1. abstract data types built on other ADTs
Graphs
abstract data types built on other ADTs

2. Graphs in computing
typically interested in both vertices and edges as objects (e.g., networks)
objects with properties

3. 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

4. Graph definitions(2) simple path cycle simple cycle
Directed Acyclic Graph DAG
edge weight (cost)
path cost (weighted path length)

5. Graph definitions(3) complete graph complete digraph |E| = |V|2
dense digraph |E| = O(|V|2)
sparse digraph |E| = O(|V|)

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

7. Example graph: mine tunnels
DATA
vertex:
key id
3 coordinates
edge:
two vertex id’s

8. Graphs as Collections
linear
trees
graphs

9. Graphs as Collections graph traversals
no obvious order of traversal (like trees)
no obvious starting point (no root)
traversals may not reach every vertex by following edges (connectedness)
traversals may return to a vertex (cycles)

10. Graph implementation(1)
adjacency matrix – ideal for dense digraph
n vertices, space: O(n2)
Graph g A B
char[] v A B C D
boolean[][] e to D C f t t f t t f t
from f t f t f f f f

11. Graph implementation(2)
adjacency list – ideal for sparse digraph
n vertices, k edges, space: O(n+k)
Graph g A B
char[] v A B C D
node[] e to D C 1 2 1 3
from 1 3

12. Sparse Directed Graph data 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 }

13. Operations on graphs collection class operations: paths and traversals
access, insert, delete, update for vertices and edges
edges are easy
vertices may impact edges also
paths and traversals
path lengths
weighted path lengths
specialized algorithms
e.g – path through mine

14. Operations on graphs e.g. delete edge algorithm:
remove edge from adjacency list

15. Operations on graphs
e.g. delete vertex  delete edges to/from the vertex also
algorithm:
delete adjacency list of vertex
search other adjacency lists and delete edges to this vertex
delete vertex

16. Path algorithms shortest path (number of edges)
shortest weighted path (more edges may be better)
negative edge weights/costs

17. Example Graph in JAVA class 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;

18. 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; }

19. Graph in JAVA  Graph g vertexMap name A adj A dist dest prev X cost
key/ vertex
adj A
dist
dest
prev X
scratch
cost
temp

20. 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 ) ); }

21. 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;

22. Graph in JAVA A B 1 A C 1 D B 1 D A 1 B D 1 C B 1 A B A B C A B C D A
sample file:
A B 1
A C 1
D B 1
D A 1
B D 1
C B 1 A B C A B C D A B C D A B A B C D C D

23. Graph in JAVA  Graph g vertexMap name A adj A dist dest prev X cost
key/ vertex
adj A
dist
dest
prev X
scratch
cost
temp