Minimum Spanning Trees

1. Minimum Spanning Trees GHS Algorithm

2. Weighted Graph

3. (MST) Minimum weight spanning tree The sum of the weights is minimized For MST : is minimized

4. Spanning tree fragment: Any sub-tree of a MST

5. Minimum weight outgoing edge (MWOE) The adjacent edge to the fragment with the smallest weight that does not create a cycle

6. Two important properties for building MST The union of a fragment and the MWOE is a fragment Property 1: Property 2: If the weights are unique then the MST is unique

7. The union of a fragment and the MWOE is a fragment Property 1: Proof: Basic idea Examine if the new fragment is part of a MST

8. Fragment MWOE Spanning tree If then is fragment

9. Fragment MWOE Spanning tree If then is fragment

10. Fragment MWOE Spanning tree then add to If and delete

11. Fragment MWOE Spanning tree then add to If and delete

12. Fragment MWOE Spanning tree Since otherwise, wouldn’t be MST

13. Fragment MWOE Spanning tree thus is fragment END OF PROOF

14. Property 2: If the weights are unique then the MST is unique Proof: Basic Idea: Suppose there are two MST Then there is another MST of smaller weight Contradiction!

15. Suppose there are two MST

16. Take the smallest weight edge not in intersection

17. Cycle in RED MST

18. Cycle in RED MST Not in BLUE MST (since blue tree is acyclic)

19. Cycle in RED MST Since is not in intersection, (the weight of is the smallest)

20. Cycle in RED MST Delete and add in RED MST We obtain a new tree with smaller weight Contradiction! END OF PROOF

21. Prim’s Algorithm Start with a node as an initial fragment Repeat Augment fragment with the MWOE Until no other edge can be added to (Assume unique IDs)

22. Fragment

23. Fragment MWOE

24. Fragment MWOE

25. Fragment MWOE

26. Fragment

27. Theorem: Prim’s algorithm gives an MST Proof: Use Property 1 repeatedly END OF PROOF

28. Kruskal’s Algorithm Initially, each node is a fragment Repeat Find the smallest MWOE of all fragments Merge the two fragments adjacent to Until there is one fragment (Assume unique IDs)

29. Initially, every node is a fragment

30. Find the smallest MWOE

31. Merge the two fragments

32. Find the smallest MWOE

33. Merge the two fragments

34. Resulting MST

35. Theorem: Kruskal’s algorithm gives an MST Proof: Use Properties 1 and 2 repeatedly Property 2 guarantees that the merged trees are fragments END OF PROOF

36. GHS Algorithm Distributed version of Kruskal’s Algorithm Initially, each node is a fragment (A Synchronous Phase) Repeat in parallel: Each fragment finds its MWOE Merge fragments adjacent to MWOE’s Until there is one fragment

37. Phase 0: Initially, every node is a fragment Every node is a root of a fragment

38. Phase 1: Find the MWOE for each fragment

39. Phase 1: Merge the fragments Root Root Root Root symmetric MWOE Asymmetric MWOE The new root is adjacent to a symmetric MWOE

40. Phase 1: New fragments

41. Phase 2: Find the MWOE for each fragment

42. Phase 2: Merge the fragments Root Root

43. Phase 2: New fragments

44. Phase 3: Find the MWOE for each fragment

45. Phase 3: Merge the fragments Root

46. Phase 3: New fragment FINAL MST

47. Rules for selecting a Root in fragment Fragment 2 Fragment 1 root root MWOE

48. Rules for selecting a Root in fragment Merged Fragment root Higher ID Node on MWOE

49. Rules for selecting a Root in fragment root root root root root root root Merging more than 2 fragments

50. Rules for selecting a Root in fragment Merged Fragment Root Higher ID Node on symmetric MWOE asymmetric