Download Java Classes

Graphs
Chapter 30
Slides by Steve Armstrong
LeTourneau University
Longview, TX
2007, Prentice Hall
Chapter Contents
• Some Examples and Terminology
 Road Maps
 Airline Routes
 Mazes
 Course Prerequisites
 Trees
• Traversals
 Breadth-First Traversal
 Dept-First Traversal
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
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
Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X
Road Maps
Nodes
Edges
Fig. 30-1 A portion of a road map.
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.
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
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
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.
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
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
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.
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
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
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.
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
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
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.
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
vertex
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.
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 traversal
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 depthfirst traversal
beginning at
vertex A of the
directed graph
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
Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X
Topological Order
Fig. 30-8
Fig. 30-12 Three topological orders for the
graph of Fig. 30-8.
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
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.
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.
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
Click to view algorithm
for finding shortest path
Fig. 30-16 (a) The graph in 30-15a after the shortestpath algorithm has traversed from vertex A to vertex H;
(b) the data in the vertex
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
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.
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
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
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
Click to view
algorithm for
finding cheapest
path in a weighted
graph
Fig. 30-20 The graph in Fig. 30-18a after finding the
cheapest path from vertex A to vertex H.
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 operations
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.
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
graph
Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X