1. CSC 2300 Data Structures & Algorithms March 30, 2007 Chapter 9. Graph Algorithms

2. Today – Graphs Graphs –  Overview  Definitions  Representation Topological Sort

3. Graphs – Overview

4. Definitions

5. More Definitions

6. Conventions

7. Directed Graph – Example

8. Graph Representations 1. Adjacency matrix 2. Adjacency list

9. Adjacency Matrix What does adjacency matrix look like? There are |E| edges and |V| vertices. How much space is used?

10. Adjacency List – Example There are |E| edges and |V| vertices. How much space is used?

11. Two More Examples

12. Graph Algorithms Overview

13. Topological Sort

14. Example – Course Prerequisites

15. Example – Indegrees What are the indegrees of the seven vertices? Is there a vertex with indegree = 0? Is it possible that there exists no vertex with indegree = 0?

16. Topological Sort Scan the array of vertices to look for a vertex with indegree 0 that has not been assigned a topological number. There are |E| edges and |V| vertices. What is the running time of this algorithm? Is it possible to do better?

17. Improved Topological Sort Why always scan through all the vertices? Only a few vertices have their indegrees updated during each iteration. We keep all (unassigned) vertices of indegree 0 in a queue. When a vertex v is removed from the queue, all vertices adjacent to v will have their indegrees decremented. A vertex is placed in the queue if its indegree has fallen to 0. There are |E| edges and |V| vertices. What is the running time of this new approach?

18. Improved Topological Sort

19. Topological Sort – Example How many iterations? Big Oh of what?