Maximum Team Size with an Overlapping Hub

There are `n` employees. Employee `i` works during the closed interval `[startTime[i], endTime[i]]`. Two employees can interact if their working intervals overlap. Intervals that share an endpoint count as overlapping. A team is valid if at least one member can interact with every other member of that team. Other pairs of team members do not need to interact, and the entire team does not need to share one common time. Implement `getMaximumTeamSize(startTime, endTime)` and return the largest possible valid team size. ### Input and Output - `startTime` and `endTime` are integer arrays of the same length `n`. - Return an integer representing the maximum number of employees in a valid team. - Employees are distinct even when their intervals are identical. ### Constraints & Assumptions - `1 <= n <= 2 * 10^5` - `1 <= startTime[i] <= endTime[i] <= 10^9` - Both endpoints are included. A zero-length interval is valid. - The employee who overlaps everyone else is part of the team and counts toward its size. ### Examples **Example 1** ```text startTime = [1, 6, 4, 3, 1] endTime = [2, 7, 5, 8, 2] Output: 3 ``` Employee `3`, with interval `[3, 8]`, overlaps employees `1` and `2`. Thus employees `[1, 2, 3]` form a valid team. Employees `1` and `2` do not overlap each other, but that is allowed. No employee can serve as the hub of a larger team. **Example 2** ```text startTime = [2, 5, 6, 8] endTime = [5, 6, 10, 9] Output: 3 ``` Employee `1`, with interval `[5, 6]`, overlaps employee `0` at time `5` and employee `2` at time `6`. The team `[0, 1, 2]` is valid, even though those three intervals have no common intersection. **Example 3** ```text startTime = [1, 2, 8, 8, 10] endTime = [4, 8, 12, 9, 11] Output: 4 ``` Employee `1`, with interval `[2, 8]`, overlaps employees `0`, `2`, and `3`. These four employees form a valid team. No single employee overlaps all five intervals. ### Clarifying Questions to Ask - Does a team require one member who overlaps everyone, or must every pair overlap? Here, only the first condition is required. - Are endpoint-only intersections valid? Yes, the intervals are closed. - Should equal intervals be deduplicated? No, they belong to distinct employees. ```hint Fix one employee as the hub. Which other intervals are definitely unable to join that employee's team? ``` ### What a Strong Answer Covers The solution should count all intervals intersecting each possible hub, including the hub itself, and maximize that count. It should handle inclusive endpoints and repeated intervals, avoid confusing this problem with maximum simultaneous attendance, and use an approach suitable for two hundred thousand employees. ### Follow-up Questions - How would you return the members of one optimal team after finding its hub? - Which comparison operators would change if touching endpoints did not count as overlap? - How would the task change if every pair of team members had to overlap?