Unique Paths

Q:

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.

Solution:

1.A easy way by using recusion, it is right but time complexity is large.:

public class Solution { public int uniquePaths(int m, int n) { // Start typing your Java solution below // DO NOT write main() function if(n==1)return 1; if(m==1)return 1; return uniquePaths(m-1,n)+uniquePaths(m,n-1); }}

2. DP Solution:

public class Solution { public int uniquePaths(int m, int n) { // Start typing your Java solution below // DO NOT write main() function int[][] res = new int[m][n]; for(int i = 0; i<m; i++){ res[i][0]=1; } for(int i =0 ; i<n;i++){ res[0][i] = 1; } for(int i =1 ; i<m ; i++){ for(int j =1; j<n ; j++){ res[i][j] = res[i-1][j]+res[i][j-1]; } } return res[m-1][n-1]; }}