Compute servers needed for daily recurring jobs

You operate a cluster of identical servers that run recurring **daily** jobs. For a single day, you are given a list of job execution intervals. Each interval has: - `job_id` (string or integer), - `start_time` (timestamp within that day), - `end_time` (timestamp within that day), with `start_time < end_time`. Interpret each interval as a half-open time range `[start_time, end_time)` during which that job must be running on some server **on that day**. Rules and constraints: - Each physical server can execute at most **one job at a time**. - However, if multiple intervals in the input correspond to the **same `job_id`**, they may overlap in time **on the same server** (they are considered parts or retries of the same job and do not consume extra capacity). - If a job's true execution would extend past midnight into the next day, it is conceptually **truncated at the end of the current day** for scheduling purposes; for each day you only see the portion that falls within that day’s 24-hour window. ### Tasks 1. Given all job intervals for a single day (potentially with repeated `job_id`s and overlapping intervals), compute the **minimum number of servers** required to schedule all the jobs under the above rules. 2. Describe the time and space complexity of your algorithm in terms of the number of intervals \(m\). ### Follow-up Some rare jobs are extremely long-running and conceptually span **multiple days** without restart. For such a multi-day job, you decide to reserve a **dedicated server** that is not shared with any other jobs, but which may still handle different intervals of that same `job_id`. Explain how you would extend or adapt your solution to: - Detect which jobs require a dedicated server based on their multi-day duration or configuration. - Account for these dedicated servers when computing the total number of servers needed for the system.