Maximum and Minimum Sums of at Most Size K Subarrays

Use a monotonic stack to track current maximum and minimum values while iterating through the array. Maintain sliding window logic to efficiently handle subarrays up to size k. Aggregate the sums dynamically to avoid recomputation, ensuring O(n) or O(nk) time depending on implementation.

Problem Statement

Given an integer array nums and a positive integer k, compute the total sum of the maximum and minimum elements for every subarray whose length is at most k. The array may contain negative numbers, and each subarray's contribution must be included exactly once.

Return the sum of all maximum and minimum elements across these subarrays. For example, if nums = [1,2,3] and k = 2, calculate all subarrays of length 1 and 2, sum their maxima and minima, and output the total sum.

Examples

Example 1

Input: nums = [1,2,3], k = 2

Output: 20

The subarrays of nums with at most 2 elements are: The output would be 20.

Example 2

Input: nums = [1,-3,1], k = 2

Output: -6

The subarrays of nums with at most 2 elements are: The output would be -6.

Constraints

• 1 <= nums.length <= 80000
• 1 <= k <= nums.length
• -106 <= nums[i] <= 106

Solution Approach

Monotonic Stack for Maximums

Maintain a decreasing stack to efficiently track maximum values within the current window. Push elements while popping smaller values to ensure the stack top always represents the maximum of the subarray. Aggregate maximums for all subarrays up to size k during traversal.

Monotonic Stack for Minimums

Use a similar increasing stack to track minimum values for each subarray. Pop larger elements while adding new elements to maintain the stack property. Sum minimums dynamically alongside maximums to compute the total efficiently without recomputing every subarray.

Sliding Window Aggregation

Iterate through nums while maintaining both stacks for maximums and minimums. For each index, consider subarrays ending at that index of lengths up to k. Add the corresponding max and min from the stacks to a running total, leveraging monotonic properties to avoid repeated comparisons.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on the stack operations. Each element is pushed and popped at most once from each monotonic stack, leading to O(n) amortized time. Space complexity is O(k) for maintaining the stacks, though worst-case can reach O(n) if k equals n.

What Interviewers Usually Probe

• Look for the candidate using monotonic stacks rather than brute force.
• Expect clear handling of subarrays of varying lengths up to k.
• Check if maximum and minimum sums are aggregated without redundant computation.

Common Pitfalls or Variants

Common pitfalls

• Trying to compute maxima and minima by iterating over all subarrays, leading to TLE.
• Not correctly limiting subarray lengths to at most k.
• Confusing stack operations, losing the monotonic property and miscalculating sums.

Follow-up variants

• Compute only the sum of maximum elements for subarrays of size exactly k.
• Find subarray sums for at most k elements but only for positive numbers.
• Calculate the product instead of sum of maximum and minimum values for subarrays up to size k.

FAQ

What is the key pattern for solving Maximum and Minimum Sums of at Most Size K Subarrays?

The key pattern is stack-based state management, specifically using monotonic stacks to track maximums and minimums efficiently.

Can I solve this problem without a stack?

Yes, but a naive approach will have O(n*k) complexity, which may be too slow for large arrays.

How do I handle subarrays of different lengths up to k?

Iterate through each index and consider subarrays ending at that index of all valid lengths, using the monotonic stacks to extract max and min efficiently.

What are common mistakes when implementing this solution?

Common mistakes include mismanaging the monotonic stack, exceeding k in subarray length, or double-counting subarray sums.

What is the expected time and space complexity?

Amortized O(n) time using monotonic stacks and O(k) space for the stacks, worst-case O(n) if k equals the array length.