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62. Unique Paths ​

题目 ​

There is a robot on an m x n grid. The robot is initially located at the top-left corner (i.e., grid[0][0]). The robot tries to move to the bottom-right corner (i.e., grid[m - 1][n - 1]). The robot can only move either down or right at any point in time.

Given the two integers m and n, return the number of possible unique paths that the robot can take to reach the bottom-right corner.

The test cases are generated so that the answer will be less than or equal to 2 * 109.

 

Example 1:

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

Example 2:

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 -> Down -> Down
2. Down -> Down -> Right
3. Down -> Right -> Down

 

Constraints:

  • 1 <= m, n <= 100
Related Topics
  • 数学
  • 动态规划
  • 组合数学

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    • 思路 ​

    推导递归方程

    grid[i][j] = grid[i - 1][j] + grid[i][j-1]

    解法 ​

    py
    # leetcode submit region begin(Prohibit modification and deletion)
class Solution:
    def uniquePaths(self, m: int, n: int) -> int:
        grid = [ [1 for _ in range(n)]  for _ in range(m)]

        for i in range(1, m):
            for j in range(1,n):
                grid[i][j] = grid[i - 1][j] + grid[i][j-1]

        return grid[m-1][n-1]
# leetcode submit region end(Prohibit modification and deletion)

    复杂度分析 ​

    • 时间复杂度 O(MN)
    • 空间复杂度 O(MN)