distributed systems basic algorithms · network as a graph • network is a graph : g = (v,e) •...

DistributedSystems

BasicAlgorithmsRikSarkar

UniversityofEdinburgh

2016/2017

DistributedComputaEon

•  Howtosendmessagestoallnodesefficiently•  Howtocomputesumsofvaluesatallnodesefficiently

•  Networkasagraph•  BroadcasEngmessages•  CompuEngsumsinatree•  CompuEngtreesinanetwork•  CommunicaEoncomplexity

DistributedSystems,Edinburgh,2016

Ref:NL

Networkasagraph•  Networkisagraph:G=(V,E)•  Eachvertex/nodeisacomputer/process•  EachedgeiscommunicaEonlinkbetween2nodes•  EverynodehasaUniqueidenEfierknowntoitself.

–  OVenused1,2,3,…n•  Everynodeknowsitsneighbors–thenodesitcanreach

directlywithoutneedingothernodestoroute–  Edgesincidentonthevertex–  Forexample,inLANorWLAN,throughlisteningtothebroadcastmedium

–  Orbyexplicitlyasking:Everyonethatreceivesthismessage,pleasereportback

•  Butanodedoesnotknowtherestofthenetwork


Example:Unitdiskgraphs

•  Supposeallnodesarewireless•  Eachcancommunicatewithnodeswithindistancer.

•  Say,r=1

•  UDGisamodel•  Notperfect

•  Ingeneral,networkscanbeanygraph


Directedgraphs

•  WhenAcansendmessagetoB,butBcannotsendmessagetoA

•  Forexample,inwirelesstransmission,ifBisinA’srange,butAisnotinB’srange


BA

Directedgraphs

•  WhenAcansendmessagetoB,butBcannotsendmessagetoA

•  OrifprotocolortechnologylimitaEonspreventBfromcommunicaEngwithA


BA

Directedgraphs

•  Protocolsmorecomplex•  Needsmoremessages


Networkasagraph•  Distance/costbetweennodespandqinthenetwork– Numberofedgesontheshortestpathbetweenpandq(whenalledgesaresame:unweighted)

•  SomeEmes,edgescanbeweighted–  Eachedgee=(a,b)hasaweightw(e)– w(e)isthecostofusingthecommunicaEonlinke(maybelengthe)

– Distance/costbetweenpandqistotalweightofedgesonthepathfromptoqwithleastweight


Networkasagraph

•  Diameter–  Themaximumdistancebetween2nodesinthenetwork

•  Radius–  Halfthediameter

•  Spanningtreeofagraph:–  Asubgraphwhichisatree,andreachesallnodesofthegraph

–  Ifnetworkhasnnodes•  Howmanyedgesdoesaspanningtreehave?


CompuEngsumsinatree

•  Supposerootwantstoknowsumofvaluesatallnodes


root

CompuEngsumsinatree•  Supposerootwantstoknowsumofvaluesatallnodes

•  Itsends“compute”messagetoallchildren

•  Theyforwardthemessagetoalltheirchildren(unlessitisaleafnode)

•  Thevaluesmoveupwardfromleaves

•  Eachnodeaddsvaluesfromallchildrenanditsownvalue

•  Sendsittoitsparent


root

CompuEngsumsinatree

•  Whatcanyoucomputeotherthansums?

•  Howmanymessagesdoesittake?

•  HowmuchEmedoesittake?


root

CommunicaEoncomplexity

•  UsedtorepresentcommunicaEoncostforgeneralscenarios

•  CalledCommunicaEonComplexityorAsymptoEccommunicaEoncomplexity

•  UsebigohnotaEon:O


Bigoh–upperbounds•  Forasystemofnnodes,•  CommunicaEoncomplexityc(n)isO(f(n))means:– ThereareconstantsaandN,suchthat:

– Forn>N:c(n)<a*f(n)

f(n)

c(n)

NAllowingsomeiniEalirregularity,‘c(n)’isnotbiggerthanaconstantEmes‘f(n)’Inthelongrun,c(n)doesnotgrowfasterthanf(n)

a*f(n)


Examples

•  3n=O(?)•  1000n=O(?)•  n2/5=O(?)•  10logn=O(?)•  2n3+n+logn+200=O(?)•  15=O(?)


Examples

•  3n=O(n)•  1000n=O(n)•  n2/5=O(n2)•  10logn=O(logn)•  2n3+n+logn+200=O(n3)•  15oranyotherconstant=O(1)


Example1

•  ‘Star’network•  CompuEngsumofallvalues•  CommunicaEoncomplexity:O(n)

Server


Example2a•  ‘Chain’topologynetwork•  Simpleprotocolwhereeveryonesendsvaluetoserver

•  CommunicaEoncomplexity:?

Server


Example2a•  ‘Chain’topologynetwork•  Simpleprotocolwhereeveryonesendsvaluetoserver

•  CommunicaEoncomplexity:1+2+…+n=O(n2)

Server


Example2b•  ‘Chain’network•  Protocolwhereeachnodewaitsforsumofpreviousvaluesandsends

•  CommunicaEoncomplexity:1+1+…+1=O(n)

Server


CompuEngsumsinatree

•  Howmanymessagesdoesittake?

•  HowmuchEmedoesittake?


root

GlobalMessagebroadcast•  Messagemustreachallnodesinthenetwork– DifferentfrombroadcasttransmissioninLAN– Allnodesinalargenetworkcannotbereachedwithsingletransmission


Source

GlobalMessagebroadcast•  Messagemustreachallnodesinthenetwork– DifferentfrombroadcasttransmissioninLAN– Allnodesinalargenetworkcannotbereachedwithsingletransmissions


Source

FloodingforBroadcast

•  ThesourcesendsaFloodmessagetoallneighbors

•  Themessagehas– TypeFlood– Uniqueid:(sourceid,messageseq)– Data


FloodingforBroadcast

•  ThesourcesendsaFloodmessage,withauniquemessageidtoallneighbors

•  Everynodepthatreceivesafloodmessagem,doesthefollowing:–  Ifm.idwasseenbefore,discardm– Otherwise,Addm.idtolistofpreviouslyseenmessagesandsendmtoallneighborsofp


Floodingforbroadcast

•  Storage– Eachnodeneedstostorealistoffloodidsseenbefore

–  Ifaprotocolrequiresxfloods,theneachnodemuststorexids•  (thereisawaytoreducethis.Think!)


AssumpEons

•  Weareassuming:– NodesareworkinginsynchronouscommunicaDonrounds(e.g.transmissionsoccurinintervalsof1secondexactly)

– MessagesfromallneighborsarriveatthesameEme,andprocessedtogether

–  Ineachround,eachnodecansuccessfullysend1messagetoeachneighbor

– AnynecessarycomputaEoncanbecompletedbeforethenextround



•  Thethemessage/communicaEoncomplexityis:



•  Thethemessage/communicaEoncomplexityis:– O(|E|)



•  Thethemessage/communicaEoncomplexityis:– O(|E|)– Worstcase:O(n2)


ReducingCommunicaEoncomplexity(slightly)

•  Nodepneednotsendmessagemtoanynodefromwhichithasalreadyreceivedm– Needstokeeptrackofwhichnodeshavesentthemessage

– Savessomemessages– DoesnotchangeasymptoEccomplexity


Timecomplexity

•  Thenumberofroundsneededtoreachallnodes:diameterofG


CompuEngTreefromanetwork

•  BFStree– TheBreadthfirstsearchtree– Withaspecifiedrootnode


BFSTree

•  Breadthfirstsearchtree– Everynodehasaparentpointer– Andzeroormorechildpointers

– BFSTreeconstrucEonalgorithmsetsthesepointers


BFSTreeConstrucEonalgorithm•  Breadthfirstsearchtree–  Theroot(source)nodedecidestoconstructatree– Usesfloodingtoconstructatree–  Everynodepongepngthemessageforwardstoallneighbors

– AddiEonally,everynodepstoresparentpointer:nodefromwhichitfirstreceivedthemessage•  IfmulEpleneighborshadfirstsentpthemessageinthesameround,chooseparentarbitrarily.E.g.nodewithsmallestid

–  pinformsitsparentoftheselecEon•  Parentcreatesachildpointertop


BFSTree

•  Property:BFStreeisashortestpathtree– Forsourcesandanynodep– TheshortestpathbetweensandpiscontainedintheBFStree


Time&messagecomplexity

•  AsymptoEcallySameasFlooding


root

Treebasedbroadcast

•  Sendmessagetoallnodesusingtree– BFStreeisaspanningtree:connectsallnodes

•  Floodingonthetree

•  Receivemessagefromparent,sendtochildren


root

Treebasedbroadcast

•  Simplerthanflooding:sendmessagetoallchildren

•  CommunicaEon:Numberofedgesinspanningtree:n-1


AggregaEon:Findthesumofvaluesatallnodes

•  WithBFStree

•  Startfromleafnodes– Nodeswithoutchildren– Sendthevaluetoparent

•  Everyothernode:– Waitforallchildrentoreport– Sumvaluesfromchildren+ownvalue– Sendtoparent


AggregaEon

•  Withoutthetree•  Floodfromallnodes:– O(|E|)costpernode– O(n*|E|)totalcost:expensive– Eachnodeneedstostorefloodidsfromnnodes

•  RequiresΩ(n)storageateachnode– Goodfaulttolerance

•  IfafewnodesfailduringoperaEon,alltherestsEllgetsomevalue


AggregaEon

•  WithTree

•  AlsocalledConvergecast


AggregaEon•  WithTree

•  Oncetreeisbuilt,anynodecanuseforbroadcast–  Justfloodonthetree

•  Anynodecanuseforconvergecast–  FirstfloodamessageonthetreerequesEngdata– Nodesstoreparentpointer–  Thenreceivedata

•  WhatisthedrawbackoftreebasedaggregaEon?


AggregaEon•  WithTree

•  Oncetreeisbuilt,anynodecanuseforbroadcast–  Justfloodonthetree

•  Anynodecanuseforconvergecast–  FirstfloodamessageonthetreerequesEngdata–  Nodesstoreparentpointer–  Thenreceivedata

•  Faulttolerancenotverygood–  Ifanodefails,themessagesinitssubtreewillbelost–  WillneedtorebuildthetreeforfutureoperaEons


BFStreescanbeusedforrouEng•  Fromeachnode,createaseparateBFStree•  EachnodestoresaparentpointercorrespondingtoeachBFStree

•  ActsasrouEngtable


1 4

2 4

3 4

4 4

1 3

2 3

3 3

5 5

1 1

2 2

4 4

5 4

2 2

3 3

4 3

5 31

2

4

3

5

1 1

3 3

4 3

5 3 DistributedSystems,Edinburgh,2014 45

BFStreescanbeusedforrouEng•  Fromeachnode,createaseparateBFStree•  EachnodestoresaparentpointercorrespondingtoeachBFStree

•  ActsasrouEngtable•  O(n*|E|)messagecomplexityincompuEngrouEngtable

DistributedSystems,Edinburgh,2014 46DistributedSystems,Edinburgh,2016

ObservaEononcomplexity

•  Supposec(n)=n– Thenc(n)isO(n)andalsoO(n2)–  Although,whenweaskforthecomplexity,wearelookingfortheEghtestpossiblebound,whichisO(n)


BigΩ–lowerbounds•  Forasystemofnnodes,•  CommunicaEoncomplexityc(n)isΩ(f(n))means:– ThereareconstantsaandN,suchthat:

– Forn>N:b*f(n)<c(n)

f(n)c(n)

NAllowingsomeiniEalirregularity,‘c(n)’isnotsmallerthanaconstantEmes‘f(n)’Inthelongrun,f(n)doesnotgrowfasterthanc(n)

b*f(n)


Bigθ–Eghtbounds:bothOandΩ•  Forasystemofnnodes,•  CommunicaEoncomplexityc(n)isθ(f(n))means:– Thereareconstantsa,bandN,suchthat:

– Forn>N:b*f(n)<c(n)<a*f(n)

f(n)c(n)

NAllowingsomeiniEalirregularity,c(n)andf(n)areWithinconstantfactorsofeachother.Inthelongrun,c(n)growsatsamerateasf(n),uptoconstantfactors.

b*f(n)

a*f(n)


BitcomplexityofcommunicaEon•  WehaveassumedthateachcommunicaEonis1message,

andwecountedthemessages•  SomeEmes,communicaEonisevaluatedbybitcomplexity

–thenumberofbitscommunicated•  Thisisdifferentfrommessagecomplexitybecausea

messagemayhavenumberofbitsthatdependonnor|E|•  Forexample,nodeidsinmessagehavesizeΘ(logn)

•  InpracEcethisismaynotbecriEcalsincelognismuchsmallerthanpacketsizes,soitdoesnotchangethenumberofpacketscommunicated

•  ButdependingonwhatotherdatathealgorithmiscommunicaEng,sizesofmessagesmaymaxer


Sizeofids

•  Inanetworkofnnodes•  EachnodeidneedsΘ(logn)(thatis,bothO(logn)andΩ(logn))bitsforstorage– ThebinaryrepresentaEonofnneedslog2nbits

•  Ω–sinceweneedatleastthismanybits– Mayvarybyconstantfactorsdependingonbaseoflogarithm


CompuEngTrees:

•  Whatiftheedgeshaveweights?


AggregaEonusingTrees:

•  Whatiftheedgeshaveweights?•  ThecostmaynotbeO(n)sinceweightscanbehigher

•  Howtogetthebestperformance?


Minimumspanningtreeis

•  Aspanningtree(reachesallnodes)•  Withminimumpossibletotalweight

•  Howcanwecomputeaminimumspanningtreeefficientlyinadistributedsystem?

•  (remember,anodeknowsonlyitsneighborsandedgeweights)


distributed systems basic algorithms · network as a graph • network is a graph : g = (v,e) •...

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