Minimum Array Changes to Make Differences Equal

Problem Description

You are given an integer array nums of size n where n is even, and an integer k. You can perform some changes on the array, where in one change you can replace any element in the array with any integer in the range from 0 to k. You need to perform some changes (possibly none) such that the final array satisfies the following condition: There exists an integer X such that abs(a[i] - a[n - i - 1]) = X for all (0 <= i < n). Return the minimum number of changes required to satisfy the above condition.

Key Insights

• The problem requires ensuring that the absolute differences between pairs of elements are consistent across the array.
• The pairs to consider are (nums[i], nums[n - i - 1]) for all valid i.
• The maximum and minimum values of each pair play a crucial role in determining the required changes.
• The allowable replacements are constrained to the range [0, k].

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve this problem, we can use a two-pointer technique to examine pairs of elements from both ends of the array towards the center. For each pair (nums[i], nums[n - i - 1]), we will determine the minimum changes required to make the absolute difference equal to a consistent value X. The possible values of X will be determined based on the values of nums[i] and nums[n - i - 1], as well as the boundaries given by 0 and k. We will count how many changes are necessary for each pair to meet the conditions for X.

Code Solutions