Maximum XOR of Subsequences
Time O(nlogr) · Space O(r) · Official statement on LeetCode
Solutions
// Time: O(nlogr), r = max(nums)
// Space: O(r)
// bitmasks, greedy
class Solution {
public:
int maxXorSubsequences(vector<int>& nums) {
const auto& bit_length = [](int x) {
return (x ? std::__lg(x) : -1) + 1;
};
const int l = bit_length(ranges::max(nums));
const auto& max_xor_subset = [&](const vector<int>& nums) { // Time: O(nlogr)
vector<int> base;
for (auto x : nums) {
for (const auto& b : base) { // gaussian elimination over GF(2)
if ((x ^ b) < x) {
x ^= b;
}
}
if (x) {
base.emplace_back(x);
}
}
int max_xor = 0;
for (const auto& b : base) { // greedy
if ((max_xor ^ b) > max_xor) {
max_xor ^= b;
}
}
return max_xor;
};
return max_xor_subset(nums);
}
};
// Time: O(nlogr), r = max(nums)
// Space: O(r)
// bitmasks, greedy
class Solution2 {
public:
int maxXorSubsequences(vector<int>& nums) {
const auto& bit_length = [](int x) {
return (x ? std::__lg(x) : -1) + 1;
};
const int l = bit_length(ranges::max(nums));
const auto& max_xor_subset = [&](const vector<int>& nums) { // Time: O(nlogr)
vector<int> base(l);
for (auto x : nums) { // gaussian elimination over GF(2)
for (int i = l - 1; i >= 0; --i) {
if (!(x & (1 << i))) {
continue;
}
if (base[i] == 0) {
base[i] = x;
break;
}
x ^= base[i];
}
}
int max_xor = 0;
for (int i = l - 1; i >= 0; --i) { // greedy
if ((max_xor ^ base[i]) > max_xor) {
max_xor ^= base[i];
}
}
return max_xor;
};
return max_xor_subset(nums);
}
};
Beginner Explanation
What is Maximum XOR of Subsequences?
Maximum XOR of Subsequences (LeetCode #3681) is a Hard problem that primarily trains greedy.
How to think about it
- Restate the goal in your own words before coding.
- Work a tiny example by hand so the invariant becomes obvious.
- Identify the pattern — this problem aligns with bit manipulation and greedy.
- Only then translate the idea into code.
Why this problem matters
Hard problems force you to combine patterns and prove complexity carefully — interview gold. Official solution notes mention: Bitmasks, Greedy.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Maximum XOR of Subsequences
Opening (30–60 seconds)
- Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
- State a brute force so the interviewer knows you can solve it naively.
- Propose the optimal direction tied to bit manipulation and greedy.
Core solution narrative
- Define the state you track (pointers, DP cell, set membership, stack top, etc.).
- Explain the transition when you process the next element.
- Call out time (O(nlogr)) and space (O(r)) before coding.
- Code cleanly; narrate variable names.
What interviewers listen for
- Correctness on edge cases
- Complexity honesty
- Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)
Follow-up questions they may ask
- Can you solve it with less memory?
- What if the input stream is infinite / doesn't fit in RAM?
- How would tests look for adversarial inputs?
Optimized Approach
Optimized solution notes
The reference solutions on AlgoForge target O(nlogr) time and O(r) space.
Pattern focus: bit manipulation and greedy
Use the pattern as a checklist:
- bit manipulation — confirm the invariant holds after each step
- greedy — confirm the invariant holds after each step
Multiple methods appear in the source solutions — compare them and explain when each is preferable.
Implementation tips
- Prefer readable names over micro-optimizations in interviews.
- Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
- After AC-level logic, re-scan for off-by-one and null checks.
Complexity Analysis
Complexity
| Measure | Bound |
|---|---|
| Time | O(nlogr) |
| Space | O(r) |
How to justify this in an interview
- Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
- Space: include hash maps, recursion stack, and output allocation when the problem asks for it.
If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.
Common Mistakes
Common mistakes on Maximum XOR of Subsequences
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for bit manipulation and greedy — updating state too early or too late.
- Mutating input unexpectedly when the problem forbids it.
- Off-by-one in windows, ranges, or binary search bounds.
- Ignoring overflow / precision for integer arithmetic problems.
- Overengineering — jumping to an advanced structure when a simpler approach works.
Alternative Approaches
Alternatives
The source file includes more than one method. Compare:
- Primary optimized path — best complexity for typical interviews.
- Secondary approach — often brute force, sorting-based, or space-optimized variant.
Practice articulating when you would pick each (constraints, readability, follow-ups).
Edge Cases
Edge cases checklist
- Minimum input size
- Maximum input size / time limits
- Duplicates and already-sorted input
- Negative numbers / zeros (if applicable)
- Disconnected structures (graphs/trees)
- Single path vs branching recursion depth
Pattern Recognition
Spotting this pattern
Signal phrases that point to bit manipulation and greedy:
- Sorted input or ability to sort without changing the answer class
- Need for contiguous subarray / substring → consider sliding window
- Need for O(1) membership → hash set/map
- Optimal substructure + overlapping subproblems → DP
- Connectivity / components → graph DFS/BFS or Union-Find
Primary topics: greedy.
Follow-up Interview Questions
Follow-ups
- How does the solution change if the input is a stream?
- Can you solve it in-place?
- What if duplicates must be handled differently?
- How would you parallelize the approach?
- Design tests that would break a buggy implementation.
Practice Recommendations
What to practice next
- Re-solve Maximum XOR of Subsequences in a second language (cpp, python).
- Drill 3–5 more problems tagged greedy.
- Teach the solution out loud in under 5 minutes.
- Add this problem to your revision calendar in 3 days and 14 days.
Visualization
Study checklist
- Read the official problem statement on LeetCode
- Solve on paper / whiteboard first
- Implement the bit manipulation and greedy approach
- Verify edge cases from the checklist
- State time and space complexity aloud
- Compare with the AlgoForge reference solution
- Schedule a revision session
Revision notes
Maximum XOR of Subsequences (#3681) — Hard. Pattern: bit manipulation and greedy. Complexity: O(nlogr) time / O(r) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Maximum XOR of Subsequences?+
The reference solutions aim for O(nlogr) time and O(r) space. Always re-derive complexity from the code you write in the interview.
What pattern does Maximum XOR of Subsequences use?+
It primarily maps to bit manipulation and greedy, within the broader topic of greedy.
Is Maximum XOR of Subsequences good for interviews?+
Yes — as a Hard problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.
Where can I read the official statement?+
Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/maximum-xor-of-subsequences/