How to Solve the Climbing Stairs Problem in Coding Interviews

The Climbing Stairs problem is a common question asked during technical interviews to test your understanding of dynamic programming, recursion, and problem-solving skills. The problem is straightforward to understand but requires a good approach to optimize and implement efficiently.

The Problem Statement

Imagine you are climbing a staircase that has n steps. You can take either 1 step or 2 steps at a time. The question is: How many different ways can you climb to the top of the staircase?

For example, if n = 3, the different ways to climb are:

• 1 step + 1 step + 1 step
• 1 step + 2 steps
• 2 steps + 1 step

The answer would be 3.

Why is this problem important?

This problem illustrates several key concepts:

• Recursion: breaking down the problem into smaller subproblems.
• Dynamic Programming: storing intermediate results to avoid redundant calculations.
• Optimization: reducing exponential time complexity to linear.

Understanding the Problem

Let's analyze the problem to find a pattern. For smaller values of n:

• If n = 1, there's only 1 way: (1)
• If n = 2, there are 2 ways: (1+1), (2)
• If n = 3, the ways are 3: (1+1+1), (1+2), (2+1)

Notice how the number of ways to reach step n depends on the ways to reach steps n-1 and n-2 because:

• To reach step n, you could have taken a single step from n-1.
• Or, you could have taken a double step from n-2.

Based on this, the problem has the following recursive relation:

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Implementing a Basic Recursive Solution

The simplest way is using recursion directly based on the relation above:

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This approach works but is slow for larger n due to repeated calculations, leading to exponential time complexity.

Optimizing with Dynamic Programming

To improve efficiency, we can use dynamic programming by storing intermediate results:

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This version runs in linear time and uses extra space proportional to n. It’s a big improvement over the recursive version.

Further Optimization with Space Efficiency

Notice that, at each step, we only need the last two results. Instead of keeping an entire array, we can just track the last two counts:

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This last version is both time and space-efficient, running in linear time with constant space.

The climbing stairs problem demonstrates fundamental concepts of recursion, dynamic programming, and optimizing code efficiency. Starting from a naive recursive solution, we move to more optimized implementations by using dynamic programming and space-efficient methods. Understanding this problem helps build skills to solve more complex questions involving similar concepts.

Common Interview Tips

• Clearly explain your approach before coding.
• Start with the simplest solution and improve it.
• Use comments to explain logic.
• Consider edge cases like n = 0 or very large n.

This problem is a good example of how to think about problems recursively and then optimize your solutions to be more efficient.