Let's pivot to Problem 2, which involves modeling processes with dependencies using Directed Acyclic Graphs (DAGs).

  • A Directed Acyclic Graph (DAG) is a directed graph that contains no cycles, meaning you cannot return to a previously visited node.
  • DAGs are essential for modeling systems where one event or task must precede another, establishing a strict partial order.
  • They are widely used in Task Scheduling (e.g., project management), compiler design, and dependency resolution (e.g., package managers).
  • The absence of cycles ensures that the dependencies are logically sound; you cannot have Task A requiring B, and B requiring A simultaneously.
  • Finding a valid sequence of tasks that respects all dependencies is the goal of Topological Sort, a key algorithm for DAGs.
DAG Prerequisite Definition
1# Prerequisite Definition in a DAG
2
3def is_prerequisite(u, v, graph):
4    # An edge (u, v) exists in the graph
5    if v in graph.get(u, []):
6        # u must be completed before v can start.
7        return True
8    return False
9
10# Example: A -> C (A is prerequisite for C)