Apply Operations on Array to Maximize Sum of Squares

Problem Description

You are given a 0-indexed integer array nums and a positive integer k. You can perform operations on the array where you choose any two distinct indices i and j, updating nums[i] to (nums[i] AND nums[j]) and nums[j] to (nums[i] OR nums[j]). Your task is to select k elements from the final array and calculate the maximum sum of their squares, returning the result modulo 10^9 + 7.

Key Insights

• The operations allow us to manipulate the values in the array by combining them using bitwise AND and OR.
• After performing operations, the larger values are often favored for maximizing the sum of squares.
• Sorting the array and selecting the top k largest elements after any operations can lead to the maximum sum of squares.
• The operations can potentially allow us to produce larger values from smaller ones, especially when combining the largest elements.

Space and Time Complexity

Time Complexity: O(n log n), where n is the number of elements in the nums array (due to sorting). Space Complexity: O(1) if sorting in place, otherwise O(n) for storing a sorted copy.

Solution

To solve this problem, we can leverage sorting and selection strategies. We observe that after performing operations, the largest elements will give the highest contributions to the sum of squares. Therefore, we can simply sort the array, select the top k elements, and calculate the sum of their squares.

1. Sort the nums array in descending order.
2. Select the first k elements from the sorted array.
3. Compute the sum of squares of these k elements.
4. Return the result modulo 10^9 + 7.

Code Solutions