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Trade-off Scheme for Fault Tolerant Connected Dominating Sets on Size and Diameter in Wireless Ad-hoc Networks

Permanent Link: http://ufdc.ufl.edu/UFE0022554/00001

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Title: Trade-off Scheme for Fault Tolerant Connected Dominating Sets on Size and Diameter in Wireless Ad-hoc Networks
Physical Description: 1 online resource (64 p.)
Language: english
Creator: Zhang, Ning
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

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Subjects / Keywords: Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: Connected Dominating Set (CDS) has been a well known approach for constructing a virtual backbone to alleviate the broadcasting storm in Wireless Ad-hoc Network. Current research has focused on minimizing the size or diameter, or improving the fault tolerance of CDS. However, to our best knowledge, no existing research has considered these three important factors together in a single model. In this work, we introduce the fault tolerant model studying a joint optimization problem in which the objective is to minimize the CDS size as well as the diameter, leading to the decrease in network latency. This model also addresses the tradeoffs between the three objective functions. Simulation results show that our solutions can gain good tradeoffs between the three factors, which coincide with theoretical analysis. Moreover, our solutions could obtain a better performance than others work.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ning Zhang.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Thai, My Tra.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022554:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022554/00001

Material Information

Title: Trade-off Scheme for Fault Tolerant Connected Dominating Sets on Size and Diameter in Wireless Ad-hoc Networks
Physical Description: 1 online resource (64 p.)
Language: english
Creator: Zhang, Ning
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Connected Dominating Set (CDS) has been a well known approach for constructing a virtual backbone to alleviate the broadcasting storm in Wireless Ad-hoc Network. Current research has focused on minimizing the size or diameter, or improving the fault tolerance of CDS. However, to our best knowledge, no existing research has considered these three important factors together in a single model. In this work, we introduce the fault tolerant model studying a joint optimization problem in which the objective is to minimize the CDS size as well as the diameter, leading to the decrease in network latency. This model also addresses the tradeoffs between the three objective functions. Simulation results show that our solutions can gain good tradeoffs between the three factors, which coincide with theoretical analysis. Moreover, our solutions could obtain a better performance than others work.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ning Zhang.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Thai, My Tra.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022554:00001


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TRADE-OFFSCHEMEFORFAULTTOLERANTCONNECTEDDOMINATING SETSONSIZEANDDIAMETERINWIRELESSAD-HOCNETWORKS By NINGZHANG ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2008 1

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c r 2008NingZhang 2

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Tomyparents,HepingShiandQuanguiZhang 3

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ACKNOWLEDGMENTS Mysincerestgratitudegoestomyadvisor,ProfessorMyT.Th ai,forherinvaluable guidanceandsupportthroughoutmymaster'sresearchwork. Duringtheinitialseveral monthsofmygraduatework,ProfessorThaiwasextremelypat ientandalwaysledme towardtherightdirectionwheneverIwouldwaver.Heracute insightintotheresearch problemsweworkedonsetanexcellentexampleandprovidedm eimmensemotivation. Shehasalwaysemphasizedtheimportanceofhigh-qualityte chnicalwritingandhasspent severalpainstakinghoursreadingandcorrectingmytechni calmanuscripts.Shehasbeen thebestmentorIcouldhavehopedfor,andIshallalwaysrema inindebtedtoherfor shapingmycareerandmoreimportantly,mythinking. IamalsoverythankfultoProfessorDing-ZhuDuatUT-Dallas ,forhisguidance duringmygraduatestudy.Hisenthusiasmandconstantwilli ngnesstohelphasalways amazedme.ThanksarealsoduetoProfessorJereyHo,forhis supportduringmy graduatestudy.ItakethisopportunitytothankProfessors ShigangChenandYeXia,for takingthetimetoserveonmycommitteeandfortheirhelpful suggestions. ItwasapleasureworkingwithIncheolShinonvariouscollab orativeresearchprojects. SeveralinterestingtechnicaldiscussionswithRezaMahjo urian,BoLi,VishakSivakumar andYingXuanprovidedastimulatingworkenvironmentinthe AppliedOptimization group. Thisworkwouldnothavebeenpossiblewithouttheconstante ncouragementand supportofmyfamily.Myparents,HepingShiandQuanguiZhan g,alwaysencouragedme tofocusonmygoalsandpursuethemagainstallodds.Mygrand fatherandgrandmother, RongfuShiandGuiyingLu,havealwaysplacedtrustinmyabil itiesandhavebeenideal examplestofollowsincemychildhood. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 1.1WirelessAd-hocNetwork ............................ 10 1.2ConnectedDominatingSet(CDS):AVirtualBackboneinWi relessAd-hoc Network ..................................... 10 1.3WorkOverview ................................. 10 2NETWORKMODEL,CDSQUALITYISSUES .................. 12 2.1NetworkModel ................................. 12 2.2TheQualityIssuesofCDS ........................... 14 2.3TheContributionsofWork ........................... 15 3RELATEDWORK .................................. 17 3.1BasicResults .................................. 17 3.1.1GeneralGraph .............................. 17 3.1.2UnitDiskGraph ............................ 17 3.1.3DiskGraphswithBidirectionallinks .................. 18 3.2RecentWork .................................. 18 4JOINTOPTIMIZATIONMODEL ......................... 20 4.1FaultTolerantModel .............................. 21 4.1.1ProblemDenition ........................... 21 4.1.2Notations ................................ 21 4.1.3AlgorithmDescription ......................... 22 4.1.4CorrectnessofFTAA .......................... 23 4.1.5TheoreticalAnalysis ........................... 26 4.1.5.1Theanalysisonsize ..................... 26 4.1.5.2Theanalysisondiameter ................... 34 4.1.5.3Timecomplexity ....................... 35 4.2BasicModel ................................... 36 4.2.1ProblemDenition ........................... 36 4.2.2BasicDistributedApproximationAlgorithm ............. 37 4.2.2.1Algorithmdescription .................... 37 5

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4.2.2.2Theoreticalanalysis ..................... 38 4.2.2.3Timecomplexity ....................... 39 4.2.3ProgressiveDistributedApproximationAlgorithm .......... 39 4.2.3.1Algorithmdescription .................... 40 4.2.3.2Theoreticalanalysis ..................... 42 4.2.3.3Timecomplexity ....................... 46 5SIMULATIONRESULTS .............................. 47 5.1NetworkParameters .............................. 47 5.2EectsofNumberofNodes .......................... 47 5.2.1SimulationsforBDAAandPDAA ................... 47 5.2.2SimulationsforCDS-BDandPDAA .................. 49 5.2.3SimulationsBasedonDierent ................... 51 5.2.4Simulationsfor k m -CDS ........................ 53 5.3EectsofTransmissionRatio ......................... 54 5.4EectsofNetworkDensity ........................... 56 5.5Summarization ................................. 56 6CONCLUSIONANDFUTUREWORK ...................... 59 6.1Conclusion .................................... 59 6.2FutureWork ................................... 59 REFERENCES ....................................... 60 BIOGRAPHICALSKETCH ................................ 64 6

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LISTOFTABLES Table page 5-1Runningtime(ms)forBDAA,BDAA-MidandPDAA ............... 50 7

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LISTOFFIGURES Figure page 2-1Adiskgraphwithbidirectionallinks ........................ 13 2-2Representationofawirelessnetworkwithtennodesasag raph ......... 13 2-3AnexampleforthediameterofCDS ........................ 15 4-1Allintermediatenodesareneighborsofthenodesinsepa ratingset ....... 31 4-2 s isnodeinseparatingset,node u and v arebothneighborsofnode s ...... 32 4-3 s isnodeinseparatingset, u isinits k -leafblockisaneighborof s ,while v is notinthe k -leafblockand v isnotaneighborof s ................ 33 4-4 s isnodeinseparatingset, u isinits k -leafblockwhile v isnotinits k -leafblock andneither u and v isareneighborsof s ...................... 34 4-5AllthenodesintheringareaCDSwithdiameterof8 .............. 43 5-1SimulationsforBDAAandPDAA ......................... 48 5-2SimulationsforCDS-BDandPDAA ........................ 51 5-3Simulationsbasedondierent ........................... 52 5-4Simulationsfor k m -CDS ............................... 53 5-5Eectsoftransmissionratio ............................. 55 5-6Eectsofnetworkdensity .............................. 57 8

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AbstractofThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience TRADE-OFFSCHEMEFORFAULTTOLERANTCONNECTEDDOMINATING SETSONSIZEANDDIAMETERINWIRELESSAD-HOCNETWORKS By NingZhang August2008 Chair:MyT.ThaiMajor:ComputerEngineering ConnectedDominatingSet(CDS)hasbeenawellknownapproac hforconstructinga virtualbackbonetoalleviatethebroadcastingstorminWir elessAd-hocNetwork.Current researchhasfocusedonminimizingthesizeordiameter,ori mprovingthefaulttolerance ofCDS.However,toourbestknowledge,noexistingresearch hasconsideredthesethree importantfactorstogetherinasinglemodel.Inthiswork,w eintroducethefaulttolerant modelstudyingajointoptimizationprobleminwhichtheobj ectiveistominimizethe CDSsizeaswellasthediameter,leadingtothedecreaseinne tworklatency.Thismodel alsoaddressesthetradeosbetweenthethreeobjectivefun ctions.Simulationresults showthatoursolutionscangaingoodtradeosbetweentheth reefactors,whichcoincide withtheoreticalanalysis.Moreover,oursolutionscouldo btainabetterperformancethan others'work. 9

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CHAPTER1 INTRODUCTION 1.1WirelessAd-hocNetwork Awirelessad-hocnetworkisadecentralizedwirelessnetwo rk.Thenetworkisad-hoc becauseeachnodeiswillingtoforwarddataforothernodes, andsothedeterminationof whichnodesforwarddataismadedynamicallybasedonthenet workconnectivity.This isincontrasttowirednetworksinwhichroutersperformthe taskofrouting.Itisalsoin contrasttomanagedwirelessnetworks,inwhichaspecialno deknownasanaccesspoint managescommunicationamongothernodes. Asweknow,thekeytoimproveperformanceofcomputernetwor ksistoorganize theclientsinnetworkintohierarchy.However,duetothela ckoftopologyofnetwork, wirelessad-hocnetworkareratinnature.Inordertoachiev ehighperformance,some algorithmshaveappearedthatrelyonavirtualbackbone,wh ichgroupsallclientsinto ahierarchy.Thevirtualbackboneistherstapplicationof ConnectedDominatingSet (CDS)inwirelessad-hocnetwork[ 37 ]. 1.2ConnectedDominatingSet(CDS):AVirtualBackboneinWi reless Ad-hocNetwork CDShasbeenappliedinwirelessad-hocnetworksperforming alotoffunctions. Forexample,theCDSworkedasvirtualbackbonetoincludeme diaaccesscoordination [ 1 { 3 ],unicast[ 4 { 6 ],multicast/broadcast[ 7 { 13 ],andlocation-basedrouting[ 14 ];energy conservation[ 15 { 19 ];andtopologycontrol[ 16 20 ].CDScanalsobeemployedtoachieve thediscoveryofresourceinmobilead-hocnetwork[ 21 22 ]. InSection 3.1 and 3.2 ,wearegoingtosurveytheCDSconstructionalgorithmsin previousresearchwork. 1.3WorkOverview Thisworkbeginswithadescriptionofnetworkmodel,CDSapp lication,qualityissues ofCDSandcontributionsofworkinChapter2,thenfollowedb yasurveyofrelatedwork inChapter3.Thedetailedanalysisofajointoptimizationm odelthatsolvesourproblem, 10

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arediscussedinSection4.Theresultsofexperimentsfromt heproposedsolutionsanda evaluationoftheproposedsolutionsfollowinChapter5.Fi nally,wepresentconclusions andpotentialfutureworkinChapter6. 11

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CHAPTER2 NETWORKMODEL,CDSQUALITYISSUES Thefollowingsectionsprovidesbackgroundinformationfo rtheCDSinwireless ad-hocnetworks.Werstpresentamathematicalmodelforth enetworksunder considerationandintroduceusefulterminologyanddenit ionsfromgraphtheory.Then wesketchtheapplicationofCDSinwirelessad-hocnetworka ndtalkaboutthequality issuesofCDS. 2.1NetworkModel Previousworkfocusedoncomputingtheminimumconnecteddo minatingset(will bedenedlater)undertheassumptionthatthetransmission rangeofeachnodeisequal. However,inpractice,thetransmissionrangesofallnodesa renotnecessarilyequal. Nodesinanetworkmayhavedierentpowersduetodierences infunctionalities.The node'stransmissionrangemaybeadjusteddierentlybased onthenodedistribution inanetworkandtherequirementsoftheapplications.Inthi scase,awirelessad-hoc networkcanbemodeledusingadirectedgraph G =( V;E ).Thenodesin V arelocated inthetwodimensionalEuclideanplaneandeachnode v i 2 V hasatransmission range r i 2 [ r min ;r max ].Adirectededge( v i ;v j ) 2 E ifandonlyif d ( v i ;v j ) r i where d ( v i ;v j )denotestheEuclideandistancebetween v i and v j .Suchgraphsarecalled Disk Graphs (DG).Anedge( v i ;v j )isbidirectionalifboth( v i ;v j )and( v j ;v i )arein E ,i.e., d ( v i ;v j ) min f r i ;r j g .Inthiswork,westudythefaulttolerantCDSproblemindisk graphswherealltheedgesinthenetworkarebidirectional, calledDiskGraphswith Bidirectionallinks(DGB).Inthiscase, G isundirected.Fig. 2-1 givesanexampleofDBG representinganetwork.InFig. 2-1 ,thedottedcirclesrepresentthetransmissionranges andtheblacknodesrepresentaCDS,wewillgivethedenitio nofCDSinthefollowing. Anumberofdenitionsfromgraphtheoryareusedinthissect ion.Fig. 2-2 canhelp toillustratethefollowingconcepts: 12

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Figure2-1.Adiskgraphwithbidirectionallinks c b a e fgh d ij Figure2-2.Representationofawirelessnetworkwithtenno desasagraph IndependentSet(Neighbor) ,isasubsetof V suchthatnotwoverticeswithintheset areadjacentin V .Forexample, a;b;f;h;j isanindependentsetinFig. 2-2 MaximalIndependentSet(MIS) ,isanindependentsetsuchthataddinganyvertex notinthesetbreakstheindependencepropertyoftheset.Th us,anyvertexoutsideof themaximalindependentsetmustbeadjacenttosomenodesin theset.Theprevious independentset a;b;f;h;j musthavenode d addedtobecomeanMIS. DominatingSet(DS) S ,isdenedasasubsetof V suchthateachnodein V S is adjacenttoatleastonenodein S .Thus,everyMISisadominatingset.However,since nodesinadominatingsetmaybeadjacenttoeachother,notev erydominatingsetisan MIS.Findingaminimum-sizeddominatingsetisNP-Hard[ 29 ]. 13

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ConnectedDominatingSet(CDS) C ,isaDSof G whichinducesaconnected subgraphof G .OneapproachtoconstructaCDSistondMIS,andthenaddadd itional nodestoconnectthenodesintheMIS.ACDSinFig. 2-2 is c;d;e;g MinimumConnectedDominatingSet(MCDS) istheCDSwithminimumcardinality. GiventhatndingminimumsizedDSisNP-Hard,itshouldnotb esurprisingthatnding theMCDSisalsoNP-Hard[ 29 ].InFig. 2-2 c;j;g isaMCDS. k -ConnectedGraph isagraph G ifitisconnectedandremovingany k 1nodesfrom G willnotpartition G ,i.e, G isstillconnected. FaultTolerantConnectedDominatingSet( k m -CDS) isasubset C V which satisfyingthefollowingtwoconditions:(i)thesubgraphi nducedby C ,i.e., G [ C ],isa k -connectedgraph,and(ii)eachnodenotin C isdominated(adjacent)byatleast m nodesin C .InFig. 2-2 c;d;j;e;g isa2-1-CDS. Diameter ofaconnectedgraph G isthenumberofhopcountsinthelongestshortest pathbetweenanypairofnodesin G .Denote d ij asthenumberofhopsintheshortest pathbetweennode i andnode j .ThenthediameterofaCDS d ( CDS )= max ( d ij ),where i and j areanytwonodesinCDS.Forexample,inFig. 2-2 ,thediameterofthegraphis 4. 2.2TheQualityIssuesofCDS ThemajorgoalofmanyworksistodetermineaminimumCDS(MCD S)inunitdisk graph(UDG)[ 23 ].Thus,themajorgoalistocalculatetheperformanceratio .Minimizing thecardinalityoftheCDScandecreasethecontroloverhead sincebroadcastingforroute discovery[ 24 25 ]andtopologyupdates[ 26 ]isrestrictedtoasmallsetofnodes[ 23 ]. Therefore,broadcastingstormproblem[ 27 ]inherenttoroodingcanbegreatlyreleased. SincethenodesinaCDSneedtocarryothernodestrac, faulttolerance mustbe alsoconsidered.Unfortunately,CDSisoftenveryvulnerab leduetofrequentnodefailure andlinkfailure,whichisinherentinwirelessadhocnetwor ks.Therefore,constructinga 14

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faulttolerantCDSthatcontinuestofunctionduringnodeor linkfailureisanimportant researchproblem. ATheCDSbyGK-algorithm BTheCDSwithsmallerdiameter Figure2-3.AnexampleforthediameterofCDS ACDSwithlargediameteroftenleadstoanincreaseinthepro pagationerror.On theotherhand,aCDSwithsmalldiameteriscertainlyprefer redforreliablemessage deliveryandshortdelayinthiscase.Forexample,asdiscus sedin[ 45 ],aCDS,shownin Fig. 2-3 (a),producedbythewellknownGK-algorithm[ 28 ]hasarelativelysmallersize butlargerdiameter.However,thenetworkadmitsaCDSwiths mallerdiameterasshown inFig. 2-3 (b). 2.3TheContributionsofWork SinceaCDSproblemisNP-hard[ 29 ]anditiseasytoreducetheCDSproblemtoour modelinpolynomialtime.Therefore,weexpectthatourmode lisalsoNP-hard. Astoourbestknowledge,noexistingresearchhasconsidere dthesethreeimportant factorstogetherinasinglemodel.Inthiswork,werststud ytheproblemofconstructing aCDSbyconsideringthethreefactorstogether.Theapproxi mationalgorithmfor k -Connected m -DominatingSets( k m -CDS,alsoknownasfaulttolerantCDS)with boundeddiameterispresented.Weminimizethesizeandthed iameterof k m -CDSwhile maintainingitsfaulttolerance.Thetradeosbetweenobje ctivefunctionsareshownby provingits'approximationratios.Meanwhile,as1-Connec tedDominatingSets(1-CDS) 15

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withboundeddiameteristheprerequisiteofourmodel,twod istributedalgorithmsare addressedforconstructing1-CDS,whichconsiderthesizea nddiameterofitatthe sametime.Solidproofsfortheirapproximationratiosarep resented,whichindicate thetradeosonsizeanddiameteralsoexistsin1-CDS.Moreo ver,ourmodelallows user-denedinputstobalancethesizeanddiameterof1-CDS .Intheend,weevaluateour proposedalgorithmsthroughtheexperiments. Thecontributionsofthisworkareasfollows:Anapproximationalgorithmfor k m -CDSwithboundeddiameterisproposed.Three factorsareoptimizedatthesametimeandthetradeosbetwe enobjectivefunctionsare presentedintheoreticalanalysisandsimulation.Toourbe stknowledge,itistherstwork tostudythismodel. Twoapproximationalgorithmstominimizethesizeanddiame terof1-CDSindisk graphsarepresentedindistributedmannerforourmodelasa whole.Thebenetsof proposedalgorithmareeitherfeaturedwithlowtimecomple xityoreectiveinminimizing thesizeanddiameterof1-CDS. Theperformanceofourmodelisadjustablebytheuser-dene dinput.Through extensivesimulation,weverifythisfactandtheresultssh owouralgorithmswillhave goodtradeosbetweenthethreefactors,whichcoincidewit htheoreticalanalysis. ComparingwithCDS-BDalgorithmproposedin[ 47 ],thesimulationresultsshowthat ouralgorithmsoutperformCDS-BDunderthesamenetworkcon dition. 16

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CHAPTER3 RELATEDWORK Inthissection,wedescribethemainideasofmanyrelatedwo rkonconstructinga CDSwiththeirtheoreticalanalysisresults. 3.1BasicResults 3.1.1GeneralGraph Severalworkhavebeenstudiedingeneralgraph.In[ 28 ],twopolynomialtime algorithmstoconstructaCDSinageneralgraphisproposedb ytheauthors.Therst algorithmhasperformanceratioof2( H ( )+1),where H isaharmonicfunctionand is themaximumdegreeof G .Theideaoftherstalgorithmistoidentifythenodewitha maximumdegreeastherootandthenbuildaspanningtree T attheroot,grow T until allnodesareaddedto T .Then,allleafnodesarecutoandtheremainingnodesin T are aCDS.Thesecondalgorithmisaprogressoftherstalgorith m.Thesecondalgorithm consistsoftwosteps.Therststepistoconstructadominat ingsetandthesecondstep istoconnectthedominatingsetwithaSteintertree.Withsu chimprovement,thesecond algorithmhasabetterperformancefactorof H ( )+2.Later,thetwoalgorithmswere simulatedbyDas etal. in[ 4 30 31 ]. 3.1.2UnitDiskGraph InUDG,mostofproposedalgorithmsaretondanMISandthenc onnecttheMIS withminimumnumberofnodes.In[ 32 { 34 ],theauthorspresentedadistributedalgorithm withaconstantperformanceratioof8.Later,Cardei etal. presentedanotherdistributed algorithmin[ 35 ].Thisalgorithmhasthesameperformanceratioasprevious work. However,themessagecomplexityislowerthanthatof[ 32 ]. Asweknowthatdistributedalgorithmhasabetterperforman cethanlocalized algorithms.Inthelocalizedalgorithms,in[ 36 ],Alzoubi etal. proposedalocalized algorithmswithaperformanceratioof192.Althoughtheper formanceof[ 36 ]can notcompetewiththatof[ 32 ]and[ 35 ].Theiralgorithmonlyneedonehopneighbors 17

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information.Therefore,onceanodeknowsthatithasthesma llestIDamongitneighbors, itbecomesadominator.Then,thedominatorscanbeconnecte dbytheintermediate nodesinthenextstep.In[ 38 ],Li etal. proposedanotherlocalizedalgorithmwitha performanceratioof172,whichisbetterthan[ 36 ]. 3.1.3DiskGraphswithBidirectionallinks SincethespecicgeographicalcharacteristicsofDGB,not allCDSconstruction algorithmsthatareapplicableinUDGcanbeappliedtoDGB.A sfarasweknow, thealgorithmsin[ 32 35 ]areapplicableinDGB.In[ 39 ],Thai etal. rstproposedthe performanceratioofCDSonsizeinDGBandthetwoproposedal gorithmscanbe implementedbydistributedways.However,theonlydieren cebetweentwoalgorithms isthestrategytoselectMIS,therstalgorithmemployedWa n'salgorithm[ 32 ]tochoose thenodesinMIS,whilethesecondalgorithmusedthegreedys trategy,thatistoinclude theminimumnumberofnodesinMIS,thusleadingtoabetterpe rformancethantherst algorithm. 3.2RecentWork Themaingoalofaboveworkistominimizethesize(thenumber ofnodes)ofCDS. Inthefollowing,wewilldiscusstherecentworkonqualityi ssuesofCDS: faulttolerance and diameter In[ 40 ],Wang etal. introducedtheproblemofconstructing k m -CDSinUDGand proposedaconstantapproximationalgorithm.However,Wan g etal. onlystudieda specialcasewhere k =2and m =1,whichisnotgeneralforthe k m -CDSproblem.In [ 41 ],Shang etal. studiedthe k m -CDSproblemwith k =2and m isanarbitrarynumber. Theresultsin[ 41 ]arestillnotgeneral,since k isrestrictedtoaconstantnumber.Based ontheextensionof[ 40 ],Thai etal. independentlyproposedanapproximationratioof k m -CDSingeneralcase[ 42 ],where k and m arenotboundedtoconstantnumbers. Besidestheapproximationalgorithms,severaldistribute dalgorithmwithout performanceratiosarealsoproposedin[ 43 44 ]forfaulttolerantCDS.In[ 44 ],Dai et 18

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al. addresstheproblemofconstructing k -connected k -dominatingvirtualbackbonewhich is k -connectedandeachnodenotinthebackboneisdominatedbya tleast k nodesinthe backbone.Theyproposethreelocalizedalgorithms.Twoalg orithms, k -Gossipalgorithm andcolorbased k -CDSalgorithm,areprobabilistic.In k -Gossipalgorithm,eachnode decidesitsownbackbonestatuswithaprobabilitybasedont henetworksize,deploying areasize,transmissionrange,and k .Colorbased k -CDSalgorithmproposesthateach noderandomlyselectsoneofthe k colorssuchthatthenetworkisdividedinto k -disjoint subsetsbasedonnodecolors.Foreachsubsetofnodes,aCDSi sconstructedand k -CDS istheunionof k CDSs.Thedeterministicalgorithm, k -Coveragecondition,onlyworks inverydensenetworkandnoupperboundonthesizeofresulta ntbackboneisanalyzed. In[ 43 ],Wu etal. proposedonecentralizedheuristicalgorithmCGAandtwodi stributed algorithm,DDAwhichisdeterministicandDPAwhichisproba bilistic,toconstructa k m -CDSforgeneral k and m ThediameterofCDShasnotbeenstudiedextensivelyuptilln ow.Mohammed et al. mentionedtheproblemofconstructingCDSwithsmalldiamet er[ 45 ].theymodied thesteppingofGK-algorithminbreadthrstmannermimicki ngtheconstructionofthe breadthrstsearch(BFS)tree[ 46 ].Thealgorithmstartstheconstructionofthetree fromaselectednodeastheroot.Thenodesareprocessedinth eorderofincreasinghop distancefromtheroot.However,theydidnotgiveaguarante edperformanceintheir model.In[ 47 ],Li etal. studiedtheCDSproblemwithboundeddiameterinUDGand proposedaconstantapproximationalgorithm,calledCDS-B D.ThemainideaofCDS-BD istoselectthenodesinMISlevelbylevelandthenconnectth eMISbyconstantnumber ofnodesandhops.Thelevelofeachnodeisdenedasthenumbe rofhopsfromtheroot. However,theiralgorithmiscentralizedandcanonlybeappl iedtoUDG. 19

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CHAPTER4 JOINTOPTIMIZATIONMODEL Astoourbestknowledge,noexistingresearchhasconsidere dthesethreeimportant factors, size,faulttoleranceanddiameterofCDS ,togetherinasinglemodel.Inthis work,werststudytheproblemofconstructingaCDSbyconsi deringthethreefactors together.Theapproximationalgorithmfor k m -CDS(faulttolerantCDS)withbounded diameterandsizeispresented.Weminimizethesizeandthed iameterof k m -CDSwhile maintainingitsfaulttolerance.Thetradeosbetweenobje ctivefunctionsareshownby provingits'approximationratios.Meanwhile,as1-Connec tedDominatingSets(1-CDS) withboundeddiameteristheprerequisiteofourmodel,twod istributedalgorithmsare addressedforconstructing1-CDS,whichconsiderthesizea nddiameterofitatthe sametime.Solidproofsfortheirapproximationratiosarep resented,whichindicate thetradeosonsizeanddiameteralsoexistsin1-CDS.Moreo ver,ourmodelallows user-denedinputstobalancethesizeanddiameterof1-CDS .Intheend,weevaluateour proposedalgorithmsthroughtheexperiments. Thecontributionsofthisworkareasfollows:Anapproximationalgorithmfor k m -CDSwithboundeddiameterisproposed.Three factorsareoptimizedatthesametimeandthetradeosbetwe enobjectivefunctionsare presentedintheoreticalanalysisandsimulation.Toourbe stknowledge,itistherstwork tostudythismodel. Twoapproximationalgorithmstominimizethesizeanddiame terof1-CDSindisk graphsarepresentedindistributedmannerforourmodelasa whole.Thebenetsof proposedalgorithmareeitherfeaturedwithlowtimecomple xityoreectiveinminimizing thesizeanddiameterof1-CDS. Theperformanceofourmodelisadjustablebytheuser-dene dinput.Through extensivesimulation,weverifythisfactandtheresultssh owouralgorithmswillhave goodtradeosbetweenthethreefactors,whichcoincidewit htheoreticalanalysis. 20

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ComparingwithCDS-BDalgorithmproposedin[ 47 ],thesimulationresultsshowthat ouralgorithmsoutperformCDS-BDunderthesamenetworkcon dition. 4.1FaultTolerantModel Inthissection,weintroduceourproblemandprovideasolut ionfor k m -CDSwith boundedsizeanddiameter,undertheconditionthat1 k m +1. 4.1.1ProblemDenition Beforeweintroducethedenitionoftheproblem,weneedtog ivethefollowing denitions:A separatingsetorcut-vertex ofagraph G =( V;E )isaset S V ,suchthat G S hasmorethanonecomponent.When j S j =1, S iscalledacutvertex.A k -block of agraphisamaximal k -connectedsubgraphof G thathasnoseparatingset.If G itselfis k -connectedandhasnoseparatingset,then G isa k -block.The k m -CDSwithbounded diameterproblemcouldbeformallydenedasfollows: Denition1. k m -CDSProblemwithBoundedDiameter: GivenaDGB G = ( V;E ) representinganetworkandtwopositiveintegers k and m ,ndasubset C km V satisfyingthefollowingthreeconditions:(1)thesubgrap hinducedby C km ,i.e., G [ C km ] ,is k -connected,and(2)eachnodenotin C km isdominated(adjacent)byatleast m nodesin C km .(3)thesizeanddiameterof C km arebounded. 4.1.2Notations Somenotationsforbetterunderstandingofthealgorithmde scriptionarelistbelow. I i beanyMaximalIndependentSet(MIS)in G ( I 1 [ I 2 [ ::: [ I i 1 ) N ID ( u )betheindependentneighborhoodofnode u V ( L )denotethevertexsetofsubgraph L r bethetransmissionrangeratio,i.e. r = r max r min Inthepreviouswork[ 39 ],Thai etal. haveprovidedtheupperboundonthesizeof theindependentneighborhoodofanynode u inDGBasfollows: Lemma1. [ 39 ]InaDGB,thesizeof N ID ( u ) isboundedby K ,i.e, j N ID ( u ) j K where K =5 if r =1 ,otherwise, K =10( d ln( r ) ln(2cos( 5 )) e +1) 21

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Notethatwhen r =1,aDGBisaUDG.Hencealltheanalysisinthisworkarealso appliedforaUDG.4.1.3AlgorithmDescription Foraboveproblem,wewillproposeageneralsolution,where 1 k m +1,called FaultTolerantApproximationAlgorithm(FTAA).Themainid eaofFTAAisasfollows. Givenanetwork G =( V;E ),thealgorithmconsistsofvemainsteps: 1.UsetheCDSMIS(tobedescribedlater)algorithmtoconstr ucta1-Connected m -DominatingSet(1m -CDS)of G 2.Computeallthe k 0 -blockin1m -CDS.Initially, k 0 =2. 3.Ifthereismorethanone k 0 -blockin1m -CDS,ndtheshortestpathintheoriginal graphthatsatisesthetworequirements:(i)thepathcanco nnecta k 0 -leafblockin 1m -CDStootherportionof1m -CDS.(ii)thepathdoesnotcontainanynodesin 1m -CDSexceptthetwoendpoints.Thenaddallintermediatenod esinthispathto 1m -CDS. 4.Repeatstep3)untilthereisonlyone k 0 -blockin1m -CDS. 5.Increase k 0 by1ateachiterationandrepeatStep3)until C is k m -CDS. Basedonaninputof1-Connected1-DominatingSet(1-CDS),w erstbuilda 1m -CDSbycallingthefunctionCDSMIS.Thenmergeallthe k 0 -blocksin1m -CDSinto onlyone k 0 -blockbyaddingextranodes,where k 0 =2initially.Then,weincrease k 0 by1 andrepeattheaboveoperationuntil k 0 = k .TheformalpresentationofFTAAisshownin Algorithm 2 Asjustmentionedearlier,wewillemployafunctioncalledC onnectedDominatingSet byMaximalIndependentSets(CDSMIS)tobuilda1m -CDS.ThemainstepsofCDSMIS is:(i)givena1-CDSwithaboundedsizeanddiameter.(ii)it erativelyaddnodesinto aCDStomakeita1m -CDS.AsshowninAlgorithm 1 ,theCDSMIShavetwostages. Attherststage,wecanuseany1-CDSwithboundedsizeanddi ameterastheinput ofAlgorithm 1 suchthatthe1-CDSincludesanMIS I 1 of G .However,inordertomake 22

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thesolutionadjustablebytheuser,an( ; )-CDS,tobeintroducedinSection 4.2.2 ,is preferredtobeaninputofAlgorithm 1 .Then,weremove I 1 from G atthisstage. Atthesecondstage,weiterativelyndanMIS I i intheremaininggraph G ( I 1 [ I 2 [ ::: [ I i 1 )bychoosingabiggesttransmissionrangeanddeleteitsnei ghborsuntilthereis nonodesexists,thenadd I i intothe1-CDSatcurrentiteration.Afterrunningthesecon d stage( m 1)times,theresultingCDSis1m -CDS. Algorithm1 ConnectedDominatingSetbyMaximalIndependentSetAlgori thm (CDSMIS)for1m -CDS 1: INPUT:An m -connectedDGB G =( V;E )anda1-CDS C 11 withboundeddiameter andsize, C 11 mustincludeanMIS I 1 of G 2: OUTPUT:A1m -CDS C 1 m ofG 3: C 1 m = C 11 4: Remove I 1 fromthegraph 5: for i =2to m do 6: ConstructanMIS I i in G ( I 1 [ I 2 [ ::: [ I i 1 ) 7: C 1 m = C 1 m [ I i 8: endfor 9: Return C 1 m where C 1 m isthe1m -CDS Theintuitionofthisalgorithmisthata k m -CDSisalsoa1-CDS,thusbygiven a1-CDSwithboundedsizeanddiameter,wedonotintroducean yunnecessarynodes. Moreover,weonlyaddnodesthatarenecessarytomakethe k -connectedand m -dominating portionlarger.Intotal,thesizeanddiameterof k m -CDScanbebounded,wewillprove thisinthefollowingsections.4.1.4CorrectnessofFTAA Intherststep,weemployCDSMIStobuilda1m -CDS C 1 m of G .Notethatthis guarantees C 1 m isa m -dominatingset.Therefore,toprovethecorrectnessofFTA A algorithm,weneedtoprove C 1 m is k -connectedafterthealgorithmterminatesandthe algorithmrunsinboundedtime. Lemma2. Inline8(ofFTAA),a k 0 -leafblockalwaysexistswhen C 1 m isnot k 0 m -CDS. Proof: Bycontradiction,supposethereisno k 0 -leafblockin C 1 m .Accordingtothe denitionsofseparatingsetand k -leafblock,theremovalofany( k 0 1)nodeswill 23

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Algorithm2 FaultTolerantApproximationAlgorithm(FTAA) 1: INPUT:A k -connectedgraph G =( V;E ) 2: OUTPUT:A k m -CDS C km withboundeddiameterandsize 3: Constructa1m -CDS C 1 m bycallingCDSMIS 4: Initialize k 0 =2; 5: B=ComputekBlock( C 1 m ;k 0 );/* B isalistofall k 0 -blocksin C 1 m */ 6: while k 0 k do 7: while B containsmorethanone k 0 -block do 8: L=ndkLeafBlock( B;k 0 );/* L isone k 0 -leafblock*/ 9: for eachnode v 2 V ( L )and v isnotanodeinseparatingset do 10: for eachnode u 2 C V ( L ) do 11: Construct G 0 from G bydeletingallnodesin C 1 m (except u and v )andall edgesincidenttothosenodes; 12: if thereexistatleastone uv -pathin G 0 then 13: P uv =shortestPath( v;u;G 0 );/* P uv istheshortestuv-pathcontainingonly thosenodesnotin C 1 m astheintermediatenodes*/ 14: endif 15: P = P [ P uv 16: endfor 17: endfor 18: P ij =thepathwithshortestlengthamongallpathsin P ; 19: C 1 m = C 1 m [ intermediatenodeson P ij ; 20: B=ComputekBlock( C 1 m ;k 0 ); 21: endwhile 22: k 0 ++ 23: endwhile 24: Return C 1 m where C 1 m isa k m -CDS C km partition C 1 m .Therefore, C 1 m isatleast k 0 m -CDS,whichcontradictstothefactthat C 1 m isnot k 0 m -CDS. 2 Lemma3. Inline18,theshortestpath P ij alwaysexists. Proof: Consider C 1 m suchthat C 1 m isnot k 0 -connected.Toprove P ij alwaysexists,we provethattherealwaysexistsapathwithonlythosenodesno tin C 1 m toconnecta k 0 -leaf blocktoa k 0 -blockof C 1 m .Thisistruebecause G is k -connectedand k 0 k .Ifwedelete theseparatingsetof C 1 m ,theremustexistapath P intheoriginalgraph G connecting the k 0 -leafblocktoa k 0 -block.Ontheotherhand,thereare( k 0 1)disjointpathswhich 24

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connectthe k 0 -leafblocktoother k 0 -blocksthroughtheseparatingsetwithsizeof( k 0 1) in C 1 m ,thusallnodesexceptthetwoendpointson P arenotin C 1 m 2 Lemma4. [ 48 ]Two k 0 -blocksin ( k 0 1) m -CDSshareatmost ( k 0 1) nodes. Whentwo k 0 -blocksof( k 0 1)m -CDSshare( k 0 1)nodes,itisstraightthattheset of( k 0 1)nodesmustbeaseparatingsetwiththesizeof( k 0 1). Lemma5. Fromline7to21(theinnerwhileloop),a k 0 -leafblockismergedintoa k 0 -blockthroughpath P ij toformalarger k 0 -blockwithoutgeneratinganynew k 0 -block. Proof: Considerasubgraph H ,whichiscomposedofa k 0 -leafblockanda k 0 -block. Atthisstep, H is( k 0 1)-connected,withaseparatingset S .Let H 0 beasubgraph obtainedfrom H byaddinganewpath P ij .Wearguethataseparatingset S 0 of H 0 must havesizeatleast k 0 ,whichmeans H 0 isatleasta k 0 -block.Therearetwopossibilities:(i) oneormorenodeson P ij existin S 0 .(ii)nonodeon P ij fallsinto S 0 Forcase(i),thewaytoseparate H 0 withminimumsizeofseparatingsetisto separate H .Becauseifweseparateanynodeon P ij ,says x ,fromtheremainingpartof H 0 j S 0 j k 0 +1,since x isdominatedbyatleast k 0 dierentnodesin H andatleastone nodeon P ij existsin S 0 .Ifweseparate H ,fromLemma 4 ,wehave j S j k 0 1.Thus, j S 0 jj S j +1 k 0 1+1= k 0 Forcase(ii),sincenonodeon P ij belongsto S 0 andassume y isanynodein P ij ,if N ( y ) S 0 ,then j S 0 j k ,becauseatthistime y isnotin H ,thus, y musthaveatleast k 0 dierentneighborsin H .Otherwise, y and N ( y ) S 0 lieinasinglecomponentof H 0 S 0 Thus S 0 mustseparate H 0 and j S 0 j k 0 .Therefore, H 0 isa k 0 -block. 2 FromLemma 5 ,ateachiterationintheinnerloop,thenumberof k 0 -blockis decreasedatleastbyone.Thus,whentheinnerloopterminat es, C 1 m mustbe k 0 m -CDS. Intheouterloop,weincrease k 0 by1andconstructa k 0 m -CDS.Therefore,baseonthe inductionmethod,weconcludethatourobtained C 1 m isa k m -CDSattheend. 25

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4.1.5TheoreticalAnalysis Inthissection,werstdiscusstheperformanceboundonsiz eof C km constructedby FTAA,thentheanalysisofthediameterof C km isalsoproposedaswell. 4.1.5.1Theanalysisonsize Itiseasytoseethattheunionof1m -CDSandthenumberofnodeswewilladd inthe1m -CDSsoastomakeit k -connected,willbetheupperboundof k m -CDS. Therefore,wewillpresenttheupperboundsofthetwopartss eparatelyandaddthem togetherintheend. The1m -CDSinCDSMISistheunionof m MISsandaset B ,where B isasetof nodesthatmake I 1 connected.Notethattheunionof I 1 and B isaninputof1-CDS withboundedsizeanddiameter.Toanalyzetheapproximatio nratioofCDSMIS,werst comparethesizeofanyMISin G withthesizeofanoptimal1m -CDS.Let I denotesan MISofaremaininggraph G atanystep, C 1 m beoursolutionobtainedfromCDSMIS, D m beanoptimal m -dominatingsetin G ,and C 1 m ,and C 11 betheoptimalsolutionsof 1m -CDSand1-CDSonsizerespectively.Wehavethefollowingle mma: Lemma6. Let G =( V;E ) beanyDGBwithboundedtransmissionrangeratio r ,then j I j ( K m +1) j DS m j Proof: Letusconsider I DS m ,therearetwopossibilities:(i) I DS m = ; ,thatis I DS m ,and(ii) I DS m 6 = ; Case(i):Because I DS m ,wehave: j I jj DS m j Case(ii):Forall u 2 I DS m ,let D u = j DS m \ N ( u ) j .As DS m isan m -dominating setof G D u m foreach u 2 I DS m andwehave: X u 2 I DS m D u m j I DS m j 26

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Forall v 2 DS m ,let d v = j ( I DS m ) \ N ( v ) j .FromLemma 1 ,forall v 2 DS m there areatmost K independentnodesinitsneighborhoodand d v K .Therefore,wehave: K j DS m j X v 2 DS m d v However,notethat: P u 2 I DS m D u = j ( u;v ) 2 E j u 2 I DS m ;v 2 DS m j = P v 2 DS m d v Fromtheabove,wehave: m j I DS m j P u 2 I DS m D u = P v 2 DS m d v K j DS m j Therefore, m j I DS m j K j DS m j Thusitfollowsthat: j I j ( K m +1) j DS m j Thereforeintwocases(i)and(ii),weconcludethat: j I j ( K m +1) j DS m j 2 Lemma7. Thenumberofnodesaddedin m MISsisatmost ( K + m ) j C 1 m j Proof: Thenodesaddedin m MISsare j I 1 [ I 2 [ ::: [ I m j .Clearly, j I i jj I j andfrom Lemma 6 ,wehave: j I 1 [ I 2 [ ::: [ I m j m j I j m ( K m +1) j DS m j ( K + m ) j DS m j ( K + m ) j C m j 2 27

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Supposewealreadyhaveaninputof1-CDSwith -approximationratioonsize. Then,wewillhavethefollowingimportantconclusion. Theorem1. Ifgivenaninputof1-CDSwith -approximationonitssize,CDSMIScan producea 1 m -CDSwithaperformanceboundof ( + K + m 1) Proof: FromLemma 7 ,wehave: j C 1 m j = j B j + j I 1 [ I 2 [ ::: [ I m j j C 11 j +( K + m 1) j C 1 m j ( + K + m 1) j C 1 m j 2 Theabovetheoremconcludesthatifthetransmissionranger atio r isboundedand theinputof1-CDShasconstantapproximationonsize,thenC DSMIScanconstructa 1m -CDSwithapproximationfactorof O (1). Inthefollowingcontext,wewillfocusonthediscussiononh owtocalculatetheupper boundofthenumberofnodeswehavetoaddin1m -CDS. FromLemma 4 ,wehaveknownthatiftwo k 0 -blocksshareatmost( k 0 1)nodes,the setoftheshared( k 0 1)nodesmustbeaseparatingsetwiththesizeof( k 0 1).Now,we provethefollowinglemma: Lemma8. A k 0 -blockanda k 0 -leafblockhaveonlyoneseparatingsetwiththesizeof ( k 0 1) Proof: Bythecontradictionmethod,assumethatwehaveanothersep aratingset S 2 withthesizeof( k 0 1)besidestheseparatingset S 1 ,whichisexactlythesetofall( k 0 1) sharednodesbythetwoblocks B 1 and B 2 .Whenwedelete( k 0 1)nodesfrom S 2 ,what remainsisconnected,becauseweretainapathforanytwonod esin B 1 [ B 2 .Thus,we mustdeleteatleast k 0 nodesin S 2 tocausepartitionofthesubgraphcomposedby B 1 and B 2 .So, j S 2 j k 0 ,contradictingourassumptionthat j S 2 j =( k 0 1). 2 28

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Asmentionedbefore,thevalueof k isintherangefrom1to m +1.Then,givenany positivenumberfor k and m ,where k m +1,wewilldivideourdiscussionintotwocase: (1) k m and,(2) k = m +1.ForCase(1): Lemma9. When k m ,atmost2newnodesareaddedinto C 1 m ateachaugmenting step.Thatis,theshortestpath P ij hasatmost2nodesnotin C 1 m Proof: Eachnodein P ij mustbe m -dominatedbecausewerstbuilda1m -CDSat therststep.Fromlemma 8 ,wehaveonlyoneseparatingset S withsizeof( k 0 1). Therefore,eachnodeon P ij mustbedominatedbyatleast1nodein C 1 m butnotin S Hence, P ij hasatmost2nodesnotin C 1 m 2 Lemma10. Weemployatmost j C 1 m 1 j augmentingstepsineachiterationtochange C 1 m from k 0 -connectedto k 0 +1 -connected. Proof: Supposethecurrent C 1 m is k 0 -connectedandFTAAhastorun atleast j C 1 m j augmentingstepstomake C 1 m k 0 +1-connected.Wehavetouseatmost j C 1 m 1 j stepstomakeallnodesin C 1 m tobe k 0 +1-connected.Thatresultsinthealgorithmwill employatleastonesteptoconnectanodenotin C 1 m toa k 0 +1-leafblocktomakethem k 0 +1-connected.However,bythedenitionof1m -CDS,eachnodenotin C 1 m mustbe dominatedbyatleast m nodesin C 1 m ,where k 0 +1 m .Thus,eachnodenotin C 1 m hasbeen m -connectedwiththerestofthegraphandthealgorithmwilln otruntheextra augmentingstepstoconnectthenodesnotin C 1 m withtherestofgraph.Therefore,we onlyneedatmost j C 1 m 1 j augmentingstepsineachiteration. 2 Theorem2. When k m ,theFTAAalgorithmproducesa k m -CDSwithsizebounded by ( + K + m 1)(2 k 1) inaDGB. Proof: Let C km beanoptimalsolutionfor k m -CDSand C km bethesolutionobtained fromourFTAAalgorithm.Notethattheconstructionof C km canbedividedintotwo parts:buildinga1m -CDSandaddingsomepaths P ij withatmost2nodesinit.Since wealreadyhavetheboundon1m -CDS(theorem 1 ),wejustneedtoanalyzethenumber 29

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ofintermediatenodeswewilladdinto C 1 m inthesecondpart.FromLemma 9 and Lemma 10 ,atmost2nodesareaddedineachstepandweemployatmost j C 1 m j 1steps tochange C 1 m from( k 0 1)m -CDSto k 0 m -CDS.Inordertoconstructa k m -CDS,we havetorepeatthisprocedure( k 1)times.Therefore,wehave: j C km jj C 1 m j +2( k 1)( j C 1 m j 1) (2 k 1) j C 1 m j ( + K + m 1)(2 k 1) j C 1 m j ( + K + m 1)(2 k 1) j C km j 2 Now,wewillconsidertheCase2indetails.InLemma 9 ,when k m ,weprovethatatmost2newnodesareaddedinto C 1 m ateachaugmentingstep.However,when k = m +1,ateachaugmentingstepinthelast outeriteration,thenumberofaddednodesisquitedierent InCase2, m isonesmallerthan k .Let C k 0 m bethe k 0 m -CDSattheiteration k 0 = k 1(aniterationrightbeforethelastiteration,atthistime k 0 = m ).Thus,werst builda1m -CDS,andthenrunFTAAtomakebecomea k 0 m -CDS.Notethateachnode notin C km willbedominatedbyatleast( k 1)dierentnodesin C km .Therefore,Lemma 9 maynotholdatthelastiterationwhere k 0 = k .Instead,wehavethefollowinglemma: Lemma11. Atthelastiterationwhen k 0 = k ,weaddatmost 2( K +1) nodesto C k 0 m at eachaugmentingstepinordertomakeitbecomea k m -CDS,where k = m +1 Proof: Supposewemarkthenodesin C k 0 m withBLACKandtheremainingnodes withGRAY.Suppose L isa k -leafblockof C k 0 m and S istheseparatingset.Suppose nodes u and v ,where u 2 V ( L )and v 2 C k 0 m V ( L ),arethetwoblacknodesconnected bytheshortestpathwithoutanyblacknodes,therearethree possibilitiesthatnodes u and v areconnected,thatis u and v areconnectedbyoneconnector,twoconnector,and morethantwoconnectors.Fig. 4-1A illustratesthescenarioofexistingmorethantwo connectorsintheshortestpath. 30

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Weclaimthatiftheshortestpathbetween u and v ,called P uv hasmorethantwo intermediatenodes,allintermediatenodesintheshortest pathexcept x and y mustbe dominatedbyallnodesinseparatingset S .(SeeFig. 4-1A )Thisistruebecause:the C k 0 m is( k 1)-dominating,suppose P uv is u;x;:::::;y;v andoneoftheintermediatenodes,let's saynode z isdominatedbyonenodeinseparatingset S (asillustratedinSeeFig. 4-1B ), z musthaveanotherblacknodeneighbor p orelse z isnotdominatedby( k 1)dierent nodesinthe C k 0 m ,whichiscontrarytothefactthatwebuild1-( k 1)-CDSatrst.Ifso, thepathbetween pu or pv hasashorterdistancethan P uv ,whichcontradictsthat P uv has theshortestdistance. k-block k-leaf block u x y v z S AScenariosformorethantwoconnectors k-block k-leaf block u y v z S p x BPath pv issmallerthanuv Figure4-1.Allintermediatenodesareneighborsofthenode sinseparatingset Weshowthatthereexistsapathconnectinga k -leafblocktoanother k -blockwitha limitednumberofintermediatenodes.Thepositionofnodes u and v hasfourpossibilities: (i)nodes u and v arebothneighborsofonenode,says s ,inseparatingset S .(ii)node u is theneighborofnode s inseparatingset S ,butnode v isnot.(iii)node v istheneighbor ofnode s inseparatingset S ,butnode u isnot.iv)neithernode u nor v areneighborsof anynodeinseparatingset S Case(i)isillustratedinFig. 4-2 .FromLemma 1 ,weknowthatthesizeof N ID ( u )is boundedby K .Wecandividetheneighborhoodofnode s into K subregionsbyusingthe samedividingstrategyproposedin[ 39 ].Therefore,ineachsubregion,thereisatmostone 31

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independentneighborofnode s .Thedashcircleisthetransmissionrangeofnodes u and v .Alltheinterconnectingnodesaremarkedwithasmallwhite circle.Tomaximizethe numberofinterconnectingnodes,weletthetransmissionra ngeofnodes u and v be r min andthedistancebetween u and v isslightlylargerthan r min ,so u isnotaneighborof v andviceversa.Notethatiftwonodesfallinthesamesubregi on,theywillbeconnected tobeapathbecauseeachsubregioncanhaveatmostoneindepe ndentneighborofnode s .Therefore,thereareatmost2nodestoformashortestpathi neachsubregion.(If wehave3ormorenodesineachsubregion,thepathmustnotbet heshortest.)Since wedividetheneighborhoodof s into K subregionsandwithsomealgebraicstep,wecan gureoutthereareatmost K ( 2 ),where =2 arcsin r min r max =2 arcsin 1 2 r u v s b Figure4-2. s isnodeinseparatingset,node u and v arebothneighborsofnode s Case(ii)isillustratedinFig. 4-3 .Since s isthenodeinseparatingset,theremust existanotherblacknode,called a inthegraph,thatisaneighborof s ,otherwisethe CDSisnotconnectedanymore.Since P uv hastheshortestlength,thentherecouldnot existinterconnectingnodesinregion A and B ,otherwise,therepath P ua hasashorter alengththan P uv .SameasCasei),atmost2nodeswillformashortestpathinea ch subregion.Theremightbeanotherinterconnectingnodewhi chisaneighborof v butnot 32

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of s ,called y inFig. 4-3 .Thus,ifonly u isinthetransmissionrangeof s ,thereareat most K ( 2 )+1interconnectingnodes,where = =2 arcsin 1 2 r s u a y v b d B A Figure4-3. s isnodeinseparatingset, u isinits k -leafblockisaneighborof s ,while v is notinthe k -leafblockand v isnotaneighborof s Similartocase(ii),case(iii)hasthesamenumberofmaxima linterconnectingnodes. Case(iv)isillustratedinFig. 4-4 .Since s isthenodeinseparatingsetandneither u nor v isinitstransmissionrange,theremustexisttwootherblac knodes,called a and b inFig. 4-4 ,thataretwoneighborsof s .Sincepath P uv hastheshortestlength,thenthere couldnotexistinterconnectingnodesinregion B A and C ,otherwiseeitherpath P ab P ub or P va hastheshorterlengththan P uv .Again,atmost2nodeswillformashortestpath ineachsubregion.Theremightbetwootherinterconnecting nodeswhichareaneighbor of u and v butnotof s respectively,called x and y inFig. 4-4 .Thusifneither u nor v is inthetransmissionrangeof s ,thereareatmost K ( 2 2 )+2interconnectingnodes between u and v Insummary,weprovethatforallcases,atmost2( K +1)interconnectingnodesare necessarytoconnecta k -leafblocktoother k -blocks. Theorem3. Theconstructed k m -CDSisofsizeatmost (2 k +2 K 1)( K +ln K + k + 1) j C km j inDGB,where k = m +1 and C km isanoptimalsolutionof k m -CDS. 33

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v b x u d b d A B C s a y Figure4-4. s isnodeinseparatingset, u isinits k -leafblockwhile v isnotinits k -leaf blockandneither u and v isareneighborsof s Proof: Again,let C km denoteourobtainedsolutionand C 1 m bethe1m -CDS obtainedfromCDSMIS.Wehave: j C km jj C 1 m j +2( k 2)( j C 1 m j 1) +2( K +1)( j C 1 m j 1) (2 k +2 K 1) j C 1 m j (2 k +2 K 1)( + K + m 1) j C 1 m j (2 k +2 m +1)( + K + m 1) j C km j 2 Fromthediscussiononthetwocases,wecanconcludethatift hetransmissionrange ratio r isboundedandtheinputof1-CDShasconstantapproximation onsize,then FTAAhasanapproximationfactorof O (1)onsize. 4.1.5.2Theanalysisondiameter Lemma12. D D km ,where D km and D denotetheoptimaldiametersof k m -CDS and1-CDSrespectively. Proof: Itisstraightforwardthat D D km ,sincea k m -CDSisalsoa1-CDS. Suppose D >D km D isnottheoptimaldiameterof1-CDS.Thus, D D km 2 34

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Lemma13. d ( C 1 m ) d ( C 11 )+2 Proof: Sinceeachnodenotin C 11 isdominatedbyatleastonenodein C 11 Therefore,whenweaddmorenodesinto C 11 inordertomakeittobe C 1 m ,weonly increase d ( C 11 )byatmost2hops. 2 Lemma14. d ( C km ) d ( C 1 m )+2 Proof: Supposetwonodes u and v arein C km .Thepositionofnode u and v hasthree possibilities:(1) u;v 2 C 1 m .(2) u 2 C km C 1 m v 2 C 1 m .(3) u;v 2 C km C 1 m .Forcase (1),thenumberofhopsbetween u and v isboundedby d ( C 1 m ).Forcase(2), u mustbe dominatedbyanodein C 1 m .Therefore, u isonlyonehopawayfromitsdominatorin C 1 m andthenumberofhopsbetween u and v isboundedby d ( C 1 m )+1.Forcase(3), u and v aredominatedbydierentnodesin C 1 m .However, u and v areonlyonehopawayfrom theirdominators.Thus,thenumberofhopsbetween u and v isboundedby d ( C 1 m )+2. 2 Theorem4. Ifthediameterofinput1-CDSisboundedby D ,theapproximationratio oftheconstructed k m -CDSondiameteris D km +4 Proof: FromLemma 12 13 and 14 ,wehavethefollowinginequality: d ( C km ) d ( C 1 m )+2 d ( C 11 )+2+2 D +4 D km +4 2 Inconclusion,Ifgivenan( ; )-CDSastheinput,where istheapproximationratio of1-CDSonsizeand isthatondiameter,thenthe k m -CDSobtainedfromFTAAhas constantapproximationfactorsof O ( ( K + m ))onsizeand O ( )ondiameter. 4.1.5.3Timecomplexity Theorem5. Suppose n isthenumberofnodesinoriginalgraph,thetimecomplexity of FTAAis O ( kn 3 ) Proof: Timecomplexityofconstructinga1m -CDSiseasytoanalyze,since computinganMIStakes O ( n )time[ 32 ].Therefore,thetimecomplexityofCDSMIS isdominatedbynding m 1MISs,whichis O ( mn ). 35

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Inthesecondstep,thewaywecompute k 0 -blockisbasedonthedistributedstrategy. Eachnodein C 1 m collectsmulti-hop-awayneighborhoodinformationtilla k 0 -blockcan bebuilt.Thisstrategyisreasonable,althoughweneedmult i-hop-awayneighborhood information.Ifatleastone k 0 -blockexistsin C 1 m ,wecanalwaysndit.Theworstcase isthatweneedallnodes'informationin C 1 m .Then,allthenodeswhofoundthesame k 0 -blockidentifyeachotherbybroadcastingandreturnthe k 0 -blockasawhole.Therefore, thecomplexityforsecondstepis O ( n 2 ). Thetimecomplexityofthirdstepisdominatedbythe ShortestPath function,which runsin O ( n 2 ).Thesecondandthirdstepareexecutedatmost n 1(themaximum numberof k 0 -blocks)timesinthefourthstep.Therefore,thetimecompl exityoffourth stepis O ( n 3 ). Thefthsteprepeatsthesecond,thirdandfourthstep k 1times,Therefore,the totaltimecomplexityofFTAAis O ( kn 3 ). 2 Now,wehavecompletedtheanalysisforFTAA.Ifpositive k m and aregiven,we willgettheconstantperformanceguaranteeonsize. 4.2BasicModel Inthissection,weintroducetwoalgorithmsfor1-CDS(also knownasCDS)with boundedsizeanddiameter.OneiscalledBasicDistributedA pproximationAlgorithm (BDAA)andanotheriscalledProgressiveDistributedAppro ximationAlgorithm(PDAA). ThetwoalgorithmscouldbeusedasaninputofFTAA.Thebene tofBDAAisthelow timecomplexityonconstructingCDS,whilePDAAperformswe llonoptimizingthesize anddiameterofCDS.4.2.1ProblemDenition TheCDSproblemwithboundeddiametercouldbeformallyden edasfollows: Denition2. CDSProblemwithBoundedDiameter: GivenaDGB G =( V;E ) representinganetwork,constructaCDS C suchthat:(1)eachnodein V iseitherin C 36

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orhasatleastoneneighborin C ,thesubgraphinducedby C ,i.e., G [ C ] ,isconnected,(2) minimizethediameterof C ,and(3)minimizethesizeof C ToconstructaCDS,weoftenemployanMaximalIndependentSe t(MIS)whichis alsoasubsetofallthenodesinthenetwork.ThenodesinMISa repairwisenonadjacent andnomorenodescanbeaddedtopreservethisproperty.Ther efore,eachnodewhich notinMISisadjacenttoatleastonenodeinMIS.Therefore,a nMISisindeedaDS. IfthenodesinMISareconnectedbyaddingmorenodestotheMI S,aCDScanbe constructed.4.2.2BasicDistributedApproximationAlgorithm4.2.2.1Algorithmdescription ThemainideaofBDAAisasfollows:1.UsingWan'sdistributedMISalgorithm[ 32 ]toconstructanMIS.Colorallthe nodesinMISblack. 2.Randomlychooseablacknodeastheroot,andassignalevel toeachnode,whichis basedonthenumberofhopsawayfromtheroot. 3.ConnectthenodesinMISfromlowleveltohighlevelwithmi nimumnumberof hops. OneexistingdistributedalgorithmsforMIS[ 32 ]isexecutedtoobtainaDS.The obtainedMISsatisesthefollowinglemma: Lemma15. AnypairofcomplementarysubsetsofaconstructedMIShasad istanceof exactlytwohops.[ 32 ] Inordertoimplementthisalgorithmindistributedmanner, eachnodemaintains alocalstatuswhichisinitializedto unexplored andsetto explored afterproceededby thealgorithm.Eachnodealsomaintainsalocalvariablewhi chstorestheIDofmessage senderandisinitiallyempty. ThefollowingoperationsforconnectingthenodesinMISwit hminimumnumberof hopsmaybeconductedasdescribedinthealgorithmbelow: 37

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Algorithm3 BasicDistributedApproximationAlgorithm(BDAA) 1: INPUT:AconnectedDGB G =( V;E )andanMIScomputedbyWan'salgorithm[ 32 ] 2: OUTPUT:ACDS T CDS withminimumdiameter 3: EachnodemaintainsauniquenodeIDandastatusof unexplored initially 4: ColorallnodesinMISblackandcoloreverynodeadjacenttoa blacknodeingrey 5: Randomlychoosearoot r inMISandset r to explored .Eachnode y isassigneda level k suchthat k = HopCount ( r;y ),where0 k k .Suppose k isthemaximum valueof k 6: r broadcasts EXPLORE messagestoitsneighborsatlevel1,where r isatlevel0 7: Uponreceiving EXPLORE messages,an unexplored greynode z atlevel i setsitself explored andcheckifithasablackneighbor y atlevel i or i +1,iftrue,itscolorisset blue ,theIDofthemessagesenderisstored,andsends EXPLORE messagestoits blackneighborsatlevel i and i +1,ifpossible. 8: Uponreceivingan EXPLORE message,an unexplored blacknode y atlevel i ( i 2) setsitself explored andtheIDofthemessagesenderisstored,thenitemploysthe storednodeIDstotracea2-hops-awayblacknode x atlevel i 2orlevel i 1viaa bluenode z ,addthepath( x;z;y )into T CDS andthensends EXPLORE messagesto itsgreyneighborsatlevel i and i +1,ifpossible. 9: Thealgorithmstopsuntilthereisnonodechangedfromgreyt oblue. 10: TheunionofblackandbluenodesisaCDS. 4.2.2.2Theoreticalanalysis Notethatifablacknode x atlevel( i 2)donothavea2-hops-awayblacknode y atlevel i ,then x musthavea2-hops-awayblacknode y atlevel( i 1),sinceLemma 15 holds.Therefore,foreachblacknode y wecolorexactlyonegreynodeinbluetomake x and y connected.So,thenumberofnodeschangedfromgreytobluei sexactly j MIS j 1. NotethattheCDSconstructedbyBDAAistheunionofMISandas etofbluenodes thatconnectsMIS.Thus,Wehavethefollowingtheorem: Theorem6. Denote T CDS asoursolutionobtainedfromBDAA,then j T CDS j 2 K j CDS j 1 and d ( T CDS ) 4 D +4 inaDGB. Proof: ItisknownthatforanMISinaDGB, j I j K j CDS j [ 39 ].Fromthe observationthatthenumberofnodeswehavetoaddtoconnect thenodesinMISis exactly j I j 1,thus, j T CDS j 2 j I j 1 2 K j CDS j 1.FordiameterofCDS, everyblacknodeatlevel k isawayfrom r withinadistanceatmost2 k hops.Suppose G hasdiameter D ,then D k ,andtheminimumdiameterofCDSisatleast D 2. 38

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Intheworstcase,twonodesin T CDS at k levelareseparatedby2 2 k hopssince eachnodeisawayfrom r atmost2 k hops.Inaddition,wenotethatnoblacknode existsatlevel1,theblacknodesinlevel2canconnectwith r with2hops.Therefore, d ( T CDS ) 2 2( k 2)+4 4 D +4 2 4.2.2.3Timecomplexity Theorem7. TheBDAAhas O ( n ) timecomplexityand O ( n log n ) messagecomplexity. Proof: ConstructionofanMIStakes O ( n )timecomplexityandsends O ( n log n ) messages[ 32 ].Afterthat,weuselinearmessageandtakeatmostlinearti metoconnect thenodesinMIS.Overall,BDAAhas O ( n )timecomplexityand O ( n log n )message complexity. 2 4.2.3ProgressiveDistributedApproximationAlgorithm Inthissection,weintroduce( ; )-CDSintoourmodeltobetheinputofFTAA. Also,itcansolvetheCDSproblemwithboundeddiameter.Ita pproximatelysatises thesizeconstraintandthediameterconstraintbyconstruc tingaCDS.Asweintentto balancethesizeanddiameter,thedenitionof( ; )-CDSisgiveninwirelessnetworksas follows: Denition3. ( ; ) -CDS: Foraxed > 1 and 1 ,aCDS C of G meetingthe followingtworequirementsiscalledan ( ; ) -CDS. (Size)Thesizeof C isatmost timestheminimumCDSsize. (Diameter)Foranypairofvertex u and v in C d ( C ) isatmost timestheminimumdiameterofCDSplusaconstantnumber. In( ; )-CDS, isanuser-denedinput,andusually isafunctionof .Therefore, thevalueof dependsontheuser-denedinput .Inthefollowing,wewilldescribe howtogeneratean( ; )-CDSandstudythetradeobetweenthesizeanddiameter. 39

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Sincean( ; )-CDSmightbeusedasaninputofFTAA,thetradeoisstillpr eservedfor k m -CDS. 4.2.3.1Algorithmdescription ThegeneralideaofourPDAAisasfollows.1.ConstructaCDS T CDS rootat r byusingBDAA.Root r shouldlocateatthe center ofnetwork,whichisthemid-pointofthelongestshortestpa thbetweentwonodes ingraph G 2.ConstructaShortestPathTree(SPT) T SPT rootedat r ,whichonlyincludesallthe shortestpathsfrom r toeveryothernodein T CDS 3.Traverse T CDS inadepth-rstmanner.Whenvisitinganode u ,ifthenumberof hopsfrom r to u in T CDS islargerthanauser-denedthreshold timesthenumberof hopsfrom r to u in T SPT ,thenanewpathfrom r to u in T SPT isaddedin T CDS Ifwedenote D CDS ( u;v )asthenumberofhopsfrom u to v in T CDS and D SPT ( u;v )as thenumberofhopsfrom u to v in T SPT .ThedetailsofPDAAisasfollows: RootSelectionandCDSTreeConstruction :WithDijkstra'salgorithm[ 46 ],which isusedtosolvethesingle-sourceshortest-pathsproblem, andaglobalvariable,which storesthemaximumshortestpathinthegraph G ,wecouldndthemid-pointofthe longestshortestpathbyrunningDijkstra'salgorithmonea chnodeandregardtheroot r asthemid-point.Whileconstructing T CDS rootedat r byusingBDAA,eachnode u needs tomaintainapointer [ u ]foritsparentonthetree T CDS andanupperbound d [ u ]for thenumberofhopsto r .Weusethe INITIALIZE and RELAX algorithmsin[ 49 ]to initializeandmaintainbothoftheseattributes. ShortestPathTreeConstruction : T SPT rootedat r isconstructedbyusingDijkstra's algorithm.Itonlycontainsalltheshortestpathsfromther oot r toeveryothernodein T CDS DepthFirstSearch(DFS) :Traversethe T CDS inaDFSmannerbeginningfromthe root r alongthepathsfrom r toalltheothernodesin T CDS .Whennode u isreachedfor 40

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Algorithm4 ProgressiveDistributedApproximationAlgorithm(PDAA) PDAA( ) 1: Locatethecenterofnetworkandchoose r atthecenter. 2: Builda T CDS basedon r 3: UseDijkstra'salgorithmtoconstructanSPT T SPT 4: C =FIND( T CDS T SPT r ) 5: return C FIND( T CDS T SPT r ) 1: INITIALIZE( T CDS r ) 2: DFS( r ) 3: returnadesiredCDS C INITIALIZE( G r ) 1: for eachvertex v 2 T CDS do 2: d [ v ] 1 3: [ v ] NIL 4: endfor 5: d [ r ] 0 RELAX( u v ) 1: if d [ v ] >d [ u ]+ D CDS ( u;v ) then 2: d [ v ]= d [ u ]+ D CDS ( u;v ) 3: [ v ] u 4: endif DFS( u ) 1: if d [ u ] >D SPT ( r;u ) then 2: ADD-PATH( u ) 3: endif 4: for eachchild v of u inMSA do 5: RELAX( u v ) 6: DFS( v ) 7: RELAX( v u ) 8: endfor ADD-PATH( v ) 1: if d [ v ] >D SPT ( r;v )and parent SPT ( v )!=NIL then 2: ADD-PATH( parent SPT ( v )) 3: RELAX( parent SPT ( v ) ;v ) 4: endif 41

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thersttime,if d [ u ]isgreaterthan D SPT ( r;u ),thentheshortest P ru in T SPT isadded to T CDS and d [ u ]and [ u ]areupdated.Afterthis,node u 'sparent v needstobechecked iftheupdatedpathfrom r to u willresultinreducingthenumberofhopsfrom r to v .If so,then v 'sparentwillbecheckedandsoonuntiltheroot r isreached. WiththeexecutionofBDAA,distributedSPT(dSPT)(e.g.[ 50 ]),anddistributed DFS(dPFS)(e.g.[ 51 ]), T CDS T SPT andaDFStraversalordercouldbeachieved.Inthis way,witha Manager (e.g.rootnode),PDAAcouldbeeasilyinitiatedandtermina ted accordingtothedetailsillustratedinAlgorithm 4 ToevaluatethecorrectnessofthePDAA,weexaminewhethert hetwoconstraints inthedenitionhasbeensatised.Taking asanuser-denedinput,wederivea relationshipbetween and ,whichshowstherelationshipbetweenthesizeofthe constructedCDSandtheoptimalsolutionofCDSonsize.Weal soanalyzethetime complexityofthePDAA.4.2.3.2Theoreticalanalysis Dene w ( T CDS )asthetotalweightof T CDS in G ,whereweassumeeachedge hasbeenassignedtheunitweightof1.Then D SPT ( u;v )and D CDS ( u;v )areequal totheweightof T SPT ( u;v )and T CDS ( u;v )respectively.Anotherobservationisthat j T CDS j = w ( T CDS )+1,sincethenumberofnodeinatreeequalstothetotalnumb erof edges,whichalsoequalsto w ( T CDS )plus1.Meanwhile,aswementionedbefore,thelower boundofminimumdiameterofCDSis D 2.Actually,theupperboundfortheminimum diameterofCDSis D ,i.e.,allthenodesin G areinCDS,therefore, D = D DuetothespecicstructuresofCDS,wewillclassifythefol lowingproofsintotwo cases.Case(1):thediameterofSPT T rootedat r thatspansallnodesin G isequalto D andallothersituationsareclassiedintoCase(2). Lemma16. Foranypairofnodes u and v in C ,thenumberofhopsbetween u and v is atmost times ( D +2) ,when d ( T )= D 42

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Proof: Whenavertex v isvisited,if d [ v ] >D SPT ( r;v ),thenshortestpathbetween r and u isaddedinto T CDS bycalling ADD PATH .Also,weknowthatthemaximum valuefor D SPT ( r;v )istheheight h of T SPT ,wewillprovethat2 h D +2inthe following.After v isvisited, d [ v ]isatmost D SPT ( r;v ),whichislessorequalto h and subsequentlyneverincreases.For u ,thesameanalysiscanalsobeapplied.Therefore,the totalnumberhopsbetween v and u in C isatmost2 h ,thereforeatmost ( D +2). Now,weprovethat2 h D +2.First,itiseasytoseethat2 h d ( T )and d ( T )= D Thenwehavethefollowing: 2 h 2 d ( T ) 2 D 2 D Therefore,weprovethat2 h D +2. 2 Lemma17. Incase(2),foranypairofnodes u and v in C ,thenumberofhopsbetween u and v isatmost 2 times ( D +1) Proof: If d ( T ) 6 = D ,theworstcaseisthat d ( T )=2 D .Asimpleexampletoillustrate thatisa ring ,thedegreeofeachnodeintheringisonly2andallthenodesi n G are includedinCDS,seeFig. 4-5 .Therefore, h D +1,thenthemaximumnumberofhops between u and v isatmost2 h ,thatis2 ( D +1). 2 Figure4-5.AllthenodesintheringareaCDSwithdiameterof 8 Inrealwirelessnetwork,case(2)rarelyhappens,sinceitr equiresallthehosts (nodes)areuniformlydeployed,suchasaring.However,inm ostcases,theyaredeployed 43

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randomly.Therefore,thediameterofCDSreturnedbyPDAAis boundedby ( D +2)in mostcases. Lemma18. Thetotalnumberofnodesontheaddedshortestpathsisatmos t (5 ) K 1 j C 11 j + 3 Proof: Let v 0 = r and v 1 ;v 2 :::::v k betheverticesthatcausedshortestpathto beaddedduringthetraversal,intheordertheywereencount ered.Whentheshortest pathfrom r to v i ( i 1)wasadded,thenumberofhopsoftheaddedpathwas D SPT ( r;v i ).Also,thenodesonthepathto v i hasbeenrelaxedinorder,sothat d [ v i ] D SPT ( r;v i 1 )+ D CDS ( v i 1 ;v i ).Theshortestpathto v i wasaddedbecause D SPT ( r;v i )
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( 1) k X i =1 ( D SPT 1)( r;v i )+ k ( 1) 2( w ( T CDS )+1) 2 ( 1) k X i =1 ( D SPT 1)( r;v i )+ k ( 1) 2 j T CDS j 2 Here,weintendtomaximize k inordertohaveatighterboundon P ki =1 ( D SPT 1)( r;v i ),whichisthetotalnumberofnewnodesontheaddedshortest paths,Letdenote P ki =1 ( D SPT 1)( r;v i )as P size forclearrepresentation. Intuitively, k isatmost j I j sinceallblacknodesinMISof T CDS maycauseshortest pathstobeaddedduringthetraversal.However,theroot r andatleasttwoblacknodes atlevel2willnotbecountedin k .Therefore, k isatmost j I j 3. ( 1) P size 2 j T CDS j 2 ( j I j 3)( 1) 4 j I j 4 ( j I j 3)( 1) (5 ) j I j +3 7 Since j I j K j C 11 j [ 39 ],wehave: P size (5 ) K 1 j C 11 j +3 2 Theorem8. Giventhevalueof ,theapproximationratio ofCDSonsizeis ( +3) K 1 Proof: Fromtheaboveanalysis, C istheunionof T CDS andtheaddedshortestpaths. Therefore,combiningtheTheorem 6 andLemma 18 j C j = j T CDS j + P size 2 K j C 11 j 1+ (5 ) K 1 j C 11 j +3 ( +3) K 1 j C 11 j +2 2 45

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4.2.3.3Timecomplexity Theorem9. ThetimecomplexityofthePDAAalgorithmis O ( n 2 ) ,andthemessage complexityofthePDAAalgorithmis O ( n 2 ) Proof: FromTheorem 7 ,thetimecomplexityandmessagecomplexityforBDAA are O ( n )and O ( n log n )respectivelyanddSPTanddDFSrunatmost O ( n 2 )time complexityandsend O ( n 2 )messages[ 50 ][ 51 ].Now,weanalyzetheprocedureofnding thecenterofnetwork.ThedSPTisexecutedateachnode x simultaneously,afterthat, x needstobroadcastthelongestpathinSPTrootedat x andcompareitwiththelongest pathsreturnedbyothernodes.Therefore,thisprocedurene eds O ( n 2 )timecomplexity and O ( n 2 )messagecomplexity.Sinceallotheroperationonlytakeat most O ( n )time complexityand O ( n )messagecomplexity,theoverallmessagecomplexityandti me complexityofPDAAare O ( n 2 )and O ( n 2 ). 2 46

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CHAPTER5 SIMULATIONRESULTS Inthissection,weconductthesimulationexperimentstome asurethediameter andsizeofCDSconstructedbyourproposedalgorithms.More over,weareinterested incomparingtheCDSsreturnedbyCDS-BD[ 47 ]andPDAA.Sincetherunningtime forPDAAandBDAAhasbeendiscussedinSection 4.2 ,wealsowouldliketoverifythe runningtimeofthetwoalgorithmsinpractice.Inaddition, wedovariousexperiments byadjustingtheuser-denedparameter inPDAA,inordertoseehowtheCDS sizeanddiametercouldbebalanced.Atlast,weevaluatethe performanceofFTAA (Algorithm 2 )bycomparingtoPDAAsothatthetradeobetweenthethreefa ctorscould besystematicallydiscovered. 5.1NetworkParameters Wesimulatethenetworkwiththefollowingtunableparamete rsthatmayaectthe algorithmperformance: n ,thenumberofnodesinagivennetwork r ,thetransmissionrangeratio,i.e., r = r max r min Thenetworkdensity,i.e.,thenumberofnodesperarea 5.2EectsofNumberofNodes Toevaluatetheperformanceoftheproposedalgorithmsunde rdierentnumberof nodes,werandomlydeployed n nodestoaxedareaof3,000mx3,000m. n changedfrom 100to300withanincrementof10.Eachnode v i randomlychosethetransmissionrange r i 2 [ r min ;r max ]where r min =100 m and r max =300 m .Foreachvalueof n ,1,000network instanceswereinvestigatedandtheresultswereaveraged.5.2.1SimulationsforBDAAandPDAA Thepurposeofthissimulationistoevaluatetheperformanc eofourproposed algorithmsunderdierentnumberofnodesandverifytheimp ortanceofrootselectionat thesametime.Inordertohighlighttherootselection,weus eavariationofBDAA,called BDAA-Mid,asareference.ComparedtoBDAA,BDAA-Midselect sthecenterofnetwork 47

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8 10 12 14 16 18 100 150 200 250 300 Diameter of CDSNumber of nodes in the network BDAA BDAA-Mid PDAA AComparetheDiameterofCDS 60 80 100 120 140 160 180 100 150 200 250 300 Size of CDSNumber of nodes in the network BDAA BDAA-Mid PDAA BComparetheSizeofCDS Figure5-1.SimulationsforBDAAandPDAA astherootinsteadofchoosingrandomly.Also,weadmitPDAA intothissimulationand issetto1. Fig. 5-1A comparesthediameterofCDSconstructedbythethreealgori thms.It isshownthat,underdierentnumberofnodesdeployedinnet works,theCDSbuilt byPDAAhasthesmallestdiameter.Weobservethatthegapbet weenBDAAand BDAA-Midisshownclearly,whichindicatesthattheCDScoul dachievessmalldiameter withtherootlocatingatthecenterofthenetwork.Ontheoth erhand,thedierence betweenBDAA-MidandPDAAissmall,whichhighlightsanimpo rtantfactthatifthe centerofnetworkisdetected,thediameterofCDSrootedatt hecenterwillbenearly 48

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optimal,evenusinganalgorithmthatonlyguaranteesaloos eboundondiameter,such asBDAA.InordertoseehowfarthediameterofCDSreturnedby BDAA-Midfromthe optimalsolution,weset to1inPDAA.Sincewith =1,PDAAwillproduceaCDS withminimumdiametermostly. InFig. 5-1B ,wepresentthesizeofCDSobtainedfromallthreealgorithm s, dependingonthenumberofnodesdeployed.ThesizesofCDSge neratedfromthe threealgorithmsarequiteclosetoeachotherandtheyallin creasewiththeincreaseof numberofnodes.Also,consideringthesamenumberofnodes, BDAAreturnsalarger sizeofCDSthanPDAAandBDAA-Mid.Althoughthegapsbetween thesealgorithm looksmallinFig. 5-1B ,thedierencebetweenBDAAandBDAA-Midiscleartoobserve inthecomparisonofrealdata,whichillustratesthatthesi zeofCDScanbereducedby choosingthecenterofnetworkastheroot.Therefore,thece nterofnetworkappearstobe animportantissueintheconstructionofCDS. InTable 5-1 ,wepresenttherunningtimeforthethreealgorithms.Asthe complexity analysisindicates,theruntimeofBDAA-MidandPDAAaremuc hhigherthanthatof BDAA.Thisisduetothelongtimespentondetectingthecente rofnetwork.Moreover, weshowinTable 5-1 thattheBDAA-MidstillrunsfasterthanPDAA,sincePDAAnee ds tocompute T SPT toshortenthediameter.Whenthenumberofnodesincreases, PDAA andBDAA-Midspendmoretimeondetectingthecenterofnetwo rk.Therefore,itisa tradeobetweenthesize(diameter)ofCDSandrunningtimeo ftheproposedalgorithms. 5.2.2SimulationsforCDS-BDandPDAA Wealsoconductedsimulationstocomparetheperformanceof CDS-BDandPDAA. CDS-BDisanalgorithmproposedin[ 47 ]toconstructaCDSwithboundeddiameterand size.ItselectsarootrandomlyandspansaCDSfromtheroot. Theapproximationratios ofCDS-BDare11.4and3onsizeanddiameterrespectively.Fo rthepurposeoffairness, weset =3(theapproximationratioofPDAAondiameter)inPDAAanda lsochoose therootofCDSrandomly. 49

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Number BDAA BDAA-Mid PDAA ofNode Runtime Runtime Runtime 100 0.0030 1.2640 1.6460 110 0.0040 1.8270 2.3550 120 0.0035 2.6260 3.4025 130 0.0040 3.6260 4.6545 140 0.0065 4.8320 6.2470 150 0.0055 6.4380 8.3000 160 0.0080 8.3955 10.790 170 0.0095 10.732 13.783 180 0.0105 13.612 17.588 190 0.0135 16.688 21.450 200 0.0150 20.604 26.650 210 0.0185 25.342 32.406 220 0.0195 30.427 38.916 230 0.0235 41.962 54.045 240 0.0245 49.698 63.839 250 0.0280 58.496 74.494 260 0.0350 78.144 100.67 270 0.0345 90.052 116.11 280 0.0385 104.91 134.48 290 0.0460 134.32 172.55 300 0.0480 155.49 197.78 Table5-1.Runningtime(ms)forBDAA,BDAA-MidandPDAA Fig. 5-2A showsthatthediametersofCDSbuiltbythetwoalgorithmsar equite closetoeachotherandthetwocurvesintersectwitheachoth erwhendierentnumber ofnodesdeployedinthenetwork.Forexample,whenthenumbe rofnodesdeployedis 130,PDAAachievessmallerdiameterthanCDS-BD,whileat14 0,CDS-BDhassmaller value.Thereasonwhytheylookclosetoeachotheristhatthe yallguaranteeaconstant approximationratioof3ondiameter.EventhoughPDAAdoesn otalwaysoutperform CDS-BDfromthisresult,outofthe21pointsinFig. 5-2A ,PDAAoutperformsCDS-BD at16points,whichisaround76%inprobability.Sostatisti cally,ifthenumberof nodesdeployedinthenetworkiswithintherangeof100to300 ,whichisthesimulation environmentinourmodel,PDAAisstillbetterthanCDS-BDin reducingthediameter. 50

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12 13 14 15 16 17 18 19 20 21 100 150 200 250 300 Diameter of CDSNumber of nodes in the network PDAA CDS-BD AComparetheDiameterofCDS 60 80 100 120 140 160 180 200 100 150 200 250 300 Size of CDSNumber of nodes in the network PDAA CDS-BD BComparetheSizeofCDS Figure5-2.SimulationsforCDS-BDandPDAA Fig. 5-2B providestheperformancecomparisonofthetwoalgorithmso nthesizeof CDS.ItshowsPDAAalwaysconstructsaCDSwithsmallersizet henCDS-BD,whichis muchbetterthantheoreticalanalysiswegaveinSection 4.2 .Therefore,wecanconclude thatPDAAoutperformsCDS-BDonsizeandondiameterwithhig hprobabilityaswell. 5.2.3SimulationsBasedonDierent Intheabovesimulations, issetto1or3.Inthissection,weconductthesimulations withdierentvaluesof .Westudytherelationshipbetween andthesizeofCDSand therelationshipbetween anddiameterofCDS.Astherootselectionwillnotaectthe comparison,werandomlychoosetherootofCDSinthisgroupo fsimulations.Resultsare showninFig. 5-3 51

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12 14 16 18 20 22 24 26 100 150 200 250 300 Diameter of CDSNumber of nodes in the network b = 4 b = 2 b = 1 AComparetheDiameterofCDS 60 80 100 120 140 160 180 100 150 200 250 300 Size of CDSNumber of nodes in the network b = 4 b = 2 b = 1 BComparetheSizeofCDS Figure5-3.Simulationsbasedondierent InFig. 5-3A ,eachlinerepresentsthediameterofCDSbasedononeofdie rent valuesof .When issetto1,PDAAaddsashortestpathfrom v to r if D CDS ( r;v )is largerthan D SPT ( r;v ).Therefore,PDAAwith =1returnsaCDSwiththesmallest diameter.When issetto4,PDAAwillnotcausetheshortestpathstobeaddedi n, sincePDAAonlyaddsthepathfrom v to r in T CDS undertheconditionthat D CDS ( r;v ) isgreaterthan4timesof D SPT ( r;v ),however,theupperboundofPDAAondiameter is4.Thus,theCDSbyPDAAwith =4hasthelargestdiameter.For =2,the correspondinglineisinthemiddle.Therefore,asweexpect ed,thediameterofCDSbuilt byPDAAcouldbecontrolledbyadjustingthevaluesof 52

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InFig. 5-3B ,eachlinerepresentsthesizeofCDSbasedononeofdierent valuesof .When issetto1,if D CDS ( r;v )islargerthan D SPT ( r;v ),PDAAaddsashortestpath from v to r .Thisstrategywillincurmorenodestobeadded.Ontheoppos ite,when is setto4,noshortestpathisneeded,whichresultsinaCDSwit hsmallersize.For =2, thecorrespondinglineisinthemiddle,thesamesituationa sinFig. 5-3A .Inconclusion, theperformanceofPDAAcanbebalanceddependingonthevalu eof andthetradeo betweensizeanddiameterisclear.5.2.4Simulationsfor k m -CDS 11 12 13 14 15 16 17 18 19 20 100 150 200 250 300 Diameter of CDSNumber of nodes in the network PDAA Alg.1 AComparetheDiameterofCDS 60 80 100 120 140 160 180 200 100 150 200 250 300 Size of CDSNumber of nodes in the network PDAA Alg.1 BComparetheSizeofCDS Figure5-4.Simulationsfor k m -CDS Inthissection,weareinterestedinevaluatingtheperform anceofFTAA.Weintend toillustratethatFTAAimprovesthefaulttoleranceof1-CD Sbyaddingmarginal 53

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overhead(intermsofthenumberofnodesaddedinto1-CDS).W egeneratea1-CDSusing PDAAwithrandomrootselectionand issetto2here.Wetakethe1-CDSgenerated usingPDAAastheinputofFTAAafterwards,andweset k =2and m =1. Fig. 5-4A comparestheperformanceofFTAAandPDAAintermsofthediam eterof CDS.Asweexpected,thereislittledierenceonthediamete rofCDSbasedonthetwo algorithms,whichperfectlymatchesourtheoreticalanaly sisforthediameterof k m -CDS. Therefore,FTAAenhancesthefaulttoleranceofCDSwithout aectingitsdiameter greatly. Meanwhile,asobservedfromFig. 5-4B ,thesizeof k m -CDSobtainedfromFTAAis certainlylargerthan1-CDSbyPDAA.Specically,theperfo rmanceofthetwoalgorithms isrelativelyproportional.Asobservedfromourexperimen ts,thesizeof k m -CDS obtainedfromFTAAisalmost1.06timesthesizeofCDSreturn edbyPDAA.Theresults indicatethatconsideringthefaulttolerancewillincreas ethesizeoftheCDSatthesame time.However,theincreaseinsizeisstillboundedandpred ictable.Therefore,itisclear toseethetradeosbetweenthethreefactors. 5.3EectsofTransmissionRatio Wealsoconductedsimulationstocomparetheperformanceof allalgorithmswhen changingthetransmissionrangeratio r aswellastoseehowthischangeaectsthesize anddiameterofanobtainedCDS.Tochange r ,wexed r min =100 m andchanged r max from100mto300mwithanincrementof20.Inthisexperiment, werandomly deployed100nodesintoaxedareaofsize3,000mx3,000m.Ea chnoderandomlychose atransmissionrangein[ r min ;r max ].Foreachnetworkinstance,weranthetestfor1000 times. Fig. 5-5A comparestheperformanceoftheproposedalgorithmsinterm softheCDS diameter.AsshowninFig. 5-5A ,PDAAisthebest.Inparticular,theCDSdiameter obtainedfromPDAAisonly5%smallerthanthatofBDAA-Midan d27%smallerthan 54

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3 4 5 6 7 8 9 10 11 12 1 1.5 2 2.5 3 Diameter of CDSRatio BDAA BDAA-Mid PDAA AComparetheDiameterofCDS 50 52 54 56 58 60 62 64 1 1.5 2 2.5 3 Size of CDSRatio BDAA BDAA-Mid PDAA BComparetheSizeofCDS Figure5-5.Eectsoftransmissionratio thatofBDAA.Again,theseresultsrevealthatselectingthe centerofthenetworkasthe CDSrootcanreducetheCDSdiameter. Asweexpected,inFig. 5-5B ,theCDSproducedbyPDAAhasabiggersizethan BDAA-MidandBDAA,sincePDAAneedtoaddmorenodesinCDSino rdertoachieve ashorterdiameter.Apparently,BDAAperformsmuchbettert hanthetwoothers,just becauseitputslessconcernonCDSdiameter. Fig. 5-5 illustrateshowthetransmissionrangesaecttheCDSsizea nddiameter. AscanbeseeninFig. 5-5 ,threecurvesshowobvioustrendofdecrease.Inotherwords theCDSsizeanddiameterdecreasewhenthemaximumtransmis sionrangeincreases. 55

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Itisduetothefactthatthelargerthetransmissionrange,t hemorenodesanodecan dominate. 5.4EectsofNetworkDensity Simulationswerealsocarriedouttocomparetheperformanc eofallthreealgorithms whenchangingthenetworkdensityaswellastoseehowthisch angeaectstheCDS sizeanddiameter.Tochangethenetworkdensity,wexedthe numberofnodesn=100 andincreasedtheareafrom500mx500mto1,500mx1,500mwith anincrementof20. Inthisexperiment,werandomlygenerated50nodesinanarea withthesizechanging asdescribed.Eachnoderandomlychoseatransmissionrange in[ r min ;r max ]where r min =100 m and r max =300 m .Foreachnetworkinstances,weranthesimulationsfor 1000timesandtheresultswereaveraged. Fig. 5-6A providestheperformancecomparisonofthreealgorithmsin termsofthe CDSdiameter.AsrevealedbyFig. 5-6A ,PDAAstilloutperformstheothertwointhis case.AndBDAA-MidoutperformsBDAA.Specically,theCDSd iameterobtainedfrom PDAAisonly1.3%lessthanthatofBDAA-Midand15.8%lesstha nthatofBDAA. Aspredicted,Fig. 5-6B indicatesthatBDAAcanbuildupaCDSwithsmallersize thanBDAA-MidandPDAA.ThedierencebetweenBDAAandPDAAi sobviouswhen thenetworkdensityishigh.However,thethreecurvesbegin tomergeasthenetwork densitydecreases. Inaddition,Fig. 5-6 showstheobvioustrendofincreaseofthreecurves,which impliesthattheCDSsizeanddiametergetbiggerwhenthenet workdensitydecreases. Thisisbecausewhenthenetworkdensitydecreases,thenumb erofneighborsofeachnode decreasesaswell.ThustheCDSsizeanddiameterneedtobela rgertodominateallnodes inanetwork. 5.5Summarization Thesimulationresultscanbesummarizedasfollows: 56

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4 6 8 10 12 14 600 800 1000 1200 1400 Diameter of CDSArea BDAA BDAA-Mid PDAA AComparetheDiameterofCDS 46 48 50 52 54 56 58 60 600 800 1000 1200 1400 Size of CDSArea BDAA BDAA-Mid PDAA BComparetheSizeofCDS Figure5-6.Eectsofnetworkdensity TheCDSrootedatthecenterofnetworkalwayshasanearlymin imumdiameterand smallersizecomparedwithallotherCDSs,whichchoosether ootrandomly.However, thetradeobetweendiameter(size)andrunningtimeofprop osedalgorithmsexists,since timecomplexityofndingthecenterofnetworkishigh. Underthesamenetworkconditions,theresultsshowsPDAApe rformsbetterthan CDS-BDproposedin[ 47 ]inmostoftime,eventhoughweselecttherootrandomly. PDAAoersamethodtobalancethediameterandsizeofCDSthr oughan user-denedinputbyadjustingthevalueof 57

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ThesimulationresultsshowthatFTAAimprovesthefaulttol eranceof1-CDSwith onlymarginalextraoverhead.Therefore,thetradeosbetw eenthethreefactorsareshown clearly. 58

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CHAPTER6 CONCLUSIONANDFUTUREWORK 6.1Conclusion Inthiswork,weinvestigatethefaulttolerantCDSproblemw ithboundeddiameter inwirelessnetworks.Weproposeanapproximationalgorith mforageneralcaseofthe k m -CDSwithboundeddiameter,andtwoapproximationalgorith msfor1-CDS,which couldbeappliedintothesolutionofthe k m -CDSmodel.Weanalyzetheapproximation ratiosofthethesealgorithmsinDGBandtheyguaranteedcon stantratiosforthose factorsconsidered.Moreover,theproposedalgorithmsfor 1-CDScanbeimplemented indistributedmannerandtheanalysisoftimeandmessageco mplexitiesispresentedas well.Throughextensivesimulations,weverifythatourpro posedalgorithmscaneectively reducethediameterandsizeofCDSandoutperformCDS-BD[ 47 ]. 6.2FutureWork Besidesextendingourresultsinthisworktothedirectedgr aph,wearealsointerested inndinganumberofCDSstobalancetheenergyconsumption. Mostoftheresearch workontheCDSproblemfocusesonconstructingasingleCDS. However,asnodesin theCDSwillbeusedheavily,theycanquicklyrunoutofpower ,therebyshortening thenetworklifetime.Therefore,itisnaturaltondanumbe rofdierentCDSs,and dynamicallyrotatetherolesoftheseCDSstocarryoutthene tworktasks. 59

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[26] T.Clausen,P.Jacquet,A.Laouiti,P.Minet,P.Muhlethaler andL.Viennot, \OptimizedLinkStateRoutingProtocol," IETFInternetDraft ,October,2001. [27] S.Y.Ni,Y.C.Tseng,Y.S.ChenandJ.P.Sheu,\Thebroadcasts tormproblem inamobileadhocnetwork," Proc.MobiCom ,pp.151{162,Seattle,Aug.1999. [28] S.GuhaandS.Khullar,\Approximationalgorithmsforconne cteddominatingsets," Algorithmica ,vol.20,no.4,pp.374{387,1998. [29] M.R.GareyandD.S.Johnson, ComputersandIntractability:Aguidetothe theoryofNP-completeness ,Freeman,SanFrancisco1978. [30] B.Das,R.SivakumarandV.Bharghavan,\Routinginadhocnet worksusinga spine," InternationalConferenceonComputersandCommunications Networks ,Las Vegas,NV.Sept.1997. [31] R.Sivakumar,B.DasandV.Bharghavan,\Animprovedspine-b asedinfrastructure forroutinginadhocnetworks," IEEESymposiumonComputersandCommunications ,Athens,Greece.June1998. [32] P.J.Wan,K.M.Alzoubi,andO.Frieder,\Distributedconstr uctiononconnected dominatingsetinwirelessadhocnetworks," ProceedingsoftheConferenceofthe IEEECommunicationsSociety(INFOCOM) ,2002. [33] K.M.Alzoubi,P.-J.Wan,andO.Frieder,\NewDistributedal gorithmforconnected dominatingsetinwirelessadhocnetworks," Proceedingsofthe35thHawaii InternationalConferenceonSystemScicences ,Hawaii,2002. [34] K.M.Alzoubi,P.-J.Wan,andO.Frieder,\Distributedheuri sticsforconnected dominatingsetsinwirelessadhocnetworks," JournalofCommunicationsand Networks ,vol.4,no.1,March2002. [35] M.Cardei,M.X.Cheng,X.Cheng,andD.Z.Du,\Connecteddomi nationinadhoc wirelessnetworks," ProceedingsoftheSixthInternationalConferenceonCompu ter ScienceandInformatics(CSI) ,2002. [36] K.M.Alzoubi,P.-J.Wang,andO.Frieder,\Message-optimal connecteddominating setsinmobileadhocnetworks," ProceedingsoftheACMInternationalSymposium onMobileAdHocNetworkingandComputing(MOBIHOC) ,2002. [37] JeremyBlum,MinDing,AndrewThaelerandXiuzhenCheng,\Co nnected DominatingSetinSensorNetworksandMANETs," HandbookofCombinatorial Optimization ,pp.329{369,2005. [38] Y.Li,S.Zhu,M.T.Thai,andD.-Z.Du,\Localizedconstructi onofconnected dominatingsetinwirelessnetworks," NSFInternationalWorkshoponThoretical AspectsofWirelessAdHoc,SensorandPeer-to-PeerNetwork s ,2004. 62

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[39] M.T.Thai,F.Wang,D.Liu,S.ZhuandD.Z.Du,\Connecteddomi natingsetsin wirelessnetworkswithdierenttransmissionranges," IEEETransactionsonMobile Computing ,vol.6,no.7,July2007. [40] F.Wang,M.T.ThaiandD.Z.Du,\Ontheconstructionof2-conn ectedvirtual backboneinwirelessnetwork," IEEETransactionsonWirelessCommunications, acceptedwithrevisions ,2006.(Therstversionisin2005atechnicalreport) [41] W.Shang,F.F.Yao,P.J.Wan,X.Hu,\Algorithmsforminimum m -connected k -dominatingsetproblem," COCOA2007 ,pp.182{190,2007. [42] M.T.Thai,N.Zhang,R.Tiwari,andX.Xu,\Onapproximationa lgorithmsof k-connectedm-dominatingsetsindiskgraphs," JournalofTheoreticalComputer Science ,vol.385,pp.49-59,2007. [43] Y.Wu,F.Wang,M.T.ThaiandY.Li,\Constructing k -connected m -dominating setsinwirelesssensornetworks," 2007MilitaryCommunicationsConference (MILCOM2007) ,Orlando,FL,October2007 [44] F.DaiandJ.Wu,\Onconstruction k -connected k -Dominatingsetinwireless network," ParallelandDistributedProcessingSymposium,2005.Proc eedings.19th IEEEInternational ,2005. [45] K.Mohammed,L.Gewali,andV.Muthukumar,\Generatingqual itydominating setsforsensornetwork," ProceedingsoftheSixthInternationalConferenceon ComputationalIntelligenceandMultimediaApplications ,pp.204-211,August2005. [46] T.H.Cormen,C.E.LeisersonR.L.RivestandC.Stein, IntroductiontoAlgorithms 2ed. ,McGrawHill2001. [47] Y.Li,D.Kim,F.Zou,D.-Z.Du,\ConstructingConnectedDomi natingSetswith BoundedDiametersinWirelessNetworks," InternationalConferenceonWireless Algorithms,SystemsandApplications ,Chicago,IL,Auguest1-32007. [48] D.Matula,\k-blocksandUltrablocksinGraphs," JournalCombin.TheorySer.B. vol.24,pp.1-13,1978. [49] Y.Li,M.T.Thai,F.Wang,D.-Z.Du,\OntheConstructionofaS trongly ConnectedBroadcastArborescencewithBoundedTransmissi onDelay," IEEE TransactionsonMobileComputing ,vol.5,no.10,pp.1460-1470,2006. [50] L.Brim,I.Cerna,P.KrcalandR.Pelanek,\Distributedshor testpathsfordirected graphswithnegativeedgelengths," TechmicalreportFIMU-RS-2001-01 ,Facultyof Informatics,MasarykUniversity,http://www..muni.cz/ informatics/reports,2001. [51] M.B.Sharma,S.S.IyengarandN.K.Mandyam,\Anoptimaldist ributed depth-rst-searchalgorithm," InProc.oftheseventeenthannualACMconferenceonComputerscience:Computingtrendsinthe1990's ,Louisville,Kentucky, pp.287-294,1989. 63

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BIOGRAPHICALSKETCH NingZhangreceivedhisbachelorofscienceincomputerscie ncefromBeijing UniversityofTechnology,Chinainsummer2006.Aftergradu ation,hejoinedthegraduate schoolattheUniversityofFloridainfall2006,whereherec eivedhismasterofsciencein summer2008fromtheDepartmentofComputerandInformation ScienceandEngineering. Hisresearchinterestsarecenteredonthecombinatorialop timizationanditsapplications tocomputernetworks.Morespecically,thefocusofhisres earchistodesignandanalyze ecientalgorithms(mainlyapproximationalgorithms)for computationallyhardproblems inwirelesssensornetworks,wirelessnetworks.Infall200 8,hewilljointheDepartment ofComputerScienceinUniversityofWisconsinatMadison,a toptencomputerscience departmentintheU.S.,asaPhDstudent. 64