Most Used Meeting Room When Meetings Wait for a Free Room

A building has `n` meeting rooms numbered `0` to `n - 1`. You are given a list of meetings, each with a start time and an end time, and no two meetings have the same start time. Meetings are assigned to rooms by the rules below; when no room is free, a meeting is delayed rather than dropped. Return the number of the room that hosts the most meetings. The report names this as a variant of a known room-scheduling problem without saying what the variant changed; this statement uses the standard rules. ### Function Signature ```python def most_used_room(n: int, meetings: list[list[int]]) -> int: ``` ### Rules - `meetings[i] = [start, end]` needs one room for the half-open interval `[start, end)`. A room whose meeting ends at time `t` can host a meeting that starts at time `t`. - Meetings are assigned one at a time, in increasing order of their original start times. - If one or more rooms are free at a meeting's start time, the meeting takes the free room with the lowest number. - If no room is free, the meeting is delayed until the earliest time at which a room becomes free, and takes the lowest-numbered room that becomes free at that time. A delayed meeting keeps its original duration `end - start`. - Every meeting is eventually held; none is dropped. - Return the room that hosts the most meetings. If several rooms tie, return the lowest room number. ### Constraints - `1 <= n <= 100` - `1 <= len(meetings) <= 10^5` - `0 <= start < end <= 5 * 10^5` for every meeting - All `start` values are distinct; `meetings` may be given in any order. - Delays accumulate, so a delayed meeting can end as late as about `5 * 10^10`, which exceeds `2^31 - 1`. Use 64-bit integers for times; every value stays well below `2^53`. ### Examples **Example 1** ```text Input: n = 2, meetings = [[0, 10], [1, 3], [2, 4], [5, 6]] Output: 1 ``` Room 0 takes `[0, 10)` and room 1 takes `[1, 3)`. At time 2 both rooms are busy, so `[2, 4)` waits for room 1, which frees at 3, and runs as `[3, 5)`. At time 5 room 1 is free again and takes `[5, 6)`. Room 1 hosts three meetings and room 0 hosts one. **Example 2** ```text Input: n = 3, meetings = [[1, 4], [2, 3], [3, 7], [4, 5]] Output: 0 ``` `[1, 4)` takes room 0 and `[2, 3)` takes room 1. At time 3 rooms 1 and 2 are free, so `[3, 7)` takes room 1. At time 4 rooms 0 and 2 are free, so `[4, 5)` takes room 0. Rooms 0 and 1 each host two meetings, and the tie goes to room 0. **Example 3** ```text Input: n = 2, meetings = [[0, 5], [3, 6], [2, 4], [1, 5], [8, 9]] Output: 0 ``` In start order, `[0, 5)` takes room 0 and `[1, 5)` takes room 1. `[2, 4)` must wait; both rooms become free at 5, so it takes room 0 and runs as `[5, 7)`. `[3, 6)` waits for room 1, free at 5, and runs as `[5, 8)`. At time 8 both rooms are free, so `[8, 9)` takes room 0. Room 0 hosts three meetings and room 1 hosts two.