Distinct Subsequences

 
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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.

主要在于找出递推关系式：

N[i,j] = N[i-1,j] + if(S[i] == T[j])? N[i-1, j -1]:0;

代码就可以写出：

public class Solution {
    public int numDistinct(String S, String T) {
        int[][] num = new int[S.length()+1][T.length()+1];
        for(int i = 0; i<=S.length(); i++){
            num[i][0] = 0;
        }
        for(int j=0; j<=T.length(); j++){
            num[0][j] = 0;
        }
        
        for(int i = 1; i<=S.length(); i++){
            for(int j = 1; j<=i && j<=T.length(); j++){
                num[i][j] = num[i-1][j];
                if(S.charAt(i-1) == T.charAt(j-1)){
                    if(j == 1){
                        num[i][j] += 1;
                    }else{
                        num[i][j] += num[i-1][j-1];
                    }
                }
            }
        }
        return num[S.length()][T.length()];
    }
}