# 72. Edit Distance

## Description

Given two words word1 and word2, find the minimum number of operations required to convert word1 to word2.

You have the following 3 operations permitted on a word:

• Insert a character
• Delete a character
• Replace a character

Example 1:

``````Input: word1 = "horse", word2 = "ros"
Output: 3
Explanation:
horse -> rorse (replace 'h' with 'r')
rorse -> rose (remove 'r')
rose -> ros (remove 'e')
``````

Example 2:

``````Input: word1 = "intention", word2 = "execution"
Output: 5
Explanation:
intention -> inention (remove 't')
inention -> enention (replace 'i' with 'e')
enention -> exention (replace 'n' with 'x')
exention -> exection (replace 'n' with 'c')
exection -> execution (insert 'u')
``````

Tags: Math, String

## 题解

### 思路1

(动态规划) O(n2)

• `word1[i]` 删除或在`word2[j]`后面添加`word1[i]`，则其操作次数等于`f[i−1,j]+1`
• `word2[j]`删除或在`word1[i]`后面添加`word2[j]`，则其操作次数等于 `f[i,j−1]+1`
• 如果`word1[i]=word2[j]`，则其操作次数等于 `f[i−1,j−1]`
• 如果`word1[i]≠word2[j]`，则其操作次数等于 `f[i−1,j−1]+1`

• 状态数 O(n2)O(n2)，状态转移复杂度是 O(1)O(1)，所以总时间复杂度是 O(n2)O(n2)。
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