Longest Subarray with Sum at most S in O(n^2) Time Complexity using Python

Given an array of positive integers and a number S, find the longest contiguous subarray having the sum at most S.

Return the start and end indices denoting this subarray.

Example

Input: nums = [3, 2, 5, 2, 2, 1, 1, 3, 1 , 2], S = 11
Output: [3, 8]
Explanation:the subarray nums[3...8] of sum 10

Your algorithm should run in O(n^2) time and use O(1) extra space.

Understanding the Problem

The core challenge of this problem is to find the longest contiguous subarray whose sum does not exceed a given value S. This problem is significant in various applications such as resource allocation, budgeting, and data analysis where constraints are imposed on the sum of elements.

Potential pitfalls include misunderstanding the requirement for the subarray to be contiguous and not considering all possible subarrays.

Approach

To solve this problem, we can use a sliding window approach. The naive solution involves checking all possible subarrays, which is not optimal. Instead, we can use a more efficient approach:

• Initialize two pointers to represent the start and end of the subarray.
• Expand the end pointer to include more elements until the sum exceeds S.
• When the sum exceeds S, move the start pointer to reduce the sum.
• Keep track of the longest subarray found that meets the condition.

Algorithm

Here is a step-by-step breakdown of the algorithm:

1. Initialize variables to keep track of the current sum, the maximum length of the subarray, and the start and end indices of the longest subarray.
2. Use a nested loop where the outer loop represents the start of the subarray and the inner loop represents the end of the subarray.
3. For each subarray, calculate the sum and check if it is less than or equal to S.
4. If the sum is within the limit, update the maximum length and the start and end indices.
5. Return the start and end indices of the longest subarray found.

Code Implementation

def longest_subarray_with_sum_at_most_s(nums, S):
    n = len(nums)
    max_length = 0
    start_index = 0
    end_index = 0

    for start in range(n):
        current_sum = 0
        for end in range(start, n):
            current_sum += nums[end]
            if current_sum <= S:
                if end - start + 1 > max_length:
                    max_length = end - start + 1
                    start_index = start
                    end_index = end
            else:
                break

    return [start_index, end_index]

# Example usage
nums = [3, 2, 5, 2, 2, 1, 1, 3, 1, 2]
S = 11
print(longest_subarray_with_sum_at_most_s(nums, S))  # Output: [3, 8]

Complexity Analysis

The time complexity of this approach is O(n^2) because we use a nested loop to check all possible subarrays. The space complexity is O(1) as we only use a few extra variables.

Edge Cases

Consider the following edge cases:

• All elements are greater than S: The function should return an empty subarray.
• The array is empty: The function should handle this gracefully.
• S is very large: The function should return the entire array.

Testing

To test the solution comprehensively, consider the following test cases:

• Simple cases with small arrays.
• Cases where all elements are greater than S.
• Cases with large values of S.
• Edge cases with empty arrays.

Thinking and Problem-Solving Tips

When approaching such problems, it is essential to:

• Understand the problem requirements and constraints thoroughly.
• Consider both naive and optimized solutions.
• Break down the problem into smaller, manageable parts.
• Practice solving similar problems to improve problem-solving skills.

Conclusion

In this blog post, we discussed how to find the longest contiguous subarray with a sum at most S. We explored the problem definition, approach, algorithm, code implementation, complexity analysis, edge cases, and testing. Understanding and solving such problems is crucial for developing strong problem-solving skills.

Additional Resources

For further reading and practice, consider the following resources:

• LeetCode - A platform for practicing coding problems.
• GeeksforGeeks - A comprehensive resource for learning algorithms and data structures.
• Coursera - Online courses on algorithms and problem-solving.