This one took me a while to even decompose properly.
Model the problem as a list scheduling simulation with a priority queue for ready tasks and a min-heap for worker availability. Use topological sorting to track dependencies and greedily assign the longest ready task to the earliest available worker. Return the maximum completion time.
Pro tip: Clarify that LPT is a heuristic and may not yield the optimal schedule; mention that the optimal makespan for unrelated machines is NP-hard, but for identical machines with precedence constraints, list scheduling gives a 2-approximation. This shows depth and avoids overpromising.
Confirm that tasks have fixed durations, workers are identical, dependencies form a DAG, and preemption is not allowed. Ask about input size and whether the schedule must be output or just the makespan.
Build an adjacency list for the DAG, compute in-degrees, and initialize a ready queue (max-heap by duration) and a worker min-heap (keyed by next available time). Set current time to 0 and completed count to 0.
While tasks remain, if a worker is free and ready tasks exist, assign the longest ready task to the earliest free worker, update its next available time, and decrement in-degrees of successors, adding newly ready tasks to the heap. If no worker is free, advance time to the next worker availability.
After all tasks are scheduled, the makespan is the maximum next available time among workers. Return that value.
State time complexity O((V+E) log V) using heaps, and space O(V+E). Discuss that LPT is a heuristic and may not be optimal, but is efficient and often near-optimal.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Straightforward DFS cycle detection, raise an error.
Start by explaining that cycle detection in a dependency graph is typically done using DFS with recursion stack or Kahn's topological sort algorithm. Then discuss the scheduler's response: it should not execute the cycle, but instead report the cycle, potentially break it by aborting or skipping tasks, and log the error for debugging.
Pro tip: Mention that in production systems, you often want to detect cycles early (e.g., at graph build time) and provide a clear error message with the cycle path to help users fix their dependencies. Also, consider that some schedulers might allow breaking cycles by prioritizing certain tasks or using a fallback strategy.
Confirm that the dependency graph is directed and that a cycle means a set of tasks where each depends on another in a loop. Ask if the graph is static or dynamic, and if cycle detection should be online or offline.
Describe either DFS with a recursion stack (coloring nodes white/gray/black) or Kahn's algorithm (topological sort with indegree). Explain the time complexity O(V+E) and space complexity.
Walk through how the algorithm works: for DFS, mark nodes as visiting and visited; if you encounter a visiting node, a cycle exists. For Kahn's, repeatedly remove nodes with indegree 0; if some nodes remain, a cycle exists.
If a cycle is found, the scheduler should not execute any tasks in the cycle. It should report the cycle (e.g., list the tasks involved), and may choose to abort the entire schedule, skip the cyclic tasks, or attempt to break the cycle by removing an edge (with user confirmation).
Mention logging the cycle for debugging, notifying the user, and possibly providing suggestions to fix dependencies. In some systems, the scheduler might retry after a timeout or use a fallback plan.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
I fumbled the state representation at first.
Clarify the data structures representing tasks, dependencies, and statuses, then propose an efficient algorithm that filters tasks based on dependency completion and current state. Discuss trade-offs between different approaches (e.g., scanning all tasks vs. maintaining a ready queue) and consider edge cases like cycles or missing dependencies.
Pro tip: Mention that in a real scheduler, you'd likely maintain a ready queue updated incrementally as tasks complete, rather than recomputing from scratch each time, to achieve O(1) amortized per task.
Ask about the representation of tasks, dependencies, and statuses (e.g., adjacency list, task objects with status fields). Confirm whether dependencies are direct only or transitive, and whether the graph is a DAG.
A task is ready if all its dependencies have status 'finished', and its own status is neither 'running' nor 'done'. Explicitly state these conditions.
Propose an approach: iterate over all tasks, check each task's status and dependencies. For efficiency, consider maintaining a reverse dependency map or a ready queue. Discuss time and space complexity.
Address cycles, missing dependencies, tasks with no dependencies, and concurrency concerns. Suggest incremental updates if the function is called frequently.
Walk through a simple example to verify correctness, and mention potential unit tests for edge cases.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.