Minimum Spanning Trees

1. Minimum Spanning Trees Gallagher-Humblet-Spira (GHS) Algorithm

2. Weighted Graph G=(V,E), |V|=n, |E|=m

3. Spanning tree Any tree T=(V,E’) (connected acyclic graph) spanning all the nodes of G

4. (MST) Minimum-weight spanning tree A spanning tree s.t. the sum of its weights is minimized: For MST : is minimized

5. Spanning tree fragment: Any (connected) sub-tree of a MST

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

7. Two important properties for building MST The union of a fragment and any of its MWOE is a fragment of some MST (so called blue rule). Property 1: Property 2: If the weights are distinct then the MST is unique

8. The union of a fragment FT and any of its MWOE is a fragment of some MST. Property 1: Proof: • Distinguish two cases: • the MWOE belongs to T • the MWOE does not belong to T • In both cases, we can prove the claim.

9. Case 1: Fragment MWOE MST T

10. Trivially, if then is a fragment Fragment MWOE MST T

11. Case 2: Fragment MWOE MST T

12. If then add to and remove Fragment MST T

13. Fragment Obtain T’ and since But w(T’)  w(T), since T is an MST  w(T’)=w(T), i.e., T’ is an MST

14. Fragment MST T’ thus is a fragment of T’ END OF PROOF

15. Property 2: If the weights are distinct then the MST is unique Proof: Basic Idea: Suppose there are two MSTs Then there is another MST of smaller weight  contradiction!

16. Suppose there are two MSTs

17. Take the smallest weight edge not in intersection

18. Cycle in RED MST

19. Cycle in RED MST e’: any red edge not in BLUE MST ( since blue tree is acyclic)

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

21. Cycle in RED MST Delete and add in RED MST  we obtain a new tree with smaller weight  contradiction! END OF PROOF

22. Prim’s Algorithm (sequential version) Start with a node as an initial fragment Repeat Augment fragment with a MWOE Until no other edge can be added to

23. Fragment

24. Fragment MWOE

25. Fragment MWOE

26. Fragment MWOE

27. Fragment

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

29. Prim’s algorithm (distributed version) • Works by repeatedly applying the blue rule to a single fragment, to yield the MST for G • Works with both asynchronousand synchronous non-anonymous, uniform models (and also with non-distinct weights) • Algorithm (asynchronous high-level version): • Let vertex r be the root as well as the first fragment • REPEAT • r broadcasts a message on the current fragment to search for the MWOE of the fragment (each vertex in the fragment searches for its local (i.e., adjacent) MWOE) • convergecast (i.e., reverse broadcast towards r) the MWOE of the appended subfragment (i.e., the minimum of the local MWOEs of itself and its descendents) • the MWOE of the fragment is then selected by r and added to the fragment, by sending an add-edge message on the appropriate path • then, the root and nodes adjacent to that just entered in the fragment are notified the edge has been added • UNTIL there is only one fragment left

30. Local description of asynchronous Prim • Each processor stores: • The state of any of its incident edges, which can be either of {basic, branch, reject} • Its own state, which can be either {in, out} • Local MWOE • MWOE for each branch-down edge • Parent channel (route towards the root) • MWOE channel (route towards the MWOE of its appended subfragment)

31. Type of messages in asynchronous Prim • Search MWOE: coordination message initiated by the root • Test: check the status of a basic edge • Reject, Accept: response to Test • Report(weight): report to the parent node the MWOE of the appended subfragment • Add edge: say to the fragment node adjacent to the fragment’s MWOE to add it • Connect: perform the union of the found MWOE to the fragment (this changes the status of the corresponding end-node from out to in) • Connected: notify the root and adjacent nodes that connection has taken place Message complexity = O(n2)

32. Synchronous Prim • It will work in O(n2) rounds…think to it by yourself…

33. Kruskal’s Algorithm (sequential version) Initially, each node is a fragment Repeat • Find the smallest MWOE e of all fragments • Merge the two fragments adjacent to e Until there is only one fragment left

34. Initially, every node is a fragment

35. Find the smallest MWOE

36. Merge the two fragments

37. Find the smallest MWOE

38. Merge the two fragments

39. Merge the two fragments

40. Resulting MST

41. Theorem: Kruskal’s algorithm gives an MST Proof: Use Property 1, and observe that no cycle is created. END OF PROOF

42. Synchronous GHS Algorithm • Distributed version of Kruskal’s Algorithm • Works by repeatedly applying the blue rule to multiple fragments • Works with non-uniform models, distinct weights, synchronous start Initially, each node is a fragment (Synchronous Phase) Repeat in parallel: • Each fragment – coordinated by a fragment rootnode - finds its MWOE • Merge fragments adjacent to MWOE’s Until there is only one fragment left

43. Local description of syncr. GHS • Each processor stores: • The state of any of its incident edges, which can be either of {basic, branch, reject} • Identity of its fragment (the weigth of a core edge – for single-node fragments, the proc. id ) • Local MWOE • MWOE for each branching-out edge • Parent channel (route towards the root) • MWOE channel (route towards the MWOE of its appended subfragment)

44. Type of messages • New fragment(identity): coordination message sent by the root at the end of a phase • Test(identity): for checking the status of a basic edge • Reject, Accept: response to Test • Report(weight): for reporting to the parent node the MWOE of the appended subfragment • Merge: sent by the root to the node incident to the MWOE to activate union of fragments • Connect(My Id): sent by the node incident to the MWOE to perform the union

45. Phase 0: Initially, every node is a fragment… … and every node is a root of a fragment

46. Phase 1: Find the MWOE for each node

47. Phase 1: Merge the nodes and select a new root Root Root Root Root Symmetric MWOE Asymmetric MWOE The new root is adjacent to a symmetric MWOE

48. Rule for selecting a new root in a fragment Fragment 2 Fragment 1 root root MWOE Merging 2 fragments

49. Rule for selecting a new root in a fragment Merged Fragment root Higher ID Node on MWOE (non-anonymity)

50. Rule for selecting a new root in a fragment root root root root root root root Merging more than 2 fragments