05_Sum_of_All_Subset_XOR_Totals.cpp

// 1863. Sum of All Subset XOR Totals

// The XOR total of an array is defined as the bitwise XOR of all its elements, or 0 if the array is empty.

// For example, the XOR total of the array [2,5,6] is 2 XOR 5 XOR 6 = 1.
// Given an array nums, return the sum of all XOR totals for every subset of nums. 

// Note: Subsets with the same elements should be counted multiple times.

// An array a is a subset of an array b if a can be obtained from b by deleting some (possibly zero) elements of b.

 

// Example 1:

// Input: nums = [1,3]
// Output: 6
// Explanation: The 4 subsets of [1,3] are:
// - The empty subset has an XOR total of 0.
// - [1] has an XOR total of 1.
// - [3] has an XOR total of 3.
// - [1,3] has an XOR total of 1 XOR 3 = 2.
// 0 + 1 + 3 + 2 = 6
// Example 2:

// Input: nums = [5,1,6]
// Output: 28
// Explanation: The 8 subsets of [5,1,6] are:
// - The empty subset has an XOR total of 0.
// - [5] has an XOR total of 5.
// - [1] has an XOR total of 1.
// - [6] has an XOR total of 6.
// - [5,1] has an XOR total of 5 XOR 1 = 4.
// - [5,6] has an XOR total of 5 XOR 6 = 3.
// - [1,6] has an XOR total of 1 XOR 6 = 7.
// - [5,1,6] has an XOR total of 5 XOR 1 XOR 6 = 2.
// 0 + 5 + 1 + 6 + 4 + 3 + 7 + 2 = 28
// Example 3:

// Input: nums = [3,4,5,6,7,8]
// Output: 480
// Explanation: The sum of all XOR totals for every subset is 480.
 

// Constraints:

// 1 <= nums.length <= 12
// 1 <= nums[i] <= 20

class Solution
{
public:
    int subsetXORSum(vector<int> &nums)
    {
        int total = 0;
        for (int num : nums)
        {
            total |= num;
        }
        return total * (1 << (nums.size() - 1));
    }
};

/*
This solution uses a mathematical approach to calculate the sum of XOR totals for all subsets:

1. We use bitwise OR (|=) to combine all numbers in the array, stored in 'total'
2. Each number appears in exactly half of all possible subsets
3. For n elements, there are 2^(n-1) subsets containing each element
4. By multiplying total with 2^(n-1), we get the sum of all subset XOR totals

Time Complexity: O(n) where n is the length of nums array
Space Complexity: O(1) as we only use a constant amount of extra space
*/