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72. 编辑距离

题目描述

给你两个单词 word1 和 word2， 请返回将 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

方法一：动态规划

我们定义 \(f[i][j]\) 表示将 \(word1\) 的前 \(i\) 个字符转换成 \(word2\) 的前 \(j\) 个字符所使用的最少操作数。初始时 \(f[i][0] = i\), \(f[0][j] = j\)。其中 \(i \in [1, m], j \in [0, n]\)。

考虑 \(f[i][j]\)：

  • 如果 \(word1[i - 1] = word2[j - 1]\)，那么我们只需要考虑将 \(word1\) 的前 \(i - 1\) 个字符转换成 \(word2\) 的前 \(j - 1\) 个字符所使用的最少操作数，因此 \(f[i][j] = f[i - 1][j - 1]\)；
  • 否则，我们可以考虑插入、删除、替换操作，那么 \(f[i][j] = \min(f[i - 1][j], f[i][j - 1], f[i - 1][j - 1]) + 1\)。

综上，我们可以得到状态转移方程：

\[ f[i][j] = \begin{cases} i, & \textit{if } j = 0 \\ j, & \textit{if } i = 0 \\ f[i - 1][j - 1], & \textit{if } word1[i - 1] = word2[j - 1] \\ \min(f[i - 1][j], f[i][j - 1], f[i - 1][j - 1]) + 1, & \textit{otherwise} \end{cases} \]

最后，我们返回 \(f[m][n]\) 即可。

时间复杂度 \(O(m \times n)\)，空间复杂度 \(O(m \times n)\)。其中 \(m\) 和 \(n\) 分别是 \(word1\) 和 \(word2\) 的长度。

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

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

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

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

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function minDistance(word1: string, word2: string): number {
    const m = word1.length;
    const n = word2.length;
    const f: number[][] = Array(m + 1)
        .fill(0)
        .map(() => Array(n + 1).fill(0));
    for (let j = 1; j <= n; ++j) {
        f[0][j] = j;
    }
    for (let i = 1; i <= m; ++i) {
        f[i][0] = i;
        for (let j = 1; j <= n; ++j) {
            if (word1[i - 1] === word2[j - 1]) {
                f[i][j] = f[i - 1][j - 1];
            } else {
                f[i][j] = Math.min(f[i - 1][j], f[i][j - 1], f[i - 1][j - 1]) + 1;
            }
        }
    }
    return f[m][n];
}

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/**
 * @param {string} word1
 * @param {string} word2
 * @return {number}
 */
var minDistance = function (word1, word2) {
    const m = word1.length;
    const n = word2.length;
    const f = Array(m + 1)
        .fill(0)
        .map(() => Array(n + 1).fill(0));
    for (let j = 1; j <= n; ++j) {
        f[0][j] = j;
    }
    for (let i = 1; i <= m; ++i) {
        f[i][0] = i;
        for (let j = 1; j <= n; ++j) {
            if (word1[i - 1] === word2[j - 1]) {
                f[i][j] = f[i - 1][j - 1];
            } else {
                f[i][j] = Math.min(f[i - 1][j], f[i][j - 1], f[i - 1][j - 1]) + 1;
            }
        }
    }
    return f[m][n];
};

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