Repository: google-deepmind/alphaevolve_results Branch: main Commit: 4226acbf237f Files: 5 Total size: 1.3 MB Directory structure: gitextract_avi_5b8y/ ├── .gitignore ├── CONTRIBUTING.md ├── LICENSE ├── README.md └── mathematical_results.ipynb ================================================ FILE CONTENTS ================================================ ================================================ FILE: .gitignore ================================================ # Byte-compiled / optimized / DLL files __pycache__/ *.py[cod] *$py.class # Distribution / packaging .Python build/ develop-eggs/ dist/ downloads/ eggs/ .eggs/ lib/ lib64/ parts/ sdist/ var/ wheels/ share/python-wheels/ *.egg-info/ .installed.cfg *.egg MANIFEST ================================================ FILE: CONTRIBUTING.md ================================================ # How to Contribute ## Contributor License Agreement Contributions to this project must be accompanied by a Contributor License Agreement. 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We also recommend that a file or class name and description of purpose be included on the same "printed page" as the copyright notice for easier identification within third-party archives. Copyright [yyyy] [name of copyright owner] Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. ================================================ FILE: README.md ================================================ # Results of AlphaEvolve This repository accompanies the technical report > Novikov et al. [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://storage.googleapis.com/deepmind-media/DeepMind.com/Blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/AlphaEvolve.pdf). *Technical report*. 2025. Specifically, the repository contains a Google Colab notebook with the mathematical discoveries of AlphaEvolve outlined in Section 3 of the paper, as well as the corresponding code for verifying their correctness. The notebook is divided into sections, one for each of the considered mathematical problems. Note that the notebook contains only the instances where the results from AlphaEvolve outperforms the state-of-the-art. To avoid clutter, we exclude those problems where the results matched but did not outperform the best known constructions. This repository does *not* contain the code to run AlphaEvolve. ## Installation No installation required. The provided notebook can be opened and run in Google Colab. ## Usage The notebook `mathematical_results.ipynb` can be opened via [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/google-deepmind/alphaevolve_results/blob/master/mathematical_results.ipynb). ## Citing this work If you use the code or data in this package, please cite: ``` @article{alphaevolve, title={Alpha{E}volve: A coding agent for scientific and algorithmic discovery}, author={Novikov, Alexander and V\~{u}, Ng\^{a}n and Eisenberger, Marvin and Dupont, Emilien and Huang, Po-Sen and Wagner, Adam Zsolt and Shirobokov, Sergey and Kozlovskii, Borislav and Ruiz, Francisco J. R. and Mehrabian, Abbas and Kumar, M. Pawan and See, Abigail and Chaudhuri, Swarat and Holland, George and Davies, Alex and Nowozin, Sebastian and Kohli, Pushmeet and Balog, Matej}, year={2025}, journal={arXiv preprint arXiv:2506.13131} } ``` ## License and disclaimer Copyright 2025 Google LLC All software is licensed under the Apache License, Version 2.0 (Apache 2.0); you may not use this file except in compliance with the Apache 2.0 license. You may obtain a copy of the Apache 2.0 license at: https://www.apache.org/licenses/LICENSE-2.0 All other materials are licensed under the Creative Commons Attribution 4.0 International License (CC-BY). You may obtain a copy of the CC-BY license at: https://creativecommons.org/licenses/by/4.0/legalcode Unless required by applicable law or agreed to in writing, all software and materials distributed here under the Apache 2.0 or CC-BY licenses are distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the licenses for the specific language governing permissions and limitations under those licenses. This is not an official Google product. ================================================ FILE: mathematical_results.ipynb ================================================ { "cells": [ { "cell_type": "markdown", "metadata": { "id": "bC4Vo85hPHfn" }, "source": [ "Copyright 2025 Google DeepMind Technologies Limited\n", "\n", "All software is licensed under the Apache License, Version 2.0 (Apache 2.0);\n", "you may not use this file except in compliance with the Apache 2.0 license.\n", "You may obtain a copy of the Apache 2.0 license at:\n", "https://www.apache.org/licenses/LICENSE-2.0\n", "\n", "All other materials are licensed under the Creative Commons Attribution 4.0\n", "International License (CC-BY). You may obtain a copy of the CC-BY license at:\n", "https://creativecommons.org/licenses/by/4.0/legalcode\n", "\n", "Unless required by applicable law or agreed to in writing, all software and\n", "materials distributed here under the Apache 2.0 or CC-BY licenses are\n", "distributed on an \"AS IS\" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND,\n", "either express or implied. See the licenses for the specific language governing\n", "permissions and limitations under those licenses.\n", "\n", "This is not an official Google product." ] }, { "cell_type": "markdown", "metadata": { "id": "5RChGCn7d8eP" }, "source": [ "# > **[Please read me first](https://github.com/google-deepmind/alphaevolve_results/blob/main/README.md)**" ] }, { "cell_type": "markdown", "metadata": { "id": "DUBRLLV4dJdV" }, "source": [ "# Tensor decompositions (algorithms for matrix multiplication)" ] }, { "cell_type": "markdown", "metadata": { "id": "VbmY-XVldLBX" }, "source": [ "This section contains several low-rank tensor decompositions, discovered by AlphaEvolve, and the code to verify their correctness. A tensor with parameters $\\langle n,m,p\\rangle$ corresponds to multiplying an $n \\times m$ and an $m \\times p$ matrix, and the rank of the decomposition corresponds to the number of scalar multiplications needed to perform the matrix multiplication." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "JCmE5JpLeJ2B" }, "outputs": [], "source": [ "#@title Verification function\n", "import numpy as np\n", "\n", "def verify_tensor_decomposition(decomposition: tuple[np.ndarray, np.ndarray, np.ndarray], n: int, m: int, p: int, rank: int):\n", " \"\"\"Verifies the correctness of the tensor decomposition.\n", "\n", " Args:\n", " decomposition: a tuple of 3 factor matrices with the same number of columns.\n", " (The number of columns specifies the rank of the decomposition.) To\n", " construct a tensor, we take the outer product of the i-th column of the\n", " three factor matrices, for 1 <= i <= rank, and add up all these outer\n", " products.\n", " n: the first parameter of the matrix multiplication tensor.\n", " m: the second parameter of the matrix multiplication tensor.\n", " p: the third parameter of the matrix multiplication tensor.\n", " rank: the expected rank of the decomposition.\n", "\n", " Raises:\n", " AssertionError: If the decomposition does not have the correct rank, or if\n", " the decomposition does not construct the 3D tensor which corresponds to\n", " multiplying an n x m matrix by an m x p matrix.\n", " \"\"\"\n", " # Check that each factor matrix has the correct shape.\n", " factor_matrix_1, factor_matrix_2, factor_matrix_3 = decomposition\n", " assert factor_matrix_1.shape == (n * m, rank), f'Expected shape of factor matrix 1 is {(n * m, rank)}. Actual shape is {factor_matrix_1.shape}.'\n", " assert factor_matrix_2.shape == (m * p, rank), f'Expected shape of factor matrix 1 is {(m * p, rank)}. Actual shape is {factor_matrix_2.shape}.'\n", " assert factor_matrix_3.shape == (p * n, rank), f'Expected shape of factor matrix 1 is {(p * n, rank)}. Actual shape is {factor_matrix_3.shape}.'\n", "\n", " # Form the matrix multiplication tensor .\n", " matmul_tensor = np.zeros((n * m, m * p, p * n), dtype=np.int32)\n", " for i in range(n):\n", " for j in range(m):\n", " for k in range(p):\n", " matmul_tensor[i * m + j][j * p + k][k * n + i] = 1\n", "\n", " # Check that the tensor is correctly constructed.\n", " constructed_tensor = np.einsum('il,jl,kl -> ijk', *decomposition)\n", " assert np.array_equal(constructed_tensor, matmul_tensor), f'Tensor constructed by decomposition does not match the matrix multiplication tensor <{(n,m,p)}>: {constructed_tensor}.'\n", " print(f'Verified a decomposition of rank {rank} for matrix multiplication tensor <{n},{m},{p}>.')\n", "\n", " # Print the set of values used in the decomposition.\n", " np.set_printoptions(linewidth=100)\n", " print('This decomposition uses these factor entries:\\n', np.array2string(np.unique(np.vstack((factor_matrix_1, factor_matrix_2, factor_matrix_3))), separator=', '))" ] }, { "cell_type": "markdown", "metadata": { "id": "4nYzMG9peJ2C" }, "source": [ "## Rank-32 decomposition of <2,4,5> over 0.5*Z" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "LICVJaCBeJ2C" }, "outputs": [], "source": [ "#@title Data\n", "\n", "decomposition_245 = (np.array([[-0. , 0.5, -0. , 0.5, 0. , 0. , 0.5, 0. , -0.5, -0.5, -0. ,\n", " -0.5, -0. , -0. , 0.5, 0. , 0.5, -1. , -0.5, -0. , -0.5, -0. ,\n", " 0.5, -0. , -0.5, 0.5, -1. , 0. , 0.5, 0. , 0.5, -0. ],\n", " [ 0.5, -0.5, 0. , 0.5, -0. , 0.5, -0.5, -0. , 0. , -0.5, 0. ,\n", " 0. , -0.5, 0.5, 0. , 0.5, 0. , -0. , -0. , -0.5, -0. , 0. ,\n", " 0. , 0.5, -0.5, 0. , -0. , 0. , 0. , 0. , -0. , 0.5],\n", " [ 0. , -0. , -0.5, -0. , 0.5, -0.5, -0. , 0. , -0. , 0. , 0. ,\n", " 0.5, -0.5, -0. , 0.5, 0. , 0.5, -0.5, 0.5, 0. , 0. , -0.5,\n", " -0.5, 0. , -0. , 0.5, 0.5, -0. , 0. , -0. , 0. , 0. ],\n", " [-0.5, 0. , 0.5, -1. , 0.5, 0. , 0. , -0. , -0.5, 0. , -0. ,\n", " 0. , -0. , -0.5, -0. , 0.5, -0. , -0.5, -0. , -0.5, -0.5, 0.5,\n", " -0. , 0.5, -1. , 0. , 0.5, -0. , -0.5, -0. , -0.5, -0.5],\n", " [ 0. , 0.5, 0. , 0.5, 0. , 0. , -0.5, -0. , -0.5, -0. , -0. ,\n", " 0.5, -0. , -0. , 0. , -0. , -0. , 0. , 0. , -0. , -0. , -0. ,\n", " -0. , -0. , -0.5, -0.5, 0. , 0.5, 0.5, -0. , 0. , -0. ],\n", " [-0.5, -0.5, -0. , -0.5, -0. , -0.5, 0.5, -0.5, -0. , 0. , 0.5,\n", " 0. , 0.5, -0.5, -0. , -0. , 0.5, -0. , -0.5, 0.5, -0. , -0. ,\n", " -0. , -0.5, 0.5, -0. , 0. , 0.5, 0. , 0. , 0. , -0. ],\n", " [-0. , 0. , 0.5, 0. , -0. , 0.5, -0. , 0.5, 0. , 0. , 0.5,\n", " -0.5, 0.5, -0.5, -0. , 0. , -0.5, 0.5, -0.5, 0. , 0. , -0.5,\n", " -0. , 0.5, -0. , -0.5, -0.5, -0. , 0. , -0.5, -0. , -0. ],\n", " [ 0.5, -0. , -0.5, -0. , -0. , -0. , 0. , 0. , -0.5, 0. , 0. ,\n", " -0. , -0. , 0. , -0. , -0. , -0. , -0.5, -0. , 0.5, -0. , 0.5,\n", " -0. , 0. , -0. , -0. , 0.5, 0. , -0.5, -0.5, 0. , -0. ]],\n", " dtype=np.float32), np.array([[ 0.5, -0. , -0. , 0. , -0. , -0.5, 0. , 0. , 1. , -0. , -0. ,\n", " -0. , 0.5, -0.5, 0.5, 0. , -0. , -0.5, -0. , 0.5, -0.5, 0. ,\n", " -0.5, -0.5, 0. , -0. , 0.5, 0. , -1. , 0. , 0.5, 0. ],\n", " [ 0.5, 0. , 0. , 1. , 0. , 0.5, 2. , 0. , -0. , -0. , 0. ,\n", " 0. , -0.5, 0.5, -0.5, 0. , 0. , 0.5, -0. , 0.5, 0.5, 0. ,\n", " 0.5, 0.5, -1. , 0. , -0.5, -0. , 0. , -0. , -0.5, 0. ],\n", " [ 0.5, 0. , -0. , -0. , 0. , -0.5, -0. , -1. , -0. , 0. , 1. ,\n", " 1. , 0.5, -0.5, -0.5, 0. , -1. , 0.5, 1. , 0.5, -0.5, -0. ,\n", " 0.5, -0.5, -0. , 1. , -0.5, -2. , -0. , -0. , 0.5, -0. ],\n", " [-0.5, -0. , -0. , 0. , 0. , 0.5, 0. , 0. , -0. , 0. , -0. ,\n", " -1. , -0.5, 0.5, -0.5, 0. , -0. , -0.5, 0. , -0.5, 0.5, 0. ,\n", " 0.5, 0.5, -0. , -1. , 0.5, -0. , 0. , -0. , -0.5, 0. ],\n", " [ 0. , 0. , 0. , 1. , -0. , -1. , 0. , -0. , -0. , -0. , -0. ,\n", " -1. , -1. , -0. , -0. , -1. , -1. , -0. , -1. , -0. , -1. , -0. ,\n", " -0. , -0. , 1. , 1. , -0. , -0. , -0. , -0. , -1. , -1. ],\n", " [ 0.5, -2. , -0. , 1. , 0. , -0.5, 0. , 0. , 1. , 0. , -0. ,\n", " 0. , 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"outputs": [], "source": [ "#@title Verification\n", "n = 2\n", "m = 4\n", "p = 5\n", "rank = 32\n", "\n", "verify_tensor_decomposition(decomposition_245, n, m, p, rank)" ] }, { "cell_type": "markdown", "metadata": { "id": "Nx23dA23eJ2D" }, "source": [ "## Rank-45 decomposition of <2,4,7> over Z" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "Y2CpfjKHeJ2D" }, "outputs": [], "source": [ "#@title Data\n", "\n", "decomposition_247 = (np.array([[ 0., -1., -1., 0., 1., 1., 1., -1., -1., 0., -0., -0., 0.,\n", " 0., -0., 1., -0., -1., 0., 0., -0., -0., -0., -1., -0., 0.,\n", " -0., 0., 0., 0., 0., 1., -1., -0., -0., -0., -0., 1., -0.,\n", " -0., 0., -1., -1., -0., 0.],\n", " [ 0., 0., 0., 0., -0., -0., 0., -0., -0., 1., -0., 0., -1.,\n", " -0., -0., -0., -0., -0., 1., -0., 1., -0., 1., 1., -1., -1.,\n", " 0., -1., -0., 0., -0., -1., -0., -0., 1., -1., -0., -0., 0.,\n", " 0., -0., 1., 0., -0., 1.],\n", " [ 0., 0., -0., 0., 1., 1., -0., -1., 0., 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"cell_type": "markdown", "metadata": { "id": "NeghojDaeJ2G" }, "source": [ "##Rank-74 decomposition of <3,4,8> over Z\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "vFwEQiTOeJ2G" }, "outputs": [], "source": [ "#@title Data\n", "\n", "decomposition_348 = (np.array([[ 1., -0., 0., -0., 0., 1., 0., -1., 0., -1., -0., -0., 1.,\n", " 1., 0., -1., 0., -1., -0., -0., 0., 1., 0., 1., 0., 1.,\n", " -1., -1., -1., 0., -0., 0., 0., 0., 0., 0., -0., -0., -0.,\n", " 1., 1., -1., 1., 1., 0., -0., -0., 0., -0., -0., -0., 0.,\n", " -0., -1., -0., -1., 1., -1., 0., 0., -1., -1., -0., 0., -0.,\n", " 1., -1., -1., 1., -0., 0., -0., 0., -1.],\n", " [-0., -0., -0., 1., -0., 1., -0., 0., -1., 0., 0., -0., 1.,\n", " 1., -0., -1., -0., 0., -0., 0., -0., -0., -0., 1., -0., -0.,\n", " 0., 0., 0., -1., -0., 0., 0., -0., 0., 0., -0., -0., -0.,\n", " -0., -0., 0., 1., -0., -0., 0., 0., -0., -0., -0., 0., -0.,\n", " 0., -1., -0., 0., -0., -1., 0., -0., 0., 0., -0., 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],\n", " [ 0. -0.5j, 0. -0.5j, 0. -0.j , 0.5+0.j , 0. +0.5j, -0. +0.j ,\n", " -0.5-0.j , -0.5-0.j , -0. +0.5j, 0. +0.5j, -0.5-0.j , 0. +0.5j,\n", " 0. +0.j , 0. +0.j , 0. -0.5j, 0. +0.j , 0.5+0.j , 0. -0.5j,\n", " 0.5+0.j , -0. +0.j , -0. +0.j , 0. -0.5j, -0.5+0.j , 0. -0.j ,\n", " 0. -0.5j, 0. +0.j , 0. +0.j , 0.5+0.j , 0.5+0.j , -0. +0.5j,\n", " 0. -0.j , -0.5-0.j , 0. -0.j , -0.5+0.j , -0.5+0.j , 0. +0.j ,\n", " 0. -0.j , 0. -0.5j, 0. -0.j , 0. -0.5j, 0. -0.5j, -0. +0.5j,\n", " 0.5+0.j , 0. -0.5j, 0. +0.j , 0. -0.5j, -0. +0.5j, -0.5-0.j ]],\n", " dtype=np.complex64))" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "-zahaCcheJ2Q" }, "outputs": [], "source": [ "#@title Verification\n", "n = 4\n", "m = 4\n", "p = 4\n", "rank = 48\n", "\n", "verify_tensor_decomposition(decomposition_444, m, n, p, rank)" ] }, { "cell_type": "markdown", "metadata": { "id": "w1XO6FnNeJ2Q" }, "source": [ "##Rank-96 decomposition of <4,4,8> over 0.5*C\n" ] }, { "cell_type": "markdown", "metadata": { "id": "c1UE8IgWeJ2Q" }, "source": [ "This decomposition can be obtained by doubling the rank-48 decomposition of <4,4,4> provided above." ] }, { "cell_type": "markdown", "metadata": { "id": "WFMp_Z2peJ2Q" }, "source": [ "## Rank-61 decomposition of <4,4,5> over Z" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "i27DJ6EbeJ2Q" }, "outputs": [], "source": [ "#@title Data\n", "\n", "decomposition_445 = (np.array([[-0., 0., -0., 0., 0., -1., 0., 1., -0., 0., 0., 0., -0.,\n", " -0., -0., -0., -1., -0., -0., 0., -0., -0., -0., 0., 0., -0.,\n", " -0., -0., -0., -0., -0., -0., -0., 0., 0., -0., 1., -0., -1.,\n", " 1., -0., -1., -0., 0., 0., -0., 0., 0., 0., 0., 0., -0.,\n", " 0., -0., 1., 0., -1., -1., -0., 0., -1.],\n", " [ 0., 1., 0., 0., -0., 1., -1., -0., 0., 0., -1., 0., -0.,\n", " -0., 1., -0., 0., -0., -1., -1., -0., -1., -1., 0., 0., -0.,\n", " 0., 0., 1., -0., -0., -0., -0., 0., -0., -0., -0., 0., 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"rank = 61\n", "\n", "verify_tensor_decomposition(decomposition_445, n, m, p, rank)" ] }, { "cell_type": "markdown", "metadata": { "id": "gqqafvLPeJ2R" }, "source": [ "## Rank-85 decomposition of <4,4,7> over Z\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "EScB4wdveJ2R" }, "outputs": [], "source": [ "#@title Data\n", "\n", "decomposition_447 = (np.array([[ 0., -0., 0., -0., -0., 0., -0., -0., 0., -0., 1., -0., 0.,\n", " 0., -0., 0., -0., -0., 0., 0., 0., -1., -0., 0., 0., -0.,\n", " -0., 1., 0., -0., 0., 0., -1., -0., -1., -0., 0., 0., -0.,\n", " -0., 0., -0., 0., -0., -0., 0., -0., 1., 0., 0., -0., 1.,\n", " 0., 1., 1., -0., -0., -1., -1., 0., -0., 0., 0., -0., -0.,\n", " 0., 0., 1., -0., 0., -0., 0., -1., -0., -0., 0., -1., -0.,\n", " 0., 0., 0., -0., -0., -0., 1.],\n", " [ 0., 0., 0., -0., 0., -1., 0., 0., 0., 0., 1., 0., -1.,\n", " -0., 1., 1., -0., 0., -0., -0., -0., -0., 0., 0., -0., 0.,\n", " -0., -0., -0., -0., -0., -0., 0., 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-1., -0., -0., 0., -0., -1., -0., -0., -0., -0., 0., -0.,\n", " 1., -1., 0., 1., 0., 1., 1., 0., 0., 1., 0., -0., -0.,\n", " 1., -0.]], dtype=np.float32))" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "4EqBhRveeJ2R" }, "outputs": [], "source": [ "#@title Verification\n", "n = 5\n", "m = 5\n", "p = 5\n", "rank = 93\n", "\n", "verify_tensor_decomposition(decomposition_555, n, m, p, rank)" ] }, { "cell_type": "markdown", "metadata": { "id": "IBOyCNqLeJ2S" }, "source": [ "# B.1. First autocorrelation inequality\n", "For any function $f:\\mathbb{R} \\rightarrow \\mathbb{R}$, define the *autoconvolution* of $f$, written $f*f$, as\n", "$$f*f (t) := \\int_\\mathbb{R} f(t-x) f(x)\\ dx.$$\n", "\n", "Let $C_1$ denote the largest constant for which one has\n", "\\begin{equation}\n", " \\max_{-1/2 \\leq t \\leq 1/2} f*f(t) \\geq C_1 \\left(\\int_{-1/4}^{1/4} f(x)\\ dx\\right)^2\n", "\\end{equation}\n", "for all non-negative $f: \\mathbb{R} \\rightarrow \\mathbb{R}$. This problem arises in additive combinatorics, relating to the size of Sidon sets. It is currently known that\n", "$$ 1.28 \\leq C_1 \\leq 1.5098$$\n", "with the lower bound proven by [Cloninger and Steinerberger (2017)](https://www.ams.org/journals/proc/2017-145-08/S0002-9939-2017-13690-9/S0002-9939-2017-13690-9.pdf) and the upper bound achieved by [Matolcsi and Vinuesa (2010)](https://www.sciencedirect.com/science/article/pii/S0022247X10006001) via a step function construction. AlphaEvolve found a step function with 600 equally-spaced intervals on $[-1/4,1/4]$ that gives a better upper bound of $C_1 \\leq 1.5053$." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "SMN-ZRiReJ2S" }, "outputs": [], "source": [ "#@title Data\n", "\n", "step_function_heights_1 = [9.00854017233134, 4.581788824134047, 5.954983797866223, 3.7314786029786324, 4.25817159660483, 3.544987049547799, 0.08876194700959494, 0.0491697316439318, 1.7439263266999894, 3.8548164683263795, 3.621038728073569, 4.04587668218835, 4.68211946529981, 5.5904236142896675, 4.737832747546433, 3.1594093823451055, 1.902874129984629, 2.7870307391136304, 3.277574995692391, 1.8981329099596054, 1.526040859367755, 2.305128838504833, 5.17673786436095, 4.583218228762042, 3.9910761392791887, 2.784600928752006, 5.450687602543662, 6.170368723277989, 7.045569321986071e-16, 7.149948549556939e-15, 0.0, 0.0, 0.0, 0.0, 1.2580295353835013e-15, 0.0, 0.0, 0.0, 0.0, 3.873037303627252e-15, 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"\n", "convolution_1 = np.convolve(step_function_heights_1, step_function_heights_1)\n", "C_upper_bound = 2 * len(step_function_heights_1) * np.max(convolution_1) / np.sum(step_function_heights_1)**2\n", "print(f\"The discovered step function shows C1 <= {C_upper_bound}.\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "rvd1otTRMjjn" }, "outputs": [], "source": [ "#@title Visualization\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "\n", "def plot_step_function(step_heights_input: list[float], title=\"\"):\n", " \"\"\"Plots a step function with equally-spaced intervals on [-1/4,1/4].\"\"\"\n", " num_steps = len(step_heights_input)\n", "\n", " # Generate x values for plotting (need to plot steps properly).\n", " step_edges_plot = np.linspace(-0.25, 0.25, num_steps + 1)\n", " x_plot = np.array([])\n", " y_plot = np.array([])\n", "\n", " for i in range(num_steps):\n", " x_start = step_edges_plot[i]\n", " x_end = step_edges_plot[i+1]\n", " x_step_vals = np.linspace(x_start, x_end, 100) # Points within each step.\n", " y_step_vals = np.full_like(x_step_vals, step_heights_input[i])\n", " x_plot = np.concatenate((x_plot, x_step_vals))\n", " y_plot = np.concatenate((y_plot, y_step_vals))\n", "\n", " # Plot the step function.\n", " plt.figure(figsize=(8, 5))\n", " plt.plot(x_plot, y_plot)\n", " plt.xlabel(\"x\")\n", " plt.ylabel(\"f(x)\")\n", " plt.title(title)\n", " plt.xlim([-0.3, 0.3]) # Adjust x-axis limits if needed.\n", " plt.ylim([-1, max(step_heights_input) * 1.2]) # Adjust y-axis limits.\n", " plt.grid(True)\n", " plt.step(step_edges_plot[:-1], step_heights_input, where='post', color='green', linewidth=2) # Overlay with plt.step for clarity.\n", " plt.show()\n", "\n", "plot_step_function(step_function_heights_1, title=\"AlphaEvolve's discovered step function\")\n", "plot_step_function(convolution_1, title=\"Autoconvolution of the discovered step function\")" ] }, { "cell_type": "markdown", "metadata": { "id": "IgZtwj9Q6iQ3" }, "source": [ "# B.2. Second autocorrelation inequality\n", "Let $C_2$ be the smallest constant for which one has\n", "$$ \\|f*f\\|_2^2 \\leq C_2 \\|f*f\\|_1 \\|f*f\\|_\\infty$$\n", "for all non-negative $f:\\mathbb{R} \\rightarrow \\mathbb{R}$.\n", "It is known that\n", "$$ 0.88922 \\leq C_2 \\leq 1,$$\n", "with the lower bound coming from a step function construction by [Matolcsi and Vinuesa (2010)](https://www.sciencedirect.com/science/article/pii/S0022247X10006001). AlphaEvolve found a step function with 50 equally-spaced intervals on $[-1/4,1/4]$ that gives a slightly better lower bound $0.8962 \\leq C_2$." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "tgIZlbuQMjjn" }, "outputs": [], "source": [ "#@title Data\n", "import numpy as np\n", "\n", "heights_sequence_2 = [0.0022217502753395443, 0.798058737836952, 0.4369294656327977, 1.1704958412868685, 1.3413665690827143, 1.5342366222696133, 1.7690742844401723, 1.9329450122360183, 2.2225113878900893, 1.9363966992163675, 2.0382191032475467, 2.2010898310433933, 2.0229605588392388, 2.029541518023742, 2.2636974412575626, 1.9622346498507677, 2.0781053776466134, 2.9856571697702514, 3.4418422600649374, 3.3477129878607825, 3.253250196453988, 3.420135507780267, 3.2509579118114464, 3.2308578066681575, 3.4707132763246245, 2.6462657430572087, 0.9614362498214617, 0, 0.0008733532713782356, 0.00041056186458359313, 0.00029587319086208687, 5.039012949497012e-06, 0, 0.5858888998745988, 6.741440691998236, 7.934548956206666e-06, 0.00013382382526231794, 4.551621108101551e-06, 0.0008898629473865954, 1.083008496291632e-05, 0.0006121618352774956, 0.0011493704284828532, 7.157034681754761, 9.111886252846807, 3.3127569806426527, 8.556232703271356e-06, 0.00017950056213609822, 2.7122354902710758e-06, 1.4036462843158317e-05, 1.1451768709981007e-05]\n", "convolution_2 = np.convolve(heights_sequence_2, heights_sequence_2)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "gQ0rQfhqMjjn" }, "outputs": [], "source": [ "#@title Verification\n", "\n", "# Calculate the 2-norm squared: ||f*f||_2^2\n", "num_points = len(convolution_2)\n", "x_points = np.linspace(-0.5, 0.5, num_points + 2)\n", "x_intervals = np.diff(x_points) # Width of each interval\n", "y_points = np.concatenate(([0], convolution_2, [0]))\n", "l2_norm_squared = 0.0\n", "for i in range(len(convolution_2) + 1): # Iterate through intervals\n", " y1 = y_points[i]\n", " y2 = y_points[i+1]\n", " h = x_intervals[i]\n", " # Integral of (mx + c)^2 = h/3 * (y1^2 + y1*y2 + y2^2) where m = (y2-y1)/h, c = y1 - m*x1, interval is [x1, x2], y1 = mx1+c, y2=mx2+c\n", " interval_l2_squared = (h / 3) * (y1**2 + y1 * y2 + y2**2)\n", " l2_norm_squared += interval_l2_squared\n", "\n", "# Calculate the 1-norm: ||f*f||_1\n", "norm_1 = np.sum(np.abs(convolution_2)) / (len(convolution_2) + 1)\n", "\n", "# Calculate the infinity-norm: ||f*f||_inf\n", "norm_inf = np.max(np.abs(convolution_2))\n", "C_lower_bound = l2_norm_squared / (norm_1 * norm_inf)\n", "\n", "print(f\"This step function shows that C2 >= {C_lower_bound}\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "qrNWgv74Mjjn" }, "outputs": [], "source": [ "#@title Visualization\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "\n", "def plot_step_function(step_heights_input: list[float], title=\"\"):\n", " \"\"\"Plots a step function with equally-spaced intervals on [-1/4,1/4].\"\"\"\n", " num_steps = len(step_heights_input)\n", "\n", " # Generate x values for plotting (need to plot steps properly).\n", " step_edges_plot = np.linspace(-0.25, 0.25, num_steps + 1)\n", " x_plot = np.array([])\n", " y_plot = np.array([])\n", "\n", " for i in range(num_steps):\n", " x_start = step_edges_plot[i]\n", " x_end = step_edges_plot[i+1]\n", " x_step_vals = np.linspace(x_start, x_end, 100) # Points within each step.\n", " y_step_vals = np.full_like(x_step_vals, step_heights_input[i])\n", " x_plot = np.concatenate((x_plot, x_step_vals))\n", " y_plot = np.concatenate((y_plot, y_step_vals))\n", "\n", " # Plot the step function.\n", " plt.figure(figsize=(8, 5))\n", " plt.plot(x_plot, y_plot)\n", " plt.xlabel(\"x\")\n", " plt.ylabel(\"f(x)\")\n", " plt.title(title)\n", " plt.xlim([-0.3, 0.3]) # Adjust x-axis limits if needed.\n", " plt.ylim([-1, max(step_heights_input) * 1.2]) # Adjust y-axis limits.\n", " plt.grid(True)\n", " plt.step(step_edges_plot[:-1], step_heights_input, where='post', color='green', linewidth=2) # Overlay with plt.step for clarity.\n", " plt.show()\n", "\n", "plot_step_function(heights_sequence_2, title=\"AlphaEvolve's discovered step function\")\n", "plot_step_function(convolution_2, title=\"Autoconvolution of the discovered step function\")" ] }, { "cell_type": "markdown", "metadata": { "id": "0b-n93vgMjjn" }, "source": [ "# B.3. Third autocorrelation inequality\n", "\n", "Let $C_3$ be the largest constant satisfying\n", "$$ \\left|\\max_{-1/2 \\leq t \\leq 1/2} f*f(t)\\right| \\geq C_3 \\left(\\int_{-1/4}^{1/4} f(x)\\ dx\\right)^2$$\n", "for any function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$. Then there is a step function that gives the upper bound $C_3 \\leq 1.4581$, see page 75 of [Vinuesa's thesis (2009)](https://www.icmat.es/Thesis/CVinuesa.pdf). AlphaEvolve discovered a step function with 400 equally-spaced intervals on $[-1/4,1/4]$ that gives a slightly better upper bound $C_3 \\leq 1.4557$." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "id": "61tDuSjWMjjn" }, "outputs": [], "source": [ "#@title Data\n", "import numpy as np\n", "\n", "height_sequence_3 = np.array([1.2065002259, 0.4579577248, 0.5163353850, 0.4072619373, 0.3357781229, 0.2857531610, 0.2820916949, 0.2605397021, -0.0945455083, 0.0263675774, 0.0066053345, 0.4231327261, 0.1460589430, 0.2693078743, 0.0894390737, 0.1956637306, 0.2479890347, -0.0252634907, 0.1662117356, -0.2218493784, 0.0739312998, 0.2415490072, 0.1698246070, 0.3250002134, 0.0020647116, -0.0688859195, 0.0585905178, 0.0939453029, -0.3060000075, -0.1837206775, -0.2352337979, 0.0835844954, -0.0058309576, -0.0832913333, -0.0166100706, -0.0593295469, -0.0358427923, -0.0137145337, -0.2200581708, -0.0020420409, -0.0154295569, -0.1668919220, 0.1254567765, 0.0104704418, -0.1281078357, -0.0961891358, -0.0180100824, 0.0285695118, -0.1876421178, -0.0503548117, 0.0705665536, -0.0849054288, -0.1712106082, 0.1330779045, -0.0395224921, -0.1763118729, 0.1943874403, 0.3136366436, 0.2907763337, 0.0972482020, 0.2859490417, 0.0544729197, 0.2859115117, 0.1089164067, 0.0171270233, -0.0434679259, -0.1024364047, -0.1167575690, -0.0003774918, -0.0344795844, -0.0972136207, -0.1226383339, -0.0252432553, 0.0400353923, 0.0455096903, 0.0122078271, -0.0826149005, 0.0068314042, -0.1352404062, 0.1294605758, -0.0250363591, 0.0356465906, 0.0085605877, -0.0734465832, 0.0239223522, 0.0560616111, 0.1051222588, 0.1186521152, 0.2114356563, 0.0580833852, 0.1532644654, 0.0340677496, 0.0584859676, 0.0676309739, 0.0501669055, 0.0098826564, -0.0699961802, -0.0655442615, -0.2431379908, 0.0191721224, -0.0101123101, 0.0395025740, 0.0369236983, -0.0405410938, 0.0043783628, -0.0022377932, 0.1882810103, -0.0392647889, 0.1110883981, -0.0056550805, -0.0165795658, 0.0484482116, 0.1040965030, 0.0315331695, 0.0290282853, 0.1700089425, 0.0788158652, 0.3053423957, 0.1407591357, 0.1288565749, 0.0731018735, 0.1212327910, 0.0265656763, -0.0026874534, 0.1448773580, -0.1569172936, -0.0332279397, -0.0961953570, -0.1057990671, -0.0208089846, 0.0622545893, 0.0086508578, 0.1134542356, 0.0796647542, -0.0118068228, 0.1183252889, 0.0322959545, 0.1390562294, 0.1334327477, 0.0790933677, -0.0121592574, 0.0355197021, 0.0680023155, 0.1367771272, 0.2764105186, 0.0917362039, 0.1789370907, 0.0801451602, 0.1507336094, 0.1499095182, 0.1338559365, 0.0883362440, 0.0130685381, -0.0010501160, 0.0434483442, -0.0452555246, -0.0160911253, 0.0001037672, -0.0215276925, 0.0922451592, 0.0914967270, 0.1247376137, 0.0674269368, 0.1233065439, 0.0662811693, 0.1056929066, 0.1236410207, 0.1092652789, 0.1253119938, 0.1350819190, 0.0760669548, 0.0975781877, 0.3030153656, -0.0405615005, 0.0736039858, 0.1516256451, 0.2310415709, 0.0903196002, 0.1368532961, 0.1558915065, 0.1395537845, 0.1134530692, 0.1672360224, 0.1779020155, 0.1110537453, 0.3179860968, 0.2418612890, 0.1879389889, 0.1759242853, 0.1751315796, 0.0707347421, 0.1498901200, 0.0009217008, 0.1217537983, 0.0774209961, 0.1585992473, 0.0621134399, 0.1515100010, 0.0780855072, 0.0997527327, 0.1158696157, 0.0293743767, 0.0551188205, 0.0597680188, -0.2775720739, 0.0759704568, 0.1250316677, 0.2132197633, -0.0252846511, -0.2298263857, -0.1916394076, -0.0828138996, -0.4176293110, -0.7396661838, -0.5511700738, -0.5448611785, -0.1727176925, -0.1101347125, 0.2043220162, -0.1082288708, -0.1748688117, 0.0862407071, -0.0531903867, -0.0616308223, 0.0700887776, 0.2081066310, 0.0954712433, -0.2241273540, 0.0753693148, 0.0747931780, -0.0124247160, -0.0919875318, 0.0869789932, 0.0228948007, 0.0330558155, 0.0365086857, 0.0425990861, 0.0946517743, 0.0461941250, 0.0314389948, 0.1547977504, 0.0995246754, 0.0972214450, 0.0642031653, 0.1125917823, 0.0823779771, 0.1138470746, 0.0561091801, 0.0318706784, 0.0083585533, 0.0141864593, 0.0550841887, 0.0552072046, 0.0056539829, 0.0160881555, 0.0233437573, 0.0715285236, 0.0809609974, 0.0733507409, 0.0740536003, 0.0214532872, 0.0210531815, 0.0613127802, 0.1098196117, 0.0679995089, 0.0841437462, 0.0422816392, 0.0795029912, 0.1219842632, 0.0629392926, 0.0442479973, 0.1754482798, 0.1892151967, 0.0506690589, 0.1127489948, 0.0489164780, 0.1195561384, 0.0803870907, 0.1088302988, 0.0086773701, 0.0350070450, -0.0039451117, -0.0491785508, 0.0777041816, 0.1256031497, 0.0758115755, 0.0338649690, 0.0774479672, 0.1600267712, 0.1116514343, 0.0393425971, 0.0880571065, 0.1752844511, 0.0567110352, 0.0848220320, 0.1023379468, 0.1731927552, 0.1015859496, 0.0456034593, 0.1484852609, 0.2335600810, 0.2680922857, 0.0964039102, 0.1370762748, 0.1194107677, 0.0705488401, 0.0492937128, 0.0010280509, 0.0748767880, 0.0413276428, -0.0120807694, -0.1011133614, 0.0378549584, 0.0742557513, 0.0834599468, 0.0828277891, 0.1937349434, 0.0722637117, 0.0539880627, 0.1092967592, 0.1314066075, 0.2276951671, 0.0993954217, 0.0506623717, 0.0665635040, 0.1475022076, 0.1348793959, 0.1120191289, 0.2268202779, 0.1968416912, 0.1785939824, 0.0653614828, 0.1308139129, 0.1845238365, 0.0400508726, -0.0030416083, 0.0612302085, 0.0446805476, 0.0413828618, 0.0087997710, -0.0288662567, 0.0106587067, 0.0459708143, 0.0777709776, 0.1302908220, 0.1269325239, 0.1144772737, 0.1448279485, 0.0854580145, 0.0780628793, 0.1185972450, 0.1083017344, 0.1382659716, 0.1161843882, 0.1031074870, 0.1081799914, 0.0968144848, 0.1369405176, 0.1597198239, 0.1076990897, 0.0998522537, 0.0980860040, 0.0856864888, 0.1147330385, 0.1090380435, 0.0478343797, 0.1104568134, 0.0535653627, 0.0395945072, 0.0615590595, 0.0846094168, 0.1366808400, 0.0979556009, 0.1150407810, 0.1093077855, 0.0701841815, 0.1536579376, 0.0281163265, -0.0188640466, 0.1095483493, 0.2435488950, 0.2224846591, 0.0349037817, 0.0471722551, 0.1366470882, 0.1619649438, -0.0201219126, 0.0776987862, 0.1770040649, 0.1642189494, 0.1510757616, 0.2256426202, 0.1989086542, 0.1778289411, 0.1869462368, 0.2779359655, 0.2578238178, 0.3850396218, 0.4772987312, 0.5429612326])\n", "convolution_3 = np.convolve(height_sequence_3, height_sequence_3)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "id": "ERD6AvSPMjjn" }, "outputs": [], "source": [ "#@title Verification\n", "C_upper_bound = abs(2 * len(height_sequence_3) * np.max(convolution_3) / (np.sum(height_sequence_3)**2))\n", "print(f\"The discovered step function shows C3 <= {C_upper_bound}\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "id": "bB7RtaQwMjjo", "cellView": "form" }, "outputs": [], "source": [ "#@title Visualization\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "\n", "def plot_step_function(step_heights_input: list[float], title=\"\"):\n", " \"\"\"Plots a step function with equally-spaced intervals on [-1/4,1/4].\"\"\"\n", " num_steps = len(step_heights_input)\n", "\n", " # Generate x values for plotting (need to plot steps properly).\n", " step_edges_plot = np.linspace(-0.25, 0.25, num_steps + 1)\n", " x_plot = np.array([])\n", " y_plot = np.array([])\n", "\n", " for i in range(num_steps):\n", " x_start = step_edges_plot[i]\n", " x_end = step_edges_plot[i+1]\n", " x_step_vals = np.linspace(x_start, x_end, 100) # Points within each step.\n", " y_step_vals = np.full_like(x_step_vals, step_heights_input[i])\n", " x_plot = np.concatenate((x_plot, x_step_vals))\n", " y_plot = np.concatenate((y_plot, y_step_vals))\n", "\n", " # Plot the step function.\n", " plt.figure(figsize=(8, 5))\n", " plt.plot(x_plot, y_plot)\n", " plt.xlabel(\"x\")\n", " plt.ylabel(\"f(x)\")\n", " plt.title(title)\n", " plt.xlim([-0.3, 0.3]) # Adjust x-axis limits if needed.\n", " plt.ylim([-1, max(step_heights_input) * 1.2]) # Adjust y-axis limits.\n", " plt.grid(True)\n", " plt.step(step_edges_plot[:-1], step_heights_input, where='post', color='green', linewidth=2) # Overlay with plt.step for clarity.\n", " plt.show()\n", "\n", "plot_step_function(height_sequence_3, title=\"AlphaEvolve's discovered step function\")\n", "plot_step_function(convolution_3, title=\"Autoconvolution of the discovered step function\")" ] }, { "cell_type": "markdown", "source": [ "A closely related problem was studied by [Matolcsi and Vinuesa](https://www.sciencedirect.com/science/article/pii/S0022247X10006001) [1].\n", "\n", "Let $C_3'$ be the largest constant satisfying\n", "$$ \\max_{-1/2 \\leq t \\leq 1/2} \\left|f*f(t)\\right| \\geq C_3' \\left(\\int_{-1/4}^{1/4} f(x)\\ dx\\right)^2$$\n", "for any function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$. Then there is a step function that gives the upper bound $C_3' \\leq 1.4993$, see [1]. AlphaEvolve discovered a step function that gives a slightly better upper bound $C_3' \\leq 1.4688$.\n", "\n", "[1]: Matolcsi, Máté, and Carlos Vinuesa. \"Improved bounds on the supremum of autoconvolutions.\" Journal of mathematical analysis and applications 372.2 (2010): 439-447." ], "metadata": { "id": "7CTocDuQ2hMC" } }, { "cell_type": "code", "source": [ "#@title Data\n", "import numpy as np\n", "\n", "height_sequence_4 = np.array([0.6223133182, 0.3965529362, 0.5155572827, 0.4026054689, 0.3435153294, 0.3473694931, 0.1726671659, 0.3115981044, 0.1755502741, 0.1167055299, 0.1692645221, -0.0345584766, 0.0844364414, 0.0237112885, -0.0333696769, -0.1167063253, -0.0992276094, 0.1315623023, 0.0677990921, 0.1915308951, 0.2268110818, 0.2037620216, -0.1051674677, 0.0612687756, -0.0159353544, 0.1443888515, -0.0413677311, -0.0046231569, 0.0181296016, 0.1581756187, -0.0299984956, 0.0739606710, 0.0550292403, 0.1497004872, 0.1122868505, 0.0268226396, 0.0146040885, 0.0520545965, 0.0354593091, 0.0444695053, 0.2039311987, 0.2034179524, 0.2082486178, 0.0961010058, 0.2216949705, 0.2455885615, 0.1275649544, 0.1550579189, 0.1877411253, 0.0808475270, 0.1635363594, -0.0862481201, 0.2347026533, -0.3339300830, -0.2394944480, -0.6335429902, -0.6536792389, 0.0540557528, 0.0476279139, 0.0575575387, 0.0832110951, 0.0622557902, 0.0471965671, 0.0720111005, 0.0922908080, 0.0673488538, 0.0542097051, 0.0800617040, 0.1177370210, 0.0585280483, 0.0735929104, 0.1157073367, 0.1360314894, 0.0857232816, 0.1204024893, 0.1302030708, 0.1460587793, 0.0852150335, 0.1499055898, 0.1576606253, 0.1742561886, 0.1496941309, 0.1773110650, 0.1740530416, 0.2257835580, 0.2151370133, 0.1261505929, 0.2156770155, 0.2444869096, 0.1658996813, 0.0812677006, 0.0777573525, 0.0843057155, 0.3319153963, 0.3085644675, 0.3235926887, 0.3825766929, 0.2156283900, 0.2172300408, 0.1900463317, 0.2513589638, 0.1663648252, 0.0420563959, 0.2484866204, 0.0170309701, 0.1050793648, 0.0144840833, 0.0559286295, -0.0438033627, -0.2310871624, 0.0176185856, 0.1083095090, 0.2172514650, 0.3339027835, 0.1756580766, 0.0281903951, 0.0786703868, 0.0485526265, 0.1056165280, 0.1561141581, -0.0699280355, 0.1110162697, 0.1539525420, 0.1322776805, 0.0051064865, 0.1633810017, 0.1509218799, 0.2955311928, 0.0429450318, 0.0858825241, 0.0273075747, 0.0289814140, -0.0069017846, 0.1518862224, 0.2043842841, 0.1724630384, 0.0552570600, 0.1267528858, 0.2247116637, 0.0628715547, 0.1358115307, 0.0939520169, 0.0302115890, 0.1357907296, 0.0403400901, -0.1789162965, 0.0647361248, 0.3075374047, 0.4926213831, 0.9597669584])\n", "convolution_4 = np.convolve(height_sequence_4, height_sequence_4)" ], "metadata": { "cellView": "form", "id": "5H8nrPSs2cVk" }, "execution_count": null, "outputs": [] }, { "cell_type": "code", "source": [ "#@title Verification\n", "C_upper_bound = 2 * len(height_sequence_4) * np.max(abs(convolution_4) / (np.sum(height_sequence_4)**2))\n", "print(f\"The discovered step function shows C3' <= {C_upper_bound}\")" ], "metadata": { "id": "qUTdMjln4bj5" }, "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "metadata": { "id": "owt4j75LeJ2S" }, "source": [ "# B.4. An uncertainty inequality\n", "\n", "Given a function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$, define the Fourier\n", "transform $\\widehat f(\\xi) := \\int_\\mathbb{R} f(x) e^{-2\\pi i x\\xi}\\ dx$ and\n", "$$ A(f) := \\inf \\{r > 0: f(x) \\geq 0 \\hbox{ for all } |x| \\geq r \\}.$$\n", "Let $C_4$ be the largest constant satisfying\n", "$$ A(f) A(\\widehat f) \\geq C_4$$\n", "for all even $f$ with $\\max ( f(0), \\widehat f(0)) < 0$. It is known that\n", "$$ 0.2025 \\leq C_4 \\leq 0.3523;$$\n", "see Theorem 2 of\n", "[Gonçalves et al. (2017)](https://www.sciencedirect.com/science/article/pii/S0022247X17301804)\n", "(the upper bound is stated as $0.353$ in the paper, but rounding their solution to the fourth digit gives $0.3523$).\n", "We improved the upper bound to $C_4 \\leq 0.3521$ with a similar linear combination as in [Gonçalves et al. (2017)](https://www.sciencedirect.com/science/article/pii/S0022247X17301804)\n", " but with refined constants that were found by AlphaEvolve.\n", "\n", "\n", "We address the problem using two different formulations described in section B.4. These two formulations lead to known upper bounds $C_4 \\leq 0.3523$ and $C_4 \\leq 0.3284$, respectively. Using AlphaEvolve, we improve these bounds to $C_4 \\leq 0.3521$ and $C_4 \\leq 0.3216$, respectively.\n", "\n", "\n", "First, following the method in [Gonçalves et al. (2017)](https://www.sciencedirect.com/science/article/pii/S0022247X17301804), we consider test functions of the form $f(x) = P(x)e^{-\\pi x^2}$, where $P(x)$ is an even polynomial. For this form, the Fourier transform $\\hat f(\\xi)$ has the structure $P(\\xi)e^{-\\pi \\xi^2}$, which implies $A(f) = A(\\hat f)$ is the largest positive root of $P(x)$ (for functions that are positive for large $|x|$). The inequality then becomes $C_4 \\le (A(f))^2$.\n", "\n", "The polynomial $P(x)$ is chosen as a linear combination of Hermite polynomials of even order $H_{4k}(x)$, specifically $P(x) = \\sum_{k=0}^m c_k H_{4k}(x)$. The goal is to find coefficients $c_k$ that minimize the largest positive root of $P(x)$ while satisfying certain conditions on $P(x)$ (related to the $f(0)<0$ condition and $P(x)>0$ for large $|x|$). In the implementation below, the polynomial $P(x)$ is constructed with the technical constraint $P(0)=0$, meaning $P(x)$ has a factor of $x^2$. The largest positive root of $P(x)$ is then the largest positive root of $P(x)/x^2$. The upper bound on $C_4$ is given by this largest root squared, divided by $2\\pi$.\n", "\n", "AlphaEvolve found the coefficients ($c_0$, $c_1$, $c_2$) defining a polynomial $P(x) = c_0 H_0(x) + c_1 H_4(x) + c_2 H_8(x)$ that provides an improved upper bound for $C_4$." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "C_Etf9eQeJ2S" }, "outputs": [], "source": [ "#@title Data & Utils\n", "\n", "import numpy as np\n", "import sympy\n", "\n", "coefficients_sota = np.array([\n", " -113 / 100,\n", " 1 / 25,\n", " 1 / 3240,\n", "])\n", "\n", "coefficients_alphaevolve = np.array([\n", " 0.3292519302257546,\n", " -0.01158510802599293,\n", " -8.921606035407065e-05\n", "])\n", "\n", "\n", "def verify_hermite_combination(inputs: sympy.Expr):\n", " x = sympy.symbols('x')\n", " # Check the value at 0 is 0.\n", " value_at_0 = inputs.subs(x, 0)\n", " assert value_at_0 == 0, f'The value at 0 is {value_at_0} != 0.'\n", " assert sympy.limit(inputs, x, sympy.oo) > 0, 'Limit at infty is not positive.'\n", "\n", "\n", "def find_hermite_combination(coeffs: np.ndarray) -> sympy.Expr:\n", " \"\"\"Computes the Hermite combination for given coefficients.\"\"\"\n", " m = len(coeffs)\n", " rational_coeffs = [sympy.Rational(c) for c in coeffs]\n", " degrees = np.arange(0, 4 * m + 4, 4)\n", " x = sympy.symbols('x')\n", " hps = [\n", " sympy.polys.orthopolys.hermite_poly(n=i, x=x, polys=False)\n", " for i in degrees\n", " ]\n", "\n", " # All but the last coefficient.\n", " partial_fn: sympy.Expr = sum(\n", " rational_coeffs[i] * hps[i] for i in range(len(rational_coeffs))\n", " )\n", "\n", " # Impose the condition that the root at 0 should be 0.\n", " a = hps[-1].subs(x, 0)\n", " b = -partial_fn.subs(x, 0)\n", " last_coeff = b / a\n", " rational_coeffs.append(sympy.Rational(last_coeff))\n", "\n", " res_fn = sum(rational_coeffs[i] * hps[i] for i in range(len(rational_coeffs)))\n", "\n", " if sympy.limit(res_fn, x, sympy.oo) < 0:\n", " res_fn = -res_fn\n", "\n", " return res_fn\n", "\n", "\n", "def get_upper_bound(coeffs: np.ndarray) -> float:\n", " \"\"\"Computes the upper bound for the given Hermite combination.\"\"\"\n", " g_exp = find_hermite_combination(coeffs)\n", " x = sympy.symbols('x')\n", " gq_fn = sympy.exquo(g_exp, x**2)\n", " rroots = sympy.real_roots(gq_fn, x)\n", " approx_roots = list()\n", " largest_sign_change = 0\n", "\n", " for root in rroots:\n", " approx_root = root.eval_rational(n=200)\n", " approx_root_p = approx_root + sympy.Rational(1e-198)\n", " approx_root_m = approx_root - sympy.Rational(1e-198)\n", " approx_roots.append(approx_root)\n", " is_sign_change = (\n", " (gq_fn.subs(x, approx_root_p) > 0 and gq_fn.subs(x, approx_root_m) < 0)\n", " or (\n", " gq_fn.subs(x, approx_root_p) < 0\n", " and gq_fn.subs(x, approx_root_m) > 0\n", " )\n", " )\n", " if is_sign_change:\n", " largest_sign_change = max(largest_sign_change, approx_root)\n", "\n", " return float(largest_sign_change**2) / (2 * np.pi)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "W-Ig0f5xeJ2T" }, "outputs": [], "source": [ "#@title Verifier\n", "\n", "sota_exp = find_hermite_combination(coefficients_sota)\n", "alphaevolve_exp = find_hermite_combination(coefficients_alphaevolve)\n", "\n", "verify_hermite_combination(sota_exp)\n", "verify_hermite_combination(alphaevolve_exp)\n", "\n", "print(\n", " 'The construction of Gonçalves et al. (2017) is correct and gives the '\n", " 'upper bound: C4 <=',\n", " get_upper_bound(coefficients_sota)\n", ")\n", "print(\n", " 'AlphaEvolve construction is correct and gives the upper bound: C4 <=',\n", " get_upper_bound(coefficients_alphaevolve)\n", ")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "Oc7f02qSNjHk" }, "outputs": [], "source": [ "#@title Plotting\n", "\n", "import matplotlib.pyplot as plt\n", "\n", "x = sympy.symbols('x')\n", "sota_fn = sympy.lambdify(x, sota_exp, modules=[\"numpy\"])\n", "sota_total_fn = lambda x: np.exp(-np.pi * x**2) * sota_fn(\n", " np.sqrt(2 * np.pi) * x\n", ")\n", "\n", "alphaevolve_fn = sympy.lambdify(x, alphaevolve_exp, modules=[\"numpy\"])\n", "alphaevolve_total_fn = lambda x: np.exp(-np.pi * x**2) * alphaevolve_fn(\n", " np.sqrt(2 * np.pi) * x\n", ")\n", "\n", "x_vals = np.linspace(-5, 5, 1000)\n", "\n", "plt.figure(figsize=(8, 5))\n", "\n", "plt.plot(\n", " x_vals,\n", " sota_total_fn(x_vals),\n", " label=\"SOTA\",\n", " color=\"red\"\n", ")\n", "\n", "plt.plot(\n", " x_vals,\n", " alphaevolve_total_fn(x_vals),\n", " label=\"AlphaEvolve\",\n", " color=\"green\",\n", ")\n", "\n", "plt.axhline(0, color=\"black\", linestyle=\"--\", linewidth=0.5) # x-axis\n", "plt.xlabel(\"x\")\n", "plt.ylabel(\"f(x)\")\n", "plt.title(\n", " \"SOTA and AlphaEvolve's discovered function for the uncertainty inequality\"\n", ")\n", "plt.legend()\n", "plt.grid(True)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "id": "dDqWFUo4w-VG" }, "source": [ "Next, we follow the basis construction outlined by [H. Cohn and F. Gonçalves (2019)](https://link.springer.com/article/10.1007/s00222-019-00875-4).\n", "\n", "For comparison we use the data provided in Table 2, p. 821, which outlines the A+(d) constants. In particular, we are interested in A+(1) (reported as about $0.572990$) - its square also gives an upper bound on the uncertainty principle coefficient (i.e. $C_4\\leq 0.3284$).\n", "\n", "Instead of Hermite polynomials, here we use Laguerre polynomials (following H. Cohn and F. Gonçalves above). To this end, below we adjust our verifier accordingly.\n", "\n", "Below we report a configuration AlphaEvolve found with 12 roots and A+(1) of about $0.56708661892$. This result implies the new upper bound $C_4 \\leq 0.3216$.\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "CeJ20_I5xMXR" }, "outputs": [], "source": [ "# @title Utils\n", "\n", "import sys\n", "from typing import Sequence\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import scipy.special\n", "import sympy\n", "\n", "sys.set_int_max_str_digits(40000)\n", "\n", "\n", "def find_laguerre_combination(m: int, n: int, z: Sequence[float]) -> sympy.Expr:\n", "\n", " alpha = sympy.Rational(n, 2) - 1\n", " degrees = np.arange(0, 4 * m + 4, 2)\n", " x = sympy.symbols(\"x\")\n", " lps = [\n", " sympy.polys.orthopolys.laguerre_poly(n=i, x=x, alpha=alpha, polys=False)\n", " for i in degrees\n", " ]\n", "\n", " num_lps = len(lps)\n", " num_conditions = 2 * m + 2 # Root at 0, double roots at z_i\n", "\n", " if num_lps < num_conditions:\n", " raise ValueError(\n", " \"Not enough Laguerre polynomials to satisfy all conditions.\"\n", " )\n", "\n", " # Create a system of linear equations to solve for alpha_i\n", " A = sympy.Matrix(num_conditions, num_lps, lambda i, j: 0)\n", " b = sympy.Matrix(num_conditions, 1, lambda i, j: 0)\n", "\n", " b[1] = 1 # Make sure the derivative of the function at zero is 1\n", "\n", " # Condition 1: Root at 0\n", " for j in range(num_lps):\n", " A[0, j] = lps[j].subs(x, 0)\n", " A[1, j] = lps[j].diff(x).subs(x, 0)\n", "\n", " # Conditions 2 to 2m+2: Double roots at z_i (g(z_i) = 0 and g'(z_i) = 0, where\n", " # g is the function we are constructing)\n", " for i in range(0, m):\n", " zi = sympy.Rational(z[i]) # Ensure z_i is rational\n", "\n", " # g(z_i) = 0\n", " for j in range(num_lps):\n", " A[2 * i + 2, j] = lps[j].subs(x, zi)\n", "\n", " # g'(z_i) = 0 (derivative with respect to x)\n", " for j in range(num_lps):\n", " deriv_lp_j = lps[j].diff(x) # Symbolic differentiation\n", " A[2 * i + 3, j] = deriv_lp_j.subs(x, zi)\n", "\n", " # Solve the linear system Ax = b for x (coefficients alpha_i)\n", " x = A.LUsolve(b) # Use LU decomposition for efficient solving\n", " alpha_coeffs = [x[i] for i in range(num_lps)]\n", " g_fn = sum(alpha_coeffs[i] * lps[i] for i in range(len(alpha_coeffs)))\n", " return g_fn\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "giQwFWBvxRcg" }, "outputs": [], "source": [ "# @title Data\n", "\n", "# Config with twelve roots\n", "\n", "m = 12\n", "n = 1\n", "\n", "zs = [\n", " 3.6331003,\n", " 5.6714292,\n", " 33.09981679,\n", " 38.849942887744,\n", " 41.1543366,\n", " 43.18733472973041,\n", " 50.98385922,\n", " 58.63890192096693,\n", " 96.0237184433404,\n", " 111.2160645819441,\n", " 118.90258667688155,\n", " 141.44196227075895,\n", "]" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "3Zcz7eXTxXso" }, "outputs": [], "source": [ "# @title Visualization and Root Checking\n", "\n", "zs = np.sort(zs)\n", "\n", "g_fn = find_laguerre_combination(m, n, zs)\n", "\n", "x = sympy.symbols(\"x\")\n", "\n", "dg_fn = sympy.diff(g_fn, x)\n", "gn_fn = sympy.lambdify(x, g_fn, modules=[\"numpy\"])\n", "dgn_fn = sympy.lambdify(x, dg_fn, modules=[\"numpy\"])\n", "\n", "div = sympy.prod([(x - sympy.Rational(z)) ** 2 for z in zs]) * x\n", "gq_fn = sympy.exquo(g_fn, div)\n", "\n", "g_fn.subs(x, sympy.Rational(0))\n", "dg_fn.subs(x, sympy.Rational(1)) == 0\n", "\n", "for z in zs:\n", " assert (\n", " g_fn.subs(x, sympy.Rational(z)) == 0\n", " ), f\"Checking condition g({sympy.Rational(z)}) = {g_fn.subs(x, sympy.Rational(z))} failed.\"\n", " assert (\n", " dg_fn.subs(x, sympy.Rational(z)) == 0\n", " ), f\"Checking condition g'({sympy.Rational(z)}) = {dg_fn.subs(x, sympy.Rational(z))} failed.\"\n", "\n", "# Plotting\n", "x_vals = np.linspace(-0.1, 4, 200)\n", "\n", "plt.figure(figsize=(8, 5))\n", "plt.plot(\n", " x_vals,\n", " gn_fn(x_vals),\n", ")\n", "\n", "plt.axhline(0, color=\"black\", linestyle=\"--\", linewidth=0.5) # x-axis\n", "plt.xlabel(\"x\")\n", "plt.ylabel(\"g(x)\")\n", "plt.title(\"Linear combination of Laguerre polynomials found by AlphaEvolve\")\n", "plt.legend()\n", "plt.grid(True)\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "id": "BEEkrNP0xdWY" }, "outputs": [], "source": [ "# @title Verification\n", "\n", "\n", "rroots = sympy.real_roots(gq_fn, x)\n", "approx_roots = list()\n", "largest_sign_change = 0\n", "all_sign_changes = list()\n", "\n", "for root in rroots:\n", " approx_root = root.eval_rational(n=200)\n", " approx_root_p = approx_root + sympy.Rational(1e-198)\n", " approx_root_m = approx_root - sympy.Rational(1e-198)\n", " approx_roots.append(approx_root)\n", " is_sign_change = (\n", " gq_fn.subs(x, approx_root_p) > 0 and gq_fn.subs(x, approx_root_m) < 0\n", " ) or (gq_fn.subs(x, approx_root_p) < 0 and gq_fn.subs(x, approx_root_m) > 0)\n", " if is_sign_change:\n", " all_sign_changes.append(approx_root)\n", " largest_sign_change = max(largest_sign_change, approx_root)\n", "\n", "lsc = float(largest_sign_change)\n", "upper_bound = lsc / 2 / np.pi\n", "ap_estimate = np.sqrt(upper_bound)\n", "print(f\"The construction of AlphaEvolve is correct and gives the A+(1) estimate: {ap_estimate}\")\n", "print(f\"This implies the upper bound: C4 <= {upper_bound}\")\n" ] }, { "cell_type": "markdown", "metadata": { "id": "hVRD69wIeJ2a" }, "source": [ "# B.5. Erdős' minimum overlap problem\n", "\n", "Let $C_5$ be the largest constant satisfying\n", "$$ \\sup_{x \\in [-2,2]} \\int_{-1}^1 f(t) g(x+t)\\ dt\\geq C_5$$\n", "for all non-negative $f,g: [-1,1] \\to [0,1]$ with $f+g=1$ on $[-1,1]$ and\n", "$\\int f(x)\\ dx = 1$, where we extend $f,g$ by zero outside of $[-1,1]$. This\n", "constant controls the asymptotics of the Minimum Overlap Problem described by\n", "[Erdős (1955)](https://link.springer.com/article/10.1007/BF02760020). The bounds\n", "$$0.379005 \\leq C_5 \\leq 0.380927$$\n", "are known; the lower bound was obtained by\n", "[White (2023)](https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/online/115217/a-new-bound-for-erdos-minimum-overlap-problem)\n", "via convex programming methods.\n", "\n", "It is known (see [Haugland (2016)](https://arxiv.org/abs/1609.08000)) that\n", "this constant is equal to the infimum, over all step\n", "functions $h$ on $[0, 2]$ with values in $[0, 1]$ and satisfying\n", "$\n", "\\int_0^2 h(x)dx = 1\n", "$\n", "of\n", "$$\n", "\\max_k \\int h(x)(1 - h(x+k))dx\n", ".$$ The upper bound to the Erdős minimum overlap problem was then obtained by using this result, in\n", "[Haugland (2016)](https://arxiv.org/abs/1609.08000) by a step function\n", "construction.\n", "\n", "AlphaEvolve discovered an alternative step function construction that improves\n", "the upper bound to $C_5 \\leq 0.380924$." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "C4ryBKcAeJ2a" }, "outputs": [], "source": [ "#@title Verification and plotting code\n", "\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "\n", "def plot_step_function(step_heights_input: list[float]):\n", " num_steps = len(step_heights_input)\n", "\n", " # Generate x values for plotting (need to plot steps properly).\n", " step_edges_plot = np.linspace(-0.25, 0.25, num_steps + 1)\n", " x_plot = np.array([])\n", " y_plot = np.array([])\n", "\n", " for i in range(num_steps):\n", " x_start = step_edges_plot[i]\n", " x_end = step_edges_plot[i + 1]\n", " x_step_vals = np.linspace(x_start, x_end, 100) # Points within each step.\n", " y_step_vals = np.full_like(x_step_vals, step_heights_input[i])\n", " x_plot = np.concatenate((x_plot, x_step_vals))\n", " y_plot = np.concatenate((y_plot, y_step_vals))\n", "\n", " # Plot the step function.\n", " plt.figure(figsize=(8, 5))\n", " plt.plot(x_plot, y_plot)\n", " plt.xlabel(\"x\")\n", " plt.ylabel(\"f(x)\")\n", " plt.title(\n", " \"Step function found by AlphaEvolve for Erdős' minimum overlap problem\"\n", " )\n", " plt.xlim([-0.3, 0.3])\n", " plt.ylim([-1, max(step_heights_input) * 1.2])\n", " plt.grid(True)\n", " plt.step(\n", " step_edges_plot[:-1],\n", " step_heights_input,\n", " where='post',\n", " color='green',\n", " linewidth=2,\n", " ) # Overlay with plt.step for clarity.\n", " plt.show()\n", "\n", "\n", "def verify_sequence(sequence: list[float]):\n", " \"\"\"Raises an error if the sequence does not satisfy the constraints.\"\"\"\n", " # Check that all values are between 0 and 1.\n", " if not all(0 <= val <= 1 for val in sequence):\n", " raise AssertionError(\"All values in the sequence must be between 0 and 1.\")\n", " # Check that the sum of values in the sequence is exactly n / 2.0.\n", " if not np.sum(sequence) == len(sequence) / 2.0:\n", " raise AssertionError(\n", " \"The sum of values in the sequence must be exactly n / 2.0. The sum is \"\n", " f\"{np.sum(sequence)} but it should be {len(sequence) / 2.0}.\")\n", " print(\n", " \"The sequence generates a valid step function for Erdős' minimum \"\n", " \"overlap problem.\"\n", " )\n", "\n", "\n", "def compute_upper_bound(sequence: list[float]) -> float:\n", " \"\"\"Returns the upper bound for a sequence of coefficients.\"\"\"\n", " convolution_values = np.correlate(\n", " np.array(sequence), 1 - np.array(sequence), mode='full'\n", " )\n", " return np.max(convolution_values) / len(sequence) * 2" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "oIko1B3reJ2a" }, "outputs": [], "source": [ "#@title Data\n", "best_sequence = np.array([\n", " 0.0, 0.0, 0.0, 0.0, 0.0,\n", " 0.0, 3.60911302e-10, 3.62124044e-10, 4.02849974e-12,\n", " 4.47352578e-12, 4.76914172e-12, 0.506074303,\n", " 0.632046692, 0.679332798, 0.888193865,\n", " 0.889214704, 0.678231235, 2.976636922840846e-07,\n", " 0.0947643739, 0.0143926342, 0.423931858,\n", " 0.598073612, 0.803909612, 0.683098916,\n", " 0.314749384, 0.404059484, 0.858443734,\n", " 0.796503042, 0.590433152, 0.41056218,\n", " 0.270932695, 0.613384276, 0.709501647,\n", " 0.580573615, 0.803538112, 0.715263878,\n", " 0.822611331, 0.808433879, 0.683533985,\n", " 0.645719012, 0.889417725, 0.943389845,\n", " 0.841536959, 0.794505216, 0.941943428,\n", " 0.962223227, 0.961270753, 0.992409079,\n", "])\n", "reversed_sequence = best_sequence[::-1]\n", "best_sequence = np.concatenate((best_sequence[:-1], reversed_sequence))" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "_nTw_qnBeJ2a" }, "outputs": [], "source": [ "#@title Verification and plotting\n", "\n", "verify_sequence(best_sequence)\n", "plot_step_function(best_sequence)\n", "new_upper_bound = compute_upper_bound(best_sequence)\n", "print(\n", " f\"The sequence provides the following upper bound on the Erdős minimum overlap problem: C5 <= {new_upper_bound}.\"\n", ")" ] }, { "cell_type": "markdown", "metadata": { "id": "pNVwWLbreJ2a" }, "source": [ "# B.6. Sums and differences of finite sets\n", "\n", "Let $C_6$ be the largest constant for which the following statement holds: there exist arbitrarily large finite sets of integers $A,B$ with $|A+B| \\ll |A|$ and $|A-B| \\gg |A+B|^{C_6}$.\n", "(Here $A+B = \\{a+b : a \\in A, b \\in B\\}$ and $A-B = \\{a-b : a \\in A, b \\in B\\}$ denote the sumset and difference set, respectively. The notation $X \\ll Y$ means that $X \\le C Y$ for some constant $C$ independent of the sets $A,B$ (for sufficiently large sets $A,B$). The notation $X \\gg Y$ means that $X \\ge C' Y$ for some positive constant $C'$ independent of the sets $A,B$ (for sufficiently large sets $A,B$).)\n", "\n", "It is known that\n", "$$ 1.14465 \\leq C_6 \\leq \\frac{4}{3}, $$\n", "where the upper and lower bounds can be respectively found in Corollary 3 and\n", "Theorem 1 of\n", "[Gyarmati et al. (2007)](https://projecteuclid.org/journals/functiones-et-approximatio-commentarii-mathematici/volume-37/issue-1/Sums-and-differences-of-finite-sets/10.7169/facm/1229618749.full).\n", "To build the lower bound, Gyarmati et al. use the inequality\n", "$$C_6 \\geq 1 + \\frac{\\log \\frac{|U-U|}{|U+U|}}{\\log (2 \\max (U) + 1)}, $$\n", "which holds for any finite set $U$ of non-negative integers containing zero satisfying $|U-U| \\leq 2 \\max (U) + 1$.\n", "\n", "AlphaEvolve found two constructions:\n", "- A set $U_1$ of 2003 integers that improves the lower bound to $1.1479 \\leq C_6$.\n", "- A set $U_2$ of 54265 integers that further improves the lower bound to $1.1584 \\leq C_6$.\n", "\n", "For completeness, we provide both constructions below." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "oMiX8-xVeJ2b" }, "outputs": [], "source": [ "#@title Data\n", "\n", "solution_1 = [0, 110, 116, 122, 128, 134, 140, 2509, 2515, 2521, 2527, 2533, 4908, 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737935451, 737935474, 737935482, 737935513, 742831554, 742874572, 742874580, 742874611, 742917590, 742917598, 742917606, 742917629, 742917637, 742917668, 748095391, 748095399, 748095407, 748095415, 748095423, 748095430, 748095438, 748095446, 748095454, 748095469, 748095477, 748095485, 748095508, 748095516, 748095547, 752991588, 753034606, 753034614, 753034645, 753077624, 753077632, 753077640, 753077663, 753077671, 753077702, 753120642, 753120650, 753120658, 753120666, 753120681, 753120689, 753120697, 753120720, 753120728, 753120759, 758298443, 758298451, 758298459, 758298467, 758298475, 758298482, 758298483, 758298490, 758298498, 758298506, 758298514, 758298521, 758298529, 758298537, 758298545, 758298560, 758298568, 758298576, 758298599, 758298607, 758298638, 763194640, 763194648, 763194679, 763237658, 763237666, 763237674, 763237697, 763237705, 763237736, 763280676, 763280684, 763280692, 763280700, 763280715, 763280723, 763280731, 763280754, 763280762, 763280793, 773397692, 773397700, 773397708, 773397731, 773397739, 773397770, 773440710, 773440718, 773440726, 773440734, 773440749, 773440757, 773440765, 773440788, 773440796, 773440827, 773483728, 773483736, 773483744, 773483752, 773483760, 773483767, 773483775, 773483783, 773483791, 773483806, 773483814, 773483822, 773483845, 773483853, 773483884, 783600744, 783600752, 783600760, 783600768, 783600783, 783600791, 783600799, 783600822, 783600830, 783600861, 783643762, 783643770, 783643778, 783643786, 783643794, 783643801, 783643809, 783643817, 783643825, 783643840, 783643848, 783643856, 783643879, 783643887, 783643918, 793803796, 793803804, 793803812, 793803820, 793803828, 793803835, 793803843, 793803851, 793803859, 793803874, 793803882, 793803890, 793803913, 793803921, 793803952, 793846814, 793846822, 793846830, 793846838, 793846846, 793846853, 793846854, 793846861, 793846869, 793846877, 793846885, 793846892, 793846900, 793846908, 793846916, 793846931, 793846939, 793846947, 793846970, 793846978, 793847009, 804006848, 804006856, 804006864, 804006872, 804006880, 804006887, 804006888, 804006895, 804006903, 804006911, 804006919, 804006926, 804006934, 804006942, 804006950, 804006965, 804006973, 804006981, 804007004, 804007012, 804007043, 814209900, 814209908, 814209916, 814209924, 814209932, 814209939, 814209940, 814209947, 814209948, 814209955, 814209963, 814209971, 814209978, 814209979, 814209986, 814209994, 814210002, 814210010, 814210017, 814210025, 814210033, 814210041, 814210056, 814210064, 814210072, 814210095, 814210103, 814210134]" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "o-xV10udcX--" }, "outputs": [], "source": [ "#@title Verification code\n", "import itertools\n", "import numpy as np\n", "\n", "def compute_lower_bound(u: list[int]) -> float:\n", " \"\"\"Returns the lower bound obtained from the input set u, which must satisfy min(u) == 0.\"\"\"\n", " if min(u) != 0:\n", " raise AssertionError(\"Set U must be nonnegative and must contain 0; got minimum value \", min(u))\n", "\n", " max_u = max(u)\n", " u_minus_u = np.zeros(2 * max_u + 1, dtype=bool) # Store the set u - u as an array of booleans.\n", " u_plus_u = np.zeros(2 * max_u + 1, dtype=bool) # Store the set u + u as an array of booleans.\n", " u_np = np.array(u)\n", "\n", " for i in u:\n", " u_minus_u[i - u_np + max_u] = True\n", " u_plus_u[i + u_np] = True\n", "\n", " u_minus_u_size = np.sum(u_minus_u)\n", " u_plus_u_size = np.sum(u_plus_u)\n", "\n", " if u_minus_u_size > 2 * max_u + 1:\n", " raise AssertionError(\n", " \"The constraint |U - U| <= 2 max (U) + 1 is not satisfied. Got: \"\n", " f\"lhs={u_minus_u_size} but rhs={2 * max_u + 1}.\"\n", " )\n", "\n", " return np.log(u_minus_u_size / u_plus_u_size) / np.log(2 * max_u + 1) + 1.0" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "--wkXmbNSPWG" }, "outputs": [], "source": [ "#@title Verification (solution with 2003 integers)\n", "\n", "print(f\"Size of the construction: {len(solution_1)}.\")\n", "print(f\"This construction proves the lower bound {compute_lower_bound(solution_1)} <= C6.\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "jf53Alw9eJ2b" }, "outputs": [], "source": [ "#@title Verification (solution with 54265 integers)\n", "\n", "print(f\"Size of the construction: {len(solution_2)}.\")\n", "print(f\"This construction proves the lower bound {compute_lower_bound(solution_2)} <= C_6.\")" ] }, { "cell_type": "markdown", "metadata": { "id": "NOMFEm9AeJ2b" }, "source": [ "# B.7. Packing unit regular hexagons inside a regular hexagon\n", "\n", "Consider the problem of packing $n$ disjoint regular hexagons with unit side length into a larger regular hexagon, minimizing the side length of the outer\n", "hexagon. The state of the art can be found on\n", "[Erich Friedman's homepage](https://erich-friedman.github.io/packing/hexinhex/).\n", "\n", "For $n=11$ and $n=12$, the best known constructions use outer hexagons of side\n", "lengths $3.943$ and $4.0$, respectively. AlphaEvolve found packing arrangements\n", "that improve these bounds to $3.931$ and $3.942$, respectively. These arrangements are shown below." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "LP3eBY9XeJ2b" }, "outputs": [], "source": [ "#@title Verification code\n", "\n", "import numpy as np\n", "import math\n", "import matplotlib.pyplot as plt\n", "\n", "\n", "def hexagon_vertices(\n", " center_x: float,\n", " center_y: float,\n", " side_length: float,\n", " angle_degrees: float,\n", ") -> list[tuple[float, float]]:\n", " \"\"\"Generates the vertices of a regular hexagon.\n", "\n", " Args:\n", " center_x: x-coordinate of the center.\n", " center_y: y-coordinate of the center.\n", " side_length: Length of each side.\n", " angle_degrees: Rotation angle in degrees (clockwise from horizontal).\n", "\n", " Returns:\n", " A list of tuples, where each tuple (x, y) represents the vertex location.\n", " \"\"\"\n", " vertices = []\n", " angle_radians = math.radians(angle_degrees)\n", " for i in range(6):\n", " angle = angle_radians + 2 * math.pi * i / 6\n", " x = center_x + side_length * math.cos(angle)\n", " y = center_y + side_length * math.sin(angle)\n", " vertices.append((x, y))\n", " return vertices\n", "\n", "\n", "def normalize_vector(v: tuple[float, float]) -> tuple[float, float]:\n", " \"\"\"Normalizes a 2D vector.\"\"\"\n", " magnitude = math.sqrt(v[0]**2 + v[1]**2)\n", " return (v[0] / magnitude, v[1] / magnitude) if magnitude != 0 else (0., 0.)\n", "\n", "\n", "def get_normals(\n", " vertices: list[tuple[float, float]]\n", ") -> list[tuple[float, float]]:\n", " \"\"\"Gets the outward normals of a polygon's edges.\"\"\"\n", " normals = []\n", " for i in range(len(vertices)):\n", " p1 = vertices[i]\n", " p2 = vertices[(i + 1) % len(vertices)] # Wrap around to the first vertex.\n", " edge = (p2[0] - p1[0], p2[1] - p1[1])\n", " normal = normalize_vector((-edge[1], edge[0])) # Rotate edge by 90 degrees.\n", " normals.append(normal)\n", " return normals\n", "\n", "\n", "def project_polygon(\n", " vertices: list[tuple[float, float]],\n", " axis: tuple[float, float],\n", ") -> tuple[float, float]:\n", " \"\"\"Projects a polygon onto an axis and returns the min/max values.\"\"\"\n", " min_proj = float('inf')\n", " max_proj = float('-inf')\n", " for vertex in vertices:\n", " projection = vertex[0] * axis[0] + vertex[1] * axis[1] # Dot product.\n", " min_proj = min(min_proj, projection)\n", " max_proj = max(max_proj, projection)\n", " return min_proj, max_proj\n", "\n", "\n", "def overlap_1d(min1: float, max1: float, min2: float, max2: float) -> bool:\n", " \"\"\"Determines whether two 1D intervals overlap.\"\"\"\n", " return max1 >= min2 and max2 >= min1\n", "\n", "\n", "def polygons_intersect(\n", " vertices1: list[tuple[float, float]],\n", " vertices2: list[tuple[float, float]],\n", ") -> bool:\n", " \"\"\"Determines if two polygons intersect using the Separating Axis Theorem.\"\"\"\n", " normals1 = get_normals(vertices1)\n", " normals2 = get_normals(vertices2)\n", " axes = normals1 + normals2\n", "\n", " for axis in axes:\n", " min1, max1 = project_polygon(vertices1, axis)\n", " min2, max2 = project_polygon(vertices2, axis)\n", " if not overlap_1d(min1, max1, min2, max2):\n", " return False # Separating axis found, polygons are disjoint.\n", " return True # No separating axis found, polygons intersect.\n", "\n", "\n", "def hexagons_are_disjoint(\n", " hex1_params: tuple[float, float, float, float],\n", " hex2_params: tuple[float, float, float, float],\n", ") -> bool:\n", " \"\"\"Determines if two hexagons are disjoint given their parameters.\"\"\"\n", " hex1_vertices = hexagon_vertices(*hex1_params)\n", " hex2_vertices = hexagon_vertices(*hex2_params)\n", " return not polygons_intersect(hex1_vertices, hex2_vertices)\n", "\n", "\n", "def is_inside_hexagon(\n", " point: tuple[float, float],\n", " hex_params: tuple[float, float, float, float],\n", ") -> bool:\n", " \"\"\"Checks if a point is inside a hexagon (given its parameters).\"\"\"\n", " hex_vertices = hexagon_vertices(*hex_params)\n", " for i in range(len(hex_vertices)):\n", " p1 = hex_vertices[i]\n", " p2 = hex_vertices[(i + 1) % len(hex_vertices)]\n", " edge_vector = (p2[0] - p1[0], p2[1] - p1[1])\n", " point_vector = (point[0] - p1[0], point[1] - p1[1])\n", " cross_product = (\n", " edge_vector[0] * point_vector[1] - edge_vector[1] * point_vector[0]\n", " )\n", " if cross_product < 0:\n", " return False\n", " return True\n", "\n", "\n", "def all_hexagons_contained(\n", " inner_hex_params_list: list[tuple[float, float, float, float]],\n", " outer_hex_params: tuple[float, float, float, float],\n", ") -> bool:\n", " \"\"\"Checks if all inner hexagons are contained within the outer hexagon.\"\"\"\n", " for inner_hex_params in inner_hex_params_list:\n", " inner_hex_vertices = hexagon_vertices(*inner_hex_params)\n", " for vertex in inner_hex_vertices:\n", " if not is_inside_hexagon(vertex, outer_hex_params):\n", " return False\n", " return True\n", "\n", "\n", "def verify_construction(\n", " inner_hex_data: tuple[float, float, float],\n", " outer_hex_center: tuple[float, float],\n", " outer_hex_side_length: float,\n", " outer_hex_angle_degrees: float,\n", "):\n", " \"\"\"Verifies the hexagon packing construction with a rotated outer hexagon.\n", "\n", " Args:\n", " inner_hex_data: List of (x, y, angle_degrees) tuples for inner hexagons.\n", " outer_hex_center: (x, y) tuple for the outer hexagon center.\n", " outer_hex_side_length: Side length of the outer hexagon.\n", " outer_hex_angle_degrees: Rotation angle of the outer hexagon in degrees.\n", "\n", " Raises:\n", " AssertionError if the construction is not valid.\n", " \"\"\"\n", "\n", " inner_hex_params_list = [\n", " (x, y, 1, angle) for x, y, angle in inner_hex_data\n", " ] # Sets the side length to 1.\n", " outer_hex_params = (\n", " outer_hex_center[0], outer_hex_center[1],\n", " outer_hex_side_length, outer_hex_angle_degrees\n", " )\n", "\n", " # Disjointness check.\n", " for i in range(len(inner_hex_params_list)):\n", " for j in range(i + 1, len(inner_hex_params_list)):\n", " if not hexagons_are_disjoint(\n", " inner_hex_params_list[i], inner_hex_params_list[j]\n", " ):\n", " raise AssertionError(f\"Hexagons {i+1} and {j+1} intersect!\")\n", "\n", " # Containment check.\n", " if not all_hexagons_contained(inner_hex_params_list, outer_hex_params):\n", " raise AssertionError(\n", " \"Not all inner hexagons are contained in the outer hexagon!\"\n", " )\n", "\n", " print(\"Construction is valid.\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "d4FPBAabNxU-" }, "outputs": [], "source": [ "#@title Plotting code\n", "\n", "import matplotlib.pyplot as plt\n", "\n", "\n", "def plot_construction(\n", " inner_hex_data: list[tuple[float, float, float]],\n", " outer_hex_center: tuple[float, float],\n", " outer_hex_side_length: float,\n", " outer_hex_angle_degrees: float,\n", "):\n", " \"\"\"Plots the hexagon packing construction using matplotlib.\n", "\n", " Args:\n", " inner_hex_data: List of (x, y, angle_degrees) tuples for inner hexagons.\n", " outer_hex_center: (x, y) tuple for the outer hexagon center.\n", " outer_hex_side_length: Side length of the outer hexagon.\n", " outer_hex_angle_degrees: Rotation angle of the outer hexagon in degrees.\n", " \"\"\"\n", "\n", " inner_hex_params_list = [(x, y, 1, angle) for x, y, angle in inner_hex_data]\n", " outer_hex_params = (\n", " outer_hex_center[0], outer_hex_center[1],\n", " outer_hex_side_length, outer_hex_angle_degrees\n", " )\n", "\n", " _, ax = plt.subplots(figsize=(6, 6))\n", " ax.set_aspect('equal')\n", "\n", " # Plot outer hexagon (in red).\n", " outer_hex_vertices = hexagon_vertices(*outer_hex_params)\n", " outer_hex_x, outer_hex_y = zip(*outer_hex_vertices)\n", " ax.plot(\n", " outer_hex_x + (outer_hex_x[0],),\n", " outer_hex_y + (outer_hex_y[0],),\n", " 'r-',\n", " label='Outer Hexagon',\n", " )\n", "\n", " # Plot inner hexagons (in blue).\n", " for i, inner_hex_params in enumerate(inner_hex_params_list):\n", " inner_hex_vertices = hexagon_vertices(*inner_hex_params)\n", " inner_hex_x, inner_hex_y = zip(*inner_hex_vertices)\n", " ax.fill(\n", " inner_hex_x + (inner_hex_x[0],),\n", " inner_hex_y + (inner_hex_y[0],),\n", " alpha=0.7,\n", " color='blue',\n", " label='Inner Hexagons' if i == 0 else None, # Label only once.\n", " )\n", "\n", " ax.set_title('Hexagon Packing Construction')\n", " ax.legend()\n", " ax.grid(False)\n", " plt.show()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "8HoDR30keJ2b" }, "outputs": [], "source": [ "#@title Construction 1 (11 hexagons): Data\n", "\n", "inner_hex_data = [\n", " [7.227468518248726, 1.406189555916512, 64.96835531696001],\n", " [9.108934177226912, 6.282338181661271, 82.7839419155727],\n", " [9.953071339809288, 4.757920599499527, 22.75943751294596],\n", " [9.119104027817858, 1.3635244317245632, 64.98416780884727],\n", " [8.202877059101247, 4.72907294863739, 22.702381798755646],\n", " [7.272127593730819, 3.260296487997907, 202.68420631619662],\n", " [9.055201371937533, 3.2081226414835964, 82.70304387527644],\n", " [11.605104986630607, 3.9103231530709373, 120.97244798016398],\n", " [10.63107376404371, 2.224090366073967, 244.99786687183936],\n", " [6.427221005557855, 4.784445729508918, 262.6855070070916],\n", " [7.357815645136251, 6.25309652428302, 22.7012724369423],\n", "]\n", "\n", "print(\"Number of inner hexagons:\", len(inner_hex_data))\n", "outer_hex_center = [\n", " np.float64(8.675544349693055), np.float64(3.861025611329817)\n", "]\n", "outer_hex_side_length = 3.930092\n", "outer_hex_angle_degrees = np.float64(60.96370590710646)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "SGEHbemTeJ2b" }, "outputs": [], "source": [ "#@title Construction 1 (11 hexagons): Verification and plotting\n", "\n", "verify_construction(\n", " inner_hex_data, outer_hex_center,\n", " outer_hex_side_length, outer_hex_angle_degrees\n", ")\n", "print(f\"Outer hexagon side length: {outer_hex_side_length}.\")\n", "plot_construction(\n", " inner_hex_data, outer_hex_center,\n", " outer_hex_side_length, outer_hex_angle_degrees\n", ")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "Au_9fA50eJ2b" }, "outputs": [], "source": [ "#@title Construction 2 (12 hexagons): Data\n", "\n", "inner_hex_data = [\n", " [-4.189650776376798, 1.8191996226926388, 27.083656548467786],\n", " [-4.1762891342159225, 7.064090887028092, 267.1521094125118],\n", " [-2.742942219672671, 3.9055564132509595, 27.098852933212644],\n", " [-6.194948523724643, 4.243623380441444, 87.15308054130165],\n", " [-2.4735773559547236, 6.43056712077689, 27.144807633484653],\n", " [-3.8735033091694633, 5.272827562035259, 207.12012595575163],\n", " [-1.040517842157661, 3.2724339122261963, 147.11193087313353],\n", " [-1.3431542020437193, 5.063486444235776, 27.124004797633685],\n", " [-5.89197412464008, 2.452466762507662, 207.10748153590382],\n", " [-4.492206360794807, 3.610020660157929, 327.09918513988094],\n", " [-2.4402287984241076, 2.1146830723057635, 147.1073847274004],\n", " [-5.576327284388157, 5.906127413424367, 207.16788311720808],\n", "]\n", "\n", "print(\"Number of inner hexagons:\", len(inner_hex_data))\n", "outer_hex_center = [\n", " np.float64(-3.7029107170331157), np.float64(4.262832492467158)\n", "]\n", "outer_hex_side_length = 3.9419123\n", "outer_hex_angle_degrees = np.float64(219.59113156095745)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "G12ZOFApeJ2b" }, "outputs": [], "source": [ "#@title Construction 2 (12 hexagons): Verification and plotting\n", "\n", "verify_construction(\n", " inner_hex_data, outer_hex_center,\n", " outer_hex_side_length, outer_hex_angle_degrees\n", ")\n", "print(f\"Outer hexagon side length: {outer_hex_side_length}.\")\n", "plot_construction(\n", " inner_hex_data, outer_hex_center,\n", " outer_hex_side_length, outer_hex_angle_degrees\n", ")" ] }, { "cell_type": "markdown", "metadata": { "id": "kZudk097eJ2b" }, "source": [ "# B.8. Minimizing the ratio of maximum to minimum distance\n", "\n", "For any $n$ and $d$, the goal of this problem is to find $n$ points in the $d$-dimensional space so as to minimize the ratio between the maximum and minimum pairwise distances. AlphaEvolve found two new constructions improving the best known bounds:\n", "\n", "* In 2 dimensions, AlphaEvolve found 16 points with ratio $\\approx \\sqrt{12.889266112}$, improving the best known bound of $\\sqrt{12.89}$ ([reference](https://erich-friedman.github.io/packing/maxmin/)).\n", "\n", "* In 3 dimensions, AlphaEvolve found 14 points with ratio $\\approx \\sqrt{4.165849767}$, improving the best known bound of $\\sqrt{4.168}$ ([reference](https://erich-friedman.github.io/packing/maxmin3/)).\n", "\n", "The found constructions are shown below. (In the above reference, instead of the ratio itself, the square of the ratio is reported, and we use the same convention.)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "5FlU8_eJTJVO" }, "outputs": [], "source": [ "#@title Construction 1: Data\n", "import numpy as np\n", "\n", "construction_1 = np.array([(0.8470278707092158, 0.2918282514726773), (-0.23745459443195124, -0.8571978965568918), (0.3557900209463371, 0.279097289488927), (-0.8607342601440215, -0.15089717398730187), (-0.4726876231254993, 0.7174253632783496), (-0.3788383952523786, -0.05470430230958746), (-0.03757671392994971, -0.4082819539247519), (0.4800129548135765, 0.7545395596986599), (-0.000979763279861421, 0.8551512034827423), (-0.7797718797834859, 0.33379016991681987), (-0.6541792267650747, -0.5967802443605281), (0.6682256918291543, -0.6282247082504422), (0.9028949024748079, -0.1963885801084583), (0.41165705912895084, -0.2091194383890215), (0.25280396897954216, -0.8907156599623591), (-0.12757658935294616, 0.3676038779372552)])" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "iOj32lArTJVO" }, "outputs": [], "source": [ "#@title Construction 1: verification\n", "import scipy as sp\n", "\n", "print(f'Construction 1 has {len(construction_1)} points in {construction_1.shape[1]} dimensions.')\n", "pairwise_distances = sp.spatial.distance.pdist(construction_1)\n", "min_distance = np.min(pairwise_distances)\n", "max_distance = np.max(pairwise_distances)\n", "\n", "ratio_squared = (max_distance / min_distance)**2\n", "print(f\"Ratio of max distance to min distance: sqrt({ratio_squared})\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "fmHnUVMaTJVO" }, "outputs": [], "source": [ "#@title Construction 1: Visualization\n", "import matplotlib.pyplot as plt\n", "\n", "def plot_point_configuration(points: np.ndarray):\n", " \"\"\"Plots a 2D point configuration, highlighting closest and furthest pairs.\"\"\"\n", " condensed_distances = sp.spatial.distance.pdist(points)\n", " min_distance = np.min(condensed_distances)\n", " max_distance = np.max(condensed_distances)\n", " ratio = (max_distance / min_distance)**2\n", " distance_matrix = sp.spatial.distance.squareform(condensed_distances)\n", "\n", " max_distance_pairs_indices = np.where(np.isclose(distance_matrix, max_distance))\n", " min_distance_pairs_indices = np.where(np.isclose(distance_matrix, min_distance))\n", "\n", " plt.figure(figsize=(6, 6))\n", " plt.scatter(points[:, 0], points[:, 1], color='black', s=100)\n", "\n", " # Highlight max distance pairs (red).\n", " for i_index, j_index in zip(*max_distance_pairs_indices):\n", " plt.plot([points[i_index, 0], points[j_index, 0]], [points[i_index, 1], points[j_index, 1]], color='red', linewidth=2, label='maximum distance' if 'maximum distance' not in plt.gca().get_legend_handles_labels()[1] else \"\")\n", "\n", " # Highlight min distance pairs (blue).\n", " for i_index, j_index in zip(*min_distance_pairs_indices):\n", " plt.plot([points[i_index, 0], points[j_index, 0]], [points[i_index, 1], points[j_index, 1]], color='blue', linewidth=2, label='minimum distance' if 'minimum distance' not in plt.gca().get_legend_handles_labels()[1] else \"\")\n", "\n", " # Customize plot appearance.\n", " plt.title(f\"{len(points)} points on the 2D plane with ratio between maximum distance and minimum distance ~ sqrt({ratio:.9f})\")\n", " plt.axis('off')\n", " plt.gca().set_aspect('equal', adjustable='box')\n", " plt.legend()\n", " plt.show()\n", "\n", "plot_point_configuration(construction_1)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "ClVyK0aJTJVO" }, "outputs": [], "source": [ "#@title Construction 2: Data\n", "import numpy as np\n", "\n", "construction_2 = np.array([(-5.596123673583122, -5.695471111926107, 4.959047320055149), (5.116926601383272, -2.9088174983840958, -2.8844213258624003), (-6.606159121469469, 1.1976559300318508, 4.205115060194249), (0.33263279200499823, -9.151690403732108, 3.5421854577228027), (-0.43984530846509556, 0.2175383405880955, 7.387161767180721), (-1.0482338026105975, 5.032254234659363, 2.3318033382908223), (0.33760218404299963, -8.010249719722893, -3.371583323549962), (4.745850590864617, -3.9828624854138783, 5.248008874626355), (-2.144088866983676, -2.3007882626834237, -6.588159606967483), (-0.8091020398380586, -1.7732214204952712, 0.6786858286172662), (5.334738572382582, 2.1672115465961803, 1.9415133400809574), (-6.83950764442355, -4.338724821681177, -1.8023709245292203), (-5.346496436622179, 2.4644877336392232, -2.570708726021154), (1.5164799724062628, 3.0213278117227174, -3.8718307541919557)])" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "BZ0RMva9S_KT" }, "outputs": [], "source": [ "#@title Construction 2: Verification\n", "import scipy as sp\n", "\n", "print(f'Construction 2 has {len(construction_2)} points in {construction_2.shape[1]} dimensions.')\n", "pairwise_distances = sp.spatial.distance.pdist(construction_2)\n", "min_distance = np.min(pairwise_distances)\n", "max_distance = np.max(pairwise_distances)\n", "\n", "ratio_squared = (max_distance / min_distance)**2\n", "print(f\"Ratio of max distance to min distance: sqrt({ratio_squared})\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "iP3SQCuoS_KU" }, "outputs": [], "source": [ "#@title Construction 2: Visualization\n", "import matplotlib.pyplot as plt\n", "\n", "def plot_point_configuration(points: np.ndarray):\n", " \"\"\"Plots a 3D point configuration, highlighting closest and furthest pairs.\"\"\"\n", " condensed_distances = sp.spatial.distance.pdist(points)\n", " min_distance = np.min(condensed_distances)\n", " max_distance = np.max(condensed_distances)\n", " ratio = (max_distance / min_distance)**2\n", " distance_matrix = sp.spatial.distance.squareform(condensed_distances)\n", "\n", " max_distance_pairs_indices = np.where(np.isclose(distance_matrix, max_distance))\n", " min_distance_pairs_indices = np.where(np.isclose(distance_matrix, min_distance))\n", "\n", " # Create the 3D plot.\n", " fig = plt.figure(figsize=(8, 8))\n", " ax = fig.add_subplot(111, projection='3d')\n", "\n", " ax.scatter(points[:, 0], points[:, 1], points[:, 2], color='black', s=100)\n", "\n", " # Highlight max distance pairs (red).\n", " for i_index, j_index in zip(*max_distance_pairs_indices):\n", " ax.plot([points[i_index, 0], points[j_index, 0]], [points[i_index, 1], points[j_index, 1]], [points[i_index, 2], points[j_index, 2]], color='red', linewidth=2, label='maximum distance' if 'maximum distance' not in ax.get_legend_handles_labels()[1] else \"\")\n", "\n", " # Highlight min distance pairs (blue lines).\n", " for i_index, j_index in zip(*min_distance_pairs_indices):\n", " ax.plot([points[i_index, 0], points[j_index, 0]], [points[i_index, 1], points[j_index, 1]], [points[i_index, 2], points[j_index, 2]], color='blue', linewidth=2, label='minimum distance' if 'minimum distance' not in ax.get_legend_handles_labels()[1] else \"\")\n", "\n", " # Customize plot appearance.\n", " plt.title(f\"{len(points)} points in the 3D space with ratio between maximum distance and minimum distance ~ sqrt({ratio:.9f})\")\n", " plt.gca().set_aspect('equal', adjustable='box')\n", " plt.legend()\n", " plt.show()\n", "\n", "plot_point_configuration(construction_2)" ] }, { "cell_type": "markdown", "metadata": { "id": "C0Wtgl9XeJ2c" }, "source": [ "# B.9. The Heilbronn problem for triangles\n", "\n", "The goal of this problem is to find $n$ points on or inside a triangle with unit area so that the area of the smallest triangle formed by these points is maximized. The state of the art can be found on [Erich Friedman's homepage](https://erich-friedman.github.io/packing/heiltri/).\n", "\n", "For $n=11$, the SOTA was $0.036$, and AlphaEvolve found a construction with minimum area larger than $0.0365$, which is shown below." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "6zDuAcDEeJ2c" }, "outputs": [], "source": [ "#@title Data\n", "import numpy as np\n", "found_points = np.array([[0.855969059106645, 0.0], [0.2956511458813343, 0.0], [0.5084386802534074, 0.7384882411813929], [0.4328408482561001, 0.32744948893270326], [0.6757703386424172, 0.2918847665632379], [0.13872648895775305, 0.2402813272304711], [0.11466976286752831, 0.05646046982765845], [0.647825572940666, 0.609984000793226], [0.3612735110422483, 0.6257440765539699], [0.5851055464997592, 0.13484874011447245], [0.9279845295533241, 0.12473445374461754]])\n", "\n", "# Vertices of an equilateral triangle that contains the points.\n", "a = np.array([0, 0])\n", "b = np.array([1, 0])\n", "c = np.array([0.5, np.sqrt(3)/2])" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "MJ7jQiE2eJ2c" }, "outputs": [], "source": [ "#@title Verification\n", "import itertools\n", "\n", "def check_inside_triangle(points: np.ndarray):\n", " \"\"\"Checks that all points are inside the triangle with vertices (0,0), (1,0), (0.5, sqrt(3)/2).\"\"\"\n", " for (x, y) in points:\n", " assert (y >= 0) and (np.sqrt(3) * x <= np.sqrt(3) - y) and (y <= np.sqrt(3) * x), f'Point ({x}, {y}) is outside the equilateral triangle.'\n", "\n", "def triangle_area(a: np.array, b: np.array, c: np.array) -> float:\n", " return np.abs(a[0]*(b[1] - c[1]) + b[0]*(c[1] - a[1]) + c[0]*(a[1] - b[1])) / 2\n", "\n", "check_inside_triangle(found_points)\n", "print(f\"Number of points is {len(found_points)} and they lie on or inside the equilateral triangle.\")\n", "\n", "min_triangle_area = min([triangle_area(p1,p2,p3) for p1, p2, p3 in itertools.combinations(found_points, 3)])\n", "# Normalize the minimum triangle area (since the equilateral triangle is not unit).\n", "min_area_normalized = min_triangle_area / triangle_area(a, b, c)\n", "print(f\"Minimum triangle area: {min_area_normalized}.\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "5pnRAJa7Og6g" }, "outputs": [], "source": [ "#@title Visualization\n", "import matplotlib.pyplot as plt\n", "import matplotlib.patches as patches\n", "\n", "def plot_points_in_triangle(points: np.ndarray, triangle_vertices: np.ndarray):\n", " \"\"\"Plots the points and the outer equilateral triangle.\"\"\"\n", " _, ax = plt.subplots(1, figsize=(5, 5))\n", " ax.set_aspect('equal')\n", "\n", " # Plot equilateral triangle.\n", " triangle_patch = patches.Polygon(triangle_vertices, closed=True, edgecolor='black', facecolor='none')\n", " ax.add_patch(triangle_patch)\n", "\n", " # Plot points inside the triangle.\n", " ax.scatter(points[:, 0], points[:, 1], color='blue')\n", "\n", " ax.set_xlim(-0.1, 1.1)\n", " ax.set_ylim(-0.1, 1.0)\n", " ax.set_title(\"Points inside a unit-area triangle that maximize the area of the smallest formed triangle\")\n", " ax.set_axis_off()\n", "\n", " plt.grid(False)\n", " plt.show()\n", "\n", "plot_points_in_triangle(found_points, np.array([a,b,c]))" ] }, { "cell_type": "markdown", "metadata": { "id": "pVCvLQobeJ2d" }, "source": [ "# B.10. The Heilbronn problem for convex regions\n", "The goal of this problem is to find $n$ points on or inside a convex region with unit area so that the area of the smallest triangle formed by these points is maximized. The state of the art can be found on [Erich Friedman's homepage](https://erich-friedman.github.io/packing/heilconvex/).\n", "\n", "\n", "* For $n=13$, the SOTA was $0.0306$, and AlphaEvolve improved it to $0.0309$ (Construction 1).\n", "* For $n=14$, the SOTA was $0.0277$, and AlphaEvolve improved it to $0.0278$ (Construction 2).\n", "\n", "The constructions found by AlphaEvolve are shown below." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "bgpm7JYXeJ2d" }, "outputs": [], "source": [ "#@title Verification function\n", "import numpy as np\n", "def triangle_area(p1: np.ndarray, p2: np.ndarray, p3: np.ndarray) -> float:\n", " \"\"\"Calculates the area of a triangle given its vertices p1, p2, and p3.\"\"\"\n", " return abs(\n", " p1[0] * (p2[1] - p3[1])\n", " + p2[0] * (p3[1] - p1[1])\n", " + p3[0] * (p1[1] - p2[1])\n", " ) / 2" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "aY5d4RhOeJ2d" }, "outputs": [], "source": [ "#@title Visualization function\n", "import matplotlib.pyplot as plt\n", "\n", "def plot_heilbronn_construction(points: np.ndarray, min_triangle_area: float):\n", " \"\"\"Plots a set of points and their convex hull.\"\"\"\n", " plt.figure(figsize=(8, 8))\n", " plt.gca().set_aspect('equal', adjustable='box')\n", "\n", " # Plot the points.\n", " plt.plot(points[:, 0], points[:, 1], 'bo')\n", "\n", " # Plot the convex hull.\n", " for simplex in ConvexHull(points).simplices:\n", " plt.plot(points[simplex, 0], points[simplex, 1], 'k-', linewidth=2)\n", "\n", " plt.title(f'{len(points)} points on or inside a unit-area convex polygon with smallest triangle having area {min_triangle_area}')\n", " plt.axis('off')\n", " plt.show()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "1E2pBLfXeJ2g" }, "outputs": [], "source": [ "#@title Construction 1: Data\n", "import numpy as np\n", "\n", "construction_1 = np.array([[0.7262503272554176, 0.4388890568372681],[0.5700515606090156, 0.3079437775114338],[0.6327871120895331, 0.5328479759933831],[0.6416207987804122, 0.31270211238697715],[0.5148490471164644, 0.3395137443428024],[0.5679223148906077, 0.5390628289790125],[0.495131275787839, 0.41235659338159497],[0.5800095646202962, 0.46771551597629857],[0.6499329141672348, 0.3910833999601802],[0.5584150294032443, 0.38652266849736683],[0.7052096091448784, 0.48702642736130247],[0.7055856016358004, 0.3662981535556451],[0.5096822331932992, 0.49886360333134927]])" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "h-4iaoSZOnks" }, "outputs": [], "source": [ "#@title Construction 1: Verification\n", "import itertools\n", "from scipy.spatial import ConvexHull\n", "\n", "min_triangle_area = min([triangle_area(p1,p2,p3) for p1, p2, p3 in itertools.combinations(construction_1, 3)])\n", "convex_hull_area = ConvexHull(construction_1).volume\n", "construction_1_score = min_triangle_area / convex_hull_area\n", "print(f'Construction 1 has {len(construction_1)} points.')\n", "print(f'After normalizing so the points lie in a unit convex region, the minimum triangle has area {construction_1_score}.')" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "B3xTp9LQS51K" }, "outputs": [], "source": [ "#@title Construction 1: Visualization\n", "plot_heilbronn_construction(construction_1, construction_1_score)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "bDmyELp7S51K" }, "outputs": [], "source": [ "#@title Construction 2: Data\n", "construction_2 = np.array([[0.20379865808395928, 0.38641175027464647],[0.7322582251226939, 0.8023825499098627],[0.9235020285628059, 0.5636539878831605],[0.5406929358339732, 0.43236480927414933],[0.5107352242483283, 0.23351702480518888],[0.33090123099682506, 0.5368686340109353],[0.6996892191561037, 0.5984909224208574],[0.05974508135403103, 0.8227363439100202],[0.01267557937837638, 0.5684070460592283],[0.7656257377750475, 0.19333896865308486],[0.18465416566003395, 0.9316973270780137],[0.4813032999198722, 0.9273920472734931],[0.3521502387277767, 0.7716110101695554],[0.9232590363682629, 0.39434874250593155]])" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "iLiK-u0ES51K" }, "outputs": [], "source": [ "#@title Construction 2: Verification\n", "import itertools\n", "from scipy.spatial import ConvexHull\n", "\n", "min_triangle_area = min([triangle_area(p1,p2,p3) for p1, p2, p3 in itertools.combinations(construction_2, 3)])\n", "convex_hull_area = ConvexHull(construction_2).volume\n", "construction_2_score = min_triangle_area / convex_hull_area\n", "print(f'Construction 2 has {len(construction_2)} points.')\n", "print(f'After normalizing so the points lie in a unit convex region, the minimum triangle has area {construction_2_score}.')" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "oAlyfV3GS51K" }, "outputs": [], "source": [ "#@title Construction 2: Visualization\n", "plot_heilbronn_construction(construction_2, construction_2_score)" ] }, { "cell_type": "markdown", "metadata": { "id": "L00ZLkfF8iW9" }, "source": [ "# B.11 Kissing number in dimension 11\n", "The kissing problem asks how many spheres can be arranged tangent to a given sphere, if they all have the same size and their interiors do not overlap. The maximum such number in $d$ dimensions is called the $d$-dimensional kissing number.\n", "For $d=11$, the best known lower bound was 592 ([Ganzhinov (2022)](https://www.arxiv.org/abs/2207.08266)), and AlphaEvolve improved this to 593.\n", "\n", "This section contains the description of a sphere packing in dimension 11, discovered by AlphaEvolve, which shows that the kissing number in dimension 11 is at least 593. That is, it shows that one can arrange 593 nonoverlapping unit spheres in dimension 11 so that they all touch a common unit sphere. This follows from the following lemma, which is proved in our paper\n", "[Novikov et al. (2025)](https://storage.googleapis.com/deepmind-media/DeepMind.com/Blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/AlphaEvolve.pdf).\n", "\n", "\n", "**Lemma.**\n", "Let $C \\subset \\mathbb{R}^d$ be a set of points satisfying\n", "$0 \\notin C$ and\n", "\\begin{equation*}\n", " \\min \\left\\{ \\|x-y\\| : x \\neq y \\in C \\right\\} \\ge \\max \\left\\{ \\|x\\|: x \\in C \\right\\}.\n", "\\end{equation*}\n", "Then unit spheres centred at $\\left\\{\\frac{2x}{\\|x\\|}: {x\\in C}\\right\\}$ form a valid kissing configuration in dimension $d$. In particular, the kissing number in dimension $d$ is at least $|C|$." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "DI1axbjLv9CC" }, "outputs": [], "source": [ "#@title Data\n", "import numpy as np\n", "import itertools\n", "\n", "sphere_centers = np.array([[ -126140549599, -4345934944399, 470267207045, 263582420739,\n", " -214242641164, 6323834592662, -5619609064852, -1309426459793,\n", " 402713598698, 2669019955225, -434830723701],\n", " [ 4995872033121, -3686427151476, -2854162550811, 1374657172375,\n", " -1039798804579, 2439226626247, -3321352498532, -4613529709005,\n", " 1873895977070, -2351434851488, -1740061033492],\n", " [ -148878709727, 6009198311530, 3423705484214, -3908379065200,\n", " -2039242115428, 740038679089, -1903248798804, -3229808619747,\n", " -67448542119, 2608302674396, 3361875535061],\n", " [-3613653483807, -421965187481, 4386947861356, 4668066279141,\n", " 2659643185631, -793481959702, -782545296485, 3044001988734,\n", " -210044264644, 732930704465, 5250176454220],\n", " [ 2145601851578, 2286907266338, -6008630565540, -951491851263,\n", " -1128202168949, -2351035905822, 338686846605, 2773377260,\n", " -3167080097124, -2121084647622, 5631442707990],\n", " [ 2905871765916, -8266251805063, 212038997927, -422612434475,\n", " -2967768531066, 917651742589, -1199305228751, -428118279913,\n", " 1046645354339, 629591995938, 3199722319491],\n", " [ 4101374281925, 140190714949, 7791359990956, 2617604236962,\n", " 2117466506344, 1449617600436, 733498008403, 1078801552552,\n", " -1108251520223, -2353519449875, -740114970023],\n", " [-4114734847133, -711568804977, -1096919699416, -2179625341981,\n", " -3277074074174, 3846872693149, -1343845317573, -4029410886259,\n", " -760938481950, 4323960137424, 3708537789894],\n", " [ 3749853727043, -920834513083, -2932082478667, 453628989161,\n", " -3091441942608, 6398413670681, 2921263471507, 652306845488,\n", " 3309571127773, -1260515535742, -2071139339072],\n", " [-5502577676278, -1227448640831, -3806962231330, -700239558766,\n", " -634751583528, 218595111082, 36644542589, -4043060252879,\n", " -4497661105875, -2852762755592, 2840093077865],\n", " [ 6431155805256, 2472183772617, 1852141756766, -3447375436498,\n", " -2150831199996, -1035995564180, -582868441525, -7570292152,\n", " -2212911701699, 4854953687339, -1645298702536],\n", " [ 5457305326303, 1209487773311, 3838331900010, 672841580707,\n", " 640009753701, -223039211201, -29010721761, 4023147410765,\n", " 4562777370943, 2817042290572, -2858123518221],\n", " [ 947994596354, -493063270445, -891551370805, 1943900881453,\n", " 4808222709486, 4576541247433, 121556039265, -2928496070412,\n", " 2867556918429, 5676016954336, -1090838534345],\n", " [ 2487751367449, -4435069885581, 1203527162971, -7039059911283,\n", " -2789489469568, -1507124796768, 408730098019, 2199308898207,\n", " -2690789476039, -886790999066, 246907513281],\n", " [ 1258393365327, 1558236082994, -2901279533502, -4588804764838,\n", " 608788356464, 99979762711, -2072852177408, -5237300731547,\n", " 2206084299760, 5428740521977, -262884833863],\n", " [-2167736272270, -3517473061426, -2581444334945, -4875068237304,\n", " 1316469763812, -5045504207465, 1421790102552, -1042617987235,\n", " -2433031020177, 1273290801609, 3828642503948],\n", " [-1748341091839, 84738171505, 491018476370, 4552281419561,\n", " -872666527835, 3684504527423, -4225299894261, -2607643958170,\n", " 1747681580888, -38267500797, 5824587161476],\n", " [-1190649965834, -4307368301932, -590143473885, 1188487865738,\n", " 1457925043503, 5074991372962, -835020531624, 4328426802452,\n", " 1938681290116, 658478446157, 5173244484302],\n", " [ 1157198248631, -3276093261966, 4564492907322, 71866472319,\n", " 6569376668375, 2635857316126, -2535627189471, 666339223157,\n", " -370257470349, 2316182402253, 2146785163984],\n", " [-3008255020828, 3938057143766, -5277192922587, 593373194319,\n", " 313114162483, 4123287795940, 1308453155442, -2544545673675,\n", " 1482541632927, -605248563516, 4403486139737],\n", " [ 2953917709599, -440143619596, -971489179779, 2328335269090,\n", " 2051703342736, -1981036966235, 1096402607228, -8032142516776,\n", " 1008884833142, 2051765017859, -2374176269185],\n", " [ -260449853094, -1557229191143, -2449904390021, 7358451024369,\n", " -307525874663, -2110837486887, -1514682853981, -3642026943603,\n", " -341788837820, -3181174130215, -2648178657093],\n", " [-4525545210765, -4561293261761, -2207465574022, -3656150761762,\n", " -1355527319115, -1325108959607, -2606830665241, 4286163870684,\n", " -2272423058824, 2270296581664, -1181276273370],\n", " [-4989184462869, -2211501213310, -3272216512438, 2932576526381,\n", " -4053121684822, -741725216911, -1173137805746, 2181007677136,\n", " -2973820944500, -3480396984635, -2615745041603],\n", " [ 3602451243212, 1363170942462, -5335236817537, -5099315765146,\n", " 1671801957653, -733157544910, -1557322490939, 3489905697601,\n", " -2568635080802, 2482181047248, 9540974612],\n", " [ -117106507206, -1557986758662, -1613271333904, 3599924179366,\n", " -4726770521248, -122491678154, 3003684160051, -2037648443680,\n", " -1689648678287, -6017572885613, 2720160888071],\n", " [ 656948042076, 5303758392177, 1223385282487, 4208826988200,\n", " -2322770619544, -1931938705746, 194867179198, -149785416225,\n", " -975234488353, -6149937849011, -2065621641353],\n", " [-2822884624434, 1200686951368, -740362280165, 2549091748238,\n", " 6473313457564, 198867059897, -5879821576926, -497127882332,\n", " -583143250935, -1042480487756, 2313887084841],\n", " [ 2071972627918, 1689375919671, 2561773181285, -215304283110,\n", " 3678351753570, -481570249929, -4708236211267, -713440165548,\n", " -2024413488114, -6510658512889, -1821622848642],\n", " [ 1540576902788, 553319845306, 5192033407329, 3790472513109,\n", " -989994836245, -6751185485237, 63271511119, 1311050643347,\n", " -364367110835, -1981107885558, 1912214169188],\n", " [ 1043725858252, 1320396911651, -3526919864724, -4284357877757,\n", " 1470366412833, 4775346083201, 1368745214981, 3701799302088,\n", " 2580249413700, -3202483086523, 2985971881393],\n", " [ 2328981318412, 782277770357, 1613319529120, 6708838015807,\n", " 2409644663632, -164697695288, -2876391291223, 1405607425611,\n", " 2563177561828, -4390790783825, 2102566367890],\n", " [-2898694569620, -1993213074975, 4767991564340, -549183545871,\n", " -1716073975618, -1376827268341, 3962365401068, -1804381043737,\n", " 364314410734, -2366368669369, -5921165923758],\n", " [-2821980692939, -2907092870119, 526484763216, -283219412226,\n", " 1816656338961, 4630761564997, 5094816643786, 822612609763,\n", " -1628794974160, 4497378787802, -2995063685088],\n", " [ 3981893553028, 6726656929267, 4509428321078, -1718973949677,\n", " 1235696414272, -3105774701931, -561577844505, 806709024917,\n", " 670699852230, -1703554165369, -340483077176],\n", " [ 958740882, 492343935025, -4877156942860, 3610017499034,\n", " 1303862660779, 2282799096236, -4071872206188, -4319287760002,\n", " -3382516964626, 3040863514676, 320653772456],\n", " [ 2668842066670, -3328224345221, 1245442050272, 807268381271,\n", " 6338371398067, -1028063433204, 2718246800023, 1487663933151,\n", " 641493721090, 2991619189786, -4404795224958],\n", " [ 491355192733, -1231352019317, 2214609261146, -4246171762475,\n", " 3060427833343, -6012660473699, 369697486180, -1926711613647,\n", " 1222190747138, -4895424389279, -694078468931],\n", " [-3936231732764, 3646035008845, 3917837393650, -2302689585385,\n", " -631606199075, -1190907291088, -1462454097742, -1026315700574,\n", " -3403121373084, 4358068529006, -3870064655165],\n", " [ -891693194815, -3159218172959, -36815057657, -5392159737721,\n", " 2695159539664, 4846777170940, -2774317263822, 2475655478805,\n", " -3507372987730, -1040335371092, 1477389122043],\n", " [-1909886271559, -3025194791002, 273889648930, 22368016637,\n", " 1769343232414, -6711034204699, 1807271442431, 5498553429397,\n", " -2113841429687, -965335059539, -236521072887],\n", " [ 1421740442238, -4445189941322, -6335315158518, -671536406806,\n", " 2646092406700, -638731456569, -171872290986, -2000918638949,\n", " 2251948982557, 2831806435076, -3619165775254],\n", " [-3348523808774, -3469324536046, -76708109001, 1992777744163,\n", " -5049711606642, 581253431088, -3927998298964, 2296310725868,\n", " 2561198403600, 3752242898555, 2365461437834],\n", " [ 779042434547, -1416461625219, 1452341221883, 1439971242011,\n", " 1285662467259, 2679379627727, 1926099800655, -1269351586896,\n", " 2663459313927, -8084575368127, -2568350771324],\n", " [-3980831743069, 3637325407505, -1572468831509, -2438291232144,\n", " -3064062866153, -2476525715560, 4274880397145, -2716987458463,\n", " -1175083650793, 1660991539240, 4145711223394],\n", " [ 6967450268144, 2625304133625, -1401474346935, 1995863610145,\n", " 763569629212, -3804173165507, 997773279985, 1860188439504,\n", " 1442735815840, 1895879894694, 3664470120161],\n", " [ 3015840597743, 1234709422899, -6952463247052, -1298276880914,\n", " -1224429458570, -1308306783583, 972871140414, 2450262921520,\n", " 5290015394161, -560255643005, -947955712831],\n", " [ -936946977342, 1116679407669, -396124541576, -5766839382952,\n", " 2927744211058, -1495168010655, 2861812090255, 3711225318549,\n", " -3673521240929, -3839839100081, 1858038850084],\n", " [ 1308510697687, -4987098459834, -5143597325210, -1869372147752,\n", " -978771635962, 3179840980566, -3422701861744, 3615532701633,\n", " 1765018082471, -1744159035660, -1204896731728],\n", " [-1646730796381, 4421637104002, 37254831639, 4273667291949,\n", " -655020690827, -2640710658714, 1397627767521, -1309174476974,\n", " 1380616410107, -2725730878362, 6250309303605],\n", " [ -627474373499, -451185628535, 1125092239854, 1228838206359,\n", " 26093010133, -2939555208566, 3370434196692, 667250715880,\n", " -7209365367712, -1252881093849, 4757673344364],\n", " [ -797420603404, -1227566289334, 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1618110226052, -2193439357575, 1695044091886, 397656341641,\n", " 1073388847499, -1576671686419, -2843336252998, 407189794653,\n", " 3766298781170, 4112633385347, 6822906738995],\n", " [ 6164789949614, 136739143580, 2175454310590, -622114272459,\n", " 2011485522508, 3965112522977, 1210393364110, -2980277210958,\n", " 902750629627, 449521686336, -5071953818337],\n", " [ 3881218273012, 1479027175218, 2070483335377, 75560330693,\n", " 1057802184062, -1384772852312, 2238953203018, 945086030287,\n", " -776171777608, -7173099477397, 4178226133483],\n", " [ 3235814803974, 1256294864562, -650900085810, -4903438216654,\n", " -3214890352777, 1970650069612, -252075430182, 5758156457942,\n", " 1664750015810, -670281194160, -3581101717788],\n", " [ 3378074137115, -5251527924600, 3110186772230, 2979998202959,\n", " 2321062449968, -1804075580466, 996582370415, 4811856309508,\n", " 2164952889186, -282367400925, 2213709332985],\n", " [ 574905043173, -4386090861270, -1092429566069, -3352980169208,\n", " 2327865872888, 1391131336277, 3388203163344, 6943674361107,\n", " 166333751023, 709426251370, -645073660217],\n", " [-5093483115816, -172520412035, -1119927279528, -298914389540,\n", " -3684232686653, -2715581429324, -5996134939421, -2654472304753,\n", " -2448900299536, 1566386998418, -533468096138],\n", " [ -694829797925, 1077657175865, 2426418664382, -6898838565028,\n", " -630728354927, 891813183909, -3004655039870, 712908865280,\n", " 1510580937205, -5343119532124, 1820512127979],\n", " [-2431689551338, -5427049186581, -2338557330124, -815424187913,\n", " 981962290892, -3062993059108, 1128316606362, 2535891852174,\n", " 6084947796540, 617493934047, 1744232021160],\n", " [-5435703869515, -2294163621518, 1848370199428, 2371054182793,\n", " -3877945120584, 3741103307158, 738838875423, 1180167173884,\n", " -2875974256951, -1153626673599, 3946706754350],\n", " [ 8219585512141, -3454414521062, -2804725743538, -219356185092,\n", " -1174314716502, -448706854993, 982240147012, 2239687163655,\n", " -1333597836815, 384385903109, -1761603754385],\n", " [ 1591055287346, 5678683783494, -615012178719, -628567142182,\n", " -3870188383064, -669880486908, 1145281971929, 4449831881766,\n", " 3042481569158, -3394493236261, 2669963658827],\n", " [-3754248407042, 5294073804022, -296736245623, 2912063924725,\n", " -791691392173, 751000806695, -1982989567750, -407567361095,\n", " -5153438369097, 2080369983896, 3624255814498],\n", " [-1109208301621, 1105002101950, -1209577854326, -2024207507329,\n", " 7350334972180, -3488128656973, -1651140239034, 2095339824418,\n", " -2280768401335, -1030270576505, -3523292766568],\n", " [-5069180086335, -1838030708503, 3563978996722, -302815316047,\n", " 440743640453, -3187355375955, 4347216998557, -1328131210369,\n", " 353915357918, -3762804268885, 3582142309006],\n", " [-3661582264231, -11651104094, -6216800774575, -1027515731357,\n", " 2255363625886, -1074441912663, 87463515947, 5662135623408,\n", " 329269044501, -2851689332257, -582416329253],\n", " [ 56248471539, 4946765000660, 3758738313061, 4784112795759,\n", " 1394864083549, -728062773091, 3727632364237, 3905942835080,\n", " 2549768126638, -577510027537, 219804846640],\n", " [ 5587427890418, 2297669997514, -1020505798020, -6126153847699,\n", " -542119172009, -1750346906402, 3777573935242, 426662882168,\n", " 1513176033026, -1672209304069, 1426348449094],\n", " [ 259175505863, 4398571312812, -563569394904, -182986310141,\n", " 198425934183, -6310433112917, 5597938858309, 1367486286215,\n", " -595778885575, -2563939288918, 487534331752],\n", " [-4802134395185, 159246504059, 1550301624727, -4853626304168,\n", " -500450446506, -3445176305248, -1086016460941, -198494483359,\n", " 3935712232757, 4174986491713, 2166600367239],\n", " [-2955758367798, 197182840185, 5575429750308, 4821829786482,\n", " -2151221731203, 1377147185183, 1369265209861, -2672300951780,\n", " 3787808035786, -2629332593910, 296485252008],\n", " [ -907637263048, -2208932588521, -3516691104412, -1517457288318,\n", " 1659108833526, -1386977275547, -7560769135554, 467600743208,\n", " 2452516555656, 3332072477470, 670257831686],\n", " [-5343273458688, 2094742859491, 467642153205, 5934014125908,\n", " 741852808966, -4562356494190, 1823039549615, 934097259328,\n", " -2087710423337, -903735244984, -945273930686],\n", " [-7240969135778, -2639902878880, 849460636225, 414183625708,\n", " -726198001149, -2021521159959, 2097041348792, -1405395811621,\n", " -3323825049514, 4169543904047, 535356746516],\n", " [-2450513019016, 1863099990948, -2437818958899, -1649975566093,\n", " 913002479189, -2735144788138, -3826123267412, -4408375983513,\n", " 804673891768, 752627625256, 6185218403530],\n", " [-1335399479000, 3823343667766, -5717460499457, -4005933034396,\n", " 1194120926048, 434347561966, 5544143288165, 32763093488,\n", " -153238809945, -626002138389, -1446843381290],\n", " [ 3384149124219, -3775174772430, -505726264120, -482507757887,\n", " -2778435182795, -5352559874474, 4355591684268, 1038732361168,\n", " 127898700843, -1750769851358, 3782290455385],\n", " [ 3090835195085, -3054792832893, -1170619376721, 128325298949,\n", " 2636523183051, -2257212225518, -784444777154, -2644511696348,\n", " -2635072545182, -2717234652626, -6763588745282],\n", " [-3729763407548, 2337035244907, -846339999856, -4541567007554,\n", " 1974270756041, 2750510194087, 636060278562, 4678194935433,\n", " -2434295636669, 1169387614202, -4270669080080],\n", " [ -182352232618, -15639990365, -806361709654, 3736501768658,\n", " 4423778632034, -1993974011012, -4511280703919, -1620258902598,\n", " 1407442349625, 2801596965688, -5385383608572]], dtype=np.int64)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "fNADwGPxv9CC" }, "outputs": [], "source": [ "#@title Verification\n", "def compute_squared_norm(point: list[int]) -> int:\n", " \"\"\"Returns the squared norm of an integer vector using exact computation.\"\"\"\n", " return sum(pow(int(x), 2) for x in point)\n", "\n", "\n", "def verify_sphere_packing(sphere_centers: np.ndarray):\n", " \"\"\"Checks that after normalizing, the points correspond to a valid sphere packing for kissing numbers.\n", "\n", " Args:\n", " sphere_centers: the list of sphere centers, of shape [num_spheres, dimension].\n", "\n", " Raises:\n", " AssertionError: if the sphere packing is not a valid kissing configuration.\n", " \"\"\"\n", " # Rounding to integers to guarantee exact computation throughout.\n", " sphere_centers = np.around(sphere_centers).astype(np.int64)\n", " squared_norms = [compute_squared_norm(list(center)) for center in sphere_centers]\n", "\n", " # Checks that the set doesn't contain 0.\n", " min_squared_norm = min(squared_norms)\n", " assert min_squared_norm > 1e-6, f'Verification failed because the set contains 0.'\n", "\n", " # Checks that the minimum pairwise distance between centers >= the maximum norm of the centers.\n", " max_squared_norm = max(squared_norms)\n", " min_squared_distance = min(compute_squared_norm(list(a - b)) for a, b in itertools.combinations(sphere_centers, 2))\n", " assert min_squared_distance >= max_squared_norm, f'Verification failed because the minimum squared distance = {min_squared_distance} < {max_squared_norm} = maximum squared norm.'\n", "\n", "\n", "num_spheres = sphere_centers.shape[0]\n", "dimension = sphere_centers.shape[1]\n", "\n", "verify_sphere_packing(sphere_centers)\n", "print(f'Verified the sphere packing showing kissing number in dimension {dimension} is at least {num_spheres}.')" ] }, { "cell_type": "markdown", "metadata": { "id": "v3kEu5yVeJ2h" }, "source": [ "# B.12. Packing circles inside a unit square to maximize sum of radii\n", "\n", "Given a positive integer $n$, the problem is to pack $n$ disjoint circles inside a unit square so as to maximize the sum of their radii. The state of the art can be found on [Erich Friedman's homepage](https://erich-friedman.github.io/packing/cirRsqu/).\n", "\n", "* For $n=26$, the SOTA was $2.634$, and AlphaEvolve improved it to $2.635$ (Construction 1).\n", "* For $n=32$, the SOTA was $2.936$, and AlphaEvolve improved it to $2.937$ (Construction 2).\n", "\n", "The constructions found by AlphaEvolve are shown below." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "PKGX_1T8eJ2h" }, "outputs": [], "source": [ "#@title Verification function\n", "import numpy as np\n", "import itertools\n", "\n", "def verify_circles(circles: np.ndarray):\n", " \"\"\"Checks that the circles are disjoint and lie inside a unit square.\n", "\n", " Args:\n", " circles: A numpy array of shape (num_circles, 3), where each row is\n", " of the form (x, y, radius), specifying a circle.\n", "\n", " Raises:\n", " AssertionError if the circles are not disjoint or do not lie inside the\n", " unit square.\n", " \"\"\"\n", " # Check pairwise disjointness.\n", " for circle1, circle2 in itertools.combinations(circles, 2):\n", " center_distance = np.sqrt((circle1[0] - circle2[0])**2 + (circle1[1] - circle2[1])**2)\n", " radii_sum = circle1[2] + circle2[2]\n", " assert center_distance >= radii_sum, f\"Circles are NOT disjoint: {circle1} and {circle2}.\"\n", "\n", " # Check all circles lie inside the unit square [0,1]x[0,1].\n", " for circle in circles:\n", " assert (0 <= min(circle[0], circle[1]) - circle[2] and max(circle[0],circle[1]) + circle[2] <= 1), f\"Circle {circle} is NOT fully inside the unit square.\"" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "htBb_J21OS7V" }, "outputs": [], "source": [ "#@title Visualization function\n", "import matplotlib.pyplot as plt\n", "import matplotlib.patches as patches\n", "\n", "def plot_circles(circles: np.ndarray):\n", " \"\"\"Plots the circles.\"\"\"\n", " _, ax = plt.subplots(1, figsize=(7, 7))\n", " ax.set_xlim(0, 1)\n", " ax.set_ylim(0, 1)\n", " ax.set_aspect('equal') # Make axes scaled equally.\n", "\n", " # Draw unit square boundary.\n", " rect = patches.Rectangle((0, 0), 1, 1, linewidth=1, edgecolor='black', facecolor='none')\n", " ax.add_patch(rect)\n", "\n", " # Draw the circles.\n", " for circle in circles:\n", " circ = patches.Circle((circle[0], circle[1]), circle[2], edgecolor='blue', facecolor='skyblue', alpha=0.5)\n", " ax.add_patch(circ)\n", "\n", " plt.title(f'A collection of {len(circles)} disjoint circles packed inside a unit square to maximize the sum of radii')\n", " plt.show()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "yBRzd7AZeJ2h" }, "outputs": [], "source": [ "#@title Construction 1: Data\n", "\n", "construction_1 = np.array([[0.68180423, 0.90401948, 0.09598051040194801], [0.90598057, 0.49972596, 0.094019420598057], [0.08464495, 0.08464502, 0.084644941535505], [0.4818405, 0.1034156, 0.10341558965844], [0.48234279, 0.89652324, 0.10347674965232401], [0.88928295, 0.11071646, 0.11071644892835401], [0.07852351, 0.50033803, 0.07852350214764901], [0.27330428, 0.1051224, 0.10512238948776001], [0.38153556, 0.70211016, 0.115517618448237], [0.13252625, 0.70386646, 0.132047406795258], [0.59610341, 0.72720176, 0.10051153994884501], [0.59576502, 0.2725969, 0.10054788994521], [0.88902242, 0.88902318, 0.11097680890231801], [0.68087256, 0.09573004, 0.095730030426996], [0.53098174, 0.49996661, 0.13587084641291403], [0.90742967, 0.68631006, 0.09257029074297], [0.74197859, 0.40386139, 0.095943250405674], [0.08471273, 0.91528735, 0.08471263152873601], [0.76300349, 0.7598366, 0.069494603050539], [0.27358888, 0.89472069, 0.105279299472069], [0.13242993, 0.29639184, 0.132426596757339], [0.90745313, 0.31316575, 0.092546580745341], [0.76284098, 0.23948486, 0.06975189302481001], [0.38095848, 0.29797266, 0.115741008425898], [0.74207757, 0.59573774, 0.09593309040669], [0.27141024, 0.50032568, 0.114361818563817]])" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "8MQMI82ueJ2h" }, "outputs": [], "source": [ "#@title Construction 1: Verification\n", "print(f\"Construction 1 has {len(construction_1)} circles.\")\n", "verify_circles(construction_1)\n", "print(f\"The circles are disjoint and lie inside the unit square.\")\n", "sum_radii = np.sum(construction_1[:, 2])\n", "print(f\"Construction 1 sum of radii: {sum_radii}\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "uFs-GsNweJ2h" }, "outputs": [], "source": [ "#@title Construction 1: Visualization\n", "plot_circles(construction_1)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "-g_h2I9PeJ2i" }, "outputs": [], "source": [ "#@title Construction 2: Data\n", "\n", "construction_2 = np.array([[0.09076163, 0.40381803, 0.090761620923837], [0.07310993, 0.92689178, 0.07310821268917801], [0.08745017, 0.22570576, 0.087381421261857], [0.24855246, 0.30880277, 0.093428060657193], [0.4079865, 0.06300614, 0.063006133699386], [0.47646318, 0.90136179, 0.09863820013617901], [0.89604966, 0.10309934, 0.10309932969006601], [0.9066386, 0.68096117, 0.09336139066386], [0.08962002, 0.76509474, 0.0895289910471], [0.06973669, 0.06965159, 0.06965158303484101], [0.40979823, 0.21756451, 0.09156283084371601], [0.25742466, 0.88393887, 0.11606111839388701], [0.09064689, 0.58506214, 0.090482500951749], [0.90294698, 0.30231577, 0.09623644037635501], [0.57265603, 0.10585396, 0.105853949414604], [0.74007588, 0.40129314, 0.09435083056491601], [0.57539962, 0.71183255, 0.115160168483982], [0.7367635, 0.21592191, 0.09104997089500201], [0.41096972, 0.40263617, 0.093512520648747], [0.88664452, 0.88667032, 0.113317128668286], [0.57582722, 0.49961748, 0.09705531029446801], [0.24962585, 0.49417195, 0.09194421080557799], [0.90546338, 0.49309632, 0.094507120549287], [0.67381348, 0.90149423, 0.09850576014942301], [0.24310147, 0.1077195, 0.10771948922805], [0.40815297, 0.5886157, 0.09248833075116601], [0.24737889, 0.6771266, 0.090994980900501], [0.75801377, 0.7532924, 0.07192969280703], [0.73526642, 0.06243992, 0.062439303756069], [0.57415412, 0.30715219, 0.095403150459684], [0.39239379, 0.75259664, 0.07223814277618501], [0.7439361, 0.58879735, 0.093166630683336]])" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "mhoe79fPO_4Q" }, "outputs": [], "source": [ "#@title Construction 2: Verification\n", "print(f\"Construction 2 has {len(construction_2)} circles.\")\n", "verify_circles(construction_2)\n", "print(f\"The circles are disjoint and lie inside the unit square.\")\n", "sum_radii = np.sum(construction_2[:, 2])\n", "print(f\"Construction 2 sum of radii: {sum_radii}\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "Z5TP7I2AO_4R" }, "outputs": [], "source": [ "#@title Construction 2: Visualization\n", "plot_circles(construction_2)" ] }, { "cell_type": "markdown", "metadata": { "id": "7f31jbMBeJ2i" }, "source": [ "# B.13. Packing circles inside a rectangle of perimeter 4 to maximize sum of radii\n", "\n", "Given a positive integer $n$, the problem is to pack $n$ disjoint circles inside a rectangle of perimeter 4 so as to maximize the sum of their radii. The state of the art can be found on [Erich Friedman's homepage](https://erich-friedman.github.io/packing/cirRrec/).\n", "\n", "For $n=21$, the SOTA was $2.364$, and AlphaEvolve improved it to $2.365$. The construction found by AlphaEvolve is shown below." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "_trhiz5aeJ2i" }, "outputs": [], "source": [ "#@title Data\n", "import numpy as np\n", "\n", "circles = np.array([[0.12396161, 0.12396161, 0.12396159760383901], [0.0747415, 0.31647214, 0.07474149252585001], [0.12740019, 0.51163445, 0.127400177259981], [0.0747415, 0.70679675, 0.07474149252585001], [0.12396161, 0.89930729, 0.12396159760383901], [0.36284141, 0.11508312, 0.115083108491688], [0.25916864, 0.31452627, 0.10969588903041], [0.37244262, 0.51163445, 0.117642228235776], [0.25916864, 0.70874263, 0.10969588903041], [0.3628414, 0.90818577, 0.11508311849168701], [0.59083599, 0.11292169, 0.11292167870783101], [0.48083152, 0.30914434, 0.112032298796769], [0.62891008, 0.51163445, 0.138825206117478], [0.48083152, 0.71412455, 0.112032298796769], [0.59083598, 0.9103472, 0.11292168870783001], [0.67852787, 0.2919303, 0.086412051358794], [0.85766825, 0.39257159, 0.11906284809371401], [0.85766825, 0.6306973, 0.11906284809371401], [0.67852787, 0.73133859, 0.086412051358794], [0.83965981, 0.8861976, 0.13707128629287002], [0.83965981, 0.1370713, 0.13707128629287002]])" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "VaSmUodSeJ2i" }, "outputs": [], "source": [ "#@title Verification\n", "import itertools\n", "\n", "def minimum_circumscribing_rectangle(circles: np.ndarray) -> tuple[float, float]:\n", " \"\"\"Returns the width and height of the minimum circumscribing rectangle.\n", "\n", " Args:\n", " circles: A numpy array of shape (num_circles, 3), where each row is of the\n", " form (x, y, radius), specifying a circle.\n", "\n", " Returns:\n", " A tuple (width, height) of the minimum circumscribing rectangle.\n", " \"\"\"\n", " min_x = np.min(circles[:, 0] - circles[:, 2])\n", " max_x = np.max(circles[:, 0] + circles[:, 2])\n", " min_y = np.min(circles[:, 1] - circles[:, 2])\n", " max_y = np.max(circles[:, 1] + circles[:, 2])\n", " return max_x - min_x, max_y - min_y\n", "\n", "\n", "def verify_circles_disjoint(circles: np.ndarray):\n", " \"\"\"Checks that circles are disjoint.\n", "\n", " Args:\n", " circles: A numpy array of shape (num_circles, 3), where each row is of the\n", " form (x, y, radius), specifying a circle.\n", "\n", " Raises:\n", " AssertionError: if the circles are not disjoint.\n", " \"\"\"\n", " for circle1, circle2 in itertools.combinations(circles, 2):\n", " center_distance = np.sqrt((circle1[0] - circle2[0])**2 + (circle1[1] - circle2[1])**2)\n", " radii_sum = circle1[2] + circle2[2]\n", " assert center_distance >= radii_sum, f\"Circles are NOT disjoint: {circle1} and {circle2}.\"\n", "\n", "\n", "num_circles = len(circles)\n", "verify_circles_disjoint(circles)\n", "print(f\"Construction has {num_circles} disjoint circles.\")\n", "width, height = minimum_circumscribing_rectangle(circles)\n", "print(f\"Perimeter of minimum circumscribing rectangle: {2 * (width + height):.6f}\")\n", "assert (width + height) <= 2, \"The circles cannot be contained within a rectangle with perimeter 4.\"\n", "sum_radii = np.sum(circles[:, 2])\n", "print('Sum of radii:', sum_radii)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "cellView": "form", "id": "NJvCx8e_PQE_" }, "outputs": [], "source": [ "#@title Visualization\n", "import matplotlib.pyplot as plt\n", "import matplotlib.patches as patches\n", "\n", "def plot_circles_rectangle(circles: np.ndarray):\n", " width, height = minimum_circumscribing_rectangle(circles)\n", " _, ax = plt.subplots(1, figsize=(7, 7))\n", " ax.set_xlim(0, width)\n", " ax.set_ylim(0, height)\n", " ax.set_aspect('equal') # Make axes scaled equally.\n", "\n", " # Draw rectangle boundary.\n", " rect = patches.Rectangle((0, 0), width, height, linewidth=1, edgecolor='black', facecolor='none')\n", " ax.add_patch(rect)\n", "\n", " # Draw the circles.\n", " for circle in circles:\n", " circ = patches.Circle((circle[0], circle[1]), circle[2], edgecolor='blue', facecolor='skyblue', alpha=0.5)\n", " ax.add_patch(circ)\n", "\n", " plt.title(f\"{len(circles)} disjoint circles packed inside a rectangle of perimeter {round(2 * width + 2 * height,6)}\")\n", " plt.show()\n", "\n", "plot_circles_rectangle(circles)" ] } ], "metadata": { "colab": { "last_runtime": { "build_target": "//research/colab/notebook:notebook_backend_py3", "kind": "private" }, "private_outputs": true, "provenance": [ { "file_id": "1f7b28BEHukNC-y6xComZa9nzWwknb_19", "timestamp": 1746110111312 }, { "file_id": "1KXtyJor8ykra3hOk15p0KyzwYB6r-jWt", "timestamp": 1746019594363 }, { "file_id": "1rBQkQHsuhuGmC4yslrsTlqgdI4h2yN-O", "timestamp": 1745341381308 } ] }, "kernelspec": { "display_name": "Python 3", "name": "python3" }, "language_info": { "name": "python" } }, "nbformat": 4, "nbformat_minor": 0 }