Unique path -- leetcode

Question:

A robot is located at the top-left corner of a m x n grid (marked 'Start' in the diagram below).

The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).

How many possible unique paths are there?

Answer:

1.DP, using Recursion (Time limit expired)

int uniquePaths(int m,int n){

        if(m==1 || n==1) return 1;

        else return uniquePaths(m-1)+ uniquePaths(n-1);

}

2.DP, look up the recursion table directly.

class Solution {
public:
    int uniquePaths(int m, int n) {
        vector<vector<int> >table(m,vector<int>(n,1));
    
        for(int i=1;i<m;++i){
            for(int j=1;j<n;++j){
                table[i][j]=table[i-1][j]+table[i][j-1];
            }
        }
        return table[m-1][n-1];
    }
};

3.DP, Similar to 2, be more space efficient.
          int uniquePath(int m,int n){

                 vector<int> table(n,1);
                for(int i=1;i<m;++i){
                      for( int j=1;j<n;++j){
                           table[j]+= table[j-1];
                       }
                }
                return table[n-1];
           }