Find Minimum Time to Finish All Jobs

Find Minimum Time to Finish All Jobs

What is this problem about?

The Find Minimum Time to Finish All Jobs interview question is a complex resource allocation problem. You are given an array of jobs, where each value represents the time required to complete that job, and an integer $k$ representing the number of workers. You must assign every job to exactly one worker such that the "makespan"—the maximum time any single worker spends working—is minimized. This Find Minimum Time to Finish All Jobs coding problem is a classic variation of the multi-processor scheduling problem.

Why is this asked in interviews?

Companies like Microsoft and Meta ask this to evaluate a candidate's mastery of Backtracking and Dynamic Programming with optimizations. Since this is an NP-hard problem, the interview focuses on how you prune the search space. It tests your ability to implement "Backtracking interview patterns" and optimize them using techniques like symmetry breaking and state-space reduction.

Algorithmic pattern used

This problem is typically solved using Backtracking with Pruning or Dynamic Programming with Bitmask.

1. Backtracking: For each job, try assigning it to each of the $k$ workers.
2. Pruning:
   • Sort Jobs: Process larger jobs first to fail faster in invalid branches.
   • Limit search: If a worker's current workload exceeds the best makespan found so far, stop.
   • Symmetry Breaking: If multiple workers currently have 0 workload, only try assigning the job to the first one.
3. Bitmask DP: dp[i][mask] represents the minimum makespan using i workers to complete jobs represented by the bitmask mask.

Example explanation

Jobs: [3, 2, 3], $k=2$.

• Assignment 1: Worker 1 gets [3, 3], Worker 2 gets [2]. Max time: 6.
• Assignment 2: Worker 1 gets [3, 2], Worker 2 gets [3]. Max time: 5. The minimum time to finish all jobs is 5.

Common mistakes candidates make

• Simple Greedy: Assuming that always giving the next job to the least busy worker works. While intuitive, it often leads to sub-optimal results.
• No Pruning: Writing a basic recursion that explores all $k^N$ possibilities, which will time out for even small $N$ (e.g., $N=12$).
• Bitmask Errors: Mismanaging the state transitions in the DP approach, leading to incorrect makespan calculations.

Interview preparation tip

Practice "Symmetry Breaking" in recursion. If you have $k$ identical workers, assigning a job to worker A is exactly the same as assigning it to worker B if both are currently idle. This simple check is often the difference between a TLE and a passing solution.