Graphs & Networks

1. Graphs & Networks Briana B. Morrison Adapted from Alan Eugenio, and William J. Collins

2. Topics • Definitions • Storage • Matrix • Adjacency sets • Algorithms • Breadth first traversal • Depth first traversal • Spanning trees • Shortest path • Implementation Graphs

3. A graph is a pair (V, E), where V is a set of nodes, called vertices E is a collection of pairs of vertices, called edges Vertices and edges are positions and store elements Example: A vertex represents an airport and stores the three-letter airport code An edge represents a flight route between two airports and stores the mileage of the route Graph 849 PVD 1843 ORD 142 SFO 802 LGA 1743 337 1387 HNL 2555 1099 1233 LAX 1120 DFW MIA Graphs

4. V = { S, T, X, Y, Z} E = { (X, Y), (X, Z), (Y, T), (T, Z), (T, S), (S, Z)} Graphs

5. EXAMPLE: X Y Z T 5 VERTICES 6 EDGES S Graphs

6. Edge Types • Directed edge • ordered pair of vertices (u,v) • first vertex u is the origin • second vertex v is the destination • e.g., a flight • Undirected edge • unordered pair of vertices (u,v) • e.g., a flight route • Directed graph • all the edges are directed • e.g., route network • Undirected graph • all the edges are undirected • e.g., flight network flight AA 1206 ORD PVD 849 miles ORD PVD Graphs

7. Applications • Electronic circuits • Printed circuit board • Integrated circuit • Transportation networks • Highway network • Flight network • Computer networks • Local area network • Internet • Web • Databases • Entity-relationship diagram Graphs

8. V a b h j U d X Z c e i W g f Y Terminology • End vertices (or endpoints) of an edge • U and V are the endpoints of a • Edges incident on a vertex • a, d, and b are incident on V • Adjacent vertices • U and V are adjacent • Degree of a vertex • X has degree 5 • Parallel edges • h and i are parallel edges • Self-loop • j is a self-loop Graphs

9. Graphs

10. Terminology (cont.) • Path • sequence of alternating vertices and edges • begins with a vertex • ends with a vertex • each edge is preceded and followed by its endpoints • Simple path • path such that all its vertices and edges are distinct • Examples • P1=(V,b,X,h,Z) is a simple path • P2=(U,c,W,e,X,g,Y,f,W,d,V) is a path that is not simple V b a P1 d U X Z P2 h c e W g f Y Graphs

11. Terminology (cont.) • Cycle • circular sequence of alternating vertices and edges • each edge is preceded and followed by its endpoints • Simple cycle • cycle such that all its vertices and edges are distinct • Examples • C1=(V,b,X,g,Y,f,W,c,U,a,) is a simple cycle • C2=(U,c,W,e,X,g,Y,f,W,d,V,a,) is a cycle that is not simple V a b d U X Z C2 h e C1 c W g f Y Graphs

12. Graphs

13. Collection List Set ArrayList LinkedList SortedSet HashSet TreeSet Graphs

14. Graphs

15. X Y Z X, Y, T, Z, S HAS LENGTH 4. T SHORTEST PATH FROM X to S = ? S IN GENERAL, HOW CAN THE SMALLEST PATH BETWEEN 2 VERTICES BE DETERMINED? Graphs

16. Graphs

17. X Y Z X, Y, T, Z, X IS A CYCLE. T Y, T, S, T, Y IS NOT A CYCLE. IS Y, Z, T, S, Z, X, Y A CYCLE? S Graphs

18. Collection List Set ArrayList LinkedList SortedSet HashSet TreeSet THIS UNDIRECTED GRAPH IS ACYCLIC. Graphs

19. Graphs

20. Graphs

21. Graph Categories • A graph is connected if each pair of vertices have a path between them • A complete graph is a connected graph in which each pair of vertices are linked by an edge Graphs

22. E D C B A Digraphs • A digraph is a graph whose edges are all directed • Short for “directed graph” • Applications • one-way streets • flights • task scheduling Graphs

23. Graphs

24. Graphs

25. Example of Digraph Graphs

26. Digraph Application • Scheduling: edge (a,b) means task a must be completed before b can be started ics21 ics22 ics23 ics51 ics53 ics52 ics161 ics131 ics141 ics121 ics171 The good life ics151 Graphs

27. Graphs

28. Graphs

29. Graphs

30. Connectedness of Digraph • Strongly connected if there is a path from any • vertex to any other vertex. (no dead ends) • Weakly connected if, for each pair of vertices vi and vj, there is either a path P(vi, vj) or a path P(vi,vj). Graphs

31. IS THE FOLLOWING GRAPH CONNECTED? Graphs

32. Graphs

33. Graphs

34. Collection List Set ArrayList LinkedList SortedSet HashSet TreeSet Graphs

35. Graphs

36. Graphs

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38. Graphs

39. Graphs

40. Graphs

41. Graph Storage Two ways to store a graph: • Adjacency Matrix • Adjacency Set / List Graphs

42. Adjacency Matrix • An m by m matrix, called an adjacency matrix, identifies the edges. An entry in row i and column j corresponds to the edge e = (v,, vj). Its value is the weight of the edge, or -1 if the edge does not exist. Graphs

43. Adjacency Matrix (cont.) • Note that for an undirected graph, the matrix is symmetrical about the diagonal line. X Z Y T S Graphs

44. Adjacency Sets Graphs

45. Vertices and edges are positions store elements Accessor methods aVertex() incidentEdges(v) endVertices(e) isDirected(e) origin(e) destination(e) opposite(v, e) areAdjacent(v, w) Update methods insertVertex(o) insertEdge(v, w, o) insertDirectedEdge(v, w, o) removeVertex(v) removeEdge(e) Generic methods numVertices() numEdges() vertices() edges() Main Methods of the Graph ADT Graphs

46. Asymptotic Performance Graphs

47. Animation • Implementation Animation Graphs

48. Graphs

49. Graphs

50. Breadth-first search (BFS) is a general technique for traversing a graph A BFS traversal of a graph G Visits all the vertices and edges of G Determines whether G is connected Computes the connected components of G Computes a spanning forest of G BFS on a graph with n vertices and m edges takes O(n + m ) time BFS can be further extended to solve other graph problems Find and report a path with the minimum number of edges between two given vertices Find a simple cycle, if there is one Breadth-First Search Graphs