Leetcode: Distinct Subsequences in C++

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.

Solution1: Normal DP

int numDistinct(string S, string T) {
      vector<int> v1(S.size()+1,0);  
      vector<vector<int>> v2(T.size()+1,v1);
      if(S.size() < T.size()) return 0;   
      for(int i=0;i<=S.size();i++)
            v2[0][i]=1;
      for(int i=1; i<= T.size(); i++)  
        for(int j =1; j<=S.size(); j++)  
          if(S[j-1] == T[i-1])  
            v2[i][j]=v2[i-1][j-1]+ v2[i][j-1];
          else
            v2[i][j]=v2[i][j-1];
      return v2[T.size()][S.size()]; 
    }

Solution 2: Use reused array

int numDistinct(string S, string T)
     {
      vector<int> DP(T.size()+1,0);
      if(S.size() < T.size()) return 0;
      DP[0] = 1;
      for(int i=1; i<= S.size(); i ++)
        for(int j =T.size(); j>=1; j--)
          if(S[i-1] == T[j-1])
            DP[j]+= DP [j-1];
      return match[T.size()];
      }