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3630. Partition Array for Maximum XOR and AND

Description

You are given an integer array nums.

Partition the array into three (possibly empty) subsequences A, B, and C such that every element of nums belongs to exactly one subsequence.

Your goal is to maximize the value of: XOR(A) + AND(B) + XOR(C)

where:

  • XOR(arr) denotes the bitwise XOR of all elements in arr. If arr is empty, its value is defined as 0.
  • AND(arr) denotes the bitwise AND of all elements in arr. If arr is empty, its value is defined as 0.

Return the maximum value achievable.

Note: If multiple partitions result in the same maximum sum, you can consider any one of them.

 

Example 1:

Input: nums = [2,3]

Output: 5

Explanation:

One optimal partition is:

  • A = [3], XOR(A) = 3
  • B = [2], AND(B) = 2
  • C = [], XOR(C) = 0

The maximum value of: XOR(A) + AND(B) + XOR(C) = 3 + 2 + 0 = 5. Thus, the answer is 5.

Example 2:

Input: nums = [1,3,2]

Output: 6

Explanation:

One optimal partition is:

  • A = [1], XOR(A) = 1
  • B = [2], AND(B) = 2
  • C = [3], XOR(C) = 3

The maximum value of: XOR(A) + AND(B) + XOR(C) = 1 + 2 + 3 = 6. Thus, the answer is 6.

Example 3:

Input: nums = [2,3,6,7]

Output: 15

Explanation:

One optimal partition is:

  • A = [7], XOR(A) = 7
  • B = [2,3], AND(B) = 2
  • C = [6], XOR(C) = 6

The maximum value of: XOR(A) + AND(B) + XOR(C) = 7 + 2 + 6 = 15. Thus, the answer is 15.

 

Constraints:

  • 1 <= nums.length <= 19
  • 1 <= nums[i] <= 109

Solutions

Solution 1

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