Apply Operations to Maximize Score

Apply Operations to Maximize Score

What is this problem about?

The Apply Operations to Maximize Score interview question is a complex problem where you are given an array of positive integers and an integer k. You can perform k operations. In each operation, you choose a non-empty subarray, find the element with the highest "prime score" (number of distinct prime factors), and add that element to your total score. If multiple elements have the same max prime score, the one with the smallest index is chosen. Each element can be the "chosen one" only a certain number of times across all possible subarrays.

Why is this asked in interviews?

This HARD problem is typical of Google or Meta. It requires synthesizing knowledge from number theory (prime factorization), monotonic stacks (to find subarray ranges), and greedy algorithms (to pick the best elements first). It tests if a candidate can break down a massive problem into manageable sub-problems.

Algorithmic pattern used

This uses the Array, Math, Monotonic Stack, Number Theory, Sorting, Stack, Greedy interview pattern.

1. Math: Precompute prime factors for all numbers.
2. Monotonic Stack: For each element, find how many subarrays it "rules" (where it has the highest prime score).
3. Greedy: Sort elements by value descending and use up the k operations on the largest values first.

Example explanation

Consider nums = [8, 3, 7], k = 4.

• Prime scores: 8 (2) -> 1, 3 (3) -> 1, 7 (7) -> 1.
• Since scores are equal, the first occurrence in a subarray is picked.
• Element 8 is the "chosen one" for subarrays [8], [8,3], [8,3,7]. (3 times).
• Element 3 is chosen for [3], [3,7]. (2 times).
• We have k=4. We take 8 as many times as possible (3 times), then 3 once.
• Score = 8^3 * 3^1 (usually the score is a product in these types of problems, or a sum).

Common mistakes candidates make

• Recomputing Primes: Calculating prime factors inside a loop instead of using a Sieve-like approach.
• Subarray Counting: Getting the formula for the number of subarrays an element dominates wrong. The formula is (index - left_boundary) * (right_boundary - index).
• Modular Arithmetic: Forgetting to apply modulo at each multiplication step if the answer is large.

Interview preparation tip

Practice using monotonic stacks to find the nearest element to the left/right that is greater than or equal to the current element. This is a vital building block for many Hard subarray problems.