1. Graphs
Chapter 30
Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved X

2. Chapter Contents Some Examples and Terminology Traversals Road Maps
Airline Routes
Mazes
Course Prerequisites
Trees
Traversals
Breadth-First Traversal
Dept-First Traversal

3. 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. 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. Road Maps
Nodes
Edges
Fig A portion of a road map.

6. Road Maps
Fig A directed graph representing a portion of a city's street map.

7. 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. 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. Weights
Fig A weighted graph.

10. Connected Graphs A connected graph A complete 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. Connected Graphs
Fig Undirected graphs

12. Adjacent Vertices
Two vertices are adjacent in an undirected graph if they are joined by an edge
Sometimes adjacent vertices are called neighbors
Fig Vertex A is adjacent to B, but B is not adjacent to A.

13. Airline Routes Note the graph with two subgraphs
Each subgraph connected
Entire graph disconnected
Fig Airline routes

14. Mazes
Fig (a) A maze; (b) its representation as a graph

15. Course Prerequisites
Fig The prerequisite structure for a selection of courses as a directed graph without cycles.

16. Trees All trees are graphs A tree is a connected graph without cycles
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. Trees
Fig The visitation order of two traversals; (a) depth first

18. Trees
Fig The visitation order of two traversals; (b) breadth first.

19. 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 vertex

20. Breadth-First Traversal
Fig (ctd.) A trace of a breadth-first traversal for a directed graph, beginning at vertex A.

21. 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 traversal

22. Depth-First Traversal
Fig A trace of a depth-first traversal beginning at vertex A of the directed graph

23. 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. Topological Order
Fig. 30-8
Fig Three topological orders for the graph of Fig

25. Topological Order
Fig An impossible prerequisite structure for three courses as a directed graph with a cycle.
Click to view algorithm for a topological sort

26. Topological Order
Fig Finding a topological order for the graph in Fig

27. Shortest Path in an Unweighted Graph
Fig (a) an unweighted graph and (b) the possible paths from vertex A to vertex H.

28. Shortest Path in an Unweighted Graph
Click to view algorithm for finding shortest path
Fig (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

29. Shortest Path in an Unweighted Graph
Fig Finding the shortest path from vertex A to vertex H in the unweighted graph

30. Shortest Path in an Weighted Graph
Fig (a) A weighted graph and (b) the possible paths from vertex A to vertex H.

31. 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. Shortest Path in an Weighted Graph
Fig Finding the cheapest path from vertex A to vertex H in the weighted graph

33. Shortest Path in an Weighted Graph
Click to view algorithm for finding cheapest path in a weighted graph
Fig The graph in Fig a after finding the cheapest path from vertex A to vertex H.

34. Java Interfaces for the ADT Graph
Methods in the BasicGraphInterface
addVertex
addEdge
hasEdge
isEmpty
getNumberOfVertices
getNumberOfEdges
clear
View interface for basic graph operations

35. Java Interfaces for the ADT Graph
Fig A portion of the flight map in Fig

36. 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 graph