3630. Partition Array for Maximum XOR and AND
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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 inarr. Ifarris empty, its value is defined as 0.AND(arr)denotes the bitwise AND of all elements inarr. Ifarris 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) = 3B = [2], AND(B) = 2C = [], 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) = 1B = [2], AND(B) = 2C = [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) = 7B = [2,3], AND(B) = 2C = [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 <= 191 <= nums[i] <= 109
Hints
Hint 1
B.Hint 2
s = XOR of all elements not in B.Hint 3
x (a subset‐XOR of the "remaining" elements) to maximize x + (s XOR x).Hint 4
x + (s XOR x) = s + 2 * (x AND ~s).Hint 5
nums[j] & ~s for each index j not in B.Hint 6
x from that basis.