Min Cost Climbing Stairs - Python Solution and Time Complexity Analysis

On a staircase, the i-th step has some non-negative cost cost[i] assigned (0 indexed).

Once you pay the cost, you can either climb one or two steps. You need to find minimum cost to reach the top floor, given that you can either start from the step with index 0 or the step with index 1.

                             ___________
                        ___ | Final Step
                   ___ | 20
             ___ | 15
__________ | 10
First Step

Example 1:

Input: cost = [10, 15, 20]
Output: 15
Explanation: Cheapest is start on cost[1], pay that cost and go to the top.

Example 2:

Input: cost = [1, 100, 1, 1, 1, 100, 1, 1, 100, 1]
Output: 6
Explanation: Cheapest is start on cost[0], and only step on 1s, skipping cost[3].

Note:

cost will have a length in the range [2, 1000].
Every cost[i] will be an integer in the range [0, 999].

Understanding the Problem

The core challenge of this problem is to find the minimum cost to reach the top of the staircase. The significance of this problem lies in its application to dynamic programming, where we need to make optimal decisions at each step to minimize the total cost. A common pitfall is to assume that we must start from the first step, but we can start from either the first or the second step.

Approach

To solve this problem, we can use dynamic programming. The idea is to keep track of the minimum cost to reach each step and use this information to calculate the minimum cost to reach the next steps.

Naive Solution

A naive solution would involve recursively calculating the cost to reach the top from each step, but this approach would be highly inefficient due to repeated calculations.

Optimized Solution

We can optimize the solution by using a dynamic programming approach. We will maintain an array dp where dp[i] represents the minimum cost to reach step i. The recurrence relation will be:

dp[i] = cost[i] + min(dp[i-1], dp[i-2])

We can further optimize the space complexity by using two variables instead of an array to keep track of the minimum costs for the last two steps.

Algorithm

1. Initialize two variables to store the minimum cost to reach the last two steps.

2. Iterate through the cost array, updating the minimum cost for each step using the recurrence relation.

3. Return the minimum cost to reach the top, which will be the minimum of the last two steps.

Code Implementation

def minCostClimbingStairs(cost):
    # Initialize the first two steps
    first = cost[0]
    second = cost[1]
    
    # Iterate through the cost array starting from the third step
    for i in range(2, len(cost)):
        current = cost[i] + min(first, second)
        first = second
        second = current
    
    # The minimum cost to reach the top is the minimum of the last two steps
    return min(first, second)

# Example usage
cost1 = [10, 15, 20]
print(minCostClimbingStairs(cost1))  # Output: 15

cost2 = [1, 100, 1, 1, 1, 100, 1, 1, 100, 1]
print(minCostClimbingStairs(cost2))  # Output: 6

Complexity Analysis

The time complexity of this approach is O(n) where n is the length of the cost array, as we iterate through the array once. The space complexity is O(1) since we only use two variables to store the minimum costs for the last two steps.

Edge Cases

Potential edge cases include:

Cost array with only two elements.
Cost array with all elements being zero.
Cost array with maximum possible values.

Each of these cases should be tested to ensure the algorithm handles them correctly.

Testing

To test the solution comprehensively, we should include a variety of test cases:

Simple cases with small arrays.
Cases with large arrays.
Cases with varying costs.

Using a testing framework like unittest in Python can help automate and organize these tests.

Thinking and Problem-Solving Tips

When approaching such problems, it's important to:

Understand the problem constraints and requirements.
Break down the problem into smaller subproblems.
Consider both naive and optimized solutions.
Analyze the time and space complexity of your solutions.

Practicing similar problems and studying dynamic programming techniques can help improve problem-solving skills.

Conclusion

In this blog post, we discussed the problem of finding the minimum cost to climb stairs. We explored a dynamic programming approach to solve the problem efficiently and analyzed its complexity. Understanding and solving such problems is crucial for developing strong algorithmic skills.

Additional Resources

For further reading and practice, consider the following resources: