MathWorks·Software Engineer·Technical Phone Screen·Intermediate

Intermediate

Apr 2026

Summary

MathWorks software engineer interview with a meaty algorithmic problem about segment intersections on a number line. One question, but it had enough layers to keep you busy for a while.

Questions Asked (1)

Given n segments on a number line, each defined by a start and end point (inclusive), find for every segment how many other segments it intersects with. Two segments intersect if they share at least one point. Implement a solution faster than O(n^2), explain the time and space complexity, and clarify how you handle edge cases like equal endpoints or duplicate segments.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

The brute force answer basically writes itself, so of course they want you to do better.

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AI HintsAI Generated

Suggested Approach

Use a sweep-line algorithm with a Fenwick tree (BIT) to count intersections in O(n log n) time. Sort events by coordinate, process segment starts and ends, and for each segment, count active segments that intersect it. Handle edge cases by carefully ordering events at the same coordinate and deduplicating segments if necessary.

Pro tip: Clarify upfront whether duplicate segments should be counted as intersecting each other (they should, since they share all points). Also, mention that the sweep-line approach can be adapted to report the actual intersecting pairs if needed, but here we only need counts.

1. Understand the problem and constraints

Confirm that segments are inclusive, intersections include touching at endpoints, and duplicate segments are considered intersecting. Discuss input size to justify O(n log n) over O(n^2).

2. Design sweep-line algorithm

Create events for each segment's start and end. Sort events by coordinate; for ties, process starts before ends to handle inclusive endpoints correctly. Use a Fenwick tree to maintain active segments' start points.

3. Count intersections per segment

When processing a segment's start, query the Fenwick tree for active segments with start ≤ current start (since all active segments have end ≥ current start). When processing its end, remove its start from the tree.

4. Handle edge cases and duplicates

For equal endpoints, ensure events are ordered so that a segment ending at x is removed after processing starts at x. For duplicate segments, treat them as separate; they will naturally count each other as intersecting.

5. Analyze complexity and test

State time O(n log n) due to sorting and Fenwick operations, space O(n). Walk through a small example with overlapping, touching, and duplicate segments to verify correctness.

Key Points to Mention

Sweep-line algorithm with event sorting
Fenwick tree (Binary Indexed Tree) for efficient prefix sums
Event ordering for inclusive endpoints (starts before ends at same coordinate)
Handling duplicate segments: they intersect each other and should be counted
Time complexity O(n log n), space O(n)
Alternative approaches like interval trees or segment trees, but sweep-line is optimal

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