Minimum XOR Sum of Two Arrays

Problem Description

Given two integer arrays nums1 and nums2 of length n, rearrange the elements of nums2 such that the resulting XOR sum is minimized. The XOR sum is defined as (nums1[0] XOR nums2[0]) + (nums1[1] XOR nums2[1]) + ... + (nums1[n - 1] XOR nums2[n - 1]).

Key Insights

• The XOR operation has properties that can be exploited to minimize the sum.
• Brute-force approaches that check all permutations are feasible due to the small constraint (n <= 14).
• Dynamic programming or bitmasking can be used to optimize the solution by caching results of subproblems.

Space and Time Complexity

Time Complexity: O(n * 2^n)
Space Complexity: O(2^n)

Solution

The problem can be solved using a dynamic programming approach combined with bitmasking. We can represent the states of the problem using a bitmask that indicates which elements of nums2 have been used. The solution involves iterating through all possible subsets of nums2 and calculating the XOR sum for each configuration, retaining the minimum found.

1. Use a bitmask to represent the set of indices in nums2 that have been used.
2. For each unused index, calculate the XOR sum and recursively call the function to explore further combinations.
3. Cache the results to avoid recalculating for the same configuration.

Code Solutions