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

自然語言處理（NLP）中，有一個基本問題就是求兩個字串的minimal Edit Distance

1. what is minimal edit distance?

簡單地說，就是僅通過插入(insert)、刪除(delete)和替換(substitute)個操作將一個字串s1變換到另一個字串s2的最少步驟數。熟悉演算法的同學很容易知道這是個動態規劃問題。 

其實一個替換操作可以相當於一個delete+一個insert，所以我們將權值定義如下：

I  (insert)：1

D (delete)：1

S (substitute)：2

2. example:

intention->execution

Minimal edit distance：

delete i ; n->e ; t->x ; insert c ; n->u 求和得cost=8

3.calculate minimal edit distance dynamically
思路見註釋，這裡D[i,j]就是取s1前i個character和s2前j個character所得minimal edit distance

三個操作動態進行更新：

D(i,j)=min { D(i-1, j) +1, D(i, j-1) +1 , D(i-1, j-1) + s1[i]==s2[j] ? 0 : 2}；中的三項分別對應D，I，S。（詳見

 1 class Solution {
 2 public:
 3     int minDistance(string word1, string word2) {
 4         int len1 = word1.length();
 5         int len2 = word2.length();
 6         if (len1 == 0) return len2;
 7         if (len2 == 0) return len1;
 8         vector<vector<int> > dp(len1 + 1, vector<int>(len2 + 1));
 9         for (int i = 0; i <= len1; ++i) dp[i][0] = i;
10         for (int j = 0; j <= len2; ++j) dp[0][j] = j;
11         int cost;
12         for (int i = 1; i <= len1; ++i) {
13             for (int j = 1; j <= len2; ++j) {
14                 cost = (word1[i-1] == word2[j - 1]) ? 0 : 1;
15                 dp[i][j] = min(dp[i-1][j-1] + cost, min(dp[i][j-1] + 1, dp[i-1][j] + 1));
16             }
17         }
18         return dp[len1][len2];
19     }
20 };