Count Pairs of Points With Distance k

Count Pairs of Points With Distance k

What is this problem about?

The "Count Pairs of Points With Distance k interview question" defines distance between two points $(x1, y1)$ and $(x2, y2)$ using the XOR operator: $(x1 \oplus x2) + (y1 \oplus y2) = k$. You are given a list of 2D points and need to find the number of pairs $(i, j)$ that satisfy this equation. This is a unique variation of coordinate geometry problems that relies on bitwise properties.

Why is this asked in interviews?

Google uses the "Count Pairs of Points With Distance k coding problem" to test a candidate's mastery of the XOR operator and the "Hash Table interview pattern." It requires recognizing that because $k$ is usually small, you can iterate through all possible ways to split $k$ into two components, $k1$ and $k2$, such that $k1 + k2 = k$. It evaluates "Bit Manipulation interview pattern" skills.

Algorithmic pattern used

This problem follows the Complement Search with XOR pattern.

1. Split k: The equation is $(x1 \oplus x2) = k1$ and $(y1 \oplus y2) = k2$, where $k1 + k2 = k$.
2. XOR Property: If $x1 \oplus x2 = k1$, then $x2 = x1 \oplus k1$.
3. Implementation:
   • Iterate through the points and store their frequencies in a hash map.
   • For each point $(x, y)$, iterate through all possible $k1$ from $0$ to $k$.
   • Calculate $targetX = x \oplus k1$ and $targetY = y \oplus (k - k1)$.
   • Check the map for the count of $(targetX, targetY)$ and add to the result.
4. Deduplication: Use a single pass to count and then add the current point to the map to handle pairs correctly.

Example explanation

Points: [[1, 2], [4, 2]], $k=5$.

• Point [1, 2]. $k1$ can be 0 to 5.
• Let $k1=5$. $targetX = 1 \oplus 5 = 4$. $targetY = 2 \oplus (5 - 5) = 2$.
• Point [4, 2] exists! Pair found. Total pairs: 1.

Common mistakes candidates make

• Misunderstanding XOR Distance: Trying to use Euclidean or Manhattan distance logic.
• Complexity: Attempting $O(N^2)$ to check every pair. The map-based approach is $O(N imes K)$.
• Iterating too much: Forgetting that $k1$ can only be between $0$ and $k$.

Interview preparation tip

Whenever a problem involves $A \oplus B = C$, remember that $A \oplus C = B$. This property is the key to transforming a search for a pair into a search for a specific target value in a hash map.