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1835. Find XOR Sum of All Pairs Bitwise AND (Hard)

The XOR sum of a list is the bitwise XOR of all its elements. If the list only contains one element, then its XOR sum will be equal to this element.

• For example, the XOR sum of [1,2,3,4] is equal to 1 XOR 2 XOR 3 XOR 4 = 4, and the XOR sum of [3] is equal to 3.

You are given two 0-indexed arrays arr1 and arr2 that consist only of non-negative integers.

Consider the list containing the result of arr1[i] AND arr2[j] (bitwise AND) for every (i, j) pair where 0 <= i < arr1.length and 0 <= j < arr2.length.

Return the XOR sum of the aforementioned list.

 

Example 1:

Input: arr1 = [1,2,3], arr2 = [6,5]
Output: 0
Explanation: The list = [1 AND 6, 1 AND 5, 2 AND 6, 2 AND 5, 3 AND 6, 3 AND 5] = [0,1,2,0,2,1].
The XOR sum = 0 XOR 1 XOR 2 XOR 0 XOR 2 XOR 1 = 0.

Example 2:

Input: arr1 = [12], arr2 = [4]
Output: 4
Explanation: The list = [12 AND 4] = [4]. The XOR sum = 4.

 

Constraints:

• 1 <= arr1.length, arr2.length <= 105
• 0 <= arr1[i], arr2[j] <= 109

Related Topics

[Bit Manipulation] [Array] [Math]

Hints

Hint 1

Think about (a&b) ^ (a&c). Can you simplify this expression?

Hint 2

It is equal to a&(b^c). Then, (arr1[i]&arr2[0])^(arr1[i]&arr2[1]).. = arr1[i]&(arr2[0]^arr2[1]^arr[2]...).

Hint 3

Let arr2XorSum = (arr2[0]^arr2[1]^arr2[2]...), arr1XorSum = (arr1[0]^arr1[1]^arr1[2]...) so the final answer is (arr2XorSum&arr1[0]) ^ (arr2XorSum&arr1[1]) ^ (arr2XorSum&arr1[2]) ^ ... = arr2XorSum & arr1XorSum.