62. Unique Paths I

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?

Solution

Recursion way:

T: O(2^(n*m))

    public int uniquePaths(int m, int n) {
        if(m == 1 || n == 1) return 1;
        
        return uniquePaths(m-1, n) + uniquePaths(m, n-1);
    }

Dp:

class Solution {
    public int uniquePaths(int m, int n) {
        int[][] dp = new int[m][n];
        
        for(int i = 0; i < m; i++){
            dp[i][0] = 1;
        }
        for(int j = 0; j < n; j++){
            dp[0][j] = 1;
        }
        
        for(int i = 1; i < m; i++){
            for(int j = 1; j < n; j++){
                dp[i][j] = dp[i-1][j] + dp[i][j-1];           
            }
        }
        return dp[m-1][n-1];
    }
}

T: O(N*M) space: O(N*M)