This problem requires tracking the maximum number of overlapping events dynamically as new intervals are added. A direct simulation can fail due to high time complexity, so using a binary search over the valid answer space combined with segment tree or prefix sum techniques is ideal. Each new booking updates event start and end points, and querying the maximum overlap yields the correct k-booking efficiently.
Problem Statement
Design a MyCalendarThree class that can store event intervals and report the maximum number of overlapping events (k-booking) dynamically. Each event is represented as a half-open interval [startTime, endTime), where 0 <= startTime < endTime <= 10^9.
After each booking, return the highest k such that k events overlap at some moment. Implement the book method to accept a new interval, update the internal state, and return the current maximum k-booking for all added intervals.
Examples
Example 1
Input: See original problem statement.
Output: See original problem statement.
Input ["MyCalendarThree", "book", "book", "book", "book", "book", "book"] [[], [10, 20], [50, 60], [10, 40], [5, 15], [5, 10], [25, 55]] Output [null, 1, 1, 2, 3, 3, 3]
Explanation MyCalendarThree myCalendarThree = new MyCalendarThree(); myCalendarThree.book(10, 20); // return 1 myCalendarThree.book(50, 60); // return 1 myCalendarThree.book(10, 40); // return 2 myCalendarThree.book(5, 15); // return 3 myCalendarThree.book(5, 10); // return 3 myCalendarThree.book(25, 55); // return 3
Constraints
- 0 <= startTime < endTime <= 109
- At most 400 calls will be made to book.
Solution Approach
Binary Search Over the Answer Space
Treat each interval as start and end events, and use a sorted list to simulate active overlaps. Apply binary search to determine if a proposed k-booking is achievable, iteratively refining the maximum value.
Segment Tree Counting
Implement a segment tree to track event additions and range increments efficiently. Each booking updates relevant nodes, and querying the root returns the current maximum overlap, aligning with the k-booking requirement.
Prefix Sum Sweep Line
Convert all start and end times into events with +1 for start and -1 for end. Sort them and traverse cumulatively to maintain a running overlap count. The maximum encountered count represents the maximum k-booking at any time.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity depends on the approach: sweep line with sorting is O(N log N), segment tree can achieve O(N log M) where M is the number of distinct time points. Space complexity is O(N) for events storage or O(M) for segment tree nodes.
What Interviewers Usually Probe
- Asks about handling large time ranges efficiently.
- Mentions potential overlapping edge cases like fully nested intervals.
- Questions the trade-off between direct simulation and segment tree efficiency.
Common Pitfalls or Variants
Common pitfalls
- Neglecting the half-open interval semantics leading to off-by-one errors.
- Assuming a simple list scan is fast enough for 400 bookings, causing timeouts.
- Not updating both start and end counts, missing maximum k-booking correctly.
Follow-up variants
- My Calendar II: track double bookings only, simpler overlap counting.
- Counting overlaps with dynamic interval insertion and deletion, requiring interval trees.
- Generalized maximum overlap queries over multiple calendars for real-time scheduling.
How GhostInterview Helps
- Highlights the key pattern: binary search over the valid answer space for maximum k-booking.
- Provides step-by-step reasoning for using sweep line or segment tree to track overlaps efficiently.
- Guides on avoiding common pitfalls like interval off-by-one errors and inefficient naive scans.
Topic Pages
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- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the main challenge in My Calendar III?
The challenge is efficiently tracking the maximum number of overlapping intervals dynamically as new bookings are added.
How does the binary search pattern apply here?
Binary search over the valid answer space helps test if a certain k-booking is possible, refining the maximum efficiently.
Can sweep line solve this problem?
Yes, converting start and end times into +1/-1 events and processing in order provides the correct maximum overlap count.
Why not use a simple list scan for bookings?
A naive scan can be too slow for 400 bookings with large time ranges, leading to timeouts in practice.
Is segment tree always better than prefix sum here?
Segment tree is more efficient when time points are sparse and dynamic range queries are needed, while prefix sum suffices for denser or smaller ranges.
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Stay in the same pattern family
Implement a calendar that allows double bookings but prevents triple bookings, managing overlapping intervals efficiently with binary search.
Open problem page#729 My Calendar IImplement a calendar supporting non-overlapping event bookings using binary search for efficient insertion and conflict detection.
Open problem page#715 Range ModuleDesign a RangeModule to track and query half-open intervals using segment trees or ordered sets.
Open problem page