Minimum Average of Smallest and Largest Elements

Minimum Average of Smallest and Largest Elements

What is this problem about?

The Minimum Average of Smallest and Largest Elements problem is a simple array-processing task. You are given an array of numbers, and you need to perform a series of operations. In each step, you remove the smallest and the largest remaining elements, calculate their average, and store it. After repeating this until the array is empty, your goal is to find the minimum of all those calculated averages.

Why is this asked in interviews?

This is an introductory-level Minimum Average of Smallest and Largest Elements interview question often used at Microsoft and Amazon. It tests a candidate's familiarity with Sorting and Two Pointers. It's designed to see if you can quickly identify the most efficient way to access extreme values in a collection.

Algorithmic pattern used

The Two Pointers interview pattern is the most intuitive here.

1. Sort the array in non-decreasing order.
2. Place one pointer left at the start (index 0) and another right at the end (index n-1).
3. In each step, calculate (nums[left] + nums[right]) / 2.0.
4. Update your minimum average variable.
5. Increment left and decrement right.

Example explanation

Array [7, 8, 3, 4, 15, 13, 4, 1].

1. Sort: [1, 3, 4, 4, 7, 8, 13, 15].
2. Pair 1: (1, 15) / 2 = 8.0.
3. Pair 2: (3, 13) / 2 = 8.0.
4. Pair 3: (4, 8) / 2 = 6.0.
5. Pair 4: (4, 7) / 2 = 5.5. Minimum average is 5.5.

Common mistakes candidates make

• Not sorting first: Trying to find the min and max repeatedly in $O(N)$ time for each pair, which makes the whole process $O(N^2)$. Sorting makes it $O(N \log N)$.
• Integer division: Using integer division (like in Python // or C++ int / int) when the averages could be floats (e.g., (3+4)/2 = 3.5).
• Incorrect loop bounds: Forgetting that you need to perform exactly n/2 steps.

Interview preparation tip

Always think about Sorting when a problem involves "smallest and largest" or "pairs from extremes." Once sorted, the extreme values are always at the boundaries, which is the perfect setup for a Two Pointers interview pattern.