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Sum of All Subset XOR Totals

Number: 1993

Difficulty: Easy

Paid? No

Companies: Google, Amazon, Meta

Problem Description

Given an array of integers, compute the sum of the XOR totals of every possible subset of the array. The XOR total of a subset is defined as the bitwise XOR of all the elements in that subset, with the empty subset having a total of 0.

Key Insights

  • There are 2^n subsets for an array of length n.
  • A brute-force approach would involve generating every subset and computing its XOR, which takes O(2^n * n) time.
  • A key observation for optimization: each bit position in the result contributes independently to the final sum. In fact, the overall answer can be computed as (bitwise OR of all numbers) * 2^(n-1), since each bit in the OR appears in exactly half of the subsets.
  • Both the backtracking and bit manipulation approaches efficiently solve the problem, though the bit manipulation approach is more direct.

Space and Time Complexity

Time Complexity: O(n) using the bitwise OR method, or O(2^n * n) using the backtracking approach. Space Complexity: O(1) for the bitwise optimization (ignoring the input size) or O(n) space for the recursion call stack in backtracking.

Solution

We can solve this problem using two main approaches:

  1. Backtracking / Enumeration Approach:

    • Enumerate all possible subsets.
    • For each subset, compute the XOR total by iterating over its elements.
    • Sum up these XOR totals.
    • This method is intuitive but may not be optimal for larger n.

  2. Bit Manipulation Optimization:

    • Instead of generating every subset, observe that each bit's contribution is independent.
    • The crucial observation is that the sum of XOR totals for all subsets equals the bitwise OR of all numbers multiplied by 2^(n-1).
    • This is because for each bit set in any of the numbers, it will contribute to half of the subsets.
    • This method provides a more efficient O(n) solution.

In our code examples below, both approaches are demonstrated with thorough line-by-line comments for clarity.

Code Solutions

# Python solution using the bit manipulation optimization approach
def subsetXORSum(nums):
    # Initialize the bitwise OR with 0.
    bitwise_or = 0
    # Compute the bitwise OR of all elements in the nums list.
    for num in nums:
        bitwise_or |= num  # OR operation accumulates bits that are set in any number
    
    # Calculate the number of subsets each bit appears in.
    # There are len(nums) numbers, so each bit appears in 2^(n-1) subsets.
    num_subsets = 1 << (len(nums) - 1)  # Equivalent to 2^(n-1)
    
    # The final sum is the product of the bitwise OR and the number of subsets contributed.
    return bitwise_or * num_subsets

# Example usage:
print(subsetXORSum([1, 3]))  # Expected output: 6
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