leetcode 62. 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?

 

思路: 由于题目只要求把unique path的个数求出来，而不要求把所有的unique path都逐个列出来，所以可以使用DP。由于机器人只能往右，或者往下走。所以最上面和最左边的一条边上，所有的位置unique path都是1。如图所示，最后的答案是m+n，所以可以得到L12的DP公式。

class Solution(object):
    def uniquePaths(self, m, n):
        """
        :type m: int
        :type n: int
        :rtype: int
        """
        dp = [[1]*n for x in range(m)]
        
        for i in range(1,m):
            for j in range(1,n):
                dp[i][j] = dp[i-1][j] + dp[i][j-1]
        
        return dp[-1][-1] 

这个DP table其实还可以简化为一个1-D array。

class Solution(object):
    def uniquePaths(self, m, n):
        """
        :type m: int
        :type n: int
        :rtype: int
        """
        dp = [1]*m
        
        for j in range(1,n):
            for i in range(1,m):
                dp[i] = dp[i-1] + dp[i]
        
        return dp[-1]