Start by computing the maximum possible bitwise OR across all subsets, then count each subset that matches this value. Use a backtracking approach to explore inclusion or exclusion of each element while pruning branches that cannot reach the maximum. This method ensures all valid subsets are counted efficiently without redundant calculations.
Problem Statement
Given an integer array nums, identify the highest achievable bitwise OR obtainable from any non-empty subset. Return the count of distinct subsets that achieve this maximum bitwise OR. Each subset is considered unique based on the indices of elements selected.
A subset of an array is formed by removing zero or more elements from the original array without changing the order of remaining elements. The bitwise OR of a subset is calculated as the OR of all its elements. Your task is to efficiently enumerate subsets and count those reaching the maximal bitwise OR.
Examples
Example 1
Input: nums = [3,1]
Output: 2
The maximum possible bitwise OR of a subset is 3. There are 2 subsets with a bitwise OR of 3:
- [3]
- [3,1]
Example 2
Input: nums = [2,2,2]
Output: 7
All non-empty subsets of [2,2,2] have a bitwise OR of 2. There are 23 - 1 = 7 total subsets.
Example 3
Input: nums = [3,2,1,5]
Output: 6
The maximum possible bitwise OR of a subset is 7. There are 6 subsets with a bitwise OR of 7:
- [3,5]
- [3,1,5]
- [3,2,5]
- [3,2,1,5]
- [2,5]
- [2,1,5]
Constraints
- 1 <= nums.length <= 16
- 1 <= nums[i] <= 105
Solution Approach
Compute Maximum Bitwise OR First
Iterate through the array to compute the OR of all elements. This gives the target value any subset must reach. Knowing the maximum early allows pruning in the backtracking search.
Backtracking Search with Inclusion/Exclusion
Use a recursive function to explore each element by including or excluding it from the current subset. Pass the current OR along and count when it matches the maximum. This pattern directly reflects the problem's backtracking nature.
Prune Ineffective Branches
If the current OR combined with remaining elements cannot reach the maximum, skip further recursion along that path. Pruning reduces unnecessary subset generation and ensures efficient computation for arrays up to length 16.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(n \cdot \text{max}) |
| Space | O(2^{17}) |
Time complexity is O(n * max) where n is the array length and max is the number of distinct OR results, and space complexity is O(2^n) to store intermediate OR states for subsets.
What Interviewers Usually Probe
- Are you enumerating all subsets or skipping some early?
- Have you considered pruning branches that cannot reach the maximum OR?
- How does the backtracking order affect counting subsets with the same OR?
Common Pitfalls or Variants
Common pitfalls
- Counting duplicate subsets incorrectly when elements are repeated.
- Failing to prune branches leading to exponential time in the worst case.
- Calculating OR of subsets after generation instead of incrementally during recursion.
Follow-up variants
- Find the subset achieving maximum bitwise AND instead of OR.
- Return the actual subsets rather than just the count.
- Apply the same approach for arrays up to length 20 with memoization to manage state.
How GhostInterview Helps
- GhostInterview highlights which branches of the backtracking search can be pruned to reach the maximum OR efficiently.
- It provides step-by-step subset evaluation, showing exactly how to count OR matches without redundant computation.
- The tool flags common mistakes like miscounting subsets with repeated elements or skipping pruning opportunities.
Topic Pages
Related GhostInterview Pages
- 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 key strategy for solving Count Number of Maximum Bitwise-OR Subsets?
Use a backtracking search with pruning, computing the maximum OR first and counting all subsets that reach it.
How can pruning improve performance in this problem?
Pruning eliminates recursive paths where the current OR cannot possibly reach the maximum, reducing unnecessary calculations.
Is it necessary to generate all subsets explicitly?
No, you can generate subsets recursively while updating the OR incrementally and pruning early, avoiding full enumeration.
How do repeated elements affect subset counting?
Subsets are considered different if indices differ, so duplicates must be counted separately in the recursion.
What is the maximum array length efficiently handled with this approach?
Arrays up to length 16 can be handled efficiently using backtracking with pruning without exceeding memory limits.
Need direct help with Count Number of Maximum Bitwise-OR Subsets instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture Count Number of Maximum Bitwise-OR Subsets from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Determine the maximum number of good people in a group given mutual statements, using precise backtracking with pruning.
Open problem page#2212 Maximum Points in an Archery CompetitionIn the "Maximum Points in an Archery Competition" problem, Bob aims to maximize his score while respecting Alice's arrow counts.
Open problem page#1863 Sum of All Subset XOR TotalsCompute the sum of all subset XOR totals using a backtracking search with pruning for arrays of small size.
Open problem page