Google·Software Engineer·Technical Phone Screen·Intermediate

IntermediatePrefer not to say

Jun 2026

Summary

Google coding interview, one question about finding the line through the most points on a plane. Pretty classic computational geometry problem but the edge cases will get you if you're not careful.

Questions Asked (1)

Design an algorithm to find a line that passes through the maximum number of points on a 2D plane.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

I jumped straight to a brute force O(n^2) approach using slope comparisons and thought I was done.

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AI HintsAI Generated

Suggested Approach

Start by clarifying the problem constraints (e.g., input size, duplicate points, collinearity definition). Then present a solution that iterates over each point as a potential anchor, computes slopes to all other points, and uses a hash map to count the maximum number of points on a line through that anchor. Discuss time/space complexity and possible optimizations or trade-offs.

Pro tip: Mention that using a hash map with slope as a reduced fraction (dx/gcd, dy/gcd) avoids floating-point precision issues, and handle vertical lines and duplicate points explicitly. This shows attention to edge cases and numerical robustness.

1. Clarify requirements and constraints

Ask about input size, whether points are unique, if collinearity includes vertical lines, and expected time complexity. This ensures you design the right solution.

2. Outline the brute-force approach

For each pair of points, check all other points for collinearity. This is O(n^3) and sets a baseline for optimization.

3. Optimize using slope counting

For each point as anchor, compute slopes to all other points and count frequencies using a hash map. Track the maximum count. This reduces complexity to O(n^2).

4. Handle edge cases and precision

Use reduced fractions for slopes to avoid floating-point errors. Handle duplicate points and vertical lines separately.

5. Analyze complexity and trade-offs

Discuss O(n^2) time and O(n) space. Mention that for very large n, approximations or randomized algorithms might be considered, but exact solution is preferred for interviews.

Key Points to Mention

Time complexity: O(n^2) with hash map vs O(n^3) brute force.
Space complexity: O(n) for the hash map per anchor.
Using reduced fractions (dx/gcd, dy/gcd) to represent slopes exactly.
Handling duplicate points by counting them separately and adding to the max.
Handling vertical lines by using a special key (e.g., (1,0) or infinity).
Edge cases: fewer than 3 points, all points collinear, all points identical.

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