graphs chapter 30 slides by steve armstrong letourneau university longview, tx 2007, prentice hall
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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
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
Fig. 30-1 A portion of a road map.
NodesNodes
EdgesEdges
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 TraversalFig. 30-11 A
trace of a depth-first 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. 30-12 Three topological orders for the graph of Fig. 30-8.
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
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
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
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
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
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