Graphs Chapter 28 © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. Data Structures and Abstractions with Java, 4e Frank Carrano

Road Maps FIGURE 28-1 A portion of a road map © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Graphs A road map is a graph Definition of a graph:  A collection of distinct vertices and distinct edges In the map of Figure 28-1  The circles are vertices or nodes  The lines are called edges A subgraph is a portion of a graph that is itself a graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Graphs FIGURE 28-2 A directed graph representing a portion of a city’s street map © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Paths A path between two vertices in a graph is a sequence of edges A path in a directed graph must consider the direction of the edges  Called a directed path The length of a path is the number of edges that it comprises. A cycle is a path that begins and ends at the same vertex © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Weights Weighted graph, has values on its edges  Values are called either weights or costs © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. FIGURE 28-3 A weighted graph

Connected Graphs A graph that has a path between every pair of distinct vertices is connected A complete graph has an edge between every pair of distinct vertices. Undirected graphs can be  Connected  Complete or  Disconnected © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Connected Graphs FIGURE 28-4 Undirected graphs © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Adjacent Vertices FIGURE 28-5 Vertex A is adjacent to vertex B, but B is not adjacent to A © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. Vertices are adjacent in an undirected graph if they are joined by an edge

Airline Routes FIGURE 28-6 Airline routes © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Mazes FIGURE 28-7 (a) A maze; (b) its representation as a graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Course Prerequisites FIGURE 28-8 The prerequisite structure for a selection of courses as a directed graph without cycles © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Traversals FIGURE 28-9 The visitation order of two traversals: (a) depth first; (b) breadth first © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Breadth-First Traversal Algorithm that performs a breadth-first traversal of a nonempty graph beginning at a given vertex © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Breadth-First Traversal Algorithm that performs a breadth-first traversal of a nonempty graph beginning at a given vertex © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Breadth-First Traversal FIGURE A trace of a breadth-first traversal beginning at vertex A of a directed graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Depth-First Traversal Algorithm that performs a depth-first traversal of a nonempty graph, beginning at a given vertex: © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Depth-First Traversal Algorithm that performs a depth-first traversal of a nonempty graph, beginning at a given vertex: © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Depth-First Traversal FIGURE A trace of a depth-first traversal beginning at vertex A of a directed graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Depth-First Traversal FIGURE A trace of a depth-first traversal beginning at vertex A of a directed graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Topological Order FIGURE Three topological orders for the graph in Figure 28-8 © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Topological Order FIGURE An impossible prerequisite structure for three courses, as a directed graph with a cycle © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. In a topological order of the vertices in a directed graph without cycles, vertex a precedes vertex b whenever a directed edge exists from a to b.

Topological Order An algorithm that describes a topological sort © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Topological Order FIGURE Finding a topological order for the graph in Figure 28-8 © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Topological Order FIGURE Finding a topological order for the graph in Figure 28-8 © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Topological Order FIGURE Finding a topological order for the graph in Figure 28-8 © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Topological Order FIGURE Finding a topological order for the graph in Figure 28-8 © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Topological Order FIGURE Finding a topological order for the graph in Figure 28-8 © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Shortest Path in an Unweighted Graph FIGURE (a) An unweighted graph and (b) the possible paths from vertex A to vertex H © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Shortest Path in an Unweighted Graph FIGURE (a) The graph in Figure 28-15a after the shortest-path algorithm has traversed from vertex A to vertex H; (b) the data in a vertex © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Shortest Path in an Unweighted Graph Algorithm to find shortest path © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Shortest Path in an Unweighted Graph Algorithm to find shortest path © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Shortest Path in an Unweighted Graph Algorithm to find shortest path © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Shortest Path in an Unweighted Graph FIGURE A trace of the traversal in the algorithm to find the shortest path from vertex A to vertex H in an unweighted graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

The Shortest Path in a Weighted Graph FIGURE (a) A weighted graph and (b) the possible paths from vertex A to vertex H, with their weights © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

The Shortest Path in a Weighted Graph FIGURE A trace of the traversal in the algorithm to find the cheapest path from vertex A to vertex H in a weighted graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

The Shortest Path in a Weighted Graph FIGURE A trace of the traversal in the algorithm to find the cheapest path from vertex A to vertex H in a weighted graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

The Shortest Path in a Weighted Graph FIGURE The graph in Figure 28-18a after finding the cheapest path from vertex A to vertex H © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

The Shortest Path in a Weighted Graph Pseudocode for shortest path (weighted) © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

The Shortest Path in a Weighted Graph Pseudocode for shortest path (weighted) © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

The Shortest Path in a Weighted Graph Pseudocode for shortest path (weighted) © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Java Interfaces for the ADT Graph LISTING 28-1 An interface of basic graph operations © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Java Interfaces for the ADT Graph LISTING 28-1 An interface of basic graph operations © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Java Interfaces for the ADT Graph LISTING 28-1 An interface of basic graph operations © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Java Interfaces for the ADT Graph FIGURE A portion of the flight map in Figure 28-6 © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Java Interfaces for the ADT Graph LISTING 28-2 An interface of operations on an existing graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Java Interfaces for the ADT Graph LISTING 28-2 An interface of operations on an existing graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Java Interfaces for the ADT Graph LISTING 28-2 An interface of operations on an existing graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

Java Interfaces for the ADT Graph LISTING 28-3 An interface for the ADT graph © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

End Chapter 28 © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.