【LeetCode】62. Unique Paths

62. Unique Paths

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?

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

Note: m and n will be at most 100.

Example 1:

Input: m = 3, n = 2
Output: 3
Explanation:
From the top-left corner, there are a total of 3 ways to reach the bottom-right corner:
1. Right -> Right -> Down
2. Right -> Down -> Right
3. Down -> Right -> Right

Example 2:

Input: m = 7, n = 3
Output: 28

解题思路：

 （1）动态规划解法

动态规划三要素：

1. 最优子结构：我们发现除去第一行第一列，其他二维数组中的数据都是由它上面格子和左边格子数据之和。

2. 边界条件：第一行、第一列都为1。

3. 状态转移方程：dp[i][j] = dp[i-1][j] + dp[i][j-1]

已经AC的代码：

class Solution:
    def uniquePaths(self, m, n):
        '''
        :param m: int 格子的列数
        :param n: int 格子的行数
        :return:  int  多少条独一无二的路径
        '''
        if not m or not n:
            return 0
        dp = [[1 for i in range(m)] for i in range(n)]
        for i in range(1,n):
            for j in range(1,m):
                dp[i][j] = dp[i-1][j] + dp[i][j-1]
        return  dp[-1][-1]

solution = Solution()
print(solution.uniquePaths(7,3))

Reference：

【1】Unique Paths I/II 【视频】