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src=\"https://img.shields.io/npm/l/@algorithm.ts/lcs.svg\"\n      />\n    </a>\n    <a href=\"#install\">\n      <img\n        alt=\"Module Formats: cjs, esm\"\n        src=\"https://img.shields.io/badge/module_formats-cjs%2C%20esm-green.svg\"\n      />\n    </a>\n    <a href=\"https://github.com/nodejs/node\">\n      <img\n        alt=\"Node.js Version\"\n        src=\"https://img.shields.io/node/v/@algorithm.ts/lcs\"\n      />\n    </a>\n    <a href=\"https://github.com/vitest-dev/vitest\">\n      <img\n        alt=\"Tested with Jest\"\n        src=\"https://img.shields.io/badge/tested_with-vitest-6E9F18.svg\"\n      />\n    </a>\n    <a href=\"https://github.com/prettier/prettier\">\n      <img\n        alt=\"Code Style: prettier\"\n        src=\"https://img.shields.io/badge/code_style-prettier-ff69b4.svg?style=flat-square\"\n      />\n    </a>\n  </div>\n</header>\n<br/>\n\nA typescript implementation to find the **lcs** (Longest Common Subsequence).\n\nThis package provide three different implementation of lcs algorithm. To measure the complexity of\nthese algorithms, let the $N_1$ and $N_2$ be the subsequences length of two sequences respectively.\nAnd let $L$ be the length of the longest common subsequences, then the $D = 2L - N_1 - N_2$.\n\n1. `myers_lcs(N1: number, N2: number, equals: (x: number, y: number) => boolean): [x: number, y: number][]`:\n   The vanilla algorithm introduced by this paper\n   [_An O(ND) Difference Algorithm and Its Variations_](https://mailserver.org/diff2.pdf).\n   - Time complexity: $O((N_1 + N_2) \\times D)$\n   - Memory complexity: $O(N_1 \\times N_2)$\n\n2. `myers_lcs_linear_space(N1: number, N2: number, equals: (x: number, y: number) => boolean): [x: number, y: number][]`:\n   The linear space refinement algorithm from\n   [_An O(ND) Difference Algorithm and Its Variations_](https://mailserver.org/diff2.pdf).\n   - Time complexity: $O((N_1 + N_2) * D)$\n   - Memory complexity: $O(N_1 + N_2)$\n\n3. `lcs_dp(N1: number, N2: number, equals: (x: number, y: number) => boolean): [x: number, y: number][]`\n   This implementation is based on dynamic programming, and can find the minimal lexicographical\n   lcs.\n   - Time complexity: $O(N_1 \\times N_2)$\n   - Memory complexity: $O(N_1 \\times N_2)$\n\nThe following definition is quoted from Wikipedia\n(https://en.wikipedia.org/wiki/Longest_common_subsequence_problem):\n\n> The longest common subsequence (LCS) problem is the problem of finding the longest subsequence\n> common to all sequences in a set of sequences (often just two sequences). It differs from the\n> longest common substring problem: unlike substrings, subsequences are not required to occupy\n> consecutive positions within the original sequences. The longest common subsequence problem is a\n> classic computer science problem, the basis of data comparison programs such as the diff utility,\n> and has applications in computational linguistics and bioinformatics. It is also widely used by\n> revision control systems such as Git for reconciling multiple changes made to a\n> revision-controlled collection of files.\n\n## Install\n\n- npm\n\n  ```bash\n  npm install --save @algorithm.ts/lcs\n  ```\n\n- pnpm\n\n  ```bash\n  pnpm add @algorithm.ts/lcs\n  ```\n\n## Usage\n\n- Basic\n\n  ```typescript\n  import { lcs_dp, lcs_myers_size } from '@algorithm.ts/lcs'\n\n  const s1: number[] = [1, 2, 3, 4, 6, 6, 7, 8, 6]\n  const s2: number[] = [2, 3, 4, 7, 9, 8, 2, 3, 5, 2]\n\n  lcs_myers_size(s1.length, s2.length, (x, y) => s1[x] === s2[y]) // => 5\n\n  lcs_dp(s1.length, s2.length, (x, y) => s1[x] === s2[y])\n  // => [ [1, 0], [2, 1], [3, 2], [6, 3], [7, 5] ]\n  //\n  //    Here is why:\n  //\n  //          0 1 2 3 4 5 6 7 8 9\n  //      s1: 1 2 3 4 6 6 7 8 6\n  //      s2: 2 3 4 7 9 8 2 3 5 2\n  //\n  //      s1[1] <----> s2[0]\n  //      s1[2] <----> s2[1]\n  //      s1[3] <----> s2[2]\n  //      s1[6] <----> s2[3]\n  //      null  <----> s2[4]\n  //      s1[7] <----> s2[5]\n  //      null  <----> s2[6]\n  //      null  <----> s2[7]\n  //      null  <----> s2[8]\n  //      null  <----> s2[9]\n  ```\n\n## Related\n\n- [An O(ND) Difference Algorithm and Its Variations](https://mailserver.org/diff2.pdf).\n- [最长公共子序列（LCS） | 光和尘][lcs]\n- [Longest common subsequence problem | Wikipedia][wikipedia-lcs]\n\n[homepage]:\n  https://github.com/guanghechen/algorithm.ts/tree/@algorithm.ts/lcs@4.0.5/packages/lcs#readme\n[lcs]: https://me.guanghechen.com/post/algorithm/lcs/\n[wikipedia-lcs]: https://en.wikipedia.org/wiki/Longest_common_subsequence_problem\n","readmeFilename":"README.md","bugs":{"url":"https://github.com/guanghechen/algorithm.ts/issues"}}