Bitwise XOR of All Pairings

Problem Description

You are given two 0-indexed arrays, nums1 and nums2, consisting of non-negative integers. Let there be another array, nums3, which contains the bitwise XOR of all pairings of integers between nums1 and nums2 (every integer in nums1 is paired with every integer in nums2 exactly once). Return the bitwise XOR of all integers in nums3.

Key Insights

• The bitwise XOR operation has properties such as commutativity and associativity.
• Pairing every element from nums1 with every element from nums2 results in a large number of XOR operations.
• The XOR of a number with itself is zero, and the XOR of a number with zero is the number itself.
• The final result can be derived without explicitly constructing the nums3 array.

Space and Time Complexity

Time Complexity: O(n + m) where n is the length of nums1 and m is the length of nums2.
Space Complexity: O(1), as we are using only a few variables for computations.

Solution

To solve the problem, we can leverage the properties of the XOR operation. For every element in nums1, it will be paired with every element in nums2. The key insight is that each number in nums1 will appear in the XOR operation multiple times based on the length of nums2. Specifically, if nums1 has n elements and nums2 has m elements, each element in nums1 will contribute to the final XOR based on how many times it is paired with elements from nums2.

1. Calculate the XOR of all elements in nums1 and store it in a variable xor1.
2. Calculate the XOR of all elements in nums2 and store it in a variable xor2.
3. The final result will be xor1 * m XOR xor2 * n, where m is the count of elements in nums2 and n is the count of elements in nums1.

This approach avoids the need to create the large nums3 array and directly computes the result.

Code Solutions