[DP] 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?



题意分析：要求计算从Start到Finish的所有可能路径。

分析：机器人只能向右活向下走。

m * n矩阵对应数组dp[m]
，数组中的每个值对应机器人走到该处所有可能的路径。

矩阵中的单元供分为两类：

像A1(对应第二维索引为0)和A2(对应第一维索引为0)这种，机器人走到该点只存在1种路径；

像B这种，到达B的所有路径为到达B1的路径加上到达B2的路径之和，即dp[i][j] = dp[i-1][j] + dp[i][j-1]，

源码如下：

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

if (i == 0 || j == 0) {
dp[i][j] = 1;
} else {
dp[i][j] = dp[i - 1][j] + dp[i][j - 1];
}
}
}
return dp[m - 1][n - 1];
}
}