# Edit Distance

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:

1. Insert a character
2. Delete a character
3. Replace a character

## **Example**

**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')
```

## Note

Let's go further and introduce an edit distance`D[i][j]`which is an edit distance between the first`i`characters of`word1`and the first`j`characters of`word2`.

![edit\_distance](https://leetcode.com/problems/edit-distance/Figures/72/72_edit.png)It turns out that one could compute`D[i][j]`, knowing`D[i - 1][j]`,`D[i][j - 1]`and`D[i - 1][j - 1]`.

There is just one more character to add into one or both strings and the formula is quite obvious.

If the last character is the same,*i.e.*`word1[i] = word2[j]`then

`D[i][j]=1+min(D[i−1][j],D[i][j−1],D[i−1][j−1]−1)`

and if not, *i.e.* `word1[i] != word2[j]`we have to take into account the replacement of the last character during the conversion.

`D[i][j]=1+min(D[i−1][j],D[i][j−1],D[i−1][j−1])`

So each step of the computation would be done based on the previous computation, as follows:

![compute](https://leetcode.com/problems/edit-distance/Figures/72/72_compute.png)The obvious base case is an edit distance between the empty string and non-empty string that means`D[i][0] = i`and`D[0][j] = j`.

一句话：相等看对角，不相等取左右对角的最小值+1

## Code

```java
class Solution {
    public int minDistance(String word1, String word2) {
        int len1 = word1.length();
        int len2 = word2.length();
        int[][] dp = new int[len1 + 1][len2 + 1];
        for (int i = 0; i <= len1; i++) {
            dp[i][0] = i;
        }
        for (int i = 0; i <= len2; i++) {
            dp[0][i] = i;
        }

        for (int i = 1; i <= len1; i++) {
            for (int j = 1; j <= len2; j++) {
                if (word1.charAt(i - 1) == word2.charAt(j - 1)) {
                    //minus one cause from 1:len loop ahead one (can be seen form dp length)
                    dp[i][j] = dp[i - 1][j - 1];
                } else {
                    dp[i][j] = Math.min(Math.min(dp[i - 1][j],dp[i][j - 1]), 
                                        dp[i - 1][j - 1]) + 1;//choose the smallest of three operations
                }
            }
        return dp[len1][len2];
    }
}
```

```java
public int minDistance(String word1, String word2) {
    // write your code here
    int n = word1.length();
    int m = word2.length();
    int[] prev = new int[m+1];
    for(int i = 0; i <= m; i++) {
        prev[i] = i;
    }
    for(int i = 1; i <= n; i++) {
        int[] current = new int[m+1];
        current[0] = i;
        for(int j = 1; j <= m; j++) {
            if(word1.charAt(i-1)==word2.charAt(j-1)) {
                current[j] = prev[j-1];
            } else {
                current[j] = Math.min(prev[j-1], Math.min(current[j-1], prev[j]))+1;
            }
        }
        prev = current;
    }
    return prev[m];
}
```