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I went with the sweep-line difference map approach because I've used ordered maps before and felt more confident explaining the correctness argument.
Use a sweep-line algorithm with a balanced BST or sorted dictionary to track interval endpoints and a running count of active bookings. For each book(start, end), increment the count at start and decrement at end, then update the global maximum. This yields O(log n) per operation and O(n) space.
Pro tip: Mention that in Python, you can use a sorted list with bisect for O(log n) insertion and deletion, but for true O(log n) you'd need a balanced BST like a treap or a segment tree with coordinate compression. Also, clarify that the maximum is maintained incrementally, not recomputed each time.
Confirm that intervals are half-open [start, end), that book returns the maximum concurrent bookings after each call, and that up to 100,000 operations are expected. Discuss the need for O(log n) amortized time and O(n) space.
Select a balanced BST (e.g., treap, AVL) or a sorted dictionary to store the net change at each endpoint. Alternatively, use a segment tree with coordinate compression if all intervals are known in advance.
For each book(start, end), increment the count at start and decrement at end. Maintain a running sum of active bookings and update the global maximum whenever the running sum exceeds it. This ensures O(log n) per operation.
Explain that each insertion/deletion in the BST takes O(log n), and the running sum update is O(1). Space is O(n) for storing up to 2n endpoints. Compare with alternative approaches like segment trees or interval trees.
Write clean code for the class, handling edge cases like overlapping intervals, zero-length intervals, and large inputs. Test with examples to verify correctness and performance.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
The half-open interval convention is the right call here and I said so confidently, but then I second-guessed myself mid-explanation when the interviewer asked about book(10,20) followed by book(20,30).
Start by explicitly stating your boundary convention (e.g., half-open intervals [start, end)) and justify why it simplifies duplicate and nested interval handling. Then walk through the given examples to show how your convention resolves them, and finally describe unit tests that verify boundary behavior, including off-by-one cases.
Pro tip: Mention that half-open intervals are standard in many scheduling systems (e.g., Google Calendar) because they avoid ambiguity at endpoints and make merging intervals trivial. Also, emphasize that writing tests first (TDD) can catch off-by-one errors early.
Clearly define whether intervals are inclusive/exclusive at start and end. For example, use half-open [start, end) to avoid overlap ambiguity.
Describe how your convention treats duplicates (e.g., [10,20) and [10,20) as identical) and nested intervals (e.g., [10,30) contains [15,25)).
Apply the convention to book(10,20), book(20,30), and book(15,25) to show that the first two are adjacent (no overlap) and the third overlaps with both.
List specific test cases that check boundaries: booking exactly at endpoints, adjacent intervals, overlapping intervals, and nested intervals.
Briefly mention other conventions (e.g., closed intervals) and why half-open is preferable for booking systems, noting potential edge cases.
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Clarify the existing data structure for bookings and maximum-k, then design cancel and query to maintain efficiency. Use a balanced BST or segment tree with lazy propagation to support interval removal and point queries in O(log n), while updating the maximum-k structure accordingly.
Pro tip: Emphasize that cancel must also update the maximum-k structure, not just remove the interval; consider using a segment tree that tracks both coverage counts and the maximum k over time.
Ask about the existing booking system, how maximum-k is computed, and whether intervals are inclusive/exclusive. Confirm that cancel removes a specific previously booked interval and that query(t) counts active bookings at time t.
Select a data structure that supports interval add/remove and point queries in O(log n), such as a segment tree with lazy propagation or a balanced BST (e.g., interval tree). For maximum-k, maintain a segment tree that tracks the maximum coverage count over time.
Implement cancel by decrementing coverage counts over the interval [start, end) in the segment tree. Update the maximum-k value by recomputing the maximum from the segment tree's root, ensuring it reflects the removal.
Implement query(t) by traversing the segment tree to the leaf corresponding to time t, summing lazy updates along the path to get the active booking count in O(log n).
Verify that both operations run in O(log n) time and O(n) space. Discuss edge cases like cancelling non-existent intervals, overlapping intervals, and concurrent modifications.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Talked about how the difference map only stores events at actual endpoints so single-point intervals just add two entries each, keeping it proportional to the number of bookings rather than the time range.
Start by clearly stating the worst-case time and space complexity of your solution, then explain how your data structure (e.g., interval tree, segment tree, or sweep line with balanced BST) guarantees O(log n) per operation even with many overlapping single-point intervals. Finally, discuss practical modifications to handle recursion depth or memory fragmentation, such as iterative implementations or custom memory allocators.
Pro tip: Quantify the impact: e.g., 'With 1M overlapping intervals, our approach does ~20 comparisons per query vs. 1M in naive, and we can switch to an iterative version if recursion depth exceeds 1000.' This shows you think in trade-offs and scale.
Clearly specify the worst-case time and space complexity for each operation (insert, delete, query) and overall. Use Big-O notation and mention any assumptions (e.g., balanced tree).
Describe the data structure (e.g., interval tree, segment tree, augmented balanced BST) and why it avoids O(n) per operation. Highlight how it handles overlapping single-point intervals, such as by storing intervals in nodes or using lazy propagation.
Explain how the structure maintains efficiency under adversarial input like many overlapping single-point intervals. For example, in an interval tree, point queries traverse O(log n) nodes; in a segment tree, range updates/queries are O(log n) even with overlaps.
If recursion depth is a concern, propose iterative implementations (e.g., iterative segment tree) or tail recursion. For memory fragmentation, suggest pooling, custom allocators, or compact representations (e.g., arrays instead of pointers).
Conclude by summarizing the trade-offs between time, space, and implementation complexity. Mention when you might choose a simpler structure if constraints are relaxed.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.