Minimum Operations to Make Elements Within K Subarrays Equal

Start by scanning the array with a sliding window of size x and tracking element counts with a hash table. For each window, compute the number of changes needed to make all elements equal, ideally using the median to minimize operations. Select k non-overlapping windows that require the fewest changes and sum their operations for the final result.

Problem Statement

Given an integer array nums and two integers x and k, determine the minimum number of operations required so that at least k non-overlapping subarrays of size x each contain identical elements. An operation consists of changing any element to any other integer value.

Return the minimum operations needed to achieve this, where a subarray is considered valid only if all elements within it are equal. For example, with nums = [5,-2,1,3,7,3,6,4,-1], x = 3, and k = 2, the correct output is 8, as you must adjust two windows optimally to satisfy the condition.

Examples

Example 1

Input: nums = [5,-2,1,3,7,3,6,4,-1], x = 3, k = 2

Output: 8

Example 2

Input: nums = [9,-2,-2,-2,1,5], x = 2, k = 2

Output: 3

Constraints

• 2 <= nums.length <= 105
• -106 <= nums[i] <= 106
• 2 <= x <= nums.length
• 1 <= k <= 15
• 2 <= k * x <= nums.length

Solution Approach

Use Sliding Window with Hash Counting

Scan the array with a window of size x and maintain a hash table counting occurrences of each number. The minimal operations for each window are calculated as window size minus the count of the most frequent element.

Select Optimal k Windows

Once each window's cost is computed, select k non-overlapping windows that collectively have the lowest operation sum. Track positions carefully to ensure no overlap, which is the main failure mode for greedy selection.

Aggregate Results

Sum the minimal operations from the selected k windows. This ensures you meet the requirement for exactly k subarrays of size x with all equal elements while minimizing changes.

Complexity Analysis

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

Time complexity depends on array scanning and hash updates per window, roughly O(n) per window calculation. Space complexity is O(x) per hash table window, plus extra for storing window costs for selection.

What Interviewers Usually Probe

• Emphasis on using hash tables to efficiently count elements in a sliding window.
• Check for edge cases where windows overlap and require careful selection of k windows.
• Understanding that median-based transformation minimizes changes within a window is a strong hint.

Common Pitfalls or Variants

Common pitfalls

• Forgetting to prevent overlapping when choosing the k windows.
• Incorrectly calculating operations without considering the most frequent element per window.
• Attempting a brute force solution, which fails for large arrays due to time complexity.

Follow-up variants

• Change the problem to require exactly k overlapping subarrays instead of non-overlapping.
• Allow subarrays of variable size x and compute minimal operations for dynamic lengths.
• Compute minimal operations if elements must form an arithmetic progression instead of equality.

FAQ

What is the main pattern used in Minimum Operations to Make Elements Within K Subarrays Equal?

The problem uses an array scanning plus hash lookup pattern, tracking element frequencies within sliding windows.

How do you calculate the minimal operations for a window?

Compute the window size minus the frequency of the most common element; this represents the number of changes needed.

Can windows overlap when selecting k subarrays?

No, selected subarrays must be non-overlapping; failing to enforce this is a common mistake.

Why is median considered for minimal operations?

Transforming all elements to the median minimizes the sum of absolute differences, reducing the number of operations needed.

What constraints should I watch out for in large arrays?

Arrays can be up to 105 elements long, so brute force checking of all subarrays will fail; efficient window scanning with hash counting is required.