Maximize the Sum of Squares After Bitwise Pair Operations
Company: Google
Role: Software Engineer
Category: Coding & Algorithms
Difficulty: hard
Interview Round: Other
# Maximize the Sum of Squares After Bitwise Pair Operations
You are given a 0-indexed integer array `nums` and a positive integer `k`.
You may perform the following operation any number of times, including zero times. Choose two distinct indices `i` and `j`, and simultaneously replace their values using the old values:
- Set `nums[i]` to `nums[i] AND nums[j]`.
- Set `nums[j]` to `nums[i] OR nums[j]`.
Here `AND` and `OR` are bitwise operations. After completing the operations, choose exactly `k` array elements and sum their squares.
Return the maximum possible sum of squares, modulo `10^9 + 7`. The objective is the maximum actual sum; apply the modulus to that result.
### Examples
**Example 1**
```text
Input: nums = [1,2,4], k = 2
Output: 49
```
An operation with `(i,j) = (0,1)` changes the array to `[0,3,4]`. An operation with `(i,j) = (1,2)` then gives `[0,0,7]`. Choosing `7` and one zero gives `49`. Each of the three available bit positions occurs only once, so the largest possible squared-sum arrangement concentrates them in `7`.
**Example 2**
```text
Input: nums = [3,5,6], k = 2
Output: 98
```
Operations with `(i,j) = (0,1)` and then `(0,2)` produce `[1,7,6]` and `[0,7,7]`. Choose the two sevens for `49 + 49 = 98`. Each of the three bit positions occurs twice, allowing two values of `7` but no larger value.
### Constraints
- `1 <= k <= nums.length <= 10^5`.
- `1 <= nums[i] <= 10^9` initially.
### Clarifying Questions
- Do both assignments in an operation use the original pair values? Yes; they occur simultaneously.
- Must the two indices be distinct? Yes.
- Must exactly `k` positions be chosen, even if some final values are zero? Yes. Operations may create zero values despite the positive initial bounds.
- Do repeated equal values represent different selectable elements? Yes; selection is by distinct array positions.
- Are operations or selected positions required as output? No; return only the maximum sum modulo the specified modulus.
### C++ Interface
```cpp
class Solution {
public:
int maxSum(vector<int>& nums, int k) {
}
};
```
### Reasoning Checkpoints
- Identify what an operation preserves and which final values it can create.
- Justify both reachability and optimality of the selected final values.
- Respect exact selection count, simultaneous updates, and arithmetic limits before taking the modulus.
```hint Count bits and concentrate them
Count the occurrences of each set bit across the array. What happens to the sum of squares when an AND/OR operation gives the OR to the larger of two nonnegative values? Try forming one largest remaining value at a time.
```
Overview: Choose exactly k final array elements after any number of simultaneous bitwise AND and OR pair operations, maximizing their squared sum modulo the stated modulus.