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.

本题也是采用的动态规划问题来解决，和triangle不太一样的是，推理的过程从小向大推比较容易，而triangle推理的过程是从大往小推，由此可知动态规划有自顶向下和自底向上两种方法，解题过程不难，代码如下：

public class Solution {
    public int uniquePaths(int m, int n) {
        int[][] dp = new int[m][n];
        for(int i=0;i<m;i++){
            dp[i][n-1] = 1;
        }
        for(int i=0;i<n;i++){
            dp[m-1][i] = 1;
        }
        for(int i=m-2;i>=0;i--){
            for(int j=n-2;j>=0;j--){
                dp[i][j] = dp[i][j+1]+dp[i+1][j];
            }
        }
        return dp[0][0];
    }
}