DistributedSystems

MinimumspanningtreesRikSarkar

UniversityofEdinburgh

Fall2016

Minimumspanningtrees

•  DefiniBon(inanundirectedgraph):– Aspanningtreethathasthesmallestpossibletotalweightofedges

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Ref:Wiki

Minimumspanningtrees

•  Usefulinbroadcast:– UsingafloodontheMSThasthesmallestpossiblecostonthenetwork

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Minimumspanningtrees

•  UsefulinpointtopointrouBng:– Minimizesthemaxweightonthepathbetweenanytwonodes

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Property:CutopBmality

•  EveryedgeoftheMSTparBBonsthegraphintotwodisjointsets(createsacut)– EachsetisindividuallyconnectedbyMSTedges

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Property:CutopBmality

•  EveryedgeoftheMSTparBBonsthegraphintotwodisjointsets(createsacut)– EachsetisindividuallyconnectedbyMSTedges

•  NoedgeacrossthecutcanhaveasmallerweightthantheMSTedge

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Property:CutopBmality•  EveryedgeoftheMSTparBBons

thegraphintotwodisjointsets(createsacut)–  EachsetisindividuallyconnectedbyMSTedges

•  NoedgeacrossthecutcanhaveasmallerweightthantheMSTedge

•  Proof:Iftherewassuchanedge,thenwecanswapitforthecurrentedgeandgetatreeofsmallertotalweight

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Property:CycleopBmality

•  Everynon-MSTedgewhenaddedtoMSTsetcreatesacycle

•  Itmusthavemaxweightinthecycle

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MST:Notnecessarilyunique

•  Why?

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MST:Notnecessarilyunique

•  Assume:– Alledgeweightsareunique

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Prim’sAlgorithm

•  IniBalizeP={x};Q=E–  (xisanyvertexinV)

•  WhileP≠V– Selectedge(u,v)inthecut(P,V\P)

•  (attheboundaryofP)• Withsmallestweight

– AddvtoP

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Prim’sAlgorithm

•  IfwesearchfortheminweightedgeeachBme:O(n2)

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Prim’sAlgorithm

•  Ifweuseheaps:– O(mlogn)[binaryheap]– O(m+nlogn)[Fibonacciheap]

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Prim’sAlgorithm

•  CanwehaveanefficientdistributedimplementaBon?

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Prim’sAlgorithm

•  Ineveryround,weneedtofindthelowestweightboundaryedge.

•  Useaconvergecast(aggregaBontreebased)–  Ineveryround– Fornrounds

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Prim’sAlgorithm

•  WhatistherunningBme?•  WhatiscommunicaBoncomplexity?

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Prim’sAlgorithm

•  Theweakness:•  DoesnotusethedistributedcomputaBon•  Treespreadsfromonepoint,restofnetworkisidle

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Kruskal’salgorithm

•  Workswithaforest:AcollecBonoftrees•  IniBally:eachnodeisitsowntree•  Sortalledgesbyweight•  Foreachtree,– Findtheleastweightboundaryedge– Addittothesetofedges:mergestwotreesintoone

– RepeatunBlonly1treelem

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Kruskal’salgorithm•  Theproblemstep:–  “Findtheleastweightboundaryedge”

•  Howdoyouknowwhichistheboundaryedge?•  Maintainidforeachtree(storethisateverynode)

•  Easytocheckifend-pointbelongtodifferenttrees

•  Whenmergingtrees,updatetheidofoneofthetrees–  Expensive,sinceallnodesinthetreehavetobeupdated

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Kruskal’salgorithm

•  Whenmergingtrees,updatetheidofoneofthetrees– Expensive,sinceallnodesinthetreehavetobeupdated

•  SoluBon:alwaysupdatetheidofthesmallertree(theonewithfewernodes)

•  ThecostforallidupdatesisO(nlogn)

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Kruskal’salgorithm•  Claim:ThecostforallidupdatesisO(nlogn)•  Proof:(byinducBononlevels)–  Supposethefinallistofnelementswasobtainedbymergingtwolistsofhelementsandn-helementsinthepreviouslevel

–  Andh≤n/2–  ThencostofcreaBngfinallistis(forsomeconstp):

•  CostforcreaBngtwolists≤phlgh+p(n-h)lg(n-h)•  CostforupdaBnglabels≤ph•  Total≤phlgh+p(n-h)lg(n-h)+ph•  Total≤ph(lg(n/2)+1)+p(n-h)lg(n-h)•  ≤pnlgn

•  Note:KruskalalsoneedsBmetosorttheedgesiniBally

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GHSDistributedMSTAlgorithm

•  ByGallagher,HumbletandSpira•  Eachnodeknowsitsownedgesandweights

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Ref:NL

GHSDistributedMSTAlgorithm

•  Worksinlevels•  Inlevel0eachnodeisitsowntree•  Eachtreehasaleader(leaderid==treeid)•  Ateachlevelk:– AllLeadersexecuteaconvergecasttofindtheminweightboundaryedgeinitstree

–  Itthenbroadcaststhisinitstreesothatthenodethathastheedgeknows

–  Thisnodeinformsthenodeontheotherside,whichinformsitsownleader

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GHSDistributedMSTAlgorithm•  ObservaBon1:

–  WearepossiblymergingmorethantwotreesatthesameBme–  Problem:whoistheleaderofthenewtree?

•  ObservaBon2:–  Themergedtreeisatreeoftrees:itcannothaveacycle–  WecanassignadirecBontoeachedgeandeachnode(tree)hasanoutgoingedge

–  Theremustbeapairofnodes(trees)thatselecteach-other(otherwisethemergedtreeisinfinite)

–  Weselecttheedgeusedtomergethesetwotrees•  SelectthenodewithhigherIDtobeleader

–  TheleaderthenbroadcastsamessageupdaBngleaderidatallnodes.

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GHSDistributedMSTAlgorithm

•  Complexity:•  Thenumberofnodesateachlevelktreeisatleast2k

•  SincestarBngatsize1,thenumberofnodesinthesmallesttreeatleastdoubleseverylevel

•  Therefore,thereareatmostO(logn)levels

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GHSDistributedMSTAlgorithm

•  Complexity:•  Ateachlevel,ateachtree,weuseconstantnumberofbroadcastsandconvergecasts

•  EachlevelcostsO(n)Bme•  Totalcosts:O(nlogn)Bme

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GHSDistributedMSTAlgorithm

•  Complexity:•  Ateachlevel,ateachtree,weuseconstantnumberofbroadcastsandconvergecasts

•  EachlevelcostsO(n)messages•  Totalcosts:O(nlogn+|E|)messages

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DistributedMSTAlgorithm

•  Non-uniqueedgeweights•  Ifedgeshaveduplicateweights•  Wemakethemunique:– Byensuringthatforanytwoedgeseande’– Eitherwt(e)<wt(e’)orwt(e’)<wt(e)– Byusingnodeids– Eg.If(u,v)and(u’,v’)havesameweight,wedefine

•  Ifu<u’thenwt(u,v)<wt(u’v’)•  Elseifu==u’,andifv<v’thenwt(u,v)<wt(u’v’)

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