2897. Apply Operations on Array to Maximize Sum of Squares

2026, Oct 11 2 mins read

1. Description

Given an integer array nums and a positive integer k, you may repeatedly select two different positions and replace their values a and b with a AND b and a OR b, respectively. Both bitwise results use the values before the operation.
After any number of operations, including zero, select k elements. Maximize the sum of their squares and return that maximum modulo $10^9 + 7$.

2. Example

Example 1

Input: nums = [2,6,5,8], k = 2
Output: 261
Explanation: Combining 2 with 8 gives [0,6,5,10]. Combining 5 with 10 then gives [0,6,0,15]. Select 15 and 6 to obtain $15^2 + 6^2 = 261$.

Example 2

Input: nums = [4,5,4,7], k = 3
Output: 90
Explanation: Select 7, 5, and 4 without changing the array, giving $7^2 + 5^2 + 4^2 = 90$.

3. Constraints

1 <= k <= nums.length <= $10^5$
1 <= nums[i] <= $10^9$

4. Solutions

Greedy with Bit Counting

n = nums.size()
Time complexity: O(n)
Space complexity: O(1)

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// AND/OR preserves the count of set bits at each position.
// Concentrate available bits into each of the k numbers to maximize their squared sum.
class Solution {
public:
    int maxSum(vector<int> &nums, int k) {
        const long long MOD = 1e9 + 7;

        array<int, 31> bit_count{};
        for (int num : nums) {
            for (int bit = 0; bit < 31; ++bit) {
                if (num & (1 << bit)) {
                    ++bit_count[bit];
                }
            }
        }

        long long answer = 0;
        for (int i = 0; i < k; ++i) {
            long long num = 0;
            for (int bit = 0; bit < 31; ++bit) {
                if (bit_count[bit] > 0) {
                    num |= 1LL << bit;
                    --bit_count[bit];
                }
            }

            answer = (answer + num * num) % MOD;
        }

        return answer;
    }
};

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