1. Minimum Spanning Trees
GHS Algorithm

2. Weighted Graph

3. Minimum weight spanning tree
(MST)
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
Property 1:
The union of a fragment and
the MWOE is a fragment
Property 2:
If the weights are unique
then the MST is unique

7. Property 1:
The union of a fragment and
the MWOE is a fragment
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 If
then add to
and delete

11. Fragment
MWOE
Spanning tree If
then add to
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. Delete and add in RED MST
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. Prim’s algorithm gives an MST
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. Kruskal’s algorithm gives an MST
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
Repeat in parallel:
(A Synchronous Phase)
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
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
Merging more than 2 fragments

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

51. In merged fragments there is exactly one symmetric MWOE
Remark:
In merged fragments there is
exactly one symmetric MWOE
two
zero
Impossible
Impossible
Creates a fragment
with two MWOE
Creates a fragment
with no MWOE

52. The new root broadcasts to the new fragment
is the symmetric MWOE of the merged fragments
The new root broadcasts
to the new fragment
is the identity of the new fragment

53. At the end of a phase each fragment
has its own unique identity.
Root
Root
Root
Root
End of phase 1

54. At the end of a phase each fragment
has its own unique identity.
Root
Root
End of phase 2

55. At the beginning of each phase each node in fragment finds its MWOE

56. Then each node reports its MWOE to the fragment root with convergecast
(the global minimum survives in propagation)
MWOE
MWOE
MWOE
MWOE

57. The root selects the minimum MWOE

58. To discover its own MWOE, each node
broadcasts its identity to neighbors

59. Then it knows which edges are outgoing,
And selects the MWOE among them
outgoing
MWOE
outgoing

60. Complexity
Smallest Fragment size
(#nodes)
Phase

61. Maximum possible fragment size
Number of
nodes
Maximum phase:

62. Time to convergecast MWOE to root:
Time of a phase:
Time to convergecast MWOE to root:
(maximum fragment size is )
Time to connect new fragments:
(Each fragment sends one message on its MWOE)
Time of root to broadcast identity:
Total phase time:

63. Algorithm Time
Total time = Phase time X #phases =
Lower bound:

64. Messages for nodes to find MWOE:
Messages in a phase:
Messages for nodes to find MWOE:
(on each edge 2 messages)
Messages to convergecast to root:
(maximum fragment size is )
Messages to connect new fragments:
(Each fragment sends one message on its MWOE)
Messages to broadcast identity:
Total phase messages:

65. Algorithm messages
Total messages = Phase messages X #phases
Can be improved to

66. Asynchronous Version of GHS Algorithm
Simulates the synchronous version
Every fragment has a level

67. Fragment
Fragment
MWOE
If then merges to
(cost of merging proportional to )

68. The combined level is
New fragment
MWOE

69. Fragment
Fragment
MWOE
If then merges with
(cost of merging proportional to )

70. The combined level is
New fragment
MWOE

71. (cost of merging would be proportional to
for every small fragment, inefficient!!)
Fragment
Fragment
MWOE
If then waits until previous
Rules apply