Design a Calendar Scheduling System (Interval Tree)

To design a calendar system that prevents double-booking efficiently, we must prioritize sub-linear time complexity for both inserting events and checking availability. A naive approach using an unsorted list would require scanning all existing events, resulting in O(N) time per check, which becomes unacceptable as the user's schedule grows. Instead, I propose an Interval Tree, a specialized binary search tree augmented to store the maximum endpoint in each subtree. The core logic relies on the observation that if a new event [start, end] overlaps with any node in the tree, it must overlap with a node where the stored max endpoint is greater than or equal to our new start time. By maintaining this 'max' value at every node during rotations and insertions, we can prune entire branches of the tree that cannot possibly contain a conflict. For example, if the left child's max endpoint is less than our new start time, we know no overlap exists there and skip it entirely. This structure ensures that both insertion and conflict detection run in O(log N) average time, assuming a self-balancing variant like an AVL or Red-Black tree underpins the structure. This efficiency aligns perfectly with Apple's focus on fluid, responsive user experiences in apps like Calendar, ensuring that even with thousands of scheduled items, the app remains snappy. While a Segment Tree is an alternative, it requires coordinate compression if time is continuous, making the Interval Tree the more natural choice for discrete event scheduling.