Maximum Product Difference Between Two Pairs

Maximum Product Difference Between Two Pairs

1. What is this problem about?

The Maximum Product Difference Between Two Pairs interview question is a straightforward array optimization problem. You are given an array of integers. You need to pick four distinct indices $(a, b, c, d)$ such that the product difference $(nums[a] * nums[b]) - (nums[c] * nums[d])$ is maximized.

To maximize this difference, you want the first product $(nums[a] * nums[b])$ to be as large as possible and the second product $(nums[c] * nums[d])$ to be as small as possible.

2. Why is this asked in interviews?

This Maximum Product Difference Between Two Pairs coding problem is often an "icebreaker" or early-round question at companies like Apple and Amazon. It tests whether a candidate can identify the core requirements of an optimization problem without being distracted by the "pairs" wording. It's a simple test of sorting or finding extremas in an array efficiently.

3. Algorithmic pattern used

The Array, Sorting interview pattern is the most common way to solve this.

1. Sort the given array in non-decreasing order.
2. The two largest numbers will be at the very end of the array (indices $n-1$ and $n-2$).
3. The two smallest numbers will be at the very beginning (indices 0 and 1).
4. The maximum product difference is (nums[n-1] * nums[n-2]) - (nums[0] * nums[1]).

Alternatively, you can find the two largest and two smallest numbers in a single O(N) pass for better performance.

4. Example explanation

Array: [5, 6, 2, 7, 4] Sorted: [2, 4, 5, 6, 7]

• Two largest: 6 and 7. Product = 42.
• Two smallest: 2 and 4. Product = 8.
• Difference: 42 - 8 = 34.

Max Product Difference is 34.

5. Common mistakes candidates make

In the Maximum Product Difference Between Two Pairs coding problem, the most common mistake is sorting the entire array ($O(N \log N)$) when only four values are needed. While sorting passes the tests, an $O(N)$ solution (finding the top 2 and bottom 2 elements) is more impressive in a technical interview. Another minor error is picking the same index twice, but the problem specifies distinct indices.

6. Interview preparation tip

For problems where you only need a fixed number of the largest or smallest elements, always mention that while sorting is easier to implement, finding the extremas in one pass is more optimal in terms of time complexity ($O(N)$ vs $O(N \log N)$). This shows you care about performance.