Leetcode 62 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.

Solution 1: simple DP problem. Below code optimize the space to O(n), time complexity is O(mn)

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