Minimum Operations to Make All Array Elements Equal

Minimum Operations to Make All Array Elements Equal

1. What is this problem about?

This problem asks you to process multiple queries on an array, where each query provides a target value. For each query, you need to calculate the minimum number of operations required to make all elements in the array equal to that target value. An operation consists of either increasing or decreasing an element by 1. Since you need to answer multiple queries efficiently, a simple iteration for each target value would be too slow for large inputs.

2. Why is this asked in interviews?

Interviewers at J.P. Morgan and Amazon use this problem to test a candidate's proficiency with Prefix Sums and Binary Search. It requires moving beyond a naive O(N*Q) approach to an optimized O((N+Q) log N) solution. The problem effectively evaluates how well you can manipulate mathematical summations and use precomputed data to answer range-sum queries in constant time.

3. Algorithmic pattern used

The solution utilizes Sorting, Binary Search, and Prefix Sums. After sorting the array, you can use binary search (specifically bisect_left or upper_bound) to find how many elements are smaller than the target value and how many are larger. By precomputing prefix sums, you can calculate the total sum of the smaller elements and the total sum of the larger elements in O(1). The total operations for a target k is then: (k * count_smaller - sum_smaller) + (sum_larger - k * count_larger).

4. Example explanation

Array: [1, 3, 6, 8], Target Query: 5.

• Sorted array: [1, 3, 6, 8].
• Prefix Sums: [0, 1, 4, 10, 18].
• Elements smaller than 5: [1, 3] (count = 2, sum = 4).
• Elements larger than 5: [6, 8] (count = 2, sum = 14).
• Operations to increase smaller elements: (5 * 2) - 4 = 6.
• Operations to decrease larger elements: 14 - (5 * 2) = 4.
• Total operations = 6 + 4 = 10.

5. Common mistakes candidates make

The most frequent mistake is neglecting the prefix sum precomputation and trying to iterate through the array for every query, which leads to time complexity issues. Another error is miscalculating the indices when using prefix sums, often leading to "off-by-one" errors. Candidates also sometimes forget that they need to sort the array before applying binary search, or they struggle with the mathematical formula for calculating the differences for the two partitions.

6. Interview preparation tip

Prefix sums are a powerful tool for optimizing range queries. Whenever you see a problem involving "sum of elements in a range" or "total difference from a target," think about how precomputing sums can help. Practice binary search variations to find split points in sorted arrays efficiently. Also, always check if your prefix sum array needs an extra 0 at the beginning to simplify range calculations.