115. Distinct Subsequences

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

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).

Example 1:

Input: S = "rabbbit", T = "rabbit"
Output: 3
Explanation:

As shown below, there are 3 ways you can generate "rabbit" from S.
(The caret symbol ^ means the chosen letters)

rabbbit
^^^^ ^^

rabbbit

^^ ^^^^ 

rabbbit

^^^ ^^^

Example 2:

Input: S = "babgbag", T = "bag"
Output: 5
Explanation:

As shown below, there are 5 ways you can generate "bag" from S.
(The caret symbol ^ means the chosen letters)

babgbag
^^ ^

babgbag

^^ ^

babgbag

^ ^^

babgbag

^ ^^

babgbag

^^^

Approach #1: DP. [C++]

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