Problems

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

we can use $dp$ array where $dp[i]=dp[i-1]+dp[i-2]$. It can be easily analysed that $dp[i]$ is nothing but i^{th}i​th​​ fibonacci number.

$Fib(n)=Fib(n-1)+Fib(n-2)$

Now we just have to find n^{th}n​th​​ number of the fibonacci series having $1$ and $2$ their first and second term respectively i.e. $Fib(1)=1$ and $Fib(2)=2$.

Solution

var climbStairs = function(n) {
    if(n===1) { return 1; }
    var first = 1;
    var second = 2;
    for(var i=3; i<=n; i++)	{
	var third = first + second;
	first = second;
	second = third;
    }
    return second;
}

Complexity Analysis

• Time complexity : $O(n)$. Single loop upto $n$ is required to calculate n^{th}n​th​​ fibonacci number.
• Space complexity : $O(1)$. Constant space is used.