Array Partition

Array Partition

What is this problem about?

The Array Partition interview question (often called Array Partition I) gives you an array of 2n integers. Your task is to group these integers into n pairs (a1, b1), (a2, b2), ..., (an, bn) such that the sum of min(ai, bi) for all i is maximized. This Array Partition coding problem is a classic greedy optimization challenge.

Why is this asked in interviews?

This is a standard question for companies like Amazon, Apple, and Google. It tests if a candidate can deduce a simple mathematical property from a seemingly complex grouping problem. It checks for a "Greedy" intuition—the realization that to maximize the sum of minimums, you should pair numbers that are closest to each other.

Algorithmic pattern used

This problem follows the Array, Sorting, Counting Sort, Greedy interview pattern. The optimal strategy is to sort the array and then sum every second element (at indices 0, 2, 4, etc.). Since sorting is the bottleneck, O(N log N) is the standard complexity, though Counting Sort can achieve O(N) if the value range is small.

Example explanation

Suppose nums = [1, 4, 3, 2].

1. Sort the array: [1, 2, 3, 4].
2. Form pairs: (1, 2) and (3, 4).
3. Sum of minimums: min(1, 2) + min(3, 4) = 1 + 3 = 4. Any other grouping, like (1, 4) and (2, 3), would yield min(1, 4) + min(2, 3) = 1 + 2 = 3, which is smaller.

Common mistakes candidates make

• Over-complicating: Trying to use Dynamic Programming or complex backtracking when a simple sort is sufficient.
• Wrong Indexing: Summing the larger elements of the pairs instead of the smaller ones.
• Ignoring Constraints: Not considering if the numbers can be negative (though sorting still works for negatives).

Interview preparation tip

Whenever you see "minimize the maximum" or "maximize the minimum" in a grouping problem, always start by sorting the data. Sorting often reveals the optimal grouping strategy immediately.