Minimum Number of Rooms to Schedule Overlapping Meetings

You are given a list of meetings, where `meetings[i] = [start, end]` means meeting `i` occupies the half-open time interval `[start, end)`. Every meeting must be held in one room for its whole interval, and two meetings can share a room only if their intervals do not overlap. Return the minimum number of rooms needed to hold all the meetings. ### Function Signature ```python def min_meeting_rooms(meetings: list[list[int]]) -> int: ``` ### Rules - Intervals are half-open: a meeting that ends at time `t` and a meeting that starts at time `t` do not overlap and may use the same room. - Meetings are given in no particular order. Identical intervals are separate meetings, and each needs its own room. - An empty list needs `0` rooms. ### Constraints - `0 <= len(meetings) <= 10^5` - Each `meetings[i]` contains exactly two integers with `0 <= start < end <= 10^9`. - The result is an integer from `0` to `len(meetings)` inclusive. ### Examples **Example 1** - Input: `meetings = [[1, 10], [2, 6], [7, 12]]` - Output: `2` - Explanation: `[1, 10)` overlaps both other meetings, so it needs a room of its own. `[2, 6)` ends before `[7, 12)` starts, so those two can share a second room. **Example 2** - Input: `meetings = [[4, 8], [8, 12], [1, 4]]` - Output: `1` - Explanation: In time order the meetings are `[1, 4)`, `[4, 8)` and `[8, 12)`, each starting exactly when the previous one ends, so one room is enough. **Example 3** - Input: `meetings = [[1, 5], [2, 6], [3, 7], [5, 9]]` - Output: `3` - Explanation: During `[3, 5)` the first three meetings are all in progress, so at least three rooms are needed. Three are enough, because `[5, 9)` can use the room that `[1, 5)` frees at time `5`.