Graphs

abstract data types built on other ADTs

Graphs in computing

• typically interested in both vertices and edges as objects (e.g., networks)

objects with properties

Graph definitions(1)

• graph: set of vertices and set of edges connecting vertices: G(V,E)

• path: sequence of vertices connected by edges

• path length

• connected graph

• directed edge

• digraph

• directed path

Graph definitions(2)

• simple path

• cycle

• simple cycle

• directed acyclic graph DAG

• edge weight (cost)

• path cost (weighted path length)

Graph definitions(3)

• complete graph

• complete digraph |E| = |V|2

• dense digraph |E| = O(|V|2)

• sparse digraph |E| = O(|V|)

Graphs in computing

• (mainly) interested in sparse directed graphs for applications– networks for communication– transportation systems– distributed computing– java hierarchies – inheritance, instance,

message

Example graph: mine tunnels

DATA

vertex:

key id

3 coordinates

edge:

two vertex id’s

Graphs as Collections

• linear

• trees

• graphs

Graphs as Collections

graph traversals

1. no obvious order of traversal (like trees)

2. no obvious starting point (no root)

3. traversals may not reach every vertex by following edges (connectedness)

4. traversals may return to a vertex (cycles)

Graph implementation(1)

• adjacency matrix – ideal for dense digraph

• n vertices, space: O(n2)

A B

D C

Graph g

char[] v

boolean[][] eA B C D

f t t f

t t f t

f t f t

f f f f

from

to

Graph implementation(2)

• adjacency list – ideal for sparse digraph

• n vertices, k edges, space: O(n+k)

A B

D C

Graph g

char[] v

node[] eA B C D

1

from

to

2

0 3

1 3

1

0 1 2 3

Sparse Directed Graphdata structures

Vertex{• ID (key)• information about

vertex• adjacency list of edges• temporary data storage

for algorithms}

Edge

{• information about edge• destination vertex• temporary data storage

for algorithms

}

Operations on graphs

1. collection class operations:– access, insert, delete, update for vertices and edges

• edges are easy• vertices may impact edges also

2. paths and traversals– path lengths– weighted path lengths

3. specialized algorithms– e.g – path through mine

Operations on graphs

• e.g. delete edge

algorithm:

1. remove edge from adjacency list

Operations on graphs

• e.g. delete vertex delete edges to/from the vertex also

algorithm:

1. delete adjacency list of vertex

2. search other adjacency lists and delete edges to this vertex

3. delete vertex

Path algorithms

• shortest path (number of edges)

• shortest weighted path (more edges may be better)

• negative edge weights/costs

Example Graph in JAVAclass Vertex{ public String name; // Vertex name public LinkedList<Edge> adj; // edges from vertex public double dist; // Cost public Vertex prev; // Previous vertex on shortest path public int scratch; // Extra variable used in algorithm

public Vertex( String nm ) { name = nm; adj = new LinkedList<Edge>( ); reset( ); } public void reset( ) // clears values used in algorithms { dist = Graph.INFINITY; prev = null; scratch = 0; } }

Example Graph in JAVA

class Edge{ public Vertex dest; // Second vertex in Edge public double cost; // Edge cost public double temp; // used in algorithms public Edge( Vertex d, double c ) { dest = d; cost = c; }}

Graph in JAVA

Graph g

vertexMap

key/ vertex

A

name A

adj

dist

prevscratch

dest

costX

temp

Graph in JAVA

public class Graph{ public static final double INFINITY = Double.MAX_VALUE; private HashMap<String,Vertex> vertexMap = new HashMap(); // maps String to Vertex

public void addEdge( String sourceName, String destName, double cost ) { Vertex v = getVertex( sourceName ); Vertex w = getVertex( destName ); v.adj.add( new Edge( w, cost ) ); }

Graph in JAVA

private Vertex getVertex( String vertexName ) { Vertex v = (Vertex) vertexMap.get( vertexName ); if( v == null ) { v = new Vertex( vertexName ); vertexMap.put( vertexName, v ); } return v; }

D

Graph in JAVA

sample file:A B 1A C 1D B 1D A 1B D 1C B 1

A B

A B

CA B

C

DC

A B

DC

A B

DC

A B

Graph in JAVA

Graph g

vertexMap

key/ vertex

A

name A

adj

dist

prevscratch

dest

costX

temp