[ { "path": ".gitignore", "content": ".ipynb_checkpoints/\n" }, { "path": "Lecture 17.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# LCS \"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 9,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"import numpy as np\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 61,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def LCS(s1, s2):\\n\",\n \" # initialize state\\n\",\n \" n, m = len(s1) + 1, len(s2) + 1\\n\",\n \" dp = [\\n\",\n \" [ 0 for _ in range(m)] \\n\",\n \" for _ in range(n)\\n\",\n \" ]\\n\",\n \" pred = [\\n\",\n \" [ (-1,-1) for _ in range(m)]\\n\",\n \" for _ in range(n)\\n\",\n \" ]\\n\",\n \" \\n\",\n \" for i in range(1,n):\\n\",\n \" for j in range(1,m):\\n\",\n \" # CASE 1: ignore character in s1\\n\",\n \" dp[i][j], pred[i][j] = dp[i-1][j], (i-1,j)\\n\",\n \" # CASE 2: ignore character in s2\\n\",\n \" if dp[i][j] < dp[i][j-1]:\\n\",\n \" dp[i][j], pred[i][j] = dp[i][j-1], (i,j-1)\\n\",\n \" # CASE 3: extend\\n\",\n \" if s1[i-1] == s2[j-1] and dp[i][j] < dp[i-1][j-1] + 1:\\n\",\n \" dp[i][j], pred[i][j] = dp[i-1][j-1] + 1, (i-1,j-1)\\n\",\n \" \\n\",\n \" print (np.array(dp))\\n\",\n \"\\n\",\n \" # recover the solution\\n\",\n \" x,y = (n-1,m-1)\\n\",\n \" solution = []\\n\",\n \" while pred[x][y] != (-1,-1):\\n\",\n \" if pred[x][y] == (x-1, y-1):\\n\",\n \" solution.append(s1[x-1])\\n\",\n \" (x,y) = pred[x][y]\\n\",\n \" solution.reverse()\\n\",\n \" \\n\",\n \" return solution\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 62,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"[[0 0 0 0 0]\\n\",\n \" [0 1 1 1 1]\\n\",\n \" [0 1 2 2 2]\\n\",\n \" [0 1 2 2 2]\\n\",\n \" [0 1 2 3 3]]\\n\"\n ]\n },\n {\n \"data\": {\n \"text/plain\": [\n \"['b', 'a', 'b']\"\n ]\n },\n \"execution_count\": 62,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"LCS(\\\"bacb\\\", \\\"babd\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 63,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"[[0 0 0 0 0 0 0 0 0 0]\\n\",\n \" [0 0 0 1 1 1 1 1 1 1]\\n\",\n \" [0 0 0 1 2 2 2 2 2 2]\\n\",\n \" [0 1 1 1 2 2 2 2 2 2]\\n\",\n \" [0 1 2 2 2 2 2 2 2 2]\\n\",\n \" [0 1 2 3 3 3 3 3 3 3]\\n\",\n \" [0 1 2 3 4 4 4 4 4 4]\\n\",\n \" [0 1 2 3 4 4 4 5 5 5]\\n\",\n \" [0 1 2 3 4 4 4 5 6 6]\\n\",\n \" [0 1 2 3 4 4 5 5 6 7]]\\n\"\n ]\n },\n {\n \"data\": {\n \"text/plain\": [\n \"['t', 'h', 'i', 's', 'l', 'c', 's']\"\n ]\n },\n \"execution_count\": 63,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"LCS(\\\"isthislcs\\\", \\\"thisislcs\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Viterbi algorithm\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 1,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"states = ('Healthy', 'Fever')\\n\",\n \" \\n\",\n \"observations = ('normal', 'cold', 'dizzy')\\n\",\n \" \\n\",\n \"start_probability = {'Healthy': 0.6, 'Fever': 0.4}\\n\",\n \" \\n\",\n \"transition_probability = {\\n\",\n \" 'Healthy' : {'Healthy': 0.7, 'Fever': 0.3},\\n\",\n \" 'Fever' : {'Healthy': 0.4, 'Fever': 0.6}\\n\",\n \" }\\n\",\n \" \\n\",\n \"emission_probability = {\\n\",\n \" 'Healthy' : {'normal': 0.5, 'cold': 0.4, 'dizzy': 0.1},\\n\",\n \" 'Fever' : {'normal': 0.1, 'cold': 0.3, 'dizzy': 0.6}\\n\",\n \"}\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 7,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def viterbi(obs, states, start_p, trans_p, emit_p):\\n\",\n \" V = [{}]\\n\",\n \" predecessor = [{}]\\n\",\n \" \\n\",\n \" # Initialize base cases (t == 0)\\n\",\n \" for y in states:\\n\",\n \" V[0][y] = start_p[y] * emit_p[y][obs[0]]\\n\",\n \" predecessor[0][y] = None\\n\",\n \" \\n\",\n \" # Run Viterbi for t > 0\\n\",\n \" for t in range(1, len(obs)):\\n\",\n \" V.append({})\\n\",\n \" predecessor.append({})\\n\",\n \"\\n\",\n \" for y in states:\\n\",\n \" highest_probability, best_y_prev = 0.0, None\\n\",\n \" for y_prev in states:\\n\",\n \" probability_from_y_prev = V[t-1][y_prev] * trans_p[y_prev][y] * emit_p[y][obs[t]]\\n\",\n \" if probability_from_y_prev > highest_probability:\\n\",\n \" highest_probability, best_y_prev = probability_from_y_prev, y_prev\\n\",\n \" V[t][y] = highest_probability\\n\",\n \" predecessor[t][y] = best_y_prev\\n\",\n \" \\n\",\n \" print(*dptable(V), sep='')\\n\",\n \" \\n\",\n \" # Return the most likely sequence over the given time frame\\n\",\n \" n = len(obs) - 1\\n\",\n \" (prob, state) = max((V[n][y], y) for y in states)\\n\",\n \" \\n\",\n \" # Recover the path\\n\",\n \" path = []\\n\",\n \" for t in range(n,-1,-1):\\n\",\n \" path.append(state)\\n\",\n \" state = predecessor[t][state]\\n\",\n \" path.reverse()\\n\",\n \" \\n\",\n \" return (prob, path)\\n\",\n \"\\n\",\n \"# Don't study this; it just prints a table of the steps.\\n\",\n \"def dptable(V):\\n\",\n \" yield \\\" \\\"\\n\",\n \" yield \\\" \\\".join((\\\"%7d\\\" % i) for i in range(len(V)))\\n\",\n \" yield \\\"\\\\n\\\"\\n\",\n \" for y in V[0]:\\n\",\n \" yield \\\"%.5s: \\\" % y\\n\",\n \" yield \\\" \\\".join(\\\"%.7s\\\" % (\\\"%f\\\" % v[y]) for v in V)\\n\",\n \" yield \\\"\\\\n\\\"\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 8,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 0 1 2\\n\",\n \"Fever: 0.04000 0.02700 0.01512\\n\",\n \"Healt: 0.30000 0.08400 0.00588\\n\",\n \"\\n\"\n ]\n },\n {\n \"data\": {\n \"text/plain\": [\n \"(0.01512, ['Healthy', 'Healthy', 'Fever'])\"\n ]\n },\n \"execution_count\": 8,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"viterbi(observations, states, start_probability, transition_probability, emission_probability)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 3\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.4.1\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "challenge1/.gitignore", "content": "coins\n" }, { "path": "challenge1/README.md", "content": "# The coins challenge\n\nTo run c++ solution run\n```bash\ng++ coins.cpp -O2 -o coins\n./run_tests.sh ./coins\n```\n\nTo run Python solution run\n\n```bash\n./run_tests.sh python2 coins.py\n```\n" }, { "path": "challenge1/coins.cpp", "content": "#include \n#include \n\n// OVERALL COMPLEXITY: O(n lg n)\n// (due to sorting)\n\n// maximum number of coins\nconst int MAXN = 1000005;\n\n// n - number of coins\n// k - we need k as many tails as heads\nint n, k;\n// c - buffer to hold all the characters\nchar c[MAXN];\n// array of (prefix_sum[i], i) values\nstd::pair seq_and_index[MAXN];\n\nint main() {\n scanf(\"%d%d\", &n, &k);\n scanf(\"%s\", c);\n // for prefix sums it is important to add an extra element\n // for empty sequence at the beginning.\n // this is so that when we compute\n //\n // seq_and_index[n].second - seq_and_index[0].second\n //\n // we get sum of elements for indexes 1 <= i <= n\n\n // We reuse seq_and_index for two things\n // initially it just stores values k and -1\n // later we compute prefix sum\n seq_and_index[0] = std::make_pair(0,0);\n for (int i=0; i \n#include \n#include \n#define ST first\n#define ND second\nusing namespace std;\ntypedef pair PI;\n\n\nint t,n,T[3000007]; //12 MB\ndeque MX, MN; // wartosc,pozycja\n//TOTAL: 12 MB\n\nbool test(int x) {\t\n\t//printf(\"%d..\\n\", x);\n\tMX.clear(); MN.clear();\n\tfor(int i=0; i=T[i]) MN.pop_back();\n\t\tMN.push_back(i);\n\t\twhile(i>=x && MN.front()<=(i-x)) MN.pop_front();\n\t\t\n\t\twhile(!MX.empty() && T[MX.back()]<=T[i]) MX.pop_back();\n\t\tMX.push_back(i);\n\t\twhile(i>=x && MX.front()<=(i-x)) MX.pop_front();\n\t\tif(i>=x-1 && T[MX.front()]-T[MN.front()]<=t) return 1;\n\t\t//if(i>=x-1) printf(\" %d: %d, %d\\n\", i, T[MN.front()], T[MX.front()]);\n\t}\n\treturn 0;\n} \n\t\n\n\n\n\nint main() {\n\tscanf(\"%d%d\", &t,&n);\n\tfor(int i=0; i 0 and \\\n A[min_extravaganza[-1]] >= A[end]:\n min_extravaganza.pop()\n min_extravaganza.append(end)\n\n while len(max_extravaganza) > 0 and \\\n A[max_extravaganza[-1]] <= A[end]:\n max_extravaganza.pop()\n max_extravaganza.append(end)\n\n end += 1\n\n if ok():\n best_sequence_length = max(best_sequence_length,\n end - start)\n\n if min_extravaganza[0] == start:\n min_extravaganza.popleft()\n if max_extravaganza[0] == start:\n max_extravaganza.popleft()\n start += 1\n\nprint best_sequence_length\n" }, { "path": "challenge2/pilots_n2.py", "content": "# n^3 solution\n\n# read in the first line of input\nk, n = [int(x) for x in raw_input().split(' ')]\n# read in the second line of input\nA = [int(x) for x in raw_input().split(' ')]\nassert len(A) == n\n\n# check all the subsequences and keep the length\nbest_sequence_length = 0\n\nfor start in range(0, n):\n minimum_so_far = A[start]\n maximum_so_far = A[start]\n for end in range(start + 1, n + 1):\n minimum_so_far = min(minimum_so_far, A[end - 1])\n maximum_so_far = max(maximum_so_far, A[end - 1])\n # line below is O(n)\n difference = maximum_so_far - minimum_so_far\n if difference <= k:\n best_sequence_length = max(best_sequence_length,\n end - start)\n else:\n # smartness (difference will newer decrease!)\n break\n\nprint best_sequence_length\n" }, { "path": "challenge2/pilots_n3.py", "content": "# n^3 solution\n\n# read in the first line of input\nk, n = [int(x) for x in raw_input().split(' ')]\n# read in the second line of input\nA = [int(x) for x in raw_input().split(' ')]\nassert len(A) == n\n\n# check all the subsequences and keep the length\nbest_sequence_length = 0\n\nfor start in range(0, n):\n for end in range(start + 1, n + 1):\n current_seq = A[start:end]\n # line below is O(n)\n difference = max(current_seq) - min(current_seq)\n if difference <= k:\n best_sequence_length = max(best_sequence_length,\n len(current_seq))\n\nprint best_sequence_length\n" }, { "path": "challenge2/pilots_nlog2n.py", "content": "# LAZYYYYYY\n" }, { "path": "challenge2/pilots_nlogn.py", "content": "# n * log n solution\n\nfrom bintrees import FastAVLTree\n\n# read in the first line of input\nk, n = [int(x) for x in raw_input().split(' ')]\n# read in the second line of input\nA = [int(x) for x in raw_input().split(' ')]\nassert len(A) == n\n\n# check all the subsequences and keep the length\nbest_sequence_length = 0\n\n\nmin_tree = FastAVLTree()\nmax_tree = FastAVLTree()\n\ndef ok():\n if max_tree.is_empty():\n return True\n else:\n difference = max_tree.max_item()[0][0] \\\n - min_tree.min_item()[0][0]\n return difference <= k\n\nstart, end = 0, 0\n\nwhile end < len(A):\n # extend the sequence until violation reached.\n while end < len(A) and ok():\n best_sequence_length = max(best_sequence_length,\n end - start)\n min_tree.insert((A[end], end), None)\n max_tree.insert((A[end], end), None)\n end += 1\n\n if ok():\n best_sequence_length = max(best_sequence_length,\n end - start)\n\n # move starting position by 1.\n min_tree.remove((A[start], start))\n max_tree.remove((A[start], start))\n start += 1\n\nprint best_sequence_length\n" }, { "path": "challenge2/run_tests.sh", "content": "#!/bin/bash\n\n# stop script on error and print it\nset -e\n# inform me of undefined variables\nset -u\n# handle cascading failures well\nset -o pipefail\n\nSCRIPT_DIR=$( cd \"$( dirname \"${BASH_SOURCE[0]}\" )\" && pwd )\n\nfor filename in `ls $SCRIPT_DIR/tests/*.in | sort --version-sort -f`; do\n output_filename=${filename::-3}.out\n time $@ < $filename | diff -bs - $output_filename\ndone\n" }, { "path": "challenge2/url.txt", "content": "http://main.edu.pl/en/archive/oi/17/pil\n" }, { "path": "hello.txt", "content": "hi\nhow are you?\n" }, { "path": "lecture1/Fast exponentiation and fibonacci sequence.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 1,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"import numpy as np\\n\",\n \"import cProfile\\n\",\n \"from operator import mul\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Fast Exponentiation\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We can quickly compute $a^b \\\\ mod\\\\ m$ using, the following trick best illustrated by example.\\n\",\n \"\\n\",\n \"$$\\n\",\n \"5^{13} = 5 * 5^{12} = 5 * (5^6)^2 = 5 * ((5^3)^2)^2 = 5 * ((5*5^2)^2)^2 = 5 * ((5*5*5)^2)^2\\n\",\n \"$$\\n\",\n \"\\n\",\n \"In that example even though we would naively need 11 multiplications to calculate results we managed to get away with 5.\\n\",\n \"In general we can write out \\n\",\n \"\\n\",\n \"$$\\n\",\n \"fexp(a,b,m) =\\n\",\n \"\\\\begin{cases}\\n\",\n \"a & \\\\text{if}\\\\ b=1\\\\\\\\\\n\",\n \"fexp(a,b/2,m)^2 &\\\\text{if}\\\\ b\\\\ \\\\text{even}\\\\\\\\\\n\",\n \"a \\\\cdot fexp(a,b-1,m) &\\\\text{otherwise}\\n\",\n \"\\\\end{cases}\\n\",\n \"$$\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 2,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def fexp_recursive(a, b, m, mul_op=mul):\\n\",\n \" # We can easiely handle b = 0, here, but we choose not to\\n\",\n \" # this will be helpful later when we deal with matrices...\\n\",\n \" assert b >= 1\\n\",\n \" if b == 1:\\n\",\n \" return a\\n\",\n \" elif b % 2 == 0:\\n\",\n \" conquered = fexp_recursive(a, b / 2, m, mul_op=mul_op)\\n\",\n \" return mul_op(conquered, conquered) % m\\n\",\n \" else:\\n\",\n \" b_one_less = fexp_recursive(a, b - 1, m, mul_op=mul_op) \\n\",\n \" return mul_op(a, b_one_less) % m\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 3,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"64\"\n ]\n },\n \"execution_count\": 3,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"fexp_recursive(2, 6, 1000)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 4,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"4\"\n ]\n },\n \"execution_count\": 4,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"fexp_recursive(2, 6, 10)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 5,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 89 function calls (46 primitive calls) in 0.000 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 44/1 0.000 0.000 0.000 0.000 :1(fexp_recursive)\\n\",\n \" 1 0.000 0.000 0.000 0.000 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 43 0.000 0.000 0.000 0.000 {operator.mul}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"fexp_recursive(2, 10000000000, 10)\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Iterative approach\\n\",\n \"Very similar algorithm can be written out without recursion by looking at binary representation of b and noticing that if $i-th$ bit is one, then we need to multiply the result by $$a^{2^i}$$\\n\",\n \"Don't worry if you don't fully understand the code below. It is included here, to show you that there are multiple ways of approaching implementation of this kind of solution. Also iterative algorithms are sometimes preferred - we will get back to that point below.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 6,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def fexp_iterative(a, b, m, mul_op=mul):\\n\",\n \" assert b >= 1\\n\",\n \" result = a\\n\",\n \" multiplier = a\\n\",\n \" b -= 1\\n\",\n \" while b > 0:\\n\",\n \" if b % 2 == 1:\\n\",\n \" result = mul_op(result, multiplier) % m\\n\",\n \" multiplier = mul_op(multiplier, multiplier) % m\\n\",\n \" b /= 2\\n\",\n \" return result\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 7,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"64\"\n ]\n },\n \"execution_count\": 7,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"fexp_iterative(2, 6, 1000)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 8,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"4\"\n ]\n },\n \"execution_count\": 8,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"fexp_iterative(2, 6, 10)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 9,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 57 function calls in 0.000 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.000 0.000 0.000 0.000 :1(fexp_iterative)\\n\",\n \" 1 0.000 0.000 0.000 0.000 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 54 0.000 0.000 0.000 0.000 {operator.mul}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"fexp_iterative(2, 10000000000, 10)\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Fibonacci sequence\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding up the two numbers before it. I.e. 3 is found by adding the two numbers before it (1+2). Here we will explore 3 different algorithms for computing the $n^{th}$ Fibonacci number and analyze their time complexity. We denote the $n^{th}$ Fibonacci number as $F_{n}$. Code for the following 3 algorithms is in recitation1.py which is available on the Stellar site under recitation materials. \\n\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Naive Recursion\\n\",\n \"\\n\",\n \"By definition, $F_{n} = F_{n - 1} + F_{n - 2}$. As this is the ``naive'' algorithm, let's not try to be too clever and instead simply write an algorithm using only this definition!\\n\",\n \"\\n\",\n \"Now to analyze the runtime. Formally this algorithm can be analyzed by solving the recurrence, $T(n) = T(n - 1) + T(n - 2) + \\\\Theta(1)$. This is a tough recursion to solve! Let us separately find an upper and lower bound instead of a $\\\\Theta$ relation. \\n\",\n \"\\n\",\n \"It is clear that the recurrence $T(n) = 2T(n - 1) + \\\\Theta(1)$ is strictly greater than our original, so let us use it to find an upper bound. Each recursive call results in two child recursive calls until the base case is reached. Therefore, there will be $\\\\Theta(2^{i})$ recursive calls made at the $i^{th}$ level of recursion. Since, the subproblem size only decreases by one on each call, there will be $\\\\Theta(n)$ levels of recursion before the base case is reached. Therefore this recurrence solves to be $\\\\Theta(2^{n})$ and we can conclude that our algorithm is $O(2^{n})$\\n\",\n \"\\n\",\n \"The recurrence $T(n) = 2T(n - 2) + \\\\Theta(1)$ is strictly less than our original. Using similar logic as above we can see that this recurrence solves to $\\\\Theta(2^{\\\\frac{n}{2}})$ and we conclude that our algorithm is $\\\\Omega(2^{\\\\frac{n}{2}})$.\\n\",\n \"\\n\",\n \"Challenge Problem: Find a tight asymptotic bound to this algorithms runtime. Hint: Draw a tree diagraming recursive calls and look for the pattern!\\n\",\n \"\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 10,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def fibonacci_recursive_slow(n, m):\\n\",\n \" assert n >= 0\\n\",\n \" if n == 0:\\n\",\n \" return 0\\n\",\n \" elif n == 1:\\n\",\n \" return 1\\n\",\n \" else:\\n\",\n \" return (fibonacci_recursive_slow(n - 1, m) + fibonacci_recursive_slow(n - 2, m)) % m \"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 11,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"55\"\n ]\n },\n \"execution_count\": 11,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"fibonacci_recursive_slow(10, 1000)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 12,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 2692539 function calls (3 primitive calls) in 0.888 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \"2692537/1 0.888 0.000 0.888 0.888 :1(fibonacci_recursive_slow)\\n\",\n \" 1 0.000 0.000 0.888 0.888 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"fibonacci_recursive_slow(30, 1000)\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Memoized Recursion\\n\",\n \"It's often the case that we can improve the efficiency of algorithms by exploiting natural ``structures'' present in the problem. Notice in the naive algorithm that we often compute the same thing multiple times! This occurs because we have overlapping subproblems. For example, both $F_{n - 1}$ and $F_{n - 2}$ depend on the solution to $F_{n - 3}$. We can take advantage of this structure by memoizing (storing) the solutions to subproblems as we go. Therefore instead of recalculating them we can simply look them up! Look in recitation1.py for python code.\\n\",\n \"\\n\",\n \"This improved algorithm has a time complexity of $\\\\Theta(n)$. This can be seen from the fact that we in total solve for $\\\\Theta(n)$ $F_{i}$s, each of which take only constant time to compute. \\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 13,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"cache = {}\\n\",\n \"def fibonacci_recursive_fast(n, m):\\n\",\n \" if not (n,m) in cache:\\n\",\n \" assert n >= 0\\n\",\n \" if n == 0:\\n\",\n \" result = 0\\n\",\n \" elif n == 1:\\n\",\n \" result = 1\\n\",\n \" else:\\n\",\n \" result = (fibonacci_recursive_fast(n - 1, m) + fibonacci_recursive_fast(n - 2, m)) % m\\n\",\n \" cache[(n,m)] = result\\n\",\n \" return cache[(n,m)]\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 14,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"55\"\n ]\n },\n \"execution_count\": 14,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"fibonacci_recursive_fast(10, 1000)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 15,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 1783 function calls (3 primitive calls) in 0.004 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1781/1 0.004 0.000 0.004 0.004 :2(fibonacci_recursive_fast)\\n\",\n \" 1 0.000 0.000 0.004 0.004 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"fibonacci_recursive_fast(900, 1000)\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Iterative versus recursive solutions\\n\",\n \"The code below should give a runtime error on a standard Python interpreter - because its exceeding the default stack limit. This kind of limitation is why we often opt for iterative versions of the algorithm. Don't worry though, it turns out that for every recursive solution there exists an itertive equivalent. Indeed - we can emulate recursion stack with a stack datastructure. Such a solution is often tedious to implemented and constact factor of the runtime become large. There's why we often seek for *natural order of calculation*, i.e. order in which we compute the subproblems, such that by the time we need a particular result it has alredy been computed. For example in case of Fibonacci the natural order of computatation is to compute $F_1$, then $F_2$, then $F_3$ etc. Notice how resulting solution is even simpler than the recursive one!\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 16,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"# fibonacci_recursive_fast(9000, 1000)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 17,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def fibonnaci_iterative(n, m):\\n\",\n \" assert n >= 0\\n\",\n \" if n == 0:\\n\",\n \" return 0\\n\",\n \" f_current, f_previous = 1, 0\\n\",\n \" for _ in range(n - 1):\\n\",\n \" f_current, f_previous = f_current + f_previous % m, f_current\\n\",\n \" return f_current\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 18,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"55\"\n ]\n },\n \"execution_count\": 18,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"fibonnaci_iterative(10, 1000)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 19,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 4 function calls in 1.025 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.901 0.901 1.025 1.025 :1(fibonnaci_iterative)\\n\",\n \" 1 0.000 0.000 1.025 1.025 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 1 0.124 0.124 0.124 0.124 {range}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"fibonnaci_iterative(10000000, 1000)\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Matrix exponentiation\\n\",\n \"\\n\",\n \"Take a moment to think back to the recursive squaring algorithm from lecture. In a similar fashion, we can compute the $n^{th}$ Fibonacci number in logarithmic time by repeatedly squaring the matrix \\n\",\n \"\\n\",\n \"\\n\",\n \"\\\\begin{align}\\n\",\n \"\\\\begin{bmatrix}\\n\",\n \" 1 & 1 \\\\\\\\\\n\",\n \" 1 & 0\\n\",\n \"\\\\end{bmatrix}\\n\",\n \"\\\\end{align}\\n\",\n \"\\n\",\n \"In fact\\n\",\n \"\\n\",\n \"\\n\",\n \"\\\\begin{align}\\n\",\n \"\\\\begin{bmatrix}\\n\",\n \" 1 & 1 \\\\\\\\\\n\",\n \" 1 & 0\\n\",\n \"\\\\end{bmatrix} ^{n} \\n\",\n \"= \\n\",\n \"\\\\begin{bmatrix}\\n\",\n \" F_{n + 1} & F_{n} \\\\\\\\\\n\",\n \" F_{n} & F_{n - 1}\\n\",\n \"\\\\end{bmatrix}\\n\",\n \"\\\\end{align}\\n\",\n \"\\n\",\n \"To give a rough proof of why this is the case, let us use induction on $n$. Our claim is trivially true in the base case $n = 1$. Now assuming that our claim holds for this matrix to the $n^{th}$ power, we must show that our claim is also true for this matrix to the $(n + 1)^{th}$ power. \\n\",\n \"\\n\",\n \"\\n\",\n \"\\\\begin{align}\\n\",\n \"\\\\begin{bmatrix}\\n\",\n \" 1 & 1 \\\\\\\\\\n\",\n \" 1 & 0\\n\",\n \"\\\\end{bmatrix}\\n\",\n \"*\\n\",\n \" \\\\begin{bmatrix}\\n\",\n \" 1 & 1 \\\\\\\\\\n\",\n \" 1 & 0 \\n\",\n \"\\\\end{bmatrix} ^{n} \\n\",\n \"= \\n\",\n \"\\\\begin{bmatrix}\\n\",\n \" 1 & 1 \\\\\\\\\\n\",\n \" 1 & 0 \\\\\\\\\\n\",\n \"\\\\end{bmatrix}\\n\",\n \"*\\n\",\n \"\\\\begin{bmatrix}\\n\",\n \" F_{n + 1} & F_{n} \\\\\\\\\\n\",\n \" F_{n} & F_{n - 1}\\n\",\n \"\\\\end{bmatrix}\\n\",\n \" = \\n\",\n \" \\\\begin{bmatrix}\\n\",\n \" F_{n + 1} + F_{n} & F_{n} + F_{n - 1} \\\\\\\\\\n\",\n \" F_{n + 1} & F_{n}\\n\",\n \"\\\\end{bmatrix}\\n\",\n \"=\\n\",\n \"\\\\begin{bmatrix}\\n\",\n \" F_{n + 2} & F_{n+1} \\\\\\\\\\n\",\n \" F_{n + 1} & F_{n }\\n\",\n \"\\\\end{bmatrix}\\n\",\n \"\\\\end{align}\\n\",\n \"\\n\",\n \"Success! \\n\",\n \"\\n\",\n \"The runtime analysis for this algorithm is identical to that for modular exponentiation using repeated squaring. In particular we do not include cost of matrix multiply in our analysis, because matrix has constant size.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 23,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"F = np.array([[1, 1],\\n\",\n \" [1, 0]])\\n\",\n \"\\n\",\n \"def fibonnaci_matrix(n, m):\\n\",\n \" Fn = fexp_recursive(F, n, m, mul_op=np.dot)\\n\",\n \" return Fn[0][1]\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 24,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"55\"\n ]\n },\n \"execution_count\": 24,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"fibonnaci_matrix(10, 1000)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 25,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 90 function calls (47 primitive calls) in 0.001 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 44/1 0.000 0.000 0.001 0.001 :1(fexp_recursive)\\n\",\n \" 1 0.000 0.000 0.001 0.001 :4(fibonnaci_matrix)\\n\",\n \" 1 0.000 0.000 0.001 0.001 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 43 0.000 0.000 0.000 0.000 {numpy.core.multiarray.dot}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"fibonnaci_matrix(10000000000, 1000)\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Problems to think about (non-examinable, non-compulsory, strictly for fun...)\\n\",\n \"\\n\",\n \"1. Give an example of another operation besides multiplication and matrix multiply that can be efficiently composed using fast exponentiation.\\n\",\n \"\\n\",\n \"2. Compute the n-th item of tribonacci sequence using the three methods presented above:\\n\",\n \"\\n\",\n \"\\\\begin{align}\\n\",\n \"s_n =\\n\",\n \"\\\\begin{cases}\\n\",\n \"1 & \\\\text{if}\\\\ n \\\\in \\\\{ 1,2,3 \\\\} \\\\\\\\\\n\",\n \"2s_{n-1} + 2s_{n-2} + s_{n-3} & \\\\text{otherwise}\\n\",\n \"\\\\end{cases}\\n\",\n \"\\\\end{align}\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture1/Stock Exchange.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 5,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"import cProfile\\n\",\n \"import random\\n\",\n \"import time\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Stock Exchange Problem\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Consider the following problem. We have a non-empty array representing evolution of price of a particular stock over time.\\n\",\n \"```python\\n\",\n \"A = [20, 3, 19, 1, 15, 6]\\n\",\n \"```\\n\",\n \"Given this information we want to find what is the optimal profit we can make using single buy and single sell operation (here we have perfect knowledge of the prices - you can imagine that this quantity is something that quantitative traders would like to know, to compare their decision to the best possible decision given perfect knowledge of the future).\\n\",\n \"\\n\",\n \"More formally we want to find two numbers $b$, $s$, such that $$0 \\\\leq b \\\\leq s \\\\leq |A|$$ and $$A_s - A_b$$ is maximum possible. Of course we cannot sell before buying.\\n\",\n \"\\n\",\n \"For example for the array given above, the biggest profit we can make is $16$ (make sure you can see that). Below we present three different solutions to this problem.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 1,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"# here we seed the random number generator, to ensure, that we generate \\n\",\n \"# the same random instance every time we pass seed equal to a particular \\n\",\n \"# value. This way the speed comparison is fair.\\n\",\n \"def make_prices(n, seed):\\n\",\n \" \\\"\\\"\\\" Return array of n random prices. \\\"\\\"\\\"\\n\",\n \" random.seed(seed)\\n\",\n \" return [ random.random() for _ in range(n) ]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Naive solution\\n\",\n \"This solution is a direct search of values of $b$ and $s$. The complexity is $O(n^2)$ (intiutively we have two nested for loops, each of which does $O(n)$ iterations, when $n$ = len(A).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 2,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def naive(A):\\n\",\n \" \\\"\\\"\\\" return best gain on A, using naive method \\n\",\n \" running time, due to doubly-nest loop, is O(n^2)\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" n = len(A)\\n\",\n \" ans = 0\\n\",\n \" for i0 in range(n):\\n\",\n \" for j0 in range(i0,n):\\n\",\n \" ans = max(ans, A[j0]-A[i0])\\n\",\n \" return ans\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 3,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"16\"\n ]\n },\n \"execution_count\": 3,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"naive([20, 3, 19, 1, 15, 6])\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 7,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 50025009 function calls in 14.463 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.004 0.004 0.005 0.005 :1(make_prices)\\n\",\n \" 1 9.050 9.050 14.458 14.458 :1(naive)\\n\",\n \" 1 0.000 0.000 14.463 14.463 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 random.py:100(seed)\\n\",\n \" 1 0.000 0.000 0.000 0.000 {function seed at 0x7f90fbfeb578}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {len}\\n\",\n \" 50005000 5.213 0.000 5.213 0.000 {max}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 10000 0.001 0.000 0.001 0.000 {method 'random' of '_random.Random' objects}\\n\",\n \" 10002 0.195 0.000 0.195 0.000 {range}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"# slowness alert!\\n\",\n \"cProfile.run(\\\"naive(make_prices(10000, 1))\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Divide and conquer approach\\n\",\n \"Here we split our array into two simpler problems corresponding to left half of the array $L$ and right half $R$. \\n\",\n \"For example if \\n\",\n \"```python\\n\",\n \"A = [20, 3, 19, 1, 15, 6]\\n\",\n \"```\\n\",\n \"we can imagine that\\n\",\n \"```python\\n\",\n \"L = [20, 3, 19]\\n\",\n \"R = [1, 15, 6]\\n\",\n \"```\\n\",\n \"In order to reduce our problem to those simpler problems we need to consider three cases:\\n\",\n \"1. $ b,s \\\\in L$ - we can solve it by solving original problem for $L$\\n\",\n \"2. $ b,s \\\\in R$ - we can solve it by solving original problem for $R$\\n\",\n \"3. $ b \\\\in L$ and $ s \\\\in R$ - we can solve it by finding minimum in $L$ and maximum in $R$ and returning the difference\\n\",\n \"\\n\",\n \"We need not consider the case $s \\\\in L$ and $b \\\\in R$ (why?).\\n\",\n \"\\n\",\n \"This way we reduced our problem to two smaller problems. This is good - we will keep changing bigger problems into smaller problems until we get problem so small that it is trivial to solve - in this case if our array is of size 1, then the maximum profit we can make is $0$.\\n\",\n \"\\n\",\n \"The complexity of this solution is can be calculated by solving the following equation:\\n\",\n \"\\n\",\n \"\\\\begin{align}\\n\",\n \"T(1) =& 1\\\\\\\\\\n\",\n \"T(n) =& T(n / 2) + T(n / 2) + O(n)\\n\",\n \"\\\\end{align}\\n\",\n \"The three summands in the equation above come for cases 1,2,3 listed above. In particular notice that case 3 requires single read through data and therefor has complexity $O(n)$. The solution to this set of equation is $T(n) = O(n \\\\lg{n})$\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 9,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def dc(A, lo=None, hi=None):\\n\",\n \" \\\"\\\"\\\" return best gain on A[lo:hi], using divide & conquer \\n\",\n \" running time is solution to T(n) = 2*T(n/2) + Theta(n) = Theta(n log n)\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" if lo is None:\\n\",\n \" lo = 0\\n\",\n \" if hi is None:\\n\",\n \" hi = len(A)\\n\",\n \" n = hi-lo\\n\",\n \" # base case\\n\",\n \" if n == 1:\\n\",\n \" return 0\\n\",\n \" # divide and conquer\\n\",\n \" # divide into lo:mid and mid:hi\\n\",\n \" mid = (lo+hi)//2 \\n\",\n \" # recurse on left half\\n\",\n \" gain_low = dc(A, lo, mid)\\n\",\n \" # recurse on right half\\n\",\n \" gain_high = dc(A, mid, hi)\\n\",\n \" # figure out best gain for buying in left half, selling in right half\\n\",\n \" buy_price = min([ A[i] for i in range(lo, mid) ])\\n\",\n \" sell_price = max([ A[i] for i in range(mid, hi)])\\n\",\n \" gain_cross = sell_price - buy_price\\n\",\n \" # optimum is max of three cases just solved\\n\",\n \" return max(gain_low, gain_high, gain_cross)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 10,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"16\"\n ]\n },\n \"execution_count\": 10,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"dc([20, 3, 19, 1, 15, 6])\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 11,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 80001 function calls (60003 primitive calls) in 0.073 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.004 0.004 0.005 0.005 :1(make_prices)\\n\",\n \" 19999/1 0.047 0.000 0.068 0.068 :1(dc)\\n\",\n \" 1 0.000 0.000 0.073 0.073 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 random.py:100(seed)\\n\",\n \" 1 0.000 0.000 0.000 0.000 {function seed at 0x7f90fbfeb578}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {len}\\n\",\n \" 19998 0.008 0.000 0.008 0.000 {max}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 10000 0.001 0.000 0.001 0.000 {method 'random' of '_random.Random' objects}\\n\",\n \" 9999 0.004 0.000 0.004 0.000 {min}\\n\",\n \" 19999 0.008 0.000 0.008 0.000 {range}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"dc(make_prices(10000, 1))\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Solution by algorithmic thinking\\n\",\n \"Notice that if we know a $k$ such that $b \\\\leq k \\\\leq s$ then we can find $b$ and $s$. Indeed, $b$ is minimum to the left of $k$ and $s$ is maximum to the right of $k$. Since we don't know which $k$ is correct, we need to try all values. Implementing that solution naively leads to $O(n^2)$ complexity. Not happy.\\n\",\n \"\\n\",\n \"To improve on it notice that we can precompute answer to all the questions of form *minimum to the left of $k$* (and store them in the array $B$) and *maximum to the right of $k$* (and store them in array $S$) in complexity $O(n)$. Once promputed - we can just look them up in complexity $O(1)$, which we will do $n$ times - once for each value of $k$. The total complexity is sequal to:\\n\",\n \"\\n\",\n \"$$\\n\",\n \"\\\\text{work to precompute B} + \\\\text{work to precompute S} + \\\\text{work to evaluate all values of k}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Notice that all the of those have complexity $O(n)$, so the total complexity is $O(n)$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 12,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def lin(A):\\n\",\n \" \\\"\\\"\\\" return best gain, computed by linear-time alg \\n\",\n \" running time is Theta(n)\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" n = len(A)\\n\",\n \" # B[k] = min{ A[i0]: i0 <= k } for k = 0, 1, ..., n-1\\n\",\n \" # = price to buy at if you have to buy no later than k (and sell no earlier than k)\\n\",\n \" B = [A[0]] * n\\n\",\n \" for k in range(1, n):\\n\",\n \" B[k] = min(B[k-1],A[k])\\n\",\n \" # S[k] = max{ A[j0]: j0 >= k } for k = 0, 1, ..., n-1\\n\",\n \" # = price to sell at if you have to sell no earlier than k (but bought no later than k)\\n\",\n \" S = [A[n-1]] * n\\n\",\n \" for k in range(n-2, -1, -1):\\n\",\n \" S[k] = max(S[k+1], A[k])\\n\",\n \" # G[k] = S[k] - B[k] for k = 0, 1, ..., n-1\\n\",\n \" # = best gain from buying no later than k, then selling no earlier than k\\n\",\n \" G = [ S[k]-B[k] for k in range(n) ]\\n\",\n \" # opt = max { G[k]: 0 <= k < n }\\n\",\n \" # = best possible gain for given input A\\n\",\n \" opt = max(G)\\n\",\n \" return opt\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 13,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"16\"\n ]\n },\n \"execution_count\": 13,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"lin([20, 3, 19, 1, 15, 6])\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 14,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 30010 function calls in 0.026 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.004 0.004 0.006 0.006 :1(make_prices)\\n\",\n \" 1 0.015 0.015 0.020 0.020 :1(lin)\\n\",\n \" 1 0.000 0.000 0.026 0.026 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 random.py:100(seed)\\n\",\n \" 1 0.000 0.000 0.000 0.000 {function seed at 0x7f90fbfeb578}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {len}\\n\",\n \" 10000 0.003 0.000 0.003 0.000 {max}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 10000 0.001 0.000 0.001 0.000 {method 'random' of '_random.Random' objects}\\n\",\n \" 9999 0.003 0.000 0.003 0.000 {min}\\n\",\n \" 4 0.001 0.000 0.001 0.000 {range}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"lin(make_prices(10000, 1))\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"collapsed\": true\n },\n \"source\": [\n \"# Problems to think about (non-examinable, non-compulsory, strictly for fun...)\\n\",\n \"1. **Maximum sum subsequence problem** - given an array A find a contiguous subsequence of maximum sum. For example for\\n\",\n \"```python\\n\",\n \"A = [10, -2, 10, 5, -4, 14]\\n\",\n \"```\\n\",\n \"the answer is 15.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"# Hint to problem 1\\n\",\n \"cyph = lambda x: chr((ord(x) + 64) % 128)\\n\",\n \"''.join(map(cyph, '\\\\x08).4z`2%$5#%`4()3`02/\\\",%-`4/`4(%`34/#+`%8#(!.\\\\'%`02/\\\",%-'))\"\n ]\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture11/BFS.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 11,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"from collections import defaultdict\\n\",\n \"from igraph import Graph as PlottableGraph, plot, layout, Plot\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 79,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"class Graph(object):\\n\",\n \" def __init__(self):\\n\",\n \" self.adj = defaultdict(lambda: [])\\n\",\n \" \\n\",\n \" def add_edge(self, a, b):\\n\",\n \" self.adj[a].append(b)\\n\",\n \" self.adj[b].append(a)\\n\",\n \" \\n\",\n \" def draw(self, root):\\n\",\n \" graph = PlottableGraph()\\n\",\n \" figure = Plot(bbox=(800, 600))\\n\",\n \" for name in self.adj:\\n\",\n \" graph.add_vertex(name=name, label=name)\\n\",\n \" for name, neighbors in self.adj.items():\\n\",\n \" for neighbor in neighbors:\\n\",\n \" if name < neighbor:\\n\",\n \" graph.add_edge(name, neighbor, color=\\\"black\\\")\\n\",\n \" for vv in graph.vs:\\n\",\n \" if vv.attributes()[\\\"name\\\"] == root:\\n\",\n \" root = vv.index\\n\",\n \" break\\n\",\n \" \\n\",\n \" figure.add(graph, layout=graph.layout(\\\"kk\\\"), margin = 100,\\n\",\n \" vertex_label_dist=1, root=root)\\n\",\n \" return figure\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 90,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def bfs(graph, start):\\n\",\n \" node_to_depth = {start: 0}\\n\",\n \" parent = {start:None}\\n\",\n \" \\n\",\n \" next_level = [start]\\n\",\n \" while len(next_level) > 0:\\n\",\n \" print (\\\"Level %d:\\\" % (node_to_depth[next_level[0]],))\\n\",\n \" current_level = next_level\\n\",\n \" next_level = []\\n\",\n \" for node in current_level:\\n\",\n \" print \\\" \\\", node\\n\",\n \" for neighbor in graph.adj[node]:\\n\",\n \" if neighbor not in node_to_depth:\\n\",\n \" node_to_depth[neighbor] = node_to_depth[node] + 1\\n\",\n \" parent[neighbor] = node\\n\",\n \" next_level.append(neighbor)\\n\",\n \" return node_to_depth, parent\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 80,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"g = Graph()\\n\",\n \"g.add_edge(\\\"START\\\", \\\"study_hard\\\")\\n\",\n \"g.add_edge(\\\"study_hard\\\", \\\"mit\\\")\\n\",\n \"g.add_edge(\\\"study_hard\\\", \\\"giveup\\\")\\n\",\n \"g.add_edge(\\\"study_hard\\\", \\\"get_a_real_job\\\")\\n\",\n \"g.add_edge(\\\"study_hard\\\", \\\"start_a_business\\\")\\n\",\n \"g.add_edge(\\\"start_a_business\\\", \\\"marry\\\")\\n\",\n \"g.add_edge(\\\"get_a_real_job\\\", \\\"marry\\\")\\n\",\n \"g.add_edge(\\\"marry\\\", \\\"HAPPY\\\")\\n\",\n \"\\n\",\n \"g.add_edge(\\\"giveup\\\", \\\"lookforjob\\\")\\n\",\n \"g.add_edge(\\\"lookforjob\\\", \\\"mcdonalds\\\")\\n\",\n \"g.add_edge(\\\"mcdonalds\\\", \\\"low_income\\\")\\n\",\n \"\\n\",\n \"g.add_edge(\\\"mit\\\", \\\"grad_school\\\")\\n\",\n \"g.add_edge(\\\"grad_school\\\", \\\"low_income\\\")\\n\",\n \"g.add_edge(\\\"low_income\\\", \\\"unsatisfied_demands\\\")\\n\",\n \"g.add_edge(\\\"unsatisfied_demands\\\", \\\"DEPRESSION\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 81,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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cwLqfwC9nfAqzN9dBGVnG7NefwO4tvyDmAv4N685z9MTITOHSpUSf+F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\\nAAAASUVORK5CYII=\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 81,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"g.draw(root=\\\"START\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 91,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"Level 0:\\n\",\n \" START\\n\",\n \"Level 1:\\n\",\n \" study_hard\\n\",\n \"Level 2:\\n\",\n \" mit\\n\",\n \" giveup\\n\",\n \" get_a_real_job\\n\",\n \" start_a_business\\n\",\n \"Level 3:\\n\",\n \" grad_school\\n\",\n \" lookforjob\\n\",\n \" marry\\n\",\n \"Level 4:\\n\",\n \" low_income\\n\",\n \" mcdonalds\\n\",\n \" HAPPY\\n\",\n \"Level 5:\\n\",\n \" unsatisfied_demands\\n\",\n \"Level 6:\\n\",\n \" DEPRESSION\\n\"\n ]\n }\n ],\n \"source\": [\n \"node_to_depth, parent = bfs(g, \\\"START\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 94,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def get_shortest_path(node, parent):\\n\",\n \" result = []\\n\",\n \" while node is not None:\\n\",\n \" result.append(node)\\n\",\n \" node = parent[node]\\n\",\n \" return list(reversed(result))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 95,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"['START',\\n\",\n \" 'study_hard',\\n\",\n \" 'mit',\\n\",\n \" 'grad_school',\\n\",\n \" 'low_income',\\n\",\n \" 'unsatisfied_demands',\\n\",\n \" 'DEPRESSION']\"\n ]\n },\n \"execution_count\": 95,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"get_shortest_path(\\\"DEPRESSION\\\", parent)\"\n ]\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture12/DFS.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 65,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"from collections import defaultdict, deque\\n\",\n \"from igraph import Graph as PlottableGraph, plot, layout, Plot\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 22,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"class Graph(object):\\n\",\n \" def __init__(self):\\n\",\n \" self.adj = defaultdict(lambda: [])\\n\",\n \" self.nodes = set()\\n\",\n \" def add_edge(self, a, b):\\n\",\n \" self.adj[a].append(b)\\n\",\n \" self.nodes.add(a)\\n\",\n \" self.nodes.add(b)\\n\",\n \"\\n\",\n \" def draw(self, root):\\n\",\n \" graph = PlottableGraph(directed=True)\\n\",\n \" figure = Plot(bbox=(800, 600))\\n\",\n \" for name in self.nodes:\\n\",\n \" graph.add_vertex(name=name, label=name)\\n\",\n \" for name, neighbors in self.adj.items():\\n\",\n \" for neighbor in neighbors:\\n\",\n \" graph.add_edge(name, neighbor, color=\\\"black\\\")\\n\",\n \" for vv in graph.vs:\\n\",\n \" if vv.attributes()[\\\"name\\\"] == root:\\n\",\n \" root = vv.index\\n\",\n \" break\\n\",\n \" \\n\",\n \" figure.add(graph, layout=graph.layout(\\\"kk\\\"), margin = 100,\\n\",\n \" vertex_label_dist=1, root=root)\\n\",\n \" return figure\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 23,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"g = Graph()\\n\",\n \"g.add_edge(\\\"START\\\", \\\"study_hard\\\")\\n\",\n \"g.add_edge(\\\"study_hard\\\", \\\"grad_school\\\")\\n\",\n \"g.add_edge(\\\"study_hard\\\", \\\"start_a_business\\\")\\n\",\n \"g.add_edge(\\\"start_a_business\\\", \\\"happy_life\\\")\\n\",\n \"\\n\",\n \"g.add_edge(\\\"grad_school\\\", \\\"fear_of_being_expelled\\\")\\n\",\n \"g.add_edge(\\\"fear_of_being_expelled\\\", \\\"apply_for_funding\\\")\\n\",\n \"g.add_edge(\\\"apply_for_funding\\\", \\\"work_hard\\\")\\n\",\n \"g.add_edge(\\\"work_hard\\\", \\\"publish\\\")\\n\",\n \"g.add_edge(\\\"publish\\\", \\\"out_of_funding\\\")\\n\",\n \"g.add_edge(\\\"out_of_funding\\\", \\\"fear_of_being_expelled\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 24,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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nFkHcvwCYhDsCRJ0oi1A4+x/YZmUWb/XwEPEW6q7PWQiruIMPcjOxfk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\\n0kj2f53hgpk+/uATAAAAAElFTkSuQmCC\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 24,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"g.draw(root=\\\"START\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# DFS\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 92,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def dfs(graph, start):\\n\",\n \" visited = {}\\n\",\n \" def recurse(node):\\n\",\n \" print(node)\\n\",\n \" for neighbor in graph.adj[node]:\\n\",\n \" if neighbor not in visited:\\n\",\n \" visited[neighbor] = True\\n\",\n \" recurse(neighbor)\\n\",\n \" recurse(start)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 93,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"START\\n\",\n \"study_hard\\n\",\n \"grad_school\\n\",\n \"fear_of_being_expelled\\n\",\n \"apply_for_funding\\n\",\n \"work_hard\\n\",\n \"publish\\n\",\n \"out_of_funding\\n\",\n \"start_a_business\\n\",\n \"happy_life\\n\"\n ]\n }\n ],\n \"source\": [\n \"dfs(g, \\\"START\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Cycle detection\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 60,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"# now ancestors is a set of nodes on \\\"DFS tree\\\"\\n\",\n \"# path from start node to current node. \\n\",\n \"\\n\",\n \"def ensure_no_cycle(graph, start):\\n\",\n \" visited = {}\\n\",\n \" ancestors = set()\\n\",\n \" \\n\",\n \" def recurse(node):\\n\",\n \" ancestors.add(node)\\n\",\n \" print(node)\\n\",\n \" for neighbor in graph.adj[node]:\\n\",\n \" if neighbor not in visited:\\n\",\n \" visited[neighbor] = True\\n\",\n \" recurse(neighbor)\\n\",\n \" elif neighbor in ancestors:\\n\",\n \" raise Exception(\\\"Cycle\\\")\\n\",\n \" ancestors.remove(node)\\n\",\n \" recurse(start)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Proof cycle detection works\\n\",\n \"1. Back edge $\\\\Rightarrow$ cycle: trivial\\n\",\n \"2. Cycle $\\\\Rightarrow$ backedge:\\n\",\n \"\\n\",\n \"Look at the first node discovered by DFS from that cycle $u$\\n\",\n \"\\n\",\n \"Let's consider a cycle $C$. Let $v$ be the first node on $C$ visited by DFS and $u$ we see on $C$ when we follow edges from $v$. We know that $u$ is descendant of $v$ in DFS tree, because of the fact that by assumption $u$ was not yet visited when we entered $v$ and there is a path from $v$ to $u$ (and DFS will always follow unexplored paths). Also edge $u \\\\rightarrow v$ cannot be a part of DFS tree, so it must be a back edge.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 61,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"START\\n\",\n \"study_hard\\n\",\n \"grad_school\\n\",\n \"fear_of_being_expelled\\n\",\n \"apply_for_funding\\n\",\n \"work_hard\\n\",\n \"publish\\n\",\n \"out_of_funding\\n\"\n ]\n },\n {\n \"ename\": \"Exception\",\n \"evalue\": \"Cycle\",\n \"output_type\": \"error\",\n \"traceback\": [\n \"\\u001b[1;31m---------------------------------------------------------------------------\\u001b[0m\",\n \"\\u001b[1;31mException\\u001b[0m Traceback (most recent call last)\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36m\\u001b[1;34m()\\u001b[0m\\n\\u001b[1;32m----> 1\\u001b[1;33m \\u001b[0mensure_no_cycle\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mg\\u001b[0m\\u001b[1;33m,\\u001b[0m \\u001b[1;34m\\\"START\\\"\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mensure_no_cycle\\u001b[1;34m(graph, start)\\u001b[0m\\n\\u001b[0;32m 17\\u001b[0m \\u001b[0mancestors\\u001b[0m\\u001b[1;33m.\\u001b[0m\\u001b[0mremove\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mnode\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 18\\u001b[0m \\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 19\\u001b[1;33m \\u001b[0mrecurse\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mstart\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mrecurse\\u001b[1;34m(node)\\u001b[0m\\n\\u001b[0;32m 12\\u001b[0m \\u001b[1;32mif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32mnot\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 13\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m[\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m]\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mTrue\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 14\\u001b[1;33m \\u001b[0mrecurse\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 15\\u001b[0m \\u001b[1;32melif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mancestors\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 16\\u001b[0m \\u001b[1;32mraise\\u001b[0m \\u001b[0mException\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[1;34m\\\"Cycle\\\"\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mrecurse\\u001b[1;34m(node)\\u001b[0m\\n\\u001b[0;32m 12\\u001b[0m \\u001b[1;32mif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32mnot\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 13\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m[\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m]\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mTrue\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 14\\u001b[1;33m \\u001b[0mrecurse\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 15\\u001b[0m \\u001b[1;32melif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mancestors\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 16\\u001b[0m \\u001b[1;32mraise\\u001b[0m \\u001b[0mException\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[1;34m\\\"Cycle\\\"\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mrecurse\\u001b[1;34m(node)\\u001b[0m\\n\\u001b[0;32m 12\\u001b[0m \\u001b[1;32mif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32mnot\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 13\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m[\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m]\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mTrue\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 14\\u001b[1;33m \\u001b[0mrecurse\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 15\\u001b[0m \\u001b[1;32melif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mancestors\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 16\\u001b[0m \\u001b[1;32mraise\\u001b[0m \\u001b[0mException\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[1;34m\\\"Cycle\\\"\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mrecurse\\u001b[1;34m(node)\\u001b[0m\\n\\u001b[0;32m 12\\u001b[0m \\u001b[1;32mif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32mnot\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 13\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m[\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m]\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mTrue\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 14\\u001b[1;33m \\u001b[0mrecurse\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 15\\u001b[0m \\u001b[1;32melif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mancestors\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 16\\u001b[0m \\u001b[1;32mraise\\u001b[0m \\u001b[0mException\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[1;34m\\\"Cycle\\\"\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mrecurse\\u001b[1;34m(node)\\u001b[0m\\n\\u001b[0;32m 12\\u001b[0m \\u001b[1;32mif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32mnot\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 13\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m[\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m]\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mTrue\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 14\\u001b[1;33m \\u001b[0mrecurse\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 15\\u001b[0m \\u001b[1;32melif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mancestors\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 16\\u001b[0m \\u001b[1;32mraise\\u001b[0m \\u001b[0mException\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[1;34m\\\"Cycle\\\"\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mrecurse\\u001b[1;34m(node)\\u001b[0m\\n\\u001b[0;32m 12\\u001b[0m \\u001b[1;32mif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32mnot\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 13\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m[\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m]\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mTrue\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 14\\u001b[1;33m \\u001b[0mrecurse\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 15\\u001b[0m \\u001b[1;32melif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mancestors\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 16\\u001b[0m \\u001b[1;32mraise\\u001b[0m \\u001b[0mException\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[1;34m\\\"Cycle\\\"\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mrecurse\\u001b[1;34m(node)\\u001b[0m\\n\\u001b[0;32m 12\\u001b[0m \\u001b[1;32mif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32mnot\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 13\\u001b[0m \\u001b[0mvisited\\u001b[0m\\u001b[1;33m[\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m]\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mTrue\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 14\\u001b[1;33m \\u001b[0mrecurse\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 15\\u001b[0m \\u001b[1;32melif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mancestors\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 16\\u001b[0m \\u001b[1;32mraise\\u001b[0m \\u001b[0mException\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[1;34m\\\"Cycle\\\"\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mrecurse\\u001b[1;34m(node)\\u001b[0m\\n\\u001b[0;32m 14\\u001b[0m \\u001b[0mrecurse\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mneighbor\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 15\\u001b[0m \\u001b[1;32melif\\u001b[0m \\u001b[0mneighbor\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mancestors\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 16\\u001b[1;33m \\u001b[1;32mraise\\u001b[0m \\u001b[0mException\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[1;34m\\\"Cycle\\\"\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 17\\u001b[0m \\u001b[0mancestors\\u001b[0m\\u001b[1;33m.\\u001b[0m\\u001b[0mremove\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mnode\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 18\\u001b[0m \\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;31mException\\u001b[0m: Cycle\"\n ]\n }\n ],\n \"source\": [\n \"ensure_no_cycle(g, \\\"START\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 79,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def recover_cycle(graph, start):\\n\",\n \" visited = {}\\n\",\n \" ancestors = set()\\n\",\n \" path = deque()\\n\",\n \" cycle_end = [None]\\n\",\n \" \\n\",\n \" def recurse(node):\\n\",\n \" ancestors.add(node)\\n\",\n \" path.append(node)\\n\",\n \" for neighbor in graph.adj[node]:\\n\",\n \" if neighbor not in visited:\\n\",\n \" visited[neighbor] = True\\n\",\n \" recurse(neighbor)\\n\",\n \" elif neighbor in ancestors:\\n\",\n \" cycle_end[0] = neighbor\\n\",\n \" raise Exception(\\\"Cycle\\\")\\n\",\n \" path.pop(node)\\n\",\n \" ancestors.pop(node)\\n\",\n \" try:\\n\",\n \" recurse(start)\\n\",\n \" except:\\n\",\n \" while path[0] != cycle_end[0]:\\n\",\n \" path.popleft()\\n\",\n \" return path\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 78,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"deque(['fear_of_being_expelled',\\n\",\n \" 'apply_for_funding',\\n\",\n \" 'work_hard',\\n\",\n \" 'publish',\\n\",\n \" 'out_of_funding'])\"\n ]\n },\n \"execution_count\": 78,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"recover_cycle(g, \\\"START\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# BFS and DFS are close cousins really.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 86,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def bfs_from(graph, source):\\n\",\n \" q = deque()\\n\",\n \" # initially source is on the queue\\n\",\n \" distance = {source: 0}\\n\",\n \" q.appendleft(source)\\n\",\n \" # while queue is not empty\\n\",\n \" while len(q) > 0:\\n\",\n \" # consider the node that has been in the queue for\\n\",\n \" # the longest\\n\",\n \" node = q.popleft()\\n\",\n \" # for all neighbours\\n\",\n \" for neighbour in graph.adj[node]:\\n\",\n \" # if they were NOT visted yet\\n\",\n \" if neighbour not in distance:\\n\",\n \" # mark their distance and put them on queue\\n\",\n \" distance[neighbour] = distance[node] + 1\\n\",\n \" q.append(neighbour)\\n\",\n \" return distance\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 95,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def bfs_from(graph, source):\\n\",\n \" q = deque()\\n\",\n \" distance = {source: 0}\\n\",\n \" q.appendleft(source)\\n\",\n \" while len(q) > 0:\\n\",\n \" node = q.popleft()\\n\",\n \" print(node)\\n\",\n \" for neighbour in graph.adj[node]:\\n\",\n \" if neighbour not in distance:\\n\",\n \" distance[neighbour] = distance[node] + 1\\n\",\n \" q.append(neighbour)\\n\",\n \" return distance\\n\",\n \"\\n\",\n \"def dfs_from(graph, source):\\n\",\n \" q = deque()\\n\",\n \" distance = {source: 0}\\n\",\n \" q.appendleft(source)\\n\",\n \" while len(q) > 0:\\n\",\n \" node = q.pop() # only change!\\n\",\n \" print(node)\\n\",\n \" for neighbour in reversed(graph.adj[node]): # ok fine\\n\",\n \" if neighbour not in distance:\\n\",\n \" distance[neighbour] = distance[node] + 1\\n\",\n \" q.append(neighbour)\\n\",\n \" return distance\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 100,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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RapXpUqQi/cK+d6QtH3bKAVOJFQFJ5PAN5YYTulEpBfAG/Jtd0GfAj4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\\nJEmSJKmC/wPpqauSTA8gEAAAAABJRU5ErkJggg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 100,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"g.draw(root=\\\"START\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 98,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"START\\n\",\n \"study_hard\\n\",\n \"grad_school\\n\",\n \"start_a_business\\n\",\n \"fear_of_being_expelled\\n\",\n \"happy_life\\n\",\n \"apply_for_funding\\n\",\n \"work_hard\\n\",\n \"publish\\n\",\n \"out_of_funding\\n\"\n ]\n }\n ],\n \"source\": [\n \"_ = bfs_from(g, \\\"START\\\") # iterative DFS\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 96,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"START\\n\",\n \"study_hard\\n\",\n \"grad_school\\n\",\n \"fear_of_being_expelled\\n\",\n \"apply_for_funding\\n\",\n \"work_hard\\n\",\n \"publish\\n\",\n \"out_of_funding\\n\",\n \"start_a_business\\n\",\n \"happy_life\\n\"\n ]\n }\n ],\n \"source\": [\n \"_ = dfs_from(g, \\\"START\\\") # iterative DFS\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 97,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"START\\n\",\n \"study_hard\\n\",\n \"grad_school\\n\",\n \"fear_of_being_expelled\\n\",\n \"apply_for_funding\\n\",\n \"work_hard\\n\",\n \"publish\\n\",\n \"out_of_funding\\n\",\n \"start_a_business\\n\",\n \"happy_life\\n\"\n ]\n }\n ],\n \"source\": [\n \"dfs(g, \\\"START\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Topological sort\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 108,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"g = Graph()\\n\",\n \"g.add_edge(\\\"intersections.cpp\\\", \\\"geometry.o\\\")\\n\",\n \"g.add_edge(\\\"triangles.cpp\\\", \\\"geometry.o\\\")\\n\",\n \"g.add_edge(\\\"utils.cpp\\\", \\\"geometry.o\\\")\\n\",\n \"\\n\",\n \"g.add_edge(\\\"utils.cpp\\\", \\\"random.o\\\")\\n\",\n \"\\n\",\n \"g.add_edge(\\\"neural_network.cpp\\\", \\\"ai.o\\\")\\n\",\n \"g.add_edge(\\\"behavioral_model.cpp\\\", \\\"ai.o\\\")\\n\",\n \"g.add_edge(\\\"utils.cpp\\\", \\\"ai.o\\\")\\n\",\n \"g.add_edge(\\\"random.o\\\", \\\"ai.o\\\")\\n\",\n \"\\n\",\n \"g.add_edge(\\\"geometry.o\\\", \\\"game.o\\\")\\n\",\n \"g.add_edge(\\\"ai.o\\\", \\\"game.o\\\")\\n\",\n \"\\n\",\n \"g.add_edge(\\\"queue_server.cpp\\\", \\\"queue_server.o\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 109,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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d6hvheZj48klPFdkTrXiRnHDvbzUOjPTC37EOFnyapXkiRJqnj/j5Bc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\\nIwAAAABJRU5ErkJggg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 109,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"g.draw(root=\\\"\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 138,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def topological_sort(graph):\\n\",\n \" visited = {}\\n\",\n \" result = []\\n\",\n \" def recurse(node):\\n\",\n \" for neighbor in graph.adj[node]:\\n\",\n \" if neighbor not in visited:\\n\",\n \" visited[neighbor] = True\\n\",\n \" recurse(neighbor)\\n\",\n \" result.append(node)\\n\",\n \" \\n\",\n \" for potential_start in graph.nodes:\\n\",\n \" if potential_start not in visited:\\n\",\n \" visited[potential_start] = True\\n\",\n \" recurse(potential_start)\\n\",\n \" return list(reversed(result))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 139,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"['triangles.cpp',\\n\",\n \" 'behavioral_model.cpp',\\n\",\n \" 'neural_network.cpp',\\n\",\n \" 'utils.cpp',\\n\",\n \" 'random.o',\\n\",\n \" 'ai.o',\\n\",\n \" 'intersections.cpp',\\n\",\n \" 'geometry.o',\\n\",\n \" 'game.o',\\n\",\n \" 'queue_server.cpp',\\n\",\n \" 'queue_server.o']\"\n ]\n },\n \"execution_count\": 139,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"topological_sort(g)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 130,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def topological_nondfs(graph):\\n\",\n \" indegree = {node:0 for node in graph.nodes}\\n\",\n \" \\n\",\n \" for node in graph.nodes:\\n\",\n \" for neighbor in graph.adj[node]:\\n\",\n \" indegree[neighbor] += 1\\n\",\n \" \\n\",\n \" enabled = [node for node in graph.nodes\\n\",\n \" if indegree[node] == 0]\\n\",\n \" result = []\\n\",\n \" while len(enabled) > 0:\\n\",\n \" node = enabled.pop()\\n\",\n \" result.append(node)\\n\",\n \" for neighbor in graph.adj[node]:\\n\",\n \" indegree[neighbor] -= 1\\n\",\n \" if indegree[neighbor] == 0:\\n\",\n \" enabled.append(neighbor)\\n\",\n \" \\n\",\n \" return result\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 131,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"['triangles.cpp',\\n\",\n \" 'behavioral_model.cpp',\\n\",\n \" 'neural_network.cpp',\\n\",\n \" 'utils.cpp',\\n\",\n \" 'random.o',\\n\",\n \" 'ai.o',\\n\",\n \" 'intersections.cpp',\\n\",\n \" 'geometry.o',\\n\",\n \" 'game.o',\\n\",\n \" 'queue_server.cpp',\\n\",\n \" 'queue_server.o']\"\n ]\n },\n \"execution_count\": 131,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"topological_nondfs(g)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture12/Google BFS.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Google BFS question\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 1,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"from collections import deque\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Given a 2D board with emergency $E$ and some policeman $P$, what is the distance from an emergency to closest policeman. Nodes marked as $X$, are walls and cannot be traversed.\\n\",\n \"\\n\",\n \"\\n\",\n \"Think a little bit about how to solve it\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 66,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"example = [\\n\",\n \" \\\"...P...\\\",\\n\",\n \" \\\"...XXX.\\\",\\n\",\n \" \\\"EX..P..\\\",\\n\",\n \" \\\".X.....\\\",\\n\",\n \" \\\".......\\\",\\n\",\n \" \\\".......\\\",\\n\",\n \"]\\n\",\n \"\\n\",\n \"class MapGraph(object):\\n\",\n \" def __init__(self, mmap):\\n\",\n \" \\\"\\\"\\\"Graph abstraction for our map\\\"\\\"\\\"\\n\",\n \" self.dim_x, self.dim_y = len(mmap), len(mmap[0])\\n\",\n \" self.mmap = {(x,y):mmap[x][y] for x in range(self.dim_x) \\n\",\n \" for y in range(self.dim_y)}\\n\",\n \"\\n\",\n \" def neighbours(self, node):\\n\",\n \" \\\"\\\"\\\"Returns all the direct neighbors of a given node\\\"\\\"\\\"\\n\",\n \" x,y = node\\n\",\n \" # There are four directions in which we can go\\n\",\n \" for dx, dy in [[0,1], [0,-1], [-1,0], [1,0]]:\\n\",\n \" nx, ny = x + dx, y + dy\\n\",\n \" if (0 <= nx < self.dim_x and # but we cannot got through walls\\n\",\n \" 0 <= ny < self.dim_y # (at least not yet!)\\n\",\n \" and self.mmap[nx,ny] != 'X'):\\n\",\n \" yield nx, ny\\n\",\n \" \\n\",\n \" def find_all(self, letter):\\n\",\n \" \\\"\\\"\\\"Finds all the coordinates where a given letter occurs\\\"\\\"\\\"\\n\",\n \" res = []\\n\",\n \" for x in range(self.dim_x):\\n\",\n \" for y in range(self.dim_y):\\n\",\n \" if self.mmap[x,y] == letter:\\n\",\n \" res.append((x,y))\\n\",\n \" return res\\n\",\n \" \\n\",\n \" def show(self, what=None):\\n\",\n \" \\\"\\\"\\\"Displays the graph\\\"\\\"\\\"\\n\",\n \" what = what or self.mmap\\n\",\n \" for x in range(self.dim_x):\\n\",\n \" for y in range(self.dim_y):\\n\",\n \" print(what[x,y] if (x,y) in what else '?', end='')\\n\",\n \" print(' ', end='')\\n\",\n \" print('')\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 67,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \". . . P . . . \\n\",\n \". . . X X X . \\n\",\n \"E X . . P . . \\n\",\n \". X . . . . . \\n\",\n \". . . . . . . \\n\",\n \". . . . . . . \\n\"\n ]\n },\n {\n \"data\": {\n \"text/plain\": [\n \"[(2, 0)]\"\n ]\n },\n \"execution_count\": 67,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"g = MapGraph(example)\\n\",\n \"g.show()\\n\",\n \"g.find_all('E')\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"The solution is maybe somewhat counterintuitive - we start our search from emergency, not policeman\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 68,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def bfs_from(graph, source):\\n\",\n \" q = deque()\\n\",\n \" # initially source is on the queue\\n\",\n \" distance = {source: 0}\\n\",\n \" q.appendleft(source)\\n\",\n \" # while queue is not empty\\n\",\n \" while len(q) > 0:\\n\",\n \" # consider the node that has been in the queue for\\n\",\n \" # the longest\\n\",\n \" node = q.popleft()\\n\",\n \" # for all neighbours\\n\",\n \" for neighbour in graph.neighbours(node):\\n\",\n \" # if they were NOT visted yet\\n\",\n \" if neighbour not in distance:\\n\",\n \" # mark their distance and put them on queue\\n\",\n \" distance[neighbour] = distance[node] + 1\\n\",\n \" q.append(neighbour)\\n\",\n \" return distance\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 76,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \". . . P . . . \\n\",\n \". . . X X X . \\n\",\n \"E X . . P . . \\n\",\n \". X . . . . . \\n\",\n \". . . . . . . \\n\",\n \". . . . . . . \\n\",\n \"\\n\",\n \"2 3 4 5 6 7 8 \\n\",\n \"1 2 3 ? ? ? 9 \\n\",\n \"0 ? 4 5 6 7 8 \\n\",\n \"1 ? 5 6 7 8 9 \\n\",\n \"2 3 4 5 6 7 8 \\n\",\n \"3 4 5 6 7 8 9 \\n\"\n ]\n }\n ],\n \"source\": [\n \"g = MapGraph(example)\\n\",\n \"distances = bfs_from(g, g.find_all('E')[0])\\n\",\n \"g.show()\\n\",\n \"print('')\\n\",\n \"g.show(distances)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 70,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def solve_google(example):\\n\",\n \" g = MapGraph(example)\\n\",\n \" # comute distances from emergency to everywhere welse\\n\",\n \" distances = bfs_from(g, g.find_all('E')[0])\\n\",\n \" # find the minimum distance policeman\\n\",\n \" res = float('inf')\\n\",\n \" for policeman in g.find_all('P'):\\n\",\n \" res = min(res, distances[policeman])\\n\",\n \" return res\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 71,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"5\"\n ]\n },\n \"execution_count\": 71,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"solve_google(example)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Success! \\n\",\n \"\\n\",\n \"We solve the problem correctly in $O(nm)$, which is the best solution we can hoped for!\\n\",\n \"\\n\",\n \"## Follow up question\\n\",\n \"\\n\",\n \"Find the worst place for emergency (furthest from all policeman)\\n\",\n \"\\n\",\n \"Think about how to solve it.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 72,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"example2 = [\\n\",\n \" \\\"...P...\\\",\\n\",\n \" \\\"...XXX.\\\",\\n\",\n \" \\\"EX..P..\\\",\\n\",\n \" \\\".X.....\\\",\\n\",\n \" \\\".......\\\",\\n\",\n \" \\\".......\\\",\\n\",\n \"]\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"There are many suboptimal solutions, but it turns out we can still solve it in $O(nm)$, by starting BFS from all the policeman simultaneoursly!\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 73,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def bfs_from_many(graph, sources):\\n\",\n \" q = deque()\\n\",\n \" distance = {s: 0 for s in sources} # <--- this line changed\\n\",\n \" q.extendleft(sources) # <--- this line changed\\n\",\n \" while len(q) > 0:\\n\",\n \" node = q.popleft()\\n\",\n \" for neighbour in graph.neighbours(node):\\n\",\n \" if neighbour not in distance:\\n\",\n \" distance[neighbour] = distance[node] + 1\\n\",\n \" q.append(neighbour)\\n\",\n \" return distance\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 74,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \". . . P . . . \\n\",\n \". . . X X X . \\n\",\n \"E X . . P . . \\n\",\n \". X . . . . . \\n\",\n \". . . . . . . \\n\",\n \". . . . . . . \\n\",\n \"\\n\",\n \"3 2 1 0 1 2 3 \\n\",\n \"4 3 2 ? ? ? 3 \\n\",\n \"5 ? 2 1 0 1 2 \\n\",\n \"6 ? 3 2 1 2 3 \\n\",\n \"6 5 4 3 2 3 4 \\n\",\n \"7 6 5 4 3 4 5 \\n\"\n ]\n }\n ],\n \"source\": [\n \"g = MapGraph(example2)\\n\",\n \"distances = bfs_from_many(g, g.find_all('P'))\\n\",\n \"g.show()\\n\",\n \"print('')\\n\",\n \"g.show(distances)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 77,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def solve_google_hard(example):\\n\",\n \" g = MapGraph(example)\\n\",\n \" # compute distances from all the policeman\\n\",\n \" distances = bfs_from_many(g, g.find_all('P'))\\n\",\n \" # find the minimum distance policeman\\n\",\n \" return max(distances.values())\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 78,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"7\"\n ]\n },\n \"execution_count\": 78,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"solve_google_hard(example2)\"\n ]\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 3\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.4.1\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture15/Dijkstra's algorithm.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 58,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"from collections import defaultdict, deque\\n\",\n \"from Queue import PriorityQueue\\n\",\n \"from igraph import Graph as PlottableGraph, plot, layout, Plot\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 94,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"class Graph(object):\\n\",\n \" def __init__(self):\\n\",\n \" self.adj = defaultdict(lambda: [])\\n\",\n \" self.nodes = set()\\n\",\n \" \\n\",\n \" def add_edge(self, a, b, weight):\\n\",\n \" self.adj[a].append((b, weight))\\n\",\n \" self.nodes.add(a)\\n\",\n \" self.nodes.add(b)\\n\",\n \"\\n\",\n \" def draw(self):\\n\",\n \" graph = PlottableGraph(directed=True)\\n\",\n \" figure = Plot(bbox=(400, 400))\\n\",\n \" for name in self.nodes:\\n\",\n \" graph.add_vertex(name=name, label=name)\\n\",\n \" for name, neighbors in self.adj.items():\\n\",\n \" for (neighbor, weight) in neighbors:\\n\",\n \" graph.add_edge(name, neighbor, color=\\\"black\\\", weight=weight, label=str(weight))\\n\",\n \" \\n\",\n \" figure.add(graph, layout=graph.layout(\\\"kk\\\"), margin = 100,\\n\",\n \" vertex_label_dist=1)\\n\",\n \" return figure\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 95,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"g = Graph()\\n\",\n \"g.add_edge(\\\"A\\\",\\\"B\\\", 2)\\n\",\n \"g.add_edge(\\\"A\\\",\\\"C\\\", 3)\\n\",\n \"g.add_edge(\\\"B\\\",\\\"D\\\", 4)\\n\",\n \"g.add_edge(\\\"C\\\",\\\"D\\\", 1)\\n\",\n \"g.add_edge(\\\"D\\\",\\\"A\\\", 1)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 96,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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kpFMIFUx6y5AyfVss2N2xtLgUsYOPn1Em8ciFsUG/OPx3J7YzUDl/4j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\\nkiRJzeD/AxyO/k8yE0oWAAAAAElFTkSuQmCC\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 96,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"g.draw()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Dijkstra's algorithm\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 97,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def dijkstra(graph, source):\\n\",\n \" # set of already explored nodes (S from lectures)\\n\",\n \" visited = set()\\n\",\n \" # distance from source\\n\",\n \" distance = defaultdict(lambda: float('inf'))\\n\",\n \" # predecessor on the shortest path from source\\n\",\n \" predecessor = defaultdict(lambda: None)\\n\",\n \" # queue of tuples (distance, node). \\n\",\n \" # returns shortest distance first\\n\",\n \" q = PriorityQueue()\\n\",\n \" \\n\",\n \" # add source to the state\\n\",\n \" distance[source] = 0\\n\",\n \" q.put((0,source))\\n\",\n \"\\n\",\n \" # while there are still nodes in the queue\\n\",\n \" while not q.empty():\\n\",\n \" _, node = q.get()\\n\",\n \" # check that we did not already process this node\\n\",\n \" # (nodes can be inserted to the queue multiple times)\\n\",\n \" if node in visited:\\n\",\n \" continue\\n\",\n \" # mark as visited\\n\",\n \" visited.add(node)\\n\",\n \" \\n\",\n \" # for every neighbor\\n\",\n \" for neighbor, edge_len in graph.adj[node]:\\n\",\n \" # can we relax the edge?\\n\",\n \" if distance[neighbor] > distance[node] + edge_len:\\n\",\n \" # store updated distance\\n\",\n \" distance[neighbor] = distance[node] + edge_len\\n\",\n \" # add edge to the queue (maybe again...)\\n\",\n \" q.put((distance[neighbor], neighbor))\\n\",\n \" # store predecessor\\n\",\n \" predecessor[neighbor] = node\\n\",\n \"\\n\",\n \" return distance, predecessor\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 98,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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MotqQpnAdenDqJJdFBq7Pginl2SZQ8CR6YOQiXOoefTHOCF1EE0iW3A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\\nJEmSJEmSJEmSJEmSJEmSJEmSlCX/H4og5XDhLpeJAAAAAElFTkSuQmCC\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 98,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"g.draw()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 99,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"(defaultdict(>, {'A': 0, 'B': 2, 'C': 3, 'D': 4}),\\n\",\n \" defaultdict(>, {'B': 'A', 'C': 'A', 'D': 'C'}))\"\n ]\n },\n \"execution_count\": 99,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"dijkstra(g, \\\"A\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Negative edge example\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 100,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"g = Graph()\\n\",\n \"g.add_edge(\\\"A\\\",\\\"B\\\", 2)\\n\",\n \"g.add_edge(\\\"A\\\",\\\"C\\\", 3)\\n\",\n \"g.add_edge(\\\"B\\\",\\\"D\\\", 3)\\n\",\n \"g.add_edge(\\\"D\\\",\\\"C\\\", 4)\\n\",\n \"g.add_edge(\\\"C\\\",\\\"B\\\", -2)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 101,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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94KbXsP8xvvB+ZC15twCDa3zI0JJL2HiOmKdw7wmn8F/qL78RvEjoSL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\\nJEmSpEH8f2kopTwnV1FaAAAAAElFTkSuQmCC\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 101,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"g.draw()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture16/Bignums.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 1,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"class Number(list):\\n\",\n \" @staticmethod\\n\",\n \" def wrap(n):\\n\",\n \" return Number(reversed([ord(c) - ord('0') for c in str(n)]))\\n\",\n \" \\n\",\n \" def __repr__(self):\\n\",\n \" return ''.join([str(d) for d in reversed(self)])\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 2,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"([1, 1], [2, 1])\"\n ]\n },\n \"execution_count\": 2,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a = Number.wrap(11)\\n\",\n \"b = Number.wrap(12)\\n\",\n \"a, b\\n\",\n \"list(a), list(b)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Addition\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 3,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def add(a, b):\\n\",\n \" c = Number()\\n\",\n \" carry = 0\\n\",\n \" for i in range(max(len(a), len(b))):\\n\",\n \" r = carry + (a[i] if i < len(a) else 0) + (b[i] if i < len(b) else 0)\\n\",\n \" c.append(r % 10)\\n\",\n \" carry = r / 10\\n\",\n \" if carry > 0:\\n\",\n \" c.append(carry)\\n\",\n \" return c\\n\",\n \"\\n\",\n \"Number.__add__ = add\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 4,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"13\"\n ]\n },\n \"execution_count\": 4,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"Number.wrap(6) + Number.wrap(7)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 5,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"76\"\n ]\n },\n \"execution_count\": 5,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"Number.wrap(6) + Number.wrap(70)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Substraction\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 80,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def sub(a, b):\\n\",\n \" c = Number()\\n\",\n \" carry = 0\\n\",\n \" for i in range(max(len(a), len(b))):\\n\",\n \" r = carry + (a[i] if i < len(a) else 0) - (b[i] if i < len(b) else 0)\\n\",\n \" carry = 0 \\n\",\n \" while r < 0:\\n\",\n \" r += 10\\n\",\n \" carry -= 1\\n\",\n \" c.append(r % 10)\\n\",\n \" assert carry == 0, \\\"negative result\\\"\\n\",\n \" while len(c) > 1 and c[-1] == 0:\\n\",\n \" c.pop()\\n\",\n \" return c\\n\",\n \"Number.__sub__ = sub\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 81,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"1\"\n ]\n },\n \"execution_count\": 81,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"a, b = Number.wrap(7), Number.wrap(6)\\n\",\n \"sub(a,b)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 82,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"64\"\n ]\n },\n \"execution_count\": 82,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"Number.wrap(70) - Number.wrap(6)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 83,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"0\"\n ]\n },\n \"execution_count\": 83,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"Number.wrap(70) - Number.wrap(70)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Multiplication by digit\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 126,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def mul_by_digit(number, digit):\\n\",\n \" res = Number()\\n\",\n \" carry = 0\\n\",\n \" for i in range(len(number)):\\n\",\n \" r = carry + number[i] * digit\\n\",\n \" res.append(r % 10)\\n\",\n \" carry = r / 10\\n\",\n \" if carry > 0:\\n\",\n \" res.append(carry)\\n\",\n \" return res\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 127,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"1107\"\n ]\n },\n \"execution_count\": 127,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"mul_by_digit(Number.wrap(123), 9)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Karatsuba Multiplication\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 116,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def mul(a, b):\\n\",\n \" if len(a) < len(b):\\n\",\n \" a, b = b, a\\n\",\n \" # now a is longer of the two\\n\",\n \" assert len(a) >= len(b)\\n\",\n \" if len(b) == 0:\\n\",\n \" return Number.wrap(0)\\n\",\n \" elif len(b) == 1:\\n\",\n \" return mul_by_digit(a, b[0])\\n\",\n \" \\n\",\n \" mid = len(a) / 2\\n\",\n \" y1, x1 = a[:mid], a[mid:]\\n\",\n \" y2, x2 = b[:mid], b[mid:]\\n\",\n \" H = mul(x1, x2)\\n\",\n \" L = mul(y1, y2)\\n\",\n \" M = mul(add(x1,y1), add(x2,y2))\\n\",\n \" M = sub(M,H)\\n\",\n \" M = sub(M,L)\\n\",\n \" \\n\",\n \" res = Number([0] * (2 * mid) + list(H)) + Number([0] * (mid) + list(M)) + L\\n\",\n \" return res\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 124,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"1353\"\n ]\n },\n \"execution_count\": 124,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"mul(Number.wrap(123), Number.wrap(11))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 123,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"for a in range(1112):\\n\",\n \" for b in range(123):\\n\",\n \" A,B = Number.wrap(a), Number.wrap(b)\\n\",\n \" assert int(str(mul(A, B))) == a*b\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Multiplication\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 12,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def sqrt(a):\\n\",\n \" nextx, x = None, a\\n\",\n \" while True:\\n\",\n \" print(bin(x), x)\\n\",\n \" nextx = (x + (a/x)) / 2\\n\",\n \" if x == nextx:\\n\",\n \" break\\n\",\n \" x = nextx\\n\",\n \" return x\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 13,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"('0b10000011100010000100', 538756)\\n\",\n \"('0b1000001110001000010', 269378)\\n\",\n \"('0b100000111000100010', 134690)\\n\",\n \"('0b10000011100010010', 67346)\\n\",\n \"('0b1000001110001100', 33676)\\n\",\n \"('0b100000111001101', 16845)\\n\",\n \"('0b10000011110110', 8438)\\n\",\n \"('0b1000010011010', 4250)\\n\",\n \"('0b100010001100', 2188)\\n\",\n \"('0b10011000001', 1217)\\n\",\n \"('0b1100111101', 829)\\n\",\n \"('0b1011100011', 739)\\n\",\n \"('0b1011011110', 734)\\n\"\n ]\n },\n {\n \"data\": {\n \"text/plain\": [\n \"734\"\n ]\n },\n \"execution_count\": 13,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"sqrt(734 * 734)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 14,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"('0b10000011111001001011', 540235)\\n\",\n \"('0b1000001111100100110', 270118)\\n\",\n \"('0b100000111110010011', 135059)\\n\",\n \"('0b10000011111001011', 67531)\\n\",\n \"('0b1000001111101001', 33769)\\n\",\n \"('0b100000111111100', 16892)\\n\",\n \"('0b10000100001101', 8461)\\n\",\n \"('0b1000010100110', 4262)\\n\",\n \"('0b100010010010', 2194)\\n\",\n \"('0b10011000100', 1220)\\n\",\n \"('0b1100111111', 831)\\n\",\n \"('0b1011100100', 740)\\n\",\n \"('0b1011011111', 735)\\n\"\n ]\n },\n {\n \"data\": {\n \"text/plain\": [\n \"735\"\n ]\n },\n \"execution_count\": 14,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"sqrt(735 * 735 + 10)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 15,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"('0b10000011111000110111', 540215)\\n\",\n \"('0b1000001111100011100', 270108)\\n\",\n \"('0b100000111110001110', 135054)\\n\",\n \"('0b10000011111001000', 67528)\\n\",\n \"('0b1000001111100111', 33767)\\n\",\n \"('0b100000111111011', 16891)\\n\",\n \"('0b10000100001101', 8461)\\n\",\n \"('0b1000010100110', 4262)\\n\",\n \"('0b100010010010', 2194)\\n\",\n \"('0b10011000100', 1220)\\n\",\n \"('0b1100111111', 831)\\n\",\n \"('0b1011100100', 740)\\n\",\n \"('0b1011011111', 735)\\n\",\n \"('0b1011011110', 734)\\n\"\n ]\n },\n {\n \"data\": {\n \"text/plain\": [\n \"734\"\n ]\n },\n \"execution_count\": 15,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"sqrt(735 * 735 - 10)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 11,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"200000000000000000000\\n\",\n \"100000000000000000000\\n\",\n \"50000000000000000001\\n\",\n \"25000000000000000002\\n\",\n \"12500000000000000004\\n\",\n \"6250000000000000009\\n\",\n \"3125000000000000020\\n\",\n \"1562500000000000041\\n\",\n \"781250000000000084\\n\",\n \"390625000000000169\\n\",\n \"195312500000000340\\n\",\n \"97656250000000681\\n\",\n \"48828125000001364\\n\",\n \"24414062500002729\\n\",\n \"12207031250005460\\n\",\n \"6103515625010921\\n\",\n \"3051757812521844\\n\",\n \"1525878906293689\\n\",\n \"762939453212380\\n\",\n \"381469726737261\\n\",\n \"190734863630774\\n\",\n \"95367432339674\\n\",\n \"47683717218412\\n\",\n \"23841860706357\\n\",\n \"11920934547482\\n\",\n \"5960475662345\\n\",\n \"2980254608357\\n\",\n \"1490160858358\\n\",\n \"745147536028\\n\",\n \"372707969625\\n\",\n \"186622291390\\n\",\n \"93846987363\\n\",\n \"47989057987\\n\",\n \"26078337348\\n\",\n \"16873768965\\n\",\n \"14363242737\\n\",\n \"14143837480\\n\",\n \"14142135726\\n\",\n \"14142135623\\n\"\n ]\n },\n {\n \"data\": {\n \"text/plain\": [\n \"1.4142135623\"\n ]\n },\n \"execution_count\": 11,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"sqrt(2 * 10**20) / float(10**10)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture2/Document distance.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Document distance\\n\",\n \"\\n\",\n \"In this section, we analyze an algorithm for calculating the distance between two documents\\n\",\n \"in terms of the angle between them. In other words, we try to see how much overlap these\\n\",\n \"documents have in terms of their constituent words with no regard to their order. For exam-\\n\",\n \"ple, `[\\\"Steve\\\",\\\"is\\\",\\\"pretty\\\",\\\"cool\\\"]` is quite close to `[\\\"Steve\\\",\\\"is\\\",\\\"pretty\\\",\\\"awesome\\\"]` but an exact\\n\",\n \"match for `[\\\"is\\\",\\\"steve\\\",\\\"pretty\\\",\\\"cool\\\"]`. This algorithm has applications in search engines, where\\n\",\n \"we match documents based on the document distance between a query and a stored document.\\n\",\n \"In this section we walk through the algorithm line-by-line. See the \\\"Document Distance\\\"\\n\",\n \"handout for a full description of the algorithm and iterative improvements we make on it. The\\n\",\n \"improvements lead to the following. Note that $O(L)$ denotes time linear in the length of an array\\n\",\n \"$L$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 1,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"import math\\n\",\n \"import string\\n\",\n \"import sys\\n\",\n \"import cProfile\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 2,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"with open(\\\"book1.txt\\\") as f:\\n\",\n \" document1 = f.read()\\n\",\n \"with open(\\\"book2.txt\\\") as f:\\n\",\n \" document2 = f.read()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 3,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def read_file(filename):\\n\",\n \" \\\"\\\"\\\" \\n\",\n \" Read the text file with the given filename;\\n\",\n \" return a list of the lines of text in the file.\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" try:\\n\",\n \" f = open(filename, 'r')\\n\",\n \" return f.readlines()\\n\",\n \" except IOError:\\n\",\n \" print \\\"Error opening or reading input file: \\\",filename\\n\",\n \" sys.exit()\\n\",\n \"\\n\",\n \"def get_words_from_line_list(L):\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" Parse the given list L of text lines into words.\\n\",\n \" Return list of all words found.\\n\",\n \" \\\"\\\"\\\"\\n\",\n \"\\n\",\n \" word_list = []\\n\",\n \" for line in L:\\n\",\n \" words_in_line = get_words_from_string(line)\\n\",\n \" word_list = word_list + words_in_line\\n\",\n \" return word_list\\n\",\n \"\\n\",\n \"def get_words_from_string(line):\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" Return a list of the words in the given input string,\\n\",\n \" converting each word to lower-case.\\n\",\n \"\\n\",\n \" Input: line (a string)\\n\",\n \" Output: a list of strings \\n\",\n \" (each string is a sequence of alphanumeric characters)\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" word_list = [] # accumulates words in line\\n\",\n \" character_list = [] # accumulates characters in word\\n\",\n \" for c in line:\\n\",\n \" if c.isalnum():\\n\",\n \" character_list.append(c)\\n\",\n \" elif len(character_list)>0:\\n\",\n \" word = \\\"\\\".join(character_list)\\n\",\n \" word = word.lower()\\n\",\n \" word_list.append(word)\\n\",\n \" character_list = []\\n\",\n \" if len(character_list)>0:\\n\",\n \" word = \\\"\\\".join(character_list)\\n\",\n \" word = word.lower()\\n\",\n \" word_list.append(word)\\n\",\n \" return word_list\\n\",\n \"\\n\",\n \"def count_frequency(word_list):\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" Return a list giving pairs of form: (word,frequency)\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" L = []\\n\",\n \" for new_word in word_list:\\n\",\n \" for entry in L:\\n\",\n \" if new_word == entry[0]:\\n\",\n \" entry[1] = entry[1] + 1\\n\",\n \" break\\n\",\n \" else:\\n\",\n \" L.append([new_word,1])\\n\",\n \" return L\\n\",\n \" \\n\",\n \"def word_frequencies_for_file(filename):\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" Return alphabetically sorted list of (word,frequency) pairs \\n\",\n \" for the given file.\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" line_list = read_file(filename)\\n\",\n \" word_list = get_words_from_line_list(line_list)\\n\",\n \" freq_mapping = count_frequency(word_list)\\n\",\n \" return freq_mapping\\n\",\n \"\\n\",\n \"def inner_product(L1,L2):\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" Inner product between two vectors, where vectors\\n\",\n \" are represented as lists of (word,freq) pairs.\\n\",\n \"\\n\",\n \" Example: inner_product([[\\\"and\\\",3],[\\\"of\\\",2],[\\\"the\\\",5]],\\n\",\n \" [[\\\"and\\\",4],[\\\"in\\\",1],[\\\"of\\\",1],[\\\"this\\\",2]]) = 14.0 \\n\",\n \" \\\"\\\"\\\"\\n\",\n \" sum = 0.0\\n\",\n \" for word1, count1 in L1:\\n\",\n \" for word2, count2 in L2:\\n\",\n \" if word1 == word2:\\n\",\n \" sum += count1 * count2\\n\",\n \" return sum\\n\",\n \"\\n\",\n \"def vector_angle(L1,L2):\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" The input is a list of (word,freq) pairs, sorted alphabetically.\\n\",\n \"\\n\",\n \" Return the angle between these two vectors.\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" numerator = inner_product(L1,L2)\\n\",\n \" denominator = math.sqrt(inner_product(L1,L1)*inner_product(L2,L2))\\n\",\n \" return math.acos(numerator/denominator)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 4,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def document_distance(d1, d2):\\n\",\n \" sorted_word_list_1 = word_frequencies_for_file(d1)\\n\",\n \" \\n\",\n \" sorted_word_list_2 = word_frequencies_for_file(d2)\\n\",\n \" distance = vector_angle(sorted_word_list_1,sorted_word_list_2)\\n\",\n \" return distance\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 5,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"ename\": \"KeyboardInterrupt\",\n \"evalue\": \"\",\n \"output_type\": \"error\",\n \"traceback\": [\n \"\\u001b[1;31m---------------------------------------------------------------------------\\u001b[0m\",\n \"\\u001b[1;31mKeyboardInterrupt\\u001b[0m Traceback (most recent call last)\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36m\\u001b[1;34m()\\u001b[0m\\n\\u001b[1;32m----> 1\\u001b[1;33m \\u001b[0mdistance\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mdocument_distance\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[1;34m\\\"book1.txt\\\"\\u001b[0m\\u001b[1;33m,\\u001b[0m \\u001b[1;34m\\\"book2.txt\\\"\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 2\\u001b[0m \\u001b[1;32mprint\\u001b[0m \\u001b[1;34m\\\"The distance between the documents is: %0.6f (radians)\\\"\\u001b[0m\\u001b[1;33m%\\u001b[0m \\u001b[0mdistance\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mdocument_distance\\u001b[1;34m(d1, d2)\\u001b[0m\\n\\u001b[0;32m 1\\u001b[0m \\u001b[1;32mdef\\u001b[0m \\u001b[0mdocument_distance\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0md1\\u001b[0m\\u001b[1;33m,\\u001b[0m \\u001b[0md2\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m----> 2\\u001b[1;33m \\u001b[0msorted_word_list_1\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mword_frequencies_for_file\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0md1\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 3\\u001b[0m \\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 4\\u001b[0m \\u001b[0msorted_word_list_2\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mword_frequencies_for_file\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0md2\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 5\\u001b[0m \\u001b[0mdistance\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mvector_angle\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0msorted_word_list_1\\u001b[0m\\u001b[1;33m,\\u001b[0m\\u001b[0msorted_word_list_2\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mword_frequencies_for_file\\u001b[1;34m(filename)\\u001b[0m\\n\\u001b[0;32m 68\\u001b[0m \\\"\\\"\\\"\\n\\u001b[0;32m 69\\u001b[0m \\u001b[0mline_list\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mread_file\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mfilename\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 70\\u001b[1;33m \\u001b[0mword_list\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mget_words_from_line_list\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mline_list\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 71\\u001b[0m \\u001b[0mfreq_mapping\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mcount_frequency\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mword_list\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 72\\u001b[0m \\u001b[1;32mreturn\\u001b[0m \\u001b[0mfreq_mapping\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mget_words_from_line_list\\u001b[1;34m(L)\\u001b[0m\\n\\u001b[0;32m 20\\u001b[0m \\u001b[1;32mfor\\u001b[0m \\u001b[0mline\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mL\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 21\\u001b[0m \\u001b[0mwords_in_line\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mget_words_from_string\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mline\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 22\\u001b[1;33m \\u001b[0mword_list\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mword_list\\u001b[0m \\u001b[1;33m+\\u001b[0m \\u001b[0mwords_in_line\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 23\\u001b[0m \\u001b[1;32mreturn\\u001b[0m \\u001b[0mword_list\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 24\\u001b[0m \\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;31mKeyboardInterrupt\\u001b[0m: \"\n ]\n }\n ],\n \"source\": [\n \"distance = document_distance(\\\"book1.txt\\\", \\\"book2.txt\\\")\\n\",\n \"print \\\"The distance between the documents is: %0.6f (radians)\\\"% distance\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 10,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"ename\": \"KeyboardInterrupt\",\n \"evalue\": \"\",\n \"output_type\": \"error\",\n \"traceback\": [\n \"\\u001b[1;31m---------------------------------------------------------------------------\\u001b[0m\",\n \"\\u001b[1;31mKeyboardInterrupt\\u001b[0m Traceback (most recent call last)\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36m\\u001b[1;34m()\\u001b[0m\\n\\u001b[1;32m----> 1\\u001b[1;33m \\u001b[0mdistance\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mdocument_distance\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[1;34m\\\"book1_small.txt\\\"\\u001b[0m\\u001b[1;33m,\\u001b[0m \\u001b[1;34m\\\"book2_small.txt\\\"\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 2\\u001b[0m \\u001b[1;32mprint\\u001b[0m \\u001b[1;34m\\\"The distance between the documents is: %0.6f (radians)\\\"\\u001b[0m\\u001b[1;33m%\\u001b[0m \\u001b[0mdistance\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mdocument_distance\\u001b[1;34m(d1, d2)\\u001b[0m\\n\\u001b[0;32m 1\\u001b[0m \\u001b[1;32mdef\\u001b[0m \\u001b[0mdocument_distance\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0md1\\u001b[0m\\u001b[1;33m,\\u001b[0m \\u001b[0md2\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m----> 2\\u001b[1;33m \\u001b[0msorted_word_list_1\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mword_frequencies_for_file\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0md1\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 3\\u001b[0m \\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 4\\u001b[0m \\u001b[0msorted_word_list_2\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mword_frequencies_for_file\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0md2\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 5\\u001b[0m \\u001b[0mdistance\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mvector_angle\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0msorted_word_list_1\\u001b[0m\\u001b[1;33m,\\u001b[0m\\u001b[0msorted_word_list_2\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mword_frequencies_for_file\\u001b[1;34m(filename)\\u001b[0m\\n\\u001b[0;32m 68\\u001b[0m \\\"\\\"\\\"\\n\\u001b[0;32m 69\\u001b[0m \\u001b[0mline_list\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mread_file\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mfilename\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 70\\u001b[1;33m \\u001b[0mword_list\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mget_words_from_line_list\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mline_list\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 71\\u001b[0m \\u001b[0mfreq_mapping\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mcount_frequency\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mword_list\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 72\\u001b[0m \\u001b[1;32mreturn\\u001b[0m \\u001b[0mfreq_mapping\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;32m\\u001b[0m in \\u001b[0;36mget_words_from_line_list\\u001b[1;34m(L)\\u001b[0m\\n\\u001b[0;32m 20\\u001b[0m \\u001b[1;32mfor\\u001b[0m \\u001b[0mline\\u001b[0m \\u001b[1;32min\\u001b[0m \\u001b[0mL\\u001b[0m\\u001b[1;33m:\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 21\\u001b[0m \\u001b[0mwords_in_line\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mget_words_from_string\\u001b[0m\\u001b[1;33m(\\u001b[0m\\u001b[0mline\\u001b[0m\\u001b[1;33m)\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[1;32m---> 22\\u001b[1;33m \\u001b[0mword_list\\u001b[0m \\u001b[1;33m=\\u001b[0m \\u001b[0mword_list\\u001b[0m \\u001b[1;33m+\\u001b[0m \\u001b[0mwords_in_line\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0m\\u001b[0;32m 23\\u001b[0m \\u001b[1;32mreturn\\u001b[0m \\u001b[0mword_list\\u001b[0m\\u001b[1;33m\\u001b[0m\\u001b[0m\\n\\u001b[0;32m 24\\u001b[0m \\u001b[1;33m\\u001b[0m\\u001b[0m\\n\",\n \"\\u001b[1;31mKeyboardInterrupt\\u001b[0m: \"\n ]\n }\n ],\n \"source\": [\n \"distance = document_distance(\\\"book1_small.txt\\\", \\\"book2_small.txt\\\")\\n\",\n \"print \\\"The distance between the documents is: %0.6f (radians)\\\"% distance\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 8,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"The distance between the documents is: 0.667024 (radians)\\n\"\n ]\n }\n ],\n \"source\": [\n \"distance = document_distance(\\\"book1_tiny.txt\\\", \\\"book2_tiny.txt\\\")\\n\",\n \"print \\\"The distance between the documents is: %0.6f (radians)\\\"% distance\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 9,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 42760 function calls in 0.193 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 2 0.000 0.000 0.000 0.000 :1(read_file)\\n\",\n \" 2 0.001 0.001 0.024 0.012 :13(get_words_from_line_list)\\n\",\n \" 200 0.016 0.000 0.023 0.000 :25(get_words_from_string)\\n\",\n \" 2 0.066 0.033 0.066 0.033 :50(count_frequency)\\n\",\n \" 2 0.000 0.000 0.090 0.045 :64(word_frequencies_for_file)\\n\",\n \" 3 0.102 0.034 0.102 0.034 :74(inner_product)\\n\",\n \" 1 0.000 0.000 0.102 0.102 :89(vector_angle)\\n\",\n \" 1 0.000 0.000 0.193 0.193 :1(document_distance)\\n\",\n \" 1 0.000 0.000 0.193 0.193 :1()\\n\",\n \" 3965 0.001 0.000 0.001 0.000 {len}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {math.acos}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {math.sqrt}\\n\",\n \" 16663 0.002 0.000 0.002 0.000 {method 'append' of 'list' objects}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 16471 0.002 0.000 0.002 0.000 {method 'isalnum' of 'str' objects}\\n\",\n \" 2720 0.001 0.000 0.001 0.000 {method 'join' of 'str' objects}\\n\",\n \" 2720 0.001 0.000 0.001 0.000 {method 'lower' of 'str' objects}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {method 'readlines' of 'file' objects}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {open}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run('document_distance(\\\"book1_tiny.txt\\\", \\\"book2_tiny.txt\\\")')\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Optimization 1: remove list concatenation\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 11,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def get_words_from_line_list(L):\\n\",\n \" word_list = []\\n\",\n \" for line in L:\\n\",\n \" words_in_line = get_words_from_string(line)\\n\",\n \" word_list.extend(words_in_line)\\n\",\n \" return word_list\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 12,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"The distance between the documents is: 1.201624 (radians)\\n\"\n ]\n }\n ],\n \"source\": [\n \"distance = document_distance(\\\"book1_tiny.txt\\\", \\\"book2_tiny.txt\\\")\\n\",\n \"print \\\"The distance between the documents is: %0.6f (radians)\\\"% distance\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 19,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"The distance between the documents is: 1.201624 (radians)\\n\"\n ]\n }\n ],\n \"source\": [\n \"distance = document_distance(\\\"book1_small.txt\\\", \\\"book2_small.txt\\\")\\n\",\n \"print \\\"The distance between the documents is: %0.6f (radians)\\\"% distance\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 21,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 442 function calls in 0.021 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 2 0.000 0.000 0.000 0.000 :1(get_words_from_line_list)\\n\",\n \" 2 0.000 0.000 0.016 0.008 :1(read_file)\\n\",\n \" 2 0.000 0.000 0.000 0.000 :25(get_words_from_string)\\n\",\n \" 2 0.000 0.000 0.000 0.000 :50(count_frequency)\\n\",\n \" 2 0.000 0.000 0.017 0.008 :64(word_frequencies_for_file)\\n\",\n \" 3 0.000 0.000 0.000 0.000 :74(inner_product)\\n\",\n \" 1 0.000 0.000 0.000 0.000 :89(vector_angle)\\n\",\n \" 1 0.004 0.004 0.020 0.020 :1(document_distance)\\n\",\n \" 1 0.000 0.000 0.021 0.021 :1()\\n\",\n \" 37 0.000 0.000 0.000 0.000 {len}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {math.acos}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {math.sqrt}\\n\",\n \" 171 0.000 0.000 0.000 0.000 {method 'append' of 'list' objects}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {method 'extend' of 'list' objects}\\n\",\n \" 157 0.000 0.000 0.000 0.000 {method 'isalnum' of 'str' objects}\\n\",\n \" 26 0.000 0.000 0.000 0.000 {method 'join' of 'str' objects}\\n\",\n \" 26 0.000 0.000 0.000 0.000 {method 'lower' of 'str' objects}\\n\",\n \" 2 0.016 0.008 0.016 0.008 {method 'readlines' of 'file' objects}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {open}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run('document_distance(\\\"book1.txt\\\", \\\"book2.txt\\\")')\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Various smaller optimizations\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 19,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"# global variables needed for fast parsing\\n\",\n \"# translation table maps upper case to lower case and punctuation to spaces\\n\",\n \"translation_table = string.maketrans(string.punctuation+string.uppercase,\\n\",\n \" \\\" \\\"*len(string.punctuation)+string.lowercase)\\n\",\n \"\\n\",\n \"def get_words_from_text(text):\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" Parse the given text into words.\\n\",\n \" Return list of all words found.\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" text = text.translate(translation_table)\\n\",\n \" word_list = text.split()\\n\",\n \" return word_list\\n\",\n \"\\n\",\n \"def count_frequency(word_list):\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" Return a dictionary mapping words to frequency.\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" D = {}\\n\",\n \" for new_word in word_list:\\n\",\n \" if new_word in D:\\n\",\n \" D[new_word] = D[new_word]+1\\n\",\n \" else:\\n\",\n \" D[new_word] = 1\\n\",\n \" return D\\n\",\n \"\\n\",\n \"def word_frequencies_for_text(text):\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" Return dictionary of (word,frequency) pairs for the given file.\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" word_list = get_words_from_text(text)\\n\",\n \" freq_mapping = count_frequency(word_list)\\n\",\n \" return freq_mapping\\n\",\n \"\\n\",\n \"def inner_product(D1,D2):\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" Inner product between two vectors, where vectors\\n\",\n \" are represented as dictionaries of (word,freq) pairs.\\n\",\n \"\\n\",\n \" Example: inner_product({\\\"and\\\":3,\\\"of\\\":2,\\\"the\\\":5},\\n\",\n \" {\\\"and\\\":4,\\\"in\\\":1,\\\"of\\\":1,\\\"this\\\":2}) = 14.0 \\n\",\n \" \\\"\\\"\\\"\\n\",\n \" sum = 0.0\\n\",\n \" for key in D1:\\n\",\n \" if key in D2:\\n\",\n \" sum += D1[key] * D2[key]\\n\",\n \" return sum\\n\",\n \"\\n\",\n \"def vector_angle(D1,D2):\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" The input is a list of (word,freq) pairs, sorted alphabetically.\\n\",\n \"\\n\",\n \" Return the angle between these two vectors.\\n\",\n \" \\\"\\\"\\\"\\n\",\n \" numerator = inner_product(D1,D2)\\n\",\n \" denominator = math.sqrt(inner_product(D1,D1)*inner_product(D2,D2))\\n\",\n \" return math.acos(numerator/denominator)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 20,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 19 function calls in 0.336 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 2 0.226 0.113 0.226 0.113 :15(count_frequency)\\n\",\n \" 2 0.000 0.000 0.313 0.156 :27(word_frequencies_for_text)\\n\",\n \" 3 0.011 0.004 0.011 0.004 :35(inner_product)\\n\",\n \" 1 0.000 0.000 0.011 0.011 :49(vector_angle)\\n\",\n \" 2 0.000 0.000 0.087 0.043 :6(get_words_from_text)\\n\",\n \" 1 0.009 0.009 0.333 0.333 :87(document_distance)\\n\",\n \" 1 0.003 0.003 0.336 0.336 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 {math.acos}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {math.sqrt}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 2 0.076 0.038 0.076 0.038 {method 'split' of 'str' objects}\\n\",\n \" 2 0.011 0.005 0.011 0.005 {method 'translate' of 'str' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"document_distance(document1, document2)\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 16,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 509 function calls in 0.026 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 2 0.000 0.000 0.021 0.011 :1(word_frequencies_for_text)\\n\",\n \" 3 0.000 0.000 0.000 0.000 :18(inner_product)\\n\",\n \" 2 0.000 0.000 0.000 0.000 :8(insertion_sort)\\n\",\n \" 2 0.000 0.000 0.000 0.000 :1(count_frequency)\\n\",\n \" 2 0.000 0.000 0.000 0.000 :13(get_words_from_string)\\n\",\n \" 1 0.000 0.000 0.000 0.000 :77(vector_angle)\\n\",\n \" 1 0.004 0.004 0.026 0.026 :87(document_distance)\\n\",\n \" 2 0.000 0.000 0.000 0.000 :1(get_words_from_line_list)\\n\",\n \" 1 0.000 0.000 0.026 0.026 :1()\\n\",\n \" 127 0.000 0.000 0.000 0.000 {len}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {math.acos}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {math.sqrt}\\n\",\n \" 148 0.000 0.000 0.000 0.000 {method 'append' of 'list' objects}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {method 'extend' of 'list' objects}\\n\",\n \" 155 0.000 0.000 0.000 0.000 {method 'isalnum' of 'str' objects}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {method 'items' of 'dict' objects}\\n\",\n \" 26 0.000 0.000 0.000 0.000 {method 'join' of 'str' objects}\\n\",\n \" 26 0.000 0.000 0.000 0.000 {method 'lower' of 'str' objects}\\n\",\n \" 2 0.021 0.010 0.021 0.010 {method 'split' of 'str' objects}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {range}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"document_distance(document1, document2)\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture2/binary_search.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 75,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"import numpy as np\\n\",\n \"from scipy.spatial.distance import cosine as cosine_similarity\\n\",\n \"import random\\n\",\n \"import cProfile\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Binary search simulation\\n\",\n \"\\n\",\n \"To understand the kind of problems that computers need to deal with on daily bases I prepared a little simulation.\\n\",\n \"\\n\",\n \"You are given a hand of 20 sorted cards and your task is to find a particular card. Have fun!\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 76,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def generated_ordered_cards():\\n\",\n \" cards = []\\n\",\n \" cards.extend(unichr(x) for x in range(127185, 127185 + 14))\\n\",\n \" cards.extend(unichr(x) for x in range(127169, 127169 + 14))\\n\",\n \" cards.extend(unichr(x) for x in range(127153, 127153 + 14)) \\n\",\n \" cards.extend(unichr(x) for x in range(127137, 127137 + 14))\\n\",\n \" return cards\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 77,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"DOMAIN = generated_ordered_cards()\\n\",\n \"NUM_SAMPLES = 20\\n\",\n \"\\n\",\n \"class Problem(object):\\n\",\n \" def __init__(self):\\n\",\n \" self.elements = set()\\n\",\n \" while len(self.elements) < NUM_SAMPLES:\\n\",\n \" self.elements.add(random.choice(DOMAIN))\\n\",\n \" self.elements = sorted(list(self.elements), key=lambda x: DOMAIN.index(x))\\n\",\n \" self.hide_all()\\n\",\n \" self.query = random.choice(self.elements)\\n\",\n \" \\n\",\n \" def ask(self, position):\\n\",\n \" assert 0 <= position < NUM_SAMPLES\\n\",\n \" self.visible[position] = True\\n\",\n \" return self\\n\",\n \" \\n\",\n \" def hide_all(self):\\n\",\n \" self.visible = [False for _ in range(NUM_SAMPLES)]\\n\",\n \" \\n\",\n \" def _repr_html_(self):\\n\",\n \" els_html = []\\n\",\n \" for el_idx in range(len(self.elements)):\\n\",\n \" if self.visible[el_idx]:\\n\",\n \" els_html.append(\\\"%s\\\" % (self.elements[el_idx]))\\n\",\n \" else:\\n\",\n \" els_html.append(\\\"%d\\\" % (el_idx,))\\n\",\n \" header_html = u\\\"

Find %s! (♣ < ♦ < ♥ < ♠)


\\\" % (self.query,)\\n\",\n \" table_html = \\\"%s
\\\" % (\\\"\\\".join(els_html))\\n\",\n \" return header_html + table_html\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 81,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/html\": [\n \"

Find 🂷! (♣ < ♦ < ♥ < ♠)


012345678910111213141516171819
\"\n ],\n \"text/plain\": [\n \"<__main__.Problem at 0x7ff69c3e8090>\"\n ]\n },\n \"execution_count\": 81,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"p = Problem()\\n\",\n \"p\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 86,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/html\": [\n \"

Find 🂷! (♣ < ♦ < ♥ < ♠)


0123456789🃈11🂱🂶🂷🂦16171819
\"\n ],\n \"text/plain\": [\n \"<__main__.Problem at 0x7ff69c3e8090>\"\n ]\n },\n \"execution_count\": 86,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"p.ask(14)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": []\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Binary search implemetation - GLOVE vectors\\n\",\n \"\\n\",\n \"Glove vectors try to assign vectors of numbers to words in surprising ways. One interesting property they have is\\n\",\n \"ability to solve many analogies. For example:\\n\",\n \"\\n\",\n \"$$\\n\",\n \"V_{\\\\text{berlin}} - V_{\\\\text{germany}} \\\\approx V_{\\\\text{paris}} - V_{\\\\text{france}}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Today we will try to write an algorithm that can quickly find vectors from Glove database\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 92,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"class Glove(object):\\n\",\n \" def __init__(self, path):\\n\",\n \" self.word_vector = []\\n\",\n \" \\n\",\n \" with open(path) as f:\\n\",\n \" for line in f:\\n\",\n \" if len(line) < 1:\\n\",\n \" break\\n\",\n \" line = line.split(' ')\\n\",\n \" word, vector = line[0], np.array([float(x) for x in line[1:]])\\n\",\n \" self.word_vector.append((word, vector))\\n\",\n \" self.word_vector.sort()\\n\",\n \" \\n\",\n \" def __call__(self, key):\\n\",\n \" return self.find_vector(key)\\n\",\n \" \\n\",\n \" def find_vector(self, key):\\n\",\n \" for word,vector in self.word_vector:\\n\",\n \" if word == key:\\n\",\n \" return vector\\n\",\n \" raise KeyError(key)\\n\",\n \" \\n\",\n \" def find_closest_word(self, key_vector,blacklist=[]):\\n\",\n \" best_similarity = float('inf')\\n\",\n \" best_word = None\\n\",\n \" for word, vector in self.word_vector:\\n\",\n \" if word in blacklist:\\n\",\n \" continue\\n\",\n \" similarity = cosine_similarity(vector, key_vector)\\n\",\n \" if best_similarity > similarity:\\n\",\n \" best_similarity = similarity\\n\",\n \" best_word = word\\n\",\n \" return best_word\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 93,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"# Download from http://nlp.stanford.edu/projects/glove/\\n\",\n \"glove = Glove(\\\"/home/sidor/projects/Dali/data/glove/glove.6B.300d.txt\\\")\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 52,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"# this call is quite slow...\\n\",\n \"def analogy(thisword, tothis, islikethis):\\n\",\n \" key_vector = glove(tothis) - glove(thisword) + glove(islikethis)\\n\",\n \" best_word = glove.find_closest_word(key_vector, blacklist=[thisword, tothis,islikethis])\\n\",\n \" print(\\\"%s is to %s like %s to <%s>\\\" % (thisword, tothis, islikethis, best_word,))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 88,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"germany is to berlin like france to \\n\"\n ]\n }\n ],\n \"source\": [\n \"analogy(\\\"germany\\\", \\\"berlin\\\", \\\"france\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 46,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"movie is to movies like school to \\n\"\n ]\n }\n ],\n \"source\": [\n \"analogy(\\\"movie\\\", \\\"movies\\\", \\\"school\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 57,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"movie is to actor like school to \\n\"\n ]\n }\n ],\n \"source\": [\n \"analogy(\\\"movie\\\", \\\"actor\\\", \\\"school\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 55,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"smaller is to small like bigger to \\n\"\n ]\n }\n ],\n \"source\": [\n \"analogy(\\\"smaller\\\", \\\"small\\\", \\\"bigger\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 56,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"smaller is to small like big to \\n\"\n ]\n }\n ],\n \"source\": [\n \"analogy(\\\"smaller\\\", \\\"small\\\", \\\"big\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 60,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"one is to two like two to \\n\"\n ]\n }\n ],\n \"source\": [\n \"analogy(\\\"one\\\", \\\"two\\\", \\\"two\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 59,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"king is to queen like man to \\n\"\n ]\n }\n ],\n \"source\": [\n \"analogy(\\\"king\\\", \\\"queen\\\", \\\"man\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Finding a vector given a word\\n\",\n \"Notice that implementation of find_word above is quite naive:\\n\",\n \" \\n\",\n \"```python\\n\",\n \" def find_vector(self, key):\\n\",\n \" for word,vector in self.word_vector:\\n\",\n \" if word == key:\\n\",\n \" return vector\\n\",\n \" raise KeyError(key)\\n\",\n \"```\\n\",\n \"\\n\",\n \"Every query to look up a word takes about 100ms. This is not very good if we want to run millions of such queries per second!\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 94,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 4 function calls in 0.135 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.000 0.000 0.135 0.135 :14(__call__)\\n\",\n \" 1 0.135 0.135 0.135 0.135 :17(find_vector)\\n\",\n \" 1 0.000 0.000 0.135 0.135 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run('glove(\\\"zebra\\\")')\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Implementation using binary search\\n\",\n \"\\n\",\n \"Here we perform binary search algorithm. List of words `glove.word_vectors` is sorted alphabetically. Now we wish to find index `i`, such that `glove.word_vectors[i][0] == key` for some `key` word.\\n\",\n \"\\n\",\n \"Let $l$ be the lowest index in our array (zero) and $h$ be the highest index in our array (in this case 400000 - 1). To find and index on which a particular word is stored we find the middle $m = (h + l) / 2$. If key at index $m$ (denoted $k_m$) is less than desired key $k^*$ we know that $k^*$ is on some index in range $(m, h)$ otherwise it is in range $(l, m-1)$. This idea can be recursively applied to all the subranges. Notice that with each query range size is halved, so the complexity of the solution is $O(\\\\log{(h - l)})$\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 95,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def find_vector(self, key, lo=None, hi=None, debug=False):\\n\",\n \" # Make sure by default we search over entire table\\n\",\n \" if lo is None:\\n\",\n \" lo = 0\\n\",\n \" if hi is None:\\n\",\n \" hi = len(self.word_vector) - 1\\n\",\n \" \\n\",\n \" if lo > hi:\\n\",\n \" raise KeyError(key)\\n\",\n \" \\n\",\n \" mid = (hi + lo) / 2\\n\",\n \" word, vector = self.word_vector[mid]\\n\",\n \" if debug:\\n\",\n \" print(\\\"Looking for %s in range(%d, %d). Middle is %s\\\" % (key, lo, hi, word))\\n\",\n \"\\n\",\n \" if word == key:\\n\",\n \" return vector\\n\",\n \" elif key < word:\\n\",\n \" return self.find_vector(key, lo, mid - 1, debug=debug)\\n\",\n \" else: # key > word\\n\",\n \" return self.find_vector(key, mid + 1, hi, debug=debug)\\n\",\n \" \\n\",\n \"Glove.find_vector = find_vector\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 96,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"Looking for zebra in range(0, 399999). Middle is jurnal\\n\",\n \"Looking for zebra in range(200000, 399999). Middle is ramdass\\n\",\n \"Looking for zebra in range(300000, 399999). Middle is syme\\n\",\n \"Looking for zebra in range(350000, 399999). Middle is vadims\\n\",\n \"Looking for zebra in range(375000, 399999). Middle is wilbarger\\n\",\n \"Looking for zebra in range(387500, 399999). Middle is yevtushenko\\n\",\n \"Looking for zebra in range(393750, 399999). Middle is zeughaus\\n\",\n \"Looking for zebra in range(393750, 396873). Middle is zaa\\n\",\n \"Looking for zebra in range(395312, 396873). Middle is zaremba\\n\",\n \"Looking for zebra in range(396093, 396873). Middle is zehava\\n\",\n \"Looking for zebra in range(396093, 396482). Middle is zayu\\n\",\n \"Looking for zebra in range(396288, 396482). Middle is zeami\\n\",\n \"Looking for zebra in range(396386, 396482). Middle is zeder\\n\",\n \"Looking for zebra in range(396386, 396433). Middle is zebu\\n\",\n \"Looking for zebra in range(396386, 396408). Middle is zebic\\n\",\n \"Looking for zebra in range(396398, 396408). Middle is zebras\\n\",\n \"Looking for zebra in range(396398, 396402). Middle is zebra\\n\"\n ]\n }\n ],\n \"source\": [\n \"vec = glove.find_vector(\\\"zebra\\\", debug=True)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 72,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"Looking for ronrivest in range(0, 399999). Middle is jurnal\\n\",\n \"Looking for ronrivest in range(200000, 399999). Middle is ramdass\\n\",\n \"Looking for ronrivest in range(300000, 399999). Middle is syme\\n\",\n \"Looking for ronrivest in range(300000, 349998). Middle is sensationalism\\n\",\n \"Looking for ronrivest in range(300000, 324998). Middle is rs6\\n\",\n \"Looking for ronrivest in range(300000, 312498). Middle is rescue\\n\",\n \"Looking for ronrivest in range(306250, 312498). Middle is riveras\\n\",\n \"Looking for ronrivest in range(309375, 312498). Middle is romero\\n\",\n \"Looking for ronrivest in range(310937, 312498). Middle is rostekhnadzor\\n\",\n \"Looking for ronrivest in range(310937, 311716). Middle is rosaleda\\n\",\n \"Looking for ronrivest in range(310937, 311325). Middle is ronstadt\\n\",\n \"Looking for ronrivest in range(310937, 311130). Middle is ronayne\\n\",\n \"Looking for ronrivest in range(311034, 311130). Middle is rongbuk\\n\",\n \"Looking for ronrivest in range(311083, 311130). Middle is ronin\\n\",\n \"Looking for ronrivest in range(311107, 311130). Middle is ronni\\n\",\n \"Looking for ronrivest in range(311119, 311130). Middle is rono\\n\",\n \"Looking for ronrivest in range(311125, 311130). Middle is ronsard\\n\",\n \"Looking for ronrivest in range(311125, 311126). Middle is ronon\\n\",\n \"Looking for ronrivest in range(311126, 311126). Middle is ronquillo\\n\",\n \"Not found!\\n\"\n ]\n }\n ],\n \"source\": [\n \"try:\\n\",\n \" vec = glove.find_vector(\\\"ronrivest\\\", debug=True)\\n\",\n \"except KeyError:\\n\",\n \" print(\\\"Not found!\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 97,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 21 function calls (5 primitive calls) in 0.000 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.000 0.000 0.000 0.000 :14(__call__)\\n\",\n \" 17/1 0.000 0.000 0.000 0.000 :1(find_vector)\\n\",\n \" 1 0.000 0.000 0.000 0.000 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 {len}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run('glove(\\\"zebra\\\")')\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {\n \"collapsed\": true\n },\n \"source\": [\n \"### Extra problems (not graded)\\n\",\n \"1. Here's another way of thinking about binary search - we are looking for higest $x$ such that property $p(x)$ is true. For example if we are looking for index of a key in array $p(x) = k_x \\\\leq k^*$. Can you give examples of other properties that we can binary search over? What makes a property suitable for use in binary search?\\n\",\n \"\\n\",\n \"2. Binary search can **not** be used to minimize a quadratic function. Can you find a similar algorithm that can?\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"# Hint to problem 2\\n\",\n \"cyph = lambda x: chr((ord(x) + 64) % 128)\\n\",\n \"''.join(map(cyph, \\\"\\\\t.34%!$`/&`30,)44).'`2!.'%`).`47/`0)%#%3l`#/.3)$%2`30,)44).'`)4`).`4(2%%`0)%#%3n\\\"))\"\n ]\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture4/Heaps.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 10,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"class MaxHeap(object):\\n\",\n \" def __init__(self, max_size=128):\\n\",\n \" self.storage = [None for _ in range(max_size)]\\n\",\n \" self.heap_size = 0\\n\",\n \" \\n\",\n \" def display(self, width=80):\\n\",\n \" next_element = 0\\n\",\n \" for i in range(0, 1000):\\n\",\n \" level_elts = 2**i\\n\",\n \" elts = self.storage[next_element:min(next_element + level_elts, self.heap_size)]\\n\",\n \" next_element = min(next_element + level_elts, self.heap_size)\\n\",\n \" if len(elts) == 0:\\n\",\n \" break\\n\",\n \" positions = [ (i+1) * width / (level_elts + 1) for i in range(level_elts)]\\n\",\n \" output = \\\"\\\"\\n\",\n \" for j, (elt, pos) in enumerate(zip(elts, positions)):\\n\",\n \" idx = 2**i + j - 1\\n\",\n \" while len(output) <= pos:\\n\",\n \" output += \\\" \\\"\\n\",\n \" output += \\\"%d[@%d]\\\" % (elt,idx)\\n\",\n \" print(output)\\n\",\n \" print()\\n\",\n \" \\n\",\n \" def fix_down(self, index):\\n\",\n \" while index < self.heap_size:\\n\",\n \" # pick maximum child\\n\",\n \" max_child_idx = None\\n\",\n \" if 2 * index + 1 < self.heap_size:\\n\",\n \" max_child_idx = 2 * index + 1\\n\",\n \"\\n\",\n \" if 2 * index + 2 < self.heap_size and \\\\\\n\",\n \" self.storage[2 * index + 1] < self.storage[2 * index + 2]:\\n\",\n \" max_child_idx = 2 * index + 2\\n\",\n \"\\n\",\n \" if max_child_idx is None or \\\\\\n\",\n \" self.storage[index] > self.storage[max_child_idx]:\\n\",\n \" # heap property satisfied\\n\",\n \" break\\n\",\n \" \\n\",\n \" self.storage[index], self.storage[max_child_idx] = self.storage[max_child_idx], self.storage[index]\\n\",\n \" index = max_child_idx\\n\",\n \" \\n\",\n \" def fix_up(self, index):\\n\",\n \" assert index < self.heap_size\\n\",\n \" while index > 0:\\n\",\n \" parent_idx = (index - 1) // 2\\n\",\n \" if self.storage[index] >= self.storage[parent_idx]:\\n\",\n \" self.storage[index], self.storage[parent_idx] = self.storage[parent_idx], self.storage[index]\\n\",\n \" index = parent_idx\\n\",\n \" else:\\n\",\n \" break\\n\",\n \" \\n\",\n \" def insert(self, element):\\n\",\n \" self.heap_size += 1\\n\",\n \" new_index = self.heap_size - 1\\n\",\n \" self.storage[new_index] = element\\n\",\n \" self.fix_up(new_index)\\n\",\n \"\\n\",\n \" def extract_max(self):\\n\",\n \" self.storage[0], self.storage[self.heap_size - 1] = self.storage[self.heap_size -1], self.storage[0]\\n\",\n \" self.heap_size -= 1\\n\",\n \" self.fix_down(0)\\n\",\n \" return self.storage[self.heap_size]\\n\",\n \" \\n\",\n \" def heapify(self):\\n\",\n \" for i in range(self.heap_size - 1, -1, -1):\\n\",\n \" self.fix_down(i)\\n\",\n \" \\n\",\n \" @staticmethod\\n\",\n \" def wrap_list(lst):\\n\",\n \" h = MaxHeap(0)\\n\",\n \" h.storage = lst\\n\",\n \" h.heap_size = len(lst)\\n\",\n \" return h\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 13,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 3[@0]\\n\",\n \"\\n\",\n \" 4[@1] 5[@2]\\n\",\n \"\\n\",\n \" 8[@3] 6[@4] 1[@5] 10[@6]\\n\",\n \"\\n\",\n \" 9[@7] 5[@8]\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"example = [3,4,5,8,6,1,10,9,5]\\n\",\n \"h = MaxHeap.wrap_list(example)\\n\",\n \"h.display()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 14,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 3[@0]\\n\",\n \"\\n\",\n \" 8[@1] 5[@2]\\n\",\n \"\\n\",\n \" 9[@3] 6[@4] 1[@5] 10[@6]\\n\",\n \"\\n\",\n \" 4[@7] 5[@8]\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"h.fix_down(1)\\n\",\n \"h.display()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 15,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 8[@0]\\n\",\n \"\\n\",\n \" 3[@1] 5[@2]\\n\",\n \"\\n\",\n \" 9[@3] 6[@4] 1[@5] 10[@6]\\n\",\n \"\\n\",\n \" 4[@7] 5[@8]\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"h.fix_up(1)\\n\",\n \"h.display()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 16,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 10[@0]\\n\",\n \"\\n\",\n \" 8[@1] 2[@2]\\n\",\n \"\\n\",\n \" 6[@3] 3[@4] 1[@5] 2[@6]\\n\",\n \"\\n\",\n \" 4[@7] 5[@8]\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"example = [3,10,2,6,8,1,2,4,5]\\n\",\n \"h = MaxHeap.wrap_list(example)\\n\",\n \"h.heapify()\\n\",\n \"h.display()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 17,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"h = MaxHeap(128)\\n\",\n \"h.display()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 18,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 10[@0]\\n\",\n \"\\n\",\n \" 5[@1] 4[@2]\\n\",\n \"\\n\",\n \" 1[@3]\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"h.insert(1)\\n\",\n \"h.insert(4)\\n\",\n \"h.insert(5)\\n\",\n \"h.insert(10)\\n\",\n \"h.display()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 19,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"10\\n\",\n \"5\\n\",\n \"4\\n\",\n \"1\\n\"\n ]\n }\n ],\n \"source\": [\n \"while h.heap_size > 0:\\n\",\n \" print(h.extract_max())\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 22,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def heap_sort(array):\\n\",\n \" as_heap = MaxHeap.wrap_list(array)\\n\",\n \" as_heap.heapify()\\n\",\n \" while as_heap.heap_size > 0:\\n\",\n \" as_heap.extract_max()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 23,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"[1, 2, 2, 3, 4, 5, 6, 8, 10]\\n\"\n ]\n }\n ],\n \"source\": [\n \"example = [3,10,2,6,8,1,2,4,5]\\n\",\n \"heap_sort(example)\\n\",\n \"print(example)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 3\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.4.1\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture5/BST.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 38,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"from collections import namedtuple\\n\",\n \"from igraph import Graph, plot, layout\\n\",\n \"%config InlineBackend.figure_format = 'png'\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 433,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"class Node(object):\\n\",\n \" def __init__(self, key, payload=None):\\n\",\n \" # key of the three\\n\",\n \" self.key = key\\n\",\n \" self.payload = payload\\n\",\n \" self.left = None\\n\",\n \" self.right = None\\n\",\n \" self.parent = None\\n\",\n \" \\n\",\n \" def is_left_child(self):\\n\",\n \" return self.parent is not None and self.parent.left is self\\n\",\n \" \\n\",\n \" def is_right_child(self):\\n\",\n \" return self.parent is not None and self.parent.right is self\\n\",\n \"\\n\",\n \" def __repr__(self):\\n\",\n \" return '' % (self.key, self.left, self.right)\\n\",\n \" \\n\",\n \"def BST_find(where, key):\\n\",\n \" if where is None:\\n\",\n \" return None\\n\",\n \" elif where.key == key:\\n\",\n \" return where\\n\",\n \" elif key < where.key:\\n\",\n \" return BST_find(where.left, key)\\n\",\n \" elif key > where.key:\\n\",\n \" return BST_find(where.right, key)\\n\",\n \" \\n\",\n \"def BST_insert(where, key, payload=None):\\n\",\n \" if where is None:\\n\",\n \" return Node(key, payload)\\n\",\n \" elif key <= where.key:\\n\",\n \" where.left = BST_insert(where.left, key, payload)\\n\",\n \" where.left.parent = where\\n\",\n \" elif key > where.key:\\n\",\n \" where.right = BST_insert(where.right, key, payload)\\n\",\n \" where.right.parent = where\\n\",\n \" return where\\n\",\n \"\\n\",\n \"def BST_minimum(root):\\n\",\n \" current_node = root\\n\",\n \" while current_node.left is not None:\\n\",\n \" current_node = current_node.left\\n\",\n \" return current_node\\n\",\n \"\\n\",\n \"def BST_draw(node, g):\\n\",\n \" if node is None:\\n\",\n \" return\\n\",\n \" \\n\",\n \" if node.left is not None:\\n\",\n \" BST_draw(node.left, g)\\n\",\n \" left_id = str(id(node.left))\\n\",\n \" left_color = \\\"black\\\"\\n\",\n \" else:\\n\",\n \" g.add_vertex(shape=0,)\\n\",\n \" left_id = len(g.vs) - 1\\n\",\n \" left_color = \\\"#00000000\\\"\\n\",\n \" \\n\",\n \" if node.right is not None:\\n\",\n \" BST_draw(node.right, g)\\n\",\n \" right_id = str(id(node.right))\\n\",\n \" right_color = \\\"black\\\"\\n\",\n \" else:\\n\",\n \" g.add_vertex(shape=0,)\\n\",\n \" right_id = len(g.vs) - 1\\n\",\n \" right_color = \\\"#00000000\\\"\\n\",\n \"\\n\",\n \" g.add_vertex(name=str(id(node)), label=str(node.key), shape=\\\"rectangle\\\", color=\\\"lightblue\\\")\\n\",\n \" if node.left is not None or node.right is not None:\\n\",\n \" g.add_edge(str(id(node)), left_id, color=left_color)\\n\",\n \" g.add_edge(str(id(node)), right_id, color=right_color)\\n\",\n \" \\n\",\n \"def BST_succesor(node):\\n\",\n \" if node is None:\\n\",\n \" return None\\n\",\n \" elif node.right is not None:\\n\",\n \" return BST_minimum(node.right)\\n\",\n \" else:\\n\",\n \" current_node = node\\n\",\n \" while True:\\n\",\n \" if current_node.parent is None:\\n\",\n \" return None\\n\",\n \" if current_node.is_right_child():\\n\",\n \" current_node = current_node.parent\\n\",\n \" elif current_node.is_left_child():\\n\",\n \" return current_node.parent\\n\",\n \" \\n\",\n \"class BST(object):\\n\",\n \" def __init__(self):\\n\",\n \" self.root = None\\n\",\n \" \\n\",\n \" def empty(self):\\n\",\n \" return self.root is None\\n\",\n \" \\n\",\n \" def insert(self, key, payload=None): \\n\",\n \" self.root = BST_insert(self.root, key, payload)\\n\",\n \" \\n\",\n \" def find(self, key):\\n\",\n \" return BST_find(self.root, key)\\n\",\n \" \\n\",\n \" def minimum(self):\\n\",\n \" return BST_minimum(self.root)\\n\",\n \" \\n\",\n \" def succesor(self, node):\\n\",\n \" return BST_succesor(node)\\n\",\n \" \\n\",\n \" def delete(self, delete_me):\\n\",\n \" replacement = None\\n\",\n \" if delete_me.right is not None:\\n\",\n \" replacement = delete_me.right\\n\",\n \" if delete_me.left is not None:\\n\",\n \" min_right = BST_minimum(replacement)\\n\",\n \" min_right.left = delete_me.left\\n\",\n \" delete_me.left.parent = min_right\\n\",\n \" elif delete_me.left is not None:\\n\",\n \" replacement = delete_me.left\\n\",\n \" \\n\",\n \" if delete_me.parent is None:\\n\",\n \" assert delete_me is self.root\\n\",\n \" self.root = replacement\\n\",\n \" if replacement is not None:\\n\",\n \" replacement.parent = None\\n\",\n \" else:\\n\",\n \" if replacement is not None:\\n\",\n \" replacement.parent = delete_me.parent\\n\",\n \" if delete_me.is_left_child():\\n\",\n \" delete_me.parent.left = replacement\\n\",\n \" elif delete_me.is_right_child():\\n\",\n \" delete_me.parent.right = replacement\\n\",\n \" \\n\",\n \" def draw(self):\\n\",\n \" g = Graph()\\n\",\n \" if self.root is not None:\\n\",\n \" BST_draw(self.root, g)\\n\",\n \" root_id = [i for i, node in enumerate(g.vs) if g.vs[i]['name'] == str(id(self.root))]\\n\",\n \" assert len(root_id) == 1\\n\",\n \" return plot(g, layout = Graph.layout_reingold_tilford(g, mode=\\\"in\\\", root=root_id), \\n\",\n \" bbox=(300,300), vertex_size=30)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 424,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"x = BST()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 425,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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\"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 425,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"x.insert(3)\\n\",\n \"x.insert(4)\\n\",\n \"x.insert(1)\\n\",\n \"x.draw()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 387,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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\"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 362,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"x.delete(x.find(4))\\n\",\n \"x.draw()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 382,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"# def BST_check_parents(node):\\n\",\n \"# if node.left is not None:\\n\",\n \"# assert node.left.parent is node\\n\",\n \"# BST_check_parents(node.left)\\n\",\n \"# if node.right is not None:\\n\",\n \"# assert node.right.parent is node\\n\",\n \"# BST_check_parents(node.right)\\n\",\n \"# BST_check_parents(x.root)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Facebook interview question\\n\",\n \"\\n\",\n \"Given a binary tree determine if it satisfies binary search invariant...\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Event Queue\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 438,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"from threading import Thread\\n\",\n \"import time\\n\",\n \"import sys\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 440,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"class EventQueue():\\n\",\n \" def __init__(self):\\n\",\n \" self.q = BST()\\n\",\n \" \\n\",\n \" def put(self, seconds_from_now, word):\\n\",\n \" self.q.insert(time.time() + seconds_from_now, word)\\n\",\n \" \\n\",\n \" def execute(self):\\n\",\n \" while not self.q.empty():\\n\",\n \" m = self.q.minimum()\\n\",\n \" scheduled_time = m.key\\n\",\n \" if scheduled_time >= time.time():\\n\",\n \" time.sleep(scheduled_time - time.time())\\n\",\n \" print(m.payload)\\n\",\n \" sys.stdout.flush()\\n\",\n \" self.q.delete(m)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 441,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"hi\\n\",\n \"hey\\n\",\n \"...\\n\",\n \"are you there?\\n\"\n ]\n }\n ],\n \"source\": [\n \"events = EventQueue()\\n\",\n \"events.put(1,\\\"hi\\\")\\n\",\n \"events.put(2,\\\"hey\\\")\\n\",\n \"events.put(3, \\\"...\\\")\\n\",\n \"events.put(5, \\\"are you there?\\\")\\n\",\n \"events.execute()\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture6/AVL_and_more.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 1,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"from collections import namedtuple\\n\",\n \"from igraph import Graph, plot, layout, Plot\\n\",\n \"%config InlineBackend.figure_format = 'png'\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# BST implementation\\n\",\n \"\\n\",\n \"Here's a simplified implementation of a Binary Search Tree. \\n\",\n \"We will use it as a starting point for our AVL implementation.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 27,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"class Node(object):\\n\",\n \" def __init__(self, key):\\n\",\n \" # node key - this is the value we compare to determine where \\n\",\n \" # the node should be inserted.\\n\",\n \" self.key = key\\n\",\n \" # left child (can be none if no child)\\n\",\n \" self.left = None\\n\",\n \" # right child (can be none if no child)\\n\",\n \" self.right = None\\n\",\n \" \\n\",\n \"def BST_insert(where, key):\\n\",\n \" \\\"\\\"\\\"Inserts a node in a binary tree\\\"\\\"\\\"\\n\",\n \" if where is None:\\n\",\n \" # if we found a branch pointing to none, it means we\\n\",\n \" # can insert the node here\\n\",\n \" return Node(key)\\n\",\n \" elif key <= where.key:\\n\",\n \" # try inserting the node somewhere in the left subtree\\n\",\n \" where.left = BST_insert(where.left, key)\\n\",\n \" elif key > where.key:\\n\",\n \" # try inserting the node in the right subtree\\n\",\n \" where.right = BST_insert(where.right, key)\\n\",\n \" return where\\n\",\n \"\\n\",\n \"def BST_draw(node, g, display_height=False):\\n\",\n \" \\\"\\\"\\\"This method allows us to draw a binary tree.\\n\",\n \" \\n\",\n \" Don't worry about the implementation.\\\"\\\"\\\"\\n\",\n \" if node is None:\\n\",\n \" return\\n\",\n \" \\n\",\n \" if node.left is not None:\\n\",\n \" BST_draw(node.left, g, display_height)\\n\",\n \" left_id = str(id(node.left))\\n\",\n \" left_color = \\\"black\\\"\\n\",\n \" else:\\n\",\n \" g.add_vertex(shape=0,)\\n\",\n \" left_id = len(g.vs) - 1\\n\",\n \" left_color = \\\"#00000000\\\"\\n\",\n \" \\n\",\n \" if node.right is not None:\\n\",\n \" BST_draw(node.right, g, display_height)\\n\",\n \" right_id = str(id(node.right))\\n\",\n \" right_color = \\\"black\\\"\\n\",\n \" else:\\n\",\n \" g.add_vertex(shape=0,)\\n\",\n \" right_id = len(g.vs) - 1\\n\",\n \" right_color = \\\"#00000000\\\"\\n\",\n \"\\n\",\n \" label = str(node.key)\\n\",\n \" if display_height:\\n\",\n \" label = label + ' (height: %d)' % (node.height,)\\n\",\n \" \\n\",\n \" g.add_vertex(name=str(id(node)), label=label, \\n\",\n \" shape=\\\"rectangle\\\", color=\\\"lightblue\\\")\\n\",\n \" if node.left is not None or node.right is not None:\\n\",\n \" g.add_edge(str(id(node)), left_id, color=left_color)\\n\",\n \" g.add_edge(str(id(node)), right_id, color=right_color)\\n\",\n \" \\n\",\n \" \\n\",\n \"def BST_root_draw(root, display_height=False, width=500, height=300, margin=50):\\n\",\n \" figure = Plot(bbox=(width, height))\\n\",\n \" \\n\",\n \" g = Graph()\\n\",\n \" if root is not None:\\n\",\n \" BST_draw(root, g, display_height)\\n\",\n \" root_id = [i for i, node in enumerate(g.vs) if g.vs[i]['name'] == str(id(root))]\\n\",\n \" assert len(root_id) == 1\\n\",\n \" figure.add(g, layout = Graph.layout_reingold_tilford(g, mode=\\\"in\\\", root=root_id), \\n\",\n \" bbox=(margin, margin, width-margin, height - margin), vertex_size=30)\\n\",\n \" return figure\\n\",\n \" \\n\",\n \"class BST(object):\\n\",\n \" def __init__(self):\\n\",\n \" \\\"\\\"\\\"This is a BST wrapper class.\\n\",\n \" \\n\",\n \" Its main function is keeping track of the root and \\n\",\n \" initiating recursive calls.\\\"\\\"\\\"\\n\",\n \" self.root = None\\n\",\n \" \\n\",\n \" def insert(self, key): \\n\",\n \" self.root = BST_insert(self.root, key)\\n\",\n \" \\n\",\n \" def draw(self, display_height=False, width=500, height=300, margin=50):\\n\",\n \" return BST_root_draw(self.root, display_height=display_height, width=width, height=height, margin=margin)\\n\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# What's wrong with Binary Search Trees?\\n\",\n \"\\n\",\n \"AVL trees extend the Binary Search Trees, to provide better complexity guarantes. Both datastructures can implement any of insertion, deletion, successor etc. in $O(h)$. The difference is the fact that in case of BST height can be $O(n)$, while AVL guarantees height $O(lg \\\\ n)$. Let's try to understand this with an example:\\n\",\n \"\\n\",\n \"Inserting sorted sequence into BST makes it unbalanced.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 29,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": \"iVBORw0KGgoAAAANSUhEUgAAAfQAAAEsCAYAAAA1u0HIAAAABmJLR0QA/wD/AP+gvaeTAAAPoUlE\\nQVR4nO3dfbDVdZ0H8PcpDEPBkFCx2LZkV1aGTXRdy8YuPmTtphVR2SoLjqWVwFBrbFsu5DatSU5r\\nuGjllgrlY5EFuUs76fpQumgPuypK5VYMDpoCo1zEUPLuH4drPnDhXuCc7zm/83rNnJl7zz0z3/cf\\nDO/5fc7v9/0mAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA\\nAAAAAAAAAAAAAAAAAAAAAABA+6mVDsA2nZVkv0JrX5zk0UJrA7CTFHoL2mvoPquPPmnSq4ftO6Kp\\n637viks3b9rYfXiSFU1dGIBdNqh0ALbtbadMy+gxBzd1zZsWX71508bupq4JwO7xktIBAIBdp9AB\\noAIUOgBUgEIHgApQ6ABQAQodACpAoQNABSh0AKgAhQ4AFaDQAaACFHpF3PPfP8wnTz4pk8ceWDoK\\nAAUo9Ir45sUXZsrZnywdA4BCHM5SEZ/5+uLSEQAoyBU6AFSAQgeAClDoAFABCh0AKkChA0AFKHQA\\nqACPrVXEczeU6f158co1peIA0GQKvSKUN0BnM3LnWT09PUOSHFQ6BwADp9B5Vq1W25JkYZL5SfYu\\nHAeAAVDoPNdTSSYlGZ7k/iRTy8YBoL8UOi/0aOpFPiXJ7CQ3JjmkaCIAdkih05dbkkxIsiTJrTGG\\nB2hpCp3t2ZJ6kY+PMTxAS/PYWotadtXCDNt3RFPX3NTdPbiPPz2UepF3JVmQZFqSmUnua1I0AHag\\nVjoA23RWkv0KrX1x6t+j92VQkulJ5iS5Msk5STY2IRcA0ACjkixKsjrG8ADQ9rqS3BN3wwNA2xuU\\nZFaStXE3PAC0PWN4AKgQY3gAqAhjeACoEGN4AKgQY3iABrKxDM000E1pRm79fAmPJLmk0NoAA6bQ\\nKWFUknlJjkm91Bf18blxQ/Ye+pMTTzuzry1pG2LD+nW5ben1Dz7R/fjoZq4LsCvs5U4J/d4bfsjQ\\noZtPnnF2Uwt99QM/z21Lr2/mkgC7zGlrlOSIVoDdRKFTmiNaAXYDhU6r6B3DT0kyO/W74Q8qmgig\\njSh0Ws1zx/ALe5I9C+cBaAsKnVbUO4aflB5PYgD0h0KnlT1aq+XJ0iEA2oFCB4AKUOgAUAEKHQAq\\nQKEDQAXY+pVK+dlt/5Xvfu1LWfmzu7LPviPyhhPenlM++okMfvmQ0tEAGsoVOpWy9PKv5B2nfzhX\\n3HFvzr/uhjy29pF89bP/WDoWQMMpdCpl7mXX5LA3H5s9h+yV4SP3zxlzP5c7f7CsdCyAhlPoVNqG\\n9euy17B9SscAaDiFTqVd868X5LjJ7y8dA6DhFDqV9b2F/5aNjz+ed50xvXQUgIZzlzuVtOTyL+fO\\nG7+fOV+9Ki99qX/mQPX5n47Kufk738wdy27I3MuuzuA9X146DkBTKHQq5e7bb8uyq67InK9enZfv\\ntXfpOABNo9CplAtmnZFN3Rsy9S/HPu/9RXeuzF7DhhVKBdB4Cp1K+fpdK0tHACjCXe6wDT3peUUS\\nM3ugbSh02JaevCTJ/Ummlo4C0B8KHbahVqutTzIlyewkNyY5pGwigO1T6NC3W5JMSLIkya1J5scY\\nHmhRCh22b0vqRT4+yfAYwwMtSqFD/zyUepEbwwMtyWNrtLRN3d2Dr13whaauuWH9uu39uXcMPz31\\nMfyVSc5JsrHxyQD6VisdALZjZOrFWcIjSS7ZwWdGJZmX5JjUS31Ro0MBAI3TleSeGMMDQNsblGRW\\nkrVxNzwAtL1RqY/eV8fd8ADQ9ozhAaAijOEBoEKM4QGgQozhAaAijOGBhrCxDJQx0E1pJm59lXDz\\n1hfQwmz9CmX07g3flWRBkmlJZia5r4/PTxx3xBs/Pe7Io5oUr27F8tuz4q47EoUOLU+hQ1n93ht+\\n3JFH5eQZZzc13LVJb6EDLc5pa1CeI1qBXabQoXU4ohXYaQodWk/vGH5J6mP4+UleVjQR0PIUOrSm\\nF47hSx0jC7QJhQ6trXcM/+3SQYDWptChPawqHQBobQodACpAoQNABSh0AKgAhQ4AFWDrV6ioFXfe\\nnn//xuW5d/mP8vTmzRlxwKi84YS3Z9IZ0zNk6LDS8YDdzBU6VNSVF56fI459Sy5cclO+/uOVmfO1\\nq/PEhsfz+ZkfLB0NaABX6FBR51295Hm/7/eq0Tn17z6VD7750EKJgEZS6NABep55Jut++3CWXPal\\nHN51fOk4QAModKi4yWMPfPbnV71uzIuu3IFq8B06VNzilWty3b2rcuHSm/LKAw7MZefNLR0JaACF\\nDh3gpYP2yB/9ydjMnDc/d/5gWek4QAModOggTz/1VAbtsUfpGEADKHSoqHkzTs/9P16ezb97Mr/b\\n9ER+/j8/yRc/Pj3Hv+/U0tGABnBTHFTUW957aq6+6IL88u6f5mWD98yo17w2b3nflBw7+f2lowEN\\noNChog7rOi6HdR1XOgbQJEbuwI68vnQAYMcUOrAjRyW5MckhpYMAfVPowI58JcmSJLcmmZ9k77Jx\\ngG1R6MCOPJN6kY9PMjzJ/UmmFk0EvIhCB/rrodSLfEqS2TGGh5ai0IGBuiXJhBjDQ0vx2Bq0iRXL\\nb8+1Bdbsw5bUi/y6JPNSH8Ofk2RRc5IBL1QrHQDol4lbXyXcvPW1PV1JFiR5JMnMJPc1NhIA0CiD\\nksxKsjbG8ADQ9kalPnpfHXfDA0Db60pyT9wNDwBtzxgeACrEGB4AKsQYHgAqwhgeACrEGB52IxvL\\nAKUNdFOas5Ls1+hQfbg4yaOF1obtUuhAKxiUZHqSOUmuTH0b2Y3b+uBeQ/dZffRJk149bN8RTYyX\\nfO+KSzdv2th9eJIVTV0Y+sle7kArGNDe8G87ZVpGjzm4eemS3LT46s2bNnY3dU0YCKetAa3EEa2w\\nkxQ60Ioc0QoDpNCBVtU7hh+fZHjqY3h3w0MffIcOtLreMXxXkgU9Pc+MLJwHWpIrdKBd3JJkQq1W\\ne7J0EGhFCh1oJ1uS2jYfZ4NOp9ABoAIUOgBUgEIHgApQ6ABQAQod6BhrfvOrzDpxYiaPPbB0FNjt\\nFDrQEZ5+6qnMnz0jZ332C6WjQEModKAjLJz3T3njW0/MwYceXjoKNIRCByrvzh8sy29Xr8o7P/CR\\n0lGgYWz9ClTa2ofW5JqLLsi5C7+ZWq1WOg40jCt0oNIu+vuZ+eCcf86w4fuWjgIN5QodqLQVd92R\\nOX/77he9P3nsgVm8ck2BRNAYCh2otG2VtjKniozcAaACFDoAVIBCBzqOcTtVpNAB+qGnp2dIkoNK\\n54C+KHSAfqjValuSLEwyP8nehePAiyh0gP55KsmkJMOT3J9katk48HwKHaD/Hk29yKckmZ3kxiSH\\nFE0EWyl0gIG7JcmEJEuS3BpjeFqAQgfYOVtSL/LxMYanBdgpDmg7y65amGH7jmjqmpu6uwf38aeH\\nUi/yriQLkkxLMjPJfU2KBkkSRw8B7easJPsVWvvi1L9H78ugJNOTzElyZZJzkmxsQi4AoAFGJVmU\\nZHWM4QGg7XUluSfuhgeAtjcoyawka+NueABoe8bwAFAhxvAAUBHG8ABQIcbwAFAhxvDsMhvLALSG\\ngW5KM3Lr50t4JMklhdamDwodoLWMSjIvyTGpl/qiPj43bsjeQ39y4mln9rUlbUNsWL8uty29/sEn\\nuh8f3cx12TF7uQO0ln7vDT9k6NDNJ884u6mFvvqBn+e2pdc3c0n6yWlrAK3JEa0MiEIHaF2OaKXf\\nFDpA6+sdw09JMjv1u+EPKpqIlqPQAdrHc8fwC3uSPQvnoYUodID20juGn5QeTyrxBwodoD09Wqvl\\nydIhaB0KHQAqQKEDQAUodACoAIUOABVg61eADjF57IHbfH/xyjVNTkIjKHSADqK8q8vIHQAqwBU6\\nQAf5wJten40bHsu++4/KmPGH5t1nzsxr/2xc6VjsBgodoEMccewJeefpH8mY8YfmiY0b8r8/vCXn\\nfWhKzjx3Xo449oTS8dhFCh2gQ/zDJVc8+/MrBo9M1zvfk31GvDKXn3+uQq8A36EDdLCDJ/xFHl71\\n69Ix2A0UOkAH+9WKu/PKUa8qHYPdQKEDdIjPfXha7l3+o2x+clM2dW/IXTf9Zy76xKxMOmNG6Wjs\\nBr5DB+gQb/2bqbnu4gvzy7t/mj2H7JXRB/1pPnTu+Tms67jS0dgNFDpAhzis6zjlXWFG7gAMSE96\\nXpFk79I5eD6FDsDA9OQlSe5PMrV0FP5AoQMwILVabX2SKUlmJ7kxySFlE5EodAB2zi1JJiRZkuTW\\nJPNjDF+UQgdgZ21JvcjHJxkeY/iiFDoAu+qh1IvcGL4gj60BtKlN3d2Dr13whaauuWH9uu39uXcM\\nPz31MfyVSc5JsrHxyaiVDgDAThmZenGW8EiSS3bwmVFJ5iU5JvVSX9ToUABA43QluSfG8ADQ9gYl\\nmZVkbdwNDwBtb1Tqo/fVcTc8ALQ9Y3gAqAhjeACoEGN4AKgQY3gAqAhj+F1gYxkAWs1AN6WZuPVV\\nws1bX8XZ+hWAVtO7N3xXkgVJpiWZmeS+Pj4/cdwRb/z0uCOPalK8uhXLb8+Ku+5IFDoAbFe/94Yf\\nd+RROXnG2U0Nd23SW+gtwWlrALQyR7T2k0IHoB04onUHFDoA7aR3DL8k9TH8/CQvK5qoRSh0ANrN\\nC8fwpY6RbSkKHYB21TuG/3bpIK1AoQPQ7laVDtAKFDoAVIBCB4AKUOgAUAEKHQAqwNavAHSk3216\\nIt/68vzc/h9Ls+7hNRn1m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\"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 29,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = BST()\\n\",\n \"for el in [1,2,3,4,5,6]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"What we would like to see is a tree that looks more like this:\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 5,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": \"iVBORw0KGgoAAAANSUhEUgAAAfQAAAEsCAYAAAA1u0HIAAAABmJLR0QA/wD/AP+gvaeTAAAQKUlE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\\nznjDgRVNBNAZtV/o8Hw2P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\"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 5,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = BST()\\n\",\n \"for el in [4,2,1,3,6,5]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw()\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# AVL trees\\n\",\n \"\\n\",\n \"We can see from above example that changing the order of insertions can result in balanced tree. However the point of BST (and most other datastructures) is to be able to update it online, as the updates arrive, without necessarily knowing the elements we are going to insert in the future. AVLs provide best of both worlds - they support online updates, and the updates always result in balanced trees. \\n\",\n \"\\n\",\n \"### AVL invariant\\n\",\n \"\\n\",\n \"Let $X$ be arbitrary node in AVL tree. Let $X_L$, $X_R$ be left and right children of $X$ respectively. Furthermore let $H_X$ be height of subtree rooted in $X$. AVL trees satisfy the following invariant:\\n\",\n \"\\n\",\n \"$$ \\\\left| H_{X_L} - H_{X_R} \\\\right| \\\\leq 1 $$\\n\",\n \"\\n\",\n \"It is satisfied before and after all the operations that we perform on an AVL tree. In this recitation we will implement AVL insertion that is capable of maintaining that invariant. Notice that BST by default does not satisfy this invariant (make sure you understand why).\\n\",\n \"\\n\",\n \"### AVL height\\n\",\n \"\\n\",\n \"First step on our journey to implement AVL trees is to keep track of tree height. To implement it we first need to make sure that we understand what $H_X$ actually means. Let's look at a definition:\\n\",\n \"\\n\",\n \"$$\\n\",\n \"H_X=\\n\",\n \"\\\\begin{cases}\\n\",\n \"-1 &\\\\text{ if } X = NIL \\\\\\\\\\n\",\n \"max(H_L, H_R) + 1 &\\\\text{otherwise}\\n\",\n \"\\\\end{cases}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Notice that this definition implies that heigh of a leaf is $0$. Alternative way to view this definition of height is the maximum number of edges we need to traverse to get from a node its some leaf (and $-1$ is a special case for convenience).\\n\",\n \"\\n\",\n \"One way to keep track of height is to calculate it every time in $O(h)$ complexity. This is a correct solution, but as we will soon find out, we will use the information about height too often to pay the extra $O(h)$ cost. Alternative solution is for every tree to store its height. Then all we need to do is to correct it every time the tree changes.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 30,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"class Node(object):\\n\",\n \" def __init__(self, key):\\n\",\n \" self.key = key\\n\",\n \" self.left = None\\n\",\n \" self.right = None\\n\",\n \" # here it is!\\n\",\n \" self.height = 0\\n\",\n \" \\n\",\n \" def recompute_height(self):\\n\",\n \" \\\"\\\"\\\"Updates the height variable.\\n\",\n \" \\n\",\n \" Should be run every time subtree changes.\\n\",\n \" Assumes children height values are correct.\\\"\\\"\\\"\\n\",\n \" # we assure left and right already has correct height computed.\\n\",\n \" # if they don't exists we assume height is -1\\n\",\n \" left_height = self.left.height if self.left is not None else -1\\n\",\n \" right_height = self.right.height if self.right is not None else -1\\n\",\n \" # height computed using formula from above\\n\",\n \" self.height = max(left_height, right_height) + 1\\n\",\n \" \\n\",\n \" # This will be useful later!\\n\",\n \" def balance_factor(self):\\n\",\n \" \\\"\\\"\\\"Returns the difference in height of children subtrees:\\n\",\n \" H_L - H_R\\n\",\n \" \\n\",\n \" Absolute value of balance factor should be less than 1 after\\n\",\n \" each AVL operation.\\\"\\\"\\\"\\n\",\n \" left_height = self.left.height if self.left is not None else -1\\n\",\n \" right_height = self.right.height if self.right is not None else -1\\n\",\n \" return left_height - right_height\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Keeping track of height in AVL insertion\\n\",\n \"\\n\",\n \"It's really not that hard! We just need to make sure that whenever tree height could have changed we recompute the height. \"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 8,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def BST_insert(where, key):\\n\",\n \" \\\"\\\"\\\"Inserts a node in a binary tree\\\"\\\"\\\"\\n\",\n \" if where is None:\\n\",\n \" # if we found a branch pointing to none, it means we\\n\",\n \" # can insert the node here\\n\",\n \" return Node(key)\\n\",\n \" elif key <= where.key:\\n\",\n \" # try inserting the node somewhere in the left subtree\\n\",\n \" where.left = BST_insert(where.left, key)\\n\",\n \" elif key > where.key:\\n\",\n \" # try inserting the node in the right subtree\\n\",\n \" where.right = BST_insert(where.right, key)\\n\",\n \" \\n\",\n \" where.recompute_height() # <---- THIS IS THE ONLY CHANGE\\n\",\n \" \\n\",\n \" return where\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Always good to run some examples to verify our implementation\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 9,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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PhHRFJ7vQccTeO+dnT9+pYUW/9sTVH3z6x/+8q6oYQ0rZZUAeek+qEn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\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 9,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = BST()\\n\",\n \"for el in [7,5,4,6,3,10,11]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# AVL rotations\\n\",\n \"\\n\",\n \"The only weapon at disposal of the lone worrior of the AVL clan is rotation or to be more specific two rotations (the left rotation and the right rotation). \\n\",\n \"\\n\",\n \"Before we dive in the technical details of how to implement the AVL rotations let's first try to gain some intuition. In AVL we strive to keep the difference between heights of right and left subtrees of all nodes at most $1$. Insertion can increase this difference to $2$ in the worst case.\\n\",\n \"\\n\",\n \"For example imagine that we inserted an element into right subtree and now right subtree is 2 higher than left subtree. \"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 11,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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vKT32lYE59S9S2Tz7WxHBwjUk7k80yYB/jkjSpvaqun8treRo68f5i9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\\n/w+ejKjdQr450QAAAABJRU5ErkJggg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 11,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = BST()\\n\",\n \"for el in [3,1,6,5,7,8]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Intuitively what we want to do is to *rotate the tree left* - steal one node from right (and replace root by it) and add one node to the left subtree (the former root). This will result in one decrase in right subtree height and one increase in left subtree height therefore balancing the tree (in reality it is not that simple, but it is good enough for now).\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 12,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"# HINT: when trying to understand this function read it from the bottom!\\n\",\n \"\\n\",\n \"def AVL_rotate_left(node):\\n\",\n \" \\\"\\\"\\\"Rotates the tree left and returns a new root.\\\"\\\"\\\"\\n\",\n \" assert node.right is not None, \\\\\\n\",\n \" \\\"Tree can only be rotated left if the right subtree is nonempty\\\"\\n\",\n \" old_right = node.right\\n\",\n \" old_right_left = node.right.left\\n\",\n \" \\n\",\n \" # instead of old right node, old root takes old rights left subtree\\n\",\n \" node.right = old_right_left\\n\",\n \" # node's children changed - we need to recompute the height\\n\",\n \" node.recompute_height()\\n\",\n \" \\n\",\n \" # old root becomes left node of old right (which becomes new root) \\n\",\n \" old_right.left = node\\n\",\n \" # old_right's children changed - we need to recompute the height\\n\",\n \" old_right.recompute_height()\\n\",\n \" \\n\",\n \" return old_right\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 13,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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v3q5bS8+w4HD+znDy/8lrprv5mw/MjxE1gyczpbGxs4eGA/WxsbWDJz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\\ngg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 13,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree.root = AVL_rotate_left(tree.root)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Exercise: prove that AVL rotation preserves BST invariant\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### AVL right rotation\\n\",\n \"\\n\",\n \"It is analogous to left rotations (in fact it inverts the left rotation)...\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 14,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"# HINT: when trying to understand this function read it from the bottom!\\n\",\n \"\\n\",\n \"def AVL_rotate_right(node):\\n\",\n \" \\\"\\\"\\\"Rotates the tree right and returns a new root.\\\"\\\"\\\"\\n\",\n \" assert node.left is not None, \\\\\\n\",\n \" \\\"Tree can only be rotated right if the left subtree is nonempty\\\"\\n\",\n \" old_left = node.left\\n\",\n \" old_left_right = node.left.right\\n\",\n \" \\n\",\n \" node.left = old_left_right\\n\",\n \" node.recompute_height()\\n\",\n \" \\n\",\n \" old_left.right = node\\n\",\n \" old_left.recompute_height()\\n\",\n \" \\n\",\n \" return old_left\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 15,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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vKT32lYE59S9S2Tz7WxHBwjUk7k80yYB/jkjSpvaqun8treRo68f5i9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\\n/w+ejKjdQr450QAAAABJRU5ErkJggg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 15,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree.root = AVL_rotate_right(tree.root)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## AVL insertion prelude.\\n\",\n \"\\n\",\n \"Assume we just performed insertion which threw our tree off balance. We will operate under recursive assumption, that child trees are balanced and that only current node maybe be unbalanced (height of one child is height of second child + 2).\\n\",\n \"\\n\",\n \"Intuition dictates that there are two cases to consider (left child higher or right child higher), but actually as we will soon find out there are four cases. Let's see why. \"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Case 1: Right-Right case \\n\",\n \"\\n\",\n \"First let's look at a simple tree where right subtree is higher.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 17,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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vKT32lYE59S9S2Tz7WxHBwjUk7k80yYB/jkjSpvaqun8treRo68f5i9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\\n/w+ejKjdQr450QAAAABJRU5ErkJggg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 17,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = BST()\\n\",\n \"for el in [3,1,6,5,7,8]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"As we have already seen this can be fixed with a single left rotation.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 18,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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v3q5bS8+w4HD+znDy/8lrprv5mw/MjxE1gyczpbGxs4eGA/WxsbWDJz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\\ngg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 18,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree.root = AVL_rotate_left(tree.root)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Case 2: Right-Left case \\n\",\n \"Let's look at the following bizzare example:\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 19,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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4ya8Ole+fbfvEPPPme0WL6F+duACa21rQXPh6eWLAxP9XFsSyp0X0yw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\\nf78nj115qKtHAAAAAElFTkSuQmCC\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 19,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = BST()\\n\",\n \"for el in [3,1,7,6,5,8]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"OBVIOUSLY* (as 6.006 progresses you will get more and more scared of this word) you can fix this with a left rotation.\\n\",\n \"\\n\",\n \"*Not\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 20,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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CHs2bWT2aUT2LN7F527Hc+wK8fw+E/uSeDlhdoQLLh5hGB1bnPiPx/2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\\n5QAAAABJRU5ErkJggg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 20,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree.root = AVL_rotate_left(tree.root)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"The tree is still unbalanced. What happened? \\n\",\n \"\\n\",\n \"Normally, for the *perfectly* blanced subtrees (balance factor of 0) left-rotation decreases hieght of right subtree by 1 and increases the height of the right subtree by 1.\\n\",\n \"\\n\",\n \"As an additional side-effect the right-left subtree (the one rooted in 6 in the example) moves from right to left subtree. If that subtree is deepest (as in our example) it results in additional decrease by 1 for the right subtree and 1 increase in the left subtree. In which case the total balance is $-2$ for right subtree and $+2$ for left subtree, which means our tree is unblanced again.\\n\",\n \"\\n\",\n \"Notice that this situation only happens if right-left (root 6 in example) subtree is higher than right-right subtree (root 8 in our example). Is there a way to cope with such a situation? \\n\",\n \"You guessed it - we need to right-rotate the right subtree (then right-left subtree is not longer higher than right-right subtree and we land at the familiar Case 1: right right).\\n\",\n \"\\n\",\n \"Let's see this in action!\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 21,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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4ya8Ole+fbfvEPPPme0WL6F+duACa21rQXPh6eWLAxP9XFsSyp0X0yw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\\nf78nj115qKtHAAAAAElFTkSuQmCC\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 21,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = BST()\\n\",\n \"for el in [3,1,7,6,5,8]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 22,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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vKT32lYE59S9S2Tz7WxHBwjUk7k80yYB/jkjSpvaqun8treRo68f5i9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\\n/w+ejKjdQr450QAAAABJRU5ErkJggg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 22,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"# Step 1:\\n\",\n \"tree.root.right = AVL_rotate_right(tree.root.right)\\n\",\n \"tree.root.recompute_height()\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 24,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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v3q5bS8+w4HD+znDy/8lrprv5mw/MjxE1gyczpbGxs4eGA/WxsbWDJz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\\ngg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 24,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"# Step 2:\\n\",\n \"tree.root = AVL_rotate_left(tree.root)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Cases 3 & 4: left-left and right-left\\n\",\n \"\\n\",\n \"Those two cases are analogous to the cases 1 and 2, but used when the left tree is higher.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## AVL insertion \\n\",\n \"\\n\",\n \"Now we are going to put all of this ideas together and implement AVL insertion!\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 35,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def AVL_insert(where, key):\\n\",\n \" \\\"\\\"\\\"Inserts a node in an AVL tree\\\"\\\"\\\"\\n\",\n \" if where is None:\\n\",\n \" # if we found a branch pointing to none, it means we\\n\",\n \" # can insert the node here.\\n\",\n \" return Node(key)\\n\",\n \" elif key <= where.key:\\n\",\n \" # try inserting the node somewhere in the left subtree\\n\",\n \" where.left = AVL_insert(where.left, key)\\n\",\n \" elif key > where.key:\\n\",\n \" # try inserting the node in the right subtree\\n\",\n \" where.right = AVL_insert(where.right, key)\\n\",\n \" \\n\",\n \" # where's height might have changed.\\n\",\n \" where.recompute_height()\\n\",\n \" \\n\",\n \" #################################################################\\n\",\n \" ###### UP TO THIS POINT IMPLEMENTATION IS IDENTICAL TO BST ######\\n\",\n \" #################################################################\\n\",\n \" \\n\",\n \" # insertion never changes balance factor to be bigger than 2.\\n\",\n \" assert abs(where.balance_factor()) <= 2\\n\",\n \" # we recursively assume that children are balanced\\n\",\n \" assert where.left is None or abs(where.left.balance_factor()) <= 1\\n\",\n \" assert where.right is None or abs(where.right.balance_factor()) <= 1\\n\",\n \" \\n\",\n \" if where.balance_factor() == 2:\\n\",\n \" # left subtree is higher.\\n\",\n \" if where.left is not None and where.left.balance_factor() < 0:\\n\",\n \" print(\\\"Case 4\\\")\\n\",\n \" # Case 4: left-right\\n\",\n \" # left-right subtree is taller\\n\",\n \" where.left = AVL_rotate_left(where.left)\\n\",\n \" where.recompute_height()\\n\",\n \" # Case 3: left-left\\n\",\n \" print(\\\"Case 3\\\")\\n\",\n \" where = AVL_rotate_right(where)\\n\",\n \" elif where.balance_factor() == -2:\\n\",\n \" # right subtree is higher.\\n\",\n \" if where.right is not None and where.right.balance_factor() > 0:\\n\",\n \" print(\\\"Case 2\\\")\\n\",\n \" # Case 2: right-left\\n\",\n \" # right-left subtree is taller\\n\",\n \" where.right = AVL_rotate_right(where.right)\\n\",\n \" where.recompute_height()\\n\",\n \" print(\\\"Case 1\\\")\\n\",\n \" # Case 1: right-right\\n\",\n \" where = AVL_rotate_left(where)\\n\",\n \"\\n\",\n \" # assert that we indeed balanced the tree.\\n\",\n \" assert abs(where.balance_factor()) <= 1\\n\",\n \"\\n\",\n \" return where\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 36,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"class AVL(BST): # <------ WARNING: inheritance\\n\",\n \" def insert(self, key): \\n\",\n \" self.root = AVL_insert(self.root, key)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Let's verify that sorted sequence is no longer a problem\\n\",\n \"\\n\",\n \"In fact it is trivial - it only requires case 1!\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 38,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"Case 1\\n\",\n \"Case 1\\n\",\n \"Case 1\\n\"\n ]\n },\n {\n \"data\": {\n 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dnuKwmOmw0EXgnb0cAzz2Ja01KWN81j8azpxGIHGXjmEMZ+93sp25/d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\\nRK5CYII=\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 38,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = AVL()\\n\",\n \"for el in [1,2,3,4,5,6]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Make sure cases 1 and 2 work correctly\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 39,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"Case 2\\n\",\n \"Case 1\\n\"\n ]\n },\n {\n \"data\": {\n \"image/png\": 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e+TYp0UdyHh30RKsseALxN+t7w74bHz+2n/EF21zBXzCE/0Oxfolvo5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\\n5AAAAABJRU5ErkJggg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 39,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = AVL()\\n\",\n \"for el in [2,5,4,3,1]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Make sure cases 3 and 4 work correctly\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 41,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"Case 4\\n\",\n \"Case 3\\n\"\n ]\n },\n {\n \"data\": {\n \"image/png\": 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TRXfPmi70Q1V+wkt7kiU84VylmmkXuQ4t+dfepeBcwj/HGJFzIdTL/j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\\nLOKhAAAAAElFTkSuQmCC\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 41,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = AVL()\\n\",\n \"for el in [4,1,2,3,5]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw(display_height=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Pushing the limit of AVL\\n\",\n \"\\n\",\n \"During the lecture we argued that \\n\",\n \"\\n\",\n \"$$\\n\",\n \"n_h \\\\leq n_{h-1} + n_{h-2}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Where $n_h$ is the number of nodes in the tree of height $h$. It is an interesting exercise to try to come up with a tree that is the worst-possible (has maximum possible number of nodes), given the height $h$. In particular we will know that our tree is one of the worst ones if it strictly satisfies above inequality. \\n\",\n \"\\n\",\n \"How can we construct such a tree? Is it at all possible?\\n\",\n \"\\n\",\n \"In this case the answer is yes. Such tree can be constructed recursively:\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 42,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def nightmare_tree(h):\\n\",\n \" assert h >= -1\\n\",\n \" \\n\",\n \" if h == -1:\\n\",\n \" return None\\n\",\n \" elif h == 0:\\n\",\n \" return Node(0)\\n\",\n \" else:\\n\",\n \" result = Node(0)\\n\",\n \" result.left = nightmare_tree(h-1)\\n\",\n \" result.right = nightmare_tree(h-2)\\n\",\n \" result.recompute_height()\\n\",\n \" return result\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 43,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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scuQJLU8FqvMLyR8M3NT4C+wCjg37FzLkm19jpwC3AEcDawN2GWjBWE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\\nVzZtzA2XjUySNDQ0JA0NaWhoSJeuLf8v4b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\"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 43,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"x = AVL()\\n\",\n \"x.root = nightmare_tree(5)\\n\",\n \"x.draw(display_height=True, width=1000, height=500)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Exercise: prove that the `nightmare_tree` is worst possible tree. Verify that is satisfies BST and AVL invariants. \\n\",\n \"\\n\",\n \"Notice that this recursive definition of nightmare_tree size is almost fibonacci (except extra one) and has a very similar solution that is related to golden ratio. If you are curious check out the wikipedia article on AVL.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture6/Counting BST.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 60,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"from collections import namedtuple\\n\",\n \"from igraph import Graph, plot, layout, Plot\\n\",\n \"%config InlineBackend.figure_format = 'png'\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Counting BST\\n\",\n \"\\n\",\n \"In this notebook we will augment BST datastructure to allow us to answer questions like \\n\",\n \"\\n\",\n \"What is the $k$-th smallest element in the tree rooted in $X$? \\n\",\n \"\\n\",\n \"in time $O(lg\\\\ n)$. The name of the function will be `kth_element(X,k)`\\n\",\n \"\\n\",\n \"First let's recap BST code:\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 32,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"class Node(object):\\n\",\n \" def __init__(self, key):\\n\",\n \" # node key - this is the value we compare to determine where \\n\",\n \" # the node should be inserted.\\n\",\n \" self.key = key\\n\",\n \" # left child (can be none if no child)\\n\",\n \" self.left = None\\n\",\n \" # right child (can be none if no child)\\n\",\n \" self.right = None\\n\",\n \" \\n\",\n \"def BST_insert(where, key):\\n\",\n \" \\\"\\\"\\\"Inserts a node in a binary tree\\\"\\\"\\\"\\n\",\n \" if where is None:\\n\",\n \" # if we found a branch pointing to none, it means we\\n\",\n \" # can insert the node here\\n\",\n \" return Node(key)\\n\",\n \" elif key <= where.key:\\n\",\n \" # try inserting the node somewhere in the left subtree\\n\",\n \" where.left = BST_insert(where.left, key)\\n\",\n \" elif key > where.key:\\n\",\n \" # try inserting the node in the right subtree\\n\",\n \" where.right = BST_insert(where.right, key)\\n\",\n \" return where\\n\",\n \"\\n\",\n \"def BST_draw(node, g, display_count=False):\\n\",\n \" \\\"\\\"\\\"This method allows us to draw a binary tree.\\n\",\n \" \\n\",\n \" Don't worry about the implementation.\\\"\\\"\\\"\\n\",\n \" if node is None:\\n\",\n \" return\\n\",\n \" \\n\",\n \" if node.left is not None:\\n\",\n \" BST_draw(node.left, g, display_count)\\n\",\n \" left_id = str(id(node.left))\\n\",\n \" left_color = \\\"black\\\"\\n\",\n \" else:\\n\",\n \" g.add_vertex(shape=0,)\\n\",\n \" left_id = len(g.vs) - 1\\n\",\n \" left_color = \\\"#00000000\\\"\\n\",\n \" \\n\",\n \" if node.right is not None:\\n\",\n \" BST_draw(node.right, g, display_count)\\n\",\n \" right_id = str(id(node.right))\\n\",\n \" right_color = \\\"black\\\"\\n\",\n \" else:\\n\",\n \" g.add_vertex(shape=0,)\\n\",\n \" right_id = len(g.vs) - 1\\n\",\n \" right_color = \\\"#00000000\\\"\\n\",\n \"\\n\",\n \" label = str(node.key)\\n\",\n \" if display_count:\\n\",\n \" label = label + ' (count: %d)' % (node.count,)\\n\",\n \" \\n\",\n \" g.add_vertex(name=str(id(node)), label=label, \\n\",\n \" shape=\\\"rectangle\\\", color=\\\"lightblue\\\")\\n\",\n \" if node.left is not None or node.right is not None:\\n\",\n \" g.add_edge(str(id(node)), left_id, color=left_color)\\n\",\n \" g.add_edge(str(id(node)), right_id, color=right_color)\\n\",\n \" \\n\",\n \"def BST_root_draw(root, display_count=False, width=500, height=300, margin=50):\\n\",\n \" figure = Plot(bbox=(width, height))\\n\",\n \" \\n\",\n \" g = Graph()\\n\",\n \" if root is not None:\\n\",\n \" BST_draw(root, g, display_count)\\n\",\n \" root_id = [i for i, node in enumerate(g.vs) if g.vs[i]['name'] == str(id(root))]\\n\",\n \" assert len(root_id) == 1\\n\",\n \" figure.add(g, layout = Graph.layout_reingold_tilford(g, mode=\\\"in\\\", root=root_id), \\n\",\n \" bbox=(margin, margin, width-margin, height - margin), vertex_size=30)\\n\",\n \" return figure\\n\",\n \" \\n\",\n \"class BST(object):\\n\",\n \" def __init__(self):\\n\",\n \" self.root = None\\n\",\n \" \\n\",\n \" def empty(self):\\n\",\n \" return self.root is None\\n\",\n \" \\n\",\n \" def insert(self, key): \\n\",\n \" self.root = BST_insert(self.root, key)\\n\",\n \" \\n\",\n \" def draw(self, display_count=False):\\n\",\n \" return BST_root_draw(self.root, display_count)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## First step towards the `kth_element`: counting\\n\",\n \"\\n\",\n \"One thing that we will need for an efficient implementation of `kth_element` is the ability to quickly answer the questions of form - how many nodes are the in subtree rooted in $X$. We can compute it online in time proportional to the size of the subtree, but since we will need to compute it frequently, we will try to keep the information available for $O(1)$ lookup at all times.\\n\",\n \"\\n\",\n \"More formally let $count_X$ be a size of a subtree rooted in $X$. It is defined by the following recursive relation:\\n\",\n \"\\n\",\n \"$$\\n\",\n \"count_X =\\n\",\n \"\\\\begin{cases}\\n\",\n \"0 &\\\\text{ if } X = NIL \\\\\\\\\\n\",\n \"count_{X_L} + count_{X_R} + 1 &\\\\text{otherwise}\\n\",\n \"\\\\end{cases}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Let's augment the node to be able to keep track of count!\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 61,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"class Node(object):\\n\",\n \" def __init__(self, key):\\n\",\n \" self.key = key\\n\",\n \" self.left = None\\n\",\n \" self.right = None\\n\",\n \" \\n\",\n \" self.count = 1 # <---- here we store the count\\n\",\n \" \\n\",\n \" # If any of the subtrees change we need to recompute count using the recursive formula\\n\",\n \" def recompute_count(self):\\n\",\n \" left_count = self.left.count if self.left is not None else 0\\n\",\n \" right_count = self.right.count if self.right is not None else 0\\n\",\n \" self.count = left_count + right_count + 1\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Insertion keeping track of count\\n\",\n \"\\n\",\n \"Let's now modify the insertion code to keep track of count. This should be quite easy, all we need to do is recompute count if any of the children of particular node changed.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 62,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def BST_insert(where, key):\\n\",\n \" \\\"\\\"\\\"Inserts a node in a binary tree\\\"\\\"\\\"\\n\",\n \" if where is None:\\n\",\n \" # if we found a branch pointing to none, it means we\\n\",\n \" # can insert the node here\\n\",\n \" return Node(key)\\n\",\n \" elif key <= where.key:\\n\",\n \" # try inserting the node somewhere in the left subtree\\n\",\n \" where.left = BST_insert(where.left, key)\\n\",\n \" elif key > where.key:\\n\",\n \" # try inserting the node in the right subtree\\n\",\n \" where.right = BST_insert(where.right, key)\\n\",\n \" \\n\",\n \" where.recompute_count() # <------- THIS IS THE ONLY LINE THAT CHANGED\\n\",\n \" \\n\",\n \" return where\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Let's verify that the code works\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 63,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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PlDmmfec38z0LpO+v/Tp/T5f/9Pjzb3rKo6QnBhW3rHzNVJPwusI3i1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\\nuAAAAABJRU5ErkJggg==\\n\",\n \"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 63,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = BST()\\n\",\n \"for el in [4,2,1,3,5,6]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw(display_count=True)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## `kth_element` implementation\\n\",\n \"\\n\",\n \"**Observation** Let $X$ be a node and $X_{key}$ its key. If $X_{key}$ is $k$-th node in the sorted order then $ k > count_{X_L}$. \\n\",\n \"\\n\",\n \"**Explanation** All the nodes in the left subtree are smaller than $X_{key}$ (for simplicity assume all are distinct). \\n\",\n \"\\n\",\n \"\\n\",\n \"This observation is fundamental for the To implement `kth_element(X,k)` function.\\n\",\n \"\\n\",\n \"#### Case 1: Result is in the left subtree\\n\",\n \"\\n\",\n \"This case occurs if $count_{X_L} \\\\geq k$. In that case we return kth_element($X_L$, k).\\n\",\n \"\\n\",\n \"#### Case 2: Result is in the root\\n\",\n \"\\n\",\n \"This case occurs if $count_{X_L} + 1 = k$. In that case we return $X_{key}$\\n\",\n \"\\n\",\n \"#### Case 3: Result is in the right subtree\\n\",\n \"\\n\",\n \"This case occurs if $count_{X_L} + 1 < k$. In that case we return kth_element($X_R$, $k - (count_{X_L} + 1)$). \"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 64,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def BST_kth_element(node, k):\\n\",\n \" if node is None or k < 1:\\n\",\n \" return None\\n\",\n \" left_count = node.left.count if node.left is not None else 0\\n\",\n \" if left_count >= k:\\n\",\n \" return BST_kth_element(node.left, k)\\n\",\n \" elif left_count + 1 == k:\\n\",\n \" return node\\n\",\n \" elif left_count + 1 < k:\\n\",\n \" return BST_kth_element(node.right, k - (1 + left_count))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Verify on example\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 65,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"image/png\": 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\"text/plain\": [\n \"\"\n ]\n },\n \"execution_count\": 65,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"tree = BST()\\n\",\n \"for el in [42,13,11,33,54,65]:\\n\",\n \" tree.insert(el)\\n\",\n \"tree.draw(display_count=True)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 66,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"True\"\n ]\n },\n \"execution_count\": 66,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"BST_kth_element(tree.root, 0) is None\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 67,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"1-th element is 11\\n\",\n \"2-th element is 13\\n\",\n \"3-th element is 33\\n\",\n \"4-th element is 42\\n\",\n \"5-th element is 54\\n\",\n \"6-th element is 65\\n\"\n ]\n }\n ],\n \"source\": [\n \"for k in range(1, 7):\\n\",\n \" print('%d-th element is %d' % (k, BST_kth_element(tree.root, k).key))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 68,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"True\"\n ]\n },\n \"execution_count\": 68,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"BST_kth_element(tree.root, 7) is None\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 69,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"Median is 33\\n\"\n ]\n }\n ],\n \"source\": [\n \"print('Median is %d' % (BST_kth_element(tree.root, tree.root.count / 2).key))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Exercise\\n\",\n \"\\n\",\n \"Augment BST code for deletion to keep track of the count. Deletion code is available in notebook for lecture5.\"\n ]\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture7/Radix Sort Performance.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 26,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"import random\\n\",\n \"import cProfile\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Radix sort\\n\",\n \"\\n\",\n \"Let's look at the code from pervious notebook. It is slightly augmented - `radix_sort_by_ith_digit` was incorporated in `radix_sort` and there are a few minor tweaks. \"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 59,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def is_sorted(array):\\n\",\n \" for i in xrange(1, len(array)):\\n\",\n \" if array[i-1] > array[i]:\\n\",\n \" return False\\n\",\n \" return True\\n\",\n \"\\n\",\n \"def radix_sort(array, b):\\n\",\n \" assert b > 1\\n\",\n \" i = 0\\n\",\n \" while True:\\n\",\n \" if is_sorted(array):\\n\",\n \" break\\n\",\n \" \\n\",\n \" buckets = [ [] for _ in xrange(b)]\\n\",\n \" for num in array:\\n\",\n \" # we no longer use an extra function call for computing the digit\\n\",\n \" bucket_idx = (num / b**i) % b\\n\",\n \" buckets[bucket_idx].append(num)\\n\",\n \" \\n\",\n \" # we reuse original space in the array,\\n\",\n \" # rather than allocating a new one.\\n\",\n \" next_index = 0\\n\",\n \" for bucket in buckets:\\n\",\n \" for num in bucket:\\n\",\n \" array[next_index] = num\\n\",\n \" next_index += 1\\n\",\n \" \\n\",\n \" i += 1\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Verify that it works on a simple example\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 60,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[1, 2, 2, 3, 4, 5, 5, 6]\"\n ]\n },\n \"execution_count\": 60,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"example = [5,3,2,5,6,1,2,4]\\n\",\n \"radix_sort(example, 2)\\n\",\n \"example\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### We are now moving on to bigger examples\\n\",\n \"The line\\n\",\n \"```python\\n\",\n \"random.seed(1)\\n\",\n \"```\\n\",\n \"ensures that we always generate the same example given test size (for fairness\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 61,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def generate_test(test_size):\\n\",\n \" random.seed(1)\\n\",\n \" example = [ random.randint(0,2**30) for _ in range(test_size)]\\n\",\n \" return example\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Let's try it for radix sort on different bases.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 62,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 30000065 function calls in 19.327 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 31 0.153 0.005 0.153 0.005 :1(is_sorted)\\n\",\n \" 1 17.945 17.945 19.315 19.315 :7(radix_sort)\\n\",\n \" 1 0.012 0.012 19.327 19.327 :1()\\n\",\n \" 31 0.000 0.000 0.000 0.000 {len}\\n\",\n \" 30000000 1.217 0.000 1.217 0.000 {method 'append' of 'list' objects}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"# BASE 2\\n\",\n \"example = generate_test(1000000)\\n\",\n \"cProfile.run(\\\"radix_sort(example, 2)\\\")\\n\",\n \"assert is_sorted(example)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 64,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 2000009 function calls in 1.696 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 3 0.150 0.050 0.150 0.050 :1(is_sorted)\\n\",\n \" 1 1.363 1.363 1.673 1.673 :7(radix_sort)\\n\",\n \" 1 0.022 0.022 1.696 1.696 :1()\\n\",\n \" 3 0.000 0.000 0.000 0.000 {len}\\n\",\n \" 2000000 0.160 0.000 0.160 0.000 {method 'append' of 'list' objects}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"# BASE 2^16\\n\",\n \"example = generate_test(1000000)\\n\",\n \"cProfile.run(\\\"radix_sort(example, 2**16)\\\")\\n\",\n \"assert is_sorted(example)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Intuitively it makes sense. We only need two iterators for $b=2^{16}$, while we need 16 for $b=2$. We cannot really have a single iteration, as $b=2^{32}$ is more than the length of the longest array we expect to ever sort.\\n\",\n \"\\n\",\n \"Now let's try to run the standard sorting algorithm that is implemented by Python (hybrid of insertion sort and quicksort). It is a comparison based sort and is therefore $O(n\\\\ lg\\\\ n)$. Radix sort is $O(n)$. We therefore expect our code to be faster.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 65,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 3 function calls in 0.547 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.000 0.000 0.547 0.547 :1()\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 1 0.547 0.547 0.547 0.547 {method 'sort' of 'list' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"example = generate_test(1000000)\\n\",\n \"cProfile.run(\\\"example.sort()\\\")\\n\",\n \"assert is_sorted(example)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"In reality our code is 3 times slower. What a shame! Let's not give up yet...\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### A bit of bit magic.\\n\",\n \"\\n\",\n \"One of basic operations in radix sort is divide and modulo. In particular if our base is $b$, then in every $i$-th iteration of radix sort algorithm one of the most common operations is indexing:\\n\",\n \"\\n\",\n \"```python\\n\",\n \"# determine appropriate bucket.\\n\",\n \"(num / b**i) % b\\n\",\n \"```\\n\",\n \"\\n\",\n \"This is great, but it consists of of expensive modulo and division operations (they can take up multiple processor cycles). \\n\",\n \"\\n\",\n \"Let's assume that b is a power of 2, i.e. $b=2^k$. Notice that `num / b**i` is equivalent to `num >> (k * i)`. In order to understand why this is the case first convince yourself that division by 2 is equivalent to shaving off rightmost bit.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 66,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"(6, 6)\\n\",\n \"(8, 8)\\n\"\n ]\n }\n ],\n \"source\": [\n \"print(100 / 16, 100 >> 4)\\n\",\n \"print(128 / 16, 128 >> 4)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Similarly notice that for $b=2^k$ we have `num % b` equivalent to `num & (b-1)`. To understand that notice that k lowest bits of `num` correspond to the reminder mod $b$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 67,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"(4, 4)\\n\",\n \"(0, 0)\\n\",\n \"(11, 11)\\n\"\n ]\n }\n ],\n \"source\": [\n \"print(100 % 16, 100 & 15)\\n\",\n \"print(128 % 16, 128 & 15)\\n\",\n \"print(11 % 16, 11 & 15)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Both `&` and `>>` are very efficient and only take on processor cycle. We can augment implementation of radix sort from above to use them instead of `%` and `/`.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 68,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def fast_radix_sort(array, k):\\n\",\n \" \\\"\\\"\\\"Fast radix sort with base 2**k.\\n\",\n \" \\n\",\n \" This implementation uses bitwise operations\\\"\\\"\\\"\\n\",\n \" assert k > 0\\n\",\n \" i = 0\\n\",\n \" \\n\",\n \" b=2**k\\n\",\n \" b_m1 = b - 1\\n\",\n \" \\n\",\n \" while True:\\n\",\n \" if is_sorted(array):\\n\",\n \" break\\n\",\n \" shift = k * i\\n\",\n \" buckets = [ [] for _ in xrange(b)]\\n\",\n \" for num in array:\\n\",\n \" bucket_idx = (num >> shift) & b_m1\\n\",\n \" buckets[bucket_idx].append(num)\\n\",\n \" \\n\",\n \" next_index = 0\\n\",\n \" for bucket in buckets:\\n\",\n \" for num in bucket:\\n\",\n \" array[next_index] = num\\n\",\n \" next_index += 1\\n\",\n \" \\n\",\n \" i += 1\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 69,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[1, 2, 2, 3, 4, 5, 5, 6]\"\n ]\n },\n \"execution_count\": 69,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"example = [5,3,2,5,6,1,2,4]\\n\",\n \"fast_radix_sort(example, 16)\\n\",\n \"example\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 71,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 2000009 function calls in 1.413 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 3 0.154 0.051 0.154 0.051 :1(is_sorted)\\n\",\n \" 1 1.081 1.081 1.390 1.390 :1(fast_radix_sort)\\n\",\n \" 1 0.023 0.023 1.413 1.413 :1()\\n\",\n \" 3 0.000 0.000 0.000 0.000 {len}\\n\",\n \" 2000000 0.155 0.000 0.155 0.000 {method 'append' of 'list' objects}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"example = generate_test(1000000)\\n\",\n \"cProfile.run(\\\"fast_radix_sort(example, 16)\\\")\\n\",\n \"assert is_sorted(example)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"*Good news* We got 0.2s speed up Yay!\\n\",\n \"\\n\",\n \"*Bad news* We are still nowhere near the performance of `.sort`. Why is that?\\n\",\n \"\\n\",\n \"The answer is in our choice of programming language. Python is interpreted and is known to have slowdowns up to 100x compared to low level langauges like C/C++. That's why many of Python routines are secretly implemented in C, `.sort` being one of them. In order to make the comparison fair we should also be allowed to write our implementation in C. Thankfully there's a Cython python extension, which makes it easy to interface with Python and compiles python-like code to C or C++. It should be pretty straightforward, but don't worry if you don't understand the details of the implementation below.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 72,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"The Cython extension is already loaded. To reload it, use:\\n\",\n \" %reload_ext Cython\\n\"\n ]\n }\n ],\n \"source\": [\n \"%load_ext Cython\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 73,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"%%cython --cplus\\n\",\n \"\\n\",\n \"from libcpp.vector cimport vector\\n\",\n \"cimport cython\\n\",\n \"\\n\",\n \"\\n\",\n \"cdef c_is_sorted(list array):\\n\",\n \" \\\"\\\"\\\"Equivalent to implementation of is_sorted from above.\\n\",\n \" \\n\",\n \" However this one is compiled to pure C++ (thanks to cdef).\\n\",\n \" We cannot call it directly for Python.\\n\",\n \" \\n\",\n \" The reasons it is slightly different from above is the fact that\\n\",\n \" \\n\",\n \" for current_num in array\\n\",\n \" \\n\",\n \" is super-efficient in Cython.\\\"\\\"\\\"\\n\",\n \" cdef unsigned int lastnum = 0\\n\",\n \" cdef bint first_iter = True\\n\",\n \" \\n\",\n \" for current_num in array:\\n\",\n \" if not first_iter:\\n\",\n \" if lastnum > current_num:\\n\",\n \" return False\\n\",\n \" else:\\n\",\n \" first_iter = False\\n\",\n \" lastnum = current_num\\n\",\n \" return True\\n\",\n \"\\n\",\n \"def c_radix_sort(list array, int k):\\n\",\n \" assert k > 0\\n\",\n \" # Just like in C Cython requires us \\n\",\n \" # to forward declare the variables\\n\",\n \" cdef int i = 0\\n\",\n \" cdef int b = 2 ** k\\n\",\n \" cdef int b_m1 = b - 1\\n\",\n \" cdef int shift = 0\\n\",\n \" cdef int next_index = 0\\n\",\n \" cdef int num = 0\\n\",\n \" # vector[vector[int]] is a list of lists of integers.\\n\",\n \" # (actually it is very efficient dynamically resizeable\\n\",\n \" # array)\\n\",\n \" cdef vector[vector[int]] buckets\\n\",\n \" \\n\",\n \" # initialize list with b empty arrays\\n\",\n \" for _ in xrange(b):\\n\",\n \" buckets.push_back(vector[int]())\\n\",\n \"\\n\",\n \" \\n\",\n \" # The code below barely changed compared to origninal\\n\",\n \" # the only difference is the fast that we access buckets\\n\",\n \" # slightly differently to be compliant with vector API.\\n\",\n \" while True:\\n\",\n \" if c_is_sorted(array):\\n\",\n \" break\\n\",\n \" shift = i * k\\n\",\n \"\\n\",\n \" for bucket_idx in xrange(b):\\n\",\n \" buckets[bucket_idx].clear()\\n\",\n \" \\n\",\n \" for num in array:\\n\",\n \" bucket_idx = (num >> shift) & b_m1\\n\",\n \" buckets[bucket_idx].push_back(num)\\n\",\n \" \\n\",\n \" next_index = 0\\n\",\n \" for bucket_idx in xrange(b):\\n\",\n \" for in_bucket_idx in xrange(buckets[bucket_idx].size()):\\n\",\n \" array[next_index] = buckets[bucket_idx][in_bucket_idx]\\n\",\n \" next_index += 1\\n\",\n \" \\n\",\n \" i += 1\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 74,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[1, 2, 2, 3, 4, 5, 5, 6]\"\n ]\n },\n \"execution_count\": 74,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"example = [5,3,2,5,6,1,2,4]\\n\",\n \"c_radix_sort(example, 16)\\n\",\n \"example\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 75,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 3 function calls in 0.128 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.000 0.000 0.128 0.128 :1()\\n\",\n \" 1 0.128 0.128 0.128 0.128 {_cython_magic_a617e99601e7e788cc896c9cdd2003a9.c_radix_sort}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"example = generate_test(1000000)\\n\",\n \"cProfile.run(\\\"c_radix_sort(example, 16)\\\")\\n\",\n \"assert is_sorted(example)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Victory!\\n\",\n \"\\n\",\n \"Our implementation of radix sort is 4x faster that default Python sort. \\n\",\n \"\\n\",\n \"This is expected as it has lower complexity and a very low constant of proportionality. \\n\",\n \"\\n\",\n \"Bear in mind that Python sort is more general though - it would be nontrival to use radix sort to sort long strings for example.\"\n ]\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture7/Understanding Radix Sort.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Understanding radix sort\\n\",\n \"\\n\",\n \"First building block we will need is the `is_sorted` function. \"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 1,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def is_sorted(array):\\n\",\n \" \\\"\\\"\\\"Takes a sequence and returns true if an only if sequence is sorted.\\\"\\\"\\\"\\n\",\n \" # check all the n-1 pairs of adjacent elements for \\n\",\n \" # order violation\\n\",\n \" for i in xrange(1, len(array)):\\n\",\n \" if array[i-1] > array[i]:\\n\",\n \" return False\\n\",\n \" # if no violations, then by transitivity of < the sequence is sorted.\\n\",\n \" return True\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 2,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"([1, 4, 6, 7, 8], True)\\n\",\n \"([1, 4, 7, 6, 8], False)\\n\"\n ]\n }\n ],\n \"source\": [\n \"# Verify implementation on few test cases\\n\",\n \"example1 = [1,4,6,7,8]\\n\",\n \"example2 = [1,4,7,6,8]\\n\",\n \"print(example1, is_sorted(example1))\\n\",\n \"print(example2, is_sorted(example2))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Stable sorting\\n\",\n \"\\n\",\n \"Python sort is stable. It means that if we have two elements that have the same value of *sorting key* they will appear in the output in the same order that they appeared in the input.\\n\",\n \"\\n\",\n \"Let's see an example: we have a list of pairs $(a,b)$ and we want to sort them in the nondecreasing order by $a$ and by nondecreasing $b$ if $a$'s are the same\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 3,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[(3, 1), (3, 2), (1, 1), (1, 2), (2, 2), (2, 1)]\"\n ]\n },\n \"execution_count\": 3,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"example = [ (3,1),(3,2),(1,1),(1,2), (2,2),(2,1)]\\n\",\n \"example\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"We can achieve that by sorting first by $b$ and then **stable-sorting** by $a$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 4,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[(3, 1), (1, 1), (2, 1), (3, 2), (1, 2), (2, 2)]\"\n ]\n },\n \"execution_count\": 4,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"sorted_idx2 = sorted(example, key=lambda x: x[1])\\n\",\n \"sorted_idx2\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 5,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (3, 2)]\"\n ]\n },\n \"execution_count\": 5,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"sorted_idx12 = sorted(sorted_idx2, key=lambda x: x[0])\\n\",\n \"sorted_idx12\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Unstable sort example\\n\",\n \"\\n\",\n \"Sort does no have to be stable. For example merge sort isn't.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 6,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"class MaxHeap(object):\\n\",\n \" def __init__(self, array, key):\\n\",\n \" self.storage = array\\n\",\n \" self.heap_size = len(array)\\n\",\n \" self.key = key\\n\",\n \" self.heapify()\\n\",\n \" \\n\",\n \" def fix_down(self, index):\\n\",\n \" while index < self.heap_size:\\n\",\n \" # pick maximum child\\n\",\n \" max_child_idx = None\\n\",\n \" if 2 * index + 1 < self.heap_size:\\n\",\n \" max_child_idx = 2 * index + 1\\n\",\n \"\\n\",\n \" if 2 * index + 2 < self.heap_size and \\\\\\n\",\n \" self.key(self.storage[2 * index + 1]) < self.key(self.storage[2 * index + 2]):\\n\",\n \" max_child_idx = 2 * index + 2\\n\",\n \"\\n\",\n \" if max_child_idx is None or \\\\\\n\",\n \" self.key(self.storage[index]) > self.key(self.storage[max_child_idx]):\\n\",\n \" # heap property satisfied\\n\",\n \" break\\n\",\n \" \\n\",\n \" self.storage[index], self.storage[max_child_idx] = self.storage[max_child_idx], self.storage[index]\\n\",\n \" index = max_child_idx\\n\",\n \" \\n\",\n \" def fix_up(self, index):\\n\",\n \" assert index < self.heap_size\\n\",\n \" while index > 0:\\n\",\n \" parent_idx = (index - 1) // 2\\n\",\n \" if self.key(self.storage[index]) >= self.key(self.storage[parent_idx]):\\n\",\n \" self.storage[index], self.storage[parent_idx] = self.storage[parent_idx], self.storage[index]\\n\",\n \" index = parent_idx\\n\",\n \" else:\\n\",\n \" break\\n\",\n \" def extract_max(self):\\n\",\n \" self.storage[0], self.storage[self.heap_size - 1] = self.storage[self.heap_size -1], self.storage[0]\\n\",\n \" self.heap_size -= 1\\n\",\n \" self.fix_down(0)\\n\",\n \" return self.storage[self.heap_size]\\n\",\n \" \\n\",\n \" def heapify(self):\\n\",\n \" for i in range(self.heap_size - 1, -1, -1):\\n\",\n \" self.fix_down(i)\\n\",\n \" \\n\",\n \"def heap_sort(array, key=lambda x:x):\\n\",\n \" h = MaxHeap(array[:], key)\\n\",\n \" while h.heap_size > 0:\\n\",\n \" h.extract_max()\\n\",\n \" return h.storage\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 7,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[(2, 1), (1, 1), (3, 1), (2, 2), (3, 2), (1, 2)]\"\n ]\n },\n \"execution_count\": 7,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"sorted_idx2 = heap_sort(example, key=lambda x: x[1])\\n\",\n \"sorted_idx2\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 8,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"[(1, 2), (1, 1), (2, 1), (2, 2), (3, 1), (3, 2)]\\n\",\n \"Notice that secondary sorting criterion is violated.\\n\"\n ]\n }\n ],\n \"source\": [\n \"sorted_idx12 = heap_sort(sorted_idx2, key=lambda x: x[0])\\n\",\n \"print(sorted_idx12)\\n\",\n \"print(\\\"Notice that secondary sorting criterion is violated.\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Counting Sort\\n\",\n \"\\n\",\n \"In order to keep complexity at $O(n)$, we will need to divise a procude that sorts without using comparisons.\\n\",\n \"\\n\",\n \"Assume we only have elements $0, 1, ..., (k-1)$ in the array. We know that all zeros come before all ones etc. We can therefore put all the numbers in $k$ different buckets and later read them off.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 9,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def count_sort(array, k, key=lambda x: x):\\n\",\n \" \\\"\\\"\\\"Stable sorts array by using key to determine ordering of elements.\\n\",\n \" \\n\",\n \" Assumes all elements are in range(0, k)\\\"\\\"\\\"\\n\",\n \" # initialize array \\n\",\n \" buckets = [[] for _ in range(k)]\\n\",\n \" # for every key store all the elements\\n\",\n \" # with that key\\n\",\n \" for element in array:\\n\",\n \" buckets[key(element)].append(element)\\n\",\n \" output = []\\n\",\n \" # red numbers from buckets in order\\n\",\n \" for bucket in buckets:\\n\",\n \" for element in bucket:\\n\",\n \" output.append(element)\\n\",\n \" return output\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 10,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[1, 2, 2, 3, 4, 5, 5]\"\n ]\n },\n \"execution_count\": 10,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"count_sort([4,3,2,5,5,1,2], 10)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Count sort complexity analysis\\n\",\n \"\\n\",\n \"We have the following steps:\\n\",\n \"- allocate space for $b$ buckets: $O(b)$\\n\",\n \"- loop throgh all the elements in the input array and put them in buckets $O(n)$\\n\",\n \"- remove elements from the buckets $O(n)$\\n\",\n \"\\n\",\n \"Therefore the total complexity is $O(n+b)$\\n\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Radix sort idea\\n\",\n \"\\n\",\n \"Imagine that you want to compare two long numbers. For example 85823421348134214 and 85823421348452456. The algorithm you would use is to compare the first digit and if it is the same then compare the next digit etc. We can say that first digit is the primary comparison criterion, second digit is the secondary sorting criterion etc. This is almost correct, but we actually need to make sure that we add extra zeros at the beginning of the number that is shorter (because sorter numbers come before longer numbers). \\n\",\n \"\\n\",\n \"Radix sort uses this idea directly for sorting. It first sorts the numbers by last digit. The it *stable-sorts* it by the second to last digit (making second to last digit primary sorting criterion and the last digit secondary sorting criterion) and so on. At the end of that process we end up with digitst sorted in exactly the order we discussed above.\\n\",\n \"\\n\",\n \"To implement that idea let's first look at how we would obtain the digits. \"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 11,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def ith_digit(number, i):\\n\",\n \" \\\"\\\"\\\"Returns the i-th digit from the end. \\n\",\n \" \\n\",\n \" i=0 resuts the very last digit.\\\"\\\"\\\"\\n\",\n \" for _ in range(i):\\n\",\n \" number /= 10\\n\",\n \" return number % 10\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 12,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"3\\n\",\n \"2\\n\",\n \"1\\n\",\n \"0\\n\",\n \"0\\n\"\n ]\n }\n ],\n \"source\": [\n \"print(ith_digit(123, 0))\\n\",\n \"print(ith_digit(123, 1))\\n\",\n \"print(ith_digit(123, 2))\\n\",\n \"print(ith_digit(123, 3))\\n\",\n \"print(ith_digit(123, 4))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Sweet! We have a function that returns i-th digit and it even yields additional zeros in the front - just what we needed.\\n\",\n \"\\n\",\n \"\\n\",\n \"## Radix Sort using digit $i$ from the end\\n\",\n \"\\n\",\n \"The idea is to use count sort with the digit being the key. \\n\",\n \"\\n\",\n \"For example if we sort `[123, 42, 73]` by the last digit, bucket nr 2 will have one number `[42]`, while bucket number three would have two numbers `[123, 73]`, while the remaining eight buckets would be empty. It is imporant that bucket nr two has `[123, 73]` not `[73, 123]` - this way if we read out the numbers in order they appear in the buckets we will get a stable sort.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 13,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def radix_sort_by_ith_digit(array, i):\\n\",\n \" return count_sort(array, \\n\",\n \" 10, # we have 10 different digits.\\n\",\n \" key=lambda number: ith_digit(number, i)) # use i-th digit as a key.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 14,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[42, 123, 73]\"\n ]\n },\n \"execution_count\": 14,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"# sort by the last digit\\n\",\n \"pass1 = radix_sort_by_ith_digit([123,42,73], 0)\\n\",\n \"pass1\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 15,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[123, 42, 73]\"\n ]\n },\n \"execution_count\": 15,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"# sort result of previous pass by the second to last digit\\n\",\n \"pass2 = radix_sort_by_ith_digit(pass1, 1)\\n\",\n \"pass2\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 16,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[42, 73, 123]\"\n ]\n },\n \"execution_count\": 16,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"# sort result of previous pass by the third to last digit\\n\",\n \"# none of the numbers are longer than third digit, so we are done.\\n\",\n \"pass3 = radix_sort_by_ith_digit(pass2, 2)\\n\",\n \"pass3\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"What happened above is exactly radix sort! Sort iteratively by digits further and further from the end until the sequence ends up sorted.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 17,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def radix_sort(array):\\n\",\n \" \\\"\\\"\\\"Returns array sorted by i-th digit from the end.\\n\",\n \" \\n\",\n \" The sorting procedure is stable.\\\"\\\"\\\"\\n\",\n \" i = 0\\n\",\n \" while True:\\n\",\n \" if is_sorted(array):\\n\",\n \" # we stop once the array is sorted\\n\",\n \" # the latest this can happen is when \\n\",\n \" # we run the number of passes eqaul to\\n\",\n \" # the length of the longest number\\n\",\n \" break\\n\",\n \" # stable sort by i-th digit.\\n\",\n \" array = radix_sort_by_ith_digit(array, i)\\n\",\n \" i += 1\\n\",\n \" return array\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 18,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[42, 73, 123]\"\n ]\n },\n \"execution_count\": 18,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"radix_sort([123,42,73])\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 19,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[42, 73, 123, 512, 524, 5214, 123123, 142124]\"\n ]\n },\n \"execution_count\": 19,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"# harder example\\n\",\n \"radix_sort([123,42,73, 123123, 142124, 524, 512, 5214])\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Radix sort with different numeric base. \\n\",\n \"\\n\",\n \"Let's try to improve our algorithm slightly. Notice that the fact that we use digits in base $10$ is kind of arbitrary. How hard would it be to use any $b \\\\geq 2$? In theory all we should be required to do is to change the digit extraction procedure and the number of buckets. \\n\",\n \"\\n\",\n \"Let's start with the digits.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 20,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def ith_digit(number, b, i):\\n\",\n \" \\\"\\\"\\\"Returns the i-th digit from the end (base b).\\n\",\n \" \\n\",\n \" i=0 resuts the very last digit.\\\"\\\"\\\"\\n\",\n \" for _ in range(i):\\n\",\n \" number /= b # changed 10 to b\\n\",\n \" return number % b # changed 10 to b\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 21,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"7 mod 2\\n\",\n \"1\\n\",\n \"1\\n\",\n \"1\\n\",\n \"0\\n\",\n \"0\\n\"\n ]\n }\n ],\n \"source\": [\n \"print(\\\"7 mod 2\\\")\\n\",\n \"print(ith_digit(7, 2, 0))\\n\",\n \"print(ith_digit(7, 2, 1))\\n\",\n \"print(ith_digit(7, 2, 2))\\n\",\n \"print(ith_digit(7, 2, 3))\\n\",\n \"print(ith_digit(7, 2, 4))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 22,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"7 mod 3\\n\",\n \"1\\n\",\n \"2\\n\",\n \"0\\n\",\n \"0\\n\",\n \"0\\n\"\n ]\n }\n ],\n \"source\": [\n \"print(\\\"7 mod 3\\\")\\n\",\n \"print(ith_digit(7, 3, 0))\\n\",\n \"print(ith_digit(7, 3, 1))\\n\",\n \"print(ith_digit(7, 3, 2))\\n\",\n \"print(ith_digit(7, 3, 3))\\n\",\n \"print(ith_digit(7, 3, 4))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Now we are ready to augment to radix_sort.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 23,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def radix_sort_by_ith_digit(array, b, i):\\n\",\n \" \\\"\\\"\\\"Returns array sorted by i-th digit from the end (base b).\\n\",\n \" \\n\",\n \" The sorting procedure is stable.\\\"\\\"\\\"\\n\",\n \" return count_sort(array, b, key=lambda number: ith_digit(number, b, i))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 24,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def radix_sort(array, b):\\n\",\n \" \\\"\\\"\\\"Returns array sorted by i-th digit from the end.\\n\",\n \" \\n\",\n \" The sorting procedure is stable.\\\"\\\"\\\"\\n\",\n \" i = 0\\n\",\n \" while True:\\n\",\n \" if is_sorted(array):\\n\",\n \" # we stop once the array is sorted\\n\",\n \" # the latest this can happen is when \\n\",\n \" # we run the number of passes eqaul to\\n\",\n \" # the length of the longest number\\n\",\n \" break\\n\",\n \" print(\\\"Iteration %d\\\" % (i,))\\n\",\n \" # stable sort by i-th digit.\\n\",\n \" array = radix_sort_by_ith_digit(array, b, i)\\n\",\n \" i += 1\\n\",\n \" return array\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Let's try sorting in base $b=2$\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 25,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"Iteration 0\\n\",\n \"Iteration 1\\n\",\n \"Iteration 2\\n\",\n \"Iteration 3\\n\",\n \"Iteration 4\\n\",\n \"Iteration 5\\n\",\n \"Iteration 6\\n\"\n ]\n },\n {\n \"data\": {\n \"text/plain\": [\n \"[42, 73, 123]\"\n ]\n },\n \"execution_count\": 25,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"radix_sort([123,42,73], 2)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Whoah! 7 iterations? That is a lot to sort just 3 numbers. How about if we increase the base? Maybe $b=1000$ ?\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 26,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"Iteration 0\\n\"\n ]\n },\n {\n \"data\": {\n \"text/plain\": [\n \"[42, 73, 123]\"\n ]\n },\n \"execution_count\": 26,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"radix_sort([123,42,73], 1000)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Much better - we only have one iteration. Notice however that we have many more buckets than numbers - even though in theory we decrease number of iterations, now every iteration is dominated by looping through every bucket. In this example $1000$ buckets visited in one iteration are much worst than two buckets visited in $7$ iterations (total of $14$ acceses). Actually array acceses contribute another 3 operations per iteraions ($7 * 3 = 21$) adding up to total of $35$ operations, but this is still much less than $1000$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 27,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"Iteration 0\\n\",\n \"Iteration 1\\n\",\n \"Iteration 2\\n\"\n ]\n },\n {\n \"data\": {\n \"text/plain\": [\n \"[42, 73, 123]\"\n ]\n },\n \"execution_count\": 27,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"# much healthier choice\\n\",\n \"radix_sort([123,42,73], 5)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Radix sort complexity analysis\\n\",\n \"\\n\",\n \"Let $b$ be the base and $n$ size of the array. Moreover let's assume that all the numbers in the array are less than or equal $a$.\\n\",\n \"\\n\",\n \"\\n\",\n \"Single iteration of count sort is $O(n + b)$.\\n\",\n \"\\n\",\n \"How many iterations are there? At most as many as the number of digits in the longest number: O($log_b\\\\ a$)\\n\",\n \"\\n\",\n \"Therefore the total complexity of the algorithm is O($(n+b) log_b\\\\ a)$.\\n\",\n \"\\n\",\n \"In theory we often assume that both $b$ and $a$ are constants - they are after all independent of $n$ - they won't influence the run time as $n$ grows. That's why some theorists say that Radix Sort is $O(n)$.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Exercises\\n\",\n \"\\n\",\n \"1. We said that the best possible algorithm that does sorting has complexity $O(n\\\\ lg\\\\ n)$. How is it possible that radix sort takes only $O(n)$ time? \\n\",\n \"\\n\",\n \"2. Can you come up with a sorting problem where it would be hard to use Radix Sort?\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Be sure to checkout the Performance of Radix Sort notebook!\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Aside: implementation of count sort from the lectures\\n\",\n \"\\n\",\n \"This implementation has the samee time and space complexity, but is faster in practice.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 28,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def count_sort_from_the_lecture(array, k, key=lambda x: x):\\n\",\n \" # initialize array \\n\",\n \" count = [0 for _ in range(k)]\\n\",\n \" # for every key count the number of times\\n\",\n \" # it occurs\\n\",\n \" for element in array:\\n\",\n \" count[key(element)] += 1\\n\",\n \" # compute cumulative count of occurences\\n\",\n \" for i in range(1, k):\\n\",\n \" count[i] += count[i-1]\\n\",\n \" # create output array\\n\",\n \" output = [None for _ in range(len(array))]\\n\",\n \" # fill in output array computing slots using\\n\",\n \" # counts array\\n\",\n \" for i in range(len(array) - 1, -1, -1):\\n\",\n \" output[count[key(array[i])] - 1] = array[i]\\n\",\n \" count[key(array[i])] -= 1\\n\",\n \" return output\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 29,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[1, 2, 2, 3, 4, 5, 5]\"\n ]\n },\n \"execution_count\": 29,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"count_sort_from_the_lecture([4,3,2,5,5,1,2], 10)\"\n ]\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 2\",\n \"language\": \"python\",\n \"name\": \"python2\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 2\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython2\",\n \"version\": \"2.7.8\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture8/Hashing and hashtables.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"code\",\n \"execution_count\": 51,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"import cProfile\\n\",\n \"import math\\n\",\n \"\\n\",\n \"from collections import namedtuple\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Hash tables and hashing functions\\n\",\n \"\\n\",\n \"## Hashing functions\\n\",\n \"\\n\",\n \"Want a function $h(k)$ which maps every key $k$ in the universe $U$ to some index $i, 0 \\\\le i < m$. We call this function a \\\\textbf{hash function}. Given a $k$, we compute $i = h(k)$ and then use this $i$ as a basis for insertion, searching or deletion. Some functions are better than others. A good hash function has these properties:\\n\",\n \"- item want $i$ to be uniformly distributed over all $m$\\n\",\n \"- item should not hash similar/related keys to the same slot.\\n\",\n \"- item fast to compute. $O(1)$ time.\\n\",\n \"- item should not change with time. We want some key $k$ to always hash to the same $h(k)$.\\n\",\n \"\\n\",\n \"\\n\",\n \"### Simple Uniform Hashing Assumption\\n\",\n \"We want our hash functions to distribute the hashes uniformly across all $m$. We usually assume something called simple uniform hashing. It means that given any two randomly chosen keys $k_1$ and $k_2$, the probability that they have equal hashes $= \\\\frac{1}{m}$.\\n\",\n \"\\n\",\n \"$$ \\n\",\n \"P[h(k_1) = h(k_2)] = \\\\frac{1}{m}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Let's see some examples of hash functions and analyze them.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Examples of hash functions\\n\",\n \"\\n\",\n \"#### Divition method\\n\",\n \"\\n\",\n \"The division method is one way to create hash functions. The functions take the form\\n\",\n \"\\n\",\n \"$$\\n\",\n \"h(k) = k \\\\mod m\\n\",\n \"$$\\n\",\n \"\\n\",\n \"Since we’re taking a value $\\\\mod m$, $h(k)$ does indeed map the universe of keys to a slot in the hash table. \\n\",\n \"\\n\",\n \"It’s important to note that if we’re using this method to create hash functions, $m$ should not be a power of 2. If $m = 2 p$ , then the $h(k)$ only looks at the $p$ lower bits of $k$, completely ignoring the rest of the bits in $k$. \"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 9,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def hashtest(b, mod):\\n\",\n \" for i in range(2*mod):\\n\",\n \" print(\\\" %d * %d (mod %d) = %d\\\" % (b, i, mod, (b*i) % mod))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 10,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 4 * 0 (mod 8) = 0\\n\",\n \" 4 * 1 (mod 8) = 4\\n\",\n \" 4 * 2 (mod 8) = 0\\n\",\n \" 4 * 3 (mod 8) = 4\\n\",\n \" 4 * 4 (mod 8) = 0\\n\",\n \" 4 * 5 (mod 8) = 4\\n\",\n \" 4 * 6 (mod 8) = 0\\n\",\n \" 4 * 7 (mod 8) = 4\\n\",\n \" 4 * 8 (mod 8) = 0\\n\",\n \" 4 * 9 (mod 8) = 4\\n\",\n \" 4 * 10 (mod 8) = 0\\n\",\n \" 4 * 11 (mod 8) = 4\\n\",\n \" 4 * 12 (mod 8) = 0\\n\",\n \" 4 * 13 (mod 8) = 4\\n\",\n \" 4 * 14 (mod 8) = 0\\n\",\n \" 4 * 15 (mod 8) = 4\\n\"\n ]\n }\n ],\n \"source\": [\n \"hashtest(4, 8)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Is it enough to take odd numbers then? \\n\",\n \"\\n\",\n \"With a composite number you have the additional issue where you get poor performance if a disproportionate number of keys share factors with $m$.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 12,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 15 * 0 (mod 12) = 0\\n\",\n \" 15 * 1 (mod 12) = 3\\n\",\n \" 15 * 2 (mod 12) = 6\\n\",\n \" 15 * 3 (mod 12) = 9\\n\",\n \" 15 * 4 (mod 12) = 0\\n\",\n \" 15 * 5 (mod 12) = 3\\n\",\n \" 15 * 6 (mod 12) = 6\\n\",\n \" 15 * 7 (mod 12) = 9\\n\",\n \" 15 * 8 (mod 12) = 0\\n\",\n \" 15 * 9 (mod 12) = 3\\n\",\n \" 15 * 10 (mod 12) = 6\\n\",\n \" 15 * 11 (mod 12) = 9\\n\",\n \" 15 * 12 (mod 12) = 0\\n\",\n \" 15 * 13 (mod 12) = 3\\n\",\n \" 15 * 14 (mod 12) = 6\\n\",\n \" 15 * 15 (mod 12) = 9\\n\",\n \" 15 * 16 (mod 12) = 0\\n\",\n \" 15 * 17 (mod 12) = 3\\n\",\n \" 15 * 18 (mod 12) = 6\\n\",\n \" 15 * 19 (mod 12) = 9\\n\",\n \" 15 * 20 (mod 12) = 0\\n\",\n \" 15 * 21 (mod 12) = 3\\n\",\n \" 15 * 22 (mod 12) = 6\\n\",\n \" 15 * 23 (mod 12) = 9\\n\"\n ]\n }\n ],\n \"source\": [\n \"hashtest(15, 12)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"A good choice for $m$ with the division method is a prime number.\\n\",\n \"\\n\",\n \"With a prime number you will still find poor performance if you have a disproportionate number of keys which are congruent $\\\\mod m$. However\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"#### Multiplication Method\\n\",\n \"\\n\",\n \"The multiplication method is another way to create hash functions. The functions take the form\\n\",\n \"\\n\",\n \"$$\\n\",\n \"h(k) = \\\\lfloor m(kA \\\\mod 1) \\\\rfloor\\n\",\n \"$$\\n\",\n \"\\n\",\n \"where $0 < A < 1$ and $(kA \\\\mod 1)$ refers to the fractional part of $kA$. Since $0 < (kA \\\\mod 1) <\\n\",\n \"1$, the range of $h(k)$ is from $0$ to $m-1$. The advantage of the multiplication method is it works equally well with any size $m$. $A$ should be chosen carefully. Rational numbers should not be chosen for $A$.\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 43,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def hashtest_multiplication(A, m):\\n\",\n \" for k in range(2 * m):\\n\",\n \" fraction = (k * A) % 1.0 \\n\",\n \" res = math.floor(m * fraction)\\n\",\n \" print(\\\"floor(%d * (%d * %f mod 1)) = %d\\\" % \\\\\\n\",\n \" (m, k, A, res))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 49,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"floor(10 * (0 * 0.750000 mod 1)) = 0\\n\",\n \"floor(10 * (1 * 0.750000 mod 1)) = 7\\n\",\n \"floor(10 * (2 * 0.750000 mod 1)) = 5\\n\",\n \"floor(10 * (3 * 0.750000 mod 1)) = 2\\n\",\n \"floor(10 * (4 * 0.750000 mod 1)) = 0\\n\",\n \"floor(10 * (5 * 0.750000 mod 1)) = 7\\n\",\n \"floor(10 * (6 * 0.750000 mod 1)) = 5\\n\",\n \"floor(10 * (7 * 0.750000 mod 1)) = 2\\n\",\n \"floor(10 * (8 * 0.750000 mod 1)) = 0\\n\",\n \"floor(10 * (9 * 0.750000 mod 1)) = 7\\n\",\n \"floor(10 * (10 * 0.750000 mod 1)) = 5\\n\",\n \"floor(10 * (11 * 0.750000 mod 1)) = 2\\n\",\n \"floor(10 * (12 * 0.750000 mod 1)) = 0\\n\",\n \"floor(10 * (13 * 0.750000 mod 1)) = 7\\n\",\n \"floor(10 * (14 * 0.750000 mod 1)) = 5\\n\",\n \"floor(10 * (15 * 0.750000 mod 1)) = 2\\n\",\n \"floor(10 * (16 * 0.750000 mod 1)) = 0\\n\",\n \"floor(10 * (17 * 0.750000 mod 1)) = 7\\n\",\n \"floor(10 * (18 * 0.750000 mod 1)) = 5\\n\",\n \"floor(10 * (19 * 0.750000 mod 1)) = 2\\n\"\n ]\n }\n ],\n \"source\": [\n \"hashtest_multiplication(3.0 / 4.0, 10)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"\\n\",\n \"All rational numbers can be formulated as $\\\\frac{a}{b}$ for some $a,b$. Note that $h(K) = m(k\\\\frac{a}{b} \\\\mod 1)$ only permits $b$ possible values for $h(k)$. For example if $b = 4$ then the only possible fractional parts are $.0, .25, .75$. In theory that's a bit of a pickle because all the IEEE floating point numbers are rational.\\n\",\n \"\\n\",\n \"$\\\\frac{\\\\sqrt{5} - 1}{2}$ is irrational and related to the golden ratio. Chosing this value for $A$ is a special form of hashing known as fibonacci hashing. This has the property that when hashing consecutive keys, each subsequent key falls in between the two widest spaced hash values already computed.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 50,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"floor(10 * (0 * 0.618034 mod 1)) = 0\\n\",\n \"floor(10 * (1 * 0.618034 mod 1)) = 6\\n\",\n \"floor(10 * (2 * 0.618034 mod 1)) = 2\\n\",\n \"floor(10 * (3 * 0.618034 mod 1)) = 8\\n\",\n \"floor(10 * (4 * 0.618034 mod 1)) = 4\\n\",\n \"floor(10 * (5 * 0.618034 mod 1)) = 0\\n\",\n \"floor(10 * (6 * 0.618034 mod 1)) = 7\\n\",\n \"floor(10 * (7 * 0.618034 mod 1)) = 3\\n\",\n \"floor(10 * (8 * 0.618034 mod 1)) = 9\\n\",\n \"floor(10 * (9 * 0.618034 mod 1)) = 5\\n\",\n \"floor(10 * (10 * 0.618034 mod 1)) = 1\\n\",\n \"floor(10 * (11 * 0.618034 mod 1)) = 7\\n\",\n \"floor(10 * (12 * 0.618034 mod 1)) = 4\\n\",\n \"floor(10 * (13 * 0.618034 mod 1)) = 0\\n\",\n \"floor(10 * (14 * 0.618034 mod 1)) = 6\\n\",\n \"floor(10 * (15 * 0.618034 mod 1)) = 2\\n\",\n \"floor(10 * (16 * 0.618034 mod 1)) = 8\\n\",\n \"floor(10 * (17 * 0.618034 mod 1)) = 5\\n\",\n \"floor(10 * (18 * 0.618034 mod 1)) = 1\\n\",\n \"floor(10 * (19 * 0.618034 mod 1)) = 7\\n\"\n ]\n }\n ],\n \"source\": [\n \"golden_ratio = (math.sqrt(5) - 1.0) / 2.0\\n\",\n \"hashtest_multiplication(golden_ratio, 10)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## How Python implements their hashing?\\n\",\n \"\\n\",\n \"Which idea does Python employ? Surely not division method. Multiplication method? Well, we managed to show it's drawbacks. Some clever cryptographic algorithm?\\n\",\n \"\\n\",\n \"...\\n\",\n \"\\n\",\n \"... It uses division method. Which prime number are they using?\\n\",\n \"\\n\",\n \"...\\n\",\n \"\\n\",\n \"... They are using powers of two. Ba dum tsh.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 27,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def test_dict(base):\\n\",\n \" d = {}\\n\",\n \" for i in range(1000000):\\n\",\n \" d[base*i] = True\\n\",\n \" print(len(d))\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 39,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"1000000\\n\",\n \" 20 function calls in 0.257 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.240 0.240 0.240 0.240 :1(test_dict)\\n\",\n \" 1 0.017 0.017 0.257 0.257 :1()\\n\",\n \" 2 0.000 0.000 0.000 0.000 iostream.py:102(_check_mp_mode)\\n\",\n \" 2 0.000 0.000 0.000 0.000 iostream.py:207(write)\\n\",\n \" 2 0.000 0.000 0.000 0.000 iostream.py:93(_is_master_process)\\n\",\n \" 1 0.000 0.000 0.257 0.257 {built-in method exec}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {built-in method getpid}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {built-in method isinstance}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {built-in method len}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {built-in method print}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {built-in method time}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {method 'write' of '_io.StringIO' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"test_dict(2*32-1)\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 40,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"1000000\\n\",\n \" 20 function calls in 0.930 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.879 0.879 0.879 0.879 :1(test_dict)\\n\",\n \" 1 0.051 0.051 0.930 0.930 :1()\\n\",\n \" 2 0.000 0.000 0.000 0.000 iostream.py:102(_check_mp_mode)\\n\",\n \" 2 0.000 0.000 0.000 0.000 iostream.py:207(write)\\n\",\n \" 2 0.000 0.000 0.000 0.000 iostream.py:93(_is_master_process)\\n\",\n \" 1 0.000 0.000 0.930 0.930 {built-in method exec}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {built-in method getpid}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {built-in method isinstance}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {built-in method len}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {built-in method print}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {built-in method time}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \" 2 0.000 0.000 0.000 0.000 {method 'write' of '_io.StringIO' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"cProfile.run(\\\"test_dict(2**32)\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Yeah, again we managed to exploit the knowledge of Python, to achieve sigificant runtime discrepancy. It is not exactly the differenc between $O(n)$ and $O(n^2)$ as one might expect, because Python hash quite clever mechanisms in place to recover from this siliness. More details during next lecture and here:\\n\",\n \"http://stackoverflow.com/questions/9010222/how-can-python-dict-have-multiple-keys-with-same-hash\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Collisions\\n\",\n \"As we mentioned, we do not want our hash functions to have a lot collisions. But collisions are unavoidable. Think of the \\\\textbf{Pigeon-hole principle}. Usually the size of universe $U$ is much larger than our table size $m$. What do we do if we find that two keys hash to the same value? Two ways which we learn in this class are \\\\textbf{chaining} and \\\\textbf{open addressing}.\\n\",\n \"\\n\",\n \"### Chaining\\n\",\n \"Instead of just storing the elements in the slots in the table, let every slot be a linked list which contains all the elements which are in the table and map to that slot. Our operations now become:\\n\",\n \"\\n\",\n \"- `Insert` $(k,v)$: hash $k$ to an index $i$ in the table; add $k$ along with $v$ to the linked list at that location.\\n\",\n \"- `Search` $(k)$: search for $k$ in the linked list by iterating through all the list.\\n\",\n \"- `Delete` $(k)$: search for $k$ and then remove it from the list.\\n\",\n \"\\n\",\n \"These operations no longer take $O(1)$ time. Lookup on the linked lists takes $O(l)$ time where $l$ is the size of the linked list. We define $\\\\alpha = \\\\frac{n}{m}$ as the **load factor**. If we assume simple uniform hashing, then each element has equal probability to go into any slot. So after $n$ independent elements have been inserted we have and expected length of $\\\\frac{n}{m} = \\\\alpha$ for each chain by linearity of expectation. So the run time of all the above operations is time to hash + time to do these operations which is $O(1 + \\\\alpha)$.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"**Note:** It is possible to have expected $O(1 + \\\\alpha)$ runtime for these operations on any given input (i.e. input chosen to make our algorithm perform poorly). This requires more sophisticated hash functions (See \\\"Universal Hashing\\\" in CLRS 11.3.3)\\n\",\n \"\\n\",\n \"If we assume that $m = O(n)$, then $\\\\alpha = O(1)$ and we get constant time operations. But what if we want to insert more elements into the hash table and we don't know the number of elements to be inserted before hand? Stay tuned...\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 72,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"class LLNode(object):\\n\",\n \" def __init__(self, key, value, next_node):\\n\",\n \" self.key, self.value, self.next_node = \\\\\\n\",\n \" key, value, next_node\\n\",\n \" \\n\",\n \"class LinkedList(object):\\n\",\n \" def __init__(self):\\n\",\n \" \\\"\\\"\\\"Key-value storage over Linked List\\n\",\n \" \\n\",\n \" unique values per key\\\"\\\"\\\"\\n\",\n \" self.root = None\\n\",\n \" \\n\",\n \" def find(self, key):\\n\",\n \" \\\"\\\"\\\"Returns node given key\\\"\\\"\\\"\\n\",\n \" # find a node with a given key,\\n\",\n \" # by following links: O(n)\\n\",\n \" node = self.root\\n\",\n \" while node is not None:\\n\",\n \" if node.key == key:\\n\",\n \" return node\\n\",\n \" node = node.next_node\\n\",\n \" \\n\",\n \" def __getitem__(self, key):\\n\",\n \" \\\"\\\"\\\"Returns value for a given key\\\"\\\"\\\"\\n\",\n \" node = self.find(key)\\n\",\n \" if node is not None:\\n\",\n \" return node.value\\n\",\n \" else:\\n\",\n \" return None\\n\",\n \" \\n\",\n \" def __setitem__(self, key, value):\\n\",\n \" \\\"\\\"\\\"Sets key given value\\\"\\\"\\\"\\n\",\n \" node = self.find(key)\\n\",\n \" if node is not None:\\n\",\n \" # modify existing node\\n\",\n \" node.value = value\\n\",\n \" else:\\n\",\n \" # append front\\n\",\n \" self.root = LLNode(key, value, self.root)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 73,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"five None\\n\"\n ]\n }\n ],\n \"source\": [\n \"ll = LinkedList()\\n\",\n \"ll[5] = \\\"fiveo\\\"\\n\",\n \"ll[4] = \\\"four\\\"\\n\",\n \"ll[5] = \\\"five\\\"\\n\",\n \"print(ll[5], ll[6])\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 65,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def test_datastructure(ds, size=3000):\\n\",\n \" for i in range(size):\\n\",\n \" ds[i] = str(i)\\n\",\n \" for i in range(size):\\n\",\n \" temp = ds[i]\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 81,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 15004 function calls in 1.328 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.007 0.007 1.328 1.328 :1(test_datastructure)\\n\",\n \" 6000 1.311 0.000 1.311 0.000 :12(find)\\n\",\n \" 3000 0.003 0.000 0.003 0.000 :2(__init__)\\n\",\n \" 3000 0.001 0.000 0.633 0.000 :22(__getitem__)\\n\",\n \" 3000 0.006 0.000 0.688 0.000 :30(__setitem__)\\n\",\n \" 1 0.000 0.000 1.328 1.328 :1()\\n\",\n \" 1 0.000 0.000 1.328 1.328 {built-in method exec}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"ll = LinkedList()\\n\",\n \"\\n\",\n \"cProfile.run(\\\"test_datastructure(ll)\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 77,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"class HashTable(object):\\n\",\n \" def __init__(self, num_slots):\\n\",\n \" \\\"\\\"\\\"Initializes chained hash table with num_slots slots\\\"\\\"\\\"\\n\",\n \" self.num_slots = num_slots\\n\",\n \" self.table = [ LinkedList() for _ in range(self.num_slots)]\\n\",\n \" self.golden = (math.sqrt(5) - 1.0) / 2.0\\n\",\n \" \\n\",\n \" def multiplication_address(self, key):\\n\",\n \" \\\"\\\"\\\"Multiplication adressing described above\\\"\\\"\\\"\\n\",\n \" # hash(k) maps any Python object that implements it\\n\",\n \" # to a 64-bit integer. It is not necessarily in the \\n\",\n \" # range of our hashtable.\\n\",\n \" fraction = (hash(key) * self.golden) % 1.0 \\n\",\n \" return math.floor(self.num_slots * fraction)\\n\",\n \" \\n\",\n \" def __getitem__(self, key):\\n\",\n \" relevant_list = self.table[self.multiplication_address(key)]\\n\",\n \" return relevant_list[key]\\n\",\n \"\\n\",\n \" def __setitem__(self, key, value):\\n\",\n \" relevant_list = self.table[self.multiplication_address(key)]\\n\",\n \" relevant_list[key] = value\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 78,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"five None\\n\"\n ]\n }\n ],\n \"source\": [\n \"ht = HashTable(2)\\n\",\n \"ht[5] = \\\"fiveo\\\"\\n\",\n \"ht[4] = \\\"four\\\"\\n\",\n \"ht[5] = \\\"five\\\"\\n\",\n \"print(ht[5], ht[6])\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 79,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 39004 function calls in 0.066 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.011 0.011 0.066 0.066 :1(test_datastructure)\\n\",\n \" 6000 0.009 0.000 0.009 0.000 :12(find)\\n\",\n \" 3000 0.004 0.000 0.004 0.000 :2(__init__)\\n\",\n \" 3000 0.003 0.000 0.007 0.000 :22(__getitem__)\\n\",\n \" 3000 0.009 0.000 0.018 0.000 :30(__setitem__)\\n\",\n \" 3000 0.005 0.000 0.019 0.000 :16(__getitem__)\\n\",\n \" 3000 0.008 0.000 0.037 0.000 :20(__setitem__)\\n\",\n \" 6000 0.012 0.000 0.017 0.000 :8(multiplication_address)\\n\",\n \" 1 0.000 0.000 0.066 0.066 :1()\\n\",\n \" 1 0.000 0.000 0.066 0.066 {built-in method exec}\\n\",\n \" 6000 0.003 0.000 0.003 0.000 {built-in method floor}\\n\",\n \" 6000 0.001 0.000 0.001 0.000 {built-in method hash}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"ht = HashTable(500)\\n\",\n \"\\n\",\n \"cProfile.run(\\\"test_datastructure(ht)\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"### Hash table with one slot (just like linked list + overhead)\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 80,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 39004 function calls in 1.426 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.007 0.007 1.426 1.426 :1(test_datastructure)\\n\",\n \" 6000 1.375 0.000 1.375 0.000 :12(find)\\n\",\n \" 3000 0.015 0.000 0.015 0.000 :2(__init__)\\n\",\n \" 3000 0.002 0.000 0.710 0.000 :22(__getitem__)\\n\",\n \" 3000 0.006 0.000 0.688 0.000 :30(__setitem__)\\n\",\n \" 3000 0.004 0.000 0.719 0.000 :16(__getitem__)\\n\",\n \" 3000 0.005 0.000 0.699 0.000 :20(__setitem__)\\n\",\n \" 6000 0.008 0.000 0.012 0.000 :8(multiplication_address)\\n\",\n \" 1 0.000 0.000 1.426 1.426 :1()\\n\",\n \" 1 0.000 0.000 1.426 1.426 {built-in method exec}\\n\",\n \" 6000 0.002 0.000 0.002 0.000 {built-in method floor}\\n\",\n \" 6000 0.001 0.000 0.001 0.000 {built-in method hash}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"ht = HashTable(1)\\n\",\n \"\\n\",\n \"cProfile.run(\\\"test_datastructure(ht)\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Super efficient Python hashtable\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 83,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \" 4 function calls in 0.004 seconds\\n\",\n \"\\n\",\n \" Ordered by: standard name\\n\",\n \"\\n\",\n \" ncalls tottime percall cumtime percall filename:lineno(function)\\n\",\n \" 1 0.003 0.003 0.003 0.003 :1(test_datastructure)\\n\",\n \" 1 0.000 0.000 0.004 0.004 :1()\\n\",\n \" 1 0.000 0.000 0.004 0.004 {built-in method exec}\\n\",\n \" 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}\\n\",\n \"\\n\",\n \"\\n\"\n ]\n }\n ],\n \"source\": [\n \"# Implemented in C...\\n\",\n \"cProfile.run(\\\"test_datastructure({})\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## A fun annecdote from Professor Indyk\\n\",\n \"\\n\",\n \"As a student in 1999, Professor Indyk was trying to large set of webpages (i.e. group pages with similar content). He hashed all the words in the sites to reduce space usage. There were two issues.\\n\",\n \"\\n\",\n \"- His algorithm clustered one of his advisor's home page with some rather shady websites.\\n\",\n \"- His algorithm was provably correct, with probabily of failure less than $10^{-6}$\\n\",\n \"\\n\",\n \"The implementation was for a word $x$ compute $h(x) = (ax \\\\mod P) \\\\mod 2^{8}$, $P = 2^{64}-59$, randomly chosen $a$. Additionally, the only used words divisible by $8$ for speed purposes. It turns out that the language they were using computed $ax$ as $ax \\\\mod 2^{64}$ because of the word size. This means the $\\\\mod P$ operation essentially did nothing. So $x | 8 \\\\implies ax | 8 \\\\implies ax \\\\mod 2^8 | 8$. All $h(x)$ had the 3 lowest order bits as zero! The total range for $h(x)$ was $2^{5}$ instead of $2^{8}$ causing word colisions.\\n\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Practice\\n\",\n \"\\n\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Duplicate Detection\\n\",\n \"\\n\",\n \"Given an array $A$ of $n$ integers and an integer $k$, detect if there is an entry $A[i]$ that is equal to one of the $k$ previous entries $A[i-1] \\\\ldots A[i-k]$.\\n\",\n \"Your algorithm should run in time $O(n)$.\\n\",\n \"You can assume you have access to a hash function which satisfies the simple uniform hashing assumption (SUHA).\\n\",\n \"\\n\",\n \"**Example:** Given an array `A=[1,3,5,7,6,5,2]` and $k=4$, the algorithm should output YES since `A[3]=A[6]=5`.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Point Lookup\\n\",\n \"Design a data structure to support the following operations on points in a plane.\\n\",\n \"You can assume you have access to a hash function which satisfies the simple uniform hashing assumption (SUHA).\\n\",\n \"Additionally, you can assume you know $n$, an upper bound on the total number of elements to ever be inserted into the structure.\\n\",\n \"Runtimes can be worst-case or expected time.\\n\",\n \"Your data structure should use $O(n)$ space.\\n\",\n \"\\n\",\n \"\\n\",\n \"- **Query($x$)** Of all the points with $x$-coordinate equal to $x$, return the one with the lowest $y$ coordinate. This should run in $O(1)$ time.\\n\",\n \"\\n\",\n \"- **Insert($x, y$)** Insert the point $(x, y)$ into the structure. This should run in $O(\\\\log{n})$ time.\\n\",\n \"\\n\",\n \"- **Delete($x, y$)** Remove the point $(x, y)$ from the structure. This should run in $O(\\\\log{n})$ time.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": null,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": []\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 3\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.4.1\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" }, { "path": "lecture9/Hashing.ipynb", "content": "{\n \"cells\": [\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"# Rabin-Karp method\\n\",\n \"\\n\",\n \"If you think hard enough, there's nothing that diferrentiates piecs of text from numbers. You can think of letters as digits and base of the numbers as sufficently big to accomodate for all the digits. For example take the following text\\n\",\n \"\\n\",\n \"$$\\n\",\n \"babacb\\n\",\n \"$$\\n\",\n \"\\n\",\n \"it can be though of as a number base 26 (for all the english letters):\\n\",\n \"\\n\",\n \"$$\\n\",\n \"(1,0,1,0,2,1)_{26}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"We can transform this number to base 10 using the following equation.\\n\",\n \"\\n\",\n \"$$\\n\",\n \"1*26^5 + 0 * 26^4 + 1*26^3 + 0*26^2 + 2*26^1 + 1*26^0 = 11899005\\n\",\n \"$$\\n\",\n \"\\n\",\n \"From the formulation above the following property should be clear.\\n\",\n \"\\n\",\n \"$$\\n\",\n \"abba = abb * 26 + b\\n\",\n \"$$\\n\",\n \"\\n\",\n \"in general we can write $concat(word, letter) = base * word + letter$ (1).\\n\",\n \"\\n\",\n \"There's also a small technicality. When we compare numbers then $0001$ and $001$ and $1$ are equivalent. This means that we cannot map any letter to 0 if we want to be able to successfuly compare the numbers. \"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Observation (1) allows us to quickly compute hashes for all the prefixes of a given word. Just like in class we are going to use modular arithmetic for our computations.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 8,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"'A' 65\\n\",\n \"'a' 97\\n\",\n \"'b' 98\\n\",\n \"'c' 99\\n\",\n \"' ' 32\\n\",\n \"'c' - 'a' + 1 = 3\\n\"\n ]\n }\n ],\n \"source\": [\n \"# we need to map letters to numbers. Python function ord does the job\\n\",\n \"print(repr('A'), ord('A'))\\n\",\n \"print(repr('a'), ord('a'))\\n\",\n \"print(repr('b'), ord('b'))\\n\",\n \"print(repr('c'), ord('c'))\\n\",\n \"print(repr(' '), ord(' '))\\n\",\n \"\\n\",\n \"print('%s - %s + 1 = %d' % (repr('c'), repr('a'), ord('c') - ord('a') + 1))\\n\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 17,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"BIG_FAT_PRIME = 2**32 - 1\\n\",\n \"ENGLISH_BASE = 30 # in theory 27 is sufficient but better safe than sorry!\\n\",\n \"\\n\",\n \"def compute_hashes(text, base=ENGLISH_BASE, modulo = BIG_FAT_PRIME):\\n\",\n \" # \\n\",\n \" h = [None for _ in range(len(text) + 1)]\\n\",\n \" h[0] = 0 # hash of empty word is 0\\n\",\n \" for i in range(len(text)):\\n\",\n \" # we only deal with english letters so we subtract 'a'\\n\",\n \" # to normalize range. We add 0 to avoid creating zero digit.\\n\",\n \" letter_as_number = (ord(text[i]) - ord('a') + 1)\\n\",\n \" h[i + 1] = h[i] * base + letter_as_number\\n\",\n \" h[i + 1] %= modulo\\n\",\n \" # at the end of the iteration h[i+1] is the hash\\n\",\n \" # of prefix of text of lenght (i+1) which in\\n\",\n \" # Python is text[:(i+1)]\\n\",\n \" return h\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"PROTIP: If you happen to ever implemented this is lower level programming language like C or C++, be ware of integer overflows.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 18,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[0, 2, 61, 1832, 54961, 1648833, 49464992]\"\n ]\n },\n \"execution_count\": 18,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"compute_hashes(\\\"babacb\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 19,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[0, 2, 61, 1832, 54964, 1648924, 49467724]\"\n ]\n },\n \"execution_count\": 19,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"compute_hashes(\\\"babddd\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 20,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[0,\\n\",\n \" 2,\\n\",\n \" 61,\\n\",\n \" 1832,\\n\",\n \" 54964,\\n\",\n \" 1648924,\\n\",\n \" 49467721,\\n\",\n \" 1484031649,\\n\",\n \" 1571276524,\\n\",\n \" 4188622771,\\n\",\n \" 1104631594,\\n\",\n \" 3074176759,\\n\",\n \" 2030989594,\\n\",\n \" 800145691,\\n\",\n \" 2529534259]\"\n ]\n },\n \"execution_count\": 20,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"# for longer strings modulo matters\\n\",\n \"compute_hashes(\\\"babddasdasdsad\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## hashes of substrings\\n\",\n \"\\n\",\n \"Now here's a crutial observation. Let's take polynomial representation of string $babacb$ (where $X$ is the base)\\n\",\n \"\\n\",\n \"$$\\n\",\n \"b*X^5 + a * X^4 + b*X^3 + a*X^2 + c*X^1 + b*X^0 \\n\",\n \"$$\\n\",\n \"\\n\",\n \"Say we want to compute hash of $ac$ which is conveniently appears on 4-th index the string we originally hashed. Moreover we have hashes of all the prefixes - it seems like we are in good shape:\\n\",\n \"\\n\",\n \"\\\\begin{align}\\n\",\n \"\\\\text{we have }\\\\ \\\\ \\\\ & hash(babac) &=\\\\ & b*X^4 + a * X^3 + b*X^2 &+& a*X^1 + c*X^0 \\\\\\\\\\n\",\n \"\\\\text{we have }\\\\ \\\\ \\\\ & hash(bab) &=\\\\ & b*X^2 + a * X^1 + b*X^0&&\\\\\\\\\\n\",\n \"\\\\text{we WANT }\\\\ \\\\ \\\\ & hash(ac) &=\\\\ & && a*X^1 + c*X^0\\\\\\\\\\n\",\n \"\\\\end{align}\\n\",\n \"\\n\",\n \"\\n\",\n \"From above we can clearly see that:\\n\",\n \"\\n\",\n \"$$\\n\",\n \"hash(ac) = hash(babac) - X^2 * hash(bab)\\n\",\n \"$$\\n\",\n \"\\n\",\n \"We can generalize this to arbitrary substring of our hashed string. Assume we hashed string $s_0, s_1, ..., s_{n-1}$ such that $h_0 = hash(\\\\emptyset)$, $h_1 = hash(s_0)$, $h_2 = hash(s_0, s_1)$ etc. \\n\",\n \"Then we can compute $hash(s_i, ..., s_j)$ using the following formula:\\n\",\n \"\\n\",\n \"$$\\n\",\n \"hash(s_i, ..., s_{j-1}) = h_j - h_i * X ^{j - i}\\n\",\n \"$$\\n\",\n \"\\n\",\n \"This looks very close to $O(1)$ complexity hash computation if not for $X ^{j - i}$. But since there are at most $n$ different powers of $X$ that we are interested in, we can precompute them in $O(n)$ time.\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 37,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"def compute_powers(n, base=ENGLISH_BASE, modulo=BIG_FAT_PRIME):\\n\",\n \" powers = [None for _ in range(n + 1)]\\n\",\n \" powers[0] = 1\\n\",\n \" for i in range(n):\\n\",\n \" powers[i+1] = (powers[i] * base) % modulo\\n\",\n \" return powers\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 38,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[1,\\n\",\n \" 30,\\n\",\n \" 900,\\n\",\n \" 27000,\\n\",\n \" 810000,\\n\",\n \" 24300000,\\n\",\n \" 729000000,\\n\",\n \" 395163525,\\n\",\n \" 3264971160,\\n\",\n \" 3459854310,\\n\",\n \" 716414220]\"\n ]\n },\n \"execution_count\": 38,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"compute_powers(10)\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Now we can put all those observations together into efficient detastructure that allows us to compute hashes of substrings in $O(1)$\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 77,\n \"metadata\": {\n \"collapsed\": true\n },\n \"outputs\": [],\n \"source\": [\n \"class Hasher(object):\\n\",\n \" def __init__(self, word):\\n\",\n \" self.h = compute_hashes(word)\\n\",\n \" self.powers = compute_powers(len(word))\\n\",\n \" \\n\",\n \" def substring_hash(self, i, j):\\n\",\n \" result = self.h[j] - self.h[i] * self.powers[j-i]\\n\",\n \" return result % BIG_FAT_PRIME\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 78,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"TEXT = \\\"abcxabcx\\\"\\n\",\n \"h = Hasher(TEXT)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 79,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"name\": \"stdout\",\n \"output_type\": \"stream\",\n \"text\": [\n \"[ab]cxabcx 32\\n\",\n \"abcx[ab]cx 32\\n\",\n \"abc[xa]bcx 721\\n\"\n ]\n }\n ],\n \"source\": [\n \"def highlight(word, i, j):\\n\",\n \" return word[:i] + \\\"[\\\" + word[i:j] + \\\"]\\\" + word[j:]\\n\",\n \"\\n\",\n \"print(highlight(TEXT, 0, 2), h.substring_hash(0, 2))\\n\",\n \"print(highlight(TEXT, 4, 6), h.substring_hash(4, 6))\\n\",\n \"print(highlight(TEXT, 3, 5), h.substring_hash(3, 5))\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Hasher complexity analysis.\\n\",\n \"\\n\",\n \"Preprocessing (`__init__`):\\n\",\n \"- `compute_hashes` is $O(n)$\\n\",\n \"- 'compute_powers` is $O(n)$\\n\",\n \"Therefore precomputing is $O(n)$.\\n\",\n \"\\n\",\n \"Queries (`substring_hash`) is of complexity $O(1)$ - it is just a simple formula.\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"Notice that this technique is very powerful. More powerful than we need for pattern matching. It should not be a surprise that we can easily use it to solve pattern matching\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 100,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"# hasher for text\\n\",\n \"text_h = Hasher(\\\"to be or not to be\\\")\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 101,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"65\"\n ]\n },\n \"execution_count\": 101,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"# hash of the pattern\\n\",\n \"compute_hashes(\\\"be\\\")[-1]\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 102,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"65\"\n ]\n },\n \"execution_count\": 102,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"# hash of the occurence of \\\"be\\\" in original text. \\n\",\n \"text_h.substring_hash(3, 5)\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 105,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [],\n \"source\": [\n \"def compute_matches(text, pattern):\\n\",\n \" # hash of patter\\n\",\n \" pattern_hash = compute_hashes(pattern)[-1]\\n\",\n \" # hasher for text\\n\",\n \" text_h = Hasher(text)\\n\",\n \" res = []\\n\",\n \" for i in range(len(text) - len(pattern) + 1):\\n\",\n \" # i is potential match start index\\n\",\n \" # compare hash in text with hash of pattern\\n\",\n \" if text_h.substring_hash(i, i + len(pattern)) == pattern_hash:\\n\",\n \" # if matching append to result list.\\n\",\n \" res.append(i)\\n\",\n \" return res\"\n ]\n },\n {\n \"cell_type\": \"code\",\n \"execution_count\": 106,\n \"metadata\": {\n \"collapsed\": false\n },\n \"outputs\": [\n {\n \"data\": {\n \"text/plain\": [\n \"[3, 16]\"\n ]\n },\n \"execution_count\": 106,\n \"metadata\": {},\n \"output_type\": \"execute_result\"\n }\n ],\n \"source\": [\n \"compute_matches(\\\"to be or not to be\\\", \\\"be\\\")\"\n ]\n },\n {\n \"cell_type\": \"markdown\",\n \"metadata\": {},\n \"source\": [\n \"## Rabin-Karp complexity analysis\\n\",\n \"\\n\",\n \"Assume that pattern is of length $n$ and text of length $m$\\n\",\n \"\\n\",\n \"- pattern hash: $O(n)$\\n\",\n \"- text preprocessing $O(m)$.\\n\",\n \"- $n - m$ iterations of main loop with $O(1)$ hash computation in each loop\\n\",\n \"\\n\",\n \"total: $O(n+m)$\\n\",\n \"\\n\",\n \"Catch? Relies on luck.\"\n ]\n }\n ],\n \"metadata\": {\n \"kernelspec\": {\n \"display_name\": \"Python 3\",\n \"language\": \"python\",\n \"name\": \"python3\"\n },\n \"language_info\": {\n \"codemirror_mode\": {\n \"name\": \"ipython\",\n \"version\": 3\n },\n \"file_extension\": \".py\",\n \"mimetype\": \"text/x-python\",\n \"name\": \"python\",\n \"nbconvert_exporter\": \"python\",\n \"pygments_lexer\": \"ipython3\",\n \"version\": \"3.4.1\"\n }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n" } ]