Sum of All Subset XOR Totals

Problem Description

Given an array nums, return the sum of all XOR totals for every subset of nums. The XOR total of an array is defined as the bitwise XOR of all its elements, or 0 if the array is empty.

Key Insights

• Each element in the array contributes to multiple subsets.
• The XOR operation has properties that can be exploited to calculate the contribution of each bit position separately.
• For each bit position, determine how many subsets will have that bit set.
• The final result can be derived from the contributions of all bits.

Space and Time Complexity

Time Complexity: O(n * 2^n), where n is the length of the input array (due to generating all subsets). Space Complexity: O(1), as we only use a constant amount of space for variables.

Solution

To solve the problem, we can use a combinatorial approach by analyzing how many times each bit position contributes to the final XOR total across all subsets.

1. Iterate through each bit position (from 0 to 20, since nums[i] <= 20).
2. For each bit position, count how many numbers in the array have that bit set.
3. For each bit that is set in k numbers, it contributes to k * (2^(n-1)) subsets.
4. Sum the contributions for all bit positions to get the final result.

We will use a simple iteration for calculating contributions based on the above logic.

Code Solutions