70. Climbing Stairs

The Problem

Link to original problem on Leetcode.

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

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

Examples

Example 1:

Input: n = 2
Output: 2
Explanation: There are two ways to climb to the top.
1. 1 step + 1 step
2. 2 steps


Example 2:

Input: n = 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

Constraints

1 ≤ n ≤ 45

My Solution

Looking at the examples should hopefully give a clue: we're dealing with a Fibonacci sequence!

If we've got one step, then there's one way to climb it—in a single step. Two steps, two ways: step once two times, or step twice one time. Three steps, and we can step once three times, or one step one time and two steps one time and vice versa for three total unique step sequences. Below is a breakdown of this pattern for a few more numbers:

n = 0 => 0
n = 1 => 1
1 step
n = 2 => 2
1 step + 1 step
2 steps
n = 3 => 3
1 + 1 + 1
2 + 1
1 + 2
n = 4 => 5
1 + 1 + 1 + 1
1 + 1 + 2
1 + 2 + 1
2 + 1 + 1
2 + 2
n = 5 => 8
1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 2
1 + 1 + 2 + 1
1 + 2 + 1 + 1
2 + 1 + 1 + 1
2 + 2 + 1
2 + 1 + 2
1 + 2 + 1
...


In this Fibonacci sequence, the nth value in the sequence is equal to the sum of the previous two values. In mathematical terms, $F_{n} = F_{n - 1} + F_{n - 2}$. So if we know $F_{n - 1}$ and $F_{n - 2}$, we know $F_{n}$. Easy enough to calculate!

const climbStairs = n => {	// No point calculating these low numbers	if (n <= 2) return n;	// Let's assume we start counting from step 3,	// so the step before F[n - 1] (or n1) has 2 ways	// and the one before that F[n - 2] (or n2) has	// only one way.	let n1 = 2;	let n2 = 1;	// We'll be setting the value of the nth step in our loop,	// so initialize to zero for now.	let sum = 0;	// We could store the values in an array and access	// them by index, but that creates O(n) space	// complexity that we don't actually need.	for (let i = 3; i <= n; i++) {		// F[n] = F[n - 1] + F[n - 2]		sum = n1 + n2;		// Reset the step two back to be the step one back		n2 = n1;		// Reset the step one back to be this step		n1 = sum;	}	return sum;};

We can also solve this recursively (with memoization as an added bonus):

const climbStairs = (n, memo = new Map([[1, 1], [2, 2]])) => {	if (memo.has(n)) return memo.get(n);	memo.set(n, climbStairs(n - 1, memo) + climbStairs(n - 2, memo));	return memo.get(n);};