I started with the O(n^2) pairwise check which was fine, they nodded along.
Start by explaining the brute-force O(n^2) pairwise approach, then introduce the sweep-line algorithm to reduce complexity to O((n + k) log n). Emphasize handling edge cases like collinear, overlapping segments, and shared endpoints, and discuss trade-offs between simplicity and efficiency.
Pro tip: Demonstrate awareness of numerical precision issues and degenerate cases; mention that robust implementations often use exact arithmetic or epsilon comparisons. Also, relate the problem to real-world applications like VLSI design or map overlay to show practical insight.
Ask about input size, whether segments are closed/open, and if endpoints count as intersections. Clarify output format and any constraints on collinear or overlapping segments.
Describe checking all pairs of segments for intersection using orientation tests and bounding box checks. Mention O(n^2) time complexity and its simplicity.
Explain the sweep-line paradigm: sort endpoints by x-coordinate, maintain an active set of segments ordered by y, and check for intersections only between adjacent segments in the active set. Highlight O((n + k) log n) complexity where k is number of intersections.
Discuss how to detect and handle collinear overlapping segments (e.g., by merging intervals or reporting overlapping regions) and shared endpoints (e.g., by treating them as intersections or not based on problem definition).
Compare brute-force vs sweep-line in terms of implementation complexity, performance, and robustness. Mention potential optimizations like using balanced BSTs for the active set and handling numerical precision.
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