This one took me a minute to even parse correctly.
First clarify interval semantics (e.g., half-open [start, end) to avoid double-counting at boundaries) and the query model (e.g., max over all time, or at a specific instant). Then propose a sweep-line algorithm using a balanced BST or segment tree over compressed time points, supporting O(log n) insertions and O(1) or O(log n) queries for the global maximum. Analyze time and space complexity, and discuss trade-offs for different query patterns.
Pro tip: Mention that if queries are only for the global maximum, you can maintain a running maximum with a segment tree; if queries are for arbitrary instants, you need a data structure that supports range maximum queries. Also, note that the 24-hour window can be handled by pruning old intervals or using a sliding window approach.
Define whether intervals are inclusive/exclusive at endpoints, whether queries are for the global maximum or at specific times, and whether the 24-hour window is fixed or sliding. Confirm if inserts and queries are interleaved.
Use a sweep-line approach: convert intervals to events (start +1, end -1) and sort by time. For dynamic inserts, use a balanced BST (e.g., TreeMap) or segment tree over compressed coordinates to maintain counts and the maximum.
If the window is sliding, remove intervals that end before (current_time - 24h). This can be done with a queue or by periodically pruning the data structure.
For n intervals, insertion is O(log n) with a segment tree, and query for global max is O(1) if maintained. Space is O(n). Discuss alternatives like difference arrays for static data or heaps for specific queries.
Consider intervals that touch at endpoints, zero-length intervals, overlapping intervals, and queries at exact event times. Verify the chosen semantics handle these correctly.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.