]); // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; } int quicksort(VI& A, int L, int R, fun choosePivot) { if (R <= L) return 0; auto k = partition(A, L, R, choosePivot); return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot); } int run(string& filename, fun choosePivot) { VI A; fstream fin{ filename }; for (string line; fin >> line; A.push_back(stoi(line))); int N = A.size(); return quicksort(A, 0, N - 1, choosePivot); } int main() { string filename{ "problem5.6.txt" }; cout << " left: " << run(filename, pivotLeft) << endl // left: 162085 << " right: " << run(filename, pivotRight) << endl // right: 164123 << "median: " << run(filename, pivotMedian) << endl; // median: 138382 return 0; } ================================================ FILE: quick_sort/main.js ================================================ let pivotLeft = (A, L, R) => L; let pivotRight = (A, L, R) => R; let pivotMedian = (A, L, R) => { let M = L + Math.floor((R - L) / 2); let cand = [A[L], A[M], A[R]].sort((a, b) => a - b), target = cand[1]; if (target == A[L]) return L; if (target == A[M]) return M; if (target == A[R]) return R; }; let partition = (A, L, R, choosePivot) => { let i = L + 1, j = L + 1, k = choosePivot(A, L, R); [A[L], A[k]] = [A[k], A[L]]; // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] [A[i], A[j]] = [A[j], A[i]]; ++i; } ++j; } [A[L], A[i - 1]] = [A[i - 1], A[L]]; // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; }; let quicksort = (A, L, R, choosePivot) => { if (R <= L) return 0; let k = partition(A, L, R, choosePivot); return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot); }; let run = (filename, choosePivot) => { let A = []; let LineByLine = require("n-readlines"); let input = new LineByLine(filename); for (let line; line = input.next(); A.push(Number(line))); return quicksort(A, 0, A.length - 1, choosePivot); } let filename = 'problem5.6.txt'; console.log(` left: ${run(filename, pivotLeft)}`); // left: 162085 console.log(` right: ${run(filename, pivotRight)}`); // right: 164123 console.log(`median: ${run(filename, pivotMedian)}`); // median: 138382 ================================================ FILE: quick_sort/main.kt ================================================ import java.io.File typealias PivotFunc = (A: MutableList, L: Int, R: Int) -> (Int) var pivotLeft: PivotFunc = { _: MutableList, L: Int, _: Int -> L } var pivotRight: PivotFunc = { _: MutableList, _: Int, R: Int -> R } fun _pivotMedian(A: MutableList, L: Int, R: Int): Int { var M = L + (R - L) / 2 var cand = intArrayOf(A[L], A[M], A[R]) cand.sort() var target = cand[1] if (target == A[L]) return L if (target == A[M]) return M if (target == A[R]) return R return -1 } var pivotMedian: PivotFunc = { A: MutableList, L: Int, R: Int -> _pivotMedian(A, L, R) } fun partition(A: MutableList, L: Int, R: Int, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { var i = L + 1 var j = L + 1 var k = choosePivot(A, L, R) A[k] = A[L].also { A[L] = A[k] } // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] A[i] = A[j].also { A[j] = A[i] } ++i } ++j } A[L] = A[i - 1].also { A[i - 1] = A[L] } // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 } fun quicksort(A: MutableList, L: Int, R: Int, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { if (R <= L) return 0 var k = partition(A, L, R, choosePivot) return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot) } fun run(filename: String, choosePivot: (A: MutableList, L: Int, R: Int) -> (Int)): Int { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } return quicksort(A, 0, A.size - 1, choosePivot) } fun main() { var filename = "problem5.6.txt" println(" left: ${run(filename, pivotLeft)}") // left: 162085 println(" right: ${run(filename, pivotRight)}") // right: 164123 println("median: ${run(filename, pivotMedian)}") // median: 138382 } ================================================ FILE: quick_sort/main.py ================================================ def pivotLeft(A, L, R): return L def pivotRight(A, L, R): return R def pivotMedian(A, L, R): M = L + (R - L) // 2 cand = sorted([A[L], A[M], A[R]]) target = cand[1] if target == A[L]: return L if target == A[M]: return M if target == A[R]: return R def partition(A, L, R, choosePivot): i = L + 1 j = L + 1 k = choosePivot(A, L, R) A[L], A[k] = A[k], A[L] # swap pivot A[k] with first element of subarray A[L] while j <= R: if A[j] < A[L]: # maintain loop invariant A[i] < pivot < A[j] A[i], A[j] = A[j], A[i] i += 1 j += 1 A[L], A[i - 1] = A[i - 1], A[L] # swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 def quicksort(A, L, R, choosePivot): if R <= L: return 0 k = partition(A, L, R, choosePivot) return (R - L) + quicksort(A, L, k - 1, choosePivot) + quicksort(A, k + 1, R, choosePivot) def run(filename, choosePivot): A = [] with open(filename) as fin: while True: line = fin.readline() if not line: break A.append(int(line)) return quicksort(A, 0, len(A) - 1, choosePivot) filename = 'problem5.6.txt' print(f' left: {run(filename, pivotLeft)}') # left: 162085 print(f' right: {run(filename, pivotRight)}') # right: 164123 print(f'median: {run(filename, pivotMedian)}') # median: 138382 ================================================ FILE: quick_sort/package.json ================================================ { "name": "quick_sort", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "Clayton Wong", "license": "ISC", "dependencies": { "n-readlines": "^1.0.1" } } ================================================ FILE: quick_sort/problem5.6.txt ================================================ 2148 9058 7742 3153 6324 609 7628 5469 7017 504 4092 1582 9572 1542 5697 2081 4218 3130 7923 9595 6558 3859 9832 3062 6788 7578 7432 479 8439 9079 7173 2667 2770 2655 972 4264 2014 3171 4715 345 4388 3816 8887 3915 3490 2327 123 4596 4307 8737 4007 6798 6551 1627 1190 4984 2480 3404 2027 4778 2951 2795 5002 8121 8910 9593 5254 448 6237 5565 1816 392 8143 9310 9293 3138 4869 6756 872 6183 3517 3513 1676 5498 9172 5739 6108 7538 7671 5780 8666 540 9771 6837 9341 1590 5689 1605 1103 5859 1622 4371 3113 488 6676 6020 2630 6541 6893 6729 4506 3230 2290 72 1976 8259 1373 9962 782 4427 6349 9619 4456 7750 41 8687 5506 2128 578 8001 4493 9107 746 2639 1456 1361 2841 3198 6344 1450 3346 6221 4816 3800 9975 8075 2400 1353 2282 147 2821 497 9028 9448 3574 2209 138 8201 2331 3895 7803 2406 9949 1349 965 1320 8836 9562 5753 1039 2348 2184 8212 4909 3025 1690 6117 7318 6478 768 4551 1054 7000 97 6868 5870 7601 6257 5461 8029 7797 1149 1216 4359 3524 556 1451 8976 1988 7949 6444 3787 8116 2381 1465 971 2917 5502 6225 3482 2078 3013 4993 5924 8444 1271 1032 1708 5366 3841 3112 9234 4156 1849 8718 396 4231 2517 3773 4088 1006 2623 7534 7113 2218 1175 1879 3188 7698 4226 2644 2625 8237 8644 3649 7304 8242 800 7335 4589 1619 1996 699 7991 4889 6241 5444 8466 7658 9880 378 2112 2831 3737 988 2619 3354 6426 4602 6058 1068 9614 6235 2049 7151 9038 232 5336 6988 6467 306 480 716 4800 9048 7423 294 8595 8055 8542 7516 1398 490 219 2820 4777 2030 7729 5005 8544 597 8790 740 2443 5217 6178 2338 8176 8320 6521 6597 5186 6604 9467 2598 6898 3274 8254 947 1178 8770 3910 8314 543 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9776 2965 6432 4697 7196 8285 1200 8918 4393 7508 955 5904 9176 4154 2914 1188 7637 6379 3140 9687 3967 9186 9816 1261 3465 4851 5801 8662 3124 7409 9627 3632 2866 9764 9651 3040 598 5446 7208 6635 7110 2622 7741 1969 2618 6795 7573 2851 3129 3554 7144 6385 5871 3544 2168 4173 9805 6131 1719 5396 764 4862 8921 1916 4960 7802 550 5933 2009 8720 4693 2313 4434 4308 982 9168 6553 9495 5569 5737 9285 1468 4788 3821 5266 3636 7333 6079 9667 735 5785 7965 9161 6538 141 9820 9282 9328 9501 8077 945 3706 473 687 6309 4209 2602 5773 6806 2145 9759 7363 5470 410 6996 131 4354 5082 3648 3647 1982 1994 3655 9755 6204 6750 8342 6053 539 7500 3740 1993 7867 4819 9140 111 4583 5213 1981 745 826 2096 6209 7981 2556 9177 4897 9650 9023 9717 8705 2578 9838 886 7717 9272 4310 7840 9228 1604 2921 2764 4988 8355 9670 1011 9762 9777 4118 9606 6805 1245 7002 7519 8954 1559 1243 9415 2403 9729 9752 177 8069 8161 8758 8056 5019 4090 9279 8735 1602 2011 9019 2849 6284 133 8829 3635 3443 5650 1529 4509 7480 9428 7775 1062 538 5133 1212 3944 3438 9944 5146 6392 8147 3260 3038 8454 1964 2837 7629 5129 2146 5122 1900 5325 7402 7758 4621 6096 575 783 9887 2003 4613 7874 3345 6542 4499 9939 9550 344 3866 3200 9396 1899 4757 8989 3351 8543 8234 8324 2288 6097 4765 6717 6311 7365 5107 9861 6940 3847 212 6228 2392 7054 1851 6352 2326 9536 4474 3065 1617 1545 7442 9355 3152 2016 6122 7764 5670 1110 3726 2880 9369 4443 7931 2179 3592 3163 8435 3079 3754 852 4572 4488 2680 2105 8934 2894 5866 1418 9319 7957 7593 9639 6948 7789 8284 981 7907 2890 3011 8306 6443 2565 3477 185 3179 4222 2254 2776 5601 2111 7712 4097 3102 376 6073 3764 300 8570 1852 1799 9927 140 9304 665 9071 9399 9708 6529 4030 8882 7420 6301 1282 8916 9914 2566 9622 4575 2414 4241 2356 1711 4617 9368 3031 7915 932 6247 7197 2469 6435 142 1809 9104 2944 9314 6810 4721 3993 2225 2502 7569 5906 9922 77 4240 2661 4763 2509 6852 9855 9831 3384 890 3878 4253 4977 4859 6686 6273 7072 7699 6121 1094 7108 7653 36 807 49 1580 1911 1294 7081 4536 8569 1332 6106 4966 9393 7179 4235 2073 2761 5137 280 1702 9052 4165 934 7383 2357 8477 3363 3215 2417 8099 7204 1833 2182 4244 4768 5064 1620 4653 8258 8611 7320 9149 4578 5343 6457 8791 229 8203 5300 4811 8559 7218 6160 8589 5318 3242 4616 3318 7620 1561 4771 6705 1440 5214 816 8106 7364 3782 7188 8003 9482 9054 1283 5740 9490 6067 8640 7266 5792 9269 ================================================ FILE: quick_sort/problem5.6test1.txt ================================================ 2148 9058 7742 3153 6324 609 7628 5469 7017 504 ================================================ FILE: quick_sort/problem5.6test2.txt ================================================ 2148 9058 7742 3153 6324 609 7628 5469 7017 504 4092 1582 9572 1542 5697 2081 4218 3130 7923 9595 6558 3859 9832 3062 6788 7578 7432 479 8439 9079 7173 2667 2770 2655 972 4264 2014 3171 4715 345 4388 3816 8887 3915 3490 2327 123 4596 4307 8737 4007 6798 6551 1627 1190 4984 2480 3404 2027 4778 2951 2795 5002 8121 8910 9593 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y) @test expect == actual println("($x x $y)\nexpect: $expect\nactual: $actual\n") end end ================================================ FILE: rec_mat_mult/main.py ================================================ # # Standard Recursive Matrix Multiplication (RecMatMult) # # Input: n x n integer matrices X and Y # Output: Z = X * Y # Assumption: n is a power of 2 # Complexity: O(n^3) - T(n) = 8T(n/2) + O(n^2) # import numpy as np def go(X, Y): n = X.shape[0] # Base case: 1x1 matrix if n == 1: return X * Y # Divide: Partition A and B into n/2 x n/2 submatrices k = n // 2 A, B = X[:k, :k], X[:k, k:] C, D = X[k:, :k], X[k:, k:] E, F = Y[:k, :k], Y[:k, k:] G, H = Y[k:, :k], Y[k:, k:] # Combine: Reconstruct the full matrix from quadrants Z = np.zeros((n, n), dtype=X.dtype) Z[:k, :k], Z[:k, k:] = go(A, E) + go(B, G), go(A, F) + go(B, H) Z[k:, :k], Z[k:, k:] = go(C, E) + go(D, G), go(C, F) + go(D, H) return Z def pretty_print(A, label=''): col_width = max(len(str(x)) for x in A.flat) if len(label): print(label) n = A.shape[0] spaces = n - 1 print('-' * (n * col_width + spaces)) fmt = f'{{:>{col_width}d}}' for row in A: print(' '.join(fmt.format(x) for x in row)) print() def main(): n = 4 # n must be a power of 2 for this simple implementation A = np.random.randint(0, 10, (n, n)) B = np.random.randint(0, 10, (n, n)) expect = A @ B actual = go(A, B) pretty_print(A, 'A') pretty_print(B, 'B') pretty_print(expect, 'A * B (expect)') pretty_print(actual, 'A * B (actual)') assert(np.array_equal(expect, actual)) if __name__ == "__main__": main() # ➜ rec_mat_mult git:(main) ✗ uv run ./main.py # A # ------- # 9 0 6 8 # 4 6 9 2 # 0 5 9 9 # 9 1 1 3 # B # ------- # 2 3 4 6 # 5 5 4 3 # 1 6 8 0 # 5 2 4 2 # A * B (expect) # --------------- # 64 79 116 70 # 57 100 120 46 # 79 97 128 33 # 39 44 60 63 # A * B (actual) # --------------- # 64 79 116 70 # 57 100 120 46 # 79 97 128 33 # 39 44 60 63 ================================================ FILE: rec_mat_mult/pyproject.toml ================================================ [project] name = "rec-mat-mult" version = "0.1.0" description = "Recursive matrix multiplication" readme = "README.md" requires-python = ">=3.14" dependencies = [ "numpy>=2.4.1", ] ================================================ FILE: rselect/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(rselect) set(CMAKE_CXX_STANDARD 17) add_executable(rselect main.cpp) ================================================ FILE: rselect/main.cpp ================================================ #include #include #include #include using namespace std; using VI = vector; int random(int L, int R) { random_device rd; mt19937 gen{ rd() }; uniform_int_distribution dist(L, R); return dist(gen); } int partition(VI& A, int L, int R) { auto i = L + 1, j = L + 1, k = random(L, R); swap(A[L], A[k]); // swap pivot A[k] with first element of the subarray A[L] while (j <= R) { if (A[j] < A[L]) // maintain loop invariant A[i] < pivot < A[j] swap(A[i++], A[j]); ++j; } swap(A[L], A[i - 1]); // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; } int rselect(VI& A, int i, int L, int R) { auto k = partition(A, L, R); if (i == k) return A[k]; // 🎯 lucky guess if (i < k) R = k - 1; else L = k + 1; return rselect(A, i, L, R); } int run(string filename, int i, VI A = {}) { fstream fin{ filename }; for (string line; fin >> line; A.push_back(stoi(line))); int N = A.size(); return rselect(A, i - 1, 0, N - 1); // -1 for 0-based indexing } int main() { cout << "problem6.5test1.txt: " << run("problem6.5test1.txt", 5) << endl; // problem6.5test1.txt: 5469 cout << "problem6.5test2.txt: " << run("problem6.5test2.txt", 50) << endl; // problem6.5test2.txt: 4715 return 0; } ================================================ FILE: rselect/main.js ================================================ let random = (L, R) => Math.floor(Math.random() * (R + 1 - L) + L); // +1 for L..R inclusive let partition = (A, L, R) => { let i = L + 1, j = L + 1, k = random(L, R); [A[L], A[k]] = [A[k], A[L]]; // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] [A[i], A[j]] = [A[j], A[i]]; ++i; } ++j; } [A[L], A[i - 1]] = [A[i - 1], A[L]]; // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1; }; let rselect = (A, i, L, R) => { let k = partition(A, L, R); if (i == k) return A[k]; // 🎯 lucky guess if (i < k) R = k - 1; else L = k + 1; return rselect(A, i, L, R); } let run = (filename, i) => { let A = []; let LineByLine = require("n-readlines"); let input = new LineByLine(filename); for (let line; line = input.next(); A.push(Number(line))); let N = A.length; return rselect(A, i - 1, 0, N - 1); // -1 for 0-based indexing }; console.log(`problem6.5test1.txt: ${run('problem6.5test1.txt', 5)}`); // problem6.5test1.txt: 5469 console.log(`problem6.5test2.txt: ${run('problem6.5test2.txt', 50)}`); // problem6.5test2.txt: 4715 ================================================ FILE: rselect/main.kt ================================================ import java.io.File import kotlin.random.Random fun partition(A: MutableList, L: Int, R: Int): Int { var i = L + 1 var j = L + 1 var k = Random.nextInt(L, R + 1) // +1 for L..R inclusive A[L] = A[k].also { A[k] = A[L] } // swap pivot A[k] with first element of subarray A[L] while (j <= R) { if (A[j] < A[L]) { // maintain loop invariant A[i] < pivot < A[j] A[i] = A[j].also { A[j] = A[i] } ++i } ++j } A[L] = A[i - 1].also { A[i - 1] = A[L] } // swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 } fun rselect(A: MutableList, i: Int, L_: Int, R_: Int): Int { var L = L_ var R = R_ var k = partition(A, L, R) if (i == k) return A[k] // 🎯 lucky guess if (i < k) R = k - 1 else L = k + 1 return rselect(A, i, L, R) } fun run(filename: String, i: Int): Int { var A = mutableListOf() File(filename).forEachLine { A.add(it.toInt()) } var N = A.size return rselect(A, i - 1, 0 , N - 1) // -1 for 0-based indexing } fun main() { println("problem6.5test1.txt: " + run("problem6.5test1.txt", 5)) // problem6.5test1.txt: 5469 println("problem6.5test2.txt: " + run("problem6.5test2.txt", 50)) // problem6.5test2.txt: 4715 } ================================================ FILE: rselect/main.py ================================================ from random import uniform from math import floor def partition(A, L, R): i = L + 1 j = L + 1 k = floor(uniform(L, R)) A[L], A[k] = A[k], A[L] # swap pivot A[k] with first element of subarray A[L] while j <= R: if A[j] < A[L]: # maintain loop invariant A[i] < pivot < A[j] A[i], A[j] = A[j], A[i] i += 1 j += 1 A[L], A[i - 1] = A[i - 1], A[L] # swap pivot A[L] with last value less-than pivot A[i - 1] return i - 1 def rselect(A, i, L, R): k = partition(A, L, R) if i == k: return A[k] # 🎯 lucky guess if i < k: R = k - 1 else: L = k + 1 return rselect(A, i, L, R) def run(filename, i): A = [] with open(filename) as fin: while True: line = fin.readline() if not line: break A.append(int(line)) N = len(A) return rselect(A, i - 1, 0, N - 1) # -1 for 0-based indexing print('problem6.5test1.txt:', run('problem6.5test1.txt', 5)) # problem6.5test1.txt: 5469 print('problem6.5test2.txt:', run('problem6.5test2.txt', 50)) # problem6.5test2.txt: 4715 ================================================ FILE: rselect/package.json ================================================ { "name": "rselect", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "Clayton Wong", "license": "ISC", "dependencies": { "n-readlines": "^1.0.1" } } ================================================ FILE: rselect/problem6.5test1.txt ================================================ 2148 9058 7742 3153 6324 609 7628 5469 7017 504 ================================================ FILE: rselect/problem6.5test2.txt ================================================ 2148 9058 7742 3153 6324 609 7628 5469 7017 504 4092 1582 9572 1542 5697 2081 4218 3130 7923 9595 6558 3859 9832 3062 6788 7578 7432 479 8439 9079 7173 2667 2770 2655 972 4264 2014 3171 4715 345 4388 3816 8887 3915 3490 2327 123 4596 4307 8737 4007 6798 6551 1627 1190 4984 2480 3404 2027 4778 2951 2795 5002 8121 8910 9593 5254 448 6237 5565 1816 392 8143 9310 9293 3138 4869 6756 872 6183 3517 3513 1676 5498 9172 5739 6108 7538 7671 5780 8666 540 9771 6837 9341 1590 5689 1605 1103 5859 ================================================ FILE: strassen/main.py ================================================ # # Strassen's Recursive Matrix Multiplication # # Input: n x n integer matrices X and Y # Output: Z = X * Y # Assumption: n is a power of 2 # Complexity: O(n^log2(7)) # import numpy as np def go(X, Y): n = X.shape[0] # Base case: 1x1 matrix if n == 1: return X * Y # Divide: Partition A and B into n/2 x n/2 submatrices k = n // 2 A, B = X[:k, :k], X[:k, k:] C, D = X[k:, :k], X[k:, k:] E, F = Y[:k, :k], Y[:k, k:] G, H = Y[k:, :k], Y[k:, k:] # Combine: Reconstruct the full matrix from quadrants Z = np.zeros((n, n), dtype=X.dtype) P1 = go(A, F - H) P2 = go(A + B, H) P3 = go(C + D, E) P4 = go(D, G - E) P5 = go(A + D, E + H) P6 = go(B - D, G + H) P7 = go(A - C, E + F) Z[:k, :k], Z[:k, k:] = (P5 + P4 - P2 + P6), (P1 + P2) Z[k:, :k], Z[k:, k:] = (P3 + P4), (P1 + P5 - P3 - P7) return Z def pretty_print(A, label=''): col_width = max(len(str(x)) for x in A.flat) if len(label): print(label) n = A.shape[0] spaces = n - 1 print('-' * (n * col_width + spaces)) fmt = f'{{:>{col_width}d}}' for row in A: print(' '.join(fmt.format(x) for x in row)) print() def main(): n = 4 # n must be a power of 2 for this simple implementation A = np.random.randint(0, 10, (n, n)) B = np.random.randint(0, 10, (n, n)) expect = A @ B actual = go(A, B) pretty_print(A, 'A') pretty_print(B, 'B') pretty_print(expect, 'A * B (expect)') pretty_print(actual, 'A * B (actual)') assert(np.array_equal(expect, actual)) if __name__ == "__main__": main() # ➜ strassen git:(main) ✗ uv run ./main.py # A # ------- # 3 8 3 3 # 9 2 9 6 # 1 8 4 8 # 1 2 8 4 # B # ------- # 7 3 1 9 # 1 7 9 2 # 6 7 7 8 # 0 1 5 8 # A * B (expect) # --------------- # 47 89 111 91 # 119 110 120 205 # 39 95 141 121 # 57 77 95 109 # A * B (actual) # --------------- # 47 89 111 91 # 119 110 120 205 # 39 95 141 121 # 57 77 95 109 ================================================ FILE: strassen/pyproject.toml ================================================ [project] name = "strassen" version = "0.1.0" description = "Add your description here" readme = "README.md" requires-python = ">=3.14" dependencies = [ "numpy>=2.4.1", ] ================================================ FILE: strassen/variants.py ================================================ # # Strassen-family Recursive Matrix Multiplication (2x2 block form) # # Methods: # - "strassen" : classical Strassen scheme (7 recursive multiplies) # - "winograd" : Winograd rearrangement of Strassen (7 multiplies, fewer additions) # - "basis2017" : alternative-basis 7-multiply scheme (basis change + recombination) # # All methods: # - Recursively split X and Y into 2x2 blocks # - Perform 7 recursive multiplications # - Recombine into result blocks # # Assumption: n is a power of 2 # import numpy as np def go(X: np.ndarray, Y: np.ndarray, method: str = "strassen") -> np.ndarray: n = X.shape[0] # Base case: 1x1 matrix multiply if n == 1: return X * Y k = n // 2 # Partition X into 2x2 block matrix # X = [[A11 A12] # [A21 A22]] A11, A12 = X[:k, :k], X[:k, k:] A21, A22 = X[k:, :k], X[k:, k:] # Partition Y into 2x2 block matrix # Y = [[B11 B12] # [B21 B22]] B11, B12 = Y[:k, :k], Y[:k, k:] B21, B22 = Y[k:, :k], Y[k:, k:] # Allocate result matrix Z = np.zeros((n, n), dtype=X.dtype) # ============================================================ # Classical Strassen scheme # ============================================================ if method == "strassen": # 7 recursive multiplications M1 = go(A11 + A22, B11 + B22, method) M2 = go(A21 + A22, B11, method) M3 = go(A11, B12 - B22, method) M4 = go(A22, B21 - B11, method) M5 = go(A11 + A12, B22, method) M6 = go(A21 - A11, B11 + B12, method) M7 = go(A12 - A22, B21 + B22, method) # Recombine into C blocks C11 = M1 + M4 - M5 + M7 C12 = M3 + M5 C21 = M2 + M4 C22 = M1 - M2 + M3 + M6 Z[:k, :k] = C11 Z[:k, k:] = C12 Z[k:, :k] = C21 Z[k:, k:] = C22 return Z # ============================================================ # Winograd rearrangement of Strassen # Same 7 multiplies, different additive structure # ============================================================ if method == "winograd": # Core recursive products t = go(A11, B11, method) u = go(A21 - A11, B12 - B22, method) v = go(A21 + A22, B12 - B11, method) # w reuses t to reduce total additions w = t + go(A21 + A22 - A11, B11 + B22 - B12, method) # Recombine result blocks C11 = t + go(A12, B21, method) C12 = w + v + go(A11 + A12 - A21 - A22, B22, method) C21 = w + u + go(A22, B21 + B12 - B11 - B22, method) C22 = w + u + v Z[:k, :k] = C11 Z[:k, k:] = C12 Z[k:, :k] = C21 Z[k:, k:] = C22 return Z # ============================================================ # 2017 alternative-basis 7-multiply scheme # Performs a basis change on A22 and B22 blocks # ============================================================ if method == "basis2017": # Basis transformation of lower-right blocks A22p = A12 - A21 + A22 B22p = B12 - B21 + B22 # Linear combinations used by recursive multiplies t1 = A21 + A22p t2 = A22p - A12 t3 = A22p - A11 t4 = B22p - B11 t5 = B21 + B22p t6 = B22p - B12 # 7 recursive multiplications M1 = go(A11, B11, method) M2 = go(A12, B21, method) M3 = go(A21, t4, method) M4 = go(A22p, B22p, method) M5 = go(t1, t5, method) M6 = go(t2, t6, method) M7 = go(t3, B12, method) # Initial recombination C11 = M1 + M2 C12 = M5 - M7 C21 = M3 + M6 C22 = M5 + M6 - M2 - M4 # Final correction step required by basis change C12 = C12 - C22 C21 = C22 - C21 Z[:k, :k] = C11 Z[:k, k:] = C12 Z[k:, :k] = C21 Z[k:, k:] = C22 return Z raise ValueError( f"Unknown method: {method!r}. Use 'strassen', 'winograd', or 'basis2017'." ) def pretty_print(A, label=''): col_width = max(len(str(int(x))) for x in A.flat) if label: print(label) n = A.shape[0] print('-' * (n * col_width + (n - 1))) fmt = f'{{:>{col_width}d}}' for row in A: print(' '.join(fmt.format(int(x)) for x in row)) print() def main(): n = 4 # power of 2 X = np.random.randint(0, 10, (n, n), dtype=int) Y = np.random.randint(0, 10, (n, n), dtype=int) expect = X @ Y for method in ["strassen", "winograd", "basis2017"]: actual = go(X, Y, method=method) print(f"=== {method} ===") assert np.array_equal(expect, actual), f"Mismatch for {method}" pretty_print(X, "X") pretty_print(Y, "Y") pretty_print(expect, "X * Y (expect)") pretty_print(go(X, Y, "strassen"), "X * Y (strassen)") pretty_print(go(X, Y, "winograd"), "X * Y (winograd)") pretty_print(go(X, Y, "basis2017"), "X * Y (basis2017)") if __name__ == "__main__": main() # ➜ strassen git:(main) ✗ uv run strassen.py # === strassen === # === winograd === # === basis2017 === # X # ------- # 0 1 6 6 # 4 9 0 4 # 8 2 4 0 # 1 9 7 7 # Y # ------- # 5 3 4 0 # 7 2 8 0 # 5 9 3 2 # 0 1 3 8 # X * Y (expect) # --------------- # 37 62 44 60 # 83 34 100 32 # 74 64 60 8 # 103 91 118 70 # X * Y (strassen) # --------------- # 37 62 44 60 # 83 34 100 32 # 74 64 60 8 # 103 91 118 70 # X * Y (winograd) # --------------- # 37 62 44 60 # 83 34 100 32 # 74 64 60 8 # 103 91 118 70 # X * Y (basis2017) # --------------- # 37 62 44 60 # 83 34 100 32 # 74 64 60 8 # 103 91 118 70 ================================================ FILE: topo_sort/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(topo_sort) set(CMAKE_CXX_STANDARD 17) add_executable(topo_sort main.cpp) ================================================ FILE: topo_sort/main.cpp ================================================ #include #include #include #include #include #include using namespace std; using VI = vector; using AdjList = unordered_map; using Set = unordered_set; using Map = unordered_map; using Queue = queue; using fun = function; class Solution { private: AdjList adj; const int N; Map m; Set seen; int color; public: Solution(AdjList& adj) : adj{ adj }, N{ int(adj.size()) } { } void init(int start) { m.clear(); seen.clear(); color = start; } string topo_sort_bfs() { init(1); // 👉 color forward from 1..N bfs(); return to_string(); } string topo_sort_dfs() { init(N); // 👈 color reverse from N..1 (as the recursive stack unwinds) for (auto [u, _]: adj) dfs(u); return to_string(); } void bfs() { Map degree; for (auto [_, neighbors]: adj) for (auto v: neighbors) ++degree[v]; Queue q; for (auto [u, _]: adj) if (!degree[u] && seen.insert(u).second) q.push(u); while (q.size()) { auto u = q.front(); q.pop(); m[u] = color++; for (auto v: adj[u]) if (!--degree[v] && seen.insert(v).second) q.push(v); } } void dfs(char start) { fun go = [&](auto u) { if (!seen.insert(u).second) return; for (auto v: adj[u]) go(v); m[u] = color--; }; go(start); } string to_string() { ostringstream os; for (auto [u, color]: m) os << u << ": " << color << endl; return os.str(); } }; int main() { // // graph from Quiz 8.3 on page 45 of Algorithms Illuminated: Part 2 // AdjList adj{ { 's', { 'v', 'w' } }, { 'v', { 't' } }, { 'w', { 't' } }, { 't', {} } }; Solution solution{ adj }; cout << "BFS:" << endl << solution.topo_sort_bfs() << endl << "DFS:" << endl << solution.topo_sort_dfs() << endl; // BFS: // t: 4 // w: 3 // v: 2 // s: 1 // // DFS: // s: 1 // w: 2 // v: 3 // t: 4 return 0; } ================================================ FILE: topo_sort/main.js ================================================ class Solution { constructor(adj) { this.adj = adj; this.N = this.adj.size; } init(start) { this.color = start; this.seen = new Set(); this.m = new Map(); } topo_sort_bfs() { this.init(1); // 👉 color forward from 1..N this.bfs(); return this.to_string(); } topo_sort_dfs() { this.init(this.N); // 👈 color reverse from N..1 (as the recursive stack unwinds) for (let [u, _] of [...this.adj]) this.dfs(u); return this.to_string(); } bfs() { let degree = new Map(); for (let [u, _] of [...this.adj]) { degree.set(u, (degree.get(u) || 0)); for (let v of this.adj.get(u)) degree.set(v, 1 + (degree.get(v) || 0)); } let q = [...this.adj].map(([u, _]) => u).filter(u => !degree.get(u)); let seen = new Set(q); while (q.length) { let u = q.shift(); this.m.set(u, this.color++); for (let v of this.adj.get(u)) { degree.set(v, -1 + degree.get(v)); if (!degree.get(v) && !seen.has(v)) q.push(v), seen.add(v); } } } dfs(u) { if (this.seen.has(u)) return; this.seen.add(u); for (let v of this.adj.get(u)) if (!this.seen.has(v)) this.dfs(v); this.m.set(u, this.color--); } to_string() { let s = []; for (let [u, color] of [...this.m]) s.push(`${u}: ${color}`); return s.join('\n'); } } let adj = new Map(); adj.set('s', ['v', 'w']); adj.set('v', ['t']); adj.set('w', ['t']); adj.set('t', []); let solution = new Solution(adj); console.log(`BFS:\n${solution.topo_sort_bfs()}\n\nDFS:\n${solution.topo_sort_dfs()}`); // BFS: // s: 1 // v: 2 // w: 3 // t: 4 // DFS: // t: 4 // v: 3 // w: 2 // s: 1 ================================================ FILE: topo_sort/main.kt ================================================ import java.util.Queue import java.util.LinkedList class Solution(val adj: MutableMap>) { var N: Int var color: Int var m = mutableMapOf() var seen = mutableSetOf() init { N = adj.size color = 0 } fun init(start: Int) { color = start m.clear() seen.clear() } fun topoSortBFS(): String { init(1) // 👉 color forward from 1..N bfs() return toString() } fun topoSortDFS(): String { init(N) // 👈 color reverse from N..1 (as the recursive stack unwinds) adj.forEach{ (u, _) -> dfs(u) } return toString() } fun bfs() { var degree = mutableMapOf() adj.forEach{ (_, neighbors) -> neighbors.forEach{ v -> degree[v] = 1 + degree.getOrDefault(v, 0) } } var q: Queue = LinkedList(adj.map{ (u, _) -> u }.filter{ !degree.contains(it) }) while (0 < q.size) { var u = q.poll() m[u] = color++ adj[u]!!.forEach{ v -> degree[v] = degree[v]!!.minus(1) if (degree[v] == 0 && !seen.contains(v)) { q.add(v); seen.add(v) } } } } fun dfs(u: Char) { if (seen.contains(u)) return seen.add(u) adj[u]!!.forEach{ v -> dfs(v) } m[u] = color-- } override fun toString(): String { var s = mutableListOf() adj.forEach{ (u, _) -> s.add("$u: ${m[u]}") } return s.joinToString("\n") } } fun main() { var adj = mutableMapOf>( 's' to listOf('v', 'w'), 'v' to listOf('t'), 'w' to listOf('t'), 't' to listOf() ) var solution = Solution(adj) println("BFS:\n${solution.topoSortBFS()}\n\nDFS:\n${solution.topoSortDFS()}") // BFS: // s: 1 // v: 2 // w: 3 // t: 4 // DFS: // s: 1 // v: 3 // w: 2 // t: 4 } ================================================ FILE: topo_sort/main.py ================================================ from collections import deque class Solution: def __init__(self, adj): self.adj = adj self.N = len(adj) self.seen = set() self.m = {} def init(self, start): self.color = start self.seen.clear() self.m.clear() def topo_sort_bfs(self): self.init(1) # 👉 color forward from 1..N self.bfs() return self.to_string() def topo_sort_dfs(self): self.init(self.N) # 👈 color reverse from N..1 (as the recursive stack unwinds) for u, _ in self.adj.items(): self.dfs(u) return self.to_string() def bfs(self): degree = {} for _, neighbors in self.adj.items(): for v in neighbors: degree[v] = 1 + (degree[v] if v in degree else 0) q = deque(u for u, _ in self.adj.items() if u not in degree) self.seen.update(*q) while q: u = q.popleft() self.m[u] = self.color; self.color += 1 for v in adj[u]: degree[v] -= 1 if not degree[v] and v not in self.seen: q.append(v); self.seen.add(v) def dfs(self, u): if u in self.seen: return self.seen.add(u) for v in adj[u]: self.dfs(v) self.m[u] = self.color; self.color -= 1 def to_string(self): s = [] for u, color in self.m.items(): s.append(f'{u}: {color}') return '\n'.join(s) # # graph from Quiz 8.3 on page 45 of Algorithms Illuminated: Part 2 # adj = { 's': ['v', 'w'], 'v': ['t'], 'w': ['t'], 't': [] } solution = Solution(adj) print(f'BFS:\n{solution.topo_sort_bfs()}\n\nDFS:\n{solution.topo_sort_dfs()}') # BFS: # s: 1 # v: 2 # w: 3 # t: 4 # DFS: # t: 4 # v: 3 # w: 2 # s: 1 ================================================ FILE: traveling_salesman/README.md ================================================ Quiz 19.2 and 20.7 input file format: ``` [number_of_vertices] [number_of_edges] [one_endpoint_of_edge_1] [other_endpoint_of_edge_1] [edge_1_cost] [one_endpoint_of_edge_2] [other_endpoint_of_edge_2] [edge_2_cost] ``` ================================================ FILE: traveling_salesman/main.cpp ================================================ #include #include #include #include #include #include #include using namespace std; using VI = vector; using Pair = pair; using Set = unordered_set; using Adj = unordered_map; using Cost = unordered_map; class Solution { int best, start, M, N; VI best_path; Adj adj; Cost cost; string key(int u, int v) { stringstream ss; ss << u << "," << v; return ss.str(); } void init(const string& input_file, string line = {}) { best = 1e9 + 7; start = M = N = 0; best_path.clear(); adj.clear(); cost.clear(); auto i = 0; fstream fin{ input_file }; while (getline(fin, line)) { VI A; istringstream is{ line }; copy(istream_iterator(is), istream_iterator(), back_inserter(A)); if (0 < i++) { auto [u, v, w] = tie(A[0], A[1], A[2]); adj[u].insert(v); cost[key(u, v)] = w; adj[v].insert(u); cost[key(v, u)] = w; } else { N = A[0]; M = A[1]; } } } void go(int u, VI&& path, Set&& seen, int t = 0) { if (seen.size() == N) { t += cost[key(u, start)]; // connect ultimate edge of tour if (adj[u].find(start) != adj[u].end() && t < best) best = t, best_path = path; return; } for (auto v: adj[u]) { if (seen.find(v) != seen.end()) continue; path.push_back(v), seen.insert(v); go(v, move(path), move(seen), t + cost[key(u, v)]); path.pop_back(), seen.erase(v); } } public: Pair run(const string& input_file) { init(input_file); start = 1; go(start, {start}, {start}); return {best, best_path}; } }; int main() { auto s = Solution(); for (auto& input_file: {"quiz19.2.txt", "quiz20.7.txt"}) { auto [best, path] = s.run(input_file); cout << input_file << " best: " << best << " path: "; copy(path.begin(), path.end(), ostream_iterator(cout, " ")); cout << endl; } // quiz19.2.txt best: 13 path: 1 3 4 2 // quiz20.7.txt best: 23 path: 1 4 5 2 3 return 0; } ================================================ FILE: traveling_salesman/main.js ================================================ let LineByLine = require('n-readlines'); class Solution { key = (u, v) => `${u},${v}`; init(input_file) { this.best = Number(1e9 + 7); this.best_path = []; this.M = 0; this.N = 0; this.adj = new Map(); this.cost = new Map(); let [A, line, i] = [[], '', 0]; let input = new LineByLine(input_file); while (line = input.next()) { let A = String.fromCharCode(...line).trim().split(' ').map(Number); if (0 < i++) { let [u, v, w] = A; if (!this.adj.has(u)) this.adj.set(u, new Set()); if (!this.adj.has(v)) this.adj.set(v, new Set()); this.adj.get(u).add(v), this.cost.set(this.key(u, v), w); this.adj.get(v).add(u), this.cost.set(this.key(v, u), w); } else { [this.N, this.M] = A; } } } run(input_file) { this.init(input_file); this.start = 1; this.go(this.start, [this.start], new Set([this.start])); return [this.best, this.best_path]; } go(u, path, seen, t = 0) { if (seen.size == this.N) { t += this.cost.get(this.key(u, this.start)) || 0; if (this.adj.get(u).has(this.start) && t < this.best) this.best = t, this.best_path = [...path]; return; } for (let v of this.adj.get(u)) { if (seen.has(v)) continue; path.push(v), seen.add(v); this.go(v, path, seen, t + this.cost.get(this.key(u, v))); path.pop(), seen.delete(v); } } } let s = new Solution(); for (let input_file of ['quiz19.2.txt', 'quiz20.7.txt']) { let [best, path] = s.run(input_file); console.log(`${input_file} best: ${best} path: ${path}`); } // quiz19.2.txt best: 13 path: 1,2,4,3 // quiz20.7.txt best: 23 path: 1,3,2,5,4 ================================================ FILE: traveling_salesman/main.kt ================================================ import java.io.File class Solution() { private var best = (1e9 + 7).toInt() private var best_path = listOf() private var start = 0 private var M = 0 private var N = 0 private var adj = mutableMapOf>() private var cost = mutableMapOf() private var key = { u: Int, v: Int -> "$u,$v" } private fun init(input_file: String) { best = (1e9 + 7).toInt() best_path = mutableListOf() adj = mutableMapOf>() cost = mutableMapOf() var i = 0 File(input_file).forEachLine { line -> var A = line.trim().split(" ").map{ it.toInt() } if (0 < i++) { var (u, v, w) = A if (!adj.contains(u)) adj[u] = mutableSetOf() if (!adj.contains(v)) adj[v] = mutableSetOf() adj[u]!!.add(v); cost[key(u, v)] = w adj[v]!!.add(u); cost[key(v, u)] = w } else { N = A[0] M = A[1] } } } fun run(input_file: String): Pair> { init(input_file) start = 1 go(start, mutableListOf(start), mutableSetOf(start)) return Pair>(best, best_path) } fun go(u: Int, path: MutableList, seen: MutableSet, t_: Int = 0) { if (seen.size == N) { var t = t_ + (cost[key(u, start)] ?: 0) if (adj[u]!!.contains(start) && t < best) { best = t; best_path = path.toList() } return } for (v in adj[u]!!) { if (seen.contains(v)) continue path.add(v); seen.add(v) go(v, path, seen, t_ + cost[key(u, v)]!!) path.removeLast(); seen.remove(v) } } } fun main() { var s = Solution() for (input_file in listOf("quiz19.2.txt", "quiz20.7.txt")) { var (best, path) = s.run(input_file) println("$input_file best: $best path: ${path.joinToString()}") } } // quiz19.2.txt best: 13 path: 1, 2, 4, 3 // quiz20.7.txt best: 23 path: 1, 3, 2, 5, 4 ================================================ FILE: traveling_salesman/main.py ================================================ from random import randint from collections import defaultdict class Solution(): def init(self, input_file): self.best = int(1e9 + 7) self.best_path = [] self.M = 0 self.N = 0 self.adj = defaultdict(set) self.cost = defaultdict(int) with open(input_file) as input: for i, line in enumerate(input): A = [int(x) for x in line.strip().split(' ')] if 0 < i: u, v, w = A self.adj[u].add(v); self.cost[(u, v)] = w self.adj[v].add(u); self.cost[(v, u)] = w else: self.N, self.M = A def run(self, input_file): self.init(input_file) self.start = 1 self.go(self.start, [self.start], set([self.start])) return self.best, self.best_path def go(self, u, path, seen, t = 0): if len(seen) == self.N: t += self.cost[(u, self.start)] # connect ultimate edge of tour if self.start in self.adj[u] and t < self.best: self.best = t self.best_path = path[:] return for v in self.adj[u]: if v not in seen: path.append(v); seen.add(v) self.go(v, path, seen, t + self.cost[(u, v)]) path.pop(); seen.remove(v) s = Solution() for input_file in ['quiz19.2.txt', 'quiz20.7.txt']: best, path = s.run(input_file) print(f'{input_file} best: {best} path: {path}') # quiz19.2.txt best: 13 path: [1, 2, 4, 3] # quiz20.7.txt best: 23 path: [1, 3, 2, 5, 4] ================================================ FILE: traveling_salesman/package.json ================================================ { "name": "traveling_salesman", "version": "1.0.0", "description": "Quiz 19.2 and 20.7 input file format:", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "", "license": "ISC", "dependencies": { "n-readlines": "^1.0.1" } } ================================================ FILE: traveling_salesman/quiz19.2.txt ================================================ 4 6 1 2 1 1 3 4 1 4 3 2 3 5 2 4 2 3 4 6 ================================================ FILE: traveling_salesman/quiz20.7.txt ================================================ 5 10 1 2 1 1 3 4 1 4 5 1 5 10 2 3 2 2 4 6 2 5 3 3 4 7 3 5 8 4 5 9 ================================================ FILE: traveling_salesman_nearest_neighbor/README.md ================================================ Quiz 20.7 input file format: ``` [number_of_vertices] [number_of_edges] [one_endpoint_of_edge_1] [other_endpoint_of_edge_1] [edge_1_cost] [one_endpoint_of_edge_2] [other_endpoint_of_edge_2] [edge_2_cost] ``` The nearest neighbor heuristic algorithm greedily consume the minimum weight edge of each unvisited vertex: 1. Begin a tour at an arbitrary vertex 2. Repeat until all vertices have been visited: a. If the current vertex is u, proceed to the closet unvisited vertex v 3. Return to the starting vertex ================================================ FILE: traveling_salesman_nearest_neighbor/input.txt ================================================ 5 10 1 2 1 1 3 4 1 4 5 1 5 10 2 3 2 2 4 6 2 5 3 3 4 7 3 5 8 4 5 9 ================================================ FILE: traveling_salesman_nearest_neighbor/main.py ================================================ from collections import defaultdict N, M = -1, -1 adj = defaultdict(list) cost, key = {}, lambda u, v: f'{u},{v}' with open('input.txt') as input: for line in input: A = [int(x) for x in line.strip().split(' ')] if len(A) == 3: u, v, w = A adj[u].append(v); cost[key(u, v)] = w adj[v].append(u); cost[key(v, u)] = w elif len(A) == 2: N, M = A start = 1 u, seen, path = start, set([start]), [start] while len(seen) < N: best, best_v = float('inf'), -1 for v in adj[u]: if v not in seen: cand = cost[key(u, v)] if best > cand: best = cand; best_v = v u = best_v; seen.add(best_v); path.append(best_v) path.append(start) # add last edge to complete the tour t = sum(cost[key(path[i - 1], path[i])] for i in range(1, len(path))) print(f'total: {t} path: {path}') # total: 29 path: [1, 2, 3, 4, 5, 1] ================================================ FILE: weighted_independent_set/CMakeLists.txt ================================================ cmake_minimum_required(VERSION 3.16) project(weighted_independent_set) set(CMAKE_CXX_STANDARD 17) add_executable(weighted_independent_set main.cpp) ================================================ FILE: weighted_independent_set/main.cpp ================================================ /* * In this problem, each file describes the weights of vertices in a path graph and has the format: * [number_of_vertices_in_path_graph] * [weight of first vertex] * [weight of second vertex] * ... * Test case: (contributed by Logan Travis) What is the value of a maximum-weight independent set of the 10-vertex path graph described in this file, and which vertices belong to the MWIS? (Answer: 2617, and the vertices 2, 4, 7, and 10). * Challenge data set: Repeat the previous problem for the 1000-vertex path graph described in this file. */ #include #include #include #include #include using namespace std; using LL = long long; using List = vector; using Map = unordered_map; namespace TopDown { LL best(List& A, Map m = {}) { int N = A.size(); using fun = function; fun go = [&](auto i) { if (m[i]) return m[i]; // 🤔 memo if (i < 0) return m[i] = 0LL; // 🛑 empty set if (!i) return m[i] = A[0]; // 🛑 single set auto include = go(i - 2) + A[i], // ✅ include A[i] exclude = go(i - 1); // 🚫 exclude A[i] return m[i] = max(include, exclude); // 🎯 best }; return go(N - 1); } } namespace BottomUp { LL best(List& A, Map m = {}) { int N = A.size(); List dp(N + 1); // 🤔 memo dp[0] = 0LL; // 🛑 empty set dp[1] = A[0]; // 🛑 single set for (auto i{ 2 }; i <= N; ++i) { auto include = dp[i - 2] + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = dp[i - 1]; // 🚫 exclude A[i] dp[i] = max(include, exclude); // 🎯 best } return dp[N]; } } namespace BottomUpMemOpt { LL best(List& A) { int N = A.size(); LL a = 0LL, // 🤔 memo + 🛑 empty set b = A[0], // 🤔 memo + 🛑 single set c = -1; for (auto i{ 2 }; i <= N; ++i) { auto include = a + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = b; // 🚫 exclude A[i] c = max(include, exclude); // 🎯 best a = b, b = c; // 👈 slide window } return c; } } void run(const string& filename) { List A; fstream fin{ filename }; int N; fin >> N; copy_n(istream_iterator(fin), N, back_inserter(A)); auto a = TopDown::best(A), b = BottomUp::best(A), c = BottomUpMemOpt::best(A); assert(a == b && b == c); // 💩 sanity check cout << filename << ": " << a << endl; } int main() { run("problem16.6test.txt"); // problem16.6test.txt: 2617 run("problem16.6.txt"); // problem16.6.txt: 2955353732 return 0; } ================================================ FILE: weighted_independent_set/main.js ================================================ /* * In this problem, each file describes the weights of vertices in a path graph and has the format: * [number_of_vertices_in_path_graph] * [weight of first vertex] * [weight of second vertex] * ... * Test case: (contributed by Logan Travis) What is the value of a maximum-weight independent set of the 10-vertex path graph described in this file, and which vertices belong to the MWIS? (Answer: 2617, and the vertices 2, 4, 7, and 10). * Challenge data set: Repeat the previous problem for the 1000-vertex path graph described in this file. */ const assert = require('assert'); const LineByLine = require('n-readlines'); let top_down = (A, m = {}) => { let N = A.length; let go = (i = N - 1) => { if (m[i]) // 🤔 memo return m[i]; if (i < 0) return m[i] = 0; // 🛑 empty set if (!i) return m[i] = A[0]; // 🛑 single set let include = go(i - 2) + A[i], // ✅ include A[i] exclude = go(i - 1); // 🚫 exclude A[i] return m[i] = Math.max(include, exclude); // 🎯 best }; return go(); }; let bottom_up = A => { let N = A.length; let dp = Array(N + 1); // 🤔 memo dp[0] = 0; // 🛑 empty set dp[1] = A[0]; // 🛑 single set for (let i = 2; i <= N; ++i) { let include = dp[i - 2] + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = dp[i - 1]; // 🚫 exclude A[i] dp[i] = Math.max(include, exclude); // 🎯 best } return dp[N]; }; let bottom_up_memopt = A => { let N = A.length; let a = 0, // 🤔 memo + 🛑 empty set b = A[0], // 🤔 memo + 🛑 single set c = -1; for (let i = 2; i <= N; ++i) { let include = a + A[i - 1], // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = b; // 🚫 exclude A[i] c = Math.max(include, exclude); // 🎯 best a = b, b = c; // 👈 slide window } return c; }; let run = filename => { let A = []; let input = new LineByLine(filename); let line; let first = true; while (line = input.next()) { if (!first) { A.push(Number(line.toString('ascii'))); } else { first = false; } } let a = top_down(A), b = bottom_up(A), c = bottom_up_memopt(A); assert(a == b && b == c); // 💩 sanity check console.log(`${filename}: ${a}`); }; run('problem16.6test.txt'); // problem16.6test.txt: 2617 run('problem16.6.txt'); // problem16.6.txt: 2955353732 ================================================ FILE: weighted_independent_set/main.kt ================================================ /* * In this problem, each file describes the weights of vertices in a path graph and has the format: * [number_of_vertices_in_path_graph] * [weight of first vertex] * [weight of second vertex] * ... * Test case: (contributed by Logan Travis) What is the value of a maximum-weight independent set of the 10-vertex path graph described in this file, and which vertices belong to the MWIS? (Answer: 2617, and the vertices 2, 4, 7, and 10). * Challenge data set: Repeat the previous problem for the 1000-vertex path graph described in this file. */ import java.io.File fun topDown(A: MutableList): Long { var N = A.size var m = mutableMapOf() fun go(i: Int = N - 1): Long { if (m.contains(i)) // 🤔 memo return m[i]!! if (i < 0) { // 🛑 empty set m[i] = 0 return 0 } if (i == 0) { // 🛑 single set m[i] = A[0] return A[0] } var include = go(i - 2) + A[i] // ✅ include A[i] var exclude = go(i - 1) // 🚫 exclude A[i] m[i] = Math.max(include, exclude) // 🎯 best return m[i]!! } return go() } fun bottomUp(A: MutableList): Long { var N = A.size var dp = LongArray(N + 1) // 🤔 memo dp[0] = 0 // 🛑 empty set dp[1] = A[0] // 🛑 single set for (i in 2..N) { var include = dp[i - 2] + A[i - 1] // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) var exclude = dp[i - 1] // 🚫 exclude A[i] dp[i] = Math.max(include, exclude) // 🎯 best } return dp[N] } fun bottomUpMemOpt(A: MutableList): Long { var N = A.size var a: Long = 0 // 🤔 memo + 🛑 empty set var b: Long = A[0] // 🤔 memo + 🛑 single set var c: Long = -1 for (i in 2..N) { var include = a + A[i - 1] // ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) var exclude = b // 🚫 exclude A[i] c = Math.max(include, exclude) // 🎯 best a = b; b = c // 👈 slide window } return c } fun run(filename: String) { var A = mutableListOf() var first = true File(filename).forEachLine { line -> if (!first) { A.add(line.toLong()) } else { first = false } } var a = topDown(A) var b = bottomUp(A) var c = bottomUpMemOpt(A) assert(a == b && b == c) // 💩 sanity check println("$filename: $a") } fun main() { run("problem16.6test.txt") // problem16.6test.txt: 2617 run("problem16.6.txt") // problem16.6.txt: 2955353732 } ================================================ FILE: weighted_independent_set/main.py ================================================ # # In this problem, each file describes the weights of vertices in a path graph and has the format: # [number_of_vertices_in_path_graph] # [weight of first vertex] # [weight of second vertex] # ... # Test case: (contributed by Logan Travis) What is the value of a maximum-weight independent set of the 10-vertex path graph described in this file, and which vertices belong to the MWIS? (Answer: 2617, and the vertices 2, 4, 7, and 10). # Challenge data set: Repeat the previous problem for the 1000-vertex path graph described in this file. # from functools import lru_cache def top_down(A): N = len(A) @lru_cache(maxsize = None) # 🤔 memo def go(i = N - 1): if i < 0: return 0 # 🛑 empty set if i == 0: return A[0] # 🛑 single set include = go(i - 2) + A[i] # ✅ include A[i] exclude = go(i - 1) # 🚫 exclude A[i] return max(include, exclude) # 🎯 best return go() def bottom_up(A): N = len(A) dp = [0] * (N + 1) # 🤔 memo dp[0] = 0 # 🛑 empty set dp[1] = A[0] # 🛑 single set for i in range(2, N + 1): include = dp[i - 2] + A[i - 1] # ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = dp[i - 1] # 🚫 exclude A[i] dp[i] = max(include, exclude) # 🎯 best return dp[N] def bottom_up_memopt(A): N = len(A) a = 0 # 🤔 memo + 🛑 empty set b = A[0] # 🤔 memo + 🛑 single set c = -1 for i in range(2, N + 1): include = a + A[i - 1] # ✅ include A[i] (use A[i - 1] since dp[i] is offset by 1 for explicit 🛑 empty set at index 0, ie. index -1 doesn't exist) exclude = b # 🚫 exclude A[i] c = max(include, exclude) # 🎯 best a = b; b = c # 👈 slide window return c def run(filename): A = [] with open(filename) as fin: first = True while True: line = fin.readline() if not line: break x = int(line) if not first: A.append(x) else: first = False N = x a = top_down(A) b = bottom_up(A) c = bottom_up_memopt(A) assert(a == b and b == c) # 💩 sanity check print(f'{filename}: {a}') run('problem16.6test.txt') # problem16.6test.txt: 2617 run('problem16.6.txt') # problem16.6.txt: 2955353732 ================================================ FILE: weighted_independent_set/package.json ================================================ { "name": "weighted_independent_set", "version": "1.0.0", "description": "", "main": "main.js", "scripts": { "test": "echo \"Error: no test specified\" && exit 1" }, "author": "clayton wong", "license": "ISC", "dependencies": { "n-readlines": "^1.0.1" } } ================================================ FILE: weighted_independent_set/problem16.6.txt ================================================ 1000 4962786 6395702 5601590 3803402 6784626 4944482 2882725 9310662 5247184 9819854 8398364 1470063 4199696 4623136 8160902 930850 3889157 8211214 6560984 8835416 3024392 3286693 736791 3862790 1420652 9767464 6093772 2133393 358615 4537366 6655609 5551123 9039549 469060 304701 5768649 1339317 8421671 513661 6792447 3944383 4692731 4614391 9344708 4169702 4345210 9744699 9407222 6480402 7985130 4407746 4040958 7960851 5394516 4024926 6784072 1710864 6886941 7495555 5654086 2481292 8892684 2186179 1539792 1828698 4741356 3476859 327340 7634220 808031 5101226 6958744 1511709 7231864 7447240 778642 3120423 1098518 6450468 399546 7275028 1081427 7154897 2804344 9440402 2909959 2686145 5099515 3776057 5765944 8935025 6477008 3490890 1691564 2225172 8510419 9788522 9484150 7236748 7334350 3501364 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