Graphs Chapter 30 Slides by Steve Armstrong LeTourneau University Longview, TX 2007, Prentice Hall

Graphs

Chapter 30

Slides by Steve ArmstrongLeTourneau University

Longview, TX2007,Prentice Hall

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


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


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


Road Maps

Fig. 30-1 A portion of a road map.

NodesNodes

EdgesEdges


Road Maps

Fig. 30-2 A directed graph representing a portion of a city's street map.


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


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


Weights

Fig. 30-3 A weighted graph.


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


Connected Graphs

Fig. 30-4 Undirected graphs


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.


Airline Routes

• Note the graph with two subgraphs Each subgraph connected Entire graph disconnected

Fig. 30-6 Airline routes


Mazes

Fig. 30-7 (a) A maze; (b) its representation as a graph


Course Prerequisites

Fig. 30-8 The prerequisite structure for a selection of courses as a directed graph without cycles.


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


Trees

Fig. 30-9 The visitation order of two traversals; (a) depth first


Trees

Fig. 30-9 The visitation order of two traversals; (b) breadth first.


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


Breadth-First Traversal

Fig. 30-10 (ctd.) A trace of a breadth-first

traversal for a directed graph,

beginning at vertex A.


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


Depth-First TraversalFig. 30-11 A

trace of a depth-first traversal beginning at

vertex A of the directed graph


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


Topological Order

Fig. 30-12 Three topological orders for the graph of Fig. 30-8.

Fig. 30-8


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

Click to view algorithm for a topological sort


Topological Order

Fig. 30-14 Finding a topological order for the graph in

Fig. 30-8.


Shortest Path in an Unweighted Graph

Fig. 30-15 (a) an unweighted graph and (b) the possible paths from vertex A to vertex H.



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 pathClick to view algorithm for finding shortest path



Fig. 30-17 Finding the shortest path from vertex A to vertex H in the unweighted graph


Shortest Path in an Weighted Graph

Fig. 30-18 (a) A weighted graph and (b) the possible paths from vertex A to vertex H.



• 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



Fig. 30-19 Finding the cheapest path from vertex A to vertex H in the 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

Click to view algorithm for

finding cheapest path in a weighted

graph


Java Interfaces for the ADT Graph

• Methods in the BasicGraphInterface addVertex addEdge hasEdge isEmpty getNumberOfVertices getNumberOfEdges clear

• View interface for basic graph operations



Fig. 30-21 A portion of the flight map in Fig. 30-6.



• 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

Documents

Graphs Chapter 30 Slides by Steve Armstrong LeTourneau University Longview, TX 2007, Prentice Hall