Lecture 29: Edge-list & Adjacency-List based Graphs

1. CSC 213 – Large Scale Programming Lecture 29:Edge-list & Adjacency-List based Graphs

2. Today’s Goals • Briefly review graphs and vital graph terms • Begin discussion of how to implement Graph • Vertex & Edge classes will implement which ADT? • How to best go about listing classes’ generic types? • How to implement Graph? What fields required? • For an implementation, what are performance effects? • Ways to improve performance & their cost examined

3. Graphs • Mathematically, graph is pair (V, E) where • Vis collection of nodes, called vertices • Two nodes can be connected by an edge in E • Positionimplemented by Vertex & Edge classes PVD 849 ORD 1843 142 SFO 802 LGA 1743 337 1387 HNL 2555 1099 LAX 1233 DFW 1120 MIA

4. Graph Types • Graph is directed if all edges directed • All edges in graph must be directed • Examples include object hierarchies & CSC courses • Any edge allowed in undirected graphs • Can have only undirected or mix of both edges • Roadways & airline travel are examples of this Object String Amherst Humboldt

5. VertexSuperclass • Value stored at each point in the Graph • Must hold an element so implements Position • Nothing else required for default Vertex class

6. VertexSuperclass • Value stored at each point in the Graph • Must hold an element so implements Position • Nothing else required for default Vertex class classVertex<V> implements Position<V> {private Velement;publicVertex(Velem){element = elem;}// Also defines setElement& element }

7. Edge Superclass • Slightly more complicated than Vertex • Also provides only minimum fields & methods • Inheritance used to allow later specialization class Edge<E> implements Position<E> {private Eelement;private Vertex source;private Vertex target;private booleanisDirected;// Add constructor, getters, setters, & element() }

8. Edge Superclass • Slightly more complicated than Vertex • Also provides only minimum fields & methods • Inheritance used to allow later specialization class Edge<E,V> implements Position<E> {private Eelement;private Vertex<V>source;private Vertex<V>target;private booleanisDirected;// Add constructor, getters, setters, & element() }

9. Edge List Structure u • Simplest Graph • Space efficient • No change to use withdirected or undirected a c b d v w z

10. Edge List Structure u • Simplest Graph • Space efficient • No change to use withdirected or undirected • Fields • Sequenceof vertices a c b d v w z vertices z w v u

11. Edge List Structure u • Simplest Graph • Space efficient • No change to use withdirected or undirected • Fields • Sequenceof vertices • Sequenceof edges a c b d v w z vertices z w v u edges c a b d

12. EdgeListImplementation classELGraphimplements Graph {private Sequence<Vertex>vertices;private Sequence<Edge>edges;public ELGraph(){vertices = // Instantiate a Sequenceedges =// Instantiate a Sequence}// Add Graph’s methods like:public Iterable<Position> vertices() { return vertices;} }

13. EdgeListImplementation classELGraph<V,E>implements Graph {private Sequence<Vertex<V>>vertices;private Sequence<Edge<E>>edges;public ELGraph(){vertices = // Instantiate a Sequenceedges = // Instantiate a Sequence}// Add Graph’s methods like:public Iterable<Position<V>> vertices() { return vertices;} }

14. EdgeListImplementation classELGraph<V,E>implements Graph {private Sequence<Vertex<V>>vertices;private Sequence<Edge<E,V>>edges;public ELGraph(){vertices = // Instantiate a Sequenceedges = // Instantiate a Sequence}// Add Graph’s methods like:public Iterable<Position<V>> vertices() { return vertices;} }

15. EdgeListImplementation classELGraph<V,E>implements Graph<V,E>{private Sequence<Vertex<V>>vertices;private Sequence<Edge<E,V>>edges;public ELGraph(){vertices = // Instantiate a Sequenceedges = // Instantiate a Sequence}// Add Graph’s methods like:public Iterable<Position<V>> vertices() { return vertices;} }

16. Graph ADT • Accessor methods • vertices(): iterable for vertices • edges(): iterable for edges • endVertices(e): array with endpoints of edge e • opposite(v,e): e’sendpoint that is not v • areAdjacent(v,w): check if v and w are adjacent • replace(v,x): make x new element at vertex v • replace(e,x): make x new element at edge e • Update methods • insertVertex(x): create vertex storing elementx • insertEdge(v,w,x): add edge (v,w) with elementx • removeVertex(v): remove v(& incident edges) • removeEdge(e): remove e • Retrieval methods • incidentEdges(v): get edges incident to v

17. Graph ADT • Accessor methods • vertices(): iterable for vertices • edges(): iterable for edges • endVertices(e): array with endpoints of edge e • opposite(v,e):e’sendpoint that is not v • areAdjacent(v,w): check if vandware adjacent • replace(v,x): makexnew element at vertexv • replace(e,x): makexnew element at edgee • Update methods • insertVertex(x): create vertex storing elementx • insertEdge(v,w,x): add edge(v,w)with elementx • removeVertex(v): remove v(& incident edges) • removeEdge(e): remove e • Retrieval methods • incidentEdges(v): get edges incident tov

18. Edge Superclass • Much easier if array used to store endpoints • Need to track index of source &target endpoint • Never hardcode constants into code • What do the numbers 0 or 1 mean to you? • static final fields should instead be used class Edge<E,V> implements Position<E> {private static final int SOURCE = 0; private static final int TARGET = 1;private Eelement;private Vertex<V>[]endPoints;private booleanisDirected;// Add constructor, getters, setters, & element()}

19. Asymptotic Performance

20. Asymptotic Performance

21. Ideal Edge-List Implementation • Great when results not needed or RAM is limited • incidentEdgesrequiresscanning all edges • All edges scanned in removeVertex, also • Anyone here really had OutOfMemoryException? • Optimized for many vertices and few edges • Examining cities with no roads connecting them • Scheduling exams for students taking 1 class • Hermit-based networkssearched for terrorists

22. Ideal Edge-List Implementation • Great when results not needed for a few years • incidentEdgesrequiresscanning all edges • All edges scanned in removeVertex, also • Edge-list is good when memory is limited • How much do you have in your machine? • Optimized for many vertices and few edges • Examining cities with no roads connecting them • Scheduling exams for students taking 1 class • Hermit-based networkssearched for terrorists

23. Real Graph Implementations • Need to consider REALgraphs • Edges outnumber vertices often by a lot • May need multiple edges between vertices • List of incident edges stored in each Vertex • Edges still refer to their endpoints • Why would this be good?

24. Adjacency-List Implementation • Vertex has Sequence of Edges • Edges still refer to Vertex v b a u u w w

25. Adjacency-List Implementation • Vertex has Sequence of Edges • Edges still refer to Vertex • Ideas in Edge-List serve as base v b a u u w w vertices w u v a b edges

26. Adjacency-List Implementation • Vertex has Sequence of Edges • Edges still refer to Vertex • Ideas in Edge-List serve as base • Extends Vertex v b a u u w w vertices w u v a b edges

27. Adjacency-List Implementation • Vertex has Sequence of Edges • Edges still refer to Vertex • Ideas in Edge-List serve as base • Extends Vertex • Could make edgeremoval slower • How to speed up? v b a u u w w vertices w u v a b edges

28. Adjacency-List Vertex • Extend existing Vertexclass • Any code using old classes will continue to work • No need to rewrite all the existing methods • Biggest change is to add field for incident edges classALVertex<V>extendsVertex<V>{ private Sequence<Edge>incidence;// No getter & setter for incidence, but // add methods to add & remove Edges}

29. Adjacency-List Vertex • Extend existing Vertexclass • Any code using old classes will continue to work • No need to rewrite all the existing methods • Biggest change is to add field for incident edges classALVertex<V,E>extendsVertex<V>{ private Sequence<ALEdge<E,V>>incidence;// No getter & setter for incidence, but // add methods to add & remove Edges}

30. Should Edge Class Change? • Ensure that SOURCE & TARGET fields protected • Can be used in subclasses of Edge that we may need • Add references to Positions in incident lists • Not strictly necessary, but can speed some work classALEdge<E,V> extends Edge<E,V> { private Position<ALEdge<E,V>>[]incidentEnd;// incidentEnd[SOURCE] is in source’s incident Sequence // incidentEnd[TARGET] is in target’s incident Sequence }

31. Asymptotic Performance

32. For Next Lecture • No weekly assignment due this week • Req'd for graduation; keep working program #2 • Nearing the end of time – final submission due Wed. • Really good idea to check your JUnit tests work • Another Graph implementation reading up next • Why is it better than adjacency list-based Graph? • Are there any situations where adjacency-list better?