Find Minimum Time to Finish All Jobs II

Find Minimum Time to Finish All Jobs II

What is this problem about?

The Find Minimum Time to Finish All Jobs II interview question is a variation of the job scheduling problem with a simpler constraint. You are given an array of jobs and an array of workers, where each worker can only handle one job. The "time" for a worker to finish a job is the time required for the job divided by the worker's efficiency (or simply the job time if efficiencies are equal). The goal is to find the minimum number of days/time units such that all jobs are completed simultaneously.

Why is this asked in interviews?

Amazon uses the Find Minimum Time to Finish All Jobs II coding problem to test a candidate's understanding of Greedy interview patterns. Unlike the first version, which is complex, this version rewards the observation that to minimize the maximum time, you should pair the most difficult jobs with the most efficient workers. It tests your ability to identify when a simple sort-and-match strategy is optimal.

Algorithmic pattern used

This problem follows the Greedy and Sorting patterns.

1. Sort both arrays: Sort the jobs in ascending order and the worker efficiencies in ascending order.
2. Pairing: Pair the $i^{th}$ smallest job with the $i^{th}$ most efficient worker.
3. Calculation: For each pair, calculate the time: ceil(jobs[i] / workers[i]).
4. Max of Mins: The answer is the maximum time found across all pairs.

Example explanation

Jobs: [10, 20], Workers: [5, 2].

1. Sort Jobs: [10, 20]
2. Sort Workers: [2, 5]
3. Pair 1: Job 10 with Worker 2 -> Time = 5.
4. Pair 2: Job 20 with Worker 5 -> Time = 4. Result: Max(5, 4) = 5. (If you paired 10 with 5 and 20 with 2, the times would be 2 and 10, resulting in 10).

Common mistakes candidates make

• Incorrect Sorting: Sorting one array ascending and the other descending. This actually maximizes the time rather than minimizing it.
• Integer Division: Forgetting to use "ceiling" division if the time calculation requires it.
• Over-engineering: Trying to use binary search or DP when a $O(N \log N)$ sort is sufficient.

Interview preparation tip

Whenever you need to "minimize the maximum" of a set of pairs, think about sorting both sets and matching them. This is a recurring theme in Greedy interview patterns.