<%BANNER%>

Topology Reconfiguration for Systems of Networked Autonomous Vehicles with Network Connectivity Constraints

Permanent Link: http://ufdc.ufl.edu/UFE0045200/00001

Material Information

Title: Topology Reconfiguration for Systems of Networked Autonomous Vehicles with Network Connectivity Constraints
Physical Description: 1 online resource (104 p.)
Language: english
Creator: Navaravong, Leenhapat
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: algorithm -- networking -- topology -- uav
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Future systems of networked autonomous vehicles, such as unmanned aerial or ground vehicles, may rely on peer-to-peer, wireless communication to coordinate their actions. The physical formation of the network may need to be reconfigured at times based on the specified missions. However, reconfiguring the physical formation also impacts the link connectivity and, hence, the connectivity of the network. If the network is partitioned, then the autonomous vehicles can no longer coordinate their movements, and the mission may fail. In this dissertation, we develop techniques to transform the formation of a system of autonomous vehicles while preserving network connectivity. Several different approaches to address this problem are presented, with a focus on a method that utilizes ideas from routing packets in networks. We also discuss the problem of formation selection and give an example of formation optimization in which communication costs are minimized under constraints on preserving network connectivity and on the amount of movement required. In this dissertation, we first consider the problem of how to transform the network topology of a system of autonomous vehicles from an initial topology to a desired topology while maintaining network connectivity throughout the topology transformation process. We propose algorithms based on the concepts of prefix labeling and routing from the computer networking community to solve this problem when the final network topology is a tree. We present simulation results to evaluate the performance of our algorithms in terms of the amount of movement and time required to achieve the desired network topology. The algorithms we develop can be used to generate navigation functions that can be used by control systems to achieve a desired physical topology. Next, we consider a key unsolved subproblem, which is how the nodes in the initial network topology should be mapped onto the nodes in the final network topology before the network topology is reconfigured, while taking into account the needs to preserve network connectivity. We develop algorithms to solve this problem based on optimal and suboptimal graph-matching algorithms. We then apply these techniques with previously developed techniques to plan node movement to reconfigure the network topology while preserving network connectivity at all times. The performance of these techniques is evaluated via simulation. Afterward, we consider the problem of optimizing the network topology of a system of networked autonomous vehicles to minimize the aggregate network traffic required to support a given set of data flows under constraints on the total amount of movement by the autonomous vehicles. We propose a solution to this problem consisting of two steps. First, we develop algorithms to select a network tree topology from an arbitrary initial connected network topology. Second, we develop optimization algorithms to reconfigure the network tree topology found in the first step while preserving the connectivity to minimize the aggregate traffic under constraints on the total number of hops that the autonomous vehicles may move. Simulation results are presented to evaluate the performance of the algorithms. Finally, we apply networking concepts and optimization strategies to determine a feasible physical formation that reduces aggregate data traffic under constraints on the total amount of movement by the autonomous vehicles. We develop techniques that provide waypoints for use by physical control algorithms, under which the network connectivity will be ensured at all times if movement is on linear paths between the waypoints. Simulation results are presented to demonstrate that our proposed techniques can significantly reduce aggregate network traffic.
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 Leenhapat Navaravong.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Shea, John Mark.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045200:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045200/00001

Material Information

Title: Topology Reconfiguration for Systems of Networked Autonomous Vehicles with Network Connectivity Constraints
Physical Description: 1 online resource (104 p.)
Language: english
Creator: Navaravong, Leenhapat
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: algorithm -- networking -- topology -- uav
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Future systems of networked autonomous vehicles, such as unmanned aerial or ground vehicles, may rely on peer-to-peer, wireless communication to coordinate their actions. The physical formation of the network may need to be reconfigured at times based on the specified missions. However, reconfiguring the physical formation also impacts the link connectivity and, hence, the connectivity of the network. If the network is partitioned, then the autonomous vehicles can no longer coordinate their movements, and the mission may fail. In this dissertation, we develop techniques to transform the formation of a system of autonomous vehicles while preserving network connectivity. Several different approaches to address this problem are presented, with a focus on a method that utilizes ideas from routing packets in networks. We also discuss the problem of formation selection and give an example of formation optimization in which communication costs are minimized under constraints on preserving network connectivity and on the amount of movement required. In this dissertation, we first consider the problem of how to transform the network topology of a system of autonomous vehicles from an initial topology to a desired topology while maintaining network connectivity throughout the topology transformation process. We propose algorithms based on the concepts of prefix labeling and routing from the computer networking community to solve this problem when the final network topology is a tree. We present simulation results to evaluate the performance of our algorithms in terms of the amount of movement and time required to achieve the desired network topology. The algorithms we develop can be used to generate navigation functions that can be used by control systems to achieve a desired physical topology. Next, we consider a key unsolved subproblem, which is how the nodes in the initial network topology should be mapped onto the nodes in the final network topology before the network topology is reconfigured, while taking into account the needs to preserve network connectivity. We develop algorithms to solve this problem based on optimal and suboptimal graph-matching algorithms. We then apply these techniques with previously developed techniques to plan node movement to reconfigure the network topology while preserving network connectivity at all times. The performance of these techniques is evaluated via simulation. Afterward, we consider the problem of optimizing the network topology of a system of networked autonomous vehicles to minimize the aggregate network traffic required to support a given set of data flows under constraints on the total amount of movement by the autonomous vehicles. We propose a solution to this problem consisting of two steps. First, we develop algorithms to select a network tree topology from an arbitrary initial connected network topology. Second, we develop optimization algorithms to reconfigure the network tree topology found in the first step while preserving the connectivity to minimize the aggregate traffic under constraints on the total number of hops that the autonomous vehicles may move. Simulation results are presented to evaluate the performance of the algorithms. Finally, we apply networking concepts and optimization strategies to determine a feasible physical formation that reduces aggregate data traffic under constraints on the total amount of movement by the autonomous vehicles. We develop techniques that provide waypoints for use by physical control algorithms, under which the network connectivity will be ensured at all times if movement is on linear paths between the waypoints. Simulation results are presented to demonstrate that our proposed techniques can significantly reduce aggregate network traffic.
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 Leenhapat Navaravong.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Shea, John Mark.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045200:00001


This item has the following downloads:


Full Text

PAGE 1

TOPOLOGYRECONFIGURATIONFORSYSTEMSOFNETWORKEDAUTONOMOUSVEHICLESWITHNETWORKCONNECTIVITYCONSTRAINTSByLEENHAPATNAVARAVONGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

PAGE 2

c2013LeenhapatNavaravong 2

PAGE 3

Tomybelovedparents,Dr.SompongNavaravongandDr.LeenNavaravong 3

PAGE 4

ACKNOWLEDGMENTS Iwouldliketoexpressmysincerestgratitudetomyacademicadvisor,Prof.JohnM.Shea,forhisinvaluableguidanceandcontinuoussupportthroughoutmygraduatestudiesattheUniversityofFlorida.Ireallyappreciatehishelpandencouragement,whichputmeontherighttracktowardsmyPh.D.Ihavegreatlybenettedfromhiswisdom,expertise,andenthusiasm.Withouthisexcellentadvice,patience,andsupport,thisdoctoraldissertationwouldnotbepossible.IwouldliketothankProf.TanF.WongforhishelpandguidancewhenIwasworkingintheWINGTestbedgroup.Iwouldliketothankallofmycommitteemembers,Prof.WarrenE.Dixon,Prof.ColeSmith,andProf.PrabirBarooah,fortheiracademicsuggestionsandcommentsonmywork,whichhavehelpedtoimprovethisdissertation.IalsoextendmyappreciationtomybestseniorsfromtheWINGlaboratory,Dr.ByonghyokChoiandDr.ChanWongWong.Bothofthemhavealwaystreatedmeastheirownbrother.Theyalwaysgavemevaluablesuggestions,fruitfuldiscussionsandgeneroussupportduringmygraduatestudiesattheUniversityofFlorida.IwouldliketothankanotherseniorfromtheWINGlaboratory,Dr.DebdeepChatterjee,forhisusefuladviceaboutcourseworkandhissupportduringmyMastersstudy.IthankallofmyWINGlabmates,includingXingFang,ChenZhang,KeTang,BingGao,SienWu,KareemGraham,EricGraves,GokulBhat,ManishBansal,RaviTeja,ShravanBhat,VivekVijayaKumar,Sudhamshu,PriyankaSinha,DavidGreen,BenjaminLander,andTimTang,innoparticularorder.AllofthemmademydaysintheWINGlaboratorymoreenjoyable.Lastbutnotleast,Iwouldliketothankmyparents,Dr.SompongNavaravong,andDr.LeenNavaravong,fortheirlove,support,andsacrice.Theyhavealwaysbeenmyrolemodelsofhardworkandaconstantsourceofinspirationandencouragementthroughoutmylife. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 12 1.1OverviewofResearchfromControlsCommunity .............. 13 1.2NetworkingApproachestoFormationControl ................ 17 1.3OrganizationofDissertation .......................... 22 2PHYSICAL-ANDNETWORK-TOPOLOGYCONTROLFORSYSTEMSOFNETWORKEDAUTONOMOUSVEHICLES .................... 24 2.1AnonymousNetworkTopologyRecongurationAlgorithms ........ 26 2.1.1BasicAlgorithm ............................. 27 2.1.2SimultaneousTransition(ST)Algorithm ............... 30 2.2RelabelingAlgorithms ............................. 30 2.2.1BranchRelabeling(BR)Algorithm .................. 31 2.2.2NeighborRelabeling(NR)Algorithm ................. 33 2.3SimulationResults ............................... 35 2.4Summary .................................... 40 3GRAPHMATCHING-BASEDTOPOLOGYRECONFIGURATIONALGORITHMFORSYSTEMSOFNETWORKEDAUTONOMOUSVEHICLES ........ 41 3.1InitialTreeSelection(ITS)Algorithm ..................... 42 3.2SimulationResults ............................... 51 3.3Summary .................................... 55 4OPTIMIZINGNETWORKTOPOLOGYTOREDUCEAGGREGATETRAFFICINASYSTEMOFNETWORKEDAUTONOMOUSVEHICLESUNDERANENERGYCONSTRAINT .............................. 56 4.1ProblemFormulation .............................. 57 4.2NetworkTopologyRecongurationAlgorithms ................ 60 4.3NetworkTopologyOptimizationAlgorithms ................. 64 4.3.1NetworkTreeSelection(NTS)Algorithm ............... 64 4.3.1.1OptimalAlgorithm ...................... 64 4.3.1.2SimulatedAnnealing(SA)Algorithm ............ 67 4.3.2NetworkTopologyRecongurationOptimization(NTRO)Algorithm 67 4.3.2.1OptimalAlgorithm ...................... 69 5

PAGE 6

4.3.2.2SimulatedAnnealing(SA)Algorithm ............ 72 4.4SimulationResults ............................... 73 4.5Summary .................................... 76 5ROUTINGAPPROACHESTOOPTIMIZETHEPHYSICALTOPOLOGYOFASYSTEMOFNETWORKEDAUTONOMOUSVEHICLESTOREDUCEAGGREGATETRAFFIC ....................................... 78 5.1ProblemFormulation .............................. 80 5.2AdvancedNetworkTopologyRecongurationAlgorithm .......... 82 5.3PhysicalTopologyRecongurationOptimization(PTRO)Algorithms ... 88 5.3.1OptimalAlgorithm ........................... 91 5.3.2SimulatedAnnealing(SA)Algorithm ................. 92 5.4SimulationResults ............................... 92 5.5Summary .................................... 94 6CONCLUSIONSANDFUTUREWORK ...................... 96 6.1Conclusion ................................... 96 6.2FutureWork ................................... 98 REFERENCES ....................................... 99 BIOGRAPHICALSKETCH ................................ 104 6

PAGE 7

LISTOFFIGURES Figure page 1-1Anexampleofthearticialpotentialeldgeneratedforadisk-shapedworkspacewithdestinationattheoriginandanobstaclelocatedat[1,1]T. ......... 13 1-2Networktopologyreconguration. ......................... 20 1-3Conversionfromlinearcongurationtostarconguration. ............ 21 1-4Aggregateowminimizationbytopologyreconguration. ............ 22 2-1Networktopology1. ................................. 27 2-2Basicalgorithmexample. .............................. 29 2-3Simultaneoustransitionalgorithmexample. .................... 31 2-4Networktopology2. ................................. 32 2-5BranchRelabeling. .................................. 33 2-6NeighborRelabeling. ................................. 35 2-7Desiredtopology. ................................... 36 2-8Averagetotalhopover100randominitialtopologiesforeachlinknumber. ... 37 2-9Averagetotaltimeover100randominitialtopologiesforeachlinknumber. ... 38 2-10Averageoftheminimumtotalhopamongthenumberofrootconsiderationover100randominitialtopologies. ......................... 39 3-1Networktopology. .................................. 46 3-2Initialnetworktopology.Numbersoutside(inside)parenthesesrepresentthenode'slabelintheinitialtopology(roletowhichthatnodeisassignedinthenalformation). .................................... 48 3-3Initialtreeselectionalgorithmexample. ...................... 49 3-4AveragerequiredtotalnumberofhopsrequiredtoachievedesirednetworktopologyfordifferentmethodsofselectingtheinitialnetworktreeGtifromtheinitialnetworkGi,withrelabelingalgorithms. .................... 52 3-5AveragerunningtimerequiredtoachievedesirednetworktopologyfordifferentmethodsofselectingtheinitialnetworktreeGtifromtheinitialnetworkGi,withrelabelingalgorithms. ................................ 54 4-1Networktopology. .................................. 60 7

PAGE 8

4-2Labelingnetworktopology. ............................. 61 4-3Networkreconguration. ............................... 65 4-4Minimumachievableaggregatetrafcfornetworktreeselectionalgorithmsasafunctionofnetworksize. .............................. 74 4-5Minimumachievableaggregatetrafcforcombinedalgorithmsasafunctionofnetworksize. .................................... 76 5-1Physicaltopology. .................................. 79 5-2Networktopology. .................................. 83 5-3Labelingnetworktreetopology. ........................... 84 5-4Networkreconguration. ............................... 88 5-5Minimumachievableaggregatetrafcforcombinedalgorithmsasafunctionofnetworksize. .................................... 94 8

PAGE 9

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTOPOLOGYRECONFIGURATIONFORSYSTEMSOFNETWORKEDAUTONOMOUSVEHICLESWITHNETWORKCONNECTIVITYCONSTRAINTSByLeenhapatNavaravongMay2013Chair:JohnM.SheaMajor:ElectricalandComputerEngineeringFuturesystemsofnetworkedautonomousvehicles,suchasunmannedaerialorgroundvehicles,mayrelyonpeer-to-peer,wirelesscommunicationtocoordinatetheiractions.Thephysicalformationofthenetworkmayneedtobereconguredattimesbasedonthespeciedmissions.However,reconguringthephysicalformationalsoimpactsthelinkconnectivityand,hence,theconnectivityofthenetwork.Ifthenetworkispartitioned,thentheautonomousvehiclescannolongercoordinatetheirmovements,andthemissionmayfail.Inthisdissertation,wedeveloptechniquestotransformtheformationofasystemofautonomousvehicleswhilepreservingnetworkconnectivity.Severaldifferentapproachestoaddressthisproblemarepresented,withafocusonamethodthatutilizesideasfromroutingpacketsinnetworks.Wealsodiscusstheproblemofformationselectionandgiveanexampleofformationoptimizationinwhichcommunicationcostsareminimizedunderconstraintsonpreservingnetworkconnectivityandontheamountofmovementrequired.Inthisdissertation,werstconsidertheproblemofhowtotransformthenetworktopologyofasystemofautonomousvehiclesfromaninitialtopologytoadesiredtopologywhilemaintainingnetworkconnectivitythroughoutthetopologytransformationprocess.Weproposealgorithmsbasedontheconceptsofprexlabelingandroutingfromthecomputernetworkingcommunitytosolvethisproblemwhenthenalnetworktopologyisatree.Wepresentsimulationresultstoevaluatetheperformanceofour 9

PAGE 10

algorithmsintermsoftheamountofmovementandtimerequiredtoachievethedesirednetworktopology.Thealgorithmswedevelopcanbeusedtogeneratenavigationfunctionsthatcanbeusedbycontrolsystemstoachieveadesiredphysicaltopology.Next,weconsiderakeyunsolvedsubproblem,whichishowthenodesintheinitialnetworktopologyshouldbemappedontothenodesinthenalnetworktopologybeforethenetworktopologyisrecongured,whiletakingintoaccounttheneedstopreservenetworkconnectivity.Wedevelopalgorithmstosolvethisproblembasedonoptimalandsuboptimalgraph-matchingalgorithms.Wethenapplythesetechniqueswithpreviouslydevelopedtechniquestoplannodemovementtorecongurethenetworktopologywhilepreservingnetworkconnectivityatalltimes.Theperformanceofthesetechniquesisevaluatedviasimulation.Afterward,weconsidertheproblemofoptimizingthenetworktopologyofasystemofnetworkedautonomousvehiclestominimizetheaggregatenetworktrafcrequiredtosupportagivensetofdataowsunderconstraintsonthetotalamountofmovementbytheautonomousvehicles.Weproposeasolutiontothisproblemconsistingoftwosteps.First,wedevelopalgorithmstoselectanetworktreetopologyfromanarbitraryinitialconnectednetworktopology.Second,wedevelopoptimizationalgorithmstorecongurethenetworktreetopologyfoundintherststepwhilepreservingtheconnectivitytominimizetheaggregatetrafcunderconstraintsonthetotalnumberofhopsthattheautonomousvehiclesmaymove.Simulationresultsarepresentedtoevaluatetheperformanceofthealgorithms.Finally,weapplynetworkingconceptsandoptimizationstrategiestodetermineafeasiblephysicalformationthatreducesaggregatedatatrafcunderconstraintsonthetotalamountofmovementbytheautonomousvehicles.Wedeveloptechniquesthatprovidewaypointsforusebyphysicalcontrolalgorithms,underwhichthenetworkconnectivitywillbeensuredatalltimesifmovementisonlinearpathsbetween 10

PAGE 11

thewaypoints.Simulationresultsarepresentedtodemonstratethatourproposedtechniquescansignicantlyreduceaggregatenetworktrafc. 11

PAGE 12

CHAPTER1INTRODUCTIONSystemsofautonomousvehiclesundercooperativecontrol,suchasaerial,underwater,surface,orspacevehicles,provideversatileplatformsforcommercialandmilitaryapplications.Forinstance,[ 1 ]providesalistofsomeofthemainapplicationsforcooperativecontrolofmultivehiclesystems.Thislistincludes: MilitarySystems:FormationFlight,CooperativeClassicationandSurveillance,CooperativeAttackandRendezvous,andMixedInitiativeSystems; MobileSensorNetworks:EnvironmentalSamplingandDistributedApertureObserving;and TransportationSystems:IntelligentHighwaysandAirTrafcControl.Thesetypesoftasksusuallyrequirethatthevehiclescoordinatetheiractions,andthusthevehiclesmustbeabletoexchangeinformationoversomeformofcommunicationsnetwork,suchasanadhocwirelessnetwork[ 2 4 ].Thus,inmanyapplications,maintainingnetworkconnectivityduringformationcontrolwillbeanimportantissue.Formostapplications,communicationswillbeoverawirelessnetwork,inwhichthecommunicationslinksbetweenautonomousvehiclesaredependentonthepropagationofelectromagneticsignalsbetweentheautonomousvehicles.Becauseelectromagneticpowerdensitydecreaseswithdistance,thenetworktopologyishighlydependentonthephysicalformationofthesystem.Thus,formationcontroltechniquesmustbedesignedthatcanmaintainnetworkconnectivitywhileachievingthedesiredformationgoals.Inthischapter,wediscusstheproblemofformationcontrolwithnetworkconnectivity(FC+NC)constraintsorgoals,hereafterreferredtoasFC+NCproblems.Werstprovideanoverviewofthepriorresearchonthistopic,whichhasprimarilycomefromthecontrolscommunity.Thenweprovideanoverviewofourwork,whichappliesideasfromroutingincomputernetworksandposesformationcontrolproblemstominimizecommunicationcosts.Finally,weprovideanoutlineofthisdissertation. 12

PAGE 13

Figure1-1. Anexampleofthearticialpotentialeldgeneratedforadisk-shapedworkspacewithdestinationattheoriginandanobstaclelocatedat[1,1]T. 1.1OverviewofResearchfromControlsCommunityOverviewsoftechniquesforformationcontrol(withoutnecessarilyincludingnetworkconsiderations)aregivenin[ 1 5 6 ].Someofthemainapproachestoformationcontrolaregivenin[ 5 ]asleader-follower,virtualstructure,andbehavior-based.Anotherlistofapproachesisgivenin[ 1 ]asoptimizationbased,potential-eldsolutions,string-stabilitybased,andswarming.Oneofthemostwidelyusedapproachesinformationcontrolistousearticialpotentialeldstoguidethemovementoftheautonomousvehicles[ 1 7 8 ].Attractivepotentialeldsarecenteredatthegoallocations,andrepulsivepotentialeldsaregeneratedaroundobstacles.Drivenbythenegativegradientofthepotentialeld,eachautonomousvehiclewillconvergetoaminimumofthepotentialeld,whichistypicallythedesirednalposition.Anexampleofthegeneratedarticialpotentialeldisshownin Figure1-1 ,inwhichthedestinationisassignedaminimumpotentialvalueandtheobstacleisassignedamaximumpotentialvalue.InSectionIIIof[ 5 ],ChenandWangdiscusstheresearch(upto2005)ontheimpactofnetworkconnectivityontheanalysisofthestabilityandcontrollabilityof 13

PAGE 14

formationsofautonomousvehicles.ThepaperscitedinthatsectionrepresentsomeoftheearliestworkonFC+NCproblems.Muchoftheworkinthisarea,aswellaslaterworkonFC+NCproblems,utilizestoolsfromgraphtheory,andinparticular,algebraicgraphtheory.BecauseoftheimportanceofthisapproachtoFC+NCproblems,wegiveanoverviewofthistopicherebasedon[ 5 ]anditsreferences.Thenetworkconnectingtheautonomousvehiclescanberepresentedbyatime-varyinggraphG(t).Forsimplicity,weconsiderthecasewherethegraphisundirectedandsuppressthetimedependence.Thus,G=(V,E),whereVisthesetofvertices(representingtheautonomousvehicles)andEisthesetofedgesconnectingtheverticesinV.Anedge(x,y)2Eisanunorderedpairthatspeciesthatacommunicationlinkexistsbetweenxandy;xandyaresaidtobeadjacentorneighbors,andthisrelationisdenotedbyxy.Therelationbetweenthephysicalpositionsoftheautonomousvehiclesandthenetworkconnectivitydependsonthenatureofthecommunicationlinksconnectingtheautonomousvehicles.MostoftheworkonFC+NCproblemsassumesthatthecommunicationlinksfollowahomogeneousprotocolmodel,inwhichtwoautonomousvehiclescancommunicateiftheyarewithinaspeciedmaximumcommunicationrangeandcannotcommunicateiftheyareoutsideofthatrange.Weassumetheuseofthishomogeneousprotocolmodelinthisdissertation.Thismodelmayaccuratelymodelcommunicationsinmanysystemsofunmannedaerialvehicles(UAVs)workingaboveobstructions;however,shadowingandmultipathfadingmakethismodelinaccurateinmanyscenariosinvolvinggroundvehiclesoroperationsinurbanenvironments.Theedgeconnectioninformationcanbecollectedintoamatrixcalledtheadjacencymatrix,A(G),whichisdenedbyAij(G)=8>><>>:1,ij0,otherwise. 14

PAGE 15

TheconnectivityofthegraphGcanbedeterminedfromA(G)bycomputingeigenvaluesofthegraphLaplacian,L(G).Let(G)beadiagonalmatrix,inwhichthe(i,i)entryisthenumberofneighborsofvertexi(alsoknownasthedegreeorvalencyofvertexi);the(i,i)entryisthesumoftheelementsintheithroworcolumnofA(G).ThenL(G)=(G))]TJ /F6 11.955 Tf 12.3 0 Td[(A(G).Insomeworks,suchas[ 9 10 ],L(G)isdeneddifferently,whereL(G)=I)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F10 7.97 Tf 6.58 0 Td[(1(G)A(G).TheLaplacianissymmetricandpositivesemidenite.Leti(L)betheithsmallesteigenvalue.Then1(L)=0,andthemultiplicityoftheeigenvaluezeroisequaltothenumberofconnectedcomponentsinthegraph(i.e.,thenumberofconnectedsubgraphsthatdonotshareconnectionswitheachother).Thus,foranetworktobeconnected,theremustbeonlyonezeroeigenvalue.Thesecondsmallesteigenvalue,2(L),iscalledtheFiedlervalue[ 11 14 ]andgivesaquantitativemeasureofhowconnectedthegraphis.Theassociatedeigenvectorcanbeusedtodetermineasetoflinksthatifremovedwillcausethenetworktopartition[ 11 ].IntheearlyworkonFC+NCproblems,thenetworkgraphwasassumedxed,andtheimpactofthenetworkonthestabilityorcontrollabilityunderparticularcontrolstrategieswasevaluated.Forinstance(againsee[ 5 15 ]),considersystemsthatcanbemodeledusingrstorderdynamics,wherethedynamicsofagentiaregivenby_xi=ui,wherexiisthestateofagenti,andthecontrollawatagentiaveragesthevaluesfromitsneighbors,ui=)]TJ /F3 11.955 Tf 14.08 8.09 Td[(1 iiXji(xi)]TJ /F6 11.955 Tf 11.95 0 Td[(xj).Thenthecontrollabilityofthesystemofautonomousvehiclesisdeterminedbytheinterconnectiongraph.Moreover,thealgebraicrepresentationofthegraph'sinterconnectionscanbeusedtogiveasimpleformforthewholesystem'sdynamics 15

PAGE 16

as_x=)]TJ /F3 11.955 Tf 9.29 0 Td[()]TJ /F10 7.97 Tf 6.59 0 Td[(1=2L)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2x.Perhapssurprisingly,connectivitycandecreasecontrollability,andacompletegraphwiththisupdatelawcanbeshowntobeuncontrollable[ 15 ].Ontheotherhand,resultsusingalgebraicgraphtheoryindicatethatformationstabilitymaybeeasilyachievableforthecompletegraph[ 9 ].Althoughtheearliestworkthatidentiedconnectivityasacontrolobjectivewaspublishedin1999,mostoftheworkonformationcontrolwithconnectivityconstraintsorgoalshasbeenpublishedin2005andlater(cf.[ 6 12 16 27 ]).Muchofthecontrol-theoryworkonFC+NCproblemsfocusesonformationcontrolwithmaintenanceofexistingcommunicationlinks(cf.[ 1 22 23 28 ]).Thiscanbeachievedinformationcontrolsystemsusingthearticialpotentialeldapproachbytreatingnetworkconnectivityasanarticialobstacle[ 28 ].AnotherbranchofworkonFC+NCproblemsfocusesonthedesignofcontrollerstoenhancesomemeasureofnetworkconnectivityortomaintainnetworkconnectivity(whichislessrestrictivethanmaintaininglinkconnectivity)duringformationcontrol.TheIntroductionof[ 6 ]givesagoodoverviewofthiswork.Here,wegiveabriefoverviewofhowalgebraicgraphtheoryisusedintheseworks.Aspreviouslydiscussed,theFiedlervalue,2(L)givesanindicationoftheconnectivityofanetwork.Thus,optimizationtechniquescanbeusedtomaximize2(L).However,theFiedlervalueisanondifferentiablefunctionoftheLaplacianmatrix[ 6 ],whichmakesitdifculttouseincontrolstrategies.However,severalapproacheshavebeendeveloped,asdiscussedin[ 6 ].OnealternativetousingtheFiedlervalueasameasureofnetworkconnectivityistouseasumofpowersoftheadjacencymatrix,SK(G)=KXk=0A(G)k. 16

PAGE 17

Thei,jthentryofSKisthenumberofpathsoflengthKbetweeneverypairofnodesinthegraph.Thus,ifGisconnected,everyentryofSn)]TJ /F10 7.97 Tf 6.58 0 Td[(1(G)willbenon-zero,andSn)]TJ /F10 7.97 Tf 6.59 0 Td[(1(G)willbepositivedenite,wherenisthetotalnumberofagentsinthesystem.Thissum-of-powersoftheadjacencymatrixcanthenbeusedtodevelopoptimization-basedcontrollersforconnectivitymaintenance,whicharegenerallycentralized.Changesinthetopologycanbeaccommodatedthroughacontrollerthatutilizesglobaltechniques,suchasgossipandauctions,toguaranteethatlinkbreakageswillnotdisconnectthenetwork[ 19 ].Theseworksdevelopanimportantprinciplethatmaketheseproblemsmoreaccessibletonetworkresearchers:theoverallFC+NCproblemcanbedecomposedsuchthatnetworkconnectivitycontrolisperformedinthediscretespaceoverthenetworkgraph,whilemotioncontrolisperformedinthecontinuousspaceusingconventionalcontroltechniques,suchaspotentialelds.Thus,networkingresearcherscancontributetotheproblemsofformationcontrolwithouthavingtobecomeexpertsinmotioncontrol.Thisapproachisdemonstratedinthenextsection. 1.2NetworkingApproachestoFormationControlTheapproachesdescribedin Section1.1 haveseverallimitations.Theapproachespresentedin[ 5 ]areprimarilyfocusedontheimpactofagivennetworkconnectivitygraphonthecontrolalgorithm.Thosedescribedin[ 1 ]aremostlyfocusedonmaintainingnetworkconnectivityduringformationcontrol.Theapproachesin[ 6 ]aredesignedtooptimizeconnectivityoronlyallowlimitedrecongurationoftheformation.Inthesepreviousworks,theabsoluteorrelativeposesoftheagentsarepre-specied.Thus,additionalworkonFC+NCproblemsisneededtoaddressthefollowingFC+NCproblems: 1. Techniquesareneededtorecongureasystemsofautonomousvehiclesfromarbitraryinitialconnectedformationstoarbitrarynalconnectedformations,withnobreakinconnectivityduringthereconguration. 17

PAGE 18

2. Manysystemsmayuseautonomousvehiclesthathaveidenticalcapabilities.Suchsystemsshoulduseanonymousformationreconguration,inwhichtheautonomousvehiclesareidenticalandcantakeanypositioninthenaltopology1. 3. Techniquesareneededthatallowoptimizationofanal,connectedformationtooptimizemeasuressuchascommunicationcostsortask-specicutilityfunctions,whilemaintainingnetworkconnectivityandobeyingconstraintsontheamountsofmovement.In Chapter2 and Chapter3 ,weproposeanetworking-basedapproachthatcanbeusedtoaddressallthreeproblemslistedabove.Inthissection,wepresentanoverviewoftheworkin Chapter2 and Chapter3 andexplainhowitcanbeusedtosimultaneouslyaddressproblems1and2.Inthissection,wegiveanexamplethatillustrateshowthesetechniquescanbeusedtoaddressproblem3;ourexampleapplicationminimizescommunicationcostsunderconstraintsontheamountofmovement.Thekeyideafrom Chapter2 and Chapter3 isthatautonomousvehiclescanbetreatedaspacketsthatarethenroutedthroughthenetwork2.Anexampleofthisapproachisshownin Figure1-2 .Thereareseveralissuesthathavetobesolvedinimplementingthisconcept.Therstisthatautonomousvehiclescanonlybemovediftheywillnotcausethenetworktopartitionandiftheycanbemovedalongpathsthatpreservetheirconnectivitytothenetwork.Thesecondisthatwewishtoperformanonymousformationreconguration.Thatis,wedonotwishtospecifywhichnodesintheinitialtopologywilltakewhichpositionsinthenaltopology;rather,weonlycarethatthereisanautonomousvehicleineachpositionspeciedinthenaltopology.Toaddresstheseissues,werstinvoketheseparationprincipledescribedin[ 6 ]:the 1Semi-anonymoustechniquesshouldbecreatedforgeneralheterogeneousnetworksinwhichsomeautonomousvehiclesareidentical.2Ourapproachtoreconguringthenetworktopologycanbeconsideredtobepathplanning.Considerationslikevehicledynamicsandcollisionavoidanceneedtobehandledbythephysicalcontrolalgorithms. 18

PAGE 19

problemsofnetworktopologycontrolandphysicalpositionaretreatedseparately.Utilizingthenavigationfunctionofthecontrollerdevelopedin[ 28 ],thedesiredphysicalformationcanbeachievedoncethenetworktopologyisasupersetofthenetworktopologyinthedesiredphysicalformation.Thus,inthisdissertation,weassumethatthegoalistospecifytopologychangesassequencesofnodepositionsspeciedbyverticesintheinitialtopologygraphandthattherearephysicalposition-controltechniquesthatcanmoveautonomousvehiclestopositionsthatcorrespondtotheedgesandverticesoftheinitialtopology.Ingeneral,wecandivideformationrecongurationproblemsintotwoclassesbasedonhowthenalformationisdetermined: 1. Forsomeapplications,theformationtobeachievedisspeciedbytheapplication.Forinstance,forlandingUAVsonasmallrunway,alinearformationmayberequired,whilemonitoringaxedareamayrequireagridformation. 2. Inotherapplications,theformationmaybeselectedduringoperationstooptimizeperformanceofthesystem.Forexample,theformationofUAVsmaybeadaptedbasedonthepositionsofmobilegroundvehiclesbeingtrackedormaybeadaptedtominimizethecommunicationcostsneededtoexchangeinformation.Ineachoftheseclassesofformationcontrolproblems,networkingconstraintsmayplayanimportantrole.Forexample,inasearchoperation,agridformationmaybedesired.However,ifthevehicles'separationistoolarge,thennetworkconnectivitymaybreak.Thus,anotherautonomousvehiclemaybeusedtohelpmaintainnetworkconnectivityacrossthesystem.Anexampleofthisscenarioisillustratedin Figure1-3 ,inwhichagroupofUAVstakesofffromalauncharea,causingthemtoinitiallybeinalinearformation.Toconductagridsearchwhilemaintainingnetworkconnectivity,theformationistransformedtoastarformation,withthecenterUAVactingasanetworkrouter.Whenthenalformationisknown,thenthenetworktopologyrecongurationapproachin Chapter2 and Chapter3 canbecoupledwiththephysicalformationcontrolapproachin[ 28 ]toachievethedesiredformationwhilemaintainingnetworkconnectivity. 19

PAGE 20

Figure1-2. Networktopologyreconguration. 20

PAGE 21

Figure1-3. Conversionfromlinearcongurationtostarconguration. Examplesofformationoptimizationaregivenin Chapter4 and Chapter5 .Inthescenarioconsidered,theowoftrafcamongtheautonomousvehiclesisnonuniform:somepairsofautonomousvehicleshavemuchhighertrafcowsthanotherpairsofautonomousvehicles.Thissituationmayariseduringprocessessuchassensorfusionordatadisseminationinpeer-to-peernetworks.Generally,messagessentfromoneautonomousvehicletoanotherautonomousvehicleinthenetworkwillhavetoberelayedbyintermediateautonomousvehicles.Wecallthetotalamountoftransmissionandretransmissiontheaggregatetrafc.Clearly,theaggregatetrafcwillbehighlydependentonthenetworktopology,andthusthenetworktopologycanbeoptimizedtominimizetheaggregatetrafc.Forexample,considerthesystemofUAVsshownin Figure1-4 (a).TheUAVswithlabels011and02areexchangingmessages,butbecauseofthenetworktopology,theirtrafchastoberelayedbyeveryothernodeinthenetwork.Asshownin Figure1-4 ,node011canberepositionedtobeaneighborof 21

PAGE 22

Figure1-4. Aggregateowminimizationbytopologyreconguration. node02.Uponachievingitsnewpositioninthenetworktopology,node011isrelabeledasnode021,andtheaggregatetrafcisminimizedsee Figure1-4 (c)and(d). 1.3OrganizationofDissertationTherestofthedissertationisorganizedasfollows.In Chapter2 ,weproposealgorithmstosolvetheproblemofhowtotransitionthenetworkedautonomousvehicle'ssystemfromonetopologytoanothertopologywithadifferentnetworkcongurationwhilemaintainingthenetworkconnectivity.Wethenevaluatetheperformanceintermsoftheamountofmovementandtimerequiredtoachievethedesirednetworkconguration.In Chapter3 ,wedevelopgraph-matchingbasedalgorithmstosolvetheproblemofhowthenodesintheinitialnetworktopologyshouldbemappedonto 22

PAGE 23

thenodesinthedesirednetworktopologytominimizetheamountofmovementoftheautonomousvehiclesbeforewerecongurethenetwork.Wealsoevaluatetheperformanceofourdevelopedalgorithmsviasimulation.In Chapter4 ,weproposealgorithmstosolvetheproblemofoptimizingthenetworktopologytominimizethetotaltrafcinanetworkrequiredtosupportagivensetofdataowsunderconstraintsonthetotalamountofmovementpossibleoftheautonomousvehicles.Wepresentsimulationresultstoevaluatetheperformanceofouralgorithms.In Chapter5 ,wedevelopoptimizationalgorithmstosolvetheproblemofoptimizingthephysicalformationofasystemofnetworkedautonomousvehiclestoreduceaggregatedatatrafcunderconstraintsonthetotalamountofmovementbytheautonomousvehicles.Wepresentsimulationresultstodemonstratethatourproposedtechniquescansignicantlyreduceaggregatenetworktrafc.Finally,theconclusionsaredrawn,andthefuturedirectionofthisworkisgivenin Chapter6 23

PAGE 24

CHAPTER2PHYSICAL-ANDNETWORK-TOPOLOGYCONTROLFORSYSTEMSOFNETWORKEDAUTONOMOUSVEHICLESInthischapterand Chapter3 ,weconsidertheproblemofhowtotransitiontheautonomousvehiclesfromonetopologytoanothertopologywithadifferentnetworkcongurationwhilemaintainingnetworkconnectivity.Atthehighestlevel,wepartitionthisproblemintotwosub-problems: I. howtotransitionthegroupofautonomousvehiclesfromonenetworktopologytoanotherwhilepreservingnetworkconnectivity,and II. howtomovethegroupofautonomousvehiclestoachievethedesiredphysicalformationonceI.iscompleted.Previousworkonformationcontroladdressesrelatedproblems.In[ 29 ]and[ 30 ],acentralizednavigationfunctionbasedonpotentialeldsisusedtocontrolagroupofautonomousvehiclestoachieveadesiredphysicalformationandorientation,withobstacleandcollisionavoidance.However,in[ 29 30 ]itisassumedthatallautonomousvehiclescancommunicatewitheachotheratalltimes,andthusnoeffortismadetoensurenetworkconnectivitywhileachievingthedesiredphysicalformation.Inthisdissertation,weconsiderformationcontrolinwhichnetworkconnectivitymustbepreservedatalltimes.Weuseaprotocolmodeltodeterminethemappingbetweenthephysicalformationandthenetworktopology.Inthismodel,acommunicationslinkexistswhenevertwonodesarewithinaspeciedmaximumcommunicationdistance.Suchascenarioisareasonablyaccuratemodelforcodedcommunicationoveranexponentialpath-losschannelwithadditivewhiteGaussiannoiseatthereceiver.Forthismodel,previousworkinthecontrolscommunityhassolvedtheproblemofreconguringthephysicalformationofthenetworkwhilemaintainingnetworkconnectivitywhen:[ 14 28 31 33 38 39 ] themappingbetweennodesintheinitialformationandthoseinthenalformationisgiven,and 24

PAGE 25

theinitialnetworktopologyisasupersetofthedesirednetworktopology.Ifthenodesareidentical,theninmanyapplications,anynodeintheinitialtopologymaybeassignedtotakeanypositioninthespeciednalformation.Wecallthistheanonymousrecongurationproblem,andusethetermrolestorefertothenodesinthenalformation.Todoanonymousrecongurationefciently,weneedtechniquesto (1) ndagoodmappingofthenodesintheinitialtopologytotherolesinthenalformation,and (2) reconguretheinitialnetworktopologysothatitisasupersetofthedesirednetworktopology.Bothofthesearestillopenresearchproblems.Becausethebestchoiceofinitialmappingwilldependonhowthenodeswillberepositioned,wefocusonproblem(2)rstinthischapterandthenproposetechniquestosolveproblem(1)in Chapter3 .Inthischapter,weproposeasolutiontoproblem(2)forthespecialcasewherethespeciednalnetworktopologyisconstrainedtobeatree.Byformingatreefromtheinitialnetworktopology,prexroutingandlabeling[ 34 ]canthenbeusedtorouteeachautonomousvehiclethroughtheinitialnetworktoachievethedesirednetworktreetopology,whilethenetworkconnectivityofasystemisalwayspreserved.Inthischapter,wedevelopalgorithmstotransformanarbitraryconnectednetworktopologyintoadesirednetworktopologybymovingnodesinthenetworkandperformingothertasks,suchasrelabelingnodes.Ouralgorithmsarebasedontheconceptsofprexlabelingandprexroutingfrom[ 34 36 ].Webelieveourworkisuniqueintheeldofformationcontrolinthatitisonlyconcernedwithachievingadesiredtopology(whilemaintainingnetworkconnectivity)andnotinplacingparticularautonomousvehiclesintoparticularlocationsinthetopology.Weevaluatetheperformanceofouralgorithmintermsoftheamountofmovementandtimerequiredtoachievethedesiredtopology. 25

PAGE 26

2.1AnonymousNetworkTopologyRecongurationAlgorithmsWeconsiderasystemofnetworkedautonomousvehiclesthatcommunicateoverwirelesslinks.Forthepurposesofadaptingthenetworktopology,wecanrepresentthenetworkasasimplegraphG=(V,E),wheretheverticesVrepresenttheautonomousvehicles,andanedgee2Ebetweenverticesuandvindicatesthatuandvcancommunicateoverawirelesslink,Inthischapter,weassumethatalloftheautonomousvehiclesareidentical.LetGidenotetheinitialnetworktopology,andletGtfdenotethedesirednaltopology.TheverticesinGrepresentspecicvehiclesintheinitialformation,andwerefertotheseasnodes.Ontheotherhand,theverticesinGtfrepresentdesiredpositionsforthevehicles.Sincethevehiclesareassumedtobeidentical,weconsideranonymoustopologyreconguration,inwhichitisacceptableforthedesiredpositionsofthevehiclestobelledbyanyofthevehiclesinGi.Thus,werefertotheverticesinGtfasroles,andtheroleswillbelledbythenodesfromGi.WeassumethatGtfisdistributedtoallofthenodesinGibeforeouralgorithmsbegin.WeconsiderthecasethatGtfisatree[ 37 ],andwelabelGtfasaprextree,whichiscommonlycalledatrie.Ourkeyobservationisthatwecanreorganizethenetworktopologybyroutingautonomousvehiclesthroughtheexistingtopologyasneeded.Werstselectaninitialprextree,Gti,fromtheinitialgraphtopologyGithatincludesalltheverticesinGi.TheproblemofhowtoselectagoodmappingbetweenthenodesintheinitialtopologyGiandthedesirednalformationGtfisitselfachallengingproblem,andthesolutionofthisproblemwilldependonhowthenodesinthetopologywillmovetoachievethedesiredformationoncetheinitialmappingisselected.Thus,inthischapterwefocusontheproblemofhowtomovethenodestoreorganizetheformationgivenaspecicmappingfromthenodesintheinitialformationtotherolesinthedesiredformation.Aspreviouslymentioned,anapproachtooptimizethismappingisgivenin Chapter3 26

PAGE 27

AInitialnetworktopology. BDesirednetworktopology.Figure2-1. Networktopology1. Inthischapter,weassumethatthemappingbetweenthenodesintheinitialformationandtherolesinthenalformationisperformedbyrstselectinganodeinGitomaptotheroleoftherootnodeinGtf.TherootnodethenassignsprexlabelstoeachnodeinGiinbreadth-rstfashion.Wheneverprexlabelingisdone,eachnodeonlymaintainsanedgee2EithatcorrespondstoaprextreetoformGti.Theprexlabelassignedtoeachnodeservesasitsnetworkaddress.Nodesthatarechosentochangepositioninthetopologyusemaximumprexmatchinglogic[ 34 36 ]toroutethemselvesthroughthenetwork.Inthissection,weusetheinitialtrietopologyanddesiredtrietopologyshownin Figure2-1 ,asexamplestoexplainourbasicapproach.Allthenodesareassumedtoknowthedesirednetworktopology. 2.1.1BasicAlgorithmWebeginbyidentifyingnodesinGtithatdonotmatchthetopologyofGtf.Anodethatismissingchildreniscalledarequestingnode.AnodethatdoesnotmatcharoleinGtf(i.e.,whoselabeldoesnotexistinthedesirednetworktopology)iscalledanextranode.Eachrequestingnodewillsendtherootamessage,M.Req,thatincludesalistoftheaddressesofallitsmissingchildren.Acopyofthismessagewillalsobestoredatallnodeswhoarelocatedalongthepathtotheroot.Atthesametime,eachextranodewillalsosendamessage,M.Label,includingitsownlabeltotheroot. 27

PAGE 28

Aftertherootobtainsallthemessagesfromthesenodes,therootwillthensendamessage,M.Move,toeachextranode,one-by-one,startingfromanextranodewhohasthelongestlabel(i.e.,theextranodethatisdeepestinthetrie)toindicatetothatnodethatitshouldmoveuptowardtheroot.ThisextranodewillcheckforcachedM.Reqmessagesinthenodeslocatedalongthewaytotheroot.Wheneveritrstdiscoversacachedmessage,itsearchesthroughthiscacheforthemissingnodedestinationaddress.Ifthereexistsmorethanonemissingnodedestination,anextranodewilldecidetomovetotheclosestmissingnodebycomparingtheprexlabelofthemissingnodedestinationwiththenodeaddressofthenodetowhichtheextranodeiscurrentlyconnected.WheneverthisextranodereachestherequestingnodethatisitsparentinGtf,therequestingnodewillcheckthedestinationaddressoftheextranodeandthensendamessage,M.RCM,totheroot.Thismessagewilltelltherootthatithasreceivedanextranode,andthenextextranodecannowbemoved.Thismessagealsotellsallthenodesalongthepathtotheroottodeletethismissingnodedestinationaddressfromtheircache.Atthesametime,theextranodewillmoveandberelabeledtotaketheroleoftheformerlymissingnodeinthedesiredtopology.WhentherootobtainsM.RCM,itwillinitiatetheprocessagainbysendinganotherM.Movetothenextdeepestextranode.Thisprocesswillcontinueuntilthedesirednetworktopologyisachieved.Anexampletoillustratethisalgorithmisshownin Figure2-2 .Thegray-shadednodesrepresentextranodes,andthenodeswithdashedlinesrepresentmissingnodes.Therearetworeasonsforrstmovinganextranodewiththelongestprexlabel.First,wemustmaintainnetworkconnectivitythroughoutthetopologytransformationprocess.Ifanextranodeisaparentofotherextranodes,theotherextranodeswillbeorphanedandthusmaylosenetworkconnectivityiftheparentnodemovesrst.ThesecondreasonismoresubtleandisrelatedtohowouralgorithmassignsextranodestollinmissingverticesinGtf.Theproblemisthatifnodesthatareclosertotherootare 28

PAGE 29

ARequestingnodesandextranodessendmes-sagestotheroot. BRootsendmessageto011111. C011111reaches0111and0111thensendsmessagetotheroot. D011111isrelabeledas01112andfor-wardedtothedestination.Figure2-2. Basicalgorithmexample. movedrst,thentheoverallamountofmovementbynodesmayincrease.Thisisbestillustratedbyanexample.ConsideragaintheGtiandGtfshownin Figure2-1 ,butnowassumethattherootsendsM.Movetoextranode012rst.Referringto Figure2-2A ,node012willndacachedmessagefromrequestingnode0111atnode01andthuswilltraveldownthetreetobecomenode01112.Thennode011111willhavetotravelallthewayuptotherootbeforendingtheM.Reqfrom0211,afterwhichitwillroutedownthe02branchtobecomenode02112.Thetotalnumberoflevelsinthetreethatthetwonodesmustmoveis9,comparedto5whentheextranodethatisdeepestinthetreeismovedrst. 29

PAGE 30

2.1.2SimultaneousTransition(ST)AlgorithmThetimerequiredtotransitiontothenaltopologycanbereducedbyallowingsomeextranodestomovesimultaneously.Wemodifythebasicalgorithmasfollowstoallowsimultaneousmovement.TherequestingnodesandextranodessendM.ReqandM.Labeltotherootastheydointhebasicalgorithm.M.Reqwillincludeboththerequestingnodedestinationaddressandthemissingchildnodeaddress.TherootassignseachmissingnodeadestinationaccordingtoAlgorithm1,andinformseachsuchnodeofitsdestinationaddressbysendingitanM.Destmessage,whichincludestherequestingnodeaddress.AnextranodethatisaleafnodecanmovewheneveritreceivesamessageM.Dest,butothernodeshavetowaitfortheirchildrentomoveupthetreebeforetheparentcanmove.Theextranodewhoreachesthedestinationrstwillbeimmediatelyrelabeledandforwardedbytherequestingnodetofulllthedesirednetworktrie.Ifmorethanoneextranodereachesthedestinationrequestingnodeatthesametime,therequestingnodewillcheckthehopnumberofeachextranodebycomparingtheextranode'slabelwiththesharedprex.Thentherequestingnodewillassignthosenodestoitssubtreetominimizethenumberofhopseachnodemusttravel.Thatis,nodesthathavealreadytraveledthelargestnumberofhopswillbeassignedaschildrenoftherequestingnode,whereasextranodesthathavetraveledfewhopswillbeassignedastheleafnodesofthedeepestpartofthesubtree.AnexamplethatillustratesSTisgivenin Figure2-3 2.2RelabelingAlgorithmsTheperformanceofthebasicandSTalgorithmsarehighlydependentonthewaythattheinitialtrieiscreatedfromtheinitialconnectedgraphtopology,includingbothwhichedgesinGiareremovedandhowtheprexlabelsareassigned.Thus,theperformancecanbeimprovedifwemodifythesealgorithmstoconsiderotherlabelingsofthetrieandotherwaystocreateatriebydeletingedgesfromGi.Bothofthesecan 30

PAGE 31

ARootsendsamessageincludingdestinationaddressto012and011111. BNodes012and011111arriveatthedesireddestinationandarerelabeledas02112and01112respectivelytocompletethedesirednetworktopology.Figure2-3. Simultaneoustransitionalgorithmexample. Algorithm1:DestinationAssignmentAlgorithm Input: Initialnetworktree,Gti=(Vti,Eti);Desirednalnetworktree,Gtf=(Vtf,Etf);AllExtraNodes=GetExtraNode(Gti,Gtf);AllMissingNodes=GetMissingNode(Gti,Gtf);foreachu2AllExtraNodesdo foreachv2AllMissingNodesdo DistanceMatrix(u,v)=GetHopDistance(u,v); /* StoringdistanceforallpairsoftheextranodesinGtiandthemissingnodesinGtfinthematrixusedforHungarianmethod*/endendExtraNodeAndMissingNodePair=GetHungarianAssignment(DistanceMatrix); /* UsingHungarianmethodtoobtainoptimalpairsoftheextranodesinGtiandthemissingnodesinGtfthatminimizestherequiredtotalamountofmovement*/foreachpair2ExtraNodeAndMissingNodePairdo ExtraNode=GetExtraNode(pair);MissingNode=GetMissingNode(pair);RequestingNode=GetRequestingNode(MissingNode,Gti,Gtf);ExtraNode.DestinationAddress=GetPrexLabel(RequestingNode);ExtraNode.MissingNodeAddress=GetPrexLabel(MissingNode);end beconsideredwaysofrelabelingthenodesinGti.Inthediscussionbelow,weusetheinitialandnaltopologiesin Figure2-4 asexamplestoillustrateourrelabelingapproaches. 2.2.1BranchRelabeling(BR)AlgorithmInthissection,weconsiderswappingtheprexlabelsofallthenodesintwoormorebranchestoreducethetotalamountofnodemovementrequiredtoachievethe 31

PAGE 32

AInitialnetworktopology. BDesirednetworktopology.Figure2-4. Networktopology2. desirednetworktopology.Beforebranchrelabeling(BR),GiistransformedintoatreeGti,whichisprex-labeledintheusualway.Theneachleafnodewillsendamessageincludingitslabeltotheroot.Therootwillthenhavetheknowledgeofallthenodeprexlabelsinthenetwork.Startingfromtheroot,nodescanthenconsiderswappingtheprexlabelassociatedwithtwobranchesofdescendants.Inthiswork,weonlyapplybranchlabelingattheroot,asthatismostlikelytohavethelargestimpactontheamountofmovementrequiredtoachievethedesiredtopology.Therootevaluatestheamountofmovementthatisrequiredtoperformthetopologytransformationforeachpossiblebranchlabeling,andselectstheminimumsuchlabeling.ThetotalnumberofhopsforagivenbranchlabelingcanbeeasilyfoundfromthedestinationassignedtoeachextranodebyAlgorithm1.AfterBRisdone,theSTalgorithmisusedtotransformthenetworktopology.AnexamplethatillustratestheBRalgorithmisshownin Figure2-5 .Inthisexample,theprextreetopologywithinitialprexlabelingisshownin Figure2-5A .Withthisinitialprexlabeling,therearethreeextranodes:03,031,and032,andtherearethreemissingnodes:011,012,and022.Theseextranodeshavetoberepositionedtotakeplaceofthemissingnodes,andthiscanbedonebyusingsimultaneoustransition 32

PAGE 33

ABeforeperformingBR(5hopsrequired). BAfterperformingBR(1hopsrequired).Figure2-5. BranchRelabeling. algorithmgivenin Section2.1.2 toachievethedesirednetworktopologyshownin Figure2-4B .Thisrequiresatotalof5hopstorepositionalltheseextranodestoachievethedesirednetworktopologyshownin Figure2-4B .Ontheotherhand,ifBRalgorithmisusedinthisscenario,itwillswaptheprexlabelingofthenodesintheleft-mostbranchwiththoseintheright-mostbranchoftheinitialprextreetopologyshownin Figure2-5A .Thisresultsintheprextreeshownin Figure2-5B .Withthisprexlabeling,thereisonlyoneextranode:03,andonemissingnode:022.Hence,onlyonehopisrequiredtorepositiontheextranodetoachievethedesirednetworktopologyshownin Figure2-4B 2.2.2NeighborRelabeling(NR)AlgorithmReducingGitoatreeaccordingtothebreadth-rstapproachdescribedin 2.1 eliminatessomeconnectionsthatcouldreducethenumberofnodesthathavetobemoved.Inthissection,weproposeatechniquetoreducethenumberofnodesthathavetobemovedbytakingadvantageofthoseadditionallinksinGi.Wepresentaneighborrelabeling(NR)algorithmtoachievethisgoal.Firstaninitiallabelingisdone,startingfromtheroot.Allthenodesrstcheckiftheyareanextranode.Allextranodeswillresettheirlabeltonull.ThenallnodesthataremissingchildreninGtfwillcheckifany 33

PAGE 34

oftheextranodesaretheirneighborsinGi.Ifso,thenodethatismissingachildwillrelabeltheneighboringextranodebysendingitamessage,M.Relabel.Thenodethatisrelabeledwillthenrepeatthesameprocess.Thisprocesscontinuesuntilnomorerelabelingispossibleoruntilatimerexpires.AllremainingextranodeswillthenbelabeledbytheirneighborstoformGti.STwillthenbeappliedimmediatelyafterNRisdonetoachievethedesirednetworktrie.AnexampleofNRisillustratedin Figure2-6 .Inthisexample,westartwiththeinitialnetworktopologywithprexlabelingassignedbytherootshownin Figure2-4A .Threeextranodesareinitiallyidentiedasextranoes:03,031,and032;thesenodessettheirprexlabeltonull.Therequestingnodesare01and02,whoaremissingnodes011,012,and022respectively.Aftertheextranodessettheirprexlabelstonull,therequestingnodescheckiftheyhaveneighborsthatareextranodes.Therequestingnode02ndstwoextranodesconnectingtoit,anditsendsamessageM.Relabeltorelabeltheprexlabelofoneofthetwoextranodestorelabelitasitsmissingnode022.However,thereisnoextranodeasneighborfortherequestingnode01.Finally,theremainingextranodesarerelabeledbytheirneighbornodes,andeachnodedisablessomeoftheirlinkstoturntheinitialnetworktopologyintoaprextree.Finally,thesimultaneoustransitionalgorithmisusedtorepositiontheremainingextranodestoachievethedesirednetworktopology.Fordensenetworks(inwhichthenumberofedgesishigh),NRmayofferadvantagesoverBRbecauseNRcantakeadvantageofthemanynetworkconnections.Toofferthegreatestbenetacrossinitialnetworktopologies,NRandBRcanbecombined,asfollows.First,networkconnectionsamongallnodesareutilizedbyNR.Afterthisisdone,BRalgorithmwillbeperformed.Theperformancesofthesealgorithmsareillustratedinthenextsection. 34

PAGE 35

ARelabelinganextranodetobe022bynode02. BTheremainingextranodesarerelabeled. CTrienetworktopology.Figure2-6. NeighborRelabeling. 2.3SimulationResultsInthissection,weevaluatetheperformanceoftheproposednetworktopologycontrolalgorithms.Weconsideranetworkofsevennodes,wheretheinitialtopologyisgeneratedatrandomwithdifferentdensityofnetworkconnections,whichwequantifyintermsofthetotalnumberofnetworklinks.Weconsidertransformingthenetworkintothefourdifferentnaltreetopologiesofvaryingdepthshownin Figure2-7 .Weevaluatetheperformanceoftherelabelingalgorithmsbyconsideringtheaveragetotalamountofmovement(numberofhops)andthetotaltimerequiredto 35

PAGE 36

ATopologyA. BTopologyB. CTopologyC. DTopologyD.Figure2-7. Desiredtopology. achievethedesiredformation.Inthischapter,weassumethatthetimerequiredforasinglehopisthesameforeverynodeinthenetwork.Wecomparetheperformancesofthedifferentalgorithmsfordifferentamountsofconnectivityintheinitialgraph,whichismeasuredintermsofthetotalnumberofcommunicationlinks(jEij,thenumberofedgesinGi).ForeachvalueofjEij,100randomtopologiesGiwereinstantiated.ForeachGi,therootofGtiisrandomlyselectedandtheprexlabelsareassignedaccordingly.Theresultsin Figure2-8 showthetotalamountofrequiredmovementasafunctionofthenumberoflinksinthenetwork.Itcanbeseenfrom Figure2-8 thattherelabeling 36

PAGE 37

Figure2-8. Averagetotalhopover100randominitialtopologiesforeachlinknumber. algorithmscanreducetheamountofmovementrequired,exceptforthecaseofthedesirednetworkshownin Figure2-7A .Sincethenetworkof Figure2-7A hasdepthofone,theonlymissingnodeswillbetheroot'schildnodes,andtherootlabelsallofitsneighborsaschildnodeswhenformingGti.Thus,therearenoopportunitiesforrelabeling.Fortheothernalnetworktopologies,weobservethattheBRalgorithmoutperformstheNRalgorithmwhenthenetworkisverysparse,andviceversa.Thisisbecausewhenthenetworkissparse,therewillbefewerredundantlinksthattheNRalgorithmcanutilize.Itcanalsobeseenthatthedensityofthenetworkaffectstheamountofmovementrequiredtoachievethedesiredtopology.Forinstance,inthecaseofnorelabeling,whenthenetworkissparse,thedesiredtopologyin Figure2-7A requiresmorenodemovementthanthetopologyshownin Figure2-7B and Figure2-7C becausetheroot 37

PAGE 38

Figure2-9. Averagetotaltimeover100randominitialtopologiesforeachlinknumber. mayhavefewerchildnodesandtherewillbemoreextranodesthathavetocomeuptotheroottofullltheplaceoftheroot'smissingchildnodes.Whenthenetworkisverydense,therootwillbeabletoassignallthenodesinthenetworkasitschildrentoformtheinitialnetwork.Hence,theinitialtopologywillalreadybeintheformofthedesiredtopologygivenin Figure2-7A .Thusforadensenetwork,topologyArequiresnonodemovement.Asexpected,thecombinedBRandNRalgorithmsprovidesthebestperformance.Next,weconsidertheaveragetotaltimerequiredtoformthedesiredtopologiesgivenin Figure2-7 .WeusetheSTalgorithm,forwhichtheamountoftimerequiredtotransformthetopologyisnotalwaysmonotonicwiththeamountofmovementrequired.Forexample,iftheGiissparse,theinitialnetworkislikelytobesimilartoanarrowtreethatisdeepwithfewchildnodesateachlevel.IftopologyAisdesired,therewill 38

PAGE 39

Figure2-10. Averageoftheminimumtotalhopamongthenumberofrootconsiderationover100randominitialtopologies. bemanyextranodes,andthetotalamountofmovementwillbemorethanifthedesiredtopologyhadmoredepth,suchasintopologiesBorC.However,fortopologyA,sincethemissingnodesaretheroot'schildnodes,whichareonlyonelevelbelowtheroot,thefurthestanyextranodehastomoveisuptotheroot.Thus,asseenin Figure2-9 ,theaveragetimetotransformthetopologyissmallerfortopologyAthanfortheothertopologies.Finally,weconsidertheeffectofthechoiceofrootnode(whichnodeinGithatischosentobetherootofGti)ontheperformanceofthetopologyrecongurationalgorithm.Foreachtopologyconsidered,weconsidertheperformanceifweselectmultiplerootsatrandomandchoosetherootthatresultsintheminimumamountofmovement,whereacombinationoftheBRandNRalgorithmsareused. 39

PAGE 40

Theaveragenumberofhopsthatnodesmustmovefor100randominitialtopologiesisshownin Figure2-10 .Asthenumberofrootsconsideredincreases,therequiredamountofmovementsignicantlydecreasesforallnaltopologies.Thus,thechoiceofrootsignicantlyaffectstheamountofmovementtorecongurethenetwork,whichmotivatesourstudyofthisissuein Chapter3 2.4SummaryInthischapter,wedevelopedanewapproachtorecongurethenetworktopologyofasystemofnetworkedautonomousvehicles.Wedevelopnewalgorithmstoroutenodesthroughthenetworkbyusingprexlabelingandrouting.Wedevelopedseveralapproachestoenhancetheperformanceofthetopologyrecongurationalgorithm.Theresultsshowthatthesimultaneoustransitionalgorithmwiththeadditionofbothrelabelingalgorithms,branchandneighborrelabeling,achievesasignicantimprovementintermsoftotalmovementandtimerequiredtoformthedesirednetworktree.Finally,appropriaterootselectionfromasetofnodesispossibletoachievebetterperformanceintermsofthetotalnumberofhopsrequiredtoachievethedesirednetworktrieconguration. 40

PAGE 41

CHAPTER3GRAPHMATCHING-BASEDTOPOLOGYRECONFIGURATIONALGORITHMFORSYSTEMSOFNETWORKEDAUTONOMOUSVEHICLESAsdiscussedin Chapter2 ,todoanonymousrecongurationefciently,weneedtechniquesto (1) ndagoodmappingofthenodesintheinitialtopologytotherolesinthenalformation,and (2) reconguretheinitialnetworktopologysothatitisasupersetofthedesirednetworktopology.In Chapter2 ,weproposeasolutiontoproblem(2)forthespecialcasewherethespeciednalnetworktopologyisconstrainedtobeatree.Byformingatreefromtheinitialnetworktopology,prexroutingandlabeling[ 34 ]canthenbeusedtorouteeachautonomousvehiclethroughtheinitialnetworktoachievethedesirednetworktreetopology,whilethenetworkconnectivityofasystemisalwayspreserved.Inthischapter,weproposetechniquestosolveproblem(1).Wewishtochooseamappingbetweenthenodesintheinitialnetworktopologyandtherolesinthespeciednalnetworktopology,suchthatthetotalamountofmovementrequiredtorepositionthenodesintheinitialnetworktoachievethedesirednetworktopologyisminimized.Inthecasethatthegraphsareisomorphic,thisisagraph-matchingproblem,whichisknowntobeNP-hard,andtheoptimalsolutionisprovidedbyUllmann'salgorithm[ 40 ].Whenthegraphsarenotisomorphic,thenwecanndanapproximatesolutionbysolvingaweightedgraph-matchingproblem,forwhichsuboptimalsolutionsarepossibleusingalgorithmsthathavereasonablecomputationalcomplexity.Inparticular,weapplyanapproachbasedonspectralgraphtheorytothisproblem.Wethenapplytherelabelingalgorithmsproposedin Chapter2 tofurtherreducethetotalamountofnode'smovementrequiredtoreconguretheinitialnetworktreetoachievethedesirednetworktreetopology.Weevaluatetheperformanceoftheproposedalgorithmsintermsof 41

PAGE 42

complexityandtherequiredamountofmovementtoachievethedesirednaltopology,whichisevaluatedthroughsimulation. 3.1InitialTreeSelection(ITS)AlgorithmWemodelthenetworktopologyofasystemofautonomousvehiclesbyanordinaryundirectedgraphG=(V,E),whereVisasetofN=jVjvertices,whichrepresentthevehicles,andEV2=VVisasetofedgesthatrepresentthecommunicationslinksbetweenpairsofvehicles.TheinitialnetworktopologyofasystemisrepresentedbyGi=(Vi,Ei).ThedesirednetworktopologyisrepresentedbyGtf=(Vtf,Etf).WeconsiderthecasewhereGtfisatree.Wewishtochooseamapping:Vti!VtffromthenodesinGtitotheroles(vertices)inGtfthatminimizestheamountofmovement(intermsofnumberofedgesthatanodesmusttraverse)thatwillberequiredtomovethenodesintothedesirednaltopology.Weutilizetheapproachin Chapter2 torecongurethetopologyaroundatreethatisasubgraphoftheinitialgraphwhilemaintainingnetworkconnectivity.LetTSdenotethesetoftreesthataresubgraphsoftheinitialgraph.Thenthebestmappingbetweenthenodesintheinitialgraphandtherolesinthenalgraphisgivenby =argmin:Vti!Vtf,G2TS(Gi)Xu2VidhopG!Gtf(u,(u)),(3)wheredhopG!Gtf(u,)isadistancefunctionwhichgivetherequiredamountofmovement,intermsofnumberofedgesthatmustbetraversed,torepositionnodeufromitspositioninGtorolevinGtf.Theselectionofissimilartoagraphmatchingproblem,whichisknowntobeNP-hard[ 41 42 ].IfthereisasubtreeGtiGithatisisomorphictoGtf;i.e.,thereexistone-to-onecorrespondenceVti!Vtfsuchthat(u,v)2Eti()(Vti!Vtf(u),Vti!Vtf(v))2Etf,thennomovementwillberequired.Inthiscase,Ullmann'salgorithm[ 40 ]canbedirectlyusedtoobtain.However,iftheredoesnotexistasubtreeofGithatisisomorphictoGtf,Ullmann'salgorithmwillabsolutelyfailtogiveGti.Inaddition,the 42

PAGE 43

complexityofUllmann'salgorithmisO(NN).Thismotivatesustoconsidertechniquesthatcanprovideapproximatesolutionsto( 3 )atlowercomplexityandthatcanprovidesolutionswhenthereisnosubtreeofGithatisisomorphictoGf.Inthesecases,thedistancecomputationsin( 3 )dependonthechoiceoftreeinT(Gi),and( 3 )doesnotdirectlymapontoagraphmatchingproblem.ThusweproposeaheuristicsolutiontochoosingGtithatcanbesolvedasanapproximateweightedgraph-matchingproblem.Fortheapproximateweightedgraph-matchingapproach,weproposetochoosethatminimizesacostfunctionbasedonthedifferencesbetweenthedesirednalgraphtopologyandtheinitialgraphtopology, J()=X(u,v)2V2iwi(u,v))]TJ /F6 11.955 Tf 11.95 0 Td[(wtf[(u),(v)]2.(3)Herewi()andwtf()areweightingfunctionsthattakeasinputanedge(speciedbytheverticesitconnects)andthatspecifytheimportanceofedgesintheinitialandnalgraphs,respectively.Thisisthesameformasin[ 43 ],inwhichafastalgorithm(thathascomplexityO(N3))isdeveloped.Thekeyistoreformulate( 3 )byreplacingwithapermutationmatrixP,yielding[ 43 ] J(P)=kPAiPT)]TJ /F23 11.955 Tf 11.96 0 Td[(Afk,(3)whereAiandAfareweightedadjacencymatricesforGiandGtf.Notethatifcorrespondstoaone-to-oneisomorphismbetweenGiandGtf,thenJ(P)=0andPAiPT=Af.However,ndingtheoptimalsolutionto( 3 )forothercasesrequiresexponentialcomplexity.ByrelaxingthedomainofPfromthesetofpermutationmatricestothesetoforthogonalmatrices,( 3 )canbesolvedapproximatelyusingthegraphspectra.LettheeigendecompositionsofAiandAfbeAi=UiiUTiandAf=UffUTf.Furthermore,let Uiand UfbematricesforwhicheachelementistheabsolutevalueofthecorrespondingelementinUiandUf,respectively.Lettingdenotethesetof 43

PAGE 44

permutationmatrices,theapproximatesolutionto( 3 )isgivenby[ 43 ] argmaxP2trPT Uf UTi,(3)whichcanbesolvedbytheHungarianmethodinO(N3)time.ThereareseveralproblemsthatmustbeaddressedinordertoapplyUmeyama'smethod[ 43 ]tondagoodsolutionfor( 3 ).Thechoiceofweightfunctionsin( 3 )(whichdeterminetheweightedadjacencymatricesin( 3 ))greatlyaffectthesolutionfor.Inaddition,themayresultinagraphGithatdoesnotpreservealloftheedgesinGtf,anddependingonwhichedgesinGtfaremissing,differentamountsofmovementwillberequiredin( 3 ).Considertheexamplenetworktopologyillustratedin Figure3-2 ,basedonGiandGtfshownin Figure3-1 .ThenumberinparenthesisoneachnodeinGiin Figure3-2 representstheroleinGtftowhichthatnodeismapped.Weobservethatinordertondamappingthatminimizestheamountofnodemovementrequiredin( 3 ),unequalprioritiesshouldbeassignedtotheedgesinGtf.Forinstance,anedgeinGtfthatisclosertotherootnodeandhasmanydescendantsshouldhaveagreaterprioritytobeinGithanedgesthatarefurtherdownthetreeandhavefewerdescendants.Weachievethisthroughtheweightingfunctionswiandwtf.WerstdenetheweightingfunctionwtfoftheedgesinGtf,exceptfortheedgesattachedtoleafnodesinGtf,as wtf(u,v)=jcvj[maxdepth(cv))]TJ /F3 11.955 Tf 11.96 0 Td[((depth(v))]TJ /F3 11.955 Tf 11.96 0 Td[(1)],(3)where u,v2Vtfdenoteaparentnodeandchildnode,respectively,inVtf, cvdenotesasetofthechildrenofnodevinGtf, maxdepth(cv)denotesafunctionofthesetcvofthedescendantsofnodevthatprovidesthemaximumdepthofthenodesincv,and 44

PAGE 45

depth(v)denoteafunctionofthenodev2Vtfthatprovidesthedepthofnodev2Vtf.EdgesattachedtoleafnodesinGtfwillbeassignedaweightof1.AftertheedgesinGtfareallcompletelyassignedtheweight,theweightofedgesinGiisassignedaccordingtotheweightingfunction wi(u,v)=max(u,v)2Etfwtf(u,v),(3)Hence,theweightsofalledgesinGiareassignedthemaximumweightoftheedgesinGtf.Next,startingfromthepermutationgivenfrom( 3 ),webuildtheinitialtreeGtiGi.Theprocessstartsfromthenodeinvi2VithatmapsontotherootnodeinGtf.Gtiisgrowninbreadth-rstfashionandworksinparallelonGtiandGtf,wherewetrytogrowedgesforGtiinGiwhereverpossible.Therootnodevfrstchecksifithaschildrenvcf2VtfinGtf.Ifitdoes,viwillalsocheckifithasneighbornodesvni2ViinGi.Ifitdoes,viwillrstselectaneighbornodevniwithVi!Vtf(vni)=vcftoformtheedge(vi,vni)2EiforGtiwhichcorrespondstotheedge(vf,vcf)2Etf.However,avnithatsatisesVi!Vtf(vni)=vcfmaynotexist,inwhichcaseoneoftheunassignedneighborsvniwillbeassignedtoformtheedge(vi,vni)2EiforGtiwhichcorrespondstotheedge(vf,vcf)2Etfbasedon min(vcf,vni)dGtf(vcf,Vi!Vtf(vni)).(3)HeredGtf(u,v)isafunctionofthedistanceintermsofhopbetweennodeu2Vtfandv2VtfinGtf.ThedistancebetweentwonodesinGtfcouldbefoundbyapplyingthebreadth-rstsearchalgorithm[ 44 ].Thealgorithmwillproceedtomaptheneighborsvnitoallofthechildrenoftheroot,vcfinthesameway,andthenwillproceedtomappingthenodesatconsecutivelydeeperlevelsofthetree.IfitisnolongerpossibletokeepformingedgesforGtiinGiandtherestillexistsaremainingnodevri2ViinGithatisdisconnectedfromtheexistingGti,vri 45

PAGE 46

AInitialnetworktopology. BDesirednetworktopology.Figure3-1. Networktopology. willbeconnectedtooneofthenodesvti2VtiintheexistingGtibyformingtheedge(vri,vti)2EiinGioneedgeatatime.Theedge(vri,vti)2EiwillbeformforexistingGtioneedgeatatimebyconnectingvriwithvtiinexistingGtibasedon min(vri,vti)2EidGtf(Vi!Vtf(vri),Vi!Vtf(vti)).(3)ThepseudocodeoftheInitialTreeSelection(ITS)algorithmcanlogicallybepartitionedintotwoparts,whichareillustratedinAlgorithm1and2,wherepartIisaboutinitiallyformingGtibasedon( 3 ),andpartIIisaboutformingthecompletedGtiafterobtainingGtifrompartIbyconnectingvritoanexistingGtibasedon( 3 ). AnexampleofhowGtiisformedisillustratedin Figure3-3 basedonGiandGtfshownin Figure3-1 ,andalsothenode'smappingbetweenViandVtf,shownin Figure3-2 .Inthisexample,vji2Viandvjf2VtfthedenotejthnodesinGiandGtf,respectively.Thealgorithmstartsattherootnodev1f2Vtfandthecorrespondingroletowhichthatnodeismapped,v3i2Vi.Inthisexample,nodev1fhastwochildren,v2fandv3f.Nodev3ihasthreeunmarkedneighbornodesv2i,v5i,andv6i,whosemappinginGtfarev2f,v4f,andv5frespectivelyasshownin Figure3-3A .ThesenodesshouldbeusedtoformedgesinGtithatwill 46

PAGE 47

Algorithm1:InitialTreeSelectionAlgorithm(PartI) Input: Initialnetwork,Gi=(Vi,Ei);Desirednalnetworktree,Gtf=(Vtf,Etf);Node'smappingobtainedfromtheinexactgraphmatchingalgorithm, Output: Initialnetworktree,Gti=(Vti,Eti);Vti=;;Eti=;;DesiredTreeRootNode=GetRootNode(Gtf);InitialGraphRootNode=)]TJ /F22 5.978 Tf 5.76 0 Td[(1(DesiredTreeRootNode);DesiredTreeQ=DesiredTreeRootNode;InitialGraphQ=InitialGraphRootNode;MarkNode(DesiredTreeRootNode);MarkNode(InitialGraphRootNode);Vti=InitialGraphRootNode;while(InitialGraphQ6=;)&&(DesiredTreeQ6=;)do DesiredTreeQTemp=;;InitialGraphQTemp=;;QueueLength=GetQueueLength(DesiredTreeQ);fori=1:QueueLengthdo DesiredTreeChildNodes=GetUnmarkedChildNodes(DesiredTreeQ[i],Gtf);InitialGraphNeighborNodes=GetUnmarkedNeighborNodes(InitialGraphQ[i],Gi);while(DesiredTreeChildNodes6=;)&&(InitialGraphNeighborNodes6=;)do MinimumDistance=1;foreachu2DesiredTreeChildNodesdo foreachv2InitialGraphNeighborNodesdo Distance=GetDistance(u,[v],Gtf);ifDistance
PAGE 48

Algorithm1:InitialTreeSelectionAlgorithm(PartII) Input: Initialnetwork,Gi=(Vi,Ei);Desirednalnetworktree,Gtf=(Vtf,Etf);Initialnetworktree(obtainedfrompartI),Gti; Output: Initialnetworktree,Gti=(Vti,Eti);UnMarkedNode=GetUnMarkedNode(Vi);whileUnMarkedNode6=;do MinimumDistance=1;foreachu2UnMarkedNodedo MarkedNeighborNodes=GetMarkedNeighborNodes(u,Gi);foreachv2MarkedNeighborNodesdo Distance=GetDistance((u),(v),Gtf);ifDistance
PAGE 49

AGiwithnode'smappingbetweennodesinGiandGtf. BTheedgebetweennodes3and2inGiisformedforGticorre-spondingtotheedgebetweennode1and2inGtf. CTheedgebetweennodes3and6inGiisformedforGti. DThemappingsofthenodes,6and4,inGiareswappedsothattheedgebetweennodes3and6inGiisnowcor-respondingtotheedgebetweenthenodes,1and3,inGtf. ETheedgebetweenthenodes,2and5,inGiisformedforGtithatcorrespondstotheedgebetweenthenodes,2and4,inGtf. FTheedgebetweenthenodes,2and1,inGiisformedforGti. GThemappingsofthenodes,1andnode4,inGi,areswappedsothattheedgebetweenthenodes,2and1,inGiisnowcorrespondingtotheedgebetweenthenodes,2and5,inGtf. HTheedgebetweenthenodes,6and7,inGiisformedforGtithatcorrespondstotheedgebetweenthenodes,3and6,inGtf. IBuildingthecorre-spondingedgeofGtfforGtiisnolongerpossible.Theedgebetweenthenodes,4and7,inGiisformedforGti. JInitialnetworktreeGtiisachieved.Figure3-3. Initialtreeselectionalgorithmexample. 49

PAGE 50

desirednalnetworktreeGtf,isconsideredanditismappedtothenodev2i2ViintheinitialnetworkGi.Thenodev2fhastwochildren,v4fandv5f,inGtf,andthemappednodev2ihastwounmarkedneighbornodes,v1iandv5i,inGiwhosemappingarethenodes,v7fandv4f,inGtfrespectively.Proceedingasbefore,weconsiderformingedgesinGtithatcorrespondto(v2f,v4f)and(v2f,v5f)inGtf.Basedon( 3 ),thenodesv5iandv1iareusedtoformtheedges(v2i,v5i)and(v2i,v1i),whichcorrespondto(v2f,v4f)and(v2f,v5f),respectively.Thealgorithmthenperformsthenecessaryswapsinthemappingandproceedstotheremaininglevels,asillustratedin Figure3-3G and Figure3-3H .However,afterthecreationoftheedge(v6i,v7i)inGti,whichcorrespondsto(v3f,v6f)inGtf,itisnolongerpossibletoformanycorrespondingedgee2EtfofthedesirednalnetworktreeGtffortheinitialnetworktreeGtiGi.However,thereisstillaremainingnodev4i2ViintheinitialnetworkGithatisdisconnectedfromtheinitialnetworktreeGti.TocompleteGtithenodev4iconsidersformingtheedgebetweenitselftooneofthenodesintheexistingGti.Using( 3 ),thebestedgewillbeformedfromv4itov7i,asillustratedin Figure3-3I .Finally,theinitialnetworktreeGtiisachieved.TheinitialnetworktreeGtiGianditsmappingareshownin Figure3-3J ComplexityofITSAlgorithm.Theworst-casecomplexityoccurswhenGtfhasdepthone,whichmeansthatallnodesexcepttherootnodeinGtfarethechildrenoftherootnode,andalsowhenallthenodesinGiexceptthemappedrootnodeareconnectedtothemappedrootnode.ThisalgorithmwilltrytomatchoneofedgesinGtftooneoftheedgesattachedtothemappedrootnodeinGioneedgeatatimeuntilalledgesinbothGiandGtfareuniquelymatched.EachtimethealgorithmcomparesalltheedgesinGtftoalltheedgesattachedtothemappedrootnodeinGianddecidestomatchonepairofedgesfrombothGiandGtfbasedon( 3 ).SincethenumberofnodesconnectingtorootnodeinGtfandthenumberofnodesconnectingtothemappedrootnodeinGiareN)]TJ /F3 11.955 Tf 12.71 0 Td[(1,whereNisthetotalnumbernodesinGi,thenthenumber 50

PAGE 51

ofedgestobeconsideredformatchingineachtopologyisequaltoN)]TJ /F3 11.955 Tf 12.36 0 Td[(1.Hence,theworstcaserunningtimeT(N)forthisalgorithmisgivenby T(N)=N)]TJ /F10 7.97 Tf 6.58 0 Td[(1Xi=1(N)]TJ /F6 11.955 Tf 11.95 0 Td[(i)2
PAGE 52

Figure3-4. AveragerequiredtotalnumberofhopsrequiredtoachievedesirednetworktopologyfordifferentmethodsofselectingtheinitialnetworktreeGtifromtheinitialnetworkGi,withrelabelingalgorithms. fromthismethodisdissimilartoGtfandthususuallyrequiresthatmorenodesinGtiberepositionedtoachieveGtf.Thus,thetotalamountofnodemovementmaybehigh.WealsoconsideredcombiningUllmann'salgorithmwiththeITSalgorithmbyrstapplyingUllmannalgorithmtoobtainGtifromGithatisisomorphictoGtfifthereexistssuchanisomorphism.IfthereisnosubgraphthatisisomorphictoGtfinGi,wethenapplytheITSalgorithmtoobtainGtithatisclosetoGtf.TheresultsofcombiningUllmann'salgorithmwiththeITSalgorithmsarealsoshownin Figure3-4 .Notethatforthisapproach,weonlyobtainedtheresultsforthenetworks10orfewernodesbecauseofthehighcomputationalcomplexityofUlmann'salgorithm.WeobservedthatthecombinedalgorithmscanfurtherreducethetotalamountofmovementrequiredtorepositionthenodesinGtitoachieveGtfincomparisontousingITSalone.However,the 52

PAGE 53

relativelysmalladditionaladvantageofthecombinedUlmann'sandITSalgorithmmaynotjustifytheadditionalO(NN)complexityoverusingthepolynomialcomplexityITSalgorithm.Alsoshownin Figure3-4 areresultsforcombiningthevariousalgorithmspreviouslydiscussedwiththetherelabelingalgorithmsproposedin Chapter2 .In Chapter2 ,weproposedtworelabelingalgorithms,neighborrelabeling(NR)andbranchrelabeling(BR)algorithmstorelabeltheprexlabelingofGtitomakeitsprexlabelingbettermatchthatofGtf(intermsofreducingthenumberofnodesthatneedtobemovedtotransformGtiintoGtf).However,BRalgorithmproposedin Chapter2 usesanexhaustivesearchtoobtaintheoptimalrelabelingpatternthatminimizesthetotalamountofnode'smovementrequiredtotransformGtiintoGtf,whichhasveryhighcomplexity.Thus,weinsteadapplythesimulatedannealingalgorithm[ 46 ],whichismetaheuristicsearchingalgorithmthatcanbeusedtoobtainanapproximatesolutionthatisclosetotheoptimalsolution.ThefollowingparametersareusedforBRalgorithmbasedonsimulatedannealingalgorithm:initialtemperatureTi=1,naltemperatureTf=0.01,coolingrate=0.97,BoltzmannconstantKc=0.1,andneighborhoodradius=1.Weapplybothrelabelingalgorithms,NRandBRalgorithms,toGtiobtainedfrom3differentmethods,includingBFS,ITS,andcombiningUllmannwithITS,andtheresultsarepresentedin Figure3-4 .Weobservedthatrelabelingalgorithmscanfurtherimprovethetotalamountofnode'smovementtorecongureGtiobtainedfromthesethreedifferentmethods.Tobetterillustratetherelativecomplexityofthealgorithms,theresultsin Figure3-5 showtheaveragerunningtimeinsecondsforeachalgorithm.Notethatthey-axisisinlogscalebecauseofthehugerangeinresults,whichisattributabletotheexponentialcomplexityofUllmmann'salgorithm.Forallofthealgorithms,theaveragerunningtimeincreasesasthenetworksizeincreases.TheaveragerunningtimeofBFSmethodislessthanthatoftheITSmethod,buttheaveragerunningtimeofcombiningBFSwithNRandBRiscomparabletoITSwithNRandBRbecausetheNRandBRalgorithms 53

PAGE 54

Figure3-5. AveragerunningtimerequiredtoachievedesirednetworktopologyfordifferentmethodsofselectingtheinitialnetworktreeGtifromtheinitialnetworkGi,withrelabelingalgorithms. dominatethecomplexity.TheaveragerunningtimeofcombiningUllmann'salgorithmwithITSislowwhenthenetworksizeissmallbecausewhenevertheUllmannalgorithmcanndthesubgraphintheinitialtreeGtithatisisomorphictothedesiredtreeGtf,itisnotnecessarytoruntheITSalgorithm(includingtheHungarianalgorithmtomatchextranodeswithmissingnodes).However,asthenetworksizeincreasestoeightormorenodes,theaveragerunningtimeofUllmann'salgorithmexceedsthecomplexityoftheotheralgorithmsandeventuallymakesthisapproachimpracticalbecauseofitsexponentialcomplexity. 54

PAGE 55

3.3SummaryInthispaper,wedevelopedalgorithmstoefcientlyperformanonymousrecongurationofthenetworktopologyofasystemofnetworkedautonomousvehicles.Wepartitiontheproblemintomappingtheinitialnetworktopologyontothedesirednetworktreetopology,selectingacorrespondinginitialtreetopology,andthenroutingthenodesthroughthenetworktopologyusingthetree-basedalgorithmsdescribedin Chapter2 .Theproblemofdeterminingamappingfromthenodesintheinitialtopologytotherolesinthenaltopologyisachievedbymappingontoaweightedgraphmatchingproblem,forwhichanapproximatesolutioncanbefoundusinggraphspectraatrelativelylowcomplexity.Weightingfunctionsareprovidedthatprovideunequalweightacrossthegraphtominimizetheamountofmovementthatwillberequiredtotransformbetweentheinitialandnaltopologies.Thenabreadth-rstsearchisusedtoformaninitialtreetopologythatmatchesthedesiredtreetopologybeforethetree-basedtopologyrecongurationalgorithmsproposedin Chapter2 areapplied.Theresultsshowthatthealgorithmsdevelopedinthischaptercansignicantlyreducethetotalamountofnodemovementrequiredtorecongurethesystemfromanarbitraryinitialnetworktopologytoadesirednetworktreetopology. 55

PAGE 56

CHAPTER4OPTIMIZINGNETWORKTOPOLOGYTOREDUCEAGGREGATETRAFFICINASYSTEMOFNETWORKEDAUTONOMOUSVEHICLESUNDERANENERGYCONSTRAINTInthischapter,weconsiderreconguringtheformationofasystemofautonomousvehiclestoreducethetotalamountoftrafcgeneratedinsendinginformationacrossthenetwork.Thetotalamountoftrafcwilldependbothontheinformationowstobetransmitted,aswellasthetopologyofthenetwork.Thelatterconsiderationisbecauseoftheneedforintermediatenodestorelayinformationbetweenasourceanddestination.Thus,theaggregatedatatrafc,whichincludesallofthedatatransmissionsfromsourcesandrelayswillgenerallybemuchlargerthanthetotaltrafcowfromthesources.Inthischapter,wefocusondatatrafconlyanddonotconsidertheimpactofcontroltrafc.Forconvenience,weusethetermaggregatetrafcinplaceofaggregatedatatrafcfromhereon.Inthischapter,weconsidertrafcatthenetworklayeranddonotconsidertheimpactoftheinteractionamongowsatthelinklayer.Sincetheautonomousvehiclesaremobile,theaggregatetrafccanbereducedbyreconguringthenetworktopologytomovesomeofthecommunicatingautonomousvehiclesclosertogether.Weconsidernetworksinwhichthenetworkconnectivitymustbemaintainedatalltimes,andanymovementschememusttakethisintoaccount.In Chapter2 and Chapter3 ,wepresentedtechniquesthatallowtheformationofasystemofvehiclestobereconguredwhilepreservingnetworkconnectivitybasedonroutingthevehiclesthroughatreetopology.Inthosechapters,weconsidertheproblemofanonymousformationreconguration,inwhicheachvehiclemaytakeanyroleinthenalformation.Thenalformationisspeciedaheadoftime.Bycontrast,inthischapter,weconsiderascenarioinwhichsomesubsetofthevehicleshaveuniquetrafcows,andhencethesevehiclescannotbetreatedanonymously.However,westillleveragetheideaofroutingthevehiclesthroughatreetopologytoensurethatnetworkconnectivityispreservedduringtheformationreconguration. 56

PAGE 57

Wealsoassumethattheautonomousvehiclesmayhaveniteenergythatlimitstheextentoftheirmovementormaybeotherwiseconstrainedintheirmovementbecauseoftheirotherduties,suchassensing.Thus,weconsidertheproblemofoptimizingthenetworktopologytominimizetheaggregatetrafcinanetworktosupportagivensetofdataows,underconstraintsonthetotalamountofmovementallowedbytheautonomousvehicles.Inthecasethattheautonomousvehiclesdonothaveanyenergyconstraintsandtheshapeofthenalnetworktopology(agraphconsistingofsetsofedgesandvertices,butnottheassignmentofautonomousvehiclestovertices)isalreadydened,thisproblemfallsintheclassofresourceallocationproblemsknownasquadraticassignmentproblems[ 47 48 ].Unfortunately,evenforthissimplersubclassofproblems,theproblemisNP-Hard,andthustherearenoknownsolutionsthatruninpolynomialtime.Inthischapter,weproposesuboptimalstrategiestorecongurethenetworktoreducetrafcbyagainpartitioningtheproblemintotwosteps,aswasdonein Chapter2 and Chapter3 .Intherststep,theinitialgraphisreducedtoatreetopologyinsuchawaythattheincreaseintrafcfromeliminatingsomelinksisminimized.Thenweproposetechniquestooptimizethetreeformationtoreducenetworktrafcunderthemovementconstraints.Theperformanceofthealgorithmsarecomparedandevaluatedusingsimulation. 4.1ProblemFormulationWeconsiderasystemofnetworkedautonomousvehiclesthatcommunicateoverwirelesslinkswithlimitedcommunicationdistance.WerepresenttheinducednetworktopologyasasimplegraphG=(V,E),whereverticesVrepresenttheautonomousvehicles,andanedgee2Ebetweenverticesuandvindicatesthatuandvcancommunicateoverawirelesslink.LetF=ff(u,v):(u,v)2G2gbethesetofdataows,wheref(u,v)denotetheamountoftrafcfromsourceutodestinationvandG2denotestheCartesianproductofGwithitself.ThentheaggregatetrafcovernetworktopologyG 57

PAGE 58

is X(u,v)2G2f(u,v)dhopG(u,v),(4)wherethedistancefunctiondhopG(u,v)isthenumberofedgesintheshortestpathbetweenverticesuandvinG.Inthischapter,weconsidertheproblemofrepositioningautonomousvehiclestotransformtheirnetworktopologywhilemaintainingnetworkconnectivityandworkingunderaconstraintonthetotalamountofmovementallowed.Todothis,weutilizepreviousworkonreconguringnetworkedautonomousvehiclesthatareinatreetopologygivenin Chapter2 and Chapter3 whilemaintainingnetworkconnectivity.Inthischapter,weallowthenetworktostartfromanarbitraryconnectednetworktopology.Thus,wedecomposethepresentproblemintotwosub-problems(theoptimalsolutiontothetwosub-problemsmaynotbeoptimalforthecombinedproblem).First,weselectatreetopologyfromtheinitialconnectednetworkGibyselectinganetworktreeGtGisuchthatitsaggregatetrafc( 4 )isminimum.NotethatsincesomelinksinGiareeliminatedinformingGt,theaggregatetrafcofGtwillbegreaterthanorequaltotheaggregatetrafcofGi.LetT(Gi)bethesetofalltreetopologiesthataresubsetsofGiandcontainallverticesofGi.ThentheproblemofchoosingthenetworktreetopologyGtthatminimizestheaggregatetrafccanbeformulatedas:Gtmin=argminG2T(Gi)X(u,v)2G2fuvdhopG(u,v). (4)AfterweobtainGtminfrom( 4 ),arootnodeischoseninGtminandprexlabelingisappliedstartingattheroottogiveinitialprextreeGtiortrie.Thedistancebetweentwonodesinthetriecanbesimplydeterminedfromtheirprexlabels.Werstndthelargestprexthatiscommontothelabelsofbothnodes.Thisistheprexlabelofacommonparentofbothnodesinthetree.Thentheshortestpathbetweentwonodesisuptothecommonparentandthenbackdowntotheothernode.Hencethe 58

PAGE 59

totaldistanceissumofthedistancesfromeachofthemtotheircommonparent.Letudenotetheprexlabelassignedtonodeu.LetL(u)denotethelengthoftheprexlabelofnodeu,andL(u,v)denotethemaximumlengthprexincommontotheprexlabelsofnodesuandv.Then dhopGt(u,v)=[L(u))]TJ /F3 11.955 Tf 11.96 0 Td[(L(u,v)]+[L(v))]TJ /F3 11.955 Tf 11.96 0 Td[(L(u,v)].(4)Secondly,weconsideroptimizingtheaggregatetrafcbyrepositioningtheautonomousvehicleswhilepreservingnetworkconnectivityandundertheconstraintthatanautonomousvehiclessystemhasniteenergy.Ascanbeseenfrom( 4 ),thelargerthedistancebetweentwonodesthatshareadataow,thegreatertheaggregatetrafcinthenetwork,sincethesamemessagewillberelayedateveryintermediatenodebetweenthem.Tominimizetheaggregatetrafcintheabsenceofanyenergyconstraints,anynalconnectedgraphtopologyGtfispossible.LetC(G)betheconnectivityfunction,whichtakesonthevalue1whenthenaltopologyisconnectedand0otherwise.ThenwewishtondGtfthatsatisesGtf=argminGX(u,v)2G2fuvdhopG(u,v) (4)subjecttoC(G)=1.Now,ifweconstrainthatanautonomousvehiclessystemhaslimitedenergy,thensomenalgraphtopologiesmaynolongerbepossible.Moreover,theconstraintthatthenetworkbeconnectedatalltimeswillalsolimitwhichnaltopologiesarepossibleinthisscenario.Forinstance,ifanodeistomoveupthetree,thenallofitschildrenmusthavesufcientenergytoatleastmoveuptoconnectwiththatnode'sparent.LethTdenotethetotalnumberofverticesinthegraphthatallnodesmaymovebeforethetotalenergybudgetisexpended.LetF(Gti,Gtf,hT)beafeasibilityfunction,whichtakesonthevalue 59

PAGE 60

AInitialnetworktopology. BDesirednetworktopology.Figure4-1. Networktopology. 1whenthenaltopologyisfeasibleundertheenergyconstraintand0otherwise.Thentheaggregatetrafcminimizationundertheenergyandnetworkconnectivityconstraintscanbeformulatedas:Gtf=argminGX(u,v)2G2fuvdhopG(u,v) (4)subjecttoC(G)=1F(Gti,G,hT)=1.WedetermineF(Gti,G,hT)basedontransformingthetopologyusingaprex-routingapproachthatisbasedonthatin Chapter2 and Chapter3 ,asdescribedinthenextsection. 4.2NetworkTopologyRecongurationAlgorithmsBeforeaddressingtechniquestosolveboth( 4 )and( 4 ),wedescribehowthenetworktopologycontrolmethodgivenin Chapter2 and Chapter3 canbeutilizedinthisapplication,inwhichnodesarenotidentical.Inthissection,weassumethatboththeinitialtopologyGtiandthenaltopologyGtfareknown.Webeginbychoosinganode 60

PAGE 61

AInitialnetworktopology. BDesirednetworktopology.Figure4-2. Labelingnetworktopology. intheinitialtopologytoserveastherootofthetree.Inthischapter,weassumethattherootischosenatrandom.Asanalternative,therootmayalsobechosenaccordingtosomecriteria;thedesignofroot-selectionalgorithmsisanareaforfuturework.Asmentionedin Section4.1 ,therootthenassignsuniqueprexlabelstoeachofitschildren,whichassignuniqueprexlabelstotheirchildren,etc.,untiltheentiretreehasprexlabels.Inprexlabels,thelabelofavertex'sparentnodeisaprexofthatnode'slabel.TheinitialtreetopologyGtibecomesaprextree,ortrie[ 37 44 49 ].Theprexlabelassignedtoeachnodeservesasitsnetworkaddress.Weexplaintheprex-routingapproachtonetworktopologycontrolusingtheexampletopologiesshownin Figure4-1 .NodeAhasbeenselectedtobetheroot.Prexlabelsarethenassignedtoallnodesintheinitialnetworkstartingfromaroot,asshownin Figure4-2A .Afterprexlabelassignmentisdonefortheinitialnetworktree,eachnodesendsamessageincludingitsownprexlabelandidentitytotheroot.Aftertherootobtainsallmessagesfromeachnode,itwillhaveaknowledgeoftheinitialnetworkgraph.Therootwillthenlabelallthenodesinthedesirednetworktreewiththeprexlabelassignedtothesamenodeintheinitialtree.Thedesirednetworktreeafterlabelassignmentiscompletedisshownin Figure4-2B 61

PAGE 62

Therootsearchesfornodesthatneedtomovebetweentheinitialandnaltopologies,startingfromthetoptothebottomofthetree,inabreadth-rstmanner.Thenodesthatmustmovearethosewhoseprexlabeldoesnotcorrespondtothepositionwhereitislocatedinthedesiredtree.Thelabelforanodeshouldalwaysbeoftheform =parentl,(4)wheredenotetheprexlabelofparent'schildren,istheconcatenateoperator,andlistheuniquesufx.Nodesthatdonothavethecorrectprexlabelmustmovefromtheirpositionintheinitialtopology,andhencearecalledmovingnodes.Nodesthathavethecorrectprexlabelandthathavenotbeenpreviouslyassignedtobeamovingnode(seemorebelow)arenonmovingnodes.Fortheexamplenetwork,allofthenodesthatareoneedgeawayfromtheroothavethecorrectprexlabelandthusarenonmovingnodes.Next,therootconsidersallnodesthataretwoedgesaway(it'schildren'sdescendants).Asshownin Figure4-2B ,thenodewithlabel021hasacorrectprexlabel,butthenodewithlabel011doesnothaveacorrectprexlabel.Thus,node011willbeamovingnode.Ifaparentmoves,itwillcausenetworkconnectivitytobreakforitschildren,soallofthedescendantsofamovingnodemustalsobemovingnodes.Forinstance,since011isamovingnode,itschild0111mustalsobeamovingnode.So,eventhough0111initiallyhasaprexlabelthatmatchesitsparentinFigure Figure4-2B ,itisstillamovingnode.Foreachmovingnode,therootrecordstwolabels.(1)Itsanchor-nodelabelisthelabelofthenon-movingnodethatwillbethemovingnode'sdestination.(2)Itsdesiredlabelisthenewlabelofthemovingnodeuponarrivalatthedestinationinthedesiredtopology.Whentheroothasalreadyconsideredallnodesinthedesiredtree,therootwillsendamessageM.Destincludingbothlabelstoeachmovingnode.Amovingnodethenrstmovestothenodewhoseprexlabelistheanchor-nodelabel.Whenamovingnodearrivesatanon-movingnode,thenon-movingnoderstchecksthemovingnode's 62

PAGE 63

anchor-nodelabeltoseeifitmatchesthenon-movingnode'sprexlabel.Ifitdoes,thenon-movingnodewillserveastheanchornodeforthatmovingnode,anditthenusesthedesiredlabelofthatmovingnodetoforwardthemovingnodetotherightpositioninthedesiredgraph.Themovingnodewillberelabeledtomatchthedesiredlabelonceitreachesitsnalposition,whichwillmakeitsprexlabelcorrespondtoitspositioninthedesirednetworktopology.Thedesiredlabelofamovingnodecanbesimplydeterminedfromitsparentinthedesiredtopologyasgivenin Figure4-2B .Ifitsparentisamovingnode,itsparentmustalreadybeassignedthedesiredlabelbytheroot,andthemovingnode'sdesiredlabelisdeterminedfromthedesiredlabelofitsparent.Ifitsparentisanonmovingnode,thedesiredlabelisdeterminedfromitsparentprexlabel.Whenthemovingnodes011and0111receiveM.Destmessagefromtherootincludingboththeanchor-nodeanddesiredlabels,theywillmovethroughtheinitialnetworktowardanchornode02byusingmaximumprexmatchinglogic[ 34 36 ].Whenmovingnodes011and0111areabletoconnecttotheanchornode02,anchornode02willlookattheiranchorlabelstocheckif02istheiranchornode.Oncenode02determinesthatitistheanchornodefor011and0111,node02willcheckthedesiredlabelofbothofthenodes.Thedesiredlabelsare022and0221,respectively,and02willusetheselabelstoforwardnodes011and0111totherightpositionsinthedesiredtopology.Afterbothnodesarriveatthedesiredposition,theirlabelswillbechangedtothedesiredlabels,whichwillmaketheirprexlabelscorrespondtotheirpositioninthedesirednetworktopology.Considernowhownodesshouldbemovefromtheirpositionsintheinitialtopologytotheirpositionsinthedesiredtopologywithoutbreakingnetworkconnectivity.Generally,iftherearemultiplemovingnodesintheinitialtopology,whenevertheyreceiveamessagefromaroot,theycanstartmovingsimultaneously.However,amovingnodethatisnotaleafnodehastowaitforitsdescendantstomoveuptoit 63

PAGE 64

beforeitcanstartmoving.Otherwise,thenode'sdescendantswillbedisconnectedfromthenetwork.Forexample,consideragainnodes011and0111in Figure4-2A .Node011cannotmoverst,sincethatwouldbreaknetworkconnectivitytonode0111;ingeneral,aparentnodecannotmoveallofitschildrenmustmoversttomakeitaleafnode.Thus,node0111rsthastomoveuptonode011untilitisabletoconnecttonode01.Thenbothnodes011and0111cancontinuemovinguptotheroot,passingbynode01,untiltheyareabletoconnectwithnode02.Oncenode011connectswith02(atthetimeitreachestheroot),itwillbeimmediatelyrelabeledas022tomakethelabelofthenode011conformtotheprextree.Thennode0111willmovetowardnode02untilitisabletoconnecttonode011,whichhasalreadybeenrelabeledas022.Finally,node0111willberelabeledas0221toachievethedesiredtopology.Anexampleofthismethodisshownin Figure4-3 ,andthepseudocodeofalabelassignmentalgorithmtobeexecutedattherootisgiveninAlgorithm1. 4.3NetworkTopologyOptimizationAlgorithmsInthissection,wepresenttechniquestosolveboth( 4 )and( 4 ).WerstpresentmethodstoselecttheoptimalnetworktreetopologyGtintermsofaggregateowfromGibasedonsolving( 4 ).Wethenprovideapproachestorecongurethenetworktominimizetheaggregateowbasedonsolving( 4 ). 4.3.1NetworkTreeSelection(NTS)Algorithm 4.3.1.1OptimalAlgorithmTosearchforanoptimalnetworktreetopologyfromaninitialnetworkGi,Char'salgorithm[ 50 ]isusedtoenumeratealltreesGtGi.Char'salgorithmassociateseachsubgraphG0c=fV0c,E0cgofGibyvertexsequencec=fc1,c2,...,cn)]TJ /F10 7.97 Tf 6.59 0 Td[(1g,wherecidenoteavertexadjacenttovertexiandndenotethenumberofverticesinGi.EachccanbetranslatedintoasubgraphG0c,whereV0c=f1,2,3,...,ng,andE0c=f(1,c1),(2,c2),...,(n)]TJ /F3 11.955 Tf 11.96 0 Td[(1,cn)]TJ /F10 7.97 Tf 6.59 0 Td[(1)g. 64

PAGE 65

ATherootnode,0,sendsmessageM.Destincludingtheanchor-nodeanddesiredlabelstothemovingnodes,011and0111. BNodes011and0111useprexroutingtomovethroughthetreeuntiltheyareabletoconnecttotheirdesignatedanchornode,02. CNode011isforwardedtoanappropri-atepositioninthedesiredtopologyandrelabeledas022. DNode0111movestowardtheanchornode02,whichallowsittoconnecttonode022. ENode0111isrelabeledas0221toachievethedesiredprextreetopology.Figure4-3. Networkreconguration. 65

PAGE 66

Algorithm1:LabelAssignment Input: Gtf=fV,Efg /* desiredgraphwithcorrespondingprefixlabelfromGti*/ Output: Gtfwithmovingindicatorsandanchor-nodeanddesiredlabelsroot=GetRoot(V);root.moving=false;Q=frootg;Qtmp=;;ConsideredNodes=Q;whilejConsideredNodesj
PAGE 67

isthecomplexityofbreadth-rstsearchtoobtainaninitialtreesequencefromGi,andthecomplexityoftheaggregateowcalculationforeachtreegeneratedbyChar'salgorithmisO(n2). 4.3.1.2SimulatedAnnealing(SA)AlgorithmThecomplexityofChar'salgorithmdependsonthenumberofgeneratedtreeandnon-treesubgraphsinGiwhichcouldbeveryhighwhenGiislarge.Thismotivatesustosearchforanoptimizationmethodthatgiveasuboptimalsolutionwithlowercomplexity.SinceChar'salgorithmassociateseachsubgraphG0cGiwithavertexsequencec,simulatedannealing[ 46 ]canbeappliedtosearchfortheglobalsolutionwithlowercomplexity.Simulatedannealingstartswithanarbitraryinitialtreesequencecandsearchesthroughitsneighborforatreethatgivesabettersolutionintermsofaggregateowanddecidestomovetotheneighborhoodprobabilistically.Here,theneighborhoodofsequencecisdenedby N(c)=fcN:jcNj)]TJ /F8 11.955 Tf 17.93 11.36 Td[(Xijc[i]\cN[i]j,jcNj=jcjg,(4)wheredenotesaneighborhoodradius.ThisprocessisdoneiterativelystartingfromapredeterminedinitialtemperatureKiuntilitreachesthepredeterminednaltemperatureKf.Thecomplexityofsimulatedannealingmainlydependsonthenumberofiterations,whichisdeterminedbythechoiceofKi,Kf,andthecoolingrate.ThecomplexityisO(n+m+n2log(Kf=Ki))[ 46 ]. 4.3.2NetworkTopologyRecongurationOptimization(NTRO)AlgorithmAfterselectinganetworktreeGtmintominimizeaggregateow,wecanthenconsiderrepositioninganodeinGtmintoformanewtreethatminimizesaggregateowbysolving( 4 ).Weapplytheprex-labeling-basednetworkrecongurationapproachfrom Section4.2 torecongurethenetworkaroundaselectedroottominimizetheaggregateow.TherootrstlabelseachnodeinGtmintoobtainaninitialprextreeGtiortrie.Therootisnowabletocontrolthenetworktopologyandwishestoselectanaltopologythat 67

PAGE 68

minimizestheaggregatetrafc,underaconstraintonthetotalamountofmovementofthenodes.Thedistancethatanodemustmovetotransitionfromtheinitialtothenaltopologyis dhopGti!Gtf(v)=dhopGti(v,av)+dhopGtf(v,av))]TJ /F3 11.955 Tf 11.96 0 Td[(2,(4)wherevisthemovingnodeandavistheanchornodeofv.Forinstanceintheexampleof Figure4-1 and Figure4-2 ,nodeDmustmovetowithinonehopofitsnewparent,whichisCor02.Thus,nodeDmovesuptotheroot,atwhichpointitiswithinonehopof02andcanthusberelabeled022toachievethedesiredpositionbyonlymovingtwohops.Withtheadditionalconstraintthatthetopologyrecongurationoccursaroundtherootandusingtheconstraintsonthetotalamountofmovementofallnodes,theoptimizationproblemcanbeformulatedasGtf=argminGX(u,v)2G2fuvdhopG(u,v) (4)subjecttoC(G)=1Xv2VidhopGti!G(v)hTdhopGti!G(vroot)=0.HeredhopGti!G(v)denotesthedistance,intermsofnumberofedgesinthegraph,thatanodevmustmovetotransitionfromGtitoG.Beforepresentingalgorithmstosolvethisproblem,werstconsiderthenecessaryscopeofthesearchbyevaluatingwhichnodesmayneedbemovedbetweenGtiandGtf.Thenodesarepartitionedintoactivenodes,whichhaveadataowtoorfromothernodes,andpassivenodes,whichdonothaveadataowtoorfromothernodes.However,passivenodesmaystillactasrelaysforothernodes'dataows.Sincethereis 68

PAGE 69

aconstraintonthetotalamountofnodemovement,itisbestnottomovepassivenodesunlessthatisrequiredtoallowactivenodestomove.Allactivenodesareconsideredforrepositioningandareplacedintheactivemovingnodeset,AM.Howeveranodecannotmovewhileitstillhaschildren.Hence,itisnecessarytoalsomoveapassivenodewhoisachildofanodeinAM,andsuchnodesaremembersofthepassivemovingnodeset,PM.ThisnodeclassicationalgorithmisformalizedinAlgorithm2. Algorithm2:MovingNodeSelection Input: Theinitialgraph,Gti=fV,Eig Output: Theactivemovingset,AMandthepassivemovingset,PMCheckFlow(v)=Su2Vnv(fuv>0[fvu>0);AM=;;PM=;;foreachv2Vdo if(CheckFlow(v)==true)then AM=AM[v;C=GetChildren(v);ifC6=;then foreachc2Cdo ifCheckFlow(c)==falsethen PM=PM[c;endendendendend Finally,aftertherootobtainsbothAMandPM,asubgraphG0GtiisformedbyremovingallverticesinAMandPM,alongwithallassociatededges.AllthenodesfrombothsetscanmoveandwillbecomedescendantsofatleastonenodeinG0accordingtothealgorithmsdescribedbelow. 4.3.2.1OptimalAlgorithmForaroottoachieveanoptimalachievabletopology,itessentiallyhastoconsiderallpossibletreetopologiesandselectthenaltopologybasedontheachievabletreethatgivestheminimumaggregatedatatrafc.Weprovidedetailsabouthowtheoptimalsolutioncanbefound,subjecttotheconstraintthatthetopologyisreorganizedaroundapre-selectedroot. 69

PAGE 70

Asdescribedbefore,therootrstobtainsAMandPM,aswellasthesubgraphG0ofnon-movingnodes.Therootusesabrute-forcesolutiontoconsiderallnodesinbothAMandPMandtoevaluatetheaggregatedataforallachievabletreetopologiesGtsuchthatG0Gt:(8Gt).ThiscanbedonebyapplyingtheconceptofPrufercodesequences[ 51 52 ]toobtainallachievabletreetopologies.Eachlabeledtreewithnnodescanbeencodedintocodesequencepoflengthn)]TJ /F3 11.955 Tf 12.27 0 Td[(2calledaPrufersequence,whereeachelementinpisobtainedfromthesetofthennodelabels.Therearenn)]TJ /F10 7.97 Tf 6.59 0 Td[(2possiblespanningtreeswithnlabelednodesandtheyallcanbeencodedintonn)]TJ /F10 7.97 Tf 6.59 0 Td[(2codesequences.TosearchforallachievabletreetopologiesGtsuchthatG0Gt:(8Gt),werstcontractthesubgraphG0toasinglenodeV0.Next,wegenerateasetPcontainingallpossiblePrufersequencesfromasetofnodesincludingAM,PM,andV0.Foreachp2P,itcanbedecodedtoobtainauniquetreeGtp.NeverthelesstheremayexistmorethanonepossibleuniquetreeforeachPrufersequencepsinceagraphGtpobtainedfromdecodingpalwayscontainsanodeV0andithastobeextractedasG0toachieveatleastoneuniquegraphGtG0.ToobtainalluniquetreesfromeachGtp,werstconsideranedgesetEpV0=fe=fV0,vg:v2(AM[PM),e2EpgcontainingtheedgethatisincidenttoV0inGtp,andthereforeeachvwilldenotearootofeachsubtreeGtsubGtpandeachGtsubisconnectedtoV0inGtpbyitsassociatededgee2EpV0.Hence,jEpV0jdenotesthenumberofsubtreesGtsubGtpconnectedtoV0inGtp.ThesesubtreeshavetobeattachedtoatleastonenodeofthesubgraphG0obtainedfromextractinganodeV0toformauniquespanningtreebyconnectingarootofeachGtsubtoanynodev2V0,whereV0denotesthesetofnodesofG0.Hence,therearejV0jjEpV0jpossiblewaystoattachallsubtreesGtsubtoG0andhenceeachPrufersequencepgivesjV0jjEpV0juniquetreetopologies. Thebranch-and-boundapproachisusedbasedonadepth-rsttreesearchacrossallpossibleuniquetreetopologiestoreducethecomplexityofthebrute-force 70

PAGE 71

Algorithm1:OptimalAlgorithmwithBranch-and-BoundforjVinV0j=2 Input: Non-movingnodesubgraph,G0=fV0,E0g;initialtree,Gti=fVi,EigRemainingFlows(G)=Pu2(VinVG)Pv2Vi(fuv+fvu)Tmin=AggregateTraffic(Gti);Gtf=Gti;P=GetAllPruferSequence(Gti,G0);foreachp2Pdo Gtp=PruferDecoding(p, V0);Gtsub=GetAllSubTree(Gtp);foreachv2V0do G01=AttachSubTree(Gtsub[1],v,G0);T=AggregateTraffic(G01);ifjGtsubj=2then BL=T+RemainingFlows(G01);foreachv2V0do if(Pv2G01dhopGti!G01(v)hT)&&(BL
PAGE 72

depth-rstmanner.Ateachnodetheminimumaggregatetrafccanbelowerboundedbytheaggregatetrafcfromthesubtreesthathavealreadybeenassignedpositionsplusthesumoftheremainingdataows.Onepossiblesolutiontotheminimumaggregatetrafcisobtainedoncethedepth-rstsearchhasreachedaleafnode,andthiswillbeusedasanupperboundonthebestminimumaggregatetrafcoverallnodes.Thenasthesearchproceedsdownotherbranches,abranchwillbeprunedwheneverthelowerboundforthatbranchexceedstheupperboundontheoptimalsolution,whichisgivenbythebestsolutionfoundsofar,orwhenevertherequiredtotalamountofmovementatparticularbranchalreadyexceedshT.Wheneverthesearchreachesaleafofthetree,theaggregatetrafcwillbecheckedandcomparedwiththebestsolutionfound.Ifthisvalueisbetterthanthebestsolution,itwillthenberecordedasanewbestfeasiblecompletesolutionandthiscompletesolutionGcompletewillalsoberecordedasthebestpossiblesolutionfound.Theoptimalsolutionisfoundwhenallnodeshavebeenconsideredorpruned.Thebranch-and-boundapproachismosteasilyimplementedusingrecursion,sinceitusesdepth-rstsearch,andweomitthedetailedalgorithmhere.Togiveanideaofhowthisalgorithmworks,wegiveanonrecursiveformofthealgorithmforjVinV0j=2inAlgorithm1.Ifallnodesaremovingnodes,thereexistnn)]TJ /F10 7.97 Tf 6.58 0 Td[(2possibleuniquetreetopologies.Hence,thecomplexityofthisalgorithmisO(nn)sincetheoptimalalgorithmtoencodeanddecodeaPrufersequenceofalabeledtreeisO(n)[ 52 ]. 4.3.2.2SimulatedAnnealing(SA)AlgorithmThecomplexityofsearchingfortheoptimalsolutioncanstillbeveryhigheventhoughbranch-and-boundisusedtoreducethenumberofsolutionsthatmustbeevaluated.Again,simulatedannealingcanbeappliedtosolve( 4 ).TheinitialtopologyGtiisrstencodedintoPrufercodepbyconsideringG0asasinglenodeV0.Fromthere,simulatedannealingstartssearchingforotherfeasiblesolutionsbyconsideringtheneighborhoodofpanddecodingittoachieveonepossibleuniquetreeandnally 72

PAGE 73

Algorithm2:SimulatedAnnealingAlgorithm Input: Non-movingnodesubgraph,G0=fV0,E0g;initialtree,Gti=fVi,Eig;InitialTemperature,Ki;Fi-nalTemperature,Kf;Boltzmannconstant,Kc;CoolingRate,;Neighborhoodradius,K=Ki;Tmin=AggregateTraffic(Gti);Gtf=Gti;Tcurrent=AggregateTraffic(Gti);pcurrent=PruferEncoding(Gti,G0);whileK>Kfdo pnext=GetNeighborhood(pcurrent,);Gtpnext=PruferDecoding(pnext, V0);Gtsub=GetAllSubTree(Gtpnext);G00=G0;achiveable=true;foreachGt2Gtsubdo v=GetRandomNode(V0);G00=AttachSubTree(Gt,v,G00);if(Pv2G00dhopGti!G00(v)>hT)then Tnext=inf;achiveable=false;break;endendifachiveable==truethen Tnext=AggregateTraffic(G00);endifTnextRandomNumber([0,1]))then pcurrent=pnext;Tcurrent=Tnext;endK=K;end decidingwhethertomovetotheneighborhoodprobabilistically.Theneighborhoodisgivenby N(p)=fpN:jpNj)]TJ /F8 11.955 Tf 17.93 11.36 Td[(Xijp[i]\pN[i]j,jpNj=jpjg(4)ThecomplexityofthisalgorithmisagainhighlydependentontheinitialandnaltemperatureKi,Kfaswellasthecoolingrate.HencethecomplexityofthisalgorithmisO(log(Kf=Ki)n2).ThesimulatedannealingalgorithmisgiveninAlgorithm2. 4.4SimulationResultsWesimulatedtheperformanceoftheproposedalgorithmsforrandomlyselectedgraphtopologieswiththreetofteennodes.Foranetworkwithnnodes,weselected 73

PAGE 74

Figure4-4. Minimumachievableaggregatetrafcfornetworktreeselectionalgorithmsasafunctionofnetworksize. thenumberofpairsofnodesthatshareaowatrandomuniformlyfrom1tod)]TJ /F4 7.97 Tf 5.48 -4.38 Td[(n2=2e,andatotalof1Mbpsisrandomlydistributedamongallthesedataows.Foreachsizenetwork,werandomlygenerate150differentcombinationsofinitialnetworktopologiesandowallocations.Wepresentresultsfortheoptimalalgorithmsandforthesimulatedannealingalgorithms.Fortheoptimalalgorithms,thecomputationalcomplexitylimitedoursearchestonetworkswith7nodesorfewer.Forthesimulatedannealingalgorithms,weusedthefollowingparameters:initialtemperatureKi=1,naltemperatureKf=0.01,coolingrate=0.97,BoltzmannconstantKc=0.1,andneighborhoodradius=1.Werstinvestigatetheperformance,intermsofaggregatetrafc,ofthenetworktreeselectionalgorithms.Wecomparetheperformanceofsimulatedannealingtothat 74

PAGE 75

ofthetrueoptimalsolution,aswellasatreethatisselectedrandomlyinabreadth-rstfashion.Theresultsareillustratedin Figure4-4 .Alsoincludedinthisgrapharetheaggregatetrafcoftheinitialnetwork,aswellasaggregatetrafcfromselectingatreeatrandominabreadth-rstfashion.Theaggregateowoftheinitialnetworkissmallerthanforanyofthenetworktreeselectionmethods,sincetherearemorealternativepathsforeachtrafcow,especiallywhentheinitialnetworkisdense.Itcanbeobservedthatwhenthenetworksizegrowslarger,thesimpletreeselectionmethodbyusingbreadth-rstsearchgivestheworstperformance.Italsocanbeseenthatthenetworktreeselectionwithsimulatedannealingofferssignicantlybetterperformancethanthebreadth-rsttreeselectiontechniqueandthattheaggregatetrafcfromsimulatedannealingisclosetooptimal,whilerequiringmuchlowercomplexity.In Figure4-5 ,weconsidertheaggregatetrafcafterselectingthenetworktreeandthenoptimizingthetopologybyrepositioningnodestoreduceaggregatetrafc.Theresultsforsimulatedannealingandthesolutionthatisoptimaltoeachsub-problemareillustratedin Figure4-5 ,alongwiththeaggregatetrafcfortheinitialnetwork.Theresultsareshownfortwovaluesoftheconstraintonthetotalamountofnodemovement,hT=nandhT=3n,wherendenotesthenumberofnodesinthenetwork.Itcanrstbeobservedthatboththeoptimalandsimulatedannealingcombinedalgorithmscanimprovetheaggregatetrafcsignicantlybyreconguringatreeobtainedfromthenetworktreeselectionalgorithm.Whenthenetworksizeissmall,bothalgorithmsprovidesimilarperformance,sincetherearealimitednumberoffeasiblecandidatetreetopologiesandthenodesinthesmallnetworkarealreadyclosetoeachother.Thus,itisdifculttoreducetheaggregatetrafcforasmallnetwork.Asthenetworksizegrowslarger,thetotalamountofenergyofasystemthatisavailableforrepositioningplaysanimportantroleinthenalaggregatetrafc.However,largehTalsotranslatesintomorefeasiblenetworktreetopologies,whichcanslowtheexecution 75

PAGE 76

Figure4-5. Minimumachievableaggregatetrafcforcombinedalgorithmsasafunctionofnetworksize. oftheoptimizationalgorithm.Itcanalsobeobservedthatthecombinedalgorithmwithsimulatedannealingprovidesperformanceclosetotheoptimalalgorithmbutwithlowercomplexity. 4.5SummaryInthischapter,wedevelopedapproachestorecongurethetopologyofagroupofnetworkedautonomousvehicleswiththegoalofreducingaggregatetrafc,whilemaintainingnetworkconnectivityandlimitingthetotalamountofmovementbasedonanenergyconstraint.Wepartitionedthisproblemintotwosub-problems.First,weselectfromtheinitialconnectedgraphatreetopologythatminimizesaggregatetrafc.Second,werepositiontheautonomousvehiclesinthetreetoreduceaggregatetrafcwhilemaintainingnetworkconnectivityandlimitingthetotalamountofmovementtobe 76

PAGE 77

lessthanagivenconstraint.Weinvestigatethecomplexityandperformanceforthesealgorithms.Theresultsshowthatthecombinationofnetworktreeselectionandnetworktopologyrecongurationoptimizationalgorithmscanreduceaggregatedatatrafc. 77

PAGE 78

CHAPTER5ROUTINGAPPROACHESTOOPTIMIZETHEPHYSICALTOPOLOGYOFASYSTEMOFNETWORKEDAUTONOMOUSVEHICLESTOREDUCEAGGREGATETRAFFICInthischapter,weaimtodeveloptechniquesthatcanguidetherecongurationofthephysicalformationofasystemofnetworkedautonomousvehiclestominimizeaggregatenetworktrafc,underconstraintsonnetworkconnectivityandtotalautonomousvehicle'smovement.Inthisscenario,thenetworktopologycanbedeterminedandcontrolledthroughthephysicalformationofthenetworkedautonomousvehicles.Weconsidertheprotocolmodelforcommunications,inwhichtwoautonomousvehiclescancommunicateiftheyarewithinacertaincommunicationrange.ThismodelisaccurateformostsystemscommunicatingoverachannelthatsuffersfromexponentialpathlossandwhiteGaussiannoise.Thus,theaggregatetrafccanbereducedbyreconguringthephysicalformationoftheautonomousvehiclessothatsourcesanddestinationsareclosertoeachotherinthenetworktopology.Ourapproachtoreconguringtheformationwhilepreservingnetworkconnectivityutilizesthetree-basedroutingapproachdevelopedin Chapter2 and Chapter3 .In Chapter4 ,wedevelopedalgorithmstorecongureanetworktopologytoreduceaggregatenetworktrafc.However,thisworkdidnottakeintoaccountthelimitationsandadvantagesthatcomewhenthenetworktopologyisinducedfromaphysicalformation.Ourapproachprovidesaseriesofwaypointsthroughwhichtheautonomousvehiclescanmovetoachieveanewtopologywhilepreservingnetworkconnectivity,underaconstraintonthetotalamountofautonomousvehicle'smovement.Thenewapproachutilizesconstraintsonmovementbasedonphysicaldistance,andthefullnetworktopologythatisinducedbythenalphysicaltopologyisusedtodeterminethetotalaggregatenetworktrafcafteroptimization.Simulationresultsarepresentedtodemonstratetheeffectivenessoftheproposedalgorithms. 78

PAGE 79

ANodesarerandomlydeployedoveraworkspace. BLinksareformedbetweenautonomousvehiclesthatarelocatedwithineachothersensingrangeR. CGraphrepresentationofphysicaltopologywithedgesbetweenautonomousvehiclesthatshareacommunicationslink.EdgeweightscorrespondtophysicaldistancesforasystemofnetworkedautonomousvehicleswithcommunicationlinksdeterminedbytheprotocolmodelFigure5-1. Physicaltopology. 79

PAGE 80

5.1ProblemFormulationInthischapter,weconsiderasystemofnetworkedautonomousvehiclesinan-dimensionalspace,wherecommunicationisoverwirelesslinkswithlimitedcommunicationdistance.Letqu2Rndenotethepositionoftheautonomousvehicleuinandimensionalspace.AutonomousvehiclesareabletocommunicatewitheachotheriftheyarenomorethandistanceRapart.(HereRcanbeconsideredtobethecommunicationradiusofthenodes.)WeabstractsomeessentialinformationaboutthephysicalformationintoasimplegraphG=(V,E),whereverticesVrepresenttheautonomousvehicles,andedgesE=f(u,v)2VVjduvRg,representcommunicationslinksbetweeneachautonomousvehicleuandv,whereduv=kqu)]TJ /F6 11.955 Tf 11.96 0 Td[(qvkdenotethedistancebetweenautonomousvehiclesuandvthatarelocatedatpositionqiandqjrespectively.Forconvenience,werefertothisasagraphofthephysicaltopol-ogy.Anexampleofhowthegraphofaphysicaltopologyisdeterminedisshownin Figure5-1 .Thisexampleshowshowthephysicaltopologyofasystemofnetworkedautonomousvehiclesisdeterminedfromthephysicalformationofasystem.Thereare6nodesrandomlydistributedoveratwo-dimensionalspace.Eachnodewillformacommunicationlinkbetweeneachotherifitislocatedwithineachothercommunicationrangetoformthephysicaltopologyofasystem.NowconsidertheaggregatetrafcinG,wherethistrafcisevaluatedasthedatatrafcmeasuredatthenetworklayer.ThesetofdataowsisgivenbyF=ff(u,v):(u,v)2G2g,wheref(u,v)denotestheamountoftrafcfromsourceutodestinationvandG2denotestheCartesianproductofGwithitself.ThentheaggregatetrafcovernetworktopologyGis X(u,v)2G2f(u,v)dhopG(u,v),(5)wherethedistancefunctiondhopG(u,v)isthenumberofedgesintheshortestpathbetweenverticesuandvinG. 80

PAGE 81

Inthischapter,weconsidertheproblemofrepositioningtheautonomousvehiclesintheformationtominimizetheaggregatenetworktrafcwhilemaintainingnetworkconnectivity,underaconstraintonthetotalamountofmovementoftheautonomousvehicles.Wethendecomposethisproblemintotwosub-problems,wheretheoptimalsolutiontothetwosub-problemsmaynotbeoptimalforthecombinedproblem.First,anetworktreetopologyisobtainedfromanarbitraryinitialphysicaltopologyGibyselectingatreeGtGisuchthatitsaggregatetrafc( 5 )isminimum.ThesetofalltreetopologiesthataresubsetsofGiandcontainsallverticesofGiisgivenbyT(Gi).ThenwecanformulatetheproblemofselectingthenetworktreetopologyGtthatminimizestheaggregatetrafcas:Gtmin=argminG2T(Gi)X(u,v)2G2fuvdhopG(u,v). (5)AselectedrootwilllabeleachnodeinGtmintoobtaininitialprextreeGtiafterGtminisobtainedfrom( 5 ).Secondly,weattempttooptimizetheaggregatetrafcbyrepositioningtheautonomousvehicleswhilemaintainingnetworkconnectivityundertheconstraintthatasystemofnetworkedautonomousvehicleshasniteenergy.WeletC(G)betheconnectivityfunction,whichtakesonthevalue1whenGisconnectedand0otherwise.LetETrepresentthetotalphysicaldistanceinGtithatallnodesmaymovebeforethetotalenergybudgetisexpended.LetF(Gti,Gtf,ET)beafeasibilityfunction,whichtakesonthevalue1whenthenaltreetopologyGtfcanbeachievedwithoutbreakingnetworkconnectivityundertheenergyconstraintand0otherwise.Toreducetheaggregatetrafcinasystem,wewishtondGtfthatminimizestheaggregatetrafc.Thenwecanformulatetheaggregatetrafcminimizationundertheenergyandnetworkconnectivityconstraintsas: 81

PAGE 82

Gtf=argminGX(u,v)2G2fuvdhopG(u,v) (5)subjecttoC(G)=1F(Gti,G,ET)=1.F(Gti,G,ET)canbedeterminedbasedontransformingthetopologyusinganapproachthatisbasedonthatin Chapter4 ,asdescribedinthenextsection.Theaggregatetrafccanbefurtherimprovedbyconsideringthephysicaltopologyofasystem.AfterGtfisobtainedbysolving( 4 ),nodesthatarephysicallyseparatedbyadistanceofnomorethanRwillformacommunicationlinktoobtainnalphysicaltopologyGf.Thus,theaggregatetrafcofGfisgivenby X(u,v)2G2f(u,v)dhopGf(u,v),(5)wheredhopGf(u,v)denotesthenumberedgesintheshortestpathbetweenverticesuandvinGf. 5.2AdvancedNetworkTopologyRecongurationAlgorithmTheinitialprextreeGticanbeobtainedbyutilizingtheNetworkTreeSelection(NTS)Algorithmproposedin Chapter4 tosolve( 5 ).Inthissectionwedescribehowthenetworktopologyofasystemofnetworkedautonomousvehiclescanbereconguredfromtheinitialtreewhilemaintainingthenetworkconnectivity.Inthissection,werepresentanovelmethodtorecongurethenetworktreetopologybasedonthenetworktopologyrecongurationalgorithmin Chapter4 .First,anodeintheinitialtopologyisrandomlychosentoserveastherootofthetree.Aftertherootisselected,itthenassignsuniqueprexlabelstoeachofitschildrenuntiltheentiretreehasprexlabels.Theprexlabelassignedtoeachnodeservesasitsnetworkaddress. 82

PAGE 83

AInitialnetworktopology. BDesirednetworktopology.Figure5-2. Networktopology. TheinitialtreetopologyGibecomesaprextree,ortrie[ 37 44 ].Exampletopologiesaregivenin Figure5-2 ,wheretheinitialnetworktopologyshownin Figure5-2A isobtainedfromthephysicaltopologyshownin Figure5-1C byselectingatreetopologyusingtheNTSalgorithm.Forthisexample,nodeAisselectedtobetheroot,andprexlabelsareassignedtoallnodesintheinitialnetworktree.Similarly,eachnodeinadesirednetworktreeislabeledwithaprexlabelassignedtothesamenodeintheinitialnetworktree.Theprextreetopologiesareshownin Figure5-3 .Thesearchingfornodesthatneedtomovebetweentheinitialandnaltopologiescanbedoneinbothdepth-rstandbreadth-rstmannerfromthetoptothebottomofthedesiredtreetopologystartingfromtheroot.Thenodesthatmustmovearethosewhoseprexlabelsdonotcorrespondtothepositionswheretheyarelocatedinthedesiredtree.Allnodesinthetopologyarerstclassiedintotwocategories,asin Chapter4 .First,movingnodesarethosenodesthatdonothavethecorrectprexlabelandmustmovefromtheirpositionintheinitialtopology.Second,nonmovingnodesarethosenodesthathavethecorrectprexlabelandthathavenotbeenpreviously 83

PAGE 84

AInitialnetworktopology. BDesirednetworktopology.Figure5-3. Labelingnetworktreetopology. assignedtobemovingnodes.Forinstance,in Figure5-3 allofthenodesexceptfor011and0111arenonmovingnodes.Inthischapter,therootconsidersthreelabelsforeachmovingnode.(1)Itsanchor-nodelabelisthelabelofthenon-movingnodewhoistherstparentofthemovingnodeinGti.(2)Itsdestination-nodelabelisthelabelofthenodeinGtithatwillbethemovingnode'sdestination.(3)Itsdesiredlabelisthenewlabelofthemovingnodeuponarrivalatthedestinationinthedesiredtopology.TherootwillsendamessageM.Destincludingallthesethreelabelstoeachmovingnodewhentheroothasconsideredallnodesinthedesiredtree.AftereachmovingnodeobtainsamessageM.Dest,itstartsmovingtothenodewhoseprexlabelisthedestination-nodelabelusingmaximumprexmatchinglogic[ 35 ].Notethateachmovingnodewillmovetowardanodewhoseprexlabelisitsanchor-nodelabeluntilitreachesthenodewhoseprexlabelisitsdestination-nodelabel.Whenamovingnodearrivesatadestinationnode,itwillmaintainthenetworkconnectivitywiththeneighboringnodethatisamovingnode'snextnodeonthepathto 84

PAGE 85

theanchornodeuntilitisabletoconnectwithitsparentnodeinthenaltreetopologyGtf.Movingnodescanalsonotdependonnetworkconnectionstootherpartsofthenetworkthatarenotonitspathtotheanchornode,asthenodesintheseotherpartsofthenetworkmayalsobemovingnodeswhosemovementwouldcausethemovingnodetolosenetworkconnectivity.Toovercomethisissue,amovingnodewillnotmakenetworkconnectionswithothernodesinthenetworkthatarenotonitspathtotheanchornodeuntilitarrivesatitsdestinationnodeandconrmsthatthedestinationnodehasachieveditspositioninthenaltreetopology,Gtf.ToverifywhetheritsneighborisitsparentinGtf,eachmovingnodechecksthedesiredlabelofitsneighbortoseeifitmatchestheprexlabelofitsparentinthedesiredtopology.Theprexlabelofamovingnode'sparentinthedesiredtreetopologycanbesimplydeterminedfromthemovingnode'sdesiredlabel.Foranon-movingnode,thedesiredlabelwillbethesameasitsinitialprexlabel.Afterallthemovingnodesarriveattheirdestinationsandareabletoconnectwiththeirparentsinthedesiredgraph,theirprexlabelwillberelabeledastheirdesiredlabeltoachievethedesiredprextree.Alsonotethatmovingnodesthatareleafnodescanstartmovingsimultaneously.Topreservenetworkconnectivity,amovingnodethatisnotaleafnodeintheinitialtopologyhastowaitforitsdescendantstomoveuptoitbeforeitcanstartmoving.Otherwise,thenode'sdescendantswillbedisconnectedfromthenetwork.Anexampleofthisapproachisshownin Figure5-4 .Inthisexample,therootnodesendsmessageM.Desttoeachofmovingnodes011and0111,asshownin Figure5-4A .Afterbothmovingnodesreceivethismessage,movingnode0111isrstmovingtowarditsdestinationnode01sinceitisaleafnodewhilemovingnode011hastowaituntilmovingnode0111reachesittopreservethenetworkconnectivity,asshownin Figure5-4B .Next,bothmovingnodescanmovetogethertowardtheirdestination,andbothmovingnodesnowarriveatnode01whichisthedestinationnodeofamovingnode0111asshownin Figure5-4C .Movingnode0111willmaintain 85

PAGE 86

networkconnectivitywiththerootnode,whichisitsnextneighbornodealongthepathtoitsanchornode02.Movingnode011nowmovestowarditsdestinationnode,0,andnallyarrivesatitsdestinationasshownin Figure5-4D .Nowmovingnode0111willconnectwithmovingnode011,whichisitsparentinthenaltreetopologyshownin Figure5-3B ,andmovingnode011willalsoconnectwithitsparentinthesamenaltreetopology.Finally,movingnode011and0111willrelabeltheirowninitialprexlabelto022and0221,respectively,asshownin Figure5-4E .Thedestination-nodelabelisanadditiontothemessagessentinourpreviousworkon Chapter4 thatcanreducetheamountofmovementrequiredtoachievethedesirednetworktopology.Theintroductionofthedestination-nodelabelisbasedontheobservationthatanodeachievescommunicationwithanothernodewheneveritreachesanyofthatnode'sone-hopneighbors.In Chapter4 ,themovingnodes011and0111in Figure5-4 movetowardtheiranchor-node02,asshownin Figure5-4A through Figure5-4D .However,sincemovingnode0111doesnotknowitsdestination-node'slabel,itcontinuestomovetoitsanchornode02toachievecommunicationwithmovingnode011.Inthealgorithmproposedhere,node0111doesnotneedtomoveafteritachievesthepositionshownin Figure5-4D becauseitsdesiredparentnode(011)isinthedesiredpositionrelativetonode02,andithasachievedcommunicationwithitsdesiredparentnode.Itiseasytodeterminethedesiredlabelforeachmovingnodesinceitcanbesimplydeterminedfromitsparentinthedesiredtopology,asshownin Figure5-3B .Forexample,ifitsparentisamovingnode,itsparentmustalreadybeassignedthedesiredlabelbytheroot,andthemovingnode'sdesiredlabelisdeterminedfromthedesiredlabelofitsparent.Nevertheless,ifitsparentisanon-movingnode,thedesiredlabelisdeterminedfromitsparent'sprexlabel.Thedesiredlabelassignmentapproachin Chapter4 isusedinthischapter.Thus,weprovidethepseudocodeforthedestination-nodeandanchor-nodelabelassignmentexecutedattherootinAlgorithm1. 86

PAGE 87

Algorithm1:Anchor-NodeandDestination-NodeLabelAssignmentAlgorithm Input: initialtree,Gti=fVi,Eigdesiredtree,Gtf=fVf,EfgCheckNeighborhood(u.nodelabel,v.nodelabel,Gt)=(true:(hu,vi2E)[(u=v)false:(hu,vi=2E)\(u6=v)Stack=;;Root=GetRootNode(Gti);Root.moving=false;Stack.push(Root);Root.visited=true;whileStack6=;do v=Stack.GetTopNode;u=GetUnvisitedChildNode(v,Gtf);ifu6=;then Stack.Push(u);u.visited=true;ifv.moving==falsethen v.destination-nodelabel=v.nodelabel;ifu.ParentLabel==v.nodelabelthen u.moving=false;else u.moving=true;u.anchornode=v;endelse u.moving=true;u.anchornode=v.anchornode;endifu.moving==truethen u.destination-nodelabel=u.nodelabel;while(CheckNeighborhood(u.destination-nodelabel,u.anchor-nodelabel,Gti)==falsedo if(CheckNeighborhood(u.destination-nodelabel,v.destination-nodelabel,Gti)==true)then break;endu.destination-nodelabel=MoveOneHopCloserToAnchorNode(u.destination-nodelabel,u.anchor-nodelabel,Gti);endifCheckNeighborhood(u.destination-nodelabel,v.destination-nodelabel,Gti)==falsethen Stack1=Stack;u1=GetTopNode(Stack1);v1=GetSecondTopNode(Stack1);whilev16=u.anchornodedo whileCheckNeighborhood(v1.destination-nodelabel,u1.destination-nodelabel,Gti)==falsedo v1.destination-nodelabel=MoveOneHopCloserToAnchorNode(v1.destination-nodelabel,v1.anchor-nodelabel,Gti);endC=GetChildren(v1,Gtf);ifjCj>1then Stack2=;;PushNode(v1,Stack2);V=u1;whileStack26=;do u2=Stack2.GetTopNode;v2=GetVisitedChildNode(u2,GtfnV);ifv26=;then Pushnode(v2,Stack2);V=V[v2;whileCheckNeighborhood(v2.destination-nodelabel,u2.destination-nodelabel,Gti)==falsedo v2.destination-nodelabel=MoveOneHopCloserToAnchorNode(v2.destination-nodelabel,v2.anchor-nodelabel,Gti);endelse Stack2.Pop;endendendStack1.Pop;u1=Stack1.GetTopNode;v1=Stack1.GetSecondTopNode;endendendelse Stack.Pop;endend 87

PAGE 88

ATherootnode,0,sendsmes-sageM.Desttothemovingnodes,011and0111. BNode0111movetowarditsdestinationnode01. CNode0111reachesitsdesti-nationnode01whilenode011movestowarditsdestinationnode0. DNode011reachesitsdestina-tionnode0.Itisabletoconnectwithitsanchor-node02andthedesiredtreetopologyisachieved. ENode011and0111relabeltheirownprexlabelas022and0221respectivelytoachievethedesiredprextreetopology.Figure5-4. Networkreconguration. 5.3PhysicalTopologyRecongurationOptimization(PTRO)AlgorithmsTominimizetheaggregatetrafc,werstselectanetworktreetopologythathasminimumaggregatetrafcfromtheinitialconnectednetworkGibyapplyingtheNetworkTreeSelection(NTS)Algorithmsfrom Chapter4 tosolve( 5 )toobtainGtminGi.In Chapter4 twoapproachesarepresentedforNTS,theoptimalandsimulatedannealingalgorithms. 88

PAGE 89

AftertheNTSalgorithmisusedtoobtainGtmin,therootwilllabeleachnodeinGtmintoobtaintheinitialprextreeGti.WecannotdirectlyapplytheNetworkTopol-ogyRecongurationOptimization(NTRO)Algorithmproposedin Chapter4 tosolve( 5 )toobtainthenetworktreetopologythatgivesminimumaggregatetrafc.ThisisbecausetheNTROAlgorithmdoesnotconsiderthephysicaltopologyofthesystemandassumesthatthedistancebetweenneighboringverticesinGtiisuniform.Inordertosolve( 5 )withtheconstraintonthelimitedamountofmovementpossiblewhenthedistancebetweentheneighborsintheinitialprextreeGtiisnotuniform,wedevelopanalgorithmbasedontheNTROAlgorithmcalledthePhysicalTopologyRecongurationOptimization(PTRO)Algorithm.PTROAlgorithmisdifferentfromNTROAlgorithminthatthePTROAlgorithmisusedtorecongurethephysicaltopologyofasystemofnetworkedautonomousvehiclesthattakeintoaccounttheconstraintonthetotalphysicalmovementinsteadofthetotalamountofmovementintermsofthenumberofedgesofthegraphoverwhichtheautonomousvehiclesmusttraveltoachievethedesiredconguration.ThedetailsofthePTROalgorithmareprovidedbelow.AfterweselectanetworktreeGtminbyusingtheNTSalgorithm,wesearchforasolutionto( 5 )thatrepositionsthenodesinGtmintominimizetheaggregatetrafc.Inthischapter,theNTRalgorithmprovidedin Section5.2 isusedtorecongurethenetworkaroundaselectedroottominimizetheaggregatetrafc.TherootrstlabelseachnodeinGtmintoobtainaninitialprextreeGti.Therootisnowabletocontrolthetopologyofthenetworkandwishestochooseanaltopologythatgivesminimumaggregatetrafc,underaconstraintonthetotalamountofmovementofthenodes.Sincethereareconstraintsonthetotalamountofmovementofallnodesandthetopologyrecongurationoccursaroundtheroot,theoptimizationproblemcanbeformulatedas 89

PAGE 90

Gtf=argminGX(u,v)2G2fuvdhopG(u,v) (5)subjecttoC(G)=1Xv2VidGti!G(v)ETdGti!G(vroot)=0.HeredGti!G(v)denotesthedistancethatanodevmustmovetotransitionfromGtitoG,whichisgivenby dGti!G(v)=dGti(v,vdest).(5)Herevdestdenotesthenodewhoselabelisthedestinationlabelofthenodev,whichcanbeobtainedfromtheadvancednetworktopologyrecongurationalgorithmgivenin Section5.2 .dGti(v,vdest)denotetheshortestdistancebetweennodevandvdestinGti.Anodeclassicationmethodisgivenin Section4.3.2 todeterminewhichnodesmayneedtobemovedbetweenGtiandGtf.Thenodesarepartitionedintoactivenodes,whichhaveadataowtoorfromothernodes,andpassivenodes,whichdonothaveadataowtoorfromothernodesbutwhichmaystillactasrelaysforothernode'dataows.Allactivenodesareplacedintheactivemovingnodeset,AMandareconsideredforrepositioning.Furthermore,itisbestnottomovepassivenodesexceptasisrequiredtoallowactivenodestomove,sincethereisaconstraintonthetotalamountofnodemovement.However,itisnecessarytoalsomoveapassivenodewhoisachildofanodeinAMsinceanodecannotmovewhileitstillhaschildren,andsuchnodesareplacedinthepassivemovingnodeset,PM. 90

PAGE 91

Finally,asubgraphG0GtiisformedbyremovingallverticesinAMandPM,alongwithallassociatededgesaftertherootobtainsbothAMandPM.AllthenodesfrombothsetscanmoveandwillbecomedescendantsofatleastonenodeinG0accordingtothealgorithmsdescribedbelow. 5.3.1OptimalAlgorithmWedevelopanoptimalalgorithmbasedonthatgivenin Chapter4 .Thebasicapproachistoconsiderallpossibletreetopologiesandselectthenaltreetopologythatisachievableandgivestheminimumaggregatetrafc.Prufercodesequences[ 52 ]areusedtoobtainallachievabletreetopologies.Eachlabeledtreewithnnodescanberepresentedbyacodesequencepoflengthn)]TJ /F3 11.955 Tf 12.89 0 Td[(2calledaPrufersequence.In Chapter4 ,theachievabilityoftheselectedtreetopologyisveriedbycheckingthetotalamountofmovementrequiredtorepositionallthenodesinGtiintermsofthenumberofedgesthatnodeshavetotraverseinGtitoachievethedesiredtreetopology.Sinceweconsiderthephysicaltopologyofasysteminthischapter,wedevelopedanoptimalalgorithmtoobtainthenaltreetopologythatgivesminimumaggregatetrafcbasedontheoptimalalgorithmin Chapter4 ,exceptthattheachievabilityofthetreetopologyisveriedbyobtainingtheamountofmovementrequiredtorepositionallnodesinGtiintermsofthephysicaldistancethatthenodehastotraversethroughGti,andthiscanbeobtainedbyrunningbreadth-rstsearchalgorithmoverallverticesinGti.Sinceourdevelopedalgorithmisbasedonthatin Chapter4 exceptthatbreadth-rstsearchalgorithmisrunoverallverticesinGtitoobtainallpairshortestpathsinGtithatwillbeusedtoobtainthetotalamountofnode'smovement.In Chapter4 ,wendthetotalamountofmovementofthenodesintermsofthenumberofhops(i.e.,verticesthatmustbetraversed)bythemovingnodes.Thenumberofhopscanbecalculatedusingprexlabels,asgivenin( 4 ).Thecomplexityofrunningbreadth-rstsearchalgorithmoverallverticesisO(n2),andthusthecomplexityofthisoptimalalgorithmisthesame 91

PAGE 92

asthatgivenin Chapter4 ,whichisgivenbyO(nn),wherendenotethenumberofnodesinGi. 5.3.2SimulatedAnnealing(SA)AlgorithmSincethecomplexityofsearchingfortheoptimalsolutionisveryhigh,weconsiderdevelopinganalgorithmthathaslowercomplexitybutcanndasolutionclosetotheoptimalsolutionof( 5 ).Asin Chapter4 ,weuseasimulatedannealingalgorithmtondanapproximatesolutionto( 5 ).TheinitialtopologyGtiisencodedintoaPrufercodesequencep.Fromthere,thealgorithmstartssearchingforotherpossiblesolutionsbyconsideringtheneighborhoodofpanddecodingittoobtainafeasibletreetopologyGtG0anddecidingwhethertomovetotheneighborhoodbyprobabilitybasedonachievabilityandaggregatetrafcthattreecanachieve.TheachievabilityisveriedbythephysicaldistancethateachnodehastotraversethroughtheinitialprextreeGtitoachievethedesiredtreetopology.ThecomplexityofthisalgorithmisgivenbyO(log(Kf=Ki)n2+n2),wherendenotesthenumberofverticesinGti.Thiscomplexityisobtainedbecauseouralgorithmisbasedonasimulatedannealingalgorithmgivenin Chapter4 ,whichhasthecomplexityofO(log(Kf=Ki)n2),andouralgorithminitiallyrunsbreadth-rstsearchalgorithmforallverticesinGtiinordertoobtainallpairshortestpathontheinitialtreeGtithatwillbeusedtoverifythenetworktree'sachievabilitywhichleadstocomplexityofO(n2).Tofurtherimprovetheaggregatetrafcofthesystem,thenalphysicaltopologyGfisformedbylettingnodeswhoarelocatedwithinthecommunicationsrange,R,formacommunicationslinkafterGffthathasminimumaggregatetrafcisobtainedfromPTROalgorithms. 5.4SimulationResultsInthissection,wepresentresultsforthecombinedalgorithmsthatincludetheNTSandPTROalgorithmswithboththeoptimalandsimulatedannealingvariants.Theperformanceofthecombinedalgorithmswassimulatedforrandomlyselected 92

PAGE 93

connectedgraphtopologieswiththreetofteennodes.Foranetworkwithnnodes,thenumberofpairsofnodesthatshareaowwasselectedrandomlyfrom1tod)]TJ /F4 7.97 Tf 5.47 -4.38 Td[(n2=2e,andatotaldataowof1Mbpsisrandomlydistributedamongallthesedataows.150differentcombinationsofinitialconnectednetworktopologiesandowallocationsarerandomlygeneratedforeachnetworksize.Werandomlygeneratedeachconnectednetworktopologybyrandomlyplacingeachnodeoveratwo-dimensionalsquareregionwithalengthofp nmeter.Thesensingrangeforeachnodeissettobe1meter.Forthecombinedoptimalalgorithm,welimitedoursearchestonetworkswith7nodesorfewerbecauseofaveryhighcomputationalcomplexityoftheoptimalalgorithm.Forthecombinedalgorithmwithsimulatedannealing,thefollowingparameterswereused:initialtemperatureKi=1,naltemperatureKf=0.01,coolingrate=0.97,BoltzmannconstantKc=0.1,andneighborhoodradius=1.Theperformanceisinvestigatedforbothcombinedalgorithmsintermsofaggregatetrafcundertwovaluesoftheconstraintonthetotalamountofnodemovement,ET=nR=3andET=nR.Forbothcombinedalgorithms,theNTSalgorithmwasrstusedtoselectatreetopologythatgivesminimumaggregatetrafc,andthenthePTROalgorithmwassubsequentlyusedtorepositionnodesinthetreetopologyobtainedfromtheNTSalgorithmtoreduceaggregatetrafcunderthedifferentvaluesofET.ThephysicaltopologyisthenformedbylettingnodesthatarewithindistanceRofeachotherformcommunicationslinksafterthedesirednetworktopologythathasminimumaggregatetrafcformisachieved.Itcanbeobservedfrom Figure5-5 thattheaggregatetrafcisimprovedsignicantlybyusingthecombinedoptimizationalgorithms.Theoptimalandsimulatedannealingvariantsprovidesimilarperformancewhenthenetworksizeissmallsincethenodesinthesmallnetworkarealreadyclosertoeachotherandtherearelimitednumberofcandidatetopologiesfortheroottobeconsidered.Hence,itisdifculttoreducetheaggregatetrafcinasmallnetwork.Weobservethatthetotalamountofenergyof 93

PAGE 94

Figure5-5. Minimumachievableaggregatetrafcforcombinedalgorithmsasafunctionofnetworksize. asystemETthatisavailableforrepositioningnodesplaysacrucialroleinthenalaggregatetrafcasthenetworksizegrowslargersincelargeETtranslatesintomorefeasiblenetworktreetopologies.However,largeETcanalsoslowdowntheexecutionofthealgorithms,especiallyfortheoptimalalgorithms.Itcanalsobeobservedthatthecombinedalgorithmwithsimulatedannealingprovidessimilarperformancetothatofthecombinedoptimalalgorithmbutwithlowercomplexity. 5.5SummaryInthischapter,weconsideredtheproblemofoptimizingthephysicalformationofasystemofnetworkedautonomousvehiclestominimizetheaggregatenetworktrafcunderconstraintsonconnectivityandtotalmovementoftheautonomousvehicles.Weusetheconceptofprextreesandprexroutingtosimplifytheproblemtomovingnodesalongatreestructureinsuchawaythatnetworkconnectivityisensuredwhen 94

PAGE 95

theautonomousvehiclescanmovelinearlybetweenthepositionsofautonomousvehicles.Thenweapplyoptimizationtechniquestosearchforanaltreecongurationthatisachievableunderthegivenconstraints.Theperformanceandcomplexityofoptimalandsimulatedannealingvariantsoftheproposedalgorithmsarecompared,andsimulationresultsshowthattheformationrecongurationalgorithmscansignicantlyreducetheaggregatenetworktrafc. 95

PAGE 96

CHAPTER6CONCLUSIONSANDFUTUREWORK 6.1ConclusionInthisdissertation,weconsiderproblemsofformationcontrolwithnetworkconnectivity(FC+NC)constraintsorgoals,withaspecialfocusontheuseofnetworkingtechniquesinformationrecongurationwithnetworkingconstraints.SolutionstoFC+NCproblemsmayoftenbedecomposedintopartsthataddressnetworkconnectivitymaintenanceandpartsthatcontrolthephysicalpositions.Networkconnectivitycanbetreatedusinggraphs.Wepresentedourworkonusingtheconceptsofprexroutingtorecongurethenetworktopology.In Chapter2 ,wedevelopedanovelapproachtoroutenodesthroughthenetworkwhilemaintainingnetworkconnectivitybyusingtheconceptsofprexroutingtorecongurethenetworktopology.Inthischapter,aninitialtreeisbuiltfromarandomlyselectedrootnodeintheinitialgraph,andtechniquesaredevelopedtomovethenodesfromtheinitialtreeformationtoadesiredtreeformation.Sincetheinitialtreeisselectedrandomly,wedevelopedtwotechniquestoprovidebettermatchesbetweentheinitialandnaltopologies.Thesearethebranchandneighborrelabelingalgorithms,whichcanreducetheamountofmovementrequiredtorecongurethenetworkformation.Thesimulationresultsconrmthatthenetworktopologyrecongurationalgorithmtogetherwiththerelabelingalgorithmscanfurtherimprovetheperformanceintermsoftotalmovementandtimerequiredtoachievethedesirednetworkconguration.Thesimulationresultsalsoconrmthatselectingtheappropriaterootintheinitialgraphcanimprovetheperformancesignicantlyintermsoftotalnumberofmovementrequiredtoformthedesirednetworkconguration.In Chapter3 ,wedevelopedanovelgraphmatching-basedtechniquetofurtherimprovetheperformanceofthenetworktopologyrecongurationalgorithmsproposedin Chapter2 intermsoftheamountofnodemovement.Wedevelopedalgorithmsto 96

PAGE 97

obtaintheinitialnetworktreetopologyfromanarbitraryinitialnetworkthatminimizestheamountofmovementrequiredtorepositionthenodestoachievethedesirednetworktreetopology.Thesimulationresultsshowedthatthealgorithmswedevelopedcanessentiallyimprovetheamountofnodemovementrequiredtorecongureanarbitraryinitialnetworktoachievethedesirednetworktreetopology.In Chapter4 ,wedevelopedapproachestoreducetheaggregatetrafcinasystembyreconguringthenetworktopologyunderconstraintsonthetotalamountofmovementofthenetworkedautonomousvehicleswhilepreservingthenetworkconnectivity.Wedevelopedthenetworktreeselectionalgorithmtoselectaninitialnetworktreewithlowaggregatetrafcfromanarbitraryinitialconnectednetworktopology.Wethendevelopedthenetworktopologyrecongurationoptimizationalgorithmtorepositiontheautonomousvehiclesinthetreeobtainedfromnetworktreeselectionalgorithmtoreducetheaggregatetrafcwhilemaintainingthenetworkconnectivityunderconstraintsonthetotalamountofmovementoftheautonomousvehiclesintermsofthenumberofedgesinthegraphthattheytraverse.Wealsoprovidedacomparisonofouralgorithmsintermsofperformanceandcomplexity.Thesimulationresultsshowedthatthecombinationofnetworktreeselectionandnetworktopologyrecongurationoptimizationalgorithmscanimprovetheaggregatetrafc.In Chapter5 ,weproposedtechniquestooptimizethephysicalformationofasystemofnetworkedautonomousvehiclestoreducetheaggregatetrafcunderconstraintsonthetotalamountofmovementoftheautonomousvehiclesandnetworkconnectivity.Wedevelopedformationrecongurationalgorithmsbasedonthatin Chapter4 toobtainanaltreecongurationthatachievestheminimumaggregatetrafcunderconstraintsonthetotalamountofmovementofautonomousvehicles.Theperformanceandcomplexityofourdevelopedalgorithmswerealsoinvestigated.Simulationresultswerepresentedtodemonstratetheperformanceofourdevelopedalgorithmintermsofhowmuchtheycanimprovetheaggregatetrafcfromthatinthe 97

PAGE 98

initialformation.Simulationsresultsshowedthatformationrecongurationalgorithmscansignicantlyimprovetheaggregatetrafc.Toconclude,thetopologyrecongurationtechniquesthatwedevelopedbasedonprexroutinghaveseveralnicefeaturesincomparisontopreviouslyproposedtechniques.Theycanachievearbitrarytransformationofthenetworktopology,areperfectlydesignedforanonymousreconguration,andareeasilyusedinsystemswherethenaltopologyistobeoptimizedunderconstraintsonconnectivityandmovement. 6.2FutureWorkOneareaoffutureworkwillfocusonprovidingperformanceimprovements(forinstance,minimizingtherequiredmovement)intheprocessoftransformingfromoneconnectedinitialtopologytoanotherconnectedinitialtopology.Path-planningtechniquesareneededthatallowmobilevehiclestotravelthroughthephysicalspacetoplaceswherethereisnocurrentautonomousvehiclebutthatmaintainnetworkconnectivity.Thiswillrequireacouplingamoresophisticatedgraph-basedalgorithmswithaphysical-positioncontrolalgorithmtomovetheautonomousvehiclesinanefcientway.Anotherareainwhichfutureworkisneededistoincorporatemorerealismintothephysicalandcommunicationsmodels,includingtakingintoaccountthedynamicsoftheautonomousvehiclesandtheeffectsofthoseoncommunications,andincorporatingmorerealisticchannelmodels.Forinstance,itisdesirabletoextendthisworktoincludemorerealisticphysical-layermodels,suchasthosewithfadingand/orshadowing,whichmaycauseintermittentconnectivity.Inaddition,theabilitytoutilizepowercontrolfortopologycontrol[ 54 56 ]toreducetheamountofmovementrequiredortoovercomefadingorshadowingisanotherapproachtheshouldbeinvestigated[ 57 58 ]. 98

PAGE 99

REFERENCES [1] R.M.Murray,Recentresearchincooperativecontrolofmultivehiclesystems,J.DynamicSystems,Measurement,andControl,vol.129,p.571,2007. [2] J.A.FreebersyserandB.Leiner,Adhocnetworking.Boston,MA,USA:Addison-WesleyLongmanPublishingCo.,Inc.,2001,ch.ADoDperspectiveonmobileAdhocnetworks,pp.29.[Online].Available: http://dl.acm.org/citation.cfm?id=374547.374549 [3] F.Giulietti,L.Pollini,andM.Innocenti,Autonomousformationight,IEEEControlSyst.Mag.,vol.20,pp.34,2000. [4] V.Kompella,J.Pasquale,andG.Polyzos,Multicastroutingformultimediacommunication,IEEE/ACMTransactionsonNetworking(TON),vol.1,1993. [5] Y.Q.ChenandZ.Wang,Formationcontrol:areviewandanewconsideration,inProc.IEEE/RSJInt.Conf.onIntelligentRobotsandSystems,Aug.2005,pp.3181. [6] M.Zavlanos,M.Egerstedt,andG.Pappas,Graph-theoreticconnectivitycontrolofmobilerobotnetworks,Proc.oftheIEEE,vol.99,no.9,pp.1525,Sept.2011. [7] M.ZavlanosandG.Pappas,Potentialeldsformaintainingconnectivityofmobilenetworks,IEEETrans.Robotics,vol.23,no.4,pp.812,Aug.2007. [8] N.LeonardandE.Fiorelli,Virtualleaders,articialpotentialsandcoordinatedcontrolofgroups,inDecisionandControl,2001.Proceedingsofthe40thIEEEConferenceon,vol.3,2001,pp.2968vol.3. [9] J.A.FaxandR.M.Murray,GraphLaplaciansandstabilizationofvehicleformations,CDSTechnicalReport01-007.CaliforniaInstituteofTechnology.Submittedto2002IFACWorldCongress.[Online].Available: http://resolver.caltech.edu/CaltechCDSTR:2001.01-007 [10] J.FaxandR.Murray,Informationowandcooperativecontrolofvehicleformations,IEEETrans.Autom.Control,vol.49,no.9,pp.1465,Sept.2004. [11] M.Fiedler,Algebraicconnectivityofgraphs,CzechoslovakMath.J.,vol.23,no.98,pp.298,1973. [12] S.Boyd,ConvexoptimizationofgraphLaplacianeigenvalues,inProc.Int.CongressofMathematicians,vol.3,2006,pp.1311. [13] M.Fiedler,Absolutealgebraicconnectivityoftrees,LinearandMultilinearAlgebra,vol.26,no.1-2,pp.85,1990. 99

PAGE 100

[14] M.MesbahiandM.Egerstedt,Graphtheoreticmethodsinmultiagentnetworks.PrincetonUnivPr,2010. [15] H.Tanner,Onthecontrollabilityofnearestneighborinterconnections,inProc.IEEEConf.onDecisionandControl,vol.3,Dec.2004,pp.2467. [16] A.GhoshandS.Boyd,Growingwell-connectedgraphs,inProc.IEEEConf.onDecisionandControl,Dec.2006,pp.6605. [17] Y.Wan,S.Roy,X.Wang,A.Saberi,T.Yang,M.Xue,andB.Malek,Onthestructureofgraphedgedesignsthatoptimizethealgebraicconnectivity,inProc.IEEEConf.onDecisionandControl,Dec.2008,pp.805. [18] M.DeGennaroandA.Jadbabaie,Decentralizedcontrolofconnectivityformulti-agentsystems,inProc.IEEEConf.onDecisionandControl(CDC),Dec.2006,pp.3628. [19] M.ZavlanosandG.Pappas,Distributedconnectivitycontrolofmobilenetworks,IEEETrans.Robotics,vol.24,no.6,pp.1416,Dec.2008. [20] G.Notarstefano,K.Savla,F.Bullo,andA.Jadbabaie,Maintaininglimited-rangeconnectivityamongsecond-orderagents,inProc.Am.ControlConf.,June2006,p.6pp. [21] K.SrivastavaandM.Spong,Multi-agentcoordinationunderconnectivityconstraints,inProc.Am.ControlConf.,June2008,pp.2648. [22] D.DimarogonasandK.Kyriakopoulos,Connectednesspreservingdistributedswarmaggregationformultiplekinematicrobots,IEEETrans.Robotics,vol.24,no.5,pp.1213,Oct.2008. [23] F.Knorn,R.Stanojevic,M.Corless,andR.Shorten,Aframeworkfordecentralisedfeedbackconnectivitycontrolwithapplicationtosensornetworks,Int.J.Control,vol.83,no.11,pp.2411,Nov.2009. [24] P.Yang,R.Freeman,G.Gordon,K.Lynch,S.Srinivasa,andR.Sukthankar,Decentralizedestimationandcontrolofgraphconnectivityinmobilesensornetworks,inAm.ControlConf.,June2008,pp.2678. [25] M.Zavlanos,A.Jadbabaie,andG.Pappas,Flockingwhilepreservingnetworkconnectivity,inProc.IEEEConf.DecisionandControl,Dec.2007,pp.2919. [26] A.CornejoandN.Lynch,Connectivityserviceformobilead-hocnetworks,inProc.IEEEInt.Conf.onSelf-AdaptiveandSelf-OrganizingSyst.,Oct.2008,pp.292. [27] Z.YaoandK.Gupta,Backbone-basedconnectivitycontrolformobilenetworks,inProc.IEEEInt.Conf.onRoboticsandAutomation,May2009,pp.1133. 100

PAGE 101

[28] Z.Kan,A.Dani,J.M.Shea,andW.E.Dixon,Ensuringnetworkconnectivityduringformationcontrolusingadecentralizednavigationfunction,inProc.IEEEMilitaryCommun.Conf.,SanJose,California,Nov.2010,pp.531. [29] H.G.TannerandA.Kumar,Towardsdecentralizationofmulti-robotnavigationfunctions,inRoboticsandAutomation,2005.ICRA2005.Proceedingsofthe2005IEEEInternationalConferenceon,Apr.2005,pp.4132. [30] A.DeGennaro,M.C.;Jadbabaie,Formationcontrolforacooperativemulti-agentsystemusingdecentralizednavigationfunctions,inAm.ControlConf.,2006,Minneapolis,MN,Jun.2006,p.6pp. [31] M.MengJi;Egerstedt,Distributedcoordinationcontrolofmultiagentsystemswhilepreservingconnectedness,inIEEETrans.Robotics,Aug.2007,pp.693. [32] M.HsiehandV.Kumar,Patterngenerationwithmultiplerobots,inRoboticsandAutomation,2006.ICRA2006.Proceedings2006IEEEInternationalConferenceon,Orlando,FL,May.2006,pp.2442. [33] Z.Kan,A.P.Dani,J.M.Shea,andW.E.Dixon,Networkconnectivitypreservingformationstabilizationandobstacleavoidanceviaadecentralizedcontroller,acceptedforpublicationinIEEETrans.AutomaticControl.[Online].Available: http://wireless.ece.u.edu/jshea/pubs/zhen tac11.pdf [34] J.J.Garcia-Luna-AcevesandD.Sampath,EfcientmulticastroutinginMANETsusingprexlabels,inProc.IEEEInt.Conf.ComputerCommun.andNetworks(ICCCN),SanFrancisco,CA,Aug.2009,pp.1. [35] D.SampathandJ.J.Garcia-Luna-Aceves,PROSE:scalableroutinginMANETsusingprexlabelsanddistributedhashing,inProc.IEEEConf.onSensorMesh,andAdHoc.Commun.(SECON),Rome,Italy,Jun.2009,pp.1. [36] J.J.Garcia-Luna-AcevesandD.Sampath,ScalableintegratedroutingusingprexlabelsanddistributedhashtablesforMANETs,inProc.IEEEInt.Conf.onMobileAdHocandSensorSyst.(MASS),MacauSAR,P.R.C.,Oct.2009,pp.188. [37] A.Drozdek,DataStructuresandAlgorithmsinC++,2nded.Brooks/Cole,2001. [38] M.JiandM.Egerstedt,Distributedcoordinationcontrolofmultiagentsystemswhilepreservingconnectedness,IEEETrans.Robotics,vol.23,no.4,pp.693,Aug.2007. [39] D.DimarogonasandK.Johansson,Boundedcontrolofnetworkconnectivityinmulti-agentsystems,ControlTheoryApplications,IET,vol.4,no.8,pp.1330,august2010. [40] J.R.Ullmann,Analgorithmforsubgraphisomorphism,J.ACM,vol.23,no.1,pp.31,Jan.1976.[Online].Available: http://doi.acm.org/10.1145/321921.321925 101

PAGE 102

[41] D.Conte,P.Foggia,C.Sansone,andM.Vento,Thirtyyearsofgraphmatchinginpatternrecognition,IJPRAI,pp.265,2004. [42] E.Bengoetxea,Inexactgraphmatchingusingestimationofdistributionalgorithms,Ph.D.dissertation,EcoleNationaleSuperieuredesTelecommunications,Paris,France,Dec2002. [43] S.Umeyama,Aneigendecompositionapproachtoweightedgraphmatchingproblems,PatternAnalysisandMachineIntelligence,IEEETransactionson,vol.10,no.5,pp.695,sep1988. [44] T.H.Cormen,C.E.Leiserson,R.L.Rivest,andC.Stein,IntroductiontoAlgorithms,2nded.McGraw-Hill,2001. [45] R.Burkard,M.Dell'Amico,andS.Martello,AssignmentProblems.Philadelphia,PA,USA:SocietyforIndustrialandAppliedMathematics,2009. [46] S.Kirkpatrick,C.D.Gelatt,andM.P.Vecchi,Optimizationbysimulatedannealing,Science,vol.220,pp.671,1983. [47] C.W.CommanderandP.M.Pardalos,Asurveyofthequadraticassignmentproblem,withapplications,TheMoreheadElectronicJournalofApplicableMathematics,2005. [48] M.R.GareyandD.S.Johnson,ComputersandIntractability:AGuidetotheTheoryofNP-Completeness.W.H.Freeman,1979. [49] E.Horowitz,S.Shani,andD.P.Metha,FundamentalsofDataStructuresinC++.Summit,NewJersey:SiliconPress,2007. [50] R.Jayakumar,K.Thulasiraman,andM.Swamy,Complexityofcomputationofaspanningtreeenumerationalgorithm,CircuitsandSystems,IEEETransactionson,vol.31,no.10,pp.853860,oct1984. [51] H.-C.ChenandY.-L.Wang,AnefcientalgorithmforgeneratingPrufercodesfromlabelledtrees,TheoryofComputingSystems/MathematicalSystemsTheory,vol.33,pp.97,2000. [52] X.Wang,L.Wang,andY.Wu,AnoptimalalgorithmforPrufercodes,J.SoftwareEng.andAppl.,vol.2.,pp.111,2009. [53] A.H.LandandA.G.Doig,Anautomaticmethodofsolvingdiscreteprogrammingproblems,Econometrica,vol.28,pp.490,1960. [54] R.RamanathanandR.Rosales-Hain,Topologycontrolofmultihopwirelessnetworksusingtransmitpoweradjustment,inProc.IEEEINFOCOM,vol.2,2000,pp.404vol.2. 102

PAGE 103

[55] B.Chen,K.Jamieson,H.Balakrishnan,andR.Morris,SPAN:Anenergy-efcientcoordinationalgorithmfortopologymaintenanceinadhocwirelessnetworks,inProc.IEEE/ACMMOBICOM,2001. [56] P.Santi,Topologycontrolinwirelessadhocandsensornetworks,ACMComput.Surv.,vol.37,pp.164,June2005. [57] S.Subramanian,J.Shea,andW.Dixon,Powercontrolforcellularcommunicationswithchanneluncertainties,inProc.2009AmericanControlConference(ACC),June2009,pp.1569. [58] ,Prediction-basedpowercontrolfordistributedcellularcommunicationnetworkswithtime-varyingchanneluncertainties,inProc.IEEEConf.onDecisionandControl,Dec.2009,pp.1998. [59] L.Navaravong,J.M.Shea,andW.E.Dixon,Physical-andnetwork-topologycontrolforsystemsofmobilerobots,inProc.IEEE/AFCEAMilitaryCommun.Conf.(MILCOM),Baltimore,MD,2011,pp.1079.[Online].Available: http://wireless.ece.u.edu/jshea/pubs/BomMILCOM2011.pdf [60] L.Navaravong,J.M.Shea,E.L.Pasiliao,G.L.Barnette,andW.E.Dixon,Optimizingnetworktopologytoreduceaggregatetrafcinsystemsofmobileagents,inModels,Algorithms,andTechnologiesforNetworkAnalysis,ser.SpringerProceedingsinMathematicsStatistics,B.Goldengorin,V.A.Kalyagin,andP.M.Pardalos,Eds.,vol.32.SpringerNewYork,2013,pp.129.[Online].Available: http://dx.doi.org/10.1007/978-1-4614-5574-5 8 [61] L.Navaravong,J.M.Shea,E.L.P.Jr,andW.E.Dixon,Optimizingnetworktopologytoreduceaggregatetrafcinasystemofmobilerobotsunderanenergyconstraint,inProc.2012IEEECommunicationsConf.,Ottawa,Canada,Jun.2012,acceptedforpublication. [62] L.Navaravong,Z.Kan,J.M.Shea,andW.E.Dixon,Formationrecongurationformobilerobotswithnetworkconnectivityconstraints,IEEENetwork,vol.26,no.4,July2012. 103

PAGE 104

BIOGRAPHICALSKETCH LeenhapatNavaravongwasbornonOctober11,1985,inBangkok,Thailand.Hereceivedhisbachelor'sdegreeintelecommunicationsengineeringwithrst-classhonoursfromShinawatraInternationalUniversity,Pathumthani,Thailand,in2007,andhismaster'sdegreeinelectricalandcomputerengineeringfromtheUniversityofFlorida,Gainesville,USA,inMay2010.InMay2013,hereceivedhisPh.D.degreeinelectricalandcomputerengineeringfromtheUniversityofFlorida,Gainesville,USA.WhilepursuinghisgraduatedegreesattheUniversityofFlorida,heworkedasaresearchandteachingassistantintheWirelessInformationNetworkingGroup(WING)LaboratoryintheDepartmentofElectricalandComputerEngineeringunderthesupervisionofProf.JohnM.Shea.HealsoworkedasaresearchinternwiththeAirForceResearchLaboratory(AFRL),EglinAirForceBase,andtheUniversityofFloridaResearchandEngineeringEducationFacility(UF-REEF)inShalimar,Floridainthesummersof2011and2012.Hisresearchinterestsarewirelesscommunicationsandnetworkingwithapplicationstosystemsofautonomousvehicles. 104