LeetCode题解：Unique Paths

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.

思路：

这道题当然可以用有向图搜索的方法来解决。但是有没有更简单的办法呢？

因为机器人只能向固定的两个方向移动，所以向右，向下移动的次数都是固定的，总共移动的次数也是固定的。一旦确定向右（向下）移动发生在哪几部，剩下的移动必然是向下（向右）的。

所以只要做一个组合计算就可以了。机器人本身在（1，1）处，所以需要去掉两步从（0，0）到（1，1）的移动。

组合计算由于包含阶乘，所以需要特别小心溢出。这里的方法选自TAOCP。

题解：

class Solution {
public:
    unsigned long long
    choose(unsigned long long n, unsigned long long k) {
        if (k > n) {
            return 0;
        }
        unsigned long long r = 1;
        for (unsigned long long d = 1; d <= k; ++d) {
            r *= n--;
            r /= d;
        }
        return r;
    }

    int uniquePaths(int m, int n) {
        if (m == 1 || n == 1)
            return 1;
            
        if (m < n) swap(m, n);
        
        return choose(m+n-2, n-1);       
    }
};