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.

一道简单的二维动态规划的问题，构造一个二维数组dp[][]，当机器人走到(i,j)点的时候，因为机器人只能向右和向下两个方向走，这时总的路径就为dp[i][j] = dp[i - 1][j] + dp[i][j - 1],(i >= 1, j >= 1)。当i = 0 或j = 0的时候都只有一种走法，dp[0][j] = 1; dp[i][0] = 1，这是初始化。上面我们已经写出了递推式，这道题就解决了。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][0] = 1;
        for(int i = 0; i < n; i++) 
            dp[0][i] = 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];
    }
}