I went for a heap-based interval approach first because that's what my brain defaults to for overlap problems.
Clarify that each event contributes its capacity during the half-open interval [start, end), then use a sweep-line algorithm: create +capacity at start and -capacity at end, sort all events, and track the running sum to find the maximum. This yields O(n log n) time and O(n) space, which is optimal for comparison-based sorting.
Pro tip: Mention the half-open interval convention explicitly to avoid double-counting at boundaries, and note that if events are already sorted or times are bounded, you can achieve O(n) with counting sort or a difference array.
Confirm whether events are inclusive/exclusive at boundaries (e.g., [start, end) vs [start, end]) and how to handle zero-duration events. This prevents off-by-one errors in the sweep.
Explain that you will convert each event into two points: (+capacity at start) and (-capacity at end). Sort all points by time, with end events processed before start events at the same timestamp if using half-open intervals.
Iterate through sorted points, maintaining a running total of capacity. After each update, compare the running total to the current maximum and update if larger.
State that sorting dominates at O(n log n) time and O(n) space. Discuss alternatives like difference arrays for bounded time ranges (O(n + T)) or if events are already sorted (O(n)).
Walk through a small example to verify correctness, and mention edge cases like no events, single event, all events overlapping, or events with zero capacity.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.