# 72. Edit Distance

#### Hard

***

Given two strings `word1` and `word2`, return *the minimum number of operations required to convert `word1` to `word2`*.

You have the following three 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')
```

 

**Constraints:**

* `0 <= word1.length, word2.length <= 500`
* `word1` and `word2` consist of lowercase English letters.

```python
class Solution:
    def minDistance(self, word1: str, word2: str) -> int:
        m = len(word1)
        n = len(word2)
        return self.edit_distance(word1, word2, m ,n)
    @lru_cache()
    def edit_distance(self, word1, word2, m, n):
        if m == 0:
            return n
        if n == 0:py
            return m
        if word1[m-1] == word2[n-1]:
            return self.edit_distance(word1, word2, m-1, n-1)
        return 1+min(self.edit_distance(word1, word2, m-1, n), self.edit_distance(word1, word2, m-1, n-1), self.edit_distance(word1, word2, m, n-1))
```