1 minimum spanning trees ghs algorithm. 2 weighted graph
TRANSCRIPT
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Minimum Spanning Trees
GHS Algorithm
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Weighted Graph
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Minimum weight spanning tree
The sum of the weights is minimized
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(MST)
For MST :T
Te
ewTw )()( is minimized
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Spanning tree fragment:
Any sub-tree of a MST
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Minimum weight outgoing edge(MWOE)
The adjacent edge to the fragment with the smallest weight that does not create a cycle
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Property 1:The union of a fragment and the MWOE is a fragment
Property 2:If the weights are uniquethen the MST is unique
Two important properties for building MST
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Property 1:The union of a fragment and the MWOE is a fragment
Proof: Basic idea
Examine if the new fragmentis part of a MST
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e)()( xwew
MWOE
Fragment
Spanning tree
If then is fragment
F
T
Te }{eF
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eMWOE
Fragment
Spanning tree
If then is fragment
}{eFF
T
Te }{eF
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ex )()( xwew
MWOE
Fragment
F
TSpanning tree
If Te then add to e Tand delete x
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ex )()( xwew
MWOE
Fragment
F
TSpanning tree
If Te then add to e Tand delete x
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ex )()( xwew
MWOE
Fragment
F
T Spanning tree
)()( TwTw Since otherwise,Twouldn’t be MST
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e)()( xwew
MWOE
Fragment
Spanning tree
thus is fragment
}{eFF
T
}{eF END OF PROOF
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Property 2:If the weights are uniquethen the MST is unique
Proof: Basic Idea:
Suppose there are two MST
Then there is another MST of smaller weight
Contradiction!
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Suppose there are two MST
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Take the smallest weight edge not in intersection
e
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e
Cycle in RED MST
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e
Cycle in RED MST
e
Not in BLUE MST
(since blue tree is acyclic)
)()( ewew
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e
Cycle in RED MST
e
)()( ewew Since is not in intersection,e
(the weight of is the smallest)e
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e
Cycle in RED MST
e
)()( ewew
Delete and add in RED MST e e
We obtain a new tree with smaller weight
Contradiction! END OF PROOF
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Prim’s Algorithm
Start with a node as an initial fragment
Augment fragment with the MWOE
Repeat
F
F
Until no other edge can be added to F
(Assume unique IDs)
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Fragment F
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Fragment F
MWOE
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Fragment F
MWOE
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Fragment F
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MWOE
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Fragment F
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Theorem: Prim’s algorithm gives an MST
Proof: Use Property 1 repeatedly
END OF PROOF
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Kruskal’s Algorithm
Initially, each node is a fragment
Find the smallest MWOE of all fragments
Merge the two fragments adjacent to
e
e
Repeat
Until there is one fragment
(Assume unique IDs)
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Initially, every node is a fragment
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Find the smallest MWOE
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Merge the two fragments
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Find the smallest MWOE
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Merge the two fragments
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Resulting MST
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Theorem: Kruskal’s algorithm gives an MST
Proof: Use Properties 1 and 2 repeatedly
END OF PROOF
Property 2 guarantees that the merged trees are fragments
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GHS Algorithm
Distributed version of Kruskal’s Algorithm
Initially, each node is a fragment
Each fragment finds its MWOE
Merge fragments adjacent to MWOE’s
Repeat in parallel:
Until there is one fragment
(A Synchronous Phase)
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Phase 0: Initially, every node is a fragment
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Every node is a root of a fragment
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Phase 1: Find the MWOE for each fragment
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Phase 1: Merge the fragments
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Root
RootRoot
Root
Asymmetric MWOE
symmetric MWOE
The new root is adjacent to a symmetric MWOE
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Phase 1: New fragments
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Phase 2: Find the MWOE for each fragment
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Phase 2: Merge the fragments
Root
Root
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Phase 2: New fragments
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Phase 3: Find the MWOE for each fragment
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Phase 3: Merge the fragments
Root
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Phase 3: New fragment
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FINAL MST
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Rules for selecting a Root in fragment
Fragment 1Fragment 2
root
rootMWOE
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Merged Fragment
root
Higher ID Node on MWOE
Rules for selecting a Root in fragment
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Rules for selecting a Root in fragment
Merging more than 2 fragments
1F2F
3F
5F
4F 6F
7Frootroot
root
root
root root
root
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Rules for selecting a Root in fragment
Higher ID Node on symmetric MWOE
Merged Fragment
Root
asymmetric
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In merged fragments there is exactly one symmetric MWOE
Remark:
Impossible Impossible
1F2F
4F3F
5F
6F
Creates a fragment with two MWOE
zero two
Creates a fragmentwith no MWOE
1F2F
4F3F
5F
6F
7F
8F
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The new root broadcasts to the new fragment
e
)(ew
e is the symmetric MWOE of the merged fragments
)(ew
)(ew)(ew
)(ew )(ew)(ew
)(ew
)(ew is the identity of the new fragment
)(ew
)(ew
)(ew )(ew
)(ew)(ew)(ew
)(ew
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At the end of a phase each fragmenthas its own unique identity.
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Root Root
Root
Root
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End of phase 1
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At the end of a phase each fragmenthas its own unique identity.
End of phase 2
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At the beginning of each phase each node in fragment finds its MWOE
MWOEMWOE
MWOE
MWOE
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Then each node reports its MWOEto the fragment root with convergecast (the global minimum survives in propagation)
MWOEMWOE
MWOE
MWOE
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The root selects the minimum MWOE
MWOE
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To discover its own MWOE, each node broadcasts its identity to neighbors
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Then it knows which edges are outgoing,And selects the MWOE among them
outgoing
outgoing
MWOE
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Smallest Fragment size (#nodes)Phase
0 1
1 2
3 4
i i2
Complexity
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Maximum phase: ni 2log
Maximum possible fragment size ni 2
Number ofnodes
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Time to convergecast MWOE to root:
)(nO
)(nO
Time to connect new fragments: )1(O(Each fragment sends one message on its MWOE)
Time of root to broadcast identity:
Time of a phase:
(maximum fragment size is ))(nO
Total phase time: )(nO
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Total time = Phase time X #phases =nlog)(nO
)log( nnO
Lower bound:
nn
log
Algorithm Time
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Messages to broadcast identity:
Messages to convergecast to root:
)(nO
)(nO
Messages to connect new fragments: )(nO(Each fragment sends one message on its MWOE)
Messages in a phase:
(maximum fragment size is ))(nO
Total phase messages: |)|( EnO
Messages for nodes to find MWOE: |)(|EO(on each edge 2 messages)
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Total messages = Phase messages X #phasesnlog|)|( EnO
)log|)|(( nEnO
Can be improved to
|)|log( EnnO
Algorithm messages
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Asynchronous Version of GHS Algorithm
Simulates the synchronous version
Every fragment has a levelF )(FL
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FragmentFragment
1F 2FMWOE
If then merges to )()( 21 FLFL 1F 2F
(cost of merging proportional to )1F
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New fragment
1F 2FMWOE
)()( 21 FLFL
The combined level is )()( 2FLFL
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MWOE
FragmentFragment
1F 2F
If then merges with )()( 21 FLFL 1F 2F
(cost of merging proportional to )1F
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New fragment
1F 2FMWOE
The combined level is 1)()( 2 FLFL
)()( 21 FLFL
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Fragment Fragment
1F 2FMWOE
If then waits until previousRules apply
)()( 21 FLFL 1F
(cost of merging would be proportional to for every small fragment, inefficient!!)
1F