Google·Software Engineer·Technical Phone Screen·Senior

SeniorPrefer not to say

May 2026Remote

Summary

Google SWE interview focused entirely on a computational geometry problem about square overlaps on a 2D plane. No coding required, just a verbal walkthrough of multiple approaches and their trade-offs. Pretty niche problem but the discussion format actually made it more interesting than I expected.

Questions Asked (1)

Given N axis-aligned squares on a 2D plane, each defined by a bottom-left corner and side length, find the maximum area covered by the largest simultaneous overlap of any subset of these squares. Discuss multiple approaches verbally, including their time and space complexity trade-offs.

Algorithms & Data StructuresTechnical Trade-offsSystem Design

Author's notes

The brute force angle was easy enough to talk through, O(N^2) pairwise intersections, nothing fancy.

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AI HintsAI Generated

Suggested Approach

Start by clarifying the problem: the maximum area covered by the largest simultaneous overlap of any subset of squares is equivalent to finding the maximum depth of overlap in the arrangement of squares. Then discuss multiple approaches: brute-force checking all subsets, sweep-line with segment trees, and plane sweep with interval trees, comparing their time and space complexities. Conclude by recommending the most efficient approach for large N, such as O(N^2 log N) or O(N log N) with advanced data structures.

Pro tip: Mention that the problem reduces to finding the maximum overlap depth in a set of rectangles, and that the optimal solution often involves a sweep-line algorithm with a segment tree that supports range add and global max queries. Also, note that the area is simply the maximum depth times the square of the side length if all squares are congruent, but if side lengths vary, the overlap region may not be a square, so careful handling is needed.

1. Clarify the problem and constraints

Ask about the range of N, whether squares can have different side lengths, and if the overlap area is defined as the area of the intersection of the subset. Confirm that we need the maximum area over all possible subsets.

2. Discuss brute-force approach

Explain that checking all subsets is exponential (2^N) and infeasible. For each subset, compute the intersection rectangle and its area, but this is too slow. Mention that for small N (e.g., N ≤ 20) it might be acceptable.

3. Propose sweep-line with segment tree

Describe a sweep-line over x-coordinates: events at left and right edges of squares. Use a segment tree over y-coordinates to maintain the number of overlapping squares at each y-interval. The maximum overlap depth times the square of the side length gives the maximum area if all squares are congruent. For varying side lengths, the area is not simply depth * side^2, so we need to track the actual intersection area, which complicates the segment tree.

4. Analyze complexity and trade-offs

For congruent squares, the sweep-line with segment tree takes O(N log N) time and O(N) space. For varying side lengths, a more complex approach is needed, such as computing the arrangement of all squares and finding the maximum depth region, which can be O(N^2 log N) or O(N^2) with plane sweep. Discuss the trade-offs between simplicity and efficiency.

5. Recommend an approach and conclude

If all squares are congruent, recommend the O(N log N) sweep-line with segment tree. If side lengths vary, suggest a plane sweep that computes the maximum overlap area by considering all critical x and y coordinates, or using a segment tree that stores the maximum area covered by at least k squares. Conclude with the chosen approach and its complexity.

Key Points to Mention

The problem is equivalent to finding the maximum depth of overlap in an arrangement of squares.
Sweep-line algorithm with a segment tree can efficiently compute the maximum overlap depth for congruent squares.
For varying side lengths, the overlap area is not simply depth times side^2; need to compute the actual intersection area.
Time complexity trade-offs: brute-force O(2^N), sweep-line O(N log N) for congruent, O(N^2 log N) for varying.
Space complexity: segment tree uses O(N) space, while storing all events also O(N).
Edge cases: squares with zero side length, no overlap, or all squares overlapping at a single point.

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