[Leetcode]Unique Paths

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?

Above is a 3 x 7 grid. How many possible unique paths are there?

Note: m and n will be at most 100.

class Solution {
public:
    /*algorithm  2-D DP
        path[m][n] = path[m-1][n]+path[m][n-1] //top-->down, left-->right
        time O(m*n) space O(m*n)
    */
    int uniquePaths(int m, int n) {//1x2
        vector<vector<int> >dp(m+1,vector<int>(n+1,0));
        for(int i = 1;i < m+1;i++){
            dp[i][1] = 1;
            for(int j = 2;j < n+1;j++){
                dp[i][j] = dp[i-1][j] + dp[i][j-1];
            }
        }
        return dp[m][n];
    }
};

class Solution {
public:
    /*algorithm  1-d DP
        path[m][n] = path[m-1][n]+path[m][n-1] //top-->down, left-->right
        /path[i] = path[i] + path[i-1]
        optimize memory to O(n)
        time O(m*n) space O(n)
    */
    int uniquePaths(int m, int n) {//1x2
        vector<int>dp(n+1,0);
        dp[1] = 1;
        for(int i = 1;i < m+1;i++){
            for(int j = 2;j < n+1;j++)
                dp[j] += dp[j-1];
        }
        return dp[n];
    }
};