The key realization is that this reduces to finding the longest path through the DAG, which took me a minute to see.
Model the problem as a longest path in a DAG where each node's weight is its duration. Compute the earliest start time for each task using topological order, then the minimum total time is the maximum earliest finish time.
Pro tip: Mention that this is equivalent to finding the critical path, and discuss how the approach scales to large graphs with parallel execution.
Confirm that tasks can run in parallel if dependencies allow, and that the goal is to minimize the makespan (total completion time).
Represent tasks as nodes with weights (durations) and dependencies as directed edges. The problem reduces to finding the longest path in this 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 each task in topological order, compute its earliest start time as the maximum earliest finish time of its dependencies. Its earliest finish time is start time plus duration.
The minimum total time is the maximum earliest finish time among all tasks. Return this value.
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