Sum of All Subset XOR Totals

class Solution {
public:
    int calculateSubsetXORSum(vector<int>& elements) {
        int combined = 0;
        for (int element : elements) {
            combined |= element;
        }
        return combined << (elements.size() - 1);
    }
};

The provided C++ code defines a function calculateSubsetXORSum that calculates the sum of all XOR combinations for all possible subsets of a given vector of integers. This function approaches the problem using bitwise manipulation.

Follow this brief explanation to understand how the code functions:

• A variable combined is initiated to zero and is used to store the combined result of a bitwise OR operation on all elements in the vector.
• The code iterates over each element in the vector elements, updating the combined variable by applying a bitwise OR with the current element. This operation accumulates all bits set in any element of the vector into combined.
• Finally, the function returns the value of combined, shifted to the left by the number (size of the vector minus one). This left shift operation corresponds to multiplying the combined result by 2 raised to the power of (elements.size() - 1), effectively calculating the sum of all subset XOR totals.

The operation of left shifting aggregates contributions of the combined value across all possible combinations of elements (subsets), thus providing the desired sum in an optimized manner.

class Solution {
    public int calculateSubsetsXOR(int[] elements) {
        int total = 0;
        for (int element : elements) {
            total |= element;
        }
        return total << (elements.length - 1);
    }
}

The provided Java code outlines a solution to calculate the sum of all subset XOR totals for a given array of integers. This solution employs a bit manipulation trick which utilizes the OR bitwise operation combined with a left shift to derive the desired result.

Here is the breakdown of the code:

• An integer total is initialized to zero, which will later accumulate the results of the bitwise OR operation.
• The code iterates over each element in the input array elements. Within the loop, the current element is combined with total using the bitwise OR operation (|=). This operation accumulates all the bits set in any element of the array.
• After processing all elements, total now contains the combined effect of all the ones in the bit positions from the entire array.
• The function finally returns total shifted to the left by (elements.length - 1) places. This left shift operation theoretically multiplies the total by 2 raised to the power of (elements.length - 1), which aligns with counting the contribution of total across all subsets, except the empty subset.

This concise approach ensures that the code is not only efficient but also capitalizes on the properties of bitwise operations to solve the problem effectively. Remember, this solution leverages the characteristic of XOR where a number XORed with itself results in zero and a number XORed with zero remains unchanged; thus, any subset where a number appears an even number of times (including zero times) in XOR operations contributes nothing to the total sum. The presented method smartly navigates these properties to achieve the desired results within computational limits.

class Solution:
    def calculateSubsetXOR(self, numbers: List[int]) -> int:
        total_xor = 0
        for number in numbers:
            total_xor |= number
        return total_xor << (len(numbers) - 1)

This Python code snippet is designed to compute the sum of all subset XOR totals for a given list of integers. The function calculateSubsetXOR within the class Solution performs the calculation based on a bit manipulation approach. The code leverages the OR bitwise operator to determine the combined effect of all elements when subjected to XOR operations across all possible subsets. The final result is then left-shifted by one less than the number of elements (i.e., len(numbers) - 1). This bitwise shift essentially multiplies the total obtained from the bitwise OR operation by two raised to the power of (len(numbers) - 1), yielding the sum of all XOR combinations of the subsets. This method efficiently computes the desired value without needing to iterate through each subset explicitly.