Climbing Stairs Algorithm in Python: A Detailed Discussion

The climbing stairs problem is a classic algorithm problem that often appears in interviews and coding challenges. The problem statement is simple: given a staircase with n steps, and you can climb either 1 or 2 steps at a time, how many distinct ways can you climb to the top of the staircase?

This problem can be solved using dynamic programming, a method for solving complex problems by breaking them down into simpler subproblems. In Python, we can implement the climbing stairs algorithm efficiently.

Python Implementation

Let’s break down the Python code to solve this problem:

pythonCopy Code
def climbStairs(n):
    if n <= 2:
        return n
    
    # Initialize the base cases
    one_step_before = 2
    two_steps_before = 1
    
    # Iterate through the staircase
    for _ in range(2, n):
        current = one_step_before + two_steps_before
        two_steps_before = one_step_before
        one_step_before = current
    
    return one_step_before

# Example usage
print(climbStairs(5))  # Output: 8

How It Works

1.‌Base Cases‌: If n is 1, there’s only one way to climb. If n is 2, there are two ways to climb: either take two steps of 1 or one step of 2.

2.‌Dynamic Programming‌: We use two variables, one_step_before and two_steps_before, to keep track of the number of ways to reach the top two steps and the top one step before the current step, respectively. For each step, the number of ways to reach the current step is the sum of the number of ways to reach the top two steps before and the top one step before.

3.‌Iteration‌: We iterate from the third step to the nth step, updating our variables according to the dynamic programming relation.

4.‌Result‌: After iterating through all steps, one_step_before holds the number of ways to reach the top of the staircase.

Complexity Analysis

–‌Time Complexity‌: O(n), where n is the number of steps. We iterate through the steps once.
–‌Space Complexity‌: O(1). We only use a fixed amount of extra space.

Conclusion

The climbing stairs algorithm is a great example of how dynamic programming can be used to solve problems efficiently. By breaking down the problem into smaller subproblems and solving those, we can find the solution to the original problem in a systematic and efficient manner.

[tags]
Python, Climbing Stairs, Algorithm, Dynamic Programming, Coding Challenge, Interview Problem