CSC 2300Data Structures & Algorithms

March 30, 2007

Chapter 9. Graph Algorithms

Today – Graphs

Graphs – Overview Definitions Representation

Topological Sort

Graphs – Overview

Definitions

More Definitions

Conventions

Directed Graph – Example

Graph Representations

1. Adjacency matrix

2. Adjacency list

Adjacency Matrix

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

Adjacency List – Example

There are |E| edges and |V| vertices. How much space is used?

Two More Examples

Graph Algorithms Overview

Topological Sort

Example – Course Prerequisites

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?

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?

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?

Improved Topological Sort

Topological Sort – Example

How many iterations? Big Oh of what?