My first instinct was to just sort by start time and do a linear scan, which works for merging intervals but I kept second-guessing whether it was enough for the transitive part.
Model the problem as finding connected components in an interval graph, where edges exist between overlapping intervals. Sort intervals by start time and use a sweep line with a min-heap of end times to efficiently identify overlapping groups, or use union-find to merge overlapping intervals. Track the size of each component to determine the largest group.
Pro tip: Clarify whether intervals are closed or half-open and whether touching endpoints count as overlap; this edge case often trips candidates. Also, mention that the optimal solution runs in O(N log N) time, which is efficient for large N.
Confirm definitions: whether intervals are inclusive, if touching endpoints count as overlap, and if input is sorted. Ask about constraints on N to guide algorithm choice.
Decide between union-find with interval sorting or sweep line with a heap. Both are O(N log N); union-find is simpler to implement, while sweep line can be more intuitive for interval problems.
For union-find: sort intervals by start, iterate and union with all overlapping intervals (using a heap or active set). For sweep line: sort events, maintain active intervals, and track component sizes.
Maintain a size array for union-find or a counter for the current component in sweep line. Update the maximum size whenever components merge or when a component closes.
State time and space complexity (O(N log N) time, O(N) space). Walk through edge cases: no overlaps, all overlap, nested intervals, and touching endpoints.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.