Climbing Stairs

Problem

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.

Solution

Typical DP Solution: O(n) time and O(n) space

public class Solution {
    public int climbStairs(int n) {
        if (n == 1) return 1;
        int[] dp = new int[n + 1];
        dp[1] = 1;
        dp[2] = 2;
        for (int i = 3; i <= n; i++) dp[i] = dp[i - 1] + dp[i - 2];
        return dp[n];
    }
}

Reduced Space DP Solution: O(n) time and O(1) space

public class Solution {
    public int climbStairs(int n) {
        if (n == 1) return 1;
        int first = 1, second = 2, res = 0;
        for (int i = 3; i <= n; i++) {
            res = first + second;
            first = second;
            second = res;
        }
        return second; //We don't return res to make n == 2 works in the codes 
    }
}

Analysis

A typical Dynamic Programming Problem
Similar to Fibonacci Number problem
We use the previous two stairs to calculate the steps for current stair