InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
StefanHoffmannandEgonWanke
Heinrich-Heine-UniversityD¨usseldorf,Germany
ABSTRACT
Thispaperconsidersthefollowing NeighborhoodBroadcast problem:Distributeamessage toallneighborsofanetworknode v undertheassumptionthat v doesnotparticipatedueto beingcorruptedordamaged.Wepresentpracticalnetworkprotocolthatcanbeusedcompletely reactive.Itisparameterizedwithapositiveinteger k ∈ N anditisproventoguaranteedeliveryfor k ≥ 2d 1,ifnode v is d-locallyconnected,whichmeansthatthesetofnodeswithdistancebetween 1 and d to v inducesaconnectedsubgraphofthecommunicationgraph.Itisalsoshownthatthe numberofparticipatingnodesisoptimalundertherestrictionto 1-hopneighborhoodinformation. Theprotocolisalsoanalyzedinsimulationsthatdemonstrateveryhighsuccessratesforverylow valuesof k
K EYWORDS
ad-hocsensornetwork,networkprotocols,faulttolerance,failurerecovery,neighborhoodbroadcast
A wirelesssensornetwork(WSN) consistsofalargenumberofsmalldevicesthatare deployedacrossageographicareatomonitorcertainaspectsoftheenvironment.These sensornodesareabletocommunicatewitheachotherthroughawirelesscommunication channel.Asaresultoftheirsizetheresourcesofthesensornodesarestronglylimitedin termsofavailableenergy,whichleadstoaverylimitedrangeoftheradiotransmitters. Thereforethenodesalsoactasrouterstorelaymessagesonmulti-hoproutingpaths, whichhavetobediscoveredandmaintainedusingaroutingprotocolsuchastheones proposedin[1,2,3,4,5,6,7,8,9].
WSNsareusuallymodeledasundirectedcommunicationgraphsthatcontainonevertex foreachsensornodeandanedgebetweentwoverticesifandonlyifthecorresponding sensornodesareabletocommunicatewitheachother.Fromatechnicalpointofview, modelingaWSNasadirectedgraphisclosertoreality,becausephysicalradiolinksarenot necessarilysymmetric[10,11].However,theusageofundirectedgraphsiswelljustifiedby theobservationthatlow-levelcommunicationprotocolsforwirelesstransmissionscurrently inuse,suchasIEEE802.11orBluetooth,requiresymmetryforreliabledatatransmission viaacknowledgements.
AtaskthatarisesnaturallyinWSNsisthefollowing NeighborhoodBroadcast:For asensornode v,aneighbor s of v hastotransmitamessagetoallotherneighborsof v undertheassumptionthat v itselfdoesnotobeyanyoftheimplementednetworkprotocols.Thisproblemoccursduringcollaborativefaultdetection[12]wheretheneighborsof asensornode v wanttoexchangetheirobservationsinordertodecidewhether v should DOI:10.5121/jgraphoc.2017.91011
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
beexcludedfromthenetworkduetofaultsormisbehavior.The NeighborhoodBroadcast problemalsooccursduringtheexclusionofamisbehavingorfaultysensornode v fromthenetwork,becauseeveryneighborof v hastobeinstructednottocommunicate with v anymoreanditcanbeapplied,forexample,tothedetectionalgorithmpresented in[13].Apractical NeighborhoodBroadcast algorithmalsohasapplicationsinreactivelyrepairingunicastroutingpathsormulticastroutingstructuresafternodefailures: Ifthereisasufficientamountofredundancywithintheroutinginformationtodetermine thesuccessor(s)ofafailednode v,theroutingtaskcanbeaccomplishedbybroadcasting themessageacrosstheneighborhoodof v.
Inthispaperwepresentthe k-HopBouncingFlood(k-HBF) networkprotocol,which isbasedonaparameter k ∈ N,todistributemessagesacrosstheneighborhoodofa sensornode.The k-HBFprotocolcanbeusedinacompletelyreactivemanner,i.e.it doesnotneedanyinitializationormaintenance.Itonlyrequiresthenodestoknow theirownneighborhood.Alternatively,anoptionalinitializationphasecanbeusedto determinetheoptimalparameter k thatguaranteessuccessfuldelivery.Bothapproaches arediscussedindetailinthispaper.Furthermoretheprotocoliseasytoimplementina distributedenvironmentandthereforereadyforpracticalapplications. k-HBFisanalyzed theoreticallyanditisproventhattheprotocolisguaranteedtodeliveramessagetoevery neighborofanode v,if k ≥ 2d 1and v is d-locallyconnected inthecommunicationgraph
G,whichmeansthatthesetofnodeswithhop-distancebetween1and d to v inducesa connectedsubgraphof G.Itisalsoshownthatanyinvocationoftheprotocolresultsin thetransmissionofatmost k ∆(G)k+1 messages,where∆(G)isthemaximumvertex degreeofthecommunicationgraph.Thismeansthat,foranyfixedparameter k,the k-HBFprotocolonlyrequirescommunicationwithinalocalizedareaofthenetwork.In fact, k-HBFdistributesthemessagetoallnodeswithhop-distanceatmost k +1to v and itisalsoproventhatthisisoptimal,i.e.nonetworkprotocolthatisrestrictedtothesame topologyinformationcanguaranteedeliverytoallneighborsofa d-locallyconnectednode v,unlesseverynodewithhop-distanceatmost2d to v receivesatleastonemessage.
Additionallywepresentalineartimealgorithmtosolvethe LocalConnectivityDistance(LCD)
problem,whichasksforthesmallestpositiveinteger d ∈ N suchthatagivenvertex v is d-locallyconnectedinagivenundirectedgraph G.Thislineartimealgorithmisbasedon globaltopologyknowledge,butitcanalsobeimplementedasadistributedalgorithmin aWSN,anoptionthatisalsodiscussedinthispaper.Thereforethisalgorithmcanbe usedtodeterminetheoptimalparameter k thatisrequiredforthesuccessofthe k-HBF protocol.
Finally,anexperimentalanalysisisperformedandthesimulationresultsdemonstrate thatthe k-HBFprotocolsuccessfullyperformsa NeighborhoodBroadcast inalmost allcasesforverylowvaluesof k.Thecomputationsareperformedonseveraldifferent networktopologymodelscommonlyusedinwirelessnetworkresearch.Asasideeffect, thisanalysisdemonstratesasignificanttopologicaldifferencebetweenthesemodels.
Regardingrelatedwork:In[12]theauthorsfocusonbuildingdataaggregationtreesthat spantheneighborhoodofasensornode v.Todiscoverandinitiallycontactallneighbors of v theyimplicitlytrytoperforma NeighborhoodBroadcast bybrieflydescribing theusageoflimitedrangeflooding.However,sendingamessagewiththetechniquethey
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
mentionisonlyguaranteedtoreachallneighborsof v,ifthefloodingrangeishighenough tofloodtheentirenetwork,evenincaseswhereitisnotnecessarytodoso.Incontrast, the k-HBFprotocolsusesmoresophisticatedextensionstothefloodingprotocolsuchas hopcounterresetsandallowingnodestorelaymessagesmultipletimes.Thegeneral multicastprobleminwirelessadhocnetworks,i.e.thetaskofdistributingmessageto aknownsetofnodes,hasreceivedtremendousattentioninthescientificcommunity andcountlessproposalsforprotocolshavebeenmade.Thereforewereferthereaderto [14,15]foranoverviewonthistopic.Whilethesemulticastprotocolsaretheoretically suitableforthe NeighborhoodBroadcast task,theyrequiretheinitiatingnodeto knowtheneighborhoodinwhichthemessageistobedistributedandthisrequiresconstant maintenance,especiallywhenthenetworktopologyissubjectedtofrequentchanges.And ofcoursetheentirenetworkcanbefloodedtodistributeamessageacrosstheneighborhood ofanode,forexamplebyusingtheprotocolsin[16,17,18].
Therestofthepaperisorganizedasfollows:Section2introducesthenecessaryterminologyandnotationsbeforethe k-HBFprotocolispresentedandevaluatedinsection3. Section4isdedicatedtosolvingthe LCD problemandtheexperimentalresultsaregiven insection5beforethepaperinconcludedinsection6.
Apair G =(V,E)isan undirectedgraph with vertexset V and edgeset E,if V isafinite setand E ⊆{{u,v}| u,v ∈ V,u = v}.The vertexdegree deg(v)ofavertex v ∈ V isgiven by deg(v):= |{e ∈ E | v ∈ e}| andthe maximumvertexdegreeof G isdenotedby∆(G).
Twodistinctvertices u,v ∈ V arecalled adjacent,if {u,v}∈ E.Avertex v ∈ V andan edge e ∈ E arecalled incident,if v ∈ e
Agraph G =(V ,E )isa subgraph ofagraph G =(V,E),if V ⊆ V and E ⊆ E.Fora subsetofvertices U ⊆ V ofanundirectedgraph G =(V,E)thegraph G|U :=(U,E|U ), E|U := {{u,v}∈ E | u,v ∈ U } iscalledthe subgraphof G inducedby U .
A pathoflength k 1 betweentwovertices v1 and vk inanundirectedgraphisasequence ofvertices v1,...,vk suchthat ∀i ∈{1,...,k 1}: {vi,vi+1}∈ E. G is connected,if thereisapathbetweeneverypairofvertices.A connectedcomponent of G isamaximal inducedsubgraphof G thatisconnected.A separationvertex isavertex v ∈ V suchthat theinducedsubgraph G|V \{v} hasmoreconnectedcomponentsthan G
Apath u1,...,uk, k ≥ 3inanundirectedgraph G =(V,E)isa cycle if {uk,u1}∈ E.A graphwithoutcyclesisa forest andaconnectedforestisa tree
The distance dG(u,v)betweentwovertices u and v inanundirectedgraph G isthe smallestinteger k suchthatthereisapathoflength k between u and v in G.Theset
N d G(v):= {u ∈ V | d(u,v)= d} iscalledthe d-hopneighborhood ofavertex v ∈ V and N d G[v]:= ∪k i=1N i G(v)isthesetofverticeswithdistancebetween1and d to v.Ifthegraph G isobviousfromthecontextthenotations d(u,v), N d(v)and N d[v]canbeusedinstead of dG(u,v), N d G(v)and N d G[v].
The diameter ofanundirectedgraph G isthemaximumdistancebetweenanytwovertices of G
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
Definition1. Let d ∈ N beapositiveinteger, G =(V,E)anundirectedgraphand v ∈ V avertex.Vertex v is d-locallyconnected in G,if N d G[v]inducesaconnectedsubgraphof G.
Definition2. Foranundirectedgraph G =(V,E)andavertex v ∈ V ,thesmallest positiveinteger dmin ∈ N suchthat v is dmin-locallyconnectedin G andnot(dmin 1)locallyconnectedin G iscalledthe localconnectivitydistance of v in G
Throughoutthispapertheterms node and vertex areusedinterchangeablywith node referringtoaphysicalinstanceofasensornode,while vertex referstothemathematical counterpartwithinthegraphmodel.
Insection2apracticalnetworkprotocolforthefollowingtaskispresentedandanalyzed.
NeighborhoodBroadcast
Given: AWSNwithundirectedcommunicationgraph G =(V,E)andtwonodes
v ∈ V and s ∈ N 1 G(v).
Task: Distributeamessageoriginatingatnode s toallnodesin N 1 G(v)without participationof v
Duringtheanalysisitisestablishedthatthis NeighborhoodBroadcast taskisclosely relatedtothelocalconnectivitydistanceandthereforethefollowing LCD problemis consideredinsection4.
LocalConnectivityDistance(LCD)
Given: Anundirectedgraph G =(V,E)andavertex v ∈ V
Task: Computethelocalconnectivitydistanceof v
Obviously,distributingamessage M toallneighbors N 1(v)of v withouttheparticipation of v itselfispossibleifandonlyif v isnotaseparationvertexofthecommunicationgraph G =(V,E).Butevenif v isnotaseparationvertex,itmightbenecessarytobroadcast M throughouttheentirenetworkinordertodistributeittoallneighbors N 1(v):Consider thecycle Cn with n verticesasanetworktopology:
Cn :=({v1,...,vn}, {{vi,vi+1 | 1 ≤ i<n}}∪{vn,v1})
Ifanyoneofthenodes v1,...,vn cannotbeusedtorelay M ,thentheonlyremaining pathbetweenbothneighborsofthisnodetraversesallothernodesandthereforeevery multicastalgorithmhastodistribute M toallnodes.However,theoreticalworstcases likethisareratherunlikelyinarealsensornetwork,whichiswhyamulticastalgorithm withabetterperformancethanfloodingtheentirenetworkisdesirableforthisscenario.
The k-HopBouncingFlood(k-HBF) protocolusestheideaofcarryingatransmission counter it inthemessageheaderthatrestrictsthenumberofretransmissionsforeach message:Theforwardingisstoppedassoonas it reachesthegivenparameter k ∈ N 4
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
M
N 1 G(
{
}.Thegreen(g)nodesreceive M duetotheir distanceto u beingatmost2in G|V \{v},theblue(b)nodesreceive M dueto x resetting thetransmissioncounter it,thepurple(p)nodesduetotheresetat y andtheyellow(y) nodesduetotheresetat w
Thisdisseminatesthemessagetoallnodesofdistanceatmost k tothenode s,who initiatedtheprotocol.Thisrangelimitationofthefloodingprocessreducesthenumber ofnodestransmittingthemessage,buttheprotocolisnolongerguaranteedtodeliver themessagetoalltargets,asdiscussedabove.Utilizingonlyinformationalreadypresent attheindividualsensornodes,thesuccessprobabilitycanbeincreasedconsiderablyby resetting it to0ateverynodethatisadjacentto v,whichleadsto M “bouncing”along theneighborsof v,seeFigure1foranexample.Theideaistodistribute M notonlytoall nodesofdistanceatmost k to s,buttoallnodeswithdistanceatmost k toanyneighbor of v or,equivalently,toeverynodewithdistanceatmost k +1to v.Toachievethisit isnotsufficienttoreset it ateveryneighborof v:Forexamplethenode r inFigure1:
Before w resetsthetransmissioncounterto0andsendsthemessagetoitsneighbors,node r alreadyreceived M withcountervalue2from s,but r isrequiredtorelaythemessage from w againinordertodeliverittotheyellownodes.Unlikeconventionalfloodingitis thereforenecessarythatnodescanrelaymessagesmultipletimesandsimplymarkinga message M as“alreadyseen”byaparticularnodetopreventinfinitetransmissionloopsby notrelaying M againbasedonthismarkisnotsufficientanymore.Inthe k-HBFprotocol, eachnode r thatreceived M keepstrackoftheminimum imin,r ofalltransmissioncounter valuescontainedincopiesof M thatreached r.Node r thenrelays M againatalater time,ifandonlyifthenewtransmissioncounterisstrictlylowerthan imin,r ,becausein thiscaseitmightreachadditionalnodes.Thismodificationguaranteesthateverynode withinthe k-hopneighborhoodofanodethatperformedatransmissioncounterresetwill eventuallyreceivethemessage,whileensuringthatthetransmissionsterminateduetothe observationthateverynodecantransmitamessageatmost k times,beforetheminimum transmissioncounterforthisnodereaches0.
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
3.1.DistributedImplementation
Oneofthebiggestchallengesforrealworlddeploymentofsensornetworksisthetesting anddebuggingofimplementedalgorithmsinarealisticenvironment,whichintroduces additionalproblemsthatareusuallynotconsideredduringsimulation.Inrecenthistory therehavebeenseveralrealworldexperimentsthatdemonstratedthisissue[19,20,21,22] andtherehasalsobeenintensiveworkontestbedenvironmentssuitabletotacklethis problem[23,24,25,26,27].
Fromthispointofview,the k-HBFprotocolofferstheadvantageofaneasyandstraightforwarddistributedimplementation,whichisdemonstratedbythepseudocodegivenin Algorithm1forthemainpartoftheprotocol.
Algorithm1 k-HBFprotocol:Distributedimplementationatnode u
1: m ← 1 Counterformessageids
2: h ← HashMap h : ID × NODE → N
3: procedure SendPacket(m,s,v,k,it,M,T )
4: Passdatatolinklayer:
5: Send(m,s,v,k,it,M )toallnodesin T ⊆ N 1(u)
6: endprocedure
7: procedure StartHBF(k,v,M )
8: m ← m +1
9: SendPacket(m,u,v,k, 0,M,N 1(u))
10: endprocedure
11: procedure ReceivePacket(m,s,v,k,it,M )
12: imin,u ←∞
13: if h contains(m,s) then
14: imin,u ← h((m,s))
15: endif
16: if it <imin,u then
17: if v ∈ N 1(u) then neighborresets it
18: SendPacket(m,s,v,k, 0,M,N 1(u))
19: h((m,s)) ←−∞
20: return
21: endif
22: if it <k then
23: SendPacket(m,s,v,k,it +1,M,N 1(u))
24: h((m,s)) ← it
25: endif
26: endif
27: endprocedure
Packetssentbythealgorithmwraparoundagivenmessage M andaddsomeheaderfields requiredfortheprotocolitselfintheformat(messageid,initiator s,target v, k, it, M ), wherethemessageid m isuniqueforeachnode s,meaningthat(m,s)providesaunique messageidentifierthroughoutthenetwork.Themessageheaderalsocontainsthenode v forwhichtheprotocolwasinitiatedaswellasthedistanceparameter k andthetransmissioncounter it
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
Twoextensionsoftheprotocolarisefromtechniquescommonlyusedincomputernetworks:
Toensurelong-termstabilityofthenetwork,thedatastructureusedtoorganizethe mappingbetweenmessageidentifiersandminimumtransmissioncountersateachnode shouldbeimplementedbasedonasoft-stateapproachinthesensethattheentriesare deletedafterareasonableamountoftimetoavoidmemoryleaks.
Inthenextsectionitisshownthat k-HBFisguaranteedtodeliver M toallneighborsof v,if k ≥ 2d 1and v is d-locallyconnected.Thereforetheoptimalchoiceof k dependson thelocalconnectivitydistanceof v,whichisusuallyunknown.Iftheapplicationisableto decidewhetherabroadcastattemptwassuccessful(whichispossibleviaacknowledgement, forexample,if |N 1(v)| isknownorthetargetisoneparticularneighborof v asinthecase ofunicastrouterepair),theprotocolcanobviouslybeextendedbysuccessivelyincreasing thedistanceparameter k aslongasthepreviousattemptwasunsuccessful.According tocommonpracticeonewoulddouble k aftereachfailedattempttoachieveexponential growthofthesearcharea,whichcompensatesforapotentiallyfastchangingnetwork topology.
Anotherpossibleextensionthatallowsdynamicadaptationofthe k-HBFprotocolafter thefirstexecutionforanode v isdiscussedattheendofsection4.
3.2.Analysis
Thissectionevaluatesthe k-HBFprotocolintermsofmessagecomplexityandsuccess guarantee.Itisshownthatthenumberofmessagestransmittedbyaninvocationof kHBFinaWSNwithcommunicationgraph G isin k ∆(G)k+1.Furthermoreasufficient conditionforthedeliveryguaranteeisproven: k-HBFisguaranteedtodeliver M to allneighborsof v,if k ≥ 2d 1and v is d-locallyconnectedin G.Finally,itisalso proventhatthesetofnodesparticipatinginthe k-HBFprotocolisoptimalwithrespect toallpossibleprotocolsfor NeighborhoodBroadcast thatarerestrictedtothesame topologyinformation.
Theorem1. Let G =(V,E)beanundirectedgraphand k ∈ N apositiveinteger.The k-HBFprotocol,initiatedbyaneighbor s ∈ N 1(v)foranode v ∈ V andamessage M , transmitsatmost k ∆(G)k+1 messages.
Proof. Everyneighbor u ∈ N 1(v)performsalimitedrangefloodingof M withrange k andthereforeatmost∆(G)k nodesreceivethemessageduetothehopcounterresetat node u.Furthermore v hasatmost∆(G)neighbors,whichmeansthatatmost∆(G)k+1 nodesreceive M duringtheexecutionofthe k-HBFprotocol.Finally,everynode w can relay M atmost k times,becauseafterwardstheminimumtransmissioncounter imin,w iseither0or −∞.Inconjunctionitfollowsthat k · ∆(G)k+1 isanupperboundforthe numberoftransmittedmessages.
TheTheoremabovedemonstratesatheoreticalworstcasefornumberofmessagestransmittedthe k-HBFprotocol,assumingthateverynodeintheconsideredpartofthenetwork actuallyhasthemaximumvertexdegree∆(G).Furthermoreitassumesthatthesetsof nodesreachedbyeachneighborof v arepairwisedisjoint,whichisnotpossible,if v has morethanoneneighbor.Thereforetheaveragenumberoftransmittedmessagesinany realworldapplicationshouldbeconsiderablylowerthanthepresentedupperbound.Of
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
coursethenumberoftransmittedpacketscanbehigherduetoretransmissions,acknowledgementsetc.
ThefollowingLemmaandthesubsequentTheoremareusedtoestablishtheconnection betweenthe k-HBFprotocolandthelocalconnectivitydistanceofthenode v itisexecuted for.
Lemma1. Let G =(V,E)beanundirectedgraphand d ∈ N apositiveinteger.Let v ∈ V beavertexthatis d-locallyconnectedin G.Then,foreverypairofvertices s,t ∈ N 1(v), thereisapath p = w1,...,wk in G|V \{v} between w1 = s and wk = t suchthatforall i ∈{1,...,k 2d +1} thepath wi,wi+1,...,wi+2d 1 oflength2d 1containsatleast onevertexof N 1(v).
Proof. Let p = u0,...,um beashortestpathbetween u0 = s and um = t in H := G|N d[v]. Since v ∈ V is d-locallyconnectedsuchapathexistsandwealsoknowthatforevery vertex u on p thedistance dG(u,v)isatmost d
Path p willnowbealteredsuccessivelytofulfilltherequiredconditionasfollows:Let l betheminimumindexsuchthatthepath ul,...,ul+2d 1 doesnotcontainavertex adjacentto v.Nowreplace ul,...,ul+2d 1 withapath ul,...,s ,...,ul+2d 1 foravertex s ∈ N 1(v)suchthatbothpaths ul,...,s and s ,...,ul+2d 1 havelengthatmost2d 1as follows.ThisprovestheLemma,becauseduringeachiterationthelengthofaconsecutive subsequenceofthecurrentpaththatviolatestherequiredconditionisstrictlyshortened.
Let dl,...,dl+2d 1 bethesequence D ofdistancestovertex v asdefinedby di := dG(v,ui) for i ∈{l,...,l +2d 1}.Furthermorelet j ∈{l +1,...,l +2d 1} betheminimumindex suchthat dj isthesecondoccurrenceofthisparticulardistancein D,meaningthatthere isanindex j <j with dj = dj .Sincealldistances di arebetween2and d bydefinition of p itfollowsthat j ≤ l + d 1.
Nowlet v,s ,q2,...,qdj 1,uj beashortestpathbetween v and uj in G.Notethat p does notcontainvertex s ,because p containsashortestpath p between ul 1 and uj such that p containsvertex uj .Therefore dH (s ,uj )= dj 1 <dH (ul 1,uj ).Thenitholds thatthelength L(q)ofthepath q = ul,...,uj ,qdj 1,...,q2,s isatmost2d 2,because
L(q)= L(ul,...,uj )+ L(uj ,qdj 1,...,q2,s ) ≤ L(ul,...,ul+d 1)+ dj 1 ≤ d 1+ dj 1 ≤ 2d 2. Furthermoreitalsoholdsthat L(q )for q = s ,q2,...,qdj 1,uj ,uj+1,...ul+2d 1 isatmost2d 1,because
L(q )= L(s ,q2,...,qdj 1,uj )+ L(uj ,uj+1,...ul+2d 1) <L(ul 1,ul,...,uj )+ L(uj ,uj+1,...,ul+2d 1)= L(ul 1,ul,...,ul+2d 1)=2d
Theorem2. Let d ∈ N beapositiveintegerand v ∈ V avertexwithlocalconnectivity distance d inanundirectedgraph G =(V,E)thatrepresentsaWSN.Thenthe(2d 1)HBFprotocol,initiatedbyaneighbor s ∈ N 1 G(v)fornode v andamessage M ,distributes M toallnodesin N 2d G (v)evenif v itselfdoesnotrelayanymessages.
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
Proof. Let t ∈ N 1 G(v)beanarbitraryneighborof v.AccordingtoLemma1thereisa path p between s and t in G|V \{v} suchthateverysub-pathoflength2d 1containsat leastoneneighborof v.Thereforethemessagesentby s isrelayedalong p,becausethe transmissioncounter it isresetafteratmost2d 1transmissionsandhence t receives M Andsinceeveryneighborof v receives M andresetsthetransmissioncounteronce, M is distributedtoallnodeswithhop-distanceatmost2d 1toanyneighborof v,i.e.toall nodeswithhop-distanceatmost2d to v
Althoughthe d-hopneighborhoodof v isconnected,itisnecessarytodistributethe messageacrossthe2d-hopneighborhoodof v,unlessthereisadditionalinformationabout thenetworktopologyavailable.ThisisshowninthenextTheorem.
Theorem3. Let G =(V,E)beanundirectedgraphthatrepresentsaWSN, d ∈ N a positiveintegerand v ∈ V avertexwithlocalconnectivitydistance d
Furthermorelet P beadistributedalgorithmsatisfyingthefollowingproperties:
1. P isinitiatedbyasinglenode s ∈ N 1 G(v)todistributeamessage M
2. P doesnotrunonnode v
3. P isdeterministicandtheonlyinformation P utilizesaboutthenetworktopologyis theinteger d andtheknowledgeaboutthe1-hopneighborhood N 1 G(u)ateachnode u
4. P terminatesafterafiniteamountofstepsandatthatpointeverynodein N 1 G(v) received M .
Then P transmitsatleastonemessagetoeverynodein
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
Let H := G|V \{v} betheinducedsubgraphof G thatdoesnotcontain v andconsiderthe graph G :=(V,E ∪{u,v}).
Vertex v is d-locallyconnectedin G :Since N 2d G (v)= s ∈N 1 G(v) N 2d 1 G (s )= s ∈N 1 G(v) N 2d 1 H (s ), thereisapath w1,...,w2d between w1 = s forsomevertex s ∈ N 1 G(v)and w2d = u in H.Andforall1 ≤ i ≤ 2d itholdsthat dG (v,wi) ≤ d,because dH (s ,wj ) ≤ d 1for j ∈{1,...,d} and dH (u,wj ) ≤ d 1for j ∈{d +1,..., 2d}
Vertex v isnot(d 1)-locallyconnectedin G :Everyvertex s ∈ N 1 G(v)satisfies dH (s ,u) ≥ 2d 1,because dG(v,u)=2d.Thereforeeverypath w1,...,wl between w1 = s forsome s ∈ N 1 G(v)and wl = u in H satisfies l ≥ 2d andthusitholdsthat dH (wd,s ) ≥ d 1 and dH (wd,u) ≥ d 1.Butthenthereisnopathbetweenneighbor u ∈ N 1 G (v)andany neighbor s ∈ N 1 G(v)in G |N d 1 G (v),meaningthat v isnot(d 1)-locallyconnectedin G .
Wewillnowcomparetheexecutionof P initiatedatnode s in G totheexecutionof P initiatedatnode s in G :Since v is d-locallyconnectedandnot(d 1)-locallyconnected bothin G and G ,theinteger d isidenticalinbothexecutionsof P andbyproperty3these twoprocessescanonlydifferfromeachotherduetoadifferenceinthe1-hopneighborhood informationatsomenode.However,the1-hopneighborhoodin G and G isidenticalat allnodesexceptfor v and u and P doesnothaveaccesstotheinformationatnode v by property2.Thereforebothexecutionsareidenticaluntil P sendsamessagetonode u in G ,whichhappensbecauseofproperty4,andthus u alsoreceivesamessagein G.
Knowingthateverynodein N 2d G (v)hastoreceiveatleastonemessage,itfollowsthat everynodein N 2d G [v]hastoreceiveatleastonemessage,becauseitispossibletogenerate anetworktopology G thatsatisfiesthepreconditionsoftheTheoremwhileforcingevery nodein N 2d 1 G [v]torelay(andthereforereceive)atleastonemessageinordertoreach allnodesin N 2d G (v): G containsallverticesandedgesfrom G andforeveryvertex w ∈ N 2d 1 G [v]withdistance x := dG(v,w)oneadditionalvertex w thatisconnectedto w viaapathoflength2d x.Then w ∈ N 2d G (v)andtherefore w hastoreceiveatleastone message,whichimpliesthat w andallnodesonthepathbetween w and w havetorelay thatmessage.
Afterhavingestablishedthatthe(2d 1)-HBFprotocolsuccessfullydistributesamessage withintheneighborhood N 1(v)ofasensornode v,ifvertex v is d-locallyconnectedin thecommunicationgraph G =(V,E),thissectioninvestigateshowtoalgorithmically determinethelocalconnectivitydistanceofavertex,i.e.theminimuminteger d> 0, suchthatagivenvertex v ofanundirectedgraphis d-locallyconnected.
The LCD problemcanobviouslybesolvedinpolynomialtimebyutilizingstandardmethodsfromgraphtheoryasfollows:
Forapositiveinteger i ∈ N theneighborhood N i[v]= i j=1 N j (v)canbedetermined intime O(|V | + |E|)byaslightlymodifiedbreadthfirstsearch,startedatvertex v, 10
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
thatkeepstrackofthedistance d(v,u)foreveryvertex u ∈ V .Afterwardstheinduced subgraph G|N i[v] canbeconstructedintime O(|V |+|E|)andtestedforconnectivityusing, forexample,anotherbreadthfirstsearch.Thelocalconnectivitydistancecannowbe computedbyperformingthistestconsecutivelyforall i ∈{1,..., |V |} inascendingorder untilthefirstconnectivitytestyieldsapositiveresult.Theoverallrunningtimeofthis straightforwardsolutionisobviouslyboundedby O(|V |· (|V | + |E|))= O(|V |2 + |V |·|E|).
Theremainderofthissectionisdedicatedtoamoreefficientsolutionforthe LCD problem thatadditionallyprovidesthepossibilityforadistributedimplementationandtherefore ismoreinterestinginthecontextofWSNs.Thealgorithmicideaisbasedonthefollowing Theoremthatisformulatedusingtheterminologyintroducedinthenextdefinition.
Definition3. Let r ∈ V beavertexinanundirectedgraph G =(V,E).A shortestpath treefor G atroot r isatree T =(V,ET ), ET ⊆ E suchthat dG(r,v)= dT (r,v)forall vertices v ∈ V
Atree T =(V ,E )offorest T |V \{r} iscalleda branch of T .The rootofbranch T isthe (uniquelydetermined)vertex v ∈ N 1 G(r)thatbelongsto T ,i.e. v ∈ V .Forallvertices v ∈ V \{r} let r(v)denotetherootofthebranchof T thatcontains v
Furthermore,anedge e = {u,v}∈ E iscalled bridgewithrespectto T ,if u and v belong todifferentbranchesof T
AlsoseeFigure3foranexampleofthetermsandnotationsintroducedinthisdefinition.
Figure3:ExampleforDefinition3: Left:Graph G =(V,E)withashortestpathtree
T =(V,ET )atroot r (thickblackedges).Thedashedrededgesarebridgeswithrespect to T . Right:Forest T |V \{r} thatdefinesfourbranchesof T ,inducedbythevertexsets {a,e,f,i}, {b}, {c,g,k,l} and {d,h}.Theroot r(e)of e isvertex a,theroot r(b)of b is vertex b,theroot r(k)of k isvertex c etc.
Theorem4. Let G =(V,E)beanundirectedgraph, r ∈ V avertexand T ashortest pathtreefor G atroot r.Furthermorelet d ∈ N beapositiveintegerand B ⊂ E theset ofbridgeswithrespectto T .
Everyedge e ∈ B isgivenapositiveintegerweightrepresentingthemaximumdistance from r tooneoftheverticesincidentto e.Formally,defineamapping f : B → N by f ({u,v}):= max{dT (r,u),dT (r,v)}
Additionally,defineasetofedges E1 betweenneighborsof r withpositiveintegerweights
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
w : E1 → N basedonthebridges B andtheirweights f asfollows:
E1 := {{r(u),r(v)}|{u,v}∈ B} w({u,v}):= min{f ({u ,v }) |{u ,v }∈ B ∧ r(u )= u ∧ r(v )= v}
Thenvertex r is d-locallyconnected,ifthereisasubset E ⊆ E1 suchthat
1.thegraph(N 1 G(r),E )isconnectedand
2. ∀e ∈ E : w(e) ≤ d
Proof. Let E ⊆ E1 beasetofedgesthatsatisfiestheproperties1.and2.aboveand w,w ∈ N 1 G(r), w = w twoarbitraryneighborsof r.Thenapath w1,...,wk in G for somepositiveinteger k ∈ N suchthat w1 = w, wk = w and ∀i ∈{1,...,k} : dG(r,wi) ≤ d canbeconstructedasfollows,meaningthat r is d-locallyconnectedin G.
Since(N 1 G(r),E )isconnected,thereisapathbetween w and w in(N 1 G(r),E ),i.e.there isasequence {v1,v1},..., {vl,vl} ofedgesof E forsomepositiveinteger l ∈ N suchthat v1 = w, vl = w and ∀i ∈{1,...,l 1} : vi = vi+1.Bydefinitionof E1,whichisasuperset of E ,inconjunctionwithproperty2.thismeansthattherealsoisasequenceofbridges {u1,u1},..., {ul,ul} satisfyingthefollowingproperties:
a) ∀i ∈{1,...,l} : {ui,ui}∈ B
b) ∀i ∈{1,...,l 1} : r(ui)= r(ui+1)
c) r(u1)= w
d) r(ul)= w
e) ∀i ∈{1,...,l} : w({ui,ui}) ≤ d
Thereforeitissufficienttoprovethat,foreverypair v,v oftwodistinctverticeswith r(v)= r(v ),thereisapath p between v and v in G suchthateveryvertex u in p satisfies dG(r,u) ≤ max{dG(r,v),dG(r,v )}.Let p bethe(unique)pathbetween v and v in T |V \{r}.Suchapathexists,because v and v belongtothesamebranchof T dueto r(v)= r(v ).Additionally,everyvertex u in p satisfies dG(r,u) ≤ max{dG(r,v),dG(r,v )}, because T isashortestpathtreefor G atroot r andofcourse p alsoisapathin G,because T isasubgraphof G.
BasedonTheoremthe LCD problemcanbesolvedbycomputingthesmallestpositive integer d ∈ N forwhichthereisasetofedges E ⊆ E1 suchthattheconditions1and2 hold.Itisnowshownthatthiscanbeachievedbycomputingaminimumspanningtree ofthegraph(N 1 G(r),E1)withedgeweights w asdefinedinTheorem. Definition4. Foranundirectedgraph G =(V,E)withpositiveedgeweights w : E → N, asubgraph(V,E )of G isa minimumspanningtreefor G,if(V,E )isconnectedand
(V,E )isconnected ⇒
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
MinimumSpanningTree(MST)
Given: Anundirectedgraph G =(V,E)withpositiveedgeweights w : E → N
Task: Computeaminimumspanningtreefor G
Althoughthe MST problemcomputesanedgesetinwhichthesumofallweightsis minimalwhilemaintainingconnectivity,itcanalsobeusedtocomputethemaximum weightnecessaryforachievingconnectivity,becausethemaximumweightofanedgeina minimumspanningtreeisindependentofthetreeitselfasshowninthefollowingLemma.
Lemma2. Let G =(V,E)beanundirectedgraphwithpositiveedgeweights w : E → N and T1 =(V,E ), T2 =(V,E )twominimumspanningtreesfor G.Then max{w(e) | e ∈ E } = max{w(e) | e ∈ E }
Proof. Let m1 := max{w(e) | e ∈ E }, m2 := max{w(e) | e ∈ E } andlet e1 ∈ E bean edgewith w(e1)= m1.Assumethat m1 = m2 and,withoutlossofgenerality,let m1 >m2 Thenthesubgraph(V,E \{e1})consistsofexactlytwoconnectedcomponentsinduced byvertexsets V1 and V2.Obviously V1 ∪ V2 = V andsince(V,E )isconnectedthereisan edge {v1,v2}∈ E with v1 ∈ V1 and v2 ∈ V2.Additionally w({v1,v2}) ≤ m2 <m1 = w(e1) andtherefore T1 isnotaminimumspanningtree,because(V, (E \{e1}) ∪{{v1,v2}})is connectedand e∈E w(e) >w({v1,v2}) w(e1)+ e∈E w(e).
AccordingtoTheoremandLemma2the LCD problemcanbesolvedusingthefollowing algorithm.
1.Computeashortestpathtree T for G atroot r
2.Determineedgeset E1 andtheirweights w asdefinedinTheorem.
3.Solvethe MST problemonthegraph(N 1 G(r),E1)withweights w
4.Returnthemaximumedgeweightthatoccursinthecomputedminimumspanning tree.
Inacentralizedalgorithmthesestepscanbeimplementedasfollows:
1.Theshortestpathtree T atroot r asdefinedinDefinition3(wherethelengthofa pathequalsthenumberofedges)canbecomputedintime O(|V | + |E|)byrunninga breadthfirstsearchonvertex r.Asimpleextensionofthisbreadthfirstsearchallows thesimultaneouscomputationofthedistance dG(r,v)andtheroot r(v)foreveryvertex v ∈ V ,whichissavedatvertex v
2.Usingtheinformationcollectedinthepreviousstep,oneiterationovertheedgeset E issufficienttodecidewhetheranedge e ∈ E isabridgeandtocomputetheweight f (e)forallofthesebridges.Let L bealistofallbridgesandforavertex w ∈ N 1 G(r) let p(r)bethepositionof w intheadjacencylistof r.Nowassigntoeverybridge {u,v} inlist L akey(p(r(u)),p(r(v)))with p(r(u)) <p(r(v)),whichisthenusedto
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
sort L inlineartimewithbucketsort.Afterwardstheedgeset E1 andtheirweights w canbeeasilyconstructedbyiteratingthesortedlistoncemore.Thereforetheoverall runningtimeforthisstepis O(|V | + |E|).
3.The MST problemasdefinedabovewithintegeredgeweightscantheoreticallybe solvedinlineartimeusingthetrans-dichotomousminimumspanningtreealgorithm presentedin[28].
4.Thisstepcanobviouslybedoneintime O(|V | + |E|).
ThediscussionaboveestablishesthefollowingTheorem. Theorem5. The LCD problemcanbesolvedinlineartime. However,thelineartimealgorithmforcomputingintegerweightminimumspanningtrees in[28]ispurelytheoreticalandnotapplicableinpractice.If,forexample,Prim’salgorithm [29]isusedforstep3,thentheoverallrunningofthealgorithmaboveisin
O(|V | + |E| +∆(G) log(∆(G))).
Additionaltothetheoreticallyachievablelineartimeimplementationbasedonglobal topologyknowledge,thisapproachissuitableforadistributedimplementationinaWSN: Ifanode r wantstodeterminetheminimumdistance d suchthatthe d-hopneighborhood N d G[r]isconnected,thenetworkneedstocooperateinordertodeterminetheedgeset E1 andtheweights w.Oncethisisdoneandthecollecteddatahasbeentransmittedtonode r,thesteps3and4canbesolvedlocallyby r itself.
Theshortestpathtree T atroot r canbebuiltbyamodifiedfloodingalgorithm,started atnode r:Everytransmittedmessage M containahop-counter i thatisincremented aftereachtransmissionandkeepstrackofthedistanceto r.Everynode v receivingsuch amessage M updatesaparentpointertoidentifyitsparent p inthetreeandnotifies p that v isachildof p,if i islowerthanthecurrentlysaveddistanceto r.Additionally M alsocontainstheneighbor u ∈ N 1(r)thatoriginallytransmittedthemessage,allowing everynode v toidentifytheroot r(v)ofitsownbranch.Basedonthisinformationit ispossibletocomputeallbridges e ∈ B andtheirweights f (e)after T hasbeenbuilt byhavingeverynode v exchangethecollecteddistanceandbranchinformationwithall neighborsin N 1(v).
Afterwards T canbeusedasadataaggregationtreetotransmitthecollectedinformation to r whilesimultaneouslycomputingtheminimumweight w(e)foreveryedge e ∈ E1: Startingattheleafsof T thenodessendalistofedgesin E1 thatresultfromtheir incidentbridgestotheirrespectiveparentin T .Everynodethatisnotaleafin T waits untilitreceivedtheselistsfromitschildrenin T ,mergesalllistsincludingitsownone bycomputingtheminimumweightforeveryedgethatiscontainedinoneofthelistsand sendstheresultinglisttoitsparent.
Toavoidfloodingtheentirenetwork,itispossibletouselimitedrangefloodingwith increasingrangestosuccessivelybuildlargersubgraphsof T until r discoversconnectivity withintheconsideredneighborhood.
Whiletheapproachaboveisveryefficientintermsoftransmittedmessages,ithasto beperformedproactively,becauseinordertodeterminethelocalconnectivitydistance
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ofanode v thenodeitselfhastoparticipateinthecomputation.Ifthisisnotviable, itisalsopossibletodeterminethelocalconnectivitydistanceofanode v withoutits participationbyusingthe k-HBFprotocol,assumingthataneighbor s ∈ N 1(v)iscapable ofdecidingwhetheranexecutionof k-HBFwassuccessful:Anadditionalcounter imax in thetransmittedmessagescankeeptrackofthemaximumvaluethatthecounter it ever reachedbeforeithasbeenreset.Ifeveryneighbor u ∈ N 1(v)computestheminimum value kmin(u)ofthe imax countersofallmessages u receivedandtransmitstheresultback totheinitiatingnode s,then s iscapableofdeterminingtheminimumvalue kmin (and thereforethelocalconnectivitydistanceof v)suchthatthe kmin-HBFprotocolsucceeds via kmin = max{kmin(u) | u ∈ N 1(v) \{s}}.Bydistributingthevalue kmin toallnodes in N 1(v),theprotocolcanbeextendedtoprovideincreasedperformanceduringfuture executionsforthesamenode v
Basedonthetheoreticalanalysisinsection,theexperimentalanalysisofthe k-HBFprotocolisperformedbysolvingthe LCD problemratherthansimulatingtheprotocolitself. Thecomputationsaredoneonrandomlygeneratedgraphusingdifferentgraphmodels thatarecommonlyusedforsimulationsinthewirelessnetworkresearchcommunity.The simulationsoftwarehasbeenimplementedinJavaandexecutedintheJavaRuntimeEnvironmentatversion8u74onasystemrunningtheKubuntu12.04LTSoperatingsystem, whoseNativePRNGwasusedforrandomnumbergeneration.
Foreverysetofparameterstheresultsarecomputedfor100graphsthataregeneratedas follows:
1.9000verticesareplacedrandomlyina3000 × 3000square.
2.Edgesareaddedaccordingtooneofthegraphmodelsdescribedbelow.
3.Theconnectedcomponentcontainingthemaximumnumberofverticesisdetermined andusedforthesimulation.
Togenerateedgesthefollowingmodelsareused:
1.UnitDiskGraph(UDG)withradius r ∈ R:Anedgebetweentwoverticesisgenerated ifandonlyiftheireuclideanisatmost r.Itiswellknownthatthisgraphmodelis unrealisticforreallifewirelessnetworksandyetitisstillusedbymanyresearchersfor simulations.Weusethismodelasapointofreference.
2.Waxmanwithparameters α ∈ [0, 1]and r ∈ R:Therandomgraphmodelintroduced byWaxmanin[30]capturesimportanteffectsthatoccurinreallifenetworks.Unlike theUDGmodelitdoesnotguaranteetheexistenceofedgesbetweenverticesthat arepositionedclosetoeachotheranditalsogenerates“long”edges.TheWaxman modelisoftenusedbyresearchers,becauseitisimplementedintheBRITEtopology generatorthatcaneasilybeusedinconjunctionwiththe ns-3 networksimulator.In thismodelanedgebetweentwovertices u,v isaddedtothegraphwithprobability P (u,v)= α exp d(u,v) 0 5 √2 r
where d(u,v)denotestheeuclideandistancebetween u and v.Thescalingfactorof
2hasbeenchosensuchthat,for α =1andthevaluesusedfor r,theaverage
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
vertexdegreeofthegeneratedgraphsissimilartotheUDGmodelwiththesame parameter r
3.Localitywithparameter r ∈ R:Thelocalitymodelthatismentionedin[31]and alsoimplementedintheBRITEtopologygeneratorpartitionstheeuclideandistances betweentwoverticesintoafiniteamountofcategoriesandassignsdifferent,constant probabilitiestoeachcategory.Basedontheparameter r andtheeuclideandistance
d(u,v)between u and v,theedge {u,v} isaddedwithprobability
P (u,v)=0.8 0.1 (i 1),
if r i 1 4 <d(u,v) ≤ r i 4 for i ∈{1,..., 8}.Edgesbetweenvertexpairs u,v with d(u,v) > 2r arenotadded.UnliketheUDGmodel,edgesbetweenverticesthatare positionedclosetoeachotherarenotguaranteedwhiletherestillisanupperbound forthedistancebetweenadjacentverticesandincontrasttotheWaxmanmodelthe probabilitydoesnotdecreaseexponentiallywiththedistance.
Foreverysetofparameters,the averagevertexdegree δ and theaveragegraphdiameter ∅ aredeterminedforcomparability.Then,foreverydistance dmin,itiscomputedhow manyofthe n vertices are dmin-locallyconnected,butnot(dmin 1)-locallyconnected, i.e.havelocalconnectivitydistance dmin.
Twospecialcasesarenotedseparatelyinthefollowingtables:The numberofseparation vertices isgiveninrow dmin = ∞,becausetheseverticesareobviouslynever d-locally connectedanditisnotpossibletoperforma NeighborhoodBroadcast forthem.
Alsothe number ∆1 ofverticeswithvertexdegree 1isgivenseparatelyduetothefact thattheyaretrivially1-locallyconnected.The∆1 verticesarealsocontainedinthe numbergivenfor dmin =1.
Figure4:Distributionoflocalconnectivitydistancesingraphswithsimilarvertexdegree:
Left: Waxman: δ =11 06,UDG: δ =11 12,Locality: δ =9 79
Right: Waxman α =0 75: δ =16.98,Waxman α =0.5: δ =17.78,UDG: δ =15.10,Locality: δ =19
Sincetheexistenceofedgesbetweenverticesthatarepositionedclosetoeachotherare guaranteedandthefactthatthemaximumlengthofanyedgeisboundedinthepurely theoreticalunitdiskgraphmodel,theTables5and6exhibitthatthemajorityofvertices inthismodelis1-locallyconnected,eveninverysparsegraphs.Allgraphsgeneratedbased ontheWaxmanmodelcontainasignificantlylowerrateof1-locallyconnectedverticesand thenumberofverticesclearlyspikesatlocalconnectivitydistance2.Inthelocalitymodel ontheotherhand,thespikegraduallyshiftsfromlocalconnectivitydistance2tolocal
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
α 1
r 50
n 899170
δ 7 63
∅ 37 38
∆1 4556
dmin #vertices(%)
1 34228(3.80)
2 797634(88 7)
3 61867(6 88)
4 6 781(0 09)
∞ 4660(0 52)
Table1:WaxmanModel
n 900000
δ 24 09
dmin #vertices(%)
1 60066(6.67)
2 839379(93 26)
3 554(0 06)
∞ 1(0 00)
α 1
r 110
n 900000
δ 35.54
∅ 12.14
∆1 0
dmin #vertices(%)
1 81543(9 10)
2 818301(90.92)
3 156(0.02)
∞ 0(0 00)
18821
dmin #vertices(%)
1 20015(2 23)
2 657711(73.41)
3 189849(21.19)
4 8373(0 93)
5 331(0 04)
6 28(0 00)
∞ 19581(2 19)
Table3:WaxmanModel
α 0 75
r 70
n 899928
δ 11.06
∅ 24.18
∆1 578
dmin #vertices(%)
1 12104(1 34)
2 872618(96.97)
3 14594(1.62)
4 30(0 00)
∞ 582(0 06)
α 0 75
r 90
n 899997
δ 18.07
∅ 16.98
∆1 30
dmin #vertices(%)
1 16325(1 81)
2 881550(97.95)
3 2092(0.23)
α
r
2 874132(97.13)
3 536(0.06)
∞ 0(0 00)
α 0 5
r 50
n 874040
δ 3.92
∅ 51.83
∆1 74781
dmin #vertices(%)
1 16385(1 87)
2 352649(40.35)
3 328613(37.60)
4 80075(9 16)
5 9 15109(1 73)
∞ 81209(9 29) 17
δ 7.38
∅ 28 04
∆
∞
r 50
n 897819
δ 7 76
d
r 60
n 899868
δ 11 12
∅ 222.63
∅ 118.13
∆1 3307
dmin #vertices(%)
1 750717(83 62)
2 67635(7 53)
3 32973(3.67)
4 17447(1.94)
5 9494(1 06)
∅ 89.56
∆1 320
dmin #vertices(%)
1 847605(94 19)
2 36412(4 05)
3 10895(1.21)
4 3153(0.35)
5 11 1254(0 14)
51 80 1568(0 20)
81 121 389(0 05)
∞ 66601(8 49)
6 10 12118(1 35)
11 21 840(0 09)
∞ 6595(0 73)
∞ 549(0 06)
r 70
n 899983
δ 15 10
∅ 73 37
∆1 39
dmin #vertices(%)
1 886854(98 54)
2 11383(1 26)
3 1470(0 16)
∞ 62(0 01)
r 90
n 900000
δ 31 16
∅ 28 91
∆1 0
dmin #vertices(%)
1 861893(95 77)
2 38107(4 23)
∞ 0(0 00) 18
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
r 30
n 790951
δ 3 75
∅ 190.64
∆1 69865
dmin #vertices(%)
1 171022(21 62)
2 365517(46.21)
3 56484(7.14)
4 10 65442(8.27)
11 40 20853(2 64)
41 102 1861(0 23)
∞ 109772(13 88)
Table7:LocalityModel
r 50
n 899844
δ 9 79
∅ 61.59
∆1 933
dmin #vertices(%)
1 407176(45 25)
2 487215(54.14)
3 4190(0.47)
4 268(0.03)
5 12(0 00)
6 7(0 00)
∞ 976(0 11)
r 70
n 899996
δ 19 00
∅ 39.14
∆1 10
dmin #vertices(%)
1 722197(80 24)
2 177787(19.75)
3 2(0.00)
∞ 10(0.00)
connectivitydistance1withincreasingvertexdegree,presumablyduetothestillexisting maximumedgelengthinthismodel.
Measuringthesuccessrateofthe k-HBFprotocolasapercentageofnon-separationvertices,theconductedsimulationsindicateasuccessrateofmorethan80%acrossallconsideredgraphsmodelsforthe5-HBFprotocol,theminimumbeingdefinedbythevery sparsegraphswithanaveragevertexdegreebelow6.Restrictedtographswithanaverage vertexdegreeofatleast7,thesuccessrateofthe5-HBFprotocolisabove95%andthe successrateofthe3-HBFprotocolisstillabove85%.
Whileonemightintuitivelypresumethatmostoftheverticeswithhigherlocalconnectivitydistanceareclosetotheborderofthegeometricregionthatgraphisplacedacross,the samplegraphstakenduringthesimulationdonotverifythisconjecture:Typicallythese verticesoffersomesortof“shortcut”throughanotherwisesparseregionofthegraph,see Figure5foranexample.
Aprotocolthatoffersthepossibilityofperforminga NeighborhoodBroadcast has applicationsinseveralareasofadhocwirelessnetworkingsuchasfaulttolerance,routing andsecurity.Thepresented k-HBFprotocoliseasytoimplementandhasbeenprovento distributeamessagesuccessfullyacrosstheneighborhoodofanon-cooperatingnode v,if theparameter k ischosentobeatleast2d 1with d beingthelocalconnectivitydistance of v andithasbeenshownthatthesetofparticipatingnodesisoptimal.
Thelocalconnectivitydistanceofavertex v canbecomputedinlineartimebasedonglobal topologyknowledgeorproactivelyinanadhocwirelessnetworkwithareasonableamount oftransmittedmessages.Thisknowledgecanbeusedtooptimizefutureexecutionsofthe k-HBFprotocolorasametricforredundancyandtheimportanceof v withinthenetwork. Alternativelythelocalconnectivitydistancecanalsobedeterminedreactivelybythe kHBFprotocolitself,whichprovidesthepossibilityofadynamicallyadaptingprotocol.
Thepresentedsimulationsdemonstratethatthe k-HBFprotocolcanbeexpectedtopro19
InternationalJournalonApplicationsofGraphTheoryinWirelessAdhocNetworksandSensor Networks(GRAPH-HOC)Vol.9,No.1,March2017
Figure5:
r
verticeshavelocalconnectivitydistance5duetotheirpositionbetweentwo“holes”.Other distancesareblue(dmin =1),green(dmin =2),yellow(dmin =3)andblack(dmin > 5). Notethatthelocalconnectivitydistanceofsomeverticesattheborderisnotverifiable basedonthisimage,becausesomeincidentedgeshavebeenremoved. Right: Topological differencebetweengraphmodels:Whiletheaveragelocalconnectivitydistancequickly approaches1withincreasingvertexdegreefortheidealizedunitdiskgraphandlocality models,themorerealisticWaxmanmodelexhibitsanaveragelocalconnectivitydistance ofabout2forreasonablydensenetworks.
videveryhighsuccessratesinrealworldnetworksforvaluesof k aslowas3to5.As asideeffect,theresultsclearlyshowasignificant,topologicaldifferencebetweengraphs generatedbytheidealizedunitdiskgraphmodelandmorerealisticnetworkmodels,furtherstrengtheningtheknownrecommendationsthatunitdiskgraphsshouldnotbeused tomodelwirelessnetworks.
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