Climbing Stairs

update Jan 17, 2018 18:40

LeetCode

You are climbing a stair case. It takes n steps to reach to the top.

Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top?

Note: Given n will be a positive integer.

Example 1:

    Input: 2
    Output:  2
    Explanation:  There are two ways to climb to the top.

    1. 1 step + 1 step
    2. 2 steps

Example 2:

    Input: 3
    Output:  3
    Explanation:  There are three ways to climb to the top.

    1. 1 step + 1 step + 1 step
    2. 1 step + 2 steps
    3. 2 steps + 1 step

Basic Idea:

使用 DP 的思路考虑，到达第 i 阶的可能步数 dp[i] = dp[i - 1] + dp[i - 2], 即等于这次通过一步或者两步到达这里的起始点的最多步数和；时间复杂度 O(n), 因为只是一次 linear scan，空间复杂度 O(n), 可以优化到 O(1);

Pyhton3 Code:

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