Count Pairs With XOR in a Range
Time O(n) · Space O(n) · Official statement on LeetCode
Solutions
// Time: O(n)
// Space: O(n)
// dp solution
class Solution {
public:
int countPairs(vector<int>& nums, int low, int high) {
return count(nums, high + 1) - count(nums, low);
}
private:
int count(const vector<int>& nums, int x) {
unordered_map<int, int> dp;
for (const auto& x : nums) {
++dp[x];
}
int result = 0;
for (; x; x >>= 1) {
unordered_map<int, int> new_dp;
for (auto const& [k, v] : dp) {
new_dp[k >> 1] += v;
if ((x & 1) == 0) {
continue;
}
if (dp.count((x ^ 1) ^ k)) {
result += v * dp[(x ^ 1) ^ k]; // current limit is xxxxx1*****, count xor pair with xxxxx0***** pattern
}
}
dp = move(new_dp);
}
return result / 2;
}
};
// Time: O(n)
// Space: O(n)
// trie solution
class Solution2 {
public:
int countPairs(vector<int>& nums, int low, int high) {
int result = 0;
Trie trie;
for (const auto& x : nums) {
result += trie.query(x, high + 1) - trie.query(x, low);
trie.insert(x);
}
return result;
}
private:
class Trie {
public:
Trie() : nodes(1) {}
void insert(int num) {
int idx = 0;
for (int i = 31; i >= 0; --i) {
int curr = (num >> i) & 1;
if (!nodes[idx][curr]) {
nodes.emplace_back();
nodes[idx][curr] = size(nodes) - 1;
}
idx = nodes[idx][curr];
++nodes[idx][2];
}
}
int query(int num, int limit) {
int result = 0, idx = 0;
for (int i = 31; i >= 0; --i) {
int curr = (num >> i) & 1;
int bit = (limit >> i) & 1;
if (bit) {
if (nodes[idx][curr]) {
result += nodes[nodes[idx][0 ^ curr]][2]; // current limit is xxxxx1*****, count xor pair with xxxxx0***** pattern
}
}
if (!nodes[idx][bit ^ curr]) {
break;
}
idx = nodes[idx][bit ^ curr];
}
return result;
}
private:
vector<array<int, 3>> nodes;
};
};
Beginner Explanation
What is Count Pairs With XOR in a Range?
Count Pairs With XOR in a Range (LeetCode #1803) is a Hard problem that primarily trains dynamic programming.
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 dynamic programming and trie.
- 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: DP, Trie.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Count Pairs With XOR in a Range
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 dynamic programming and trie.
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(n)) and space (O(n)) 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(n) time and O(n) space.
Pattern focus: dynamic programming and trie
Use the pattern as a checklist:
- dynamic programming — confirm the invariant holds after each step
- trie — 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(n) |
| Space | O(n) |
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 Count Pairs With XOR in a Range
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for dynamic programming and trie — 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 dynamic programming and trie:
- 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: dynamic programming.
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 Count Pairs With XOR in a Range in a second language (cpp, python).
- Drill 3–5 more problems tagged dynamic programming.
- 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 dynamic programming and trie 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
Count Pairs With XOR in a Range (#1803) — Hard. Pattern: dynamic programming and trie. Complexity: O(n) time / O(n) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Count Pairs With XOR in a Range?+
The reference solutions aim for O(n) time and O(n) space. Always re-derive complexity from the code you write in the interview.
What pattern does Count Pairs With XOR in a Range use?+
It primarily maps to dynamic programming and trie, within the broader topic of dynamic programming.
Is Count Pairs With XOR in a Range 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/count-pairs-with-xor-in-a-range/