From c3f80e52aff26071562cb97beb5a0de6913e0109 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 13 Jul 2023 05:27:47 +0800 Subject: [PATCH] Add the section of edit distance problem (#599) --- .../CMakeLists.txt | 3 +- .../edit_distance.cpp | 136 ++++++++++ .../edit_distance.java | 139 ++++++++++ .../edit_distance.py | 123 +++++++++ .../edit_distance_decision_tree.png | Bin 0 -> 74240 bytes .../edit_distance_dp_step1.png | Bin 0 -> 49644 bytes .../edit_distance_dp_step10.png | Bin 0 -> 53143 bytes .../edit_distance_dp_step11.png | Bin 0 -> 53575 bytes .../edit_distance_dp_step12.png | Bin 0 -> 53039 bytes .../edit_distance_dp_step13.png | Bin 0 -> 53457 bytes .../edit_distance_dp_step14.png | Bin 0 -> 53386 bytes .../edit_distance_dp_step15.png | Bin 0 -> 43157 bytes .../edit_distance_dp_step2.png | Bin 0 -> 45050 bytes .../edit_distance_dp_step3.png | Bin 0 -> 52786 bytes .../edit_distance_dp_step4.png | Bin 0 -> 52340 bytes .../edit_distance_dp_step5.png | Bin 0 -> 52607 bytes .../edit_distance_dp_step6.png | Bin 0 -> 52768 bytes .../edit_distance_dp_step7.png | Bin 0 -> 53119 bytes .../edit_distance_dp_step8.png | Bin 0 -> 52376 bytes .../edit_distance_dp_step9.png | Bin 0 -> 52893 bytes .../edit_distance_example.png | Bin 0 -> 49147 bytes .../edit_distance_state_transfer.png | Bin 0 -> 76882 bytes .../edit_distance_problem.md | 245 ++++++++++++++++++ mkdocs.yml | 1 + 24 files changed, 646 insertions(+), 1 deletion(-) create mode 100644 codes/cpp/chapter_dynamic_programming/edit_distance.cpp create mode 100644 codes/java/chapter_dynamic_programming/edit_distance.java create mode 100644 codes/python/chapter_dynamic_programming/edit_distance.py create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_decision_tree.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step1.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step10.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step11.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step12.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step13.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step14.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step15.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step2.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step3.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step4.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step5.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step6.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step7.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step8.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_dp_step9.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_example.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_state_transfer.png create mode 100644 docs/chapter_dynamic_programming/edit_distance_problem.md diff --git a/codes/cpp/chapter_dynamic_programming/CMakeLists.txt b/codes/cpp/chapter_dynamic_programming/CMakeLists.txt index 908e55db8..ed185458a 100644 --- a/codes/cpp/chapter_dynamic_programming/CMakeLists.txt +++ b/codes/cpp/chapter_dynamic_programming/CMakeLists.txt @@ -6,4 +6,5 @@ add_executable(min_cost_climbing_stairs_dp min_cost_climbing_stairs_dp.cpp) add_executable(min_path_sum min_path_sum.cpp) add_executable(unbounded_knapsack unbounded_knapsack.cpp) add_executable(coin_change coin_change.cpp) -add_executable(coin_change_ii coin_change_ii.cpp) \ No newline at end of file +add_executable(coin_change_ii coin_change_ii.cpp) +add_executable(edit_distance edit_distance.cpp) \ No newline at end of file diff --git a/codes/cpp/chapter_dynamic_programming/edit_distance.cpp b/codes/cpp/chapter_dynamic_programming/edit_distance.cpp new file mode 100644 index 000000000..647d7de1e --- /dev/null +++ b/codes/cpp/chapter_dynamic_programming/edit_distance.cpp @@ -0,0 +1,136 @@ +/** + * File: edit_distance.cpp + * Created Time: 2023-07-13 + * Author: Krahets (krahets@163.com) + */ + +#include "../utils/common.hpp" + +/* 编辑距离:暴力搜索 */ +int editDistanceDFS(string s, string t, int i, int j) { + // 若 s 和 t 都为空,则返回 0 + if (i == 0 && j == 0) + return 0; + // 若 s 为空,则返回 t 长度 + if (i == 0) + return j; + // 若 t 为空,则返回 s 长度 + if (j == 0) + return i; + // 若两字符相等,则直接跳过此两字符 + if (s[i - 1] == t[j - 1]) + return editDistanceDFS(s, t, i - 1, j - 1); + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + int insert = editDistanceDFS(s, t, i, j - 1); + int del = editDistanceDFS(s, t, i - 1, j); + int replace = editDistanceDFS(s, t, i - 1, j - 1); + // 返回最少编辑步数 + return min(min(insert, del), replace) + 1; +} + +/* 编辑距离:记忆化搜索 */ +int editDistanceDFSMem(string s, string t, vector> &mem, int i, int j) { + // 若 s 和 t 都为空,则返回 0 + if (i == 0 && j == 0) + return 0; + // 若 s 为空,则返回 t 长度 + if (i == 0) + return j; + // 若 t 为空,则返回 s 长度 + if (j == 0) + return i; + // 若已有记录,则直接返回之 + if (mem[i][j] != -1) + return mem[i][j]; + // 若两字符相等,则直接跳过此两字符 + if (s[i - 1] == t[j - 1]) + return editDistanceDFSMem(s, t, mem, i - 1, j - 1); + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + int insert = editDistanceDFSMem(s, t, mem, i, j - 1); + int del = editDistanceDFSMem(s, t, mem, i - 1, j); + int replace = editDistanceDFSMem(s, t, mem, i - 1, j - 1); + // 记录并返回最少编辑步数 + mem[i][j] = min(min(insert, del), replace) + 1; + return mem[i][j]; +} + +/* 编辑距离:动态规划 */ +int editDistanceDP(string s, string t) { + int n = s.length(), m = t.length(); + vector> dp(n + 1, vector(m + 1, 0)); + // 状态转移:首行首列 + for (int i = 1; i <= n; i++) { + dp[i][0] = i; + } + for (int j = 1; j <= m; j++) { + dp[0][j] = j; + } + // 状态转移:其余行列 + for (int i = 1; i <= n; i++) { + for (int j = 1; j <= m; j++) { + if (s[i - 1] == t[j - 1]) { + // 若两字符相等,则直接跳过此两字符 + dp[i][j] = dp[i - 1][j - 1]; + } else { + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1; + } + } + } + return dp[n][m]; +} + +/* 编辑距离:状态压缩后的动态规划 */ +int editDistanceDPComp(string s, string t) { + int n = s.length(), m = t.length(); + vector dp(m + 1, 0); + // 状态转移:首行 + for (int j = 1; j <= m; j++) { + dp[j] = j; + } + // 状态转移:其余行 + for (int i = 1; i <= n; i++) { + // 状态转移:首列 + int leftup = dp[0]; // 暂存 dp[i-1, j-1] + dp[0] = i; + // 状态转移:其余列 + for (int j = 1; j <= m; j++) { + int temp = dp[j]; + if (s[i - 1] == t[j - 1]) { + // 若两字符相等,则直接跳过此两字符 + dp[j] = leftup; + } else { + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1; + } + leftup = temp; // 更新为下一轮的 dp[i-1, j-1] + } + } + return dp[m]; +} + +/* Driver Code */ +int main() { + string s = "bag"; + string t = "pack"; + int n = s.length(), m = t.length(); + + // 暴力搜索 + int res = editDistanceDFS(s, t, n, m); + cout << "将 " << s << " 更改为 " << t << " 最少需要编辑 " << res << " 步\n"; + + // 记忆化搜索 + vector> mem(n + 1, vector(m + 1, -1)); + res = editDistanceDFSMem(s, t, mem, n, m); + cout << "将 " << s << " 更改为 " << t << " 最少需要编辑 " << res << " 步\n"; + + // 动态规划 + res = editDistanceDP(s, t); + cout << "将 " << s << " 更改为 " << t << " 最少需要编辑 " << res << " 步\n"; + + // 状态压缩后的动态规划 + res = editDistanceDPComp(s, t); + cout << "将 " << s << " 更改为 " << t << " 最少需要编辑 " << res << " 步\n"; + + return 0; +} diff --git a/codes/java/chapter_dynamic_programming/edit_distance.java b/codes/java/chapter_dynamic_programming/edit_distance.java new file mode 100644 index 000000000..89954d90b --- /dev/null +++ b/codes/java/chapter_dynamic_programming/edit_distance.java @@ -0,0 +1,139 @@ +/** + * File: edit_distance.java + * Created Time: 2023-07-13 + * Author: Krahets (krahets@163.com) + */ + +package chapter_dynamic_programming; + +import java.util.Arrays; + +public class edit_distance { + /* 编辑距离:暴力搜索 */ + static int editDistanceDFS(String s, String t, int i, int j) { + // 若 s 和 t 都为空,则返回 0 + if (i == 0 && j == 0) + return 0; + // 若 s 为空,则返回 t 长度 + if (i == 0) + return j; + // 若 t 为空,则返回 s 长度 + if (j == 0) + return i; + // 若两字符相等,则直接跳过此两字符 + if (s.charAt(i - 1) == t.charAt(j - 1)) + return editDistanceDFS(s, t, i - 1, j - 1); + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + int insert = editDistanceDFS(s, t, i, j - 1); + int delete = editDistanceDFS(s, t, i - 1, j); + int replace = editDistanceDFS(s, t, i - 1, j - 1); + // 返回最少编辑步数 + return Math.min(Math.min(insert, delete), replace) + 1; + } + + /* 编辑距离:记忆化搜索 */ + static int editDistanceDFSMem(String s, String t, int[][] mem, int i, int j) { + // 若 s 和 t 都为空,则返回 0 + if (i == 0 && j == 0) + return 0; + // 若 s 为空,则返回 t 长度 + if (i == 0) + return j; + // 若 t 为空,则返回 s 长度 + if (j == 0) + return i; + // 若已有记录,则直接返回之 + if (mem[i][j] != -1) + return mem[i][j]; + // 若两字符相等,则直接跳过此两字符 + if (s.charAt(i - 1) == t.charAt(j - 1)) + return editDistanceDFSMem(s, t, mem, i - 1, j - 1); + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + int insert = editDistanceDFSMem(s, t, mem, i, j - 1); + int delete = editDistanceDFSMem(s, t, mem, i - 1, j); + int replace = editDistanceDFSMem(s, t, mem, i - 1, j - 1); + // 记录并返回最少编辑步数 + mem[i][j] = Math.min(Math.min(insert, delete), replace) + 1; + return mem[i][j]; + } + + /* 编辑距离:动态规划 */ + static int editDistanceDP(String s, String t) { + int n = s.length(), m = t.length(); + int[][] dp = new int[n + 1][m + 1]; + // 状态转移:首行首列 + for (int i = 1; i <= n; i++) { + dp[i][0] = i; + } + for (int j = 1; j <= m; j++) { + dp[0][j] = j; + } + // 状态转移:其余行列 + for (int i = 1; i <= n; i++) { + for (int j = 1; j <= m; j++) { + if (s.charAt(i - 1) == t.charAt(j - 1)) { + // 若两字符相等,则直接跳过此两字符 + dp[i][j] = dp[i - 1][j - 1]; + } else { + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1; + } + } + } + return dp[n][m]; + } + + /* 编辑距离:状态压缩后的动态规划 */ + static int editDistanceDPComp(String s, String t) { + int n = s.length(), m = t.length(); + int[] dp = new int[m + 1]; + // 状态转移:首行 + for (int j = 1; j <= m; j++) { + dp[j] = j; + } + // 状态转移:其余行 + for (int i = 1; i <= n; i++) { + // 状态转移:首列 + int leftup = dp[0]; // 暂存 dp[i-1, j-1] + dp[0] = i; + // 状态转移:其余列 + for (int j = 1; j <= m; j++) { + int temp = dp[j]; + if (s.charAt(i - 1) == t.charAt(j - 1)) { + // 若两字符相等,则直接跳过此两字符 + dp[j] = leftup; + } else { + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1; + } + leftup = temp; // 更新为下一轮的 dp[i-1, j-1] + } + } + return dp[m]; + } + + public static void main(String[] args) { + String s = "bag"; + String t = "pack"; + int n = s.length(), m = t.length(); + + // 暴力搜索 + int res = editDistanceDFS(s, t, n, m); + System.out.println("将 " + s + " 更改为 " + t + " 最少需要编辑 " + res + " 步"); + + // 记忆化搜索 + int[][] mem = new int[n + 1][m + 1]; + for (int[] row : mem) + Arrays.fill(row, -1); + res = editDistanceDFSMem(s, t, mem, n, m); + System.out.println("将 " + s + " 更改为 " + t + " 最少需要编辑 " + res + " 步"); + + // 动态规划 + res = editDistanceDP(s, t); + System.out.println("将 " + s + " 更改为 " + t + " 最少需要编辑 " + res + " 步"); + + // 状态压缩后的动态规划 + res = editDistanceDPComp(s, t); + System.out.println("将 " + s + " 更改为 " + t + " 最少需要编辑 " + res + " 步"); + } +} diff --git a/codes/python/chapter_dynamic_programming/edit_distance.py b/codes/python/chapter_dynamic_programming/edit_distance.py new file mode 100644 index 000000000..f13923c91 --- /dev/null +++ b/codes/python/chapter_dynamic_programming/edit_distance.py @@ -0,0 +1,123 @@ +""" +File: edit_distancde.py +Created Time: 2023-07-04 +Author: Krahets (krahets@163.com) +""" + + +def edit_distance_dfs(s: str, t: str, i: int, j: int) -> int: + """编辑距离:暴力搜索""" + # 若 s 和 t 都为空,则返回 0 + if i == 0 and j == 0: + return 0 + # 若 s 为空,则返回 t 长度 + if i == 0: + return j + # 若 t 为空,则返回 s 长度 + if j == 0: + return i + # 若两字符相等,则直接跳过此两字符 + if s[i - 1] == t[j - 1]: + return edit_distance_dfs(s, t, i - 1, j - 1) + # 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + insert = edit_distance_dfs(s, t, i, j - 1) + delete = edit_distance_dfs(s, t, i - 1, j) + replace = edit_distance_dfs(s, t, i - 1, j - 1) + # 返回最少编辑步数 + return min(insert, delete, replace) + 1 + + +def edit_distance_dfs_mem(s: str, t: str, mem: list[list[int]], i: int, j: int) -> int: + """编辑距离:记忆化搜索""" + # 若 s 和 t 都为空,则返回 0 + if i == 0 and j == 0: + return 0 + # 若 s 为空,则返回 t 长度 + if i == 0: + return j + # 若 t 为空,则返回 s 长度 + if j == 0: + return i + # 若已有记录,则直接返回之 + if mem[i][j] != -1: + return mem[i][j] + # 若两字符相等,则直接跳过此两字符 + if s[i - 1] == t[j - 1]: + return edit_distance_dfs_mem(s, t, mem, i - 1, j - 1) + # 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + insert = edit_distance_dfs_mem(s, t, mem, i, j - 1) + delete = edit_distance_dfs_mem(s, t, mem, i - 1, j) + replace = edit_distance_dfs_mem(s, t, mem, i - 1, j - 1) + # 记录并返回最少编辑步数 + mem[i][j] = min(insert, delete, replace) + 1 + return mem[i][j] + + +def edit_distance_dp(s: str, t: str) -> int: + """编辑距离:动态规划""" + n, m = len(s), len(t) + dp = [[0] * (m + 1) for _ in range(n + 1)] + # 状态转移:首行首列 + for i in range(1, n + 1): + dp[i][0] = i + for j in range(1, m + 1): + dp[0][j] = j + # 状态转移:其余行列 + for i in range(1, n + 1): + for j in range(1, m + 1): + if s[i - 1] == t[j - 1]: + # 若两字符相等,则直接跳过此两字符 + dp[i][j] = dp[i - 1][j - 1] + else: + # 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1 + return dp[n][m] + + +def edit_distance_dp_comp(s: str, t: str) -> int: + """编辑距离:状态压缩后的动态规划""" + n, m = len(s), len(t) + dp = [0] * (m + 1) + # 状态转移:首行 + for j in range(1, m + 1): + dp[j] = j + # 状态转移:其余行 + for i in range(1, n + 1): + # 状态转移:首列 + leftup = dp[0] # 暂存 dp[i-1, j-1] + dp[0] += 1 + # 状态转移:其余列 + for j in range(1, m + 1): + temp = dp[j] + if s[i - 1] == t[j - 1]: + # 若两字符相等,则直接跳过此两字符 + dp[j] = leftup + else: + # 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + dp[j] = min(dp[j - 1], dp[j], leftup) + 1 + leftup = temp # 更新为下一轮的 dp[i-1, j-1] + return dp[m] + + +"""Driver Code""" +if __name__ == "__main__": + s = "bag" + t = "pack" + n, m = len(s), len(t) + + # 暴力搜索 + res = edit_distance_dfs(s, t, n, m) + print(f"将 {s} 更改为 {t} 最少需要编辑 {res} 步") + + # 记忆化搜索 + mem = [[-1] * (m + 1) for _ in range(n + 1)] + res = edit_distance_dfs_mem(s, t, mem, n, m) + print(f"将 {s} 更改为 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zYOMtZW`#3wgb)LkoNk%--B5S?8J-zDA+g){>U@KiaRy_ZkPga)msj4%5B@SM{g=klUPNY(T(g~hMDvJglxS;2VpHT=;31BUW=mXP`%9FiAI8rPoCpNQFA1w$J4i=vckjHyNxakZT;z@Q(a6BnlmKT$w z0haH4wQZaG7uadc7vN4J+u8RBUBrArQyB-exXtuk$9)Uz{{m6fkwr!(@#`X;R8%Oh!IxmA(=R#ko{ ztivh_#z=;E)R**BvMWk-xwZ9=Cr?1cKx-+d-vk<0fWmG2_(ubpeH91DoBe+!4_2MM S-x?zWK4=#&=O)LH%>M(uc}lDR literal 0 HcmV?d00001 diff --git a/docs/chapter_dynamic_programming/edit_distance_problem.md b/docs/chapter_dynamic_programming/edit_distance_problem.md new file mode 100644 index 000000000..78d2197e7 --- /dev/null +++ b/docs/chapter_dynamic_programming/edit_distance_problem.md @@ -0,0 +1,245 @@ +# 编辑距离问题 + +编辑距离,也被称为 Levenshtein 距离,是两个字符串之间互相转换的最小修改次数,通常用于在信息检索和自然语言处理中度量两个序列的相似度。 + +!!! question + + 输入两个字符串 $s$ 和 $t$ ,返回将 $s$ 转换为 $t$ 所需的最少编辑步数。 + + 你可以在一个字符串中进行三种编辑操作:插入一个字符、删除一个字符、替换字符为任意一个字符。 + +如下图所示,将 `kitten` 转换为 `sitting` 需要编辑 3 步,包括 2 次替换操作与 1 次添加操作;将 `hello` 转换为 `algo` 需要 3 步,包括 2 次替换操作和 1 次删除操作。 + +![编辑距离的示例数据](edit_distance_problem.assets/edit_distance_example.png) + +**编辑距离问题可以很自然地用决策树模型来解释**。字符串对应树节点,一轮决策(一次编辑操作)对应树的一条边。 + +如下图所示,在不限制操作的情况下,每个节点都可以派生出许多条边,每条边对应一种操作。实际上,从 `hello` 转换到 `algo` 有许多种可能的路径,下图展示的是最短路径。从决策树的角度看,本题目标是求解节点 `hello` 和节点 `algo` 之间的最短路径。 + +![基于决策树模型表示编辑距离问题](edit_distance_problem.assets/edit_distance_decision_tree.png) + +**第一步:思考每轮的决策,定义状态,从而得到 $dp$ 表** + +每一轮的决策是对字符串 $s$ 进行一次编辑操作。 + +我们希望在编辑操作的过程中,问题的规模逐渐缩小,这样才能构建子问题。设字符串 $s$ 和 $t$ 的长度分别为 $n$ 和 $m$ ,我们先考虑两字符串尾部的字符 $s[n-1]$ 和 $t[m-1]$ : + +- 若 $s[n-1]$ 和 $t[m-1]$ 相同,我们可以直接跳过它们,接下来考虑 $s[n-2]$ 和 $t[m-2]$ ; +- 若 $s[n-1]$ 和 $t[m-1]$ 不同,我们需要对 $s$ 进行一次编辑(插入、删除、替换),使得两字符串尾部的字符相同,从而可以跳过它们,考虑规模更小的问题; + +也就是说,我们在字符串 $s$ 中进行的每一轮决策(编辑操作),都会使得 $s$ 和 $t$ 中剩余的待匹配字符发生变化。因此,状态定义为当前在 $s$ , $t$ 中考虑的第 $i$ , $j$ 个字符,记为 $[i, j]$ 。 + +状态 $[i, j]$ 对应的子问题:**将 $s$ 的前 $i$ 个字符更改为 $t$ 的前 $j$ 个字符所需的最少编辑步数**。 + +至此得到一个尺寸为 $(i+1) \times (j+1)$ 的二维 $dp$ 表。 + +**第二步:找出最优子结构,进而推导出状态转移方程** + +考虑子问题 $dp[i, j]$ ,其对应的两个字符串的尾部字符为 $s[i-1]$ 和 $t[j-1]$ ,可根据不同编辑操作分为三种情况: + +1. 在 $s$ 尾部添加 $t[j-1]$ ,则剩余子问题 $dp[i, j-1]$ ; +2. 删除 $s[i-1]$ ,则剩余子问题 $dp[i-1, j]$ ; +3. 将 $s[i-1]$ 替换为 $t[j-1]$ ,则剩余子问题 $dp[i-1, j-1]$ ; + +![编辑距离的状态转移](edit_distance_problem.assets/edit_distance_state_transfer.png) + +根据以上分析,可得最优子结构:$dp[i, j]$ 的最少编辑步数等于 $dp[i, j-1]$ , $dp[i-1, j]$ , $dp[i-1, j-1]$ 三者中的最少编辑步数,再加上本次编辑的步数 $1$ 。对应的状态转移方程为: + +$$ +dp[i, j] = \min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 +$$ + +请注意,**当 $s[i-1]$ 和 $t[j-1]$ 相同时,无需编辑当前字符**,此时状态转移方程为: + +$$ +dp[i, j] = dp[i-1, j-1] +$$ + +**第三步:确定边界条件和状态转移顺序** + +当两字符串都为空时,编辑步数为 $0$ ,即 $dp[0, 0] = 0$ 。当 $s$ 为空但 $t$ 不为空时,最少编辑步数等于 $t$ 的长度,即 $dp[0, j] = j$ 。当 $s$ 不为空但 $t$ 为空时,等于 $s$ 的长度,即 $dp[i, 0] = i$ 。 + +观察状态转移方程,解 $dp[i, j]$ 依赖左方、上方、左上方的解,因此通过两层循环正序遍历整个 $dp$ 表即可。 + +=== "Java" + + ```java title="edit_distance.java" + [class]{edit_distance}-[func]{editDistanceDP} + ``` + +=== "C++" + + ```cpp title="edit_distance.cpp" + [class]{}-[func]{editDistanceDP} + ``` + +=== "Python" + + ```python title="edit_distance.py" + [class]{}-[func]{edit_distance_dp} + ``` + +=== "Go" + + ```go title="edit_distance.go" + [class]{}-[func]{editDistanceDP} + ``` + +=== "JavaScript" + + ```javascript title="edit_distance.js" + [class]{}-[func]{editDistanceDP} + ``` + +=== "TypeScript" + + ```typescript title="edit_distance.ts" + [class]{}-[func]{editDistanceDP} + ``` + +=== "C" + + ```c title="edit_distance.c" + [class]{}-[func]{editDistanceDP} + ``` + +=== "C#" + + ```csharp title="edit_distance.cs" + [class]{edit_distance}-[func]{editDistanceDP} + ``` + +=== "Swift" + + ```swift title="edit_distance.swift" + [class]{}-[func]{editDistanceDP} + ``` + +=== "Zig" + + ```zig title="edit_distance.zig" + [class]{}-[func]{editDistanceDP} + ``` + +=== "Dart" + + ```dart title="edit_distance.dart" + [class]{}-[func]{editDistanceDP} + ``` + +如下图所示,编辑距离问题的状态转移过程与背包问题非常类似,都可以看作是填写一个二维网格的过程。 + +=== "<1>" + ![编辑距离的动态规划过程](edit_distance_problem.assets/edit_distance_dp_step1.png) + +=== "<2>" + ![edit_distance_dp_step2](edit_distance_problem.assets/edit_distance_dp_step2.png) + +=== "<3>" + ![edit_distance_dp_step3](edit_distance_problem.assets/edit_distance_dp_step3.png) + +=== "<4>" + ![edit_distance_dp_step4](edit_distance_problem.assets/edit_distance_dp_step4.png) + +=== "<5>" + ![edit_distance_dp_step5](edit_distance_problem.assets/edit_distance_dp_step5.png) + +=== "<6>" + ![edit_distance_dp_step6](edit_distance_problem.assets/edit_distance_dp_step6.png) + +=== "<7>" + ![edit_distance_dp_step7](edit_distance_problem.assets/edit_distance_dp_step7.png) + +=== "<8>" + ![edit_distance_dp_step8](edit_distance_problem.assets/edit_distance_dp_step8.png) + +=== "<9>" + ![edit_distance_dp_step9](edit_distance_problem.assets/edit_distance_dp_step9.png) + +=== "<10>" + ![edit_distance_dp_step10](edit_distance_problem.assets/edit_distance_dp_step10.png) + +=== "<11>" + ![edit_distance_dp_step11](edit_distance_problem.assets/edit_distance_dp_step11.png) + +=== "<12>" + ![edit_distance_dp_step12](edit_distance_problem.assets/edit_distance_dp_step12.png) + +=== "<13>" + ![edit_distance_dp_step13](edit_distance_problem.assets/edit_distance_dp_step13.png) + +=== "<14>" + ![edit_distance_dp_step14](edit_distance_problem.assets/edit_distance_dp_step14.png) + +=== "<15>" + ![edit_distance_dp_step15](edit_distance_problem.assets/edit_distance_dp_step15.png) + +下面考虑状态压缩,将 $dp$ 表的第一维删除。由于 $dp[i,j]$ 是由上方 $dp[i-1, j]$ 、左方 $dp[i, j-1]$ 、左上方状态 $dp[i-1, j-1]$ 转移而来,而正序遍历会丢失左上方 $dp[i-1, j-1]$ ,倒序遍历无法提前构建 $dp[i, j-1]$ ,因此两种遍历顺序都不可取。 + +为解决此问题,我们可以使用一个变量 `leftup` 来暂存左上方的解 $dp[i-1, j-1]$ ,这样便只用考虑左方和上方的解,与完全背包问题的情况相同,可使用正序遍历。 + +=== "Java" + + ```java title="edit_distance.java" + [class]{edit_distance}-[func]{editDistanceDPComp} + ``` + +=== "C++" + + ```cpp title="edit_distance.cpp" + [class]{}-[func]{editDistanceDPComp} + ``` + +=== "Python" + + ```python title="edit_distance.py" + [class]{}-[func]{edit_distance_dp_comp} + ``` + +=== "Go" + + ```go title="edit_distance.go" + [class]{}-[func]{editDistanceDPComp} + ``` + +=== "JavaScript" + + ```javascript title="edit_distance.js" + [class]{}-[func]{editDistanceDPComp} + ``` + +=== "TypeScript" + + ```typescript title="edit_distance.ts" + [class]{}-[func]{editDistanceDPComp} + ``` + +=== "C" + + ```c title="edit_distance.c" + [class]{}-[func]{editDistanceDPComp} + ``` + +=== "C#" + + ```csharp title="edit_distance.cs" + [class]{edit_distance}-[func]{editDistanceDPComp} + ``` + +=== "Swift" + + ```swift title="edit_distance.swift" + [class]{}-[func]{editDistanceDPComp} + ``` + +=== "Zig" + + ```zig title="edit_distance.zig" + [class]{}-[func]{editDistanceDPComp} + ``` + +=== "Dart" + + ```dart title="edit_distance.dart" + [class]{}-[func]{editDistanceDPComp} + ``` diff --git a/mkdocs.yml b/mkdocs.yml index 2ddc0a45c..0974342d3 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -219,6 +219,7 @@ nav: - 13.3.   DP 解题思路(New): chapter_dynamic_programming/dp_solution_pipeline.md - 13.4.   0-1 背包问题(New): chapter_dynamic_programming/knapsack_problem.md - 13.5.   完全背包问题(New): chapter_dynamic_programming/unbounded_knapsack_problem.md + - 13.6.   编辑距离问题(New): chapter_dynamic_programming/edit_distance_problem.md - 14.     附录: - 14.1.   编程环境安装: chapter_appendix/installation.md - 14.2.   一起参与创作: chapter_appendix/contribution.md