DAGs and Topological Sorting

🎯 Learning Objectives

1. Define a Directed Acyclic Graph (DAG)
2. Perform topological sorting using Kahn's algorithm and DFS
3. Apply DAGs to model dependencies and scheduling
4. Detect cycles in directed graphs

📋 Prerequisites

• Graph Fundamentals — directed graphs
• DFS — graph traversal concepts

📖 Core Content

20.1 What is a DAG?

• Course prerequisites (Math 1 → Math 2 → Advanced ML)
• Build dependencies (compile A before B which depends on A)
• Git version history (commits form a DAG)
• Family tree (ancestor → descendant)

💡 Why this matters: DAGs model any system with dependencies and partial order. Understanding DAGs is essential for dependency management, build systems, and scheduling.

20.2 Properties of DAGs

• Every DAG has at least one source (vertex with in-degree 0)
• Every DAG has at least one sink (vertex with out-degree 0)
• A DAG can be topologically sorted (linear ordering respecting edge directions)
• A directed graph is a DAG iff it has a topological ordering

20.3 Topological Sorting

20.4 Kahn's Algorithm (BFS-based)

1. Compute in-degree of all vertices
2. Enqueue all vertices with in-degree 0
3. While queue not empty:
   • Dequeue vertex $v$, add to result
   • For each neighbor $w$ of $v$: decrement in-degree of $w$
   • If in-degree of $w$ becomes 0, enqueue $w$
4. If result size < number of vertices → graph has a cycle! Time complexity: $O(V + E)$

20.5 DFS-based Topological Sort

1. Run DFS on the graph
2. Add vertex to a stack after all its neighbors are processed (post-order)
3. Pop from stack to get topological order

20.6 Cycle Detection

📐 Key Formulas — Summary Table

⚠️ Common Pitfalls

Pitfall 1: Topological Sort ≠ Unique

Pitfall 2: Only DAGs Have Topological Sorts

Pitfall 3: Confusing Sources and Sinks

📝 Practice Questions

Q1: Is a DAG with a single vertex a DAG?

Yes — no cycle possible with one vertex.

$\boxed{\text{Yes}}$ Q2: Find a topological order for $1\to2$, $1\to3$, $2\to4$, $3\to4$.

$[1, 2, 3, 4]$ or $[1, 3, 2, 4]$

$\boxed{[1,2,3,4] \text{ or } [1,3,2,4]}$ Q3: Does a complete directed graph on 3 vertices contain a cycle?

Yes — it's a directed cycle of length 3.

$\boxed{\text{Yes}}$ Q4: What data structure does Kahn's algorithm use?

Queue.

$\boxed{\text{Queue}}$ Q5: In Kahn's algorithm, if result has fewer than $|V|$ elements, what does it mean?

The graph has a cycle.

$\boxed{\text{Cycle detected}}$ Q6: Give an application of topological sorting.

Course prerequisite ordering, build dependency resolution, task scheduling.

$\boxed{\text{Dependency resolution}}$ Q7: How many sources must a DAG have?

At least one.

$\boxed{\text{At least 1}}$ Q8: Does every DAG have a unique topological ordering?

No — multiple valid orderings typically exist.

$\boxed{\text{No}}$

🔗 Cross-References

• Previous: Graph Traversal
• Next: Shortest Path Algorithms
• Across courses: PDSA (DAG-based scheduling), Build systems (Make, Gradle) Join Discord Previous10.2 Graph Traversal (BFS/DFS)Next11.1 Shortest Path Algorithms