This problem asks for the sum of XOR totals across all subsets of a given array. The most direct approach is a backtracking search that explores including or excluding each element while accumulating the XOR total. Careful pruning and bit manipulation ensure that all subsets are counted correctly without redundant computation, making it feasible even for arrays up to length 12.
Problem Statement
Given an integer array nums, compute the sum of XOR totals for every possible subset of nums. Each subset's XOR total is calculated as the bitwise XOR of all its elements, and the empty subset counts as 0. Subsets with the same elements are counted separately.
Return the integer sum of all these XOR totals. For example, for nums = [1,3], the subsets are [], [1], [3], and [1,3], producing XOR totals 0, 1, 3, and 2, summing to 6.
Examples
Example 1
Input: nums = [1,3]
Output: 6
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
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
The sum of all XOR totals for every subset is 480.
Constraints
- 1 <= nums.length <= 12
- 1 <= nums[i] <= 20
Solution Approach
Backtracking over subsets
Use a recursive function that explores including or skipping each element of nums. At each step, update the current XOR total and call recursively until all elements are considered.
Prune unnecessary computations
Since XOR is associative and commutative, you can pass the running XOR total through recursion instead of recalculating from scratch for each subset. This reduces redundant operations and aligns with the backtracking pattern.
Sum XOR totals
Maintain a global sum that adds the current XOR total when reaching the end of recursion. Return this sum once all subsets have been processed, ensuring every subset contributes correctly.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(N) |
| Space | O(1) |
The algorithm visits each element once per subset, but the total number of subsets is 2^N. Due to the problem constraint N <= 12, this is feasible. Space is minimal because recursion depth is at most N and no extra data structures are needed.
What Interviewers Usually Probe
- Focus on generating all subsets without duplicates using backtracking.
- Check if intermediate XOR totals can be carried forward to avoid recomputation.
- Consider how bit manipulation can simplify combining subset elements.
Common Pitfalls or Variants
Common pitfalls
- Failing to include the empty subset or initializing XOR incorrectly.
- Recomputing XOR totals for each subset from scratch, causing inefficiency.
- Confusing subset counting leading to missed or double-counted totals.
Follow-up variants
- Compute the XOR totals of subsets for arrays containing negative numbers using the same backtracking approach.
- Find the maximum XOR total among all subsets instead of the sum.
- Count how many subsets produce an XOR total equal to a given target using recursive pruning.
How GhostInterview Helps
- Provides step-by-step guidance on constructing the backtracking recursion efficiently.
- Highlights pruning opportunities to prevent unnecessary XOR recomputation.
- Ensures all subsets are counted and summed correctly without manual debugging.
Topic Pages
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- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the best approach to sum all subset XOR totals?
A backtracking search with pruning is ideal. Carry the running XOR total and recursively include or skip each element.
Can this method handle arrays larger than length 12?
Direct backtracking becomes inefficient for large arrays due to 2^N subsets, so optimizations or different algorithms are needed.
Why does XOR allow pruning in this problem?
XOR is associative and commutative, so you can pass the accumulated total down recursion instead of recalculating from each subset.
Does the empty subset contribute to the sum?
Yes, the empty subset has an XOR total of 0 and should be included in the sum.
Is bit manipulation necessary for solving Sum of All Subset XOR Totals?
Bit manipulation simplifies combining elements efficiently and aligns with the backtracking pattern, but recursion alone can also work.
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Maximize the score after n operations by selecting pairs from the array and using their GCD with dynamic programming or backtracking techniques.
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