Recognized it as a topological sort problem pretty fast, which felt good, but then I fumbled a bit because the variable completion times per task threw me off from the version I'd practiced.
Model the tasks as a directed acyclic graph (DAG) where edges represent dependencies. The minimum total time is the length of the longest path (critical path) in the DAG, which can be computed using topological sort and dynamic programming. If tasks can be parallelized, the answer is the critical path length; if not, it's the sum of all task times.
Pro tip: Clarify upfront whether tasks can be executed in parallel. If parallel execution is allowed, the problem reduces to finding the critical path; otherwise, it's simply the sum of all completion times. This distinction shows you understand the problem's constraints and avoids solving the wrong problem.
Ask whether tasks can be executed in parallel or must be sequential, and confirm that dependencies form a DAG (no cycles). This determines the solution approach.
Represent tasks as nodes with weights equal to their completion times, and dependencies as directed edges. This transforms the problem into finding the longest path in a DAG.
Perform a topological sort to order tasks such that each task appears after its dependencies. This ensures we process tasks in a valid order for dynamic programming.
Process tasks in topological order, computing the earliest start time for each task as the maximum finish time of its dependencies. The answer is the maximum finish time across all tasks.
State that the algorithm runs in O(V+E) time and O(V) space. Discuss edge cases like no dependencies, multiple independent chains, and disconnected components.
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