#2897

Hard

auto_awesomeArray scanning plus hash lookup

LeetCode Problem Workspace

Apply Operations on Array to Maximize Sum of Squares

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Maximizing the sum of squares in an array through bitwise operations on selected elements.

array hash table greedy bit manipulation

Problem Statement

You are given an integer array nums and a positive integer k. Your task is to choose k elements from nums and calculate the sum of their squares after performing any number of bitwise operations. These operations allow you to modify the array, where for any two selected indices i and j, you can replace nums[i] with (nums[i] AND nums[j]) and nums[j] with (nums[i] OR nums[j]).

To solve this problem, you need to apply the bitwise operations strategically to select the most optimal k elements that result in the highest possible sum of their squares. You can apply these operations any number of times on the array elements. Your goal is to compute the maximum sum of squares from the final selected elements after applying the operations.

Examples

Example 1

Input: nums = [2,6,5,8], k = 2

Output: 261

We can do the following operations on the array:

Choose i = 0 and j = 3, then change nums[0] to (2 AND 8) = 0 and nums[3] to (2 OR 8) = 10. The resulting array is nums = [0,6,5,10].
Choose i = 2 and j = 3, then change nums[2] to (5 AND 10) = 0 and nums[3] to (5 OR 10) = 15. The resulting array is nums = [0,6,0,15]. We can choose the elements 15 and 6 from the final array. The sum of squares is 152 + 62 = 261. It can be shown that this is the maximum value we can get.

Example 2

Input: nums = [4,5,4,7], k = 3

Output: 90

We do not need to apply any operations. We can choose the elements 7, 5, and 4 with a sum of squares: 72 + 52 + 42 = 90. It can be shown that this is the maximum value we can get.

Constraints

1 <= k <= nums.length <= 105
1 <= nums[i] <= 109

Solution Approach

Initial Array Scanning and Selection

Start by scanning the array to find potential elements that could be transformed into higher values using bitwise operations. This involves choosing which elements to apply AND and OR operations to maximize the final result. The array should be examined for elements where bitwise operations could significantly increase their values.

Strategic Bitwise Operations

For each selected pair of elements, perform the AND and OR operations to transfer certain bits from one element to another. The goal is to transfer bits that would increase the value of the elements you are interested in selecting for the final sum. Track the changes that result in the largest values to ensure that your final selection is optimal.

Final Element Selection for Maximum Sum

After applying the bitwise operations to modify the array elements, select the k largest values from the final array. The sum of their squares will give you the desired result. Ensure that the elements you select are those that have the highest possible values after transformations.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of the solution depends on the approach for applying bitwise operations and selecting the top k elements. A brute-force solution might have higher complexity, but optimizing the selection process with efficient scanning and hash table lookups can reduce the time complexity. Space complexity is driven by the need to store modified elements and the results of bitwise operations.

What Interviewers Usually Probe

Candidates should be able to efficiently identify the most significant elements to modify using bitwise operations.
Interviewers should look for the candidate’s ability to optimize the selection of k elements from the array after transformations.
The candidate’s approach to the problem should showcase familiarity with bitwise operations and array manipulation for optimization.

Common Pitfalls or Variants

Common pitfalls

Failing to optimize the selection of elements after bitwise operations, resulting in suboptimal answers.
Overlooking the need for efficient array scanning and hash lookup when selecting the best k elements.
Inadequate handling of bitwise operation results, leading to incorrect final values for the selected elements.

Follow-up variants

Applying bitwise operations but changing the problem constraints (e.g., different k or larger array sizes).
Changing the allowed operations to include XOR or other bitwise manipulations.
Limiting the number of times bitwise operations can be applied, requiring additional optimization strategies.

FAQ

How do I maximize the sum of squares in 'Apply Operations on Array to Maximize Sum of Squares'?

The key to maximizing the sum of squares is applying bitwise AND and OR operations strategically on selected elements to increase their final values.

What bitwise operations are allowed in the 'Apply Operations on Array to Maximize Sum of Squares' problem?

You can apply the AND and OR bitwise operations between selected pairs of array elements.

What is the optimal approach to solving the 'Apply Operations on Array to Maximize Sum of Squares' problem?

The optimal approach involves scanning the array to find candidates for bitwise transformations, applying operations to maximize values, and then selecting the k largest elements for summing their squares.

How do bitwise operations affect the array in this problem?

Bitwise operations transfer specific bits between elements, increasing their values in certain cases, which allows you to maximize the sum of their squares.

What is the expected time complexity for the 'Apply Operations on Array to Maximize Sum of Squares' problem?

The time complexity depends on how efficiently you apply bitwise operations and select the k largest elements, but optimizing these steps can significantly reduce the time complexity.

terminal

Solution

Solution 1: Bitwise Operation + Greedy

According to the problem description, for an operation, we can change $nums[i]$ to $nums[i] \textit{ AND } nums[j]$, and change $nums[j]$ to $nums[i] \textit{ OR } nums[j]$. Let's consider the bits of the numbers. If two bits are both $1$ or both $0$, the result of the operation will not change the bits. If two bits are different, the result of the operation will change the bits to $0$ and $1$, respectively. Therefore, we can move $1$ bits to $0$ bits, but not vice versa.

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class Solution:
    def maxSum(self, nums: List[int], k: int) -> int:
        mod = 10**9 + 7
        cnt = [0] * 31
        for x in nums:
            for i in range(31):
                if x >> i & 1:
                    cnt[i] += 1
        ans = 0
        for _ in range(k):
            x = 0
            for i in range(31):
                if cnt[i]:
                    x |= 1 << i
                    cnt[i] -= 1
            ans = (ans + x * x) % mod
        return ans

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