[LeetCode] 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.

这道题要求的是找出m×n地图上从 (1, 1) 方格到达 (m, n)方格的所有不同路径的数量，并限制了机器人只能向右或向下运动。

这是一道很简单的动态规划的问题，因为机器人只能向右或向下走，因此任意一个方格的前一个结点只能是它的左边或是上面的方格，所以到达该方格的路径总数等于到达该方格左边和上面节点的路径数之和。我们用 map[i][j] 来记录到达 (i, j)方格的路径数，然后我们就可以得到map[i][j] = map[i - 1][j] + map[i][j - 1]，值得注意的是对于所有的 i 和 j 都有 map[i][0] = 1，map[0][j] = 1，因为显然到达第一行和第一列上方格的路径都只有一条。

class Solution {public:int map[100][100];    int uniquePaths(int m, int n) {        for (int i = 0; i < m; i++) {        map[i][0] = 1;}for (int i = 0; i < n; i++) {map[0][i] = 1;}        for (int i = 1; i < m; i++) {    for (int j = 1; j < n; j++) {    map[i][j] = map[i - 1][j] + map[i][j - 1];}}return map[m - 1][n - 1];    }};