Make K-Subarray Sums Equal

Problem Description

You are given a 0-indexed integer array arr and an integer k. The array arr is circular. In other words, the first element of the array is the next element of the last element, and the last element of the array is the previous element of the first element.

You can do the following operation any number of times:

• Pick any element from arr and increase or decrease it by 1.

Return the minimum number of operations such that the sum of each subarray of length k is equal.

Key Insights

• The problem involves making the sum of all k-length subarrays equal in a circular array.
• Each subarray of length k overlaps with its neighbors, which means changes will affect multiple subarrays.
• The goal is to minimize the number of operations required to achieve equal sums across all subarrays.
• The optimal target value for the elements can be derived from the average sum of the elements in each k-length segment.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(n)

Solution

To solve the problem, we can use a sliding window approach to consider each k-length segment in the circular array. The steps are as follows:

1. Calculate the total sum of the array and determine the desired sum for each k-length subarray.
2. Iterate through the array while maintaining a running total of the current k-length segment.
3. For each position, compute how far the current segment is from the desired sum.
4. Use a counter to keep track of the adjustments needed to bring the current segment to the desired sum.
5. Since the array is circular, ensure to consider wrapping around when indexing.
6. The final answer will be the total number of operations needed to balance all segments.

Code Solutions