Minimum Array Changes to Make Differences Equal

Minimum Array Changes to Make Differences Equal

What is this problem about?

The Minimum Array Changes to Make Differences Equal problem gives you an array of length n (where n is even) and an integer k. You can change any element to any value between 0 and k. The goal is to make the absolute difference between nums[i] and nums[n - 1 - i] equal to some value X for all i. You want to find the minimum number of changes needed to achieve this for the best possible X.

Why is this asked in interviews?

This is a sophisticated Minimum Array Changes to Make Differences Equal interview question asked at Google and Airbus. It tests Prefix Sums (Difference Arrays) and range-based optimization. It's a great example of a problem that looks like it needs a brute force over all possible X, but can be solved much faster by analyzing the "contribution" of each pair to each possible difference.

Algorithmic pattern used

The Difference Array / Prefix Sum interview pattern is the key.

1. For each pair (a, b), calculate the current difference diff = |a - b|.
2. Determine the range of differences reachable with one change: [0, max_diff_with_one_change].
3. Any difference X outside this range requires two changes.
4. Use a difference array to track how many changes are needed for each possible X from 0 to k.
5. For each pair:
   • It costs 0 changes for X = diff.
   • It costs 1 change for X in [0, max_range] (excluding diff).
   • It costs 2 changes for X in [max_range + 1, k].
6. Traverse the prefix sum array to find the minimum value.

Example explanation

Array [1, 0, 1, 2], k = 2. Pairs: (1, 2) and (0, 1).

• Pair (1, 2): diff = 1. With one change, max diff is max(2-1, 1-0) = 1? No, if we change 1 to 0, diff is |0-2|=2. Max diff with one change is max(1, 0, 2-1, 1-0, 2-2, 2-0...) = 2.
• So for X=1, 0 changes. For X in [0, 2], 1 change.
• Aggregate these costs across all pairs to find the X that minimizes the total.

Common mistakes candidates make

• Trying every X: Iterating X from 0 to k and for each X, iterating through all pairs. This is $O(K \cdot N)$, which is too slow if k is $10^5$.
• Incorrect range calculation: Miscalculating the maximum difference reachable with one change for a pair (a, b). It is max(a, b, k - a, k - b).
• Ignoring the two-change case: Forgetting that if X is very large, you might need to change both numbers in a pair to satisfy the condition.

Interview preparation tip

When you need to find an optimal "global" value (like X) and each element provides a "cost range," think about Difference Arrays. This allows you to perform range updates in $O(1)$ and find the optimal value in $O(N + K)$ time. This Prefix Sum interview pattern is essential for high-level competitive programming.