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

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved 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 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 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 X Road Maps Fig A portion of a road map. Nodes Edges

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved X Road Maps Fig 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 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 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 X Weights Fig A weighted graph.

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved 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 X Connected Graphs Fig Undirected graphs

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved 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 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 X Airline Routes Note the graph with two subgraphs  Each subgraph connected  Entire graph disconnected Fig Airline routes

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved X Mazes Fig (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 X Course Prerequisites Fig 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 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 X Trees Fig 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 X Trees Fig 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 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 vertexView algorithm

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved X Breadth-First Traversal Fig (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 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 traversalView algorithm

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved X Depth-First Traversal Fig 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 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 X Topological Order Fig Three topological orders for the graph of Fig Fig. 30-8

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved X Topological Order Fig 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 X Topological Order Fig Finding a topological order for the graph in Fig

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved X Shortest Path in an Unweighted Graph Fig (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 X Shortest Path in an Unweighted Graph Fig (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 path

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved X Shortest Path in an Unweighted Graph Fig 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 X Shortest Path in an Weighted Graph Fig (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 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 X Shortest Path in an Weighted Graph Fig 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 X Shortest Path in an Weighted Graph Fig The graph in Fig a after finding the cheapest path from vertex A to vertex H. 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 X Java Interfaces for the ADT Graph Methods in the BasicGraphInterface  addVertex  addEdge  hasEdge  isEmpty  getNumberOfVertices  getNumberOfEdges  clear View interface for basic graph operationsView interface

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved X Java Interfaces for the ADT Graph Fig A portion of the flight map in Fig

Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved 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 graphView interface