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

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


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


Graphs

FIGURE 28-2 A directed graph representing a portion of a city’s street map


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


Weights• Weighted graph, has values on its edges

Values are called either weights or costs


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


Connected Graphs

FIGURE 28-4 Undirected graphs


Adjacent Vertices

FIGURE 28-5 Vertex A is adjacent to vertex B, but B is not adjacent to A


• Vertices are adjacent in an undirected graph if they are joined by an edge

Airline Routes

FIGURE 28-6 Airline routes


Mazes

FIGURE 28-7 (a) A maze; (b) its representation as a graph


Course Prerequisites

FIGURE 28-8 The prerequisite structure for a selection of courses as a directed graph without cycles


Traversals

FIGURE 28-9 The visitation order of two traversals: (a) depth first; (b) breadth first


Breadth-First Traversal

Algorithm that performs a breadth-first traversal of a nonempty graph beginning at a given vertex



Algorithm that performs a breadth-first traversal of a nonempty graph beginning at a given vertex



FIGURE 28-10 A trace of a breadth-first traversal beginning at vertex A of a directed graph


Depth-First Traversal

Algorithm that performs a depth-first traversal of a nonempty graph, beginning at a given vertex:



Algorithm that performs a depth-first traversal of a nonempty graph, beginning at a given vertex:



FIGURE 28-11 A trace of a depth-first traversal beginning at vertex A of a directed graph



FIGURE 28-11 A trace of a depth-first traversal beginning at vertex A of a directed graph


Topological Order

FIGURE 28-12 Three topological orders for thegraph in Figure 28-8


Topological Order

FIGURE 28-13 An impossible prerequisite structure for three courses, as a directed graph with a cycle


• 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


Topological Order

FIGURE 28-14 Finding a topological order for the graph in Figure 28-8


Topological Order



Topological Order



Topological Order



Topological Order



Shortest Path in an Unweighted Graph

FIGURE 28-15 (a) An unweighted graph and (b) the possible paths from vertex A to vertex H



FIGURE 28-16 (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



Algorithm to find shortest path









FIGURE 28-17 A trace of the traversal in the algorithm to find the shortest path from vertex A to vertex H in an unweighted graph


The Shortest Path in a Weighted Graph

FIGURE 28-18 (a) A weighted graph and (b) the possible paths from vertex A to vertex H, with their weights



FIGURE 28-19 A trace of the traversal in the algorithm to find the cheapest path from vertex A to vertex H in a weighted graph



FIGURE 28-19 A trace of the traversal in the algorithm to find the cheapest path from vertex A to vertex H in a weighted graph



FIGURE 28-20 The graph in Figure 28-18a after finding the cheapest path from vertex A to vertex H



Pseudocode for shortest path (weighted)








Java Interfaces for the ADT Graph

LISTING 28-1 An interface of basic graph operations









FIGURE 28-21 A portion of the flight map in Figure 28-6



LISTING 28-2 An interface of operations on an existing graph









LISTING 28-3 An interface for the ADT graph


End

Chapter 28


Documents

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