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Minimum Spanning Trees. Gallagher-Humblet-Spira (GHS) Algorithm. Weighted Graph G=(V,E), |V|=n, |E|=m. Spanning tree. Any tree T=(V,E’) (connected acyclic graph) spanning all the nodes of G. (MST). Minimum-weight spanning tree. A spanning tree s.t. the sum of its weights is minimized:. - PowerPoint PPT Presentation
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Minimum Spanning Trees
Gallagher-Humblet-Spira (GHS) Algorithm
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Weighted Graph G=(V,E), |V|=n, |E|=m
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Spanning tree
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Any tree T=(V,E’) (connected acyclic graph) spanning all the nodes of G
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Minimum-weight spanning tree
A spanning tree s.t. the sum of its weights is minimized:
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(MST)
For MST :T
Te
ewTw )()( is minimized
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Spanning tree fragment:
Any (connected) sub-tree of a MST
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Minimum weight outgoing edge(MWOE)
An edge adjacent to the fragment with smallest weight and that does not create a cycle
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Property 1: The union of a fragment and any of its MWOE is a fragment of some MST (so called blue rule).
Property 2: If the weights are distinctthen the MST is unique
Two important properties for building MST
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Property 1: The union of a fragment FT and any of its MWOE is a fragment of some MST.
Proof: Distinguish two cases:1. the MWOE belongs to T2. the MWOE does not belong to TIn both cases, we can prove the
claim.
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eMWOE
Fragment
MST T
F
Case 1: Te
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eMWOE
Fragment}{eFF
MST T
Trivially, if then is a fragment Te }{eF
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ex )()( xwew
MWOE
Fragment
F
MST T
Case 2: Te
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ex )()( xwew
Fragment
F
MST T
If Te then add to e T and removex
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ex)()( xwew
Fragment
F
But w(T’) w(T), since T is an MST
w(T’)=w(T), i.e., T’ is an MST
Obtain T’
w(T))w(T'
and since
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e
Fragment
thus is a fragment of T’
}{eFF
}{eF END OF PROOF
MST T’
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Property 2: If the weights are distinctthen the MST is unique
Proof: Basic Idea:
Suppose there are two MSTs
Then there is another MST of smaller weight
contradiction!
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Suppose there are two MSTs
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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
e’: any red edge not in BLUE MST
( since blue tree is acyclic)
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e
Cycle in RED MST
e
)()( ewew Since is not in the 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 (sequential version)
Start with a node as an initial fragment
Augment fragment with a MWOE
Repeat
F
F
Until no other edge can be added to F
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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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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 asynchronous and 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 fragmentREPEAT
• 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
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Local description of asynchronous Prim
Each processor stores:1. The state of any of its incident edges,
which can be either of {basic, branch, reject}
2. Its own state, which can be either {in, out}
3. Local MWOE 4. MWOE for each branch-down edge5. Parent channel (route towards the root)6. MWOE channel (route towards the
MWOE of its appended subfragment)
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Type of messages in asynchronous Prim1. Search MWOE: coordination message initiated by
the root2. Test: check the status of a basic edge3. Reject, Accept: response to Test4. Report(weight): report to the parent node the
MWOE of the appended subfragment5. Add edge: say to the fragment node adjacent to
the fragment’s MWOE to add it6. Connect: perform the union of the found MWOE to
the fragment (this changes the status of the corresponding end-node from out to in)
7. Connected: notify the root and adjacent nodes that connection has taken place
Message complexity = O(n2)
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Synchronous Prim
It will work in O(n2) rounds…think to it by yourself…
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Kruskal’s Algorithm (sequential version)
Initially, each node is a fragment
• Find the smallest MWOE e of all fragments• Merge the two fragments adjacent to e
Repeat
Until there is only one fragment left
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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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Merge the two fragments
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Resulting MST
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Theorem: Kruskal’s algorithm gives an MST
Proof: Use Property 1, and observe that no cycle is created.
END OF PROOF
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Synchronous GHS AlgorithmDistributed 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
•Each fragment – coordinated by a fragment root node - finds its MWOE
•Merge fragments adjacent to MWOE’s
Repeat in parallel:
Until there is only one fragment left
(Synchronous Phase)
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Local description of syncr. GHS
Each processor stores:1. The state of any of its incident edges, which
can be either of {basic, branch, reject} 2. Identity of its fragment (the weigth of a core
edge – for single-node fragments, the proc. id )
3. Local MWOE4. MWOE for each branching-out edge5. Parent channel (route towards the root)6. MWOE channel (route towards the MWOE of
its appended subfragment)
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Type of messages
1. New fragment(identity): coordination message sent by the root at the end of a phase
2. Test(identity): for checking the status of a basic edge
3. Reject, Accept: response to Test4. Report(weight): for reporting to the parent
node the MWOE of the appended subfragment
5. Merge: sent by the root to the node incident to the MWOE to activate union of fragments
6. Connect(My Id): sent by the node incident to the MWOE to perform the union
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Phase 0: Initially, every node is a fragment…
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… and every node is a root of a fragment
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Phase 1: Find the MWOE for each node
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Phase 1: Merge the nodes and select a new root
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Root
RootRoot
Root
Asymmetric MWOE
Symmetric MWOE
The new root is adjacent to a symmetric MWOE
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Rule for selecting a new root in a fragment
Fragment 1Fragment 2
root
rootMWOE
Merging 2 fragments
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Merged Fragment
root
Higher ID Node on MWOE(non-anonymity)
Rule for selecting a new root in a fragment
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Rule for selecting a new root in a fragment
Merging more than 2 fragments
1F2F
3F
5F
4F 6F
7Frootroot
root
root
root root
root
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Rule for selecting a new root in a fragment
Higher ID Node on symmetric MWOE
Merged Fragment
Root
asymmetric
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In merged fragments there is exactly one symmetric MWOE (n-1 edges vs n arrows)
Remark:
Impossible Impossible
1F2F
4F3F
5F
6F
Creates a fragment with two MWOEs
zero two
Creates a fragmentwith no MWOE
1F2F
4F3F
5F
6F
7F
8F
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After merging has taken place, the new root broadcasts New fragment(w(e)) to the new fragment, and afterwards a new phase starts
ee 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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In our example, at the end of phase 1 each fragment has its new identity.
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Root Root
Root
Root
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End of phase 1
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At the beginning of each new phase each node in fragment finds its MWOE
MWOEMWOE
MWOE
MWOE
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To discover its own MWOE, each node sends a Test message containing its identity over its basic edge of min weight, until it receives an Accept
test()accept
test()
6reject
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10
3
Then it knows its local MWOE
MWOE
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Then each node sends a Report with the MWOE of the appended subfragment to the root with convergecast (the global minimum survives in propagation)
MWOEMWOE
MWOE
MWOE
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The root selects the minimum MWOE and sends along the appropriate path a Merge message, which will become a Connect message at the proper node
MWOE
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Phase 2 of our example: After receiving the new identity, find again the MWOE for each fragment
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Phase 2: Merge the fragments
Root
Root
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At the end of phase 2 each fragmenthas its own unique identity.
End of phase 2
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Root7
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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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Correctness• To guarantee correctness, phases must be syncronized• But at the beginning of a phase, each fragment can have a
different number of nodes, and thus the MWOE selection is potentially asynchronous…
• But each fragment can have at most n nodes, has height at most n-1, and each node has at most n-1 incident edges…
• So, the MWOE selection requires at most 3n rounds, and the Merge message requires at most n rounds.
• Then, the Connect message must be sent at round 4n+1 of a phase, and so at round 4n+2 a node knows whether it is a new root
• Finally, the New fragment message will require at most n rounds.
A fixed number of 5n+2 total rounds can be used to complete each phase (in some rounds nodes do nothing…)!
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Smallest Fragment size (#nodes)Phase
0 1
1 2
2 4
i i2
Complexity
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Maximum # phases: ni 2log
Maximum possible fragment size ni 2
Number of nodes
Algorithm Time Complexity
Total time = Phase time • #phases =nlog)(nO
)log( nnO
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Thr: Synchronous GHS requires O(m+n logn) msgs. Proof: We have the following messages:
1. Test-Reject msgs: at most 2 for each edge;
2. Each node sends/receives at most a single: New Fragment, Test-Accept, Report, Merge, Connect message for each phase.
Since from previous lemma we have at most log n phases, the claim follows.
Algorithm Message Complexity
END OF PROOF
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Asynchronous Version of GHS Algorithm
•Simulates the synchronous version•Works with uniform models, distinct weights, asynchronous start
•Every fragment F has a level L(F)≥0: at the beginning, each node is a fragment of level 0
•Two types of merges: absorption and join
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Local description of asyncr. GHS
Like the synchronous, but:1. Identity of a fragment is now given by an
edge weight plus the level of the fragment;2. A node has its own status, which can be
either of {sleeping, finding, found}
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Type of messages
Like the synchronous, but now:
1. New fragment(weight,level,status): coordination message sent just after a merge
2. Test(weight,level): to test an edge3. Connect(weight,level): to perform
the union
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FragmentFragment
1F 2FMWOE
If then is absorbed by)()( 21 FLFL 1F2F
(cost of merging proportional to )1F
Absorption
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New fragment
1F 2FMWOE
The combined level is )()( 2FLFL
and a New fragment(weight,level,status) message is broadcasted to nodes of F1 by the node of F2 on which the merge took place
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MWOE
FragmentFragment
1F 2F
If and F1 and F2 have a symmetric MWOE, then F1 joins with F2
)()( 21 FLFL
(cost of merging proportional to ) 1F 2F
Join
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New fragment
1F 2FMWOE
The combined level is 1)()( 2,1 FLFL
and a New fragment(weight,L(F2)+1,finding) message is broadcasted to all nodes of F1 and F2, where weigth is that of the edge on which the merge took place
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Fragment Fragment
1F2FTest
If L(F1)>L(F2), then F1 waits until L(F1)= L(F2) (this is obtained by letting F2 not replying to Test messages from F1 )
Remark: a joining requires that fragment levels are equal…how to control this?
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Lemma: A fragment of level L contains at least 2L nodes.
Proof: By induction. For L=0 is trivial. Assume it is true up to L=k-1, and let F be of level k. But then, either:
1. F was obtained by joining two fragments of level k-1, each containing at least 2k-1 nodes by inductive hypothesis F contains at least 2k-1 + 2k-1 = 2k nodes;
2. F was obtained after absorbing another fragment F’ of level<k apply recursively to F\F’, until case (1) applies.
Algorithm Message Complexity
END OF PROOF
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Thr: Asynchronous GHS requires O(m+n logn) msgs. Proof: We have the following messages:
1. Connect msgs: at most 2 for each edge;2. Test-Reject msgs: at most 2 for each
edge;3. Each node sends/receives at most a
single: New Fragment, Test-Accept, Merge, Report message each time the level of its fragment increases;
and since from previous lemma each node can change at most log n levels, the claim follows.
Algorithm Message Complexity (2)
END OF PROOF
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HomeworkExecute asynchronous GHS on the following graph:
assuming that system is pseudosynchronous: Start from 1 and 5, and messages sent from odd (resp., even) nodes are received after 1 (resp., 2) round(s)
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