Pure Storage·Software Engineer·Technical Phone Screen·Intermediate

Intermediate

Apr 2026

Summary

Pure Storage asked a geometry problem about counting squares from a set of 2D points, and the whole conversation was basically a walkthrough of three different algorithmic approaches back to back. Pretty focused technical screen, no fluff.

Questions Asked (1)

Given n 2D coordinates, count how many distinct squares can be formed using four of those points as corners. Walk through a brute-force solution, then optimize it step by step.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

Started with the O(n^4) brute-force which they expected, no issues there.

Create a free account to read the full note

AI HintsAI Generated

Suggested Approach

Start by clearly defining the problem and walking through a brute-force O(n^4) solution that checks all quadruples. Then optimize by using a hash set of points and iterating over pairs to find potential squares, reducing complexity to O(n^2). Discuss trade-offs between time and space, and handle edge cases like duplicate points and collinear points.

Pro tip: Mention that using a hash set for point lookup requires careful handling of floating-point coordinates or using integer hashing to avoid precision issues. Also, note that the optimized approach can be extended to count rectangles or other shapes.

1. Clarify and Define

Restate the problem: given n distinct 2D points, count distinct squares formed by four points as corners. Clarify assumptions: points are distinct, coordinates are integers, squares can be of any orientation.

2. Brute-Force Solution

Explain checking all combinations of four points (O(n^4)) and verifying if they form a square by checking equal sides and right angles. Mention this is impractical for large n.

3. Optimized Approach

Use a hash set of points. Iterate over all pairs of points (O(n^2)), treat each pair as a diagonal of a square, compute the other two corners, and check if they exist in the set. Count each square once by ensuring the diagonal is unique (e.g., only consider pairs where p1 < p2 lexicographically).

4. Complexity and Trade-offs

Analyze time and space: O(n^2) time due to pair iteration, O(n) space for the hash set. Compare with brute-force O(n^4) time and O(1) extra space. Discuss when brute-force might be acceptable (small n).

5. Edge Cases and Testing

Address duplicate points, collinear points, squares of zero area (if points can coincide), and large coordinate values. Suggest testing with small cases and random points.

Key Points to Mention

Brute-force O(n^4) checks all quadruples, verifying equal sides and right angles.
Optimized O(n^2) uses hash set and pair iteration, treating each pair as a diagonal.
Compute other corners using vector rotation: for diagonal (x1,y1)-(x2,y2), midpoints and perpendicular vectors.
Avoid double-counting by ensuring each square is counted once (e.g., only consider pairs where p1 < p2).
Handle floating-point precision by using integer arithmetic or rational numbers.
Trade-offs: O(n^2) time vs O(n) space; brute-force may be simpler for very small n.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.