1. Minimum Spanning Tree Dr. Bernard Chen Ph.D. University of Central Arkansas Fall 2008

2. Minimum Spanning Tree

3. Example of MST

4. Problem: Laying Telephone Wire Central office

5. Wiring: Naïve Approach Central office Expensive!

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

7. Growing an MST: general idea GENERIC-MST(G,w) 1. A  {} 2. while A does not form a spanning tree 3. do find an edge (u,v) that is safe for A 4. A  A U {(u,v)} 5. return A

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

9. 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

10. 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

11. Prim’s MST algorithm

12. Example A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

13. Min Edge Pick a root A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7 = in heap

14. Min Edge = 1 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

15. Min Edge = 2 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

16. A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

17. Min Edge = 3 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

18. Min Edge = 4 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

19. Min Edge = 3 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

20. Min Edge = 4 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

21. Min Edge = 6 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

22. Example II

23. 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.

24. Example A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

25. Min Edge = 1 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

26. Min Edge = 2 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

27. A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

28. Min Edge = 3 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7 Now have 2 disjoint components: ABFG and CH

29. Min Edge = 3 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

30. Min Edge = 4 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7 Two components now merged into one.

31. Min Edge = 4 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7

32. Min Edge = 5 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7 Rejected due to a cycle BFGB

33. Min Edge = 6 A B D G C F I E H 23 4 5 7 8 4 3 1 6 9 2 7