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.

算法思路：

典型的二维DP，dp[i][j]表示从左上角(0,0)到(i,j)的paths。状态转移方程：dp[i][j] = dp[i - 1][j] + dp[i][j - 1];

代码如下：

 1 public class Solution { 2     public int uniquePaths(int m, int n) { 3         if(m == 1 || n == 1) return 1; 4         int[][] dp = new int[m][n]; 5         for(int i = 0; i < n; dp[0][i++] = 1); 6         for(int i = 0; i < m; dp[i++][0] = 1); 7         for(int i = 1; i < m; i++){ 8             for(int j = 1; j < n; j++){ 9                 dp[i][j] = dp[i - 1][j] + dp[i][j - 1];10             }11         }12         return dp[m - 1][n - 1];13     }14 }