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.