Apply Operations to Maximize Frequency Score

Apply Operations to Maximize Frequency Score

What is this problem about?

The Apply Operations to Maximize Frequency Score interview question tasks you with finding the maximum possible frequency of any element in an array after performing at most k operations. An operation consists of increasing or decreasing an element by 1. To maximize frequency, you want to make as many elements as possible equal to some target value. This Apply Operations to Maximize Frequency Score coding problem is a test of range optimization.

Why is this asked in interviews?

This is a HARD level problem often seen at PhonePe or Deutsche Bank. It tests multiple advanced concepts: sorting, sliding windows, and prefix sums. Interviewers look for candidates who can recognize that the "target value" to which elements should be converted is the median of a sorted range, as the sum of absolute differences is minimized at the median.

Algorithmic pattern used

The solution utilizes the Array, Sorting, Sliding Window, Binary Search, Prefix Sum interview pattern. By sorting the array, you ensure that elements you want to make equal are close to each other. A sliding window or binary search on the answer (the frequency) can be used to check if a range of a certain size can be equalized within the budget k. Prefix sums allow for O(1) calculation of the cost to change a range of numbers to their median.

Example explanation

Suppose you have an array [1, 4, 5] and k = 2.

• Can we get a frequency of 3? To make all elements equal to the median (4), we need |1-4| + |4-4| + |5-4| = 3 + 0 + 1 = 4. Since 4 > k, we can't.
• Can we get a frequency of 2? Look at window [4, 5]. Cost to make them equal (to 4 or 5) is 1. Since 1 <= k, yes. The answer is 2.

Common mistakes candidates make

• Not Sorting: Trying to solve without sorting makes it nearly impossible to efficiently find the best elements to group.
• Inefficient Cost Calculation: Using a loop to calculate the cost for every window, leading to O(N^2) instead of using Prefix Sums for O(N).
• Wrong Target: Not realizing that for any given window, the median is the optimal value to convert all elements to.

Interview preparation tip

Master the property of the median: it minimizes the sum of absolute deviations. Also, practice combining sliding windows with prefix sums, as this is a very common pattern in HARD array problems.