Leetcode–Distinct Subsequences

The Problem:

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.

Notes:

1. Here when seeing only required to return the number of ways (not the entire list of ways), we could consider DP instead of pure recursion. use match[i][j] to represent number of distinct sub sequence that S[0-i] has for T[0-j] seems a good start.
2. As for the recurrence formula, if S[i]!=T[j], then simply means S[i] is not used for the match (match[i][j] = match[i-1][j]).  if(S[i]=T[j]), then we can consider the number of ways comes from two paths: use S[i]  (match[i][j] = match[i-1][j-1]); and  not use S[i]  (match[i][j] = match[i-1][j]).
3. Watch out for the details and edge cases.

Java Solution

public class Solution {
    public int numDistinct(String S, String T) {
        int slen = S.length();
        int tlen = T.length();
        // check length
        if(slen==0 || slen < tlen)
            return 0;
        if(tlen==0)
            return 1;
        // record dist. subseqs of t[0-j] in in s[0-i] in match[i][j]
        int[][] match = new int[slen][tlen];       
        for(int i = 0; i < slen; i++){
            for(int j = 0; j < tlen; j++){
                if(i < j)
                    match[i][j] = 0;
                else if(i==0 && j==0){
                    match[i][j] = S.charAt(0)==T.charAt(0)? 1:0;
                }
                else if(S.charAt(i)==T.charAt(j)){
                    //divide case by either use s[i] or not
                    // only consider j=0 case, i=0 already covered in previous conditions: 
                     //1. i<j  2. i==0&& j==0
                    if(j==0)
                        match[i][j] = match[i-1][j]+1;
                    else
                        match[i][j] = match[i-1][j-1] + match[i-1][j];
                }
                else{
                    match[i][j] = match[i-1][j];
                }
            }
        }
        return match[slen-1][tlen-1];
    }
}

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