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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 Any tree T=(V,Eā) (connected acyclic graph) spanning all the nodes of G

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**Minimum-weight spanning tree**

(MST) A spanning tree s.t. the sum of its weights is minimized: For MST : 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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**Two important properties for building MST**

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 distinct then the MST is unique

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

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Case 1: Fragment MWOE MST T

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**Trivially, if then is a fragment**

MWOE MST T

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Case 2: Fragment MWOE MST T

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If then add to and remove Fragment MST T

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Fragment Obtain Tā and since But w(Tā) ļ³ w(T), since T is an MST ļ w(Tā)=w(T), i.e., Tā is an MST

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Fragment MST Tā thus is a fragment of Tā END OF PROOF

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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 ļ contradiction!

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**Suppose there are two MSTs**

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**Take the smallest weight edge**

not in intersection

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Cycle in RED MST

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Cycle in RED MST eā: any red edge not in BLUE MST (ļ¤ since blue tree is acyclic)

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

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**Delete and add in RED MST**

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

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**Primās 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

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Fragment

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Fragment MWOE

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Fragment MWOE

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Fragment MWOE

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Fragment

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**Primās algorithm gives an MST**

Theorem: Primās algorithm gives an MST Proof: Use Property 1 repeatedly END OF PROOF

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**Primās 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 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

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**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)

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**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 fragmentās 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)

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Synchronous Prim It will work in O(n2) roundsā¦think to it by yourselfā¦

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**Kruskalās 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

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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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**Kruskalās algorithm gives an MST**

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

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**Synchronous GHS Algorithm**

Distributed version of Kruskalās 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 Repeat in parallel: (Synchronous Phase) Each fragment ā coordinated by a fragment root node - finds its MWOE Merge fragments adjacent to MWOEās Until there is only one fragment left

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**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 ā 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)

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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

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**Phase 0: Initially, every node is a fragmentā¦**

ā¦ 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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**Root Root Root Root Phase 1: Merge the nodes and select a new root**

Symmetric MWOE Asymmetric MWOE The new root is adjacent to a symmetric MWOE

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**Rule for selecting a new root in a fragment**

MWOE Merging 2 fragments

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**Rule for selecting a new root in a fragment**

Merged Fragment root Higher ID Node on MWOE (non-anonymity)

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**Rule for selecting a new root in a fragment**

Merging more than 2 fragments

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**Rule for selecting a new root in a fragment**

Merged Fragment Root Higher ID Node on symmetric MWOE asymmetric

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**Remark: ļ³ two zero Impossible Impossible In merged fragments there is**

exactly one symmetric MWOE (n-1 edges vs n arrows) ļ³ two zero Impossible Impossible Creates a fragment with two MWOEs Creates a fragment with no MWOE

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**is the identity of the new fragment**

After merging has taken place, the new root broadcasts New fragment(w(e)) to the new fragment, and afterwards a new phase starts e is the symmetric MWOE of the merged fragments is the identity of the new fragment

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**In our example, at the end of phase 1 each fragment has its new identity.**

Root Root Root Root End of phase 1

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**At the beginning of each new phase each **

node in fragment finds its MWOE MWOE MWOE 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(ļ”) accept test(ļ”) reject

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**Then it knows its local 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) MWOE MWOE 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 fragment**

has its own unique identity. Root Root End of phase 2

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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 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 asynchronousā¦ But each fragment can have at most n nodes, has height at most n-1, and each node has at most n-1 incident edgesā¦ 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. ļ A fixed number of 5n+2 total rounds can be used to complete each phase (in some rounds nodes do nothingā¦)!

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Complexity Smallest Fragment size (#nodes) Phase

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**ļ Algorithm Time Complexity Maximum possible fragment size**

Number of nodes Maximum # phases: ļ Total time = Phase time ā¢ #phases =

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**Algorithm Message Complexity**

Thr: Synchronous GHS requires O(m+n logn) msgs. Proof: We have the following messages: Test-Reject msgs: at most 2 for each edge; 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. 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)ā„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: Identity of a fragment is now given by an edge weight plus the level of the fragment; 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: New fragment(weight,level,status): coordination message sent just after a merge Test(weight,level): to test an edge Connect(weight,level): to perform the union

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Absorption Fragment Fragment MWOE If then is absorbed by (cost of merging proportional to )

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The combined level is New fragment MWOE 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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Join Fragment Fragment MWOE If and F1 and F2 have a symmetric MWOE, then F1 joins with F2 (cost of merging proportional to )

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The combined level is New fragment MWOE 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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**Remark: a joining requires that fragment levels are equalā¦how to control this?**

ļ 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 ) Fragment Fragment Test

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**Algorithm Message Complexity**

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: F was obtained by joining two fragments of level k-1, each containing at least 2k-1 nodes by inductive hypothesis ļ F contains at least 2k-1 + 2k-1 = 2k nodes; F was obtained after absorbing another fragment Fā of level<k ļ apply recursively to F\Fā, until case (1) applies. END OF PROOF

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**Algorithm Message Complexity (2)**

Thr: Asynchronous GHS requires O(m+n logn) msgs. Proof: We have the following messages: Connect msgs: at most 2 for each edge; Test-Reject msgs: at most 2 for each edge; 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. END OF PROOF

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**Homework Execute 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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