Unique Paths I

Question

https://leetcode.com/problems/unique-paths/?tab=Description

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?

Note: m and n will be at most 100.

Tags

• array
• dynamic programming

Analysis

This is a path searching problem of dp. We need to build a 2-d dp table and initialize the value as 1.

Since the simplest situation is on the right-bottom corner, we start from that part and fill out all the values in the table according to the recursive equation: dp[i][j] = dp[i][j + 1] + dp[i + 1][j].

Code:

class Solution(object):
    def uniquePaths(self, m, n):
        """
        :type m: int
        :type n: int
        :rtype: int
        """
        dp = [[1] * n for _ in range(m)]
        for i in xrange(m - 1):
            dp[i][-1] = 1
        for i in xrange(n - 1):
            dp[-1][i] = 1
        for i in xrange(m - 2, -1, -1):
            for j in xrange(n - 2, -1, -1):
                dp[i][j] = dp[i][j + 1] + dp[i + 1][j]
        return dp[0][0]