72. 编辑距离

题目描述

给你两个单词 word1 和 word2，请你计算出将 word1 转换成 word2 所使用的最少操作数。

你可以对一个单词进行如下三种操作：

插入一个字符
删除一个字符
替换一个字符

示例 1：

输入：word1 = "horse", word2 = "ros"
输出：3
解释：
horse -> rorse (将 'h' 替换为 'r')
rorse -> rose (删除 'r')
rose -> ros (删除 'e')

示例 2：

输入：word1 = "intention", word2 = "execution"
输出：5
解释：
intention -> inention (删除 't')
inention -> enention (将 'i' 替换为 'e')
enention -> exention (将 'n' 替换为 'x')
exention -> exection (将 'n' 替换为 'c')
exection -> execution (插入 'u')

提示：

0 <= word1.length, word2.length <= 500
word1 和 word2 由小写英文字母组成

解法

动态规划。

设 dp[i][j] 表示将 word1 前 i 个字符组成的字符串 word1[0...i-1] 转换成 word2 前 j 个字符组成的字符串 word2[0...j-1] 的最小操作次数。m, n 分别表示 word1, word2 的长度。

初始化 dp[i][0] = i（i∈[0, m]），dp[0][j] = j （j∈[0, m]）。

i, j 分别从 1 开始遍历，判断 word1[i - 1] 与 word2[j - 1] 是否相等：

若 word1[i - 1] == word2[j - 1]，则 dp[i][j] = dp[i - 1][j - 1]。
若 word1[i - 1] != word2[j - 1]，则 dp[i][j] = min(dp[i - 1][j], dp[i][j - 1], dp[i - 1][j - 1]) + 1。其中 dp[i - 1][j] + 1 对应插入操作，dp[i][j - 1] + 1 对应删除操作，dp[i - 1][j - 1] + 1 对应替换操作。取三者的最小值即可。

递推公式如下：

最后返回 dp[m][n] 即可。

Python3

class Solution:
    def minDistance(self, word1: str, word2: str) -> int:
        m, n = len(word1), len(word2)
        dp = [[0] * (n + 1) for _ in range(m + 1)]
        for i in range(m + 1):
            dp[i][0] = i
        for j in range(n + 1):
            dp[0][j] = j
        for i in range(1, m + 1):
            for j in range(1, n + 1):
                if word1[i - 1] == word2[j - 1]:
                    dp[i][j] = dp[i - 1][j - 1]
                else:
                    dp[i][j] = min(dp[i][j - 1], dp[i - 1]
                                   [j], dp[i - 1][j - 1]) + 1
        return dp[-1][-1]

Java

class Solution {
    public int minDistance(String word1, String word2) {
        int m = word1.length(), n = word2.length();
        int[][] dp = new int[m + 1][n + 1];
        for (int i = 0; i <= m; ++i) {
            dp[i][0] = i;
        }
        for (int j = 0; j <= n; ++j) {
            dp[0][j] = j;
        }
        for (int i = 1; i <= m; ++i) {
            for (int j = 1; j <= n; ++j) {
                if (word1.charAt(i - 1) == word2.charAt(j - 1)) {
                    dp[i][j] = dp[i - 1][j - 1];
                } else {
                    dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;
                }
            }
        }
        return dp[m][n];
    }
}

C++

class Solution {
public:
    int minDistance(string word1, string word2) {
        int m = word1.size(), n = word2.size();
        vector<vector<int>> dp(m + 1, vector<int>(n + 1));
        for (int i = 0; i <= m; ++i) {
            dp[i][0] = i;
        }
        for (int j = 0; j <= n; ++j) {
            dp[0][j] = j;
        }
        for (int i = 1; i <= m; ++i) {
            for (int j = 1; j <= n; ++j) {
                if (word1[i - 1] == word2[j - 1]) {
                    dp[i][j] = dp[i - 1][j - 1];
                } else {
                    dp[i][j] = min(min(dp[i - 1][j], dp[i][j - 1]), dp[i - 1][j - 1]) + 1;
                }
            }
        }
        return dp[m][n];
    }
};

Go

func minDistance(word1 string, word2 string) int {
	m, n := len(word1), len(word2)
	dp := make([][]int, m+1)
	for i := 0; i <= m; i++ {
		dp[i] = make([]int, n+1)
		dp[i][0] = i
	}
	for j := 0; j <= n; j++ {
		dp[0][j] = j
	}
	for i := 1; i <= m; i++ {
		for j := 1; j <= n; j++ {
			if word1[i-1] == word2[j-1] {
				dp[i][j] = dp[i-1][j-1]
			} else {
				dp[i][j] = min(min(dp[i-1][j], dp[i][j-1]), dp[i-1][j-1]) + 1
			}
		}
	}
	return dp[m][n]
}

func min(a, b int) int {
	if a < b {
		return a
	}
	return b
}

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