1. Graphs Chapter 30 Slides by Steve Armstrong LeTourneau University Longview, TX  2007,  Prentice Hall

2. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Chapter Contents Some Examples and Terminology  Road Maps  Airline Routes  Mazes  Course Prerequisites  Trees Traversals  Breadth-First Traversal  Dept-First Traversal

3. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Chapter Contents Topological Order Paths  Finding a Path  Shortest Path in an Unweighted Graph  Shortest Path in a Weighted Graph Java Interfaces for the ADT Graph

4. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Some Examples and Terminology Vertices or nodes are connected by edges A graph is a collection of distinct vertices and distinct edges  Edges can be directed or undirected  When it has directed edges it is called a digraph A subgraph is a portion of a graph that itself is a graph

5. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Road Maps Fig. 30-1 A portion of a road map. Nodes Edges

6. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Road Maps Fig. 30-2 A directed graph representing a portion of a city's street map.

7. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Paths A sequence of edges that connect two vertices in a graph In a directed graph the direction of the edges must be considered  Called a directed path A cycle is a path that begins and ends at same vertex  Simple path does not pass through any vertex more than once A graph with no cycles is acyclic

8. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Weights A weighted graph has values on its edges  Weights or costs A path in a weighted graph also has weight or cost  The sum of the edge weights Examples of weights  Miles between nodes on a map  Driving time between nodes  Taxi cost between node locations

9. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Weights Fig. 30-3 A weighted graph.

10. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Connected Graphs A connected graph  Has a path between every pair of distinct vertices A complete graph  Has an edge between every pair of distinct vertices A disconnected graph  Not connected

11. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Connected Graphs Fig. 30-4 Undirected graphs

12. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Adjacent Vertices Two vertices are adjacent in an undirected graph if they are joined by an edge Sometimes adjacent vertices are called neighbors Fig. 30-5 Vertex A is adjacent to B, but B is not adjacent to A.

13. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Airline Routes Note the graph with two subgraphs  Each subgraph connected  Entire graph disconnected Fig. 30-6 Airline routes

14. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Mazes Fig. 30-7 (a) A maze; (b) its representation as a graph

15. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Course Prerequisites Fig. 30-8 The prerequisite structure for a selection of courses as a directed graph without cycles.

16. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Trees All trees are graphs  But not all graphs are trees A tree is a connected graph without cycles Traversals  Preorder, inorder, postorder traversals are examples of depth-first traversal  Level-order traversal of a tree is an example of breadth-first traversal Visit a node  For a tree: process the node's data  For a graph: mark the node as visited

17. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Trees Fig. 30-9 The visitation order of two traversals; (a) depth first

18. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Trees Fig. 30-9 The visitation order of two traversals; (b) breadth first.

19. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Breadth-First Traversal A breadth-first traversal  visits a vertex and  then each of the vertex's neighbors  before advancing View algorithm for breadth-first traversal of nonempty graph beginning at a given vertexView algorithm

20. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Breadth-First Traversal Fig. 30-10 (ctd.) A trace of a breadth-first traversal for a directed graph, beginning at vertex A.

21. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Depth-First Traversal Visits a vertex, then  A neighbor of the vertex,  A neighbor of the neighbor,  Etc. Advance as possible from the original vertex Then back up by one vertex  Considers the next neighbor View algorithm for depth-first traversalView algorithm

22. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Depth-First Traversal Fig. 30-11 A trace of a depth- first traversal beginning at vertex A of the directed graph

23. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Topological Order Given a directed graph without cycles In a topological order  Vertex a precedes vertex b whenever  A directed edge exists from a to b

24. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Topological Order Fig. 30-12 Three topological orders for the graph of Fig. 30-8. Fig. 30-8

25. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Topological Order Fig. 30-13 An impossible prerequisite structure for three courses as a directed graph with a cycle. Click to view algorithm for a topological sort

26. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Topological Order Fig. 30-14 Finding a topological order for the graph in Fig. 30-8.

27. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Shortest Path in an Unweighted Graph Fig. 30-15 (a) an unweighted graph and (b) the possible paths from vertex A to vertex H.

28. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Shortest Path in an Unweighted Graph Fig. 30-16 (a) The graph in 30-15a after the shortest- path algorithm has traversed from vertex A to vertex H; (b) the data in the vertex Click to view algorithm for finding shortest path

29. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Shortest Path in an Unweighted Graph Fig. 30-17 Finding the shortest path from vertex A to vertex H in the unweighted graph

30. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Shortest Path in an Weighted Graph Fig. 30-18 (a) A weighted graph and (b) the possible paths from vertex A to vertex H.

31. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Shortest Path in an Weighted Graph Shortest path between two given vertices  Smallest edge-weight sum Algorithm based on breadth-first traversal Several paths in a weighted graph might have same minimum edge-weight sum  Algorithm given by text finds only one of these paths

32. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Shortest Path in an Weighted Graph Fig. 30-19 Finding the cheapest path from vertex A to vertex H in the weighted graph

33. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Shortest Path in an Weighted Graph Fig. 30-20 The graph in Fig. 30-18a after finding the cheapest path from vertex A to vertex H. Click to view algorithm for finding cheapest path in a weighted graph

34. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Java Interfaces for the ADT Graph Methods in the BasicGraphInterface  addVertex  addEdge  hasEdge  isEmpty  getNumberOfVertices  getNumberOfEdges  clear View interface for basic graph operationsView interface

35. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Java Interfaces for the ADT Graph Fig. 30-21 A portion of the flight map in Fig. 30-6.

36. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X Java Interfaces for the ADT Graph Operations of the ADT  Graph enable creation of a graph and  Answer questions based on relationships among vertices View interface of operations on an existing graphView interface