leetcode@ [72/115] Edit Distance & Distinct Subsequences (Dynamic Programming)

https://leetcode.com/problems/edit-distance/

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

用dp[i][j] 来表示 长度为 i 的 word1 经过 dp[i][j]次变换 可以得到长度为 j 的word2，那么我们主要考察两种情况，第一种是：word1[i] == word2[j]，那么这个问题的规模便转换成了：dp[i][j] = dp[i-1][j-1]. 第二种情况是：word1[i] != word2[j]，那么我们可以删除掉word1中的第 i 个字符，或者我们可以把 word1中的第 i 个字符换成与 word2[j] 相同的字符，或者我们还可以同时 删除 word1[i] 和 word2[j]. 于是状态转移方程为：dp[i][j] = min(dp[i-1][j-1], dp[i-1][j], dp[i][j-1]) + 1.

class Solution {
public:
    int minDistance(string word1, string word2) {
        int m = word1.length(), n = word2.length();
        
        vector<vector<int> > dp(m+1, vector<int> (n+1, 0));
        
        for(int i=0;i<=m;++i) dp[i][0] = i;
        for(int j=0;j<=n;++j) dp[0][j] = j;
        
        for(int i=1;i<=m;++i) {
            for(int j=1;j<=n;++j) {
                if(word1[i-1] == word2[j-1]) {
                    dp[i][j] = min(dp[i-1][j-1], dp[i-1][j]+1);
                }
                else {
                    dp[i][j] = min(dp[i-1][j], min(dp[i][j-1], dp[i-1][j-1])) + 1;
                }
            }
        }
        
        return dp[m][n];
    }
};

https://leetcode.com/problems/distinct-subsequences/

Given a string S and a string T, count the number of distinct subsequences of T in S.

A subsequence of a string is a new string which is formed from the original string by deleting some (can be none) of the characters without disturbing the relative positions of the remaining characters. (ie, "ACE" is a subsequence of "ABCDE" while "AEC" is not).

Here is an example:
S = "rabbbit", T = "rabbit"

Return 3.

解题报告：用 dp[i][[j] 来表示长度为 i 的 S串 包含了几个 T串。分两种情况考虑，第一种：S[i] != T[j]，那么问题转而变成求S[1,...i-1] 包含了几个 T[1...i]。因为S[i] 对解的个数不会产生影响。第二种: S[i] == T[j]，那么一种可能是 S[1...i-1] 包含了 若干个 T[1...j-1]，或者是S[1...i-1] 包含了 若干个T[1...j]。所以状态转移方程为：

dp[i][j] = dp[i-1][j] + dp[i-1][j-1] (if S[i] == T[j])

dp[i][j] = dp[i-1][j] (if S[i] != T[j])

class Solution {
public:
    int numDistinct(string s, string t) {
        int m = s.length(), n = t.length();
        vector<vector<int> > dp(m+1, vector<int>(n+1, 0));
        
        for(int i=0;i<=m;++i) dp[i][0] = 1;
        
        for(int i=1;i<=m;++i) {
            for(int j=1;j<=n;++j) {
                if(s[i-1] == t[j-1]) dp[i][j] = dp[i-1][j-1] + dp[i-1][j];
                else dp[i][j] = dp[i-1][j];
            }
        }
        
        return dp[m][n];
    }
};