Count Pairs With XOR in a Range

Count Pairs With XOR in a Range

What is this problem about?

The "Count Pairs With XOR in a Range interview question" is a difficult bitwise problem. You are given an array of non-negative integers and a range [low, high]. You need to count the number of pairs $(i, j)$ with $i < j$ such that the bitwise XOR sum nums[i] ^ nums[j] falls within the given range. This problem requires efficient bitwise prefix searching.

Why is this asked in interviews?

Google asks the "Count Pairs With XOR in a Range coding problem" to test a candidate's mastery of the Trie data structure and its application to bitwise operations. It evaluates the ability to solve a "Range" problem by breaking it into two "Prefix" problems (counts for high and low-1) and using a Binary Trie to perform $O( ext{bits})$ prefix counts. It’s a classic "Hard" level bitwise challenge.

Algorithmic pattern used

This problem follows the Binary Trie and Prefix Search patterns.

1. The Range Trick: count(low, high) = count(high) - count(low-1), where count(K) is the number of pairs with XOR sum $\leq K$.
2. Trie Construction: Build a Trie where each node stores the count of numbers that pass through it. Each bit of a number (from most significant to least) defines a path.
3. Counting $\leq K$: For each number x in the array:
   • Traverse the Trie bit by bit.
   • At each bit $i$, if the $i^{th}$ bit of $K$ is 1:
     • If we XOR x with the branch that matches x's bit, the XOR result at this bit is 0. Since $0 < 1$, all numbers in this branch satisfy the condition. Add their pre-calculated count.
     • Move down the other branch (XOR result 1) to check further bits.
   • If the $i^{th}$ bit of $K$ is 0:
     • You must move down the branch that matches x's bit (XOR result 0).
4. Iterative Update: Add the number to the Trie after counting to ensure $i < j$.

Example explanation

Numbers: [1, 4, 2, 7], $K = 3$.

• Insert 1 into Trie.
• For 4: Find numbers y in Trie such that 4 ^ y <= 3.
  • 4 is 100, 1 is 001. 4 ^ 1 = 101 (5). $5 > 3$. Count = 0.
• Insert 4 into Trie.
• For 2: 2 ^ 1 = 3. $3 \leq 3$. OK. 2 ^ 4 = 6. $6 > 3$. NO. Count = 1. Total pairs: 1.

Common mistakes candidates make

• $O(N^2)$ Brute Force: Trying all pairs, which is too slow.
• Trie Depth: Not processing enough bits (usually 15-16 bits for constraints up to $2 imes 10^4$).
• Logic Errors in Trie Traversal: Mismanaging the "less than" condition when a bit in $K$ is 1.

Interview preparation tip

Practice using Tries for bitwise problems. If a problem asks about "XOR" and "Count" or "Range," a Binary Trie is almost always the required "Trie interview pattern" solution.