LeetCode:Edit Distance（字符串编辑距离DP）

Given two words word1 and word2, find the minimum number of steps required to convert word1 to word2. (each operation is counted as 1 step.)

You have the following 3 operations permitted on a word:

a) Insert a character

b) Delete a character

c) Replace a character

思路：题意求字符串的最小编辑距离。设状态为f[i][j],表示A[0][i] B[0][j]之间的最小编辑距离。
forexamplr:str1c str2d

1、c==d f[i][j]=f[i-1][j-1]
2、c!=d
　 (1)如果将c换成d 则f[i][j]=f[i-1][j-1]+1
(2)如果在c后面添加一个d f[i][j]=f[i][j-1]+1
(3)如果将c删除 f[i][j]=f[i-1][j-1]+1

简单的状态方程为

dp[i, j] = min { dp[i - 1, j] + 1,  dp[i, j - 1] + 1,  dp[i - 1, j - 1] + (s[i] == t[j] ? 0 : 1) }

class Solution {
public:
int minDistance(string word1, string word2) {

const int n=word1.size();
const int m=word2.size();

vector<vector<int>> f(n+1,vector<int>(m+1,0));

for(int i=0;i<=n;i++)
f[i][0]=i;

for(int j=0;j<=m;j++)
f[0][j]=j;

for(int i=1;i<=n;i++)
for(int j=1;j<=m;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-1],min(f[i-1][j],f[i][j-1]))+1;

}

return f
[m];
}
};