Maximal Score After Applying K Operations

Maximal Score After Applying K Operations

What is this problem about?

In this problem, you are given an integer array and an integer k. In a single operation, you can select any element x from the array, add its value to your total score, and then replace that element in the array with ceil(x / 3). You must perform exactly k operations. Your goal is to maximize your total score.

Why is this asked in interviews?

This is a standard greedy optimization problem. Interviewers ask it to test a candidate's familiarity with Priority Queues (Max Heaps). It evaluates if you know how to repeatedly access, process, and re-insert the largest element in a dataset efficiently. It is a highly reliable screener for basic data structure competence.

Algorithmic pattern used

The problem is tailor-made for a Max Heap (Priority Queue). Since you want to maximize your score, you should always pick the absolute largest number currently available in the array.

1. Insert all array elements into a Max Heap.
2. Loop k times.
3. Pop the largest element, add it to your score.
4. Calculate ceil(value / 3.0) and push it back into the heap.
5. Return the total score.

Example explanation

Array: [10, 20, 7], k = 3. Build Max Heap: [20, 10, 7]. Score = 0.

• Operation 1: Pop 20. Score = 20. Push ceil(20 / 3) = 7. Heap: [10, 7, 7].
• Operation 2: Pop 10. Score = $20 + 10 = 30$. Push ceil(10 / 3) = 4. Heap: [7, 7, 4].
• Operation 3: Pop 7. Score = $30 + 7 = 37$. Push ceil(7 / 3) = 3. Heap: [7, 4, 3]. Operations complete. Total maximum score is 37.

Common mistakes candidates make

A very common programming error is doing integer division before applying the ceiling function. In Java or C++, Math.ceil(10 / 3) will result in 3.0 because 10 / 3 performs integer division (truncating to 3) before ceil is called! You must use floating-point math: Math.ceil(10 / 3.0). Another mistake is sorting the array and iterating backward, which fails because the reduced elements might still be larger than other elements in the array.

Interview preparation tip

When solving the Maximal Score After Applying K Operations coding problem, always use double division to ensure accurate ceiling calculations. Furthermore, recognize that extracting the maximum element dynamically over multiple steps is the textbook definition of a Priority Queue. Mentioning that Heap extraction is $O(\log N)$ while resorting an array is $O(N \log N)$ shows a strong grasp of algorithmic time complexities.