1. Graph Dr. Bernard Chen Ph.D. University of Central Arkansas

2. Graph Algorithms • Graphs and Theorems about Graphs • Graph Algorithms • minimum spanning tree

3. What can graphs model? • Cost of wiring electronic components together. • Shortest route between two cities. • Finding the shortest distance between all pairs of cities in a road atlas.

4. What is a Graph? • Informally a graph is a set of nodes joined by a set of lines or arrows. 1 2 3 1 3 2 4 4 5 6 5 6

5. Directed Graph • A directed graph is a pair ( V, E ), where the set V is a finite set and E is a binary relation on V . • The set V is called the vertex set of G and the elements are called vertices. • The set E is called the edge set of G and the elements are edges

6. Directed Graph Self loop Isolated node 1 3 7 2 4 5 6 V = { 1, 2, 3, 4, 5, 6, 7 }| V | = 7 E = { (1,2), (2,2), (2,4), (4,5), (4,1), (5,4),(6,3) }| E | = 7

7. Undirected Graph • An undirected graph G = ( V, E ) , but unlike a digraph the edge set E consist of unordered pairs. • We use the notation (a,b ) to refer to a directed edge, and { a, b } for an undirected edge.

8. Undirected Graph V = { A, B, C, D, E, F } |V | = 6 A B C E = { {A, B}, {A,E}, {B,E}, {C,F} } |E | = 4 D F E Some texts use (a, b) also for undirected edges. So ( a, b ) and ( b, a ) refers to the same edge.

9. Degree • Degree of a Vertex in an undirected graph is the number of edges incident on it. • In a directed graph , the out degree of a vertex is the number of edges leaving it and the in degree is the number of edges entering it.

10. Degree The degree of B is 2. A B C D F E

11. Degree The in degree of 2 is 2 andthe out degree of 2 is 3. 1 2 4 5

12. Weighted Graph • A weighted graph is a graph for which each edge has an associated weight, usually given by a weight functionw: E R.

13. Weighted Graph 1.2 1 3 2 .2 1.5 .5 .3 4 5 6 .5

14. Implementation of a Graph • Adjacency-list representation of a graph G = ( V, E ) consists of an array ADJ of |V | lists, one for each vertex in V. For each uV , ADJ [ u ] points to all its adjacent vertices.

15. Implementation of a Graph 2 5 1 1 2 2 1 5 3 4 3 3 2 4 5 4 4 2 5 3 5 4 1 2

16. Implementation of a Graph 2 5 1 1 2 2 5 3 4 3 3 4 5 4 4 5 5 5

17. Adjacency lists • Advantage: • Saves space for sparse graphs. Most graphs are sparse. • “Visit” edges that start at v • Must traverse linked list of v • Size of linked list of v is degree(v) • (degree(v))

18. Adjacency-matrix-representation • Adjacency-matrix-representation of a graph G = ( V, E) is a |V | x |V | matrix A = ( aij ) • such that aij = 1 (or some Object) if (i, j ) E 0 (or null) otherwise.

19. Adjacency-matrix-representation 0 1 2 3 4 0 1 2 0 1 0 0 1 0 1 2 3 4 1 0 1 1 1 4 3 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0

20. Adjacency-matrix-representation 0 1 2 3 4 0 1 0 0 1 0 1 0 1 2 3 4 0 0 1 1 1 2 0 0 0 1 0 4 0 0 0 0 1 3 0 0 0 0 0

21. Adjacency-matrix-representation • Advantage: • Saves space on pointers for dense graphs, and on • small unweighted graphs using 1 bit per edge. • Check for existence of an edge (v, u) • (adjacency [i] [j]) == true?) • So (1)

22. Graph Algorithms • Graphs and Theorems about Graphs • Graph Algorithms • minimum spanning tree

23. Minimum Spanning Tree

24. Example of MST

25. Problem: Laying Telephone Wire Central office

26. Wiring: Naïve Approach Central office Expensive!

27. Wiring: Better Approach Central office Minimize the total length of wire connecting the customers

28. Growing an MST: general idea • GENERIC-MST(G,w) • A{} • while A does not form a spanning tree • do find an edge (u,v) that is safe for A • A A U {(u,v)} • return A

29. Tricky part • How do you find a safe edge? • This safe edge is part of the minimum spanning tree

30. Algorithms for MST • Prim’s • Grow a MST by adding a single edge at a time • Kruskal’s • Choose a smallest edge and add it to the forest • If an edge is formed a cycle, it is rejected

31. Prim’s greedy algorithm • Start from some (any) vertex. • Build up spanning tree T, one vertex at a time. • At each step, add to T the lowest-weight edge in G that does not create a cycle. • Stop when all vertices in G are touched

32. Prim’s MST algorithm

33. Example C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

34. = in heap Min EdgePick a root C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

35. Min Edge = 1 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

36. Min Edge = 2 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

37. Min Edge = 2 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

38. Min Edge = 3 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

39. Min Edge = 4 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

40. Min Edge = 3 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

41. Min Edge = 4 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

42. Min Edge = 6 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

43. Example II

44. Kruskal’s Algorithm • Choose the smallest edge and add it to a forest • Keep connecting components until all vertices connected • If an edge would form a cycle, it is rejected.

45. Example C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

46. Min Edge = 1 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

47. Min Edge = 2 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

48. Min Edge = 2 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G

49. Min Edge = 3 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D Now have 2 disjoint components: ABFG and CH 9 8 G

50. Min Edge = 3 C 4 1 B F 2 7 3 3 2 E A I 4 6 7 5 H D 9 8 G