Concurrent Task Scheduling

Graph Scheduling Dp

You are given a list of tasks required to complete a project. Each task is represented as an object with the following properties:

  • id: A unique identifier for the task.
  • duration: An integer representing the time required to complete the task.
  • dependencies: A list of task IDs that must be completed before this task can begin.

Assume that:

  • There are no cyclic dependencies.
  • Tasks can run concurrently as soon as all their dependencies are met.
  • The project starts at time 0.

Task: Write a function that takes in the list of tasks and returns the minimum total time required to complete all tasks.

Example:

Consider the following tasks:

  • Task A: id = "A", duration = 3, dependencies = []
  • Task B: id = "B", duration = 2, dependencies = ["A"]
  • Task C: id = "C", duration = 4, dependencies = ["A"]
  • Task D: id = "D", duration = 1, dependencies = ["B", "C"]

A possible schedule would be:

  1. Time 0: Start Task A.
  2. Time 3: Task A finishes; start Tasks B and C concurrently.
  3. Time 5: Task B finishes (while Task C is still running).
  4. Time 7: Task C finishes; start Task D.
  5. Time 8: Task D finishes.

The total time required to complete all tasks is 8.

Your implementation should correctly compute this minimum time based on any valid list of tasks.