【LeetCode with Python】 62. Unique Paths

题目

原题页面:https://leetcode.com/problems/unique-paths/
本文地址:http://leetcode.xnerv.wang/unique-paths/
题目类型:Array, Dynamic Programming
难度评价:Medium
类似题目:(M) Unique Paths II, (M) Minimum Path Sum, (H) Dungeon Game

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?

robot_maze
Above is a 3 x 7 grid. How many possible unique paths are there?

Note: m and n will be at most 100.


分析

用数学中的排列组合来解题,从(m+n)次选择中选出m次向下走。
这道题跟“n阶台阶,每次一步或两步,总共多少种走法”一样,也可以用动态规划解题,不过这里是二维动态规划,而后者是一维动态规划。Unique Paths还比较简单,所以可用排列组合快速解题,但是Unique Paths II这道题,就只能用动态规划了。


代码

class Solution:
    # @return an integer
    def uniquePaths(self, m, n):
        if 0 == m or 0 == n:
            return 1
        up = 1
        for i in range(m + n - 2, n - 1, -1):
            up *= i
        down = 1
        for j in range(1, m):
            down *= j
        return up / down

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