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Utilitarian approaches for multi-metric optimization in VLSI circuit design and spatial clustering

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Utilitarian approaches for multi-metric optimization in VLSI circuit design and spatial clustering
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Gupta, Upavan
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Expected utility theory
Game theory
Nash equilibrium
Risk averse optimization
Process variations
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ABSTRACT: In the field of VLSI circuit optimization, the scaling of semiconductor devices has led to the miniaturization of the feature sizes resulting in a significant increase in the integration density and size of the circuits. At the nanometer level, due to the effects of manufacturing process variations, the design optimization process has transitioned from the deterministic domain to the stochastic domain, and the inter-relationships among the specification parameters like delay, power, reliability, noise and area have become more intricate. New methods are required to examine these metrics in a unified manner, thus necessitating the need for multi-metric optimization. The optimization algorithms need to be accurate and efficient enough to handle large circuits.As the size of an optimization problem increases significantly, the ability to cluster the design metrics or the parameters of the problem for computational efficiency as well as better analysis of possible trade-offs becomes critical. In this dissertation research, several utilitarian methods are investigated for variation aware multi-metric optimization in VLSI circuit design and spatial pattern clustering. A novel algorithm based on the concepts of utility theory and risk minimization is developed for variation aware multi-metric optimization of delay, power and crosstalk noise, through gate sizing. The algorithm can model device and interconnect variations independent of the underlying distributions and works by identifying a deterministic linear equivalent model from a fundamentally stochastic optimization problem.Furthermore, a multi-metric gate sizing optimization framework is developed that is independent of the optimization methodology, and can be implemented using any mathematical programming approach. It is generalized and reconfigurable such that the metrics can be selected, removed, or prioritized for relative importance depending upon the design requirements. In multi-objective optimization, the existence of multiple conflicting objectives makes the clustering problem challenging. Since game theory provides a natural framework for examining conflicting situations, a game theoretic algorithm for multi-objective clustering is introduced in this dissertation research. The problem of multi-metric clustering is formulated as a normal form multi-step game and solved using Nash equilibrium theory.This algorithm has useful applications in several engineering and multi-disciplinary domains which is illustrated by its mapping to the problem of robot team formation in the field in multi-emergency search and rescue. The various algorithms developed in this dissertation achieve significantly better optimization and run times as compared to other methods, ensure high utility levels, are deterministic in nature and hence can be applied to very large designs. The algorithms have been rigorously tested on the appropriate benchmarks and data sets to establish their efficacy as feasible solution methods. Various quantitative sensitivity analysis have been performed to identify the inter-relationships between the various design parameters.
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Dissertation (Ph.D.)--University of South Florida, 2008.
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by Upavan Gupta.
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UtilitarianApproachesforMulti-MetricOptimizationinV LSICircuitDesignandSpatialClustering by UpavanGupta Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofComputerScienceandEngineering CollegeofEngineering UniversityofSouthFlorida MajorProfessor:NagarajanRanganathan,Ph.D. DeweyRundus,Ph.D. SrinivasKatkoori,Ph.D. KandethodyM.Ramachandran,Ph.D. DateofApproval: May30,2008 Keywords:expectedutilitytheory,gametheory,nashequil ibrium,riskaverseoptimization,process variations,gatesizing,patternrecognition,searchandr escuerobotics Copyright2008,UpavanGupta

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DEDICATION Tomywonderfulparents, withallmyloveandrespect.

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ACKNOWLEDGEMENTS IwouldliketotakethisopportunitytothankProfessorNaga rajanRanganathanforproviding metheopportunitytoworkwithhimoninterestingproblems. Iammostgratefultohimforhiscontinuousencouragement,motivation,andvaluablesuggesti onsfromhisvastexperience.Hispatience, guidance,andemotionalsupportduringdifculttimeshave beeninstrumentalininspiringmetobecomeagoodresearcher,andmoreimportantly,abetterperso n.Iconsidermyselfextremelyfortunate forhavingworkedwithsuchaneminentscholarandaninspiri ngteacher.Iwouldalsoliketothank ProfessorDeweyRundus,ProfessorSrinivasKatkoori,andP rofessorKandethodyM.Ramachandran fortheirtimeandvaluablesuggestions.Mysincereacknowl edgmentstoSemiconductorResearch Corporation(grant#2007-HJ-1596)andNationalScienceFo undation(ComputingResearchInfrastructuregrant#CNS-0551621)forsupportingthisresearch inparts. Mypeersandfriendsfromthelabhavemadetheyearsspentinp ursuingthisdegreefeelshorter. Theirconstructiveideasanddiscussionshavebeenextreme lyusefulinimprovingthequalityofthis research.Onthepersonalfront,Iwouldliketothankmypare nts,whohavealwaysstoodbyme,and havebeenaconstantsourceofinspiration.Atanearlyageth eyinculcatedanattitudethatnothing isimpossibleandtherearenoshortcutsinlife.Mylovelysi sterandmybrother-in-lawhavebeen emotionallyverysupportiveallthistime.MygoodfriendKe vinFrenzelhasalwaysbeentherefor me.Iwouldalsoliketothankmyfriends,Sunny,Sonali,Piyo osh,Himanshu,Kadambari,Sanjay, Mahalingam,andShekhar,fortheirloveandsupport.Finall y,IthankmywifeShikhaforallthe effortsshehasmadethatenabledmetosailsmoothlythrough thedemandingtimes.

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NOTETOREADER Theoriginalofthisdocumentcontainscolorthatisnecessa ryforunderstandingthedata.The originaldissertationisonlewiththeUniversityofSouth FloridalibraryinTampa,Florida.

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TABLEOFCONTENTS LISTOFTABLES iv LISTOFFIGURES v LISTOFALGORITHMS vii ABSTRACT viii CHAPTER1INTRODUCTION 1 1.1Motivation 5 1.2WhyUtilitarianApproaches? 6 1.3ScopeandContributions 7 1.4OutlineofDissertation 10 CHAPTER2BACKGROUNDANDRELATEDWORK11 2.1UtilityTheory 11 2.1.1ExpectedUtilityTheoryandRiskAversion13 2.1.1.1ExpectedUtility 13 2.1.1.2RiskAversion 14 2.1.2GameTheoryandNashEquilibrium16 2.1.2.1ClassicationofGames192.1.2.2MathematicalRepresentation202.1.2.3CritiqueofGameTheory21 2.2MathematicalProgramming 22 2.3VLSICircuitOptimization 23 2.3.1ProcessVariations 26 2.4VariationAwareGateSizing 27 2.4.1OptimizationMetrics 28 2.4.2OptimizationMethods 31 2.5SpatialDataClustering 32 2.5.1ClusteringTechniques 32 CHAPTER3EXPECTEDUTILITYBASEDCIRCUITOPTIMIZATION35 3.1IssuesinCircuitOptimization 35 3.2Expected-UtilityBasedModeling 39 3.2.1DeterministicModeling 43 3.3ParametricModels 44 3.3.1Delay,ProcessVariationsandSpatialCorrelation453.3.2LeakageandDynamicPower 46 3.3.3CrosstalkNoise 47 i

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3.4StochastictoDeterministicGateSizing 49 3.4.1StochasticOptimizationProblem 49 3.4.2DeterministicEquivalentModel 52 3.5ExperimentalResults 53 3.5.1Setup 54 3.5.2SensitivityofUtilityAssuranceConstant563.5.3RiskAverseOptimizationResults573.5.4OptimizationConsideringDeviceandInterconnectVa riations59 3.6Discussion 61 CHAPTER4INTEGRATEDFRAMEWORKFORCIRCUITOPTIMIZATION64 4.1NeedforIntegratedFramework 65 4.2SingleMetricOptimizationModels 66 4.2.1UnconstrainedDelayOptimization674.2.2PowerOptimizationUnderDelayConstraints674.2.3CrosstalkNoiseOptimizationUnderDelayConstraint s68 4.3IntegratedFrameworkforVariationAwareGateSizing69 4.3.1ObjectiveFunctionModeling 69 4.3.2IntegratedFramework 71 4.3.3MathematicalProgrammingMethodology724.3.4PathstoNodes 75 4.4ExperimentalResults 76 4.4.1SimulationSetup 76 4.4.2OptimalNoiseMargins 77 4.4.3DeterminationofTimingSpecication794.4.4LeakagePower,DynamicPower,andCrosstalkNoiseOpt imization80 4.4.5SingleMetricOptimizationResults814.4.6ResultsforPriorityBasedOptimization83 4.5Discussion 84 CHAPTER5AMICROECONOMICAPPROACHTOSPATIALDATACLUSTERI NG87 5.1SpatialDataClustering 87 5.2WhyGameTheoryforClustering? 89 5.3MicroeconomicClusteringAlgorithm 90 5.3.1MathematicalPartitioning 90 5.3.2Multi-StepNormalFormGameModel92 5.3.2.1IdenticationofPlayers945.3.2.2DenitionofStrategy 95 5.3.2.3PayoffFunction 97 5.3.2.4NashEquilibriumSolution98 5.3.3EnsembleBasedGameTheoreticClustering995.3.4AnalysisofGameTheoreticAlgorithm100 5.3.4.1ComputationalComplexityAnalysis1005.3.4.2NatureofAlgorithmExecution101 5.4ExperimentalResults 101 5.4.1SimulationSetup 102 5.4.2ExperimentswithExistingDataSets1035.4.3ExperimentswithArticialDataSets105 ii

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5.4.4FairnessofClustering 107 5.4.5SensitivityAnalysis 108 5.4.5.1DataSetSimilarityMeasure1085.4.5.2NumberofPlayersandStrategies1085.4.5.3ExecutionTime 111 5.5Discussion 111 CHAPTER6GAMETHEORETICAPPROACHTOROBOTTEAMFORMATION11 5 6.1ProblemDescription 115 6.2WhyMicroeconomicsforRobotTeamFormation?1206.3Background 120 6.4MicroeconomicModeling 122 6.4.1KMeansPartitioning 122 6.4.2GameTheoreticPartitioningofRobots123 6.5ExperimentalResults 125 6.5.1SimulationToolsandSetup 126 6.5.2Analysis 126 6.6Discussion 129 CHAPTER7CONCLUSIONSANDFUTUREDIRECTIONS130REFERENCES 134 LISTOFPUBLICATIONS 142 ABOUTTHEAUTHOR EndPage iii

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LISTOFTABLES Table3.1Comparisonbetweenequallyweightedmulti-metri coptimizationof leakagepower,dynamicpowerandcrosstalknoiseforriskaw aregate sizing(RAGS)andfuzzymathematicalprogramming(FMP).62 Table3.2Comparisonbetweenequallyweightedmulti-metri coptimizationwith w = 0 : 92andsinglemetricoptimizationfordynamicpower,leakag e power,andcrosstalknoisemetrics. 63 Table4.1Improvementintheoptimizationofmetricsformul ti-metricoptimizationwithequalpriority( a = b = g = 0 : 33),ascomparedtothevalues obtainedduringunconstraineddelayoptimization.80 Table4.2Comparisonofsinglemetricdynamicpowerandcros stalknoiseoptimizationwiththeequallyweightedmulti-metricoptimizat ionvalues.84 Table5.1Notationsandterminology. 91 Table5.2PerformanceofthealgorithmsonIrisdataset.113Table5.3Fairnessoftheclusteringalgorithms. 114 Table6.1Notationsforrobotpartitioning. 123 Table6.2Parametersaffectingthegametheoreticmodel.12 9 iv

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LISTOFFIGURES Figure1.1Scopeandcontributionsofthedissertation. 8 Figure2.1Autilitybasedsystemfromtheperspectiveofara tionalagent.12 Figure2.2Afunctionrepresentingtherelationshipbetwee nexpectedutilityand riskaversion. 15 Figure2.3Asimpleexampleoftwoplayernon-cooperativeno rmalformgame.17 Figure2.4Generationofthestrategysets,identicationo fthedominantstrategies,andNashequilibriuminprisoners'dilemmagame.19 Figure2.5Variationimpactatdifferenttechnologynodes[ 1].28 Figure2.6Taxonomyofthevariationawaregatesizingworks .29 Figure3.1Utilitycurveforarandomfunction. 42 Figure3.2Acouplingstructurewithsinglevictim-aggress orpairsetting.48 Figure3.3Simulationsetupfortheriskaversegatesizingo ptimizationproblem.55 Figure3.4Improvementinthetimingyieldofthecircuitsfo rdifferentvaluesof w .57 Figure3.5Percentageimprovementintheoptimizationofth eobjectivefunction forvariousvaluesof w 58 Figure3.6Impactofinterconnectvariationsandgatesizev ariationsontheoptimizationofthemetrics. 60 Figure4.1Gatesizingframeworkformulti-metriccircuito ptimization.72 Figure4.2Flowchartforsimulationsetup. 78 Figure4.3Effectofdifferentnoisetolerancevaluesonthe optimalityoftheobjectives.79 Figure4.4Averageimprovementinthemetricsvaluesforsim ultaneousmultimetricoptimizationascomparedtothedeterministicworst casepessimisticanalysis. 81 Figure4.5Effectofsinglemetricleakagepoweroptimizati onascomparedto equallyweightedmulti-metricoptimization.82 v

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Figure4.6Theimprovementinleakagepoweranddynamicpowe rwhenoptimizedwithpriorities a = 0 : 5, b = 0 : 5,and g = 0ascomparedtothe scenariowhere a = 0 : 33, b = 0 : 33,and g = 0 : 33.85 Figure4.7Comparativestudyofleakagepoweroptimization inthreedifferent scenarios,unconstraineddelayoptimization,singlemetr icleakagepower optimization,andmulti-metricoptimization.85 Figure5.1Identicationofoptimumclustersusinggamethe oreticclustering(GTKMeans)andKMeansmethodologies. 94 Figure5.2Anexamplefordenitionofstrategy. 97 Figure5.3PerformanceofthealgorithmsontheBritishTown dataset.104 Figure5.4PerformanceofthealgorithmsontheGermanTownd ataset.105 Figure5.5Averageimprovementinthecompactionobjective fortheexperiments onarticialdatasets. 106 Figure5.6Averageimprovementintheequi-partitioningob jectivefortheexperimentsonarticialdataset. 107 Figure5.7Effectofdatasetsimilaritymeasureontheexecu tiontimeoftheGTKMeansalgorithm. 109 Figure5.8Relationshipbetweentheexecutiontimeandthen umberofclusters.110 Figure5.9Averagenumberofplayersandstrategiesfordiff erentclustersizes.110 Figure6.1Exampleofamulti-emergencysituationinasubur banarea.117 Figure6.2Anexampleofsearchandrescuerobotdeploymenti nmulti-emergency scenario. 118 Figure6.3Effectofhighcommunicationoverheadonthesear chandrescueprocess. 118 Figure6.4Partitioningofrobotssuchthattheintra-clust ercommunicationisminimized,andeachpartitionhasaheadnoderesponsibleforin ter-cluster communication. 119 Figure6.5Partitioningresultsforrobotteamformationus ingKMeansalgorithm.119 Figure6.6Identicationofoptimumsizesoftheclustersan dthelocationsofthe clustercentersusinggametheoreticalgorithm,andKMeans algorithm.127 Figure6.7Averageperformanceofalgorithmsonarticiald atasets.128 vi

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LISTOFALGORITHMS Algorithm4.1Multi-metricgatesizingalgorithm 73 Algorithm5.1KMeanspartitioning 92 Algorithm5.2Gametheoreticalgorithm 93 Algorithm5.3Generationofstrategyset 96 Algorithm5.4Payoffmatrixgeneration 98 Algorithm5.5Nashequilibriumalgorithm 99 Algorithm6.1Microeconomicrobotteamclusteringalgorit hm124 vii

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UTILITARIANAPPROACHESFORMULTI-METRICOPTIMIZATIONINV LSICIRCUIT DESIGNANDSPATIALCLUSTERING UpavanGupta ABSTRACT IntheeldofVLSIcircuitoptimization,thescalingofsemi conductordeviceshasledtothe miniaturizationofthefeaturesizesresultinginasignic antincreaseintheintegrationdensityandsize ofthecircuits.Atthenanometerlevel,duetotheeffectsof manufacturingprocessvariations,thedesignoptimizationprocesshastransitionedfromthedeterm inisticdomaintothestochasticdomain,and theinter-relationshipsamongthespecicationparameter slikedelay,power,reliability,noiseandarea havebecomemoreintricate.Newmethodsarerequiredtoexam inethesemetricsinauniedmanner, thusnecessitatingtheneedformulti-metricoptimization .Theoptimizationalgorithmsneedtobeaccurateandefcientenoughtohandlelargecircuits.Asthes izeofanoptimizationproblemincreases signicantly,theabilitytoclusterthedesignmetricsort heparametersoftheproblemforcomputationalefciencyaswellasbetteranalysisofpossibletrad e-offsbecomescritical.Inthisdissertation research,severalutilitarianmethodsareinvestigatedfo rvariationawaremulti-metricoptimizationin VLSIcircuitdesignandspatialpatternclustering. Anovelalgorithmbasedontheconceptsofutilitytheoryand riskminimizationisdevelopedfor variationawaremulti-metricoptimizationofdelay,power andcrosstalknoise,throughgatesizing.The algorithmcanmodeldeviceandinterconnectvariationsind ependentoftheunderlyingdistributions andworksbyidentifyingadeterministiclinearequivalent modelfromafundamentallystochastic optimizationproblem.Furthermore,amulti-metricgatesi zingoptimizationframeworkisdeveloped thatisindependentoftheoptimizationmethodology,andca nbeimplementedusinganymathematical programmingapproach.Itisgeneralizedandrecongurable suchthatthemetricscanbeselected, removed,orprioritizedforrelativeimportancedepending uponthedesignrequirements. viii

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Inmulti-objectiveoptimization,theexistenceofmultipl econictingobjectivesmakestheclusteringproblemchallenging.Sincegametheoryprovidesanatur alframeworkforexaminingconicting situations,agametheoreticalgorithmformulti-objectiv eclusteringisintroducedinthisdissertation research.Theproblemofmulti-metricclusteringisformul atedasanormalformmulti-stepgame andsolvedusingNashequilibriumtheory.Thisalgorithmha susefulapplicationsinseveralengineeringandmulti-disciplinarydomainswhichisillustratedby itsmappingtotheproblemofrobotteam formationintheeldinmulti-emergencysearchandrescue. Thevariousalgorithmsdevelopedinthisdissertationachi evesignicantlybetteroptimizationand runtimesascomparedtoothermethods,ensurehighutilityl evels,aredeterministicinnatureand hencecanbeappliedtoverylargedesigns.Thealgorithmsha vebeenrigorouslytestedontheappropriatebenchmarksanddatasetstoestablishtheirefca cyasfeasiblesolutionmethods.Various quantitativesensitivityanalysishavebeenperformedtoi dentifytheinter-relationshipsbetweenthe variousdesignparameters. ix

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CHAPTER1 INTRODUCTION Theadvancesinscienceandtechnologyimpacttherealmofen gineering.Themostimportant facetofthetechnologyevolutionisthatitfacilitatesthe developmentofimprovedproducts,andhelps inapplyingtheknowledgeandintelligencegainedfromoned isciplinetoadvanceotherdisciplines. Theimportantobjectivesindevelopingtheseproductsare, incorporationofenhancedfeaturesets, improvementinperformance,andminiaturization.Onewayt oachievetheseobjectivesistoscale downthedimensionsofvariousconstituentelementsorcomp onentsoftheseproductssothatmore componentscanbeintegratedonit.Improvementsinthefabr icationtechnologiesaidinachieving thesegoals.However,thetransitionfromonetechnologyle veltoanotherisnotrudimentary,andit uncoversnewconcerns.Inthecontextofverylargescaleint egratedcomputeraideddesign(VLSICAD),specicallycircuitoptimization,theseconcernsca nbeexplainedasfollows. Withtheaggressivescalingofsemiconductordevicestothe nano-meterlevel,theintegrationdensityofthecircuitsincreases.AccordingtotheInternatio nalTechnologyRoadmapforSemiconductors (ITRS)[2],thefeaturesizesforthedevicesandinterconne ctswillcontinuetoscaledownattherate of0.7xpergeneration.Thisreductioninsizesaffecttheci rcuitoptimizationprocessinseveralways. First,asthewiringdensityandconsequentlytheaspectrat iosinthemetallinesincrease,thecrosscouplingcapacitancebetweentheneighboringinterconnec tsgrows.Thismayresultinanincreasein theinterconnectcrosstalknoiseonawire,duetothecharge injectedinitduringtheswitchinginthe neighboringnets.Inthedeepsub-nanometerdesigns,suchc ouplingcapacitanceeffectsbetweenthe adjacentnetscancausefunctionalityfailurescausingrel iabilityissues[3].Thenoiseduetocrosscouplingcapacitanceisadominantcomponentamongthenois esources,andhenceisanessential considerationduringthecircuitoptimizationprocess.Se cond,thedemandforpowersensitivedevices hasgrownsignicantlyinrecentyears.Thisisattributedt otheremarkablegrowthofpersonalcomputingandmobiledevicessuchaslaptopcomputers,cellula rphones,musicplayersandotherportable 1

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devicesthatarepredominantlybatterydriven.Thesedevic esdemandhigh-speedcomputationalfunctionalitieswithlowpowerconsumption.However,astheint egrationdensityoftransistorsinadieand thefrequencyofoperationsincrease,thepowerconsumptio ninadieincreaseswitheachgeneration. Tomaintainlowpowerdissipation,supplyvoltageisscaled down.However,thescalingofsupply voltageislimitedbythehigh-performancerequirements.I nordertomaintaintheperformance,the transistorthresholdvoltageshouldbescaleddowntoachie velowswitchingenergyperdevice.Scalingofthresholdvoltagesignicantlyincreasesthesub-th resholdleakagecurrent[4],resultinginhigh leakagepowerdissipationduringstandby.Thus,atthesubnanometerlevel,powerminimizationisan importantmetricinthecircuitoptimizationprocessalong withtheperformancemetric.Hence,with thescalingoftechnology,newparadigmsthatimpacttheper formanceandreliabilityofthedesigns becomeanintegralpartofthedesignandoptimizationproce ss. Theinter-relationshipsbetweentheseoptimizationmetri cshavebecomemoreintricateinthe nano-meterregime.Optimizationofonemetricalonemayres ultinaperformanceshiftfromone metrictoanother,therebyintroducingsub-optimalityint hevaluesofothermetrics.Asasimple example,ifsomecircuitoptimizationtechniqueisemploye dwithanobjectiveofonlypowerminimization,theresultingcircuitcongurationmaypotenti allyhavehighinterconnectcrosstalknoise, andhencelowsignalreliability.Alternatively,iftheopt imizationisperformedwiththeobjectiveof crosstalknoiseminimization,theresultingdesignmaynot belowpowerdissipating,therebyaffecting theperformanceofthedevice.Addressingtheseaspectsofo ptimizationareimportantconsiderations inthenextgenerationcircuitoptimization. Astheprocesstechnologyisscaleddown,thelimitationsdu etomanufacturingprocessesandenvironmentalnoise,makethephysicalrealizationofdevice sandinterconnectsunpredictableduringthe front-enddesign.Duringthefabricationofsemiconductor devices,theexistenceofnon-uniformconditionsatthedepositionanddiffusionstages,orduetothe limitedresolutionofthephotolithographic process,theparameterslikeoxidethickness,effectivega telengthofindividualtransistorsandinterconnectwidthsmaynotfollowthespecications.Thesevari ationsmayresultindramaticchanges inthedeviceperformancecharacteristics,aswellasthere liabilityofthedesigns.Asaresult,the designandoptimizationproblemhastransitionedfromthed eterministicdomaintotheprobabilistic domain[5].Also,theseprocesstolerancesdonotscaleprop ortionally,therebyincreasingtherelative impactofprocessvariationonthedesignprocesswitheachn ewtechnologynode. 2

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Thistransitionofoptimizationprocesstothestochasticd omainaffectsthecircuitoptimization process.Since,thestochasticoptimizationtechniquesar einherentlyslowerthantheirdeterministic equivalents,theoptimizationprocessisadverselyaffect ed.Inrecentyears,thestateoftheartresearch inVLSIdesignautomationhasaddressedthisissue.Several circuitoptimizationmethodshavebeen developedwithanobjectiveofcenteringthedesignsspeci cationssuchthatmajorityofthefabricated circuitsfollowdesignandperformancespecications.Man yofthesemethodsarebasedontheassumptionsthatthevariationsourcesofthecomponentsfoll owspecicdistributions,suchasGaussian distribution,identiedduringthepreliminaryanalysis[ 6,7].However,recentresearchrefutesuch assumptions[8,9].Additionally,moresourcesofprocessv ariationarebecomingpredominantasthe levelofminiaturizationisincreasing,whichisaprincipa lconcerninthesemiconductorindustry. Asaresultoftechnologyscaling,morecomponentsareinteg ratedonthedesignarea.Consideringasimpleexample,therecentIntelItaniumprocessor,c odenamed'Tukwila',releasedin2008is atwobilliontransistorchip[10]manufacturedwith65nmte chnology.Thetotalareaofthechipis 699mm 2 ascomparedtothebilliontransistorItanium'Montecito'c hipwithadesignareaof580mm 2 Althoughthenumberoftransistorshavedoubled,thechipar eahasincreaseonlyby20%.Duetothe increaseinthenumberofcomponents,andconsequentlythep roblemsize,theoptimizationprocess becomessignicantlyslow.Inageneraloptimizationprobl em,thesizeofaproblemcanbereduced bypartitioningitintoseveralsmallerclusters,andperfo rmingoptimizationineachclusterseparately. However,theclusteringproblemisnotelementary,andanyt echniquedevelopedspecicallyforclusteringofdataobjectsinoneknowledgedisciplinemaynotbe directlyapplicableforclusteringinother disciplines. Inspatialpatternclustering,severaltechniqueshavebee ndevelopedforvariousapplicationsin awidevarietyofscienticdisciplinessuchasbiology,com putervisionandpatternrecognition,and communicationsandcomputernetworks[11,12].Thesetechn iquesarelargelyapplicationspecicand performsinglemetricoptimization.Hence,theymaynotbea pplicabletotheapplicationslikeVLSI designpartitioning,rescuerobotsdeployment,ad-hocnet worksestablishment,andmulti-emergency resourcemanagementetc.Often,multiplecompetitivemetr icsarerequiredtobetargetedforoptimizationintheseengineeringdomains.Tounderstandthisp roblem,wecanconsiderahypothetical multi-emergencyenvironmentwhereanad-hocnetworkofnod es(rescuepersonnel,resourcesetc.) performingtherescueoperationsatdifferentemergencylo cationsistobeestablishedoverawireless 3

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link.Eventhougheachnodemayhaveidenticalcapabilities ,duetothebatterypowerconstraints, asubsetofnodesarerequiredtobeidentiedthatwouldbere sponsibleforinter-andintra-cluster communication.Anoptimalclusteringmechanismmustensur ethatthenodes,aswellasthecomplete clustersdonotdropoutofthenetwork.Aclusteringperform edonthebasisofonemetric,say cluster compactness forlowpowerdissipationinintra-clustercommunication, mayresultinasituationwhere someclustersaretoolargeandsomearetoosmall.Thenon-un iformpowerdistributionamongthe clustersinthiscasemayresultinasituationwherethebatt eryofthenodesinsmallerclustersmay soongetexhausted,andthenodesdropoutofthesystem.This wouldresultinlossofcommunication fromtheemergencylocationsthesenodeswereservicing. Theexponentialnatureofsuchclusteringproblemsqualie stheapplicationofheuristicsbasedoptimizationmethodologies.However,anyheuristicapproac hmaynotbeadequateforspatialclustering inthisdomainduetosomeinherentcharacteristicsofthese problems.First,theoptimizationmetrics hereareoftencompetitiveinnature,andhencecannotbeopt imizedusingtheclassicalheuristicsbased optimizationmethodsthatperformasinglemetricoptimiza tion,suchasgeneticalgorithms,simulated annealingetc.Theclusteringproblemdescribedabove,rep resentonesuchclassofproblems.Thetwo objectives,clustercompactnessanduniformpowerdistrib utionareconictinginnatureandneedtobe optimizedsimultaneously.Second,inseveralapplication softhistype,eachobjectivetobeoptimized duringtheclusteringprocessiscritical.Intermsofthecl usteringperformance,thistranslatestoa situationwherethesuccessofaclusteringmethodologyisa scertainedbythemutualsatisfactionof theoptimizationscorrespondingtoeachobjectiveinthepr oblem.Formally,thismetricofsuccessis termedasthe socialfairness [13]ofthesystem.Aconceptwidelystudiedandusedinthee ldof economics,socialfairnessofasystemcorrespondstoasitu ationwhereeachindividual(ormetric)in thesystemissatisedwithrespecttoeveryotherindividua linthesystem,andtheoverallgoalsare achieved.Inthisexample,thesocialfairnessofthepartit ioningmechanismforthead-hocnetwork clusteringproblemismaximizedifboththeobjectives,com pactnessanduniformpowerdistribution aresatised(optimized)withrespecttoeachother.Themut ualsatisfactionensuresthatallthemetrics areconsideredwiththesameprioritylevel,andatanequili briumsolutionpoint,anyimprovementin onemetriccanonlybeachievedbyworseningtheoptimizatio nofothermetrics. Thespeedoftechnologyevolutiondecidesthelifetimeofth eproducts.Thelifespanoftheproductsisshrinkingduetorapidimprovementinthemanufactur ingtechnology.Thisentailsthedesigners 4

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toidentifyanddevelopgeneralizedoptimizationmethodsc apableofincorporatingthedesignobjectivesofthefuturegenerationproducts,andareapplicable tomultipledisciplineswithrelativeease. Theobjectivesmayincludeexaminationofadditionalmetri csduringoptimization,andinvestigation andincorporationoftheeffectofrandomnessatseverallev els.Theoptimizationframeworkscapable ofaddressingtheseissueseffectivelywouldbebenecialf orthecommunity. 1.1Motivation Theissuesdiscussedabovegiveastrongintuitionaboutthe problemsthatwillbeprevalentinthe nextgenerationcomputerengineeringresearch.Morepreci sely,intheVLSI-CAD,thetechnology trends[2]suggestthatwiththeaggressivescalingofdevic estheuncertaintiesduetoprocessvariationsareexpectedtoworseninfuture.Thedimensionalityo fthecircuitoptimizationprocesswill furtherexpandduetoanincreasingimpactofdesigncompone ntsaffectingtheperformanceandreliabilityofthecircuits.Also,themulti-foldescalationint hedesigndensityofthecircuitsisinevitable. Thus,thecircuitoptimizationmethodologiescapableofad dressingonlytheproblemsoccurringin currenttechnologygenerationmaynotscalewellwiththene xtgenerationissues.Thesinglemetricoptimizationmethodsthatresultinaperformanceshift fromoneobjectivetoanotherandarenot generalizedtoincorporateadditionalmetricsarenolonge racceptable.Hence,animportantchallenge intheVLSIcircuitoptimizationistoidentifyverticallya swellashorizontallyintegratedsolution methodologies[14]. Likewise,theexistingmethodsindataclusteringareincap ableofaddressingtheclusteringrequirementsforvariousmulti-disciplinaryengineeringapplica tions.Specically,theseapplicationsrequire methodscapableofsimultaneouslyexaminingmultiplemetr icsduringclustering.Also,aclustering methodmustsatisfythesocialfairness[13]fromtheperspe ctiveofeachclusteringcriterion.This wouldensurethateachclusteringmetricissatisedwithre specttoeveryothermetricinthesystem. Themotivationforthisdissertationistoexplorethecorei ssuesintheseproblemdomains,and developnewmulti-metricoptimizationapproachesthatexh ibitthefollowingfeatures. Aframeworkthatisgeneralizedinitsabilitytoincorporat eanynumberofoptimizationmetrics thatmaybenecessarytobeoptimizedforfeasiblesolutions totheproblems.Also,theframe5

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workshouldberecongurabletoenablerelativeprioritiza tionofthemetricstobeoptimizedas pertherequirements. Adomainindependentapproachthatiseasilyportabletosol vetheoptimizationproblemsin severalknowledgedisciplines. Anapproachthatisfast,scalabletolargerproblemsizes,a ccurateintermsofoptimizations, andfeasibletosolverealproblems. Amethodthatiscapableofaddressingtheimpactofrandomne ssatseveralavenues.Inthe contextofVLSIcircuitoptimizationthispropertyisextre melyimportantforaddressingthe impactofprocessvariationsinmultipledesigncomponents Anapproachthatiscapableofinherentlymodelingthemulti -objectiveoptimizationproblems wheretheobjectivesarecompetingorconictinginnature. Amethodologythatcanguaranteetheoptimizationisperfor medfromtheperspectiveofeach metric,andhencesatisesthesocialfairnessproperty. Severalavorsofutilitarianoptimizationmethodshavebe enwidelyappliedtosolvetheproblems intheeldofeconomicsandnance[15].Inrecentyears,com puterscientistshaveexploredthe realmofutilitarianmethodstosolvevariouscomputerscie nce[16–23]andcomputerengineering [24–26]problems.Thesuccessfulimplementationofutilit arianapproachesintheseapplicationareas hasbenetedtheengineeringresearchcommunity.Thisenco uragedustoexplorethesemethodsto solvetheproblemsinVLSI-CADandspatialpatternclusteri ng. 1.2WhyUtilitarianApproaches? Theutilitytheoreticapproachesareattractiveasoptimiz ationmethodologiesduetosomeofunique featuresandpropertiesthattheypossess.Thetwovariants oftheutilitarianmethodsaregametheoreticoptimizationtechniquesandexpectedutilitytheory basedtechniques.Gametheory[27,28]isa microeconomicapproachforvisualizingaproblemasasitua tionthatconsistsofseveralplayers,each playercompetingwithallotherplayersinthesystemandtry ingtomaximizeitsownutilityorgains fromthesystem.Inthiscompetitivesetting,anequilibriu mpointisidentiedthatmaximizesthe 6

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utilityofeachplayerwithrespecttoeveryotherplayerint hegame.Thus,theperformancecriteriaof thesystemsasawholearedeterminedbyacombinationofthep erformancecriteriaoftheindividual agents.Thesalientfeaturesofgametheorythatserveasrea sonsforapplicationtotheoptimization problemsare: Thesituationsofconictandcooperationaremosteffectiv elymodeledasgames[29–31]. Gametheoreticmodelshavesimpleandwelldenedenvironme ntsforavarietyofproblems. Amethodologylike Nashequilibrium thatidentiesasociallyfairsolution,perfectlycomplementstheproblemsmodeledasagame.Thesocialfairnessoft hesolutionisaparticularly attractivefeaturefromtheperspectiveofmulti-metricsp atialclustering. Theexpectedutilitytheory[32]wasproposedbyVonNeumann andMorgensternin1944,asasound prescriptionforrationaldecision-making.Thistheoryha sbeenwidelystudiedandappliedinthevariouseldsofscienceandengineeringlikepoliticalscienc es,nance,economics,computernetworks, anddistributedcomputing.Thesuccessofutilitytheoryis attributedtothefactthatitenablesthe designers(ordecisionmakes)tovisualizetheoptimizatio nproblemsfromadifferentperspective.As asimpleexample,letusconsiderastochasticoptimization probleminthemathematicalprogramming setting,wheretheobjectivesaretobeminimizedwhilesati sfyingtheconstraintsthatarerandomized innature.Intheexpectedutilityframework,thisoptimiza tionproblemcanbeconceivedasanoptimizationprobleminwhichtheriskoffailureofconstraints isminimizedbymaximizingtheexpected utilityoftheconstraints.Inlargescalestochasticoptim izationproblems,likethoseinVLSI-CAD, thismayhelpinsubstantiallyreducingthesizeoftheprobl em,aswellastranslatingastochastic optimizationproblemtothedeterministicequivalentunde rcertainsituations. 1.3ScopeandContributions Thisdissertationexploresthevariousoptimizationissue scurrentlyexistingintheVLSI-CAD eld,specicallyatthecircuitlevel.Italsoidentiesth econcernsforspatialdataclusteringfromthe viewpointofitsapplicationsinseveralmulti-disciplina ryareas.Weidentifythedifferentmetricsthat arerequiredtobeexaminedforpragmaticsolutionstothese problems.Thestateoftheartresearch isstudiedtoevaluatethefeasibility,portabilityandsca labilityoftheexistingsolutionmethodologies 7

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fornextgenerationtechnologiesandemergingresearchdis ciplines.Thegeneralizedmulti-metric optimizationframeworksbasedupontheutilitarianmethod saredevelopedtosolvetheseproblems. Thethemeofthisdissertationandthemajorcontributionsa resummarizedinFigure1.1.Ashort descriptionoftheresearchworksthatcontributedtothedi ssertationisasfollows. Figure1.1Scopeandcontributionsofthedissertation.The themeofthedissertationistoidentifyand developnewmulti-metricoptimizationmethodsforVLSIcir cuitoptimizationandspatialdataand patternclustering. ExpectedUtilityBasedOptimization:Multi-objectiveopt imizationofdelay,leakagepower,dynamicpowerandcrosstalknoiseinVLSIcircuitsisperforme dviagatesizingusingamethodologythatisbasedontheconceptsofexpectedutilitytheor yandconstraintriskminimization. Itidentiesadeterministicequivalentmodelofthestocha sticoptimizationproblemusingthe conceptsofboundedrationality.Themethodologyisvariat iondistributionindependent,and identiessolutionswithhighlevelsofutility,inthepres enceofscarceinformationaboutthe distributionoftheprocessvariations.Themethodiscapab leofaddressingtheimpactofprocessvariationsandrandomnessatseverallevels,bothinth eobjectivefunctionaswellasinthe 8

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constraints.Thisapproacheffectivelytriestominimizet heriskofviolationorfailureoftheconstraintsinthemodel,evaluatedandcontrolledbyanexpect edutilitymeasurethatismaximized toensurethataconstraintissatised.Thedeterministicm odelidentiedusingthisapproachis especiallyattractiveforoptimizationinlargescaleVLSI -CADproblems. IntegratedFrameworkforCircuitOptimization:Inthiswor k,anewvariationawaremultimetricgatesizingframeworkhasbeendeveloped,whichcanb eusedtoperformoptimizationof severalmetricslikedelay,leakagepower,dynamicpower,a ndcrosstalknoiseetc.Theproposed frameworkiscompletelyrecongurableandgeneralizedint ermsofitscapabilitytoincorporate newmetricsandselectivelyprioritizethemetricsdependi nguponthedesignrequirements,with minimalchangesinthemodel.Moreimportantly,anymathema ticalprogrammingapproach canbeutilizedwithinthisframework,tosolvetheoptimiza tionproblem.Theprocessvariation effectsareincorporatedasstochasticcomponentsinthede laymodel.Animportantaspectof theproposedframeworkistheidenticationoftheinter-re lationshipsbetweendynamicpower, leakagepower,andcrosstalknoiseintermsofgatesizes,an dmodelingtheminauniedmanner. AMicroeconomicApproachtoSpatialDataClustering:Anove lmulti-objectiveclusteringapproachthatisbasedontheconceptsofmicroeconomics,spec icallygametheory,hasbeen developedinthiswork.Thisapproachiscapableofsimultan eouslyoptimizingmultipleconictingobjectives.Themethodologyconsistsofthreecomp onents,aniterativehillclimbing basedpartitioningalgorithm,amulti-stepnormalformgam etheoreticformulation,andaNash equilibriumbasedsolutionmethodology.Thenormalformno n-cooperativegameconsistsof randomlyinitializedclustersasplayersthatcompetefort heallocationofresources(dataobjects).TheNashequilibriumbasedmethodologyevaluatesa solutionthatissociallyfairfor alltheplayers,andanymathematicalhillclimbingalgorit hmcanbeusedtoupdatetheclusters aftereachiterationofthegame. RobotTeamFormation:Therescuerobotteamsformationprob leminthemulti-emergency searchandrescueenvironmentsisapracticalapplicationo fthemicroeconomicspatialclustering algorithmbeingdeveloped.Intheseenvironments,robotsp erformingthesearchandrescue operationsintheeldarerequiredtobedividedintoteamss incethepowerdissipationininterrobotcommunicationandtherobottobasestationcommunica tionishigh,whiletherobotsare 9

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runningprimarilyrunningonbatteries,andeachemergency locationisrequiredtobeattended allthetime.Droppingoutofalltherobotsservicingalocal itywouldsignicantlyhamperthe rescueprocess.Thus,inthiswork,robotteamsarecreatedo nthebasisofclustercompaction anduniformpowerdistributionobjectivestoidentifydece ntralizedrobotteamswitheachrobot inateamclosesttoitscommunicationgateway,aswellaseac hteamisequallyrepresentedin termsofitsstrength(batterypower). 1.4OutlineofDissertation Theremainderofthisdissertationisorganizedinsixchapt ers.Chapter2describesthebackground andthestateoftheartresearchrelatedtotheproblemsbein gaddressedinthisdissertation.Specically,ashorttutorialoftheimportantconceptsinexpecte dutilitytheory,mathematicalprogramming, andgametheoryispresented.Also,thestateoftheartresea rchintheeldofvariationawarecircuit optimization,anddataandpatternclusteringisdescribed indetails.InChapter3,ariskaverseutilitarianapproachVLSIcircuitoptimizationunderscarcein formationabouttheprocessvariationsis presented.Thisisapostlayoutgatesizingapproachformul ti-metricoptimization.Here,theexpected utilitytheoreticmethodologyisappliedtoconvertthesto chasticoptimizationproblemtoadeterministicequivalentmodel.InChapter4,anintegratedframework isdevelopedformulti-metricoptimization ofdelay,leakagepower,dynamicpower,andcrosstalknoise consideringtheeffectofprocessvariationsinthenanoscaleVLSIcircuits.Thisgatesizingfram eworkiscompletelyrecongurableand generalizedtoincorporate,removeorprioritizethemetri cstobeoptimized.Chapter5denesthe problemofmulti-objectivespatialclusteringintheconte xtofnovelmulti-disciplinaryapplicationareas,anddevelopsanovelgametheoreticclusteringalgorit hm.Thedifferentcomponentsofthegame theoreticmodelingareexplainedindetailsandthesimulat ionsareperformedtoevaluatetheefcacy oftheproposedmethod.InChapter6,theproblemofrobottea msformationinthemulti-emergency searchandrescueenvironmentsisdescribed.Thegametheor eticclusteringalgorithmbeingdevelopedanddiscussedinChapter5isadaptedtosolvethisprobl embyformingteamsonthebasistwo optimizationobjectivesclustercompaction,anduniformp owerdistribution.Theconcludingremarks andthesuggestedfutureworkintermsofextensionstothepr oblemsaddressedinthisdissertation, andotherideasforfurtherrenementsaregiveninChapter7 10

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CHAPTER2 BACKGROUNDANDRELATEDWORK Inthischapter,wepresentabriefintroductionofthevario usconceptsthatformthebasisforthe researchdescribedinthisdissertation.Specically,wed iscusstheexpectedutilitytheoreticapproach andthegametheoryapproach.Theseapproachesareusedtoso lvethemulti-metricoptimization problemsinthecontextofVLSIcircuitdesignandspatialpa tternclusteringproblems.Theutility theoryisusedforsolvingtheVLSIcircuitdesignoptimizat ionproblem,specically,gatesizing. Gametheoryisappliedinsolvingthemulti-objectivepatte rnclusteringproblem.Since,stochastic andmathematicalprogrammingmethodsareusedinthepropos edsolution,somebackgroundonthese topicsisprovided.WebrieyintroducethevariousVLSIcir cuitoptimizationtechniquesavailablein theliterature,andpresentissueofVLSImanufacturingpro cessvariationseffectsinthenanometer regime.Adetaileddiscussionofthevariousrelatedworksf orVLSIcircuitoptimizationandspatial patternclusteringisalsopresentedinthischapter.2.1UtilityTheory Autilitariantheoryformstheethicalframeworkforeffect ivemoralaction.Inthisframework, themeasureofsatisfactionisquantiedintermsoftheutil ityofthesatisfaction,andisattemptedto bemaximizedbyanindividual.Theutilityisoftenmeasured asthehappiness,asthesatisfactionof preferences,orthepreferenceutilitarianism.Thephilos ophybehindtheutilitytheoryistoachievethe greatestgoodforthegreatestnumber.Utilitytheoryhasbe enusedasaframeworktoargueforthe valueofdifferentactions.Twoprimaryvariantsoftheutil itytheoryintermsoftheexpectedutility optimizationexistintheliterature.Intherstform,thei ndividuals,alsoknownastheagents,try toformulateandactunderguidanceofrulesthatmaximizeth eutilityiftheyweretobeconsistently followed.Alternatively,inthesecondvariant,thegoalis tominimizenegativeutilityratherthan maximizingthepositiveutility. 11

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Theutilityofanactionorastateofenvironmentmapsthesta teontoarealnumbertodescribe thedegreeofsatisfactionfromthestate[33].Thisnotiono ftheutilityhastwoimportantimplications intermsofthegoals.First,thescenarioswherethegoalsar ecompetingorconicting,theutility functionspeciestheappropriatetrade-off.Second,thes ituationswhereseveralgoalsarespecied, noneofwhichcanbeachievedwithcertainty,theutilityfun ctionmapsthelikelihoodofsuccessof eachgoalaccordingtotheweightedimportanceofthegoals. Theoverallutilitybasedsystemcanbe representedbyasimplediagramasshowninFigure2.1. Figure2.1Autilitybasedsystemfromtheperspectiveofara tionalagent.Dependinguponthesystem'scurrentstate,theagent'ssatisfactionfromthecurr entstate,andtheagent'saction,thesatisfactionoftheagentinthesystem'snextstateisidentied.The agentchoosesitsfutureactionsbasedon thechangeinitssatisfactionvalueduetoitsownactionint hepreviousstate. Onthebasisofthenumberofrationalagentsinteractingint hesystem,theutilitytheorycanbe categorizedasexpectedutilitytheoryandgametheory.Int heexpectedutilitytheory,thesystem assumesasingleagentplayingagameagainstthenature,whe reasingametheory,multipleagents interactwiththenatureandagainsteachotherinanautonom ousmanner. 12

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2.1.1ExpectedUtilityTheoryandRiskAversion Inanenvironmentwheretheagentsmaynothavecompletecont roloraccesstotheenvironmental variables,asituationofuncertaintywouldarise.Asanexa mple,inthesemiconductordevicesfabricatedwiththesub-100nmtechnologynodes,theenvironment alfactorsmayaffectthemanufacturing processsignicantly,therebycausinginconsistenciesin thefabricateddevices.TheCADengineers areunawareofthedegreeofdisparitybetweenthespecicat ionsandtheactualdesigns.Thisuncertaintychangesthewayinwhichanagent(ordesigner)makesd ecisions.Inthepresenceofuncertainty, theactionsoftheagentsshiftfromdeterministicactionst othepreferencesasafunctionoftheoutcome probabilitiesoftheactions.The expectedutilityfunction mapsthesepreferencestorealvalues. Anaction a ofanagent A intheexpectedutilityframeworkwouldhaveasetofpossibl eoutcomes (alsoknownasstates) O i ( a ) asaconsequenceofthataction.Theindex i rangesoverthesetof outcomes.Also,correspondingtoeachaction a ,theagent A assignsaprobability P ( O i ( a ) j Do ( a ) ; K ) toeachoutcome.Here, Do ( a ) isthepropositionthattheaction a resultstotheassociatedoutcome, giventheagent A 'sinformationorknowledge K oftheenvironment.The expectedutility ofanaction giventheknowledge K ofthesystemisgivenby EU ( a j K ) asshowninEquation(2.1) EU ( a j K )= i P ( O i ( a ) j Do ( a ) ; K ) U ( O i ( a )) (2.1) Here, U ( O i ( a )) correspondsthequantitativemeasureoftheutilityoftheo utcome O i fortheaction a Accordingtotheprincipleofmaximumexpectedutility,the rationalagentshouldchooseanactionthat maximizesitsexpectedutility EU .Thisnotionofutilityintermsofprobabilitiesandtheout comes wasproposedbyJohnVonNeumannandOskarMorgensterninthe ir1944book TheoryofGames andEconomicBehavior [32].Accordingtothistheory,ifanagentmaximizesautili tyfunctionthat correctlyreectstheperformancemeasurebywhichitsbeha viorisbeingjudged,thenitwillachieve thehighestpossibleperformanceforitself.2.1.1.1ExpectedUtility Theutilityfunctionmapsthestatestotherealnumbers.Hyp othetically,theutilityofastate couldbeanyrealnumberdependingupontheagent'schoice,a ndisanarbitraryfunction.However,in practice,thepreferencesoftheagentsfollowamoresystem aticapproach.Inasimpleeconomicsetup, 13

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theutilitycanbeconsideredasamonotonicpreferencefunc tionofthemonetaryvalues.Accordingto thisdenition,theutilityoftheactionmonotonicallyinc reasesasthewealthincreasesinagamblingor alotterytypeofsituation.However,theutilitymaynotbea linearfunctionoftheexpectedmonetary value.Thiscanbeexplainedwithasimpleexample.Suppose, inagameof'dealornodeal',youhave alreadywon$1,000,000.Atthisstage,thebankerasksyouif youwouldliketoopenonemorecase thatmayhave$3,000,000.Ifthecasehas$3,000,000,youwil lwinthewholeamount;otherwiseyou willgohomewithnomoneyatall.Insuchasituation,theexpe ctedmonetaryvalueofthegambleis 0 : 5 $0 + 0 : 5 $3 ; 000 ; 000 = $1 ; 500 ; 000.Thisvalueisgreaterthanyourcurrentearnings.Howev er, wouldyoubewillingtoplaysuchagamble?Thisisasubjectiv equestion,anditdependsuponseveral factors,includingyourcurrentnancialstatuswithoutth emilliondollars,theimprovementinthelife styleamilliondollarscanbring,andhowmuchyouvaluethea dditionaltwomillionsifyoualready haveamilliondollars.Thus,utilityisnotdirectlypropor tionaltotheexpectedmonetaryvalue. 2.1.1.2RiskAversion Riskaversionisintuitivelydenedassituationwhereanag ent,whenfacedwiththechoiceof comparablereturns,tendstochoosethelessriskyalternat ive[34].Inanexpectedutilityframework, thisconceptcanbeexplainedthroughtheconcavefunctiong raphshowninFigure2.2.Here, X is arandomvariablewhichcantakeontwovalues, x 1and x 2.Considering p betheprobabilitythat x 1happensand(1 p )betheprobabilitythat x 2happens.Theexpectedoutcome E ( x )= p x 1 + ( 1 p ) x 2isshownonthe X axisasaconvexcombinationof x 1and x 2.Consideringa u : beanelementaryconcaveutilityfunction,asshowninFigur e2.2,theexpectedutilityisgivenas E ( u )= p u ( x 1 )+( 1 p ) u ( x 2 ) denotedby B ,between A =( x 1 ; u ( x 1 )) and C =( z 2 ; u ( z 2 )) .Now, bycomparingpoints B and D inFigure2.2,itisidentiedthattheutilityofexpectedin come, u [ E ( x )] isgreaterthanexpectedutility E ( u ) ,givenby, u [ p x 1 +( 1 p ) x 2 ] > p u ( x 1 )+( 1 p ) u ( x 2 ) (2.2) Now,wecanconsiderthescenarioshowninFigure2.2astwolo tteriessuchthatonepays E ( x ) with certaintyandanotherpays x 1or x 2withprobabilities p and ( 1 p ) respectively.Accordingtothe VonNeumann-Morgensternutilitynotion,theutilityofthe rstlotterywouldbe U ( E ( x ))= u ( E ( x )) 14

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Figure2.2Afunctionrepresentingtherelationshipbetwee nexpectedutilityandriskaversion.Ina concaveutilityfunction,iftheaveragereturnsfortheage ntinsituationsofdeterministicdecisionsand probabilisticdecisionsarecomparable,theexpectedutil ityoftheriskaversedecisionsishigherthan theriskcentricdecisions.receivedwithcertaintyandtheutilityofthesecondlotter ywouldbe U ( x 1 ; x 2; p ; 1 p )= p u ( x 1 )+ ( 1 p ) u ( x 2 ) .Inthissituation,evenwhentheexpectedincomeinbothlot teriesissame,theobvious decisionforariskaverseagentwouldbe E ( x ) withcertainty. InaVLSIcircuitoptimizationproblemunderuncertainty,a similarsituationarises.Theoptimizationoftheperformanceobjectivescanbeimprovedbyin creasingtheriskoffailureofthetiming constraints,therebyresultinginanincreaseinunreliabi lityofthecircuit.Specically,inthisparadigm, themarginalutilitydeclinesmuchmorerapidlyascompared totheelementaryutilityfunctioncurve asshownintheFigure2.2.Thus,astricternotionof quadraticutilityfunction canbeusedinsuch scenarios.Thequadraticutilityfunction[32]isgivenas: u ( x )= a + b x g x 2 (2.3) where a b and g arethecoefcientsofabsoluteriskaversion,derivedtoev aluatetheutilityfunction. 15

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2.1.2GameTheoryandNashEquilibrium Gametheorycanbedenedasacollectionofmathematicalmod elsformulatedtostudythesituationsofconictandcooperationbetweenintelligentratio naldecision-makers.Gametheoryanalysis situationsinwhichtwoormoreindividualsmakedecisionst hatwillinuenceoneanother'swelfare. Thesedecisionmakers,alsoknownasthe players ,choosefromanitelistofalternativecoursesof actions,leadingtowelldenedoutcomesexpressedinterms ofnumericalpayoffsassociatedwiththe chosencourseofactionforeachdecisionmaker. Formally,moderngametheorybeganwiththepublicationoft heseminalbookbyVonNeumann andMorgensternin1944[35].In1951,JohnNashdescribedan equilibriumconcept[36]fornoncooperativegamesasacongurationofstrategiesthatensu resawin-winsituationforalldecision makers.Thisconceptofcooperationundernon-cooperative environmentswasphenomenal,andasa resultgametheoryhasbeensuccessfullyappliedextensive lyintheeldofeconomics,engineering [25][24][16],andseveralotherreallifesituationsofdec isionmakingunderuncertainty. Theimportantelementsofagamearecategorizedasplayers, strategies,strategysets,strategy combinations,payoffs,information,andequilibrium.The playersareasetofrationaldecisionmakers, eachhavingasetofstrategies S i = f s i g availablewiththem.Astrategy s i isarulethataplayer i usestochooseanactionateachinstanceofthegame.Corresp ondingtoeachstrategy,autilityis associated,whichisrepresentedasapayoffdenotedby P i ( s 1 ; ; s N ) that i triestomaximize.A strategycombinationisanorderedset s =( s 1 ; ; s N ) thatconsistsofonestrategyforeachof N players,andonesuchcombinationthatmaximizeseveryplay er'spayoffinthegameisidentiedas anequilibriumpoint. Theideabehindgametheorycanbeexplainedwiththeaidofan interestingandaclassicalexample of prisoners'dilemma .Considerasituationwherethepolicehasconvictedtwocom puterprogrammers RobinandDavidinacaseofcriticaldatatheftfromthedatab aseofthecompanythatemploysthem. Thepoliceisassuredthattheyareguilty,buttheycouldnot proveitsincetherearenowitnesses.So, thepoliceisdependentupontheconvicts'testimoniestoid entifywhoisguilty.Thepolicedecides tokeeptheminseparateroomsforinterrogation.Theconvic tsaregivenonlytwooptions,confess orrefuse.Thepolicehasdecidedtoassigndifferentpenalt iesfortheconvictsdependingupontheir 16

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independentresponses,aswellthecombinationoftherespo nsesofbothconvicts.Thepenaltiesfor differentscenariosareasfollows: Ifbothconvictsconfesstostealingthedata,thepunishmen tis5yearsofjailtermforeachof them. Ifoneprisonerconfessesandotherrefuses,thentheconfes sorisgiven1yearofjailtermforhis truthfulness,andtheonewhohasrefusedispenalizedfor10 yearsofjailterm. Ifbothconvictsrefusetoaccepttheirinvolvementintheth eft,thenbothofthemaresentenced for3yearsduetothelackofsufcientevidence. Now,thesituationbeforetheconvictsiscomplex,sincethe ycannotcommunicateanddecidewhat theyshouldbedoing.Also,eachofthemisafraidoftheother 'spositionorstandpoint.Thissituation canbemodeledasamatrixgameasshowninFigure2.3. Figure2.3Asimpleexampleoftwoplayernon-cooperativeno rmalformgame.Theprisoners' dilemmaintermsofthestrategies(confess,refuse)andthe differentpayoffs(1year,3years,5years or10years)areshown. Inthisexample,thetwoconvicts,DavidandRobin,arethepl ayersofthegame.Eachplayer hastwostrategies,confessandrefuse.Theelementsofthem atrixgamearethepayoffsorutilities associatedwiththestrategieschosenbytheplayers.Forex ample,ifRobinchooseshisstrategyof 17

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refusingtobeinvolvedinthetheft,hispunishmentwilldep enduponthestrategychosenbyDavid.If Davidalsorefuses,thenRobinwillget3yearsofjailterm,w hereasifDavidacceptstheirinvolvement inthetheft,thenRobinwillbesentencedfor10yearsinjail .Theinformationavailablewitheach playeristhestrategiesavailablewiththeotherplayer.As trategycombinationisatupleconsisting ofonestrategycorrespondingtoeachplayerinthegame.One suchstrategycombinationistheset (confess,confess). Thesolutionofagamemodelisidentiedusinganequilibriu mtechnique.Nashequilibrium[36] isonesuchtechniquethathasbeenwidelyusedtosolvethega metheoreticformulations.Nash equilibriuminanon-cooperativegamesettingisidentied asapoint(orstrategycombination)at whichnoplayercanimproveitsutilitybydeviatingfromtha tpoint,consideringtheotherplayersdo notdeviatefromthatpoint.ANashequilibriumintheprison ers'dilemmagamecanbeexplainedwith theaidoftheFigure2.4(a)–2.4(c). AsshowninFigure2.4(a),Robin,ifrefusestotestifythath ewasinvolvedintheft,wouldreceive atermof10yearsintheworstcasescenario,and3yearsinthe bestcasescenario.However,ifhe confesseshisinvolvementinthetheft,wouldserveatermof 5yearsintheworstcase,and1yearin thebestcasescenario.Thus,toconfesshisinvolvementist heobviousdominantstrategyforhim.This isshownastheyellowshadedregioninthegure.ConvictDav idhasthesimilarsituationasshown inthepayoffmatrixinFigure2.4(b).Withthesimilarsetof arguments,itislogicalforDavidtoalso confesshisinvolvementinthetheft. Now,ifwetakethedominantstrategiesofboththeplayers,t henalequilibriumstrategyisidentied,asshowninFigure2.4(c).Here,thepurpleshadedregio ndenotestheintersectionofthedominant strategiesofthetwoplayers.Thispointispreciselycalle dtheequilibriumpoint,andthestrategycombination(confess,confess)istheNashequilibriumstrate gy.Atthisstrategypoint,ifDavidtriesto changeitsstrategyfromconfesstorefuse,whileRobinmain taininghispositionofconfession,David willonlylooseandwillgetmoreyearsinthejailterm.Simil arsituationoccurswhenRobintries tochangehisstrategyunilaterally.Thus,attheNashequil ibriumpointeachplayerissatisedwith respecttoeveryotherplayerinthegame. 18

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(a)StrategiesandPayoffsofRobin (b)StrategiesandPayoffsofDavid (c)NashEquilibriumStrategyCombination Figure2.4Generationofstrategysets,identicationofth edominantstrategies,andNashequilibrium inprisoners'dilemmagame.In(a)and(b)therespectivestr ategiesandthepayoffsforRobinand Davidareshown.TheNashequilibriumstrategyonthebasiso fthedominantstrategiesforeach playerisshownintherightbottomboxof(c).2.1.2.1ClassicationofGames Gamescanbeclassiedonthebasisofseveraldifferentcrit eria.Someoftheimportantclassicationsofthegamesaregivenasfollows. Numberofplayers-2-player(prisoners'dilemma),N-playe r(nite),andinniteplayergames Numberofmovesandchoices-nitestrategysetandinnites trategies Degreeofopposinginterests-zero-sumgamesandgeneral-s umgames Degreeofcooperation-cooperativegamesandnon-cooperat ivegames Numberofstages-one-shotgamesandrepeatedgames 19

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Timedependence-staticgamesanddynamicgames Involvementofprobability-deterministicgamesandstoch asticgames Here,wewilldiscusssomeoftheseclassications,specic allytheonesofourinterestintermsof solvingthemulti-metricoptimizationproblems.Togetdet ailedinformationontheotherclassication criteria,thereadersmayreferto[27,28].Inaclassicati onbasedonthedegreeofcooperation, thenon-cooperativegamesconsistofrationalplayerschoo singtheirstrategiesindependently,with nominalinformationofthestrategiesavailablewiththeot herplayers.Eachplayerplaysastrategy thatisitsbestresponsetothestrategycombinationoftheo therplayers.Unlikecooperativegames, thecoordinationamongtheplayersisnotforcedexternally ,butisself-enforcing.Inmulti-player situations,whereexternalcommunicationforcooperation iscomplex,andhenceimpractical,thenoncooperativegamesarepragmatic.Non-cooperativegamesca nbefurtherclassiedasnormalform orstrategicgames,andextensiveformgames.Inthenormalf ormgames,playerssimultaneously choosetheirstrategiesandastrategycombinationthatgiv esthebestpossiblepayoffstoeveryplayer isconsideredasanequilibriumpoint.Whereas,intheexten siveformgames,theplayersmoveina sequentialorder,andtheorderofplayaffectsthenaloutc omeofthegame.Sincealltheplayers maketheirmovessimultaneouslyinanormalformgame,theyd onotgettolearneachother'sprivate information.2.1.2.2MathematicalRepresentation Anon-cooperativenormalformgameisanitegameifthestra tegysets S 1 ; ; S N arenite.Here, N isthesetofallplayersinthegame,and S i isthesetofallthestrategiesofplayer i .Thegameis beingrepresentedas: G =( S i ; p i ) ; 8 i 2 N (2.4) Here, p i representsthepayofffunctionforplayer i ,andisgivenas: p i = i N S i (2.5) Forthegame G representedbyEquations(2.4)and(2.5),the N -tupleofstrategies s 1 ; ; s N where s 1 2 S 1 ; ; s N 2 S N ,isdenedastheNashequilibriumpointof G ifEquation(2.6)satises 8 s i 2 S i 20

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and i = 1 ; ; N p i ( s 1 ; ; s i ; ; s N ) p i ( s 1 ; ; s i 1 ; s i ; s i + 1 ; s N ) (2.6) Qualitatively,Nashequilibriumisasociallyfair,goodqu alitysolutionpointatwhicheveryplayeris satisedwithrespecttoeveryotherplayer.2.1.2.3CritiqueofGameTheory Althoughgametheoryhasbeenwidelystudiedandappliedins everalimportantapplicationareas, itisoftencriticizedforsomeofitsproperties.Theprimar ycritiquesofgametheoryare: Whynon-cooperativegames?:Iftheprisoners'dilemmagame beingpresentedaboveisrevisited,anaturalquestionthatarisesistowhynotplayacoope rativegame?Thisisintuitive,since insuchascenario,playersmaycomeoutwithamoreadvantage ousstrategycombinationof (refuse,refuse).Theissuewithcooperativegamesisthati nsuchgamestheplayersneedto makepriorcommitmentsforcooperation.Theprisoners'dil emmagame(representedby c per say)canbetransformedintoacooperativegamebyamapping x ,suchthat x ( c ) isanothergame thatrepresentsthesituationexistingwhere,inadditiont othestrategysetsspeciedin c ,each playerwouldhavesomewiderangeofoptionsforbargainingw iththeotherplayerstojointly plancooperativestrategies.Insuchsituationsthestrate gysetofeachplayerwouldexplode andthegamewouldpotentiallybecomeinconceivable.Anoth erreasonfornotconsideringa cooperativegamesolutionistherequirementofimpartiala rbitratorincooperativegames,who couldperformpre-playcommunicationwithalltheplayersb eforehand.Insuchsituations,a considerableamountoftimeisrequiredforsucharbitratio ns,whichisnotpragmaticinsolving realengineeringproblems. MultipleNashequilibriumsandparetooptimality:TheNash equilibriumforagametheoretic modelconsistsofallthedominantstrategies.However,the remaybemultipleNashequilibriums inagame,anditispossiblethatseveralNashequilibriumsm aynotbe paretooptimal [28].A solutionisparetooptimal,ifthereexistnoothersolution thatcanmakeat-leastoneindividual betteroffwithoutmakinganyotherindividualinthesystem worseoff.Agoodexamplefor suchasituationisthePrisoners'dilemmasituation.Here, thedominantstrategyandtheNash 21

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equilibriumpointisthecombinationwhereboththeprisone rsconfesstheircrimes,whichis reasonablefromtheplayers'aswellasthesystem'sperspec tive,consideringthattheplayersare rationalandnon-cooperative.Asevident,thesolutionisn otparetooptimal.Theparetooptimal solutionpointis(refuse,refuse).However,theparetoopt imalitywouldrequirecooperation amongtheplayers,existenceoffocalarbitrator,andacoal itionformation,whichisinfeasible. Itisimportanttonotethatthecriterionofparetooptimali tydoesnotensurethatasolution isbyanysenseequitableandsociallyfair,whichisanimpor tantcriterioninmulti-objective optimization. 2.2MathematicalProgramming Amathematicalprogrammingproblemisanoptimizationprob lem,whereinoneseekstominimize ormaximizearealvaluedfunctionofrealorintegervariabl es,subjecttoconstraintsonthevariables. Mathematicalprogrammingstudiesthefollowingpropertie sofanoptimizationproblem: Themathematicalpropertiesoftheoptimizationproblem. Thedevelopmentandimplementationofthealgorithmstosol vetheoptimizationproblems. Theapplicationofthesealgorithmstorealworldproblems. Themathematicalprogrammingisprimarilyperformedtosol vetwotypesofproblems,continuous anddiscrete.Thecontinuousoptimizationproblemscouldb econstrainedorunconstrained.Tosolve theunconstrainedoptimizationproblems,severalmethods likenon-linearprogramming,non-linear leastsquareoptimizationmethods,non-differentiableop timizationmethodsandotherglobaloptimizationmethodsareapplied.Theconstrainedoptimizatio nproblemscouldbelinear,stochastic, non-linearlyconstrainedorboundconstrained.Severalal gorithmshavebeendevelopedtosolvesuch problems[37].Thedeterministicdiscreteoptimizationpr oblemsaresolvedusingintegerprogrammingmethods.Thestochasticoptimizationproblems,which couldbediscreteorcontinuousproblems, arehardertosolve,sincetheyinvolveuncertainty. Stochasticprogramming isaframeworkformodelingoptimizationproblemsthatinvo lveuncertainty.Stochasticprogrammingmethodstakeadvantageoft hefactthatprobabilitydistributionsgoverningthedataareknownorcanbeestimated.Thegoalhereis tondsomepolicythatisfeasiblefor 22

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all(oralmostall)thepossibledatainstancesandmaximize stheexpectationofsomefunctionofthe decisionsandtherandomvariables.Moregenerally,suchmo delsareformulated,solvedanalytically ornumerically,andanalyzedinordertoprovideusefulinfo rmationtoadecision-maker. Stochasticprogrammingisawidelystudiedandappliedopti mizationproblemtotherealworld problemssinceanyrealworldproblemalmostinvariablyinc ludessomeunknownparameters.Severalalgorithmsandsolutionmethodologieshavebeendevel opedtosolvethestochasticoptimization problems.Chanceconstrainedprogramming,twostagelinea rprogramming,multi-stagelinearprogramming,fuzzymathematicalprogrammingandgeometricpr ogrammingareafewstateoftheart methodstosolvestochasticoptimizationproblems.Ingene raltermstheoptimizationmethodsinthis disciplinecombinethepowerofmathematicalprogrammingw ithadvancedprobabilitytechniques,to attackoptimizationproblemsthatinvolveuncertainty.Ac onstraintorpresumptioninthesemethods isthattheprobabilitydistributionsoftherandomparamet ersareknown,andcannotdependonthe decisionstaken.2.3VLSICircuitOptimization Inthenanometerera,theperformanceofaVLSIcircuitisnot onlydeterminedbythethedelayor thefrequencyofthecircuitalone.Thereliability,scalab ility,powerdissipation,energytoperforma function,cost,yieldandthetime-to-marketthechipsarea lsoimportantperformancemetrics.Theoptimizationofthesemetricsisthusanessentialpartofdesi gningrobust,reliableandhighperformance circuits.Thepersistentpushforhigherperformanceandre liabilityinmuchmorecomplexdesignshas ledtoanincreasinginterestintheoptimizationtechnique s.Circuitoptimizationprimarilyinvolves tuningofvariouscomponentsofacircuittoachievedesired changesintheperformancemetrics.The componentsthatcanbetuned,includetransistors,wires,b uffers,powersupplyvoltage,andthresholdvoltageetc.[38].Inadditiontothesecontinuoustunin gtechniques,variousdiscreteoptimization methodslikebufferinsertion,reorderingofinputpins,an dchoiceofgatesfromdiscretelibrariesetc. arealsowidelystudiedinliterature.Since,inthisdisser tationmulti-metricoptimizationofdelay, powerandcrosstalknoiseisbeingperformed,themethodsth atareeffectivefortheoptimizationof thesemetricsarereviewed.Itisimportanttonotethatthef rameworkforVLSIoptimizationbeingde23

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velopedinthisresearchisindependentofthemetricsthatc anbeincorporatedforoptimization.Other performancemetricscanbeaddedinthemodelwithminimalef fort. Inacircuit,themaximumdelayisdenedasthetotaldelayof thelongestpath( criticalpath )in thedesign.Someoftheprominenttechniquesfordelayminim izationincludegatesizing,transistor ordering,deningalternativelogicstructures,bufferin sertion,reducingthevoltageswingofthegates, andinterconnectwiresizing[39,40].Inagatesizingtechn ique,thesizesofthegatesinthepathare adjustedtominimizethedelayofthepath.Thesizesofthega tesintheentirecircuitorasub-circuitare adjustedproperlyaccordingtotheircapacitiveloadsforp erformanceimprovement.Inthetransistor orderingtechnique,thetransistorsareorderedinarowand orientedinsuchawaythatthesharing ofsourceanddrainregionsismaximized.Thisaidsinreduci ngthetotaldiffusionareaandthecell widths.Delayofacircuitcanalsobereducedbycarefullyre placinglogicstructuresinacircuit. Forexample,afunctionlike F = ABCDEFGH beingimplementedusing5twoinputNANDgates,2 twoinputNORgatesandaNOTgatecanbereplacedbyaeightinp utNANDgateandaNOTgate. Anothereffectivetechniquefordelayoptimizationistoin sertbuffersinordertoisolatethefan-in fromthefan-out,therebyreducingtheloadonthecriticalp athofthecircuit.Inthebufferinsertion technique,aseriesofcascadedinvertersareinsertedonin terconnectsbetweenthegates.Inawire sizingtechnique,thewidthsoftheinterconnectwiresares izedtoreducetheinterconnectdelays.The techniqueslikemulti-VDDassignmentandthresholdvoltag escalinghavealsobeenappliedfordelay minimization. PowerdissipationinVLSIcircuitsisprimarilyduetotwoco mponents;staticpower,anddynamic power[41].Thedynamicpowerdissipationisduetotwosourc es,switchingpowerduetocharging anddischargingofloadcapacitance,andshortcircuitpowe rduetonon-zeroriseandfalltimesof inputwaveforms.Thestaticpowerorleakagepowerdissipat ionoccurswhenthedeviceisnotactive. Thethreecomponentsofleakagepoweraresub-thresholdlea kageduetocurrentfromdraintosource, directtunnelinggateleakageduetotunnelingofelectrons orholesfromthebulksiliconthroughthe gateoxidepotentialbarrierintothegate,sourceanddrain orsubstrateandsubstratereversebiasedp-n junctionleakage.Severaltechniqueshavebeenproposedto reducethesecomponentsofpowerdissipation.Fordynamicpowerreduction,gatesizing,interc onnectsizing,clockgating,supplyvoltage scalingandbufferinsertionareprimarytechniques. 24

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Dynamicpowerofthecircuitisminimizedbysizingdowntheg atesinthecircuit.However,such sizingtechniqueincreasesthedelayofthecircuit.Inorde rtooptimizebothdelayanddynamicpower, apathbasedtechniquecanbeapplied,wheregatesinthecrit icalpathsaresized-upandthegates inthenon-criticalpathsaresizeddown.Alternatively,ag lobaloptimizationcanbeperformedwith delay-powertrade-off.Wiresizingtechniquefollowsasim ilarrelationship.Ifthewidthofthewireis increased,theresistanceperunitlengthofthewiredecrea ses.However,thelinecapacitanceincreases, consequentlyincreasingtheinterconnectpower.Inaclock gatingscheme,theclockismaskedsuch thattheswitchingactivityoftheidleblocksofthecircuit isminimized,therebyreducingdynamic powerdissipation.Thistechniquealsoreducestheclockpo werdissipation.Supplyvoltagescaling minimizestheswitchingpowerdissipation.Sincesupplyvo ltagehasaquadraticdependencyonthe switchingpower,thetechniqueiseffective.Inthismethod ,eitherthesupplyvoltageofthenon-critical partofthecircuitcanbeloweredinastaticmanner,orthesu pplyvoltagecanbedynamicallylowered dependingupontheperformancedemandofthecircuit. Leakagepowerminimizationatthecircuitlevelcanbeperfo rmedbyapplyingtechniqueslikegate sizing,thresholdvoltagescaling,transistorstackingan dadaptivebodybiasing.Sincethegatesizeis directlyproportionaltotheaverageleakagepowerofthega te,sizingthegatereducestheleakage powerofthecircuit.Assignmentofhighthresholdvoltaget osometransistorsinthenon-criticalpaths canreducethesub-thresholdleakage.Thetransistorstack ingmethodinsertsextratransistors(sleep transistors)connectedintheserieswiththepull-up/pull -downpathofthegatesandturnsthem'off' duringthestandbymode.Inadaptivebodybiasing,theforwa rdbodybias(FBB)andthereversebody bias(RBB)isappliedtovarythethresholdvoltageofthetra nsistors,therebyturningthemoffduring thepassivemode. Thecouplingofaquietlinewithoneormoreswitchinglinesi nducesnoiseonthequietline.Ifthe noiseishigh,thelogicofthequietlinemayswitchcausingl ogicfailures.Thiscrosstalknoisecanbe reducedbyapplyingmethodslikewiresizing,wirespacing, wireshielding,sizingofthedrivergates ofthevictimandaggressorinterconnects,andsizingofthe receivergatesofthevictimandaggressor nets[42,43].Ifawireissizedup,theresistanceofthewire increases,therebyreducingthecoupling effectonit.Alternatively,ifthecoupledwiresarespaced farther,thecouplingcapacitancebetween themreduces,consequentlyreducingthenoiseoneachofthe m.Inthedrivergatesizing,ifthevictim net'sdrivergateissizedup,thesignalstrengthonthevict imnetincreases,resultinginadecreasein 25

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thecouplingnoiseonitself.Theimpactiscomplementary,s inceanincreaseinsignalstrengthonthe netinduceshighercouplingnoiseontheneighboringnets.S imilarly,up-sizingthereceivergateofa victimnetreducesthenoiseonthenet.However,theeffecto freceiversizingissignicantlysmaller ascomparedtothedriversizing. SeveralstateofthearttechniquesforVLSIoptimizationbe ingdiscussedhereareeffective,and havebeensuccessfullyappliedforoptimizationofeitherd elay,powerorcrosstalknoise.However, amongthesetechniques, gatesizing isparticularlyinterestingduetoseveralreasons.Gatesi zing isasimple,generalpurposepost-layoutoptimizationappr oachthatcanbeutilizedtooptimizeall theimportantmetricslikedelay,power,andcrosstalknois e.Itdoesnotrequiretheincorporation ofanyadditionalcircuitryinthedesign,andhenceincursm inimumoverhead.Gatesizingatthe post-layoutleveldoesnotrequireanycircuitre-routingt obeperformed.Also,drivergatesizingis themosteffectivetechniqueforcrosstalknoiseoptimizat ion[43].Thus,weutilizegatesizingasthe optimizationmethodologyformulti-metricVLSIcircuitop timization,consideringprocessvariations. 2.3.1ProcessVariations Theaggressivescalingofdevicesandinterconnects,theli mitationsofthemanufacturingprocesses,andtheenvironmentalnoiseaffectingthemanufact uringprocesses,havesignicantlyaffected theVLSIdesignparadigm,resultinginatransitionofthede signandoptimizationprocessfromthe deterministictotheprobabilisticdomain[5].Sucheffect sdegradethequalityofthesignalsandaffectthereliabilityofthemanufacturedcircuits.Thesepr ocessvariationsoccurprimarilyduetotwo factors. EnvironmentalFactors:Thisincludesthevariationsinthe processingduetothevariationsinenvironmentalfactorsliketemperature,powersupplyvoltag e,humidity,pressure,electromagnetic interference,cosmicraysetc. PhysicalFactors:Theseincludethevariationsintheelect ricalandthegeometricalparameters causedduetoimperfectionsinprocessingtechnologieslik ephotolithography,planarization, metaletching,polysiliconetchingetc. Thephysicalfactorscanbefurtherclassiedasdie-to-die physicalvariationsandwithin-diephysical variations.Thedie-to-diephysicalvariationscausethei nconsistenciesbetweenthedifferentdies,but 26

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arelargelyuniformwithineachdie.Duetowhich,thesevari ationsarelargelyindependentofthe designimplementationandareusuallymodeledusingworstcasedesigncorners.Thewithin-dievariationsarethevariationsinthedeviceparameterswithinas inglechip.Duetothesevariations,different devicesatdifferentlocationsonasinglediemayhavediffe rentdevicefeatures.Thevariationsingate dimensionswithinadieareanexampleofwithin-dievariati ons. Thewithin-dievariationsarecausedduetothreetypesofde fects. Randomdefects:Thedefectsthatarecausedduetointroduct ionofforeignparticlesinthewafer duringtheprocessing.Thesedefectscanbeintroducedduri nganystepinthemanufacturing process,andcanresultincreationofopensorshortsinthem anufacturedcircuits. Systematicdefects:Thesedefectsoccurduetosub-wavelen gthlithographyprocess,andcanbe controlledbyincorporatingtightercontrolduringthepro cessing,andbyapplyingtechniques likeopticalpatterncorrection. Parametricdefects:Suchdefectsoccurduetovariationsin themanufacturingprocess.As theprocesstechnologyscalesdown,withthescalingofthed eviceparameterslikegateoxide thickness,gatelength,interconnectspacingetc.,theimp actofparametricvariationsincreases rapidly.Therelativeimpactofthesedefectsfordifferent technologynodesisshowninFigure 2.5. Anotheraspectofintra-dievariationsisthatthesevariat ionsexhibit spatialcorrelations ,where thedevicesthatareclosetoeachotherhaveahigherprobabi lityofhavingsimilardeviceproperties thanthosewhichareplacedfarapart.Whencoupledwiththep rocessvariations,thesecorrelations cancauseprimereliabilityconcerns.Hence,itisessentia lforthedesigntoolstoaccountforthe uncertainties,anddesignrobustcircuitsthatareinsensi tivetotheprocessvariations. 2.4VariationAwareGateSizing Severalapproachesfortheoptimizationofdelay,leakagep ower,dynamicpower,andcrosstalk noiseinthepresenceofdeviceprocessvariationshavebeen proposedinrecentyears.Inthissection, wediscussthestateoftheartstatisticalstatictimingana lysis(SSTA)basedandmathematicalprogrammingbasedapproachesforvariationawaregatesizing, onthebasisoftheirstrengths,aswellas 27

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Figure2.5Variationimpactatdifferenttechnologynodes[ 1].Astheprocesstechnologyismoving towardlowertechnologynodes,theparametricvariationsa rebecomingadominantfactorindeterminingthetotalimpactofprocessvariations.limitationsatthecurrenttechnologynodes.Theanalysisa ndthenextgenerationVLSIdesignchallengesmakeastrongcase,foridentifyingnewmethodsformu lti-metriccircuitoptimizationofthe VLSIdesignproblems.2.4.1OptimizationMetrics Toanalyzeandoptimizemetricslikedelay,power,yield,cr osstalknoiseetc.inthepresence ofprocessvariations,severalmethodshavebeenproposedi ntheliterature.Since,thisproblemis addressedfromagatesizingperspective,thediscussionis restrictedtoreviewonlythevariationaware gatesizingmethods.Gatesizingisasimpleyeteffectivete chniqueforcircuitoptimizationatthepostlayoutlevel,where-intheobjectiveistoidentifytheopti maldrivestrengthofeachgateinthedesign. InFigure2.6,ataxonomyoftherecentworksingatesizing,c lassiedaccordingtotheoptimization metricsandthemethodologiesispresented. PowerOptimization:Severalworkscanbefoundinthelitera tureonpoweroptimizationwith gatesizing,suchasminimizingleakagepower[44–46],dyna micpower[47,48],andtotalpower 28

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Figure2.6Taxonomyofthevariationawaregatesizingworks .Thevariousworksonthevariationawaregatesizingarecla ssiedonthebasisof optimizationmetricsandoptimizationmethods.29

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[49–52].In[47],adynamicpowerminimizationmethodispro posedwithdynamicpower identiedasafunctionofthegatesizesinastochasticprog rammingmodel.Similarly,in[48], theauthorshaveproposedafuzzymathematicalprogramming basedsolutionfordynamicpower optimization.Leakagepowerminimizationunderprocessva riationsisperformedusingSSTA basedmethods[44,45],inwhichcontinuousdistributionsa repropagatedthroughthepaths insteadofthedeterministicvaluestondtheclosedformex pressionsforperformance.In[46], amethodtoestimatetheleakagecurrentvariationduetoint er-dieandintra-diegatelength variationsispresented. CrosstalkNoiseOptimization:Thepoweroptimizationmeth odsareprimarilysinglemetric modelsthatdonotconsidertheeffectofgatesizingonother metricssuchascrosstalknoise ofthecircuits.Atthepost-layoutlevel,interconnectcou plingeffectscanworsenthesignal strength,leadingtologicfailures.Severaltechniquesto reducecrosstalknoisehavebeenpresentedintherecentyears.In[42],theauthorsproposealin earprogrammingbasedformulation fortransistorsizingtominimizecrosstalknoiseincircui ts.Inanotherapproach[53,54],anyield drivenLagrangianRelaxationbasedmethodidentiestheup per-boundonnoiseforeachnetas anoiseconstraint.Thegatesareiterativelysized-uptosa tisfythetimingandnoiseconstraints, andasimplelinearmodelisevaluatedforcrosstalknoisemi nimization.Inarecentwork[55], astochasticgametheoreticalgorithmforpostlayoutdelay uncertaintyandcrosstalknoiseoptimizationconsideringspatialcorrelations[56,57]ispr oposed.Thenon-linearcrosstalknoise modelusedinthismethodisderivedfrom[58],whichaccurat elyidentiesacloserapproximationofthecrosstalknoise. DelayOptimization:Additionally,theoptimizationofoth erimportantmetricslikedelay,timing yieldandbinningyieldhavealsobeendiscussedwidely[59– 61].However,thisresearchis largelyone-dimensionalinthesensethatthesemethodstyp icallyaimatoptimizingspecic metricsandoftendonotconsiderthefactthatoptimizingon emetricmaynegativelyimpactthe optimizationofothermetrics,leadingtoaninaccurateana lysisofthecompletedesign. 30

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2.4.2OptimizationMethods Several SSTAbasedapproaches havesuccessfullybeenappliedfordelayminimizationoryi eld improvementproblems[57,62–64].Theseapproachintuitiv elymodeldelayandyieldoptimization problemsinasimplemodel.TheSSTAbasedapproaches,impro veoverthepessimisticworst-case cornerbasedmodeling[61]byperformingamean-variancean alysisforthetotalcircuitdelay.However,suchapproachesareessentiallypathbased[65],andt raditionallyappliedtooptimizeasingle parameter.AnassumptioninaSSTAbasedtechniqueisthatth ecompleteinformationaboutthevariationdistributionofthedesignparametersisknown,andth emethodologyisbasedonsuchassumptions.Severalworks[6,7]haveassumedaGaussiandistribu tion.However,globalsourcesofvariation followalog-normaldistributionmoreclosely[8,9]ascomp aredtotheGaussiandistribution. Mathematicalprogrammingbasedapproaches havebeenwidelyinvestigatedintheliteraturefor optimizingseveralmetrics.Animportantaspectofmathema ticalprogrammingapproachesforcircuit optimizationisthatanypathbasedproblemcanbeeasilycon vertedtothenodebasedequivalentwith somesub-optimalitybeingintroduced.Ageometricprogram ming(GP)approachhasbeenproposed in[66]fordelayoptimizationinthepresenceofprocessvar iations.Although,theapproachisrobust, theobjectivefunctionandtheconstraintsarerequiredtob eposynomialfunctions.Thus,modelinga generalizedoptimizationprobleminaGPframeworkrequire sconvertingeachoptimizationfunction andtheconstraintsinaposynomialform,andtheproblemcan onlybemodeledforminimizationof objectives. Inanotherapproachfordynamicpowerminimizationunderde layconstraints[47],theproblemis modeledasachanceconstrainedstochasticprogram(CCP).A lthoughCCPtechniquescantransform simpleproblemstotheirdeterministicequivalentmodels, thetransformationisextremelydifcultfor largescaleproblems.Also,themethodisboundedbycontinu ousdistributions,andrequiresanumber ofoperationstobeperformediterativelyateachnode,thus involvinghigherruntimes.However,if thevariationdistributioninformationisavailable,them ethodologycanbemodiedtoincorporate multiplemetricsforoptimization.Alternatively,thesto chasticprogrammingbasedstatisticaloptimizationtechniquesarereasonablyfast,butmoreconserva tiveintermsofyield,andhenceprovide lessersavingsintermsofobjectivefunctionoptimization s.Inarecentwork[48],thedynamicpower optimizationproblemconsideringprocessvariationshasb eenmodeledinafuzzyoptimizationframe31

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work.Here,thestochasticparametersaremodeledasfuzzyn umbers,andacrispnon-linearproblemis formulatedtomaximizethevariationresistance(toleranc e)ofthecircuit.Theproblemisthensolved usingcommerciallyavailableoptimizationsolvers.These methodologiestypicallyaimatoptimizing specicmetricsandoftendonotconsiderthefactthatoptim izingonemetriccannegativelyimpact theoptimizationofothermetrics,leadingtoaninaccurate analysisofthecompletedesign.TheLagrangianrelaxationbasedmethods[53]arelimitedtoeithe rup-sizing,ordown-sizingthegatesfor theoptimization. Ashortcomingintheproposedmethodsforgatesizingconsid eringprocessvariationsarisefrom thefactthatseveralmethods[43,47,48]incorporatetheef fectof processvariations duetoonlyone designparameter,likegatesizes(duetochannellength,an doxidethickness).Theimpactofinterconnectvariations,whichcancause12-25%variationsinth etimingofthecircuit,dependingupon thedesignandimplementation[67,68]cannotbeignoredatt hedeepsub-nanometerlevel.Theprocessvariationscanbemodeledmoreaccuratelyusingcomple xandnon-linearmodelsthatincorporate moreparameters,andhavehigheraccuracy[55].Thedisadva ntageofsuchamodelingliesinthe implementationcomplexity.2.5SpatialDataClustering Spatialdataclusteringinvolvesthegroupingofobjectsin toasetofsub-groupsinsuchamanner thatthesimilaritymeasurebetweenthedataobjectswithin asub-groupishigherthanthesimilarity measurebetweenthedataobjectsfromdifferentsub-groups .Theobjectanddataclusteringtechniques ndapplicationsinawidevarietyofscienticdisciplines suchasbiology,computervisionandpattern recognition,communicationsandcomputernetworks,andin formationsystems.Asaresult,cluster analysishasreceivedsignicantattention,andseveralcu stomizedclusteringmethodologieshavebeen developedtosatisfyspecicapplicationrequirements[11 ,12]. 2.5.1ClusteringTechniques Objectclusteringisawellresearchedproblemreportedext ensivelyintheliterature,including severaldetailedsurveypapers.Jain etal. [11],andScheunders[69]reviewclusteringmethodsfrom patternrecognitionandimagequantizationviewpoint,whi leKolatch etal. [70],andBerkhin[71] 32

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identifymethodologiesfromthedataminingperspective.S imilarly,Murtagh[72],andBaraldi[73] surveyedvarioushierarchical,andfuzzyandneuralcluste ringalgorithmsrespectively.Foradetailed discussionandsurveyofdifferentsurveys,oneisreferred to[12]. Clusteringtechniquescanbeclassiedonthebasisofsever alcriteria,suchastheprinciples,type ofdata,shapeofclusters,formofnalpartitions,distanc emeasure,andthenumberofobjectives. Here,wewilllimitthediscussiontopartitioningofdatase tsonthebasisofclusteringobjectives.The threemajorgroupsofclusteringobjectivesarecompaction ,connectedness,andspatialseparation. Thecompactionobjectiveattemptstoidentifyclusterswit hminimumintra-clustervariation.The KMeansalgorithm[74]isthesimplestandthemostwidelymat hematicalmethodusedinthiscategory. Otheralgorithmsincludeaverage-linkagglomerativeclus tering[75]andmodelbasedapproaches[76]. Clusteringwithanobjectiveofmaximizationofconnectedn essensuresthatneighboringdataitems sharethesamecluster.Thedensity-basedmethods[77],and single-linkagglomerativeclustering methods[75]implementthisprincipletoidentifyclusters witharbitraryshapes.Inspatialseparation basedmethods,theobjectiveistomaximizetheinter-clust erseparation.However,itprovideslittle guidanceduringclusteringandmayproducetrivialresults .Additionally,animportantcriterionthat hasreceivedsignicantattentionrecentlyinthedomainof dataclusteringisequipartitioningorloadsharing[78].Load-sharingmethodologieshavebeenwidely researchedintheeldofdistributed systems[79,80],butdidnotreceivemuchattentioninclust eringdomainuntilrecently.Thenew applicationdomainslikead-hocnetworks[81,82]andemerg encyresourcedeploymentrequireclusters withalmostequalnumberofdataobjectsperclustertosatis fytheconstraints. Fromtheclusteringmethodologiesperspective,severalhe uristicsbasedtechniqueshavebeendevelopedinadditiontothemathematicalclusteringmethodo logies.Thisincludessimulatedannealing[83],evolutionaryalgorithms[84–86],tabusearch[87 ],andantcolonyoptimization[88].Also, hybridapproachesthatcombinedifferentalgorithmshaveb eenproposedinliterature[85][84].Such techniquesareprimarilyusedforfeatureselectioninunsu pervisedclassication,andarelargelylimitedtosingleobjectiveoptimization.Themulti-objectiv eclusteringproblemhasbeensolvedusing thefollowingprinciples. Ensemblemethods:Here,theinitialensemblesarecreatedb yclusteringthedataeithermultipletimesusingthesamealgorithm(withdifferentinitia lizationsorusingbootstrapping)or 33

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usingcomplementaryclusteringtechniques[89].Then,the solutionsarecombinedtocreate ensemblesusingexpectationmaximizationorgraphbasedap proaches[89].However,such a posteriori integrationofsingleobjectiveclusteringresultsdonote xploittherealstrengthof simultaneousmulti-objectiveoptimization. Paretooptimization:Afeasiblesolutionisparetooptimal ifthereisnootherfeasiblesolution thatisstrictlybetter.Multi-objectiveparetooptimizat ion[86,90]performssimultaneousoptimizationofcomplementaryobjectives,andhence,isbetter thantheensemblebasedmethods. Microeconomicmethods:Thesituationsofconictingobjec tivescanbenaturallymodeledina gametheoreticsetting.Theproblemscanbemodeledinafram eworkconsistingofplayerswith conictingobjectivescompetingtooptimizetheirutiliti es[27,28].Thegameissolvedusingthe Nashequilibriumbasedmethodologythatidentiesasocial lyfairsolution.Thesocialfairness ensuresthateveryplayerissatisedwithrespecttoeveryo therplayer. Microeconomicapproacheshavebeenappliedtoawidespectr umofproblemsinthedomain ofcomputerscience.Murugavel etal. [25]developedauctiontheoreticalgorithmsinVLSIdesign automationforsimultaneousgatesizingandbufferinserti onproblem.Hanchate[24]appliedgame theoreticconceptsforsimultaneousoptimizationofinter connectdelayandcrosstalknoisethrough gatesizing,whileGuptaandRanganathan[16]implementedg ametheoryforresourceallocationand schedulingintheeldofmulti-emergencymanagement.Ingr idcomputing,negotiatingagentshave beenusedforleasingofresourcesusingsuchmodels[21,22] .Similarly,Grosu etal. [23,91]used cooperativegamesandtheNashbargainingsolutionsforloa dbalancingindistributedsystems,and Lazar[20]implementedauctionsforoptimalbandwidthallo cationinwiredandwirelessnetworks. 34

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CHAPTER3 EXPECTEDUTILITYBASEDCIRCUITOPTIMIZATION Aggressivetechnologyscalinghasadverselyaffectedthec ircuitoptimizationprocessintwoimportantways.Theimpactofprocessvariationsinseveralco mponents,coupledwithmulti-foldincreaseinthedesigncomplexityhasresultedinasituationt hatrequiresthecircuitoptimizationtechniquestopossessimportantfeatureslikeaccuracyofoptim ization,incorporationofprocessvariation effectsduetovarioussourcesinasinglemodel,andfastexe cutiontime.Also,incontrasttothe optimizationtechniquesthatarebasedonspecicparametr icvariationdistributions(likeGaussian), thesecircuitoptimizationtechniquesshouldbevariation distributionindependent.Inthischapter,we presentanovelapproachforcircuitoptimizationinthepre senceof scarceinformation aboutthedistributionoftheprocessvariations.Thisalgorithmrelies upontheconceptsofutilitytheoryandrisk minimizationformulti-metricoptimizationofdelay,dyna micpower,leakagepower,andcrosstalk noise,throughthegatesizingtechnique.Animportantcont ributionofthisworkistheidenticationofa deterministiclinearequivalent modelfromafundamentallystochasticoptimizationproble m, ensuringhighlevelsofexpectedutility.Thealgorithmach ievessignicantspeedupintheoptimizationprocessforlargecircuits.Thisalgorithmcanaddress theimpactofprocessvariationsatseveral levelsincludingdevicevariations,interconnectvariati onsetc.,andisindependentoftheunderlying variationdistribution.Usingtheconceptsofboundedrati onality,thismethodminimizestheriskof constraintshortfallinalinearprogrammingsetup.Theexp erimentalresultsindicatethatthealgorithm isefcient,andacomparativestudywithanexistinggatesi zingtechniqueshowsthatourmethodis multi-foldfasteraswellascomparableintermsoftheoptim izationresults. 3.1IssuesinCircuitOptimization Thescalingofprocesstechnologyinsub-nanometerregime, andtheappositionofMoore'slaw [92]hasaffectedtherealmofCMOSdesignandoptimizationp rocess.Duetotheaggressivetech35

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nologyscaling,theimpactofdeviceprocessvariationsont hedesignprocesshasaggravated,and consecutively,reliabilityandperformanceofthefabrica tedcircuitshavedegraded.Onereasonfor suchaneffectisthat,atthelowertechnologynodes,thepar ametricvariationsinotherdesignparametershavesizableimpactonthecircuitperformance.Forexa mple,insub-65nmdesigns,inadditionto thegatesizevariations(oxidethickness,channellength) ,thevariationsduetointerconnectsandvias havesizableimpactonthedesign.Duetothesevariations,t heVLSIdesignoptimizationprocesshas switchedfromthedeterministicdomaintothestochasticdo maininthesensethatthesizesofgates, wiresetc.arenolongeradeterministicquantity,butrathe r,adistribution.Thestateoftheartresearch inrecentyearshasaddressedthecircuitoptimizationproc essprimarilythroughthestatisticalstatic timinganalysis(SSTA)basedapproachesandmathematicalp rogrammingapproaches. VariousSSTAbasedapproachesarevariationdistributiond ependent,andseveralworkshaveconsideredthevariationsourcesofcomponentsasGaussiandis tribution[6,7].However,thisassumption hasbeeninvalidatedbysomerecentanalyses[8],according towhichtheprocessvariationsdueto differentdesignparametersfollowdifferentdistributio ns.Forexample,in[93]theauthorshaveidentiedthattheglobalsourcesofvariationfollowalog-norm aldistributionmorecloselyascompared totheGaussiandistribution.Hence,newmethodsforcircui toptimizationthatareindependentofthe underlyingvariationdistributionsneedtobeexplored. Anotheraspectoftherapidprogressinthefabricationtech nologyisthemulti-foldincreasein thedensityoftheVLSIcircuits,resultinginlargerandmor ecomplexdesigns.Thisissue,although independent,hasacouplingeffectwiththeprocessvariati onimpactinthesensethatitfurtherworsens thecircuitoptimizationprocess.Thestochasticoptimiza tiontechniquesareinherentlyslowerthan theirdeterministicequivalentsforobviousreasons.This isaggravatedbytheever-growingsizeofthe designs,andpresentsthedesignerswithachallengeofiden tifyingoptimizationmethodologiesthat arefaster,canaddresstheeffectsofprocessvariations,a ndareyieldefcient. Inthecircuitoptimizationdomain,theoptimizationofasi nglemetricmayintroducesomesuboptimalityinthevaluesofothermetrics.Although,athigh ertechnologynodes,theimpactmay benegligible,however,suchassumptionsarenottrueforna no-scaledesigns.Thus,asinglemetric optimizationthatresultsinaperformanceshiftfromoneme trictoanotherisnotpracticalatthislevel. Asasimpleexample,ifanoptimizationisperformedwithano bjectiveofcrosstalknoiseminimization, 36

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theresultingdesignmaynotbelowpowerdissipating.Thus, anotherchallengeincircuitoptimization isthequestformethodsandsolutionsthatareverticallyas wellashorizontallyintegrated[14]. Thestudyoftheexistingresearchincircuitoptimizationa tthepostlayoutlevel,asdiscussedin Section2.4,raisesimilarissuesthatneedtobeaddressedi ndevelopingnextgenerationoptimization methods.Thesecanbesummarizedasfollows: Mostoftheworksperformasinglemetricoptimizationofeit herdelay,power,orcrosstalknoise. However,suchoptimizationsarenolongeradequatefornano meterdesigns,andnewmodeling techniquesformulti-metricoptimizationarerequiredtob edeveloped. Severalmethodsassumetheprocessvariationstofollowcer taindistributionsandaredeveloped toworkspecicallywiththosedistributions.However,suc hassumptionsarenotvalidforseveralvariationssources.Animportantaspectofthenextgen erationVLSIoptimizationisto identifymethodsthatarevariationdistributionindepend ent. Therecentanalysisonthevariationdistributionofthevar iousvariationsourcesidentifythatthe variationsdonotfollowthesamedistributionsaswereiden tiedinthepreliminaryanalysis. Thus,themethodsthatarecapableofperformingoptimizati onunderscarceinformationabout variationdistributionaredesirable. DuetotheincreasingcomplexityandsizeoftheVLSIcircuit s,thecircuitoptimizationprocess hasbecomeslower.Duetotheprocessvariations,theoptimi zationprocesshastransitionedfrom certaintydomaintotheuncertaintydomain,adverselyaffe ctingtheoptimizationtime.Thus,an accurateandfaststochasticoptimizationtechniquethatc ouldincorporatetheimpactofprocess variationsthroughasimpleyeteffectivemodelingisrequi redtobedeveloped. Withtheincreasingintegrationdensity,thesizesoftheci rcuitsareincreasingsignicantly.A fastoptimizationmethodisthusrequiredforpracticalsol utionstothelargescaleVLSIdesign optimizationproblems. Withthescalingoftechnology,processvariationsinother componentsofdesignarerapidly becomingevident.Amodelingtechniquethatcanaddressthe impactofprocessvariations atvariouslevels,withoutcomplicatingthemodelingwould scalewellforthenextgeneration 37

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circuitoptimizationproblems.Agenericcircuitoptimiza tionmodelcapableofincorporating theimpactofparametricvariationsduetoseveralfactorsi sthusdesirable. Inthiswork,wedevelopanovelexpected-utilitytheorybas edmethodologyforoptimizationofmultipleperformancemetricsthroughgatesizingtechnique.T hisapproacheffectivelytriestominimize theriskofviolationorfailureoftheconstraintsinthemod el,evaluatedandcontrolledbyanexpected utilitymeasurethatismaximizedtoensurethataconstrain tissatised.Themodelingassumesthe availabilityoflimitedinformationaboutthesystem,i.e. onlythemean,andstandarddeviationof theprocessvariationparametersisavailable,andnotthea ctualdistribution.Alinearprogramming modelisidentiedusingthesevalues,andissolvedforopti malsolution.Thismethodologyiscapable ofcopingwiththescantinformation,evaluatesadetermini sticequivalentmodelwhichisimportant forlargescaleproblems,andcanaddressthevariabilityin severalmodelingparameters.Thekey contributionsofthealgorithmare: Usingtheconceptsofconstraintriskaversionandminimiza tion,ityieldsadeterministicequivalentoftheinherentlystochasticoptimizationproblem,w hileensuringhighutilitylevels. Performsoptimizationinthepresenceofscarceinformatio naboutthevariationdistribution.In termsofscarceinformation,onlythemeanandthestandardd eviation,andnotthecomplete informationabouttheunderlyingdistributionarerequire d. Performssimultaneousoptimizationofmultiplemetrics.T hemetricsconsideredinthiswork aredelay,leakagepower,dynamicpowerandcrosstalknoise .Theinter-relationshipbetween thesemetricsintermsofgatesizesisidentiedandmodeled inamathematicalprogramming model. Incorporatestheimpactofprocessvariationsduetogatesi zesaswellasinterconnects. Theresultingdeterministicproblemissignicantlyfaste rthanthecorrespondingstochasticproblem, andachieveshightimingyields.Also,highlevelofutility isobtainedbycontrollingtheriskfrom eachconstraintinthemodel.Theprocessvariationeffects andtherandomnesscanbeincorporatedin themodelatvariouslevelsincludingthevariationsintheg atesizeswithinthedelaymodels,theinterconnectvariations,aswellasthevariationsinobjectivef unctions.Theimpactofspatialcorrelationis alsomodeledintheoptimizationmethodologyusingagridba sedcorrelationmodel[62]. 38

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Therestofthechapterisorganizedasfollows.Thedetailed descriptionoftheexpectedutility baseddeterministicmodelingofageneralstochasticoptim izationproblemispresentedinsection3.2. Insection3.3,delay,power,andcrosstalknoisemodelsuse dinthisworkarebrieyvisited.The modelsfordelayandpowerhavebeenadaptedfromliterature ,whileanovelcrosstalknoisemodel hasbeendevelopedinthisresearch,andisdiscussed.Also, arelationshipbetweenthemodelsin termsofgatesizesisderivedinthissection.Section3.4pr esentsthedetailsofthetransformationof stochasticgatesizingproblemtotheequivalentdetermini sticmodel.Experimentalresultsfordifferent scenariosandsensitivityanalysisofthealgorithmparame tersarediscussedinsection3.5. 3.2Expected-UtilityBasedModeling Inthissection,themethodologytoconvertastochasticopt imizationproblemtoalineardeterministicequivalentusingtheconceptsofexpectedutilitymax imizationispresented.Inthisalgorithm, differentpossiblescenariosforarandomconstraintsatis factionareanalyzedintermsofthequadratic utilityfunction.Theproblemisthenconvertedtoautility maximizationconstraineddeterministic model. Ageneralstochasticoptimizationproblemisgivenby(3.1) ,subjecttotherandomconstraint(3.2), alongwiththesetofnon-randomconstraints,andthenon-ne gativityconditions. minZ = n j = 1 z j s j ; 8 s j 2 S (3.1) s : t : a i = n j = 1 a ij s j b i ; 8 i 2 M (3.2) Here, s j isthe j thdesignparametertobeoptimized, z j istheweight(unitcost)of s j ,whichbyitself couldbearandomvalue. a ij s j isarandom i thconstraintcorrespondingtotheparameter s j and, b i is therandomconstraintsatisfactionvalue. S isthetotalnumberofdesignparameterstobeoptimized, and M isthetotalnumberofrandomconstraintsintheproblem. Now,from(3.2),thecriticalrandomvariablefortherandom constraint i canbedenedas: h i =( a i b i ) = b i ; 8 i 2 M (3.3) 39

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where, b i isthemeanvalueofall b i 's.Takingtherstandsecondmomentsof h i ,wegetthemeanfor theconstraint i as: h i =( n j = 1 a ij s j b i ) = b i (3.4) andthevarianceas: s 2i ( 1 = b i ) 2 ( n j = 1 s ij s j + s b i ) 2 (3.5) where, s ij and s b i aretherespectivestandarddeviationsof a ij and b i Aneffectivewayofcontrollingtheriskoffailureofaconst raintisbymaximizingtheexpected utilityoftheconstraint.Theassumptionofscarceinforma tionstatesthatonlythemeanandthe standarddeviationvalues,andnotthecompleteinformatio naboutthedistributionforeachrandom variableareavailable.Also,inthecontextofscarceinfor mation,itisassumedthatthe riskoffailure signicantlyexists ,i.e.thenegativevalueof h i canoccurwithsignicantprobability,andthusour goalistominimizethatbymaximizingtheutilityvalue.Int hecontextofgatesizingproblem,it correspondstothesituationswherethedelayconstraintsa renotmetduetothevariationeffects,and consequentlyaffectingtheyield.Forthejusticationsan dthedetaileddescriptionsofthetechnical informationthatfollows,pleaserefertoBallestero'spap er[94]. Thedecisionmaker'sutilitycanbegivenbythestandardqua draticVonNeumannandMorgenstern utilityfunction[32]for h i as: U i ( h i )= a + 2 bh i gh 2i (3.6) h i > 0; 8 i 2 M where a b ,and g aretheparameterstobedetermined.In(3.6),therearethre epossibleconditions: Ashortfall ,where h i < 0andtheconstraintisnotsatised Azeroshortfall ,where h i = 0,andreectsacriticalsituationwheretheconstraintmay ormay notsatisfydependingupontherandomnessof h i Asurplus ,where h i > 0,andtheconstraintissafelysatised. Thesepossibilitiesandtheirutilitiesarepictoriallyde scribedinFigure3.1. 40

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Now,forashortfall,theutility(3.6)decreasesrapidlyas the h increases,butthevalueisstill positivetill a > 2 bh i gh 2i .Thisintervalisshownas'bearableshortfall'regioninFi gure3.1.So,the greatestbearableshortfall h i ,whichisaverysmallvalue,canbeexpressedasafunctionof themean valueof h i ( h i ),givenby U i ( h i )= a + 2 bh i gh 2 i (3.7) a + 2 bl i h i gl 2i h i 2 = 0 where l i isapositiveparameterclosetozero,sincethegreatestbea rableshortfallisaverysmallvalue. Fromtherstderivativeofutility(3.6)withrespectto h i ,weget U 0 i ( h i )= 2 ( b gh i ) > 0(3.8) Astheutilitymonotonicallyincreaseswith h i (whichisnegative),lessshortfallispreferredtomore shortfall. Incaseofzeroshortfall,theutilityisgivenby a .Atthiscriticalpoint,therandomnessof h i decidesiftheconstraintismetornot.Asurplusisthuspref erredbythedecisionmakerssincethe zeroshortfallisarandomvalueattheedge.So,a securitymargin intermsofsmallsurplusispreferred. However,alargesurplusvalueisnotgoodsinceitcanadvers elyaffecttheachievementoftheobjective goals.Thesecuritymarginisshownastheshadedregioninth eFigure3.1.Autilitymaximization functionisderivedbysubstitutingthevaluesof a b ,and g in(3.6). Now,iftherstderivativeofutilitygivenin(3.8)isequat edtozeroformaximalvalueof h i ,we get, h imax = y i = b = g (3.9) Substitutingthevaluesin(3.7),weget b = g = l 2i h i 2 + 2 y i l i h i (3.10) a = b =( l 2i h i 2 + 2 y i l i h i ) = y i (3.11) 41

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Figure3.1Utilitycurveforarandomfunction.Thepossible scenariosofshortfall,zeroshortfalland thesurplusareshownhere.Theshadedregionshowsthesecur itymarginforrandomfunction h whosevalueismaximizedwhentherstderivativeoftheexpe ctedutilityfunctionisequatedto0. Thesevaluescorrespondtothegreatestbearableshortfall ,themean,andthesurplusofminimum utility.Now,theoverallmaximumutilityvaluecanbederiv edwithrespecttothemeanvalueas U imax = K i h i h i (3.12) ThevalueofKisirrelevanthere,sincetheutilityvalueisj ustanindex.Now,Puttingthevaluesof a b ,and g in(3.6),wegetthemaximumutilityas, h i a a = y 2i ( l 2i h i 2 + 2 y i l i h i ) (3.13) Now,foraprobleminwhichtheshortfallisunacceptable,su chasgatesizing,theutilityfor shortfallis0.Thus,inFigure3.1,thebearableshortfalla realimitsto0,andhence a = l i h i .Now, 42

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substitutingthisvaluein(3.13),weget y i 2 h i ; as l i 0 : (3.14) Intheoptimizationproblem,theexpectedutility(EU)fore achrandomconstraint a i istobekept atahighleveltoassurethatthesolutionpointsareidenti edbysatisfyingtheconstraints.Thisis mathematicallyexpressedintermsofrstandsecondderiva tivesas: EU i ( h i )= U i ( h i )+ 0 : 5 U 00 i ( h i ) s 2i (3.15) = a +( 2 y i h i h i 2 ) g s 2i g Aparameter w ,symbolizingtheutilityvalue(anindex)canbeintroduced heresuchthat: a +( 2 y i h i h i 2 ) g s 2i g > w b a +( 2 y i h i h i 2 ) g c (3.16) where w isclosetounity.Theconstraint(3.16)ensuresthattheexp ectedutilityoftheconstraintis closetounity,andissatised. Now,solving(3.16)for s 2i ,weget, s 2i < ( 1 w )( 3 + l 2i + 4 l i ) h i 2 (3.17) 3 ( 1 w ) h i 2 ; h i > 0 Thisequationgivesaclearrelationshipbetweenthevarian ceandthemeanintermsofexpectedutility. Theserelationshipsarethenutilizedinidentifyingadete rministicmodel,asdiscussednext. 3.2.1DeterministicModeling Inamean-varianceapproachfortheexpectedutilitymaximi zationproblem,thegeneralminimizationproblemdescribedin(3.1)isconvertedintoanequival entmaximizationfunctionoftheexpected utility,subjecttotheparametricvarianceconstraints.T heexpectedutilitymaximizationfunctioncan 43

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begivenas: max L = C Z = C n j = 1 z j s j (3.18) Here,Cisalargepositiveconstant.Also, Z = n j = 1 z j s j > 0,sincenoresourceisfree.Thus,the equivalentminimizationproblembecomes, min n j = 1 z j s j (3.19) subjecttothefollowingconstraints, Constraints(3.17),asdevelopedearlier, n j = 1 s ij s j + s b i < q ( 1 w )( 3 + l 2i + 4 l i )( n j = 1 a ij s j b i ) (3.20) < p 3 ( 1 w )( n j = 1 a ij s j b i ) ; 8 i 2 m Parametricvarianceconstraintcorrespondingtothemeanv alueobjectivefunction, n j = 1 s z j s j r n j = 1 z j s j (3.21) Theparameter r correspondstothevalueof coefcientofvariation ,ifthatinformationisavailable.Thisinformationisrequiredonlyinsituationswhent heobjectivefunctionitselfhasrandomparameters. Thesetofnon-randomconstraints,andthenon-negativityc onditionsintheoriginaloptimization problem. 3.3ParametricModels Inthissection,wewouldpresentthemodelscorrespondingt oeachoptimizationmetric,delay, leakagepower,dynamicpowerandcrosstalknoise.Thedelay andpowermodelshavebeenadapted fromtheliterature,whereasanovelcrosstalknoisemodelh asbeendevelopedaspartofthiswork. Thiscrosstalkmodelidentiesnoiseasalinearfunctionof thesizesofthedrivergates.Thedis44

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cussionfocusesonidentifyingarelationshipbetweenthes emetrics,andformulatesthemathematical programmingmodelsrequiredforoptimization.Also,thede vicelevelandinterconnectprocessvariationarebrieyvisited.3.3.1Delay,ProcessVariationsandSpatialCorrelation Reducingthesizeofagate(say s i ),reducestheintrinsicgatecapacitanceofgate i ,thepower consumption,andthefan-inloadcapacitancesof i .Inthelineardelaymodel[95],thedelayismodeled asafunctionofthegatesizes,asshownin(3.22). d i = a i b i s i + c i j 2 fo i s j (3.22) where, d i isthedelayofgate i s i isthesizeofgate i ,and s j correspondstothesizesofallthefanoutgatesof i .Thecoefcients a i ; b i ; c i areempiricallydeterminedbyextensiveSPICEsimulations foreachgateinthestandardcelllibraryforallcombinatio nsofsizesandfan-out.Specically, b i correspondstotheimpactofchannellength( L eff )onthedelayofagate,and c i correspondstothe impactofoxidethickness( t ox )onthedelay.Thus,thedelaymodelincorporatestheimpact ofdevice processvariations. Theuncertaintyduetoparametervariationsingatesizesis modeledaccordingto(3.23),whichis expressedintermsofnominaldelay( d i ),andrandomparameters X j and X r ,determiningthecorrelated andindependentvariationsrespectively. D = d i + n j = 1 d j X j + d r X r (3.23) Here, X j modelstheprincipalcomponentsofcorrelatedrandomvaria bleswiththecorresponding d j valuesevaluatingthesensitivityofdelay. X r N ( 0 ; 1 ) modelstherandomcomponentofvariationsin allprocessparameterslumpedintoasingleterm,and d r isthestandarddeviationindelayduetothese randomvariations.Themagnitudeof d j and d r isdeterminedbyextensivesimulations. Inthelineardelaymodel,onlytheprocessvariationeffect sinthegatesizes( L eff and t ox )areincorporated,butnottheinterconnectvariations.Theintercon nectvariationsintoday'sgigahertzdesigns arehigh,andcancauseupto25%variationintheclockskew,a sshowninapaperbyLiu etal. [68]. 45

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Also,thesevariationscannotbeincorporatedinasimpleno minal-worstcasetypeofanalysis.So, inthismodel,theeffectoftheinterconnectvariationsare addressedinamean-varianceapproachat thetimingconstraintslevel.Theoptimaldelaysidentied throughanunconstraineddelayminimizationasarststepintheoptimizationprocessareusedascon straintsforsimultaneousoptimizationof powerandcrosstalknoise.Weincorporateaconservative10 %variancearoundthemeanofthebest casetimingvalues,correspondingtotheinterconnectvari ationeffects. Thespatialcorrelationsaremodeledusingagridbasedcorr elationmodelproposedin[62].Accordingtothismodeling,thecompletedesignisdividedint odifferentnumberofregions.Thegates thatareinsameregionarehighlycorrelatedandthevariati oneffectsonallofthemaresimilar,whereas thevariationeffectsonthegatesthatareindifferentregi onsaredifferentandarelesscorrelated.These effectsareincorporatedin(3.22)toevaluatethevaluesof b i and c i ,approximatingthevariationsin channellengthandgateoxidethicknessrespectively.3.3.2LeakageandDynamicPower Thepowermodelsproposedin[51]fordynamicandleakagepow erareadaptedinouroptimization formulation.Thedynamicpowerdissipation P d ( i ) ofagate i ineachclockcycle t c dependsuponthe transitionprobability tp i ,thepowersupplyvoltage V dd ,andtheloadcapacitance L i .Loadcapacitance onagate i isgivenby(3.24). L i = W i + k = fo i C i ( k ) s k (3.24) where W i isthewirecapacitance,and C i ( k ) isthegatecapacitanceoftheunitsizedfan-outgate.The equationfordynamicpowerdissipationcanbegivenas(3.25 ). P d ( i )= tp i k = fo i C i ( k ) s k V 2 dd = ( 2 t c ) (3.25) Theaverageleakagepowerdissipation P l ( i ) ofgate i asafunctionofitssize s i ,andthetransition probabilitiesaregivenby(3.26). P l ( i )=[ r P ( i ; r ) I l ( i ; r )] s i V dd (3.26) 46

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where, I l ( i ; r ) isgivenastheleakagecurrentofgate i fortheinputpattern r P ( i ; r ) istheleakage powerforgate i correspondingtotheinputpattern r I l ( i ; r ) istheleakagecurrentfortheunitsized gate i andinputpattern r s i isthesizeofgate i ,and V dd ispowersupplyvoltage. From(3.25)and(3.26),itisidentiedthattheleakagepowe rofagateisdirectlyproportionalto itssize,andthedynamicpowerisproportionaltothesumoft hesizesofitsfan-outgates. 3.3.3CrosstalkNoise Thecouplingcapacitanceeffectsinthecircuitsaresubsta ntial,andpresentamajorthreattothe reliabilityofthedesigns.Theyinducecrosstalknoiseont hecouplednetsleadingtotimingyield failures.Although,theeffectofcrosstalknoiseonanetca nbereducedbyusingtechniqueslikewire sizing,wireshielding,wirespacing,driversizing(victi mandaggressor),andreceiversizing(victim andaggressor),themosteffectivetechniqueforreducingt hecrosstalknoiseatthepost-layoutlevel isprimarilydriversizing.Inthistechnique,thedrivingg ateofthenet(oftenreferredasthe victim net ),andallotherdrivergatesofthenetsthathaveacouplinge ffectonthatvictimnet(referredas aggressornets )aresized.Theup-sizingofthegatesincreasesthesignals trengthoncorresponding net,andhencereducesthecouplingeffects. Figure3.2showsasimpleexampleofthenoiseonanetinasing levictim-aggressorpairsetting. Here,G1,G2,G3,andG4arethegatessizes,C1andC2aretheco uplingcapacitancesbetweenthe wires,andNet1andNet2aretheinternalresistances.Asdis cussedin[43],themajorcontributorsto thecrosstalknoiseonanetarethesizesofthevictimdriver (G1),andaggressordrivergates(G2). Ifthesizeofthedrivinggateofthenetisincreased,thesig nalstrengthonthenetincreases,thereby reducingthecouplingnoiseonthenet.Also,ifthesizeofth edrivergateofthecouplednetis decreased,thenoiseonthevictimnetdecreases.However,s uchdown-sizingwillincreasethenoise onthecouplednet.Similarly,theup-sizingofthevictimdr iverwillincreasethenoiseonthecoupled nets.Thus,thesizinghassymmetriceffectsonthecoupledn ets,andagateisanaggressoraswellas avictimatthesametime. 47

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Figure3.2Acouplingstructurewithsinglevictim-aggress orpairsetting.Here,G1,G2,G3,andG4 arethefourgates,Net1andNet2arethetwonets,andC1andC2 correspondtothetwohalf-coupling capacitancesbetweenthenets.Forthevictimnet,Net1,G1i sthevictimdrivergateandG2isthe victimreceivergate.G3istheaggressordrivergateforNet 1andG4istheaggressorreceivergatefor Net1. Therelationshipbetweenthesizesofthedrivinggatesofth ecouplednetsisincorporatedin formulatingasimplecrosstalknoisemodel.Here,thecross talknoiseonanet N i isgivenas(3.27) N i = F i ( j 2 coupleds i ( s i s j )) 8 s i 2 n (3.27) where, s i isthesizeofthedrivinggateofthenet, j isthesetofallthecouplednetscorrespondingto i ,and s j isthesizeofthedrivergateofthecouplednetinan n -gatedesign.Hence,foreverygate,the noiseonitsfanoutnetisafunctionofthetotalcross-coupl ingcapacitanceonthenet. Theeffectofcrosstalknoiseinacircuitcanalsobeminimiz edatanotherlevelbyincorporating thenoisemarginconstraintsinthemodel.Theseconstraint scontrolthatthemaximumnoiseanetcan 48

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tolerate.Inthepresentsetting,themaximumtolerablenoi seonnet j isgivenby(3.28). U j = H ( k = fo j ( l j u k )) 8 s k 2 coupled ( s j ) (3.28) where, l j and u k correspondtotheminimumandthemaximumsizeofthegatesav ailableinthecell library.3.4StochastictoDeterministicGateSizing Inthissection,arelationshipbetweenthedynamicpower,l eakagepower,andcrosstalknoise,is rstidentiedasafunctionofgatesizes,andagatesizinga pproachisformulatedinamathematical programmingmodel.Thesemetricsareincorporatedintheob jectivefunction,withthedelayandnoise toleranceastheconstraints.Theimpactofprocessvariati onsindifferentdesignparameterssuchas gatesize,andinterconnectsisaddressedinthedelayconst raints.Oncetheproblemisformulatedasa stochasticmodel,itisconvertedtothelineardeterminist icmean-varianceequivalentmodelusingthe resultsfromthemethodologydiscussedinSection3.2.3.4.1StochasticOptimizationProblem Toformulatetheobjectivefunctionforthemulti-metricop timizationproblem,arelationshipbetweentheleakagepower,thedynamicpower,andthecrosstal knoiseisderivedasafunctionofthe sizeofgatesinthedesign.Insection3.3,wederivedtheser elationshipsindependently,whichcanbe summarizedasfollows: InEquation3.25,itisidentiedthatthedynamicpowerdiss ipationofagateisprimarilyaffected bythetotalsizeofitsfan-outgatesinthecircuit.Thus,fo ragate(say i ),thetotalnumberof gatesitsfan-innetsareconnectedtodeterminetheimpacto fthegate i onthedynamicpower ofthegatesthatareinitsfan-in.Hence,theweightforsizi ngthisgate i isproportionaltothe numberofgatesthatitisconnectedtointheirfan-out. FromEquation3.26,itisshownthattheleakagepowerofagat eisdirectlyproportionaltothe sizeofthegate,andhencehasadirectimpact.So,increasin gthesizeofthegatewouldincrease theleakagepowerdissipationofthecircuit. 49

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Thecrosstalknoiseonanethasaninverserelationshipwith thesizeofitsdrivergate.Ifthe drivergateissizedup,thesignalstrengthonthenetincrea ses,andhencethecrosstalknoiseon thenetreduces.However,theup-sizingofthegatehasanadv erseeffectonthecouplednets. Byup-sizingthedrivergatesofthecouplednets,thenoiseo ntheircorrespondingoutputnets canbereduced. Theseperformancemetricsarenowmodeledinasingleobject ivefunction,whichisoptimizedinthe presenceofdelayconstraints.Theimpactoftheseparamete rsonthesizeofthegateisincorporatedby multiplyingthedimensionlessnormalizedcoefcients k n ,and x ,referringtotheimpactofagatesize ontheleakagepower,dynamicpower,andcrosstalknoiseres pectively.Thecoefcient k isdirectly proportionaltothesizeofthegate,andcoefcient n isafunctionofthenormalizedimpactofthe gatesizeonthegatesthatareinitsfan-in.Soifagate i isinthefan-outofalargenumberofgates, theimpactofup-sizing i willbehigherforthecircuit.However, x isinverselyproportionaltothe sizeofthegate,anditsnormalizedvalueisafunctionofthe maximumcouplingcapacitanceofits correspondingnetwiththeaggressornets. Theobjectivefunctionfortheoptimizationproblemisgive nbythefollowingequation: MinimizeGS = n i = 1 ( k s i + n s i x s i ) (3.29) where, s i isthesizeofthegate i ,and n isthetotalnumberofgatesinthedesign. Now,toderivethedelayconstraintsforthemulti-metricop timizationproblem,thedeterministic bestcasedelaysforeachgateinthepathsareidentiedbype rformingalinearprogramming(LP) optimizationunderthepathdelayconstraints(3.22),andt henoisemarginconstraints(3.28).Since delayoptimizationistheprimaryobjectiveinanycircuito ptimizationtechnique,thedelayisoptimized asapre-processingstep.Asshownin(3.30),thedelay( t spec )istheobjectivefunction,whichis minimizedtoidentifythebestpossiblecircuitdelay.Thed esignconstraintsintermsofthenode delaysinthepathsformtheconstraintsfortheproblem.Spe cically,theconstraintsensurethatagate i + 1,thatisconnectedinthefan-outofanothergate i ,hasadelaygreaterthanthetotaldelayofthepath tillgate i andtheinternalgatedelayof i .Thenominal(meanvalues)casedelaycoefcientsareused duringthisoptimization.Thisdeterministicoptimizatio nsolutiongeneratesthedelayspecications 50

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forallthepathsinthedesign,whichareusedasconstraints inthenextsteps. mint spec (3.30) s : t : at i ( p )+ d i at i + 1 ( p ) 8 i 2 n ; 8 p 2 P d i = i 2 p ( a i b i s i + c i j 2 fo i s j ) l i s i u i 8 i 2 n where, at i ( p ) isthearrivaltimeatthegate i inpath p d i istheinternalgatedelayof i ,and at i + 1 ( p ) isthearrivaltimeatthenextgate i + 1inthepath p .Thevalues b i ,and c i arethemeanparameter coefcientvalues,and P isthesetofallthepathsinthedesign. Afterthedelaysarecalculated,themulti-metricoptimiza tionproblemisformulatedforsimultaneousoptimizationofcrosstalknoise,leakagepower,anddyn amicpowerunderdelayandnoisemargin constraints.Thestochasticmulti-metricoptimizationpr oblemisgivenas(3.31). minGS = n i = 1 ( k i + n i x i ) s i (3.31) s : td p t spec 8 p 2 P d p = i 2 p ( a i b i s i + c i j 2 fo i s j ) N i U i 8 i 2 n ; l i s i u i 8 i 2 n Thisoptimizationproblemisstochasticinnature,since d p containstheparametricvariationcoefcients b i and c i correspondingtothegatesizevariations.Also, t spec isarandomparametersincethe variationsduetointerconnectsarenotaccounted.Theinte rconnectvariationscanbeincorporatedin themodelbyconsidering t spec asadistributionratherthananumber. Thenextstepintheproblemistoconvertthisstochasticpro blemintoanequivalentdeterministic problembasedupontheconceptsofexpectedutilitymaximiz ationandriskminimization,discussed inSection3.2. 51

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3.4.2DeterministicEquivalentModel Iftheobjectivefunctioncontainsrandomvariables,theex pectedutilityapproachcanbeusedto transformtheoriginalstochasticoptimizationproblemto thedeterministicequivalentmodel,eitherby minimizingthevariancesubjecttomeanvalueconstraints, orbymaximizingthemeanvaluesubject totheparametricvarianceconstraints,asdiscussedinthe Equations(3.18–3.21).Inthegatesizing problem,thiscorrespondstotherandomnessintheparamete rs k n and x .Theequivalentobjective functionforsuchmodelingthenbecomes(correspondingtoE quation(3.19)), min n i = 1 ( k i + n i x i ) s i (3.32) where k i n i x i correspondtothemeanvaluesoftheparameters.Incaseofno n-random k n and x thedeterministicvalueswillcorrespondtothemeanvalues Theconstraintsforthenewgatesizingproblemunderthesca rceinformation,andutilitymaximizationscenarioarederivedasfollows: Thelinearproxyconstraintsintermsofvarianceoftherand omparameters,correspondingto theconstraintsderivedinEquation(3.20).Pleasenotetha tinthisworkwehaveconsidereda 10%standarddeviationinthe t spec valuefromitsmeanvalueidentiedduringtherststepof theoptimizationprocess.Theapproximationismoreconser vativethantheonederivedin[68]. Amoreaccurateestimatecanbeincorporatedwithoutanymod icationsinthemodel. s d p + s t spec < p 3 ( 1 w )( d p t spec ) ; 8 p 2 P (3.33) s d p i 2 p ( s a i s b i + s c i j 2 fo i s i ) d p i 2 p ( a i b i + c i j 2 fo i s i ) Thevalueoftheutilitymaximizationparameter w canbeexperimentallydetermined,andcan bekeptatahighlevelof0.9,0.92oreven0.95. Theparametricvarianceconstraintcorrespondingtotheme anvaluedobjectivefunctionisgiven by, n i = 1 ( s k i + s n i s x i ) s i r n i = 1 ( k i + n i x i ) s i (3.34) 52

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where r isthecoefcientofvariation.Thisconstraintisrequired onlyinsituationswherethe objectivefunctionalsohasrandomparameters(sayrandomc ostparameters)init.However,in thiswork,themodelingdoesnotconsidertherandomnessint heobjectivefunctionparameters, andhencethisconstraintisnotutilizedatall.Ifthecoef cientsintheobjectivefunctionare consideredasrandomduringthemodeling,thenexperimenta tionwithdifferentvaluesof r providesacompletefrontieroftheoptimalsolutions. TheNon-randomconstraints,andtheconditionscorrespond ingtotheavailablegatesizeranges aremaintained. N i U i ; 8 i 2 n (3.35) l i s i u i ; 8 i 2 n (3.36) Thus,thestochasticproblemdenedinEquation(3.31),isc onvertedintoadeterministicequivalentmulti-metricmodelgivenbyEquations(3.32),(3.33) ,(3.34),(3.35)and(3.36)whichcanbe solvedusinganylinearprogrammingoptimizationtool.3.5ExperimentalResults Inthissection,wepresentthesimulationresultstoverify theefciency,accuracyandefcacy ofthismethodology.First,asensitivityanalysisisperfo rmedtoevaluatethesensitivityoftheexpectedutilityassuranceconstant( w )withreferencetothetimingyieldandtheoptimizationoft he metrics.Theanalysishelpsinidentifyingtheoptimumvalu eof w correspondingtotheoptimization requirements.Theriskaverseoptimizationalgorithmisth enevaluatedonthebenchmarkcircuitsfor optimizationofdifferentmetricsandtheexecutiontime.A lso,theapproachiscomparedwitharecentlyproposeddevicevariationawaremathematicalprogr ammingbasedapproach.Thealgorithmis thenusedtoperformsinglemetricoptimization,andtheres ultsarecomparedwiththemulti-metricoptimizationvaluestoevaluatetherelativeimpactofasingl emetricoptimizationonthesub-optimality introducedinothermetrics.Finally,theimpactofincorpo ratinginterconnectvariationsinaddition tothegatesizevariationsinthealgorithmisanalyzed.Iti simportanttonotethatthismethodhas beentransformedfromapathbasedapproachtoanequivalent nodebasedapproachaccordingtothe techniqueproposedin[47].Thiscontrolsthesizeofthepro blemtofurtherimprovetheruntimeand 53

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feasibilityforlargecircuits.Thesub-optimalityintrod ucedduetothetransformationiscloseto2% forcircuitswith20levelsoflogic.3.5.1Setup Themulti-metricoptimizationalgorithmforgatesizingun derprocessvariationswasrigorously testedontheITC'99benchmarkcircuits.Thesetupconsiste dofthreesteps,aslistedbelow. TheRTLlevelVHDLnet-listsofthebenchmarkcircuitsweree xtractedforgeneratinggate levelVeriloglesusingthe SynopsysDesignCompiler tool.ThesegatelevelVerilogles andtheTSMC180nmStandardCelllibraries(LEF,TLF,DBFile setc.)werethenusedto placeandroutethedesignsandgeneratetheDEFles,cellde layinformationetc.usingthe CadenceDesignEncounter toolkit.ThebenchmarkcircuitsweresynthesizedusingTSM C 180nmlibrariessincethelowerlevellibrarieswerenotava ilabletous. Theparasiticresistanceandcapacitanceinformation(SPE Fle)wasextractedfromtherouted designsusingtheCadence FireN'Ice RCextractor.Thisinformationwasutilizedforextracting thecouplingcapacitancefromtheroutedcircuits.APERLsc riptwaswrittentoextractthe couplingcapacitanceinformationofeachnetwithitstopth reecouplednetsfromtheSPEFle. Thedelaycoefcientsforavailablegatesizes(1x-6x)andf an-outsofthestandardcellsinthe TSMC180nmstandardcelllibrarywerecharacterizedusingt heHSPICEsimulations.Also, thevariationsingatesizingparameterswereassumedtobe2 5%ofthenominalvalues,which wereappropriatelytranslatedtothecoefcients a b and c inEquation3.22.Aconservative estimateof10%variancefortheinterconnectvariationsin thebestcasedelayconstraintswas incorporatedinthemodel.Thebestcasedelayswereidenti edinapre-processingstep,through theunconstrainedlinearprogrammingoptimization. Afterthedelays,couplingnoiseetc.werecalculatedforth ebenchmarkcircuits,thestochasticgate sizingoptimizationproblemwasformulated.Thealgorithm forformulatingthestochasticlinearprogrammingmodelwasprogrammedinClanguage.Next,thestoch asticproblemwasconvertedinto adeterministiclinearprogrammingequivalentmodelthrou ghtheexpectedutilitymaximizationapproach.Thelinearprogramwasthenconvertedtothestandar dAMPLformat,whichwassolvedusing 54

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therobustKNITRO[96]optimizationsolver.KNITROusesint eriorpointandactivesetmethodsfor optimizationandiscapableofutilizingmultipleprocesso rs.Thissolverisspecicallydesignedto solveproblemswithlargedimensionality.Adetailedsyste mowisshowninFigure3.3. Figure3.3Simulationsetupfortheriskaversegatesizingo ptimizationproblem.TheSynopsysand CadencetoolsareusedtogenerateDEF,CAP,VerilogandSPEF lesforeachbenchmarkcircuit. Theseles,andthedelaycoefcients( a ; b ; c )determinedbytheextensiveSPICEsiimulationsare utilizedinformulatingtheunconstrainedlinearprogramm ingproblem,whichissolvedtoevaluatethe bestdelaysforeachbenchmark.Themulti-metricstochasti coptimizationproblemisthenformulated andconvertedtotheequivalentriskaversedeterministicp roblem.Theselinearprogrammingproblems arethensolvedusingtheKNITROoptimizationsolver. 55

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3.5.2SensitivityofUtilityAssuranceConstant Theexpectedutilityconstrainedoptimizationprocessinv olvesidenticationofappropriatevalue fortheparameter w ,whichdeterminestheutilityofsatisfyingeachconstrain tintheoptimization problem.Inthedomainofcircuitoptimization,eachconstr aintiscritical,andhencethesatisfaction oftheconstraintsiscentraltotheoptimizationprocess.T hus,thegreatestbearableshortfalliskeptas zero.Theexpectedutilityassuranceconstant w evaluatestheutilityofeachconstraintinthemodel. Intuitively,ahighvalueof w wouldensurethateachconstraintintheproblemiscritical ,andthus needstobesatised.However,suchahighvalueof w mayresultintheunder-achievementofthe optimizationgoals,whereasalowvaluewouldresultinunre liabilityoftheoptimizationprocessand consequentlytheyield.Toevaluatetheconsistencyofthis intuition,andtoidentifytheoptimumrange ofvaluesfortheutilityassuranceconstant,experimentsw ereperformedonthebenchmarkcircuitsto determinetheaveragechangeintimingyieldandthemetrico ptimizationvalues. Thetimingyieldofacircuitdeterminestheprobabilitytha tthecircuitsatisestimingconstraints. Inthismodel,thebesttimingspecication( t spec )isidentiedduringtheunconstraineddelayoptimization,andisusedasaconstraintduringthemulti-metricopt imization.Intheriskaversemathematical programmingformulationwithdelayconstraintscorrespon dingtoeachnode,ahightimingyieldis obtainedsinceeachtimingconstraintissatised.However ,atimingvalueofthecriticalgatescloser tothe t spec valueisat occam'srazor ,withahighprobabilityoffailureduetoprocessvariation seffects. Thetimingyieldofthecircuitishigherifthedifferencebe tweenthemaximumdelayofthecircuit afteroptimizationandthetimingspecication( t spec )islarge.Thusweevaluatetheimprovementin thetimingyieldofthebenchmarkcircuitsforvariousvalue sof w as ( t spec t max ) 100 = t max ,where t max isthemaximumdelayofthecircuitafterthemulti-metricop timizationisperformed.Thegraph showningure3.4displaysthetimingyieldimprovementval uesforthebenchmarkcircuits.The resultsendorsetheintuitionthatasthe w valuedecreases,thetimingyieldofthecircuitdecreases, sincethesatisfactionofeachconstraintdoesnothaveahig hutilityvalue,andthusmaynotsatisfy, whichisthecaseforsomedesigns(b11,b12,andb20)atlowut ilityvalues. Theeffectof w ontheobjectivefunctionoptimizationfollowsacountertr end.Asshowninthe Figure3.5,theobjectivefunctionoptimizationvaluesfor differentutilityassuranceconstants( w )as comparedtothevaluescorrespondingto w = 0 : 99consistentlyimproveforeachbenchmarkcircuit. 56

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Figure3.4Improvementinthetimingyieldofthecircuitsfo rdifferentvaluesof w .Thetimingyield improvementisgivenasthepercentageincreaseinthediffe rencebetween t spec andthemaximum delay( t max )ofthecircuitafterthemulti-metricoptimizationisperf ormed. Thisalsofollowstheintuition,sincetheutilityofeachde layconstraintforalowervalueof w isless, thereisgreatermarginforoptimizationatthecostofdissa tisfactionoftheconstraint.Animportant stepinthisriskaverseoptimizationprocessistoidentify theoptimum w valuesbasedupontheoptimizationrequirements.Agoodoptioninthegeneralcircuit optimizationdomainistousea w value thatprovidesat-least98-99%timingyield,aswellasident iesgoodsolutionpoints.Inthiswork,we haveperformedmostoftheanalysiswiththevaluesof w = 0 : 95,0 : 92,and0 : 90. 3.5.3RiskAverseOptimizationResults Toevaluatetheoptimizationvaluesfordynamicpower,leak agepower,andcrosstalknoiseinan equallyweightedmulti-metricoptimizationsettingofthi sriskaversegatesizing(RAGS)methodology,acomparativeanalysiswithanexistingmulti-metricf uzzymathematicalprogrammingmethodology(FMP)[97,98]hasbeenperformed.Itisimportanttono tethatsincethefuzzymathematical programmingmethodonlyincorporatestheeffectofprocess variationsduetogatesizes(constants a i b i ,and c i ),theriskaversegatesizingmethodwasalsoimplementedwi thonlydevicevariations forafaircomparison.Thefuzzymathematicalprogramminga pproachisathreestepprocessthat 57

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Figure3.5Percentageimprovementintheoptimizationofth eobjectivefunctionforvariousvaluesof w .Theobjectivefunctionimprovementsarecomparedwiththe valuesobtainedfor w = 0 : 99. rstcalculatestheworstcaseandnominalcasegatesizeval ues,withtheoptimaldelaysincorporated asconstraintsinasimilarfashionasthisapproach.Thesev aluesarethenusedtoformulateafuzzy non-linearprogramthatissolvedforoptimization.Theres ultsfortwovaluesofutilityassurance constant w = 0 : 90and w = 0 : 95areshownintheTable3.1.Theresultsindicatethattheop timizationvaluesobtainedusingtheriskaversegatesizingmetho discomparabletothefuzzymathematical programmingresults.Onaverage,theimprovementinmetric optimizationvaluesforeachmetricis approximatelyequaltotheFMPcounter-partsfor w = 0 : 95.TheRAGSperformedslightlybetter thanFMPintermsofmetricsoptimizationsfor w = 0 : 90.However,animportantaspectoftherisk aversegatesizingapproachistheexecutiontimeofthealgo rithm.Asshowninlastthreecolumnsof Table3.1,thealgorithmexecutiontimeofourmethodologyi ssignicantlylessascomparedtothe FMPmethod.TheRAGSis5.85timesfasterthantheFMPmethodf or w = 0 : 90andmorethan6.4 timesfasterfor w = 0 : 95.ThisisattributedtothefactthattheRAGSmethodisasin glesteplinear programmingmethodascomparedtoFMPmethodwhichrequires threesteps,andthelaststepisa non-linearprogram.Thisissignicantlyimportantforlar gercircuitslikeb17whichhavemorethan 21000gates. 58

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Thisalgorithmcanbeutilizedtoperformsinglemetricopti mizationofthemetricsdependingupon thedesignrequirements.Themetrictobeoptimizedcanbepr ioritizedbyassigningahighweightvectortoit.Forexample,ifthedesignerintendstooptimizeon lyleakagepower,thentheweightvector correspondingto k isassignedas1whereastheothermetrics x and n intheobjectivefunctionare assignedas0and0respectively.However,duringsuchoptim izationtheimprovementintheoptimizationvaluesofthecorrespondingmetriccomesatthecostofi ntroducingsub-optimalityinthevalues ofothermetrics.Wehavecomparedtheresultsofsinglemetr icoptimizationsfordynamicpower, leakagepowerandcrosstalknoise,withtheequallyweighte dsimultaneousmulti-metricoptimization ofallthreemetrics.Theresultsoftheaveragechangeinthe optimizationvaluesforthemetricsas comparedtomulti-metricoptimizationinallthreecasesis showninTable3.2. Thedynamicpowerdissipationforsinglemetricoptimizati onislowerthanthemulti-metricoptimization(asshownincolumn2ofTable3.2).However,itis interestingtonotethatonaverage, dynamicpowerdissipationisreducedatthecostofleakagep owerandnotcrosstalknoise.Thistrend occursduetothefactthatduringthedynamicpowerminimiza tion,fewergatesareresizedfromthe sub-optimalsizes(afterdelayoptimization)ascomparedt oleakagepoweroptimization.Thisresults inadecreaseindynamicpower,buttheleakagepowerislarge lyunaffected.Whenonlyleakagepower isoptimized,theoptimizationintroducessub-optimality primarilyincrosstalknoisemetric.Thisis intuitive,sincetheleakagepowerisdirectlyproportiona ltothegatesizes,where-asthecrosstalk noisehasaninverserelationshipwiththegatesizes.Whent hesinglemetricoptimizationforcrosstalk noiseisperformed,thegatesizingproblemtranslatesinto amaximizationproblem.Thereduction incrosstalknoiseascomparedtotheequallyweightedmulti -metricoptimizationisnotable(almost 47%).Thisisduetothefactthatthemaximizationproblemsa tisesthedelayconstraintsmucheasily ascomparedtotheminimizationproblem.Increasingthegat esizesreducesthecrosstalknoise,as wellasthegatedelays.However,thisincreasesthepowerdi ssipationofthedesignby30%. 3.5.4OptimizationConsideringDeviceandInterconnectVa riations Inthissetofexperiments,bothdevicelevel(gatesize)var iationsaswellasinterconnectvariationsareincorporatedinthemodel,andtheimpactofinterc onnectvariationsontheoptimizationof objectivefunctionandthetimingyieldisanalyzed.Theres ultsarecomparedtothescenariowhere 59

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multi-metricoptimizationisperformedconsideringonlyg atesizevariations.Theoverallvariations duetointerconnectsareassumedtobe10%ofthenominalcase values,andareincorporatedinthe delayconstraintsbyaddingavarianceof10%totheoptimald elayvalues( t spec ).Theutilityassurance constant w iskeptas0 : 92.AsshowninFigure3.6,theriskaversegatesizingmethod ologyensures thatthetimingyieldofthecircuitisnotsacriced,eventh oughtheoptimizationresultsareaffected. Itisinterestingtonotethattheinthepresenceofintercon nectvariations,thetimingyieldformost casesactuallyimproves,sincethemethodologyisriskaver se,andintendstosatisfytheconstraints withhighutility.Duetothis,theoptimizationisadversel yaffected. Figure3.6Impactofinterconnectvariationsandgatesizev ariationsontheoptimizationofthemetrics.Intheseexperiments,theinterconnectvariations(o f10%variationfromthemeanvalue)are incorporateinadditiontothegatesizevariations.Theper centagechangeintheyieldimprovement forthetwocases(gatesizevariationsandgatesizeandinte rconnectvariationsrespectively),andthe correspondingpercentagechangeintheobjectivefunction achievementforthelattercase(gatesize andinterconnectvariations)ascomparedtotheformer(onl ygatesizevariations)areplottedinthe graph.Thevalueforexpectedutilityassuranceiskeptat w = 0 : 92. 60

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3.6Discussion Inthiswork,anewalgorithmforsimultaneousmulti-metric optimizationofdelay,dynamicpower, leakagepowerandcrosstalknoiseinpresenceofprocessvar iationsandscarceinformationhasbeen developed.Thealgorithmisindependentoftheunderlyingv ariationdistribution,andcanhandlethe impactofvariationsatseverallevelsincludingvariation sduetogatesizesandduetointerconnects. Thisexpectedutilitymaximizationbasedmethodologymode lstheprobleminamean-variancebased deterministiclinearprogrammingmodel,whichoptimizest heobjectiveswhileensuringhighlevels ofexpectedutilityforconstraintssatisfaction.Theexpe rimentsconductedontheITC'99benchmarks suggestthatthealgorithmismulti-foldfasterthantheexi stingmathematicalprogrammingalgorithms availableintheliterature,andensurescomparableoptimi zationresults.Goodoptimizationresultsand timingyieldsareobtainedwhentheutilityassuranceconst ant w iskeptatthelevelsbetween0 : 95and 0 : 90.Acomparativestudybetweenthesinglemetricandthemul ti-metricoptimizationrevealsthatthe improvementsinasinglemetricarelargelyachievedbyinco rporatingsub-optimalityinothermetrics. Thismethodiscapableofincorporatingmoreoptimizationm etricslikesecurityandreliabilityetc., ifthemetriccanbeexpressedasafunctionofgatesizes.Ing eneral,themethodologydevelopedin thisworkisafast,accurate,andefcienttoolfornano-lev elpost-layoutgatesizingoptimizationfor complexcircuits,wherethecompletevariationdistributi oninformationisunavailable. 61

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Table3.1Comparisonbetweenequallyweightedmulti-metri coptimizationofleakagepower,dynamicpowerandcrosstal knoiseforriskawaregate sizing(RAGS)andfuzzymathematicalprogramming(FMP). ComparisonforExpectedUtilityAssuranceConstant w = 0 : 90 ITC'99 Number Optimal †PerformanceofRAGSasComparedtoFMP Exec.Time(ET)(secs) Speed-up Benchmark ofgates Delay(ns) DynamicPower LeakagePower CrosstalkNoise RAGS FMP ofRAGS b11 385 0.71 15.65% 18.75% -5.91% 0.53 2.35 4.43x b12 834 0.36 5.86% 7.52% 9.44% 9.63 38.65 4.01x b13 249 0.26 -3.95% -6.05% -2.69% 0.143 0.848 5.93x b14 4232 2.5 -5.42% -5.59% 9.31% 23 177 7.7x b15 4585 3.43 -1.58% -1.26% 5.98% 54 213 3.94x b20 8900 3.59 8.89% 9.75% -4.58% 97 713 7.35x b22 12128 2.63 3.69% 4.08% 1.48% 145 978 6.74x b17 21191 2.68 3.89% 1.79% -3.99% 349 2338 6.7x PercentageChange 3.38% 3.62% 1.13% 5.85x ComparisonforExpectedUtilityAssuranceConstant w = 0 : 95 b11 385 0.71 13.40% 16.28% -7.49% 0.51 2.35 4.6x b12 834 0.36 4.03% 5.33% 14.56% 8.38 38.65 4.61x b13 249 0.26 -4.82% -7.59% 2.17% 0.131 0.848 6.47x b14 4232 2.5 4.66% 4.95% -9.20% 19 177 9.31x b15 4585 3.43 -6.78% -6.46% 13.21% 49 213 4.34x b20 8900 3.59 -9.12% -9.36% 0.82% 88 713 8.1x b22 12128 2.63 1.50% 1.75% -4.66% 141 978 6.93x b17 21191 2.68 2.53% 1.21% -2.34% 330 2338 7.08x PercentageChange 0.68% 0.77% 0.88% 6.43x †:ThePercentageChangeintheoptimizationofeachmetrici scomputedusingtheformula ( ET FMP ET RAGS ) 100 = ET RAGS 62

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Table3.2Comparisonbetweenequallyweightedmulti-metri coptimizationwith w = 0 : 92andsinglemetricoptimizationfordynamicpower,leakag e power,andcrosstalknoisemetrics. ITC'99 †DynamicPowerOptimization ‡LeakagePowerOptimization CrosstalkNoiseOptimization Benchmark DP LP Noise DP LP Noise DP LP Noise b11 33.05% -7.21% -9.57% -5.90% 36.94% -22.37% -11.73% -13.09% 29.49% b12 35.87% -9.03% -20.79% -8.29% 37.36% -33.76% -11.20% -12.09% 28.28% b13 1.18% -15.54% 3.01% -13.09% 1.83% 12.29% -18.51% -18.99% 77.61% b14 17.90% -3.54% 1.98% -2.87% 19.25% -26.84% -7.54% -7.77% 16.57% b15 40.47% -8.67% -2.82% -7.97% 41.20% -21.88% -21.37% -21.30% 42.91% b20 9.15% -14.17% 13.76% -13.15% 10.07% -29.70% -18.03% -18.75% 76.01% b22 18.07% -11.49% 6.45% -10.83% 19.33% -19.25% -13.24% -13.84% 57.59% Average 22.24% -9.95% -1.12% -8.87% 23.71% -20.21% -14.52% -15.12% 46.92% †:PercentageChangeinmetricvalueswhensinglemetricopt imizationforDynamicPowerisperformed ‡:PercentageChangeinmetricvalueswhensinglemetricopt imizationforLeakagePowerisperformed :PercentageChangeinmetricvalueswhensinglemetricopti mizationforCrosstalkNoiseisperformed :DynamicPower :LeakagePower 63

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CHAPTER4 INTEGRATEDFRAMEWORKFORCIRCUITOPTIMIZATION Inthenanometerregime,thetransitionoftheprocesstechn ologyfromonegenerationtothenext iscontributingtowardtheidenticationofnewmetricstha tcansignicantlyaffecttheperformance andreliabilityofthedesigns.Thus,circuitoptimization techniquesdevelopedtoaddressthecurrent generationVLSIoptimizationissuesmaynotbeapplicablet othefuturegenerationoptimizationrequirements.Also,awidespectrumofdevicesincorporateVL SIcircuitsasanintegralpartofthe design.Thedesignrequirementsofthesedevicesvarywidel y.Forexample,mobiledevicesprimarilyrequirelowpowerdissipatingdesign,whilemissioncri ticaldevicesmustensurethatthedesign isreliable.Thus,thereisaneedfornewhorizontallyandve rticallyintegratedcircuitoptimization solutionsthatarecompletelyrecongurableintermsofthe metricstobeoptimized,theoptimization methodologytobeutilized,andtherelativeprioritieswit hwhichthemetricsareoptimized.Thus,in thischapter,aframeworktooptimizemultipleperformance metricsinauniedmannerisdeveloped. Inthisvariationawareoptimizationmodel,arelationship betweentheoptimizationmetrics(likedynamicpower,leakagepower,andcrosstalknoise)asafuncti onofgatesizesisincorporatedinthe objectivefunction.Thedelayvaluesobtainedfromunconst raineddelayoptimization,andnoisemarginsobtainedfromthecouplingcapacitanceinformationfo rmtheconstraintsformtheoptimization problem,whichisthensolvedforsimultaneousoptimizatio nofmultiplemetrics.Theframeworkis independentoftheoptimizationmethodology,andcanbeimp lementedusinganymathematicalprogrammingapproach.Itiscompletelyrecongurableandgene ralizedsuchthatmetricscanbeselected, removed,orprioritizedforrelativeimportancedepending uponthedesignrequirements.Thisframeworkisimplemented,andtestedonITC'99benchmarksfordif ferentcombinationsofmulti-metric andsinglemetricoptimizationsofdelay,dynamicpower,le akagepower,andcrosstalknoise.The resultsindicatethattheapproachidentiesgoodsolution points,andisanefcientmechanismfor post-layoutoptimizationviagatesizing. 64

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4.1NeedforIntegratedFramework Inthenano-meterregime,theincreaseindensityandcomple xityoftheVLSIcircuitshasaffected thecircuitoptimizationprocessinseveralways.First,th einter-relationshipsbetweenthemetricslike delay,power,andcrosstalknoisehavebecomemoreintricat einsuchamannerthattheoptimization ofonemetricmayworsentheoptimalityofothermetrics.Thu s,theoptimizationofasinglemetric maynolongerbeadequate.Second,duetoaggressivescaling ,thewiringdensityandconsequently theaspectratiosinmetallineshaveincreased,there-byma gnifyingtheimpactofcouplingcapacitance betweenthenets.Thecrosstalknoiseinducedbetweentheco uplednetscouldcausefunctionalfailures inthecircuits.Asasimpleexample,ifagatesizingtechniq ueisaimedatonlypowerminimization ofagivencircuit,basedontimingconstraints,theresulti nggatesizecongurationcouldpotentially haveahighinterconnectcrosstalknoise. Accordingtothetechnologytrends[99],theseeffectsandu ncertaintiesareexpectedtoworsenin future,andoptimizationmethodologieswhichresultinape rformanceshiftfromoneobjectivetoanotherwillnotbeacceptable.FromtheVLSIcircuitoptimiza tionperspective,animportantchallenge istoidentifyverticallyaswellashorizontallyintegrate dsolutionmethodologies[14].Thisnecessitatestheexaminationofnewapproachesthatcansimultaneo uslyoptimizemultipledesignparameters forfeasiblesolutionstocircuitdesignproblems. Anotherimportantaspectinnano-levelVLSIdesignandtheo ptimizationprocessistoaddressthe effectofprocessvariations,whichintroduceuncertainty inthegeometriesofdeviceslikegatesizes (gatelength,oxidethicknessetc.)ofthefabricatedcircu its.AsshowninFigure2.5,atsub-100nm levels,theintra-dieparametricandsystematicvariation sarecomparabletotherandomvariations[1]. Theeffectsoflayoutschematicsaswellasparametricvaria tionsincreasesignicantlyduetothe shrinkinggeometries.Theuncertaintyduetothesemanufac turingvariationsimpactstheperformance characteristicsandthereliabilityofthecircuits.Anopt imizationmodelthatdoesnotincorporatethe impactofprocessvariationscouldresultininaccurateana lysis. Inthiswork,wepresentanewvariationawaremulti-metricg atesizingframeworkthatcanbeused toperformoptimizationofseveralmetricslikedelay,leak agepower,dynamicpower,andcrosstalk noiseetc.Thisapproachiscompletelyrecongurableandge neralizedintermsofitscapabilityto incorporatenewmetricsandselectivelyprioritizethemet ricsdependinguponthedesignrequirements, 65

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withminimalchangesinthemodel.Moreimportantly,anymat hematicalprogrammingapproachcan beutilizedwithinthisframework,tosolvetheoptimizatio nproblem.Animportantaspectofthis approachistheidenticationoftheinter-relationshipsb etweendynamicpower,leakagepower,and crosstalknoiseintermsofgatesizes,andmodelingthemina uniedframework. Inthisframework,sincedelayistheprimaryobjectiveinan ycircuitoptimizationprocess,itis optimizedwithhighestpriorityasarststepintheprocess .Thedelayvaluesobtainedfromunconstraineddelayoptimizationarethenusedasconstraintsdu ringthesimultaneousoptimizationofother metrics:dynamicpower,leakagepowerandcrosstalknoise. Theprocessvariationeffectsduetogate sizes(channellength,oxidethickness)areincorporatedi nthemodel,andagridbasedmodelisused toaddressthespatialcorrelationeffects[62]. ThestateoftheartresearchinVLSIdesignoptimizationtha tconsidertheimpactsofprocess variationshasbeendiscussedindetailsintheSection2.4o fChapter3.Themethodsimplementedfor optimizingvariousmetricshasbeenstudiedandcompared.A lso,thedifferentmathematicalprogrammingtechniquesavailableintheliterature,theirpropert iesandlimitationsarereviewedinthatsection. Oneofthese,oranyothermathematicalprogrammingmethodo logymaybeusedastheoptimization toolinthisframework,withoutanyapparentmodicationin themodeling.Thereadersmightwantto visitSections2.4and3.3againtofollowthematerialprese ntedintherestofthischapter. Therestofthechapterisorganizedasfollows.InSection4. 2,singlemetricmathematicalprogrammingoptimizationmodelsforleakagepower,dynamicpo wer,andcrosstalknoisearederived onthebasisoftheparametricmodelsderivedintheSection3 .3ofChapter3.Therelationshipbetweenthedesignparametersintermsofgatesizes,andthest epsinvolvedinmodelingtheproblemin amulti-metricoptimizationframeworkarepresentedinsec tion4.3.Also,amathematicalprogrammingapproachthatisutilizedforoptimizationinthiswork isbrieydiscussed.InSection4.4,the simulationsetup,experimentalresultsforvariousoptimi zationscenarios,andanalysisarepresented, followedbyasummaryanddiscussioninSection4.5.4.2SingleMetricOptimizationModels Animportantstepinmodelinganyprobleminamathematicalp rogrammingframeworkisto identifytherelationshipsbetweenthedesignparameters. Amodelthatrelatestheparametersina 66

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simpleyetaccuratefashionimprovestheoptimizationproc essbothintermsoftheefciencyand theapplicabilityofthemodel.Inthissection,wepresentt hegeneralmathematicalprogramming formulationsforoptimizationofeachmetric,delay,power ,andcrosstalknoise. 4.2.1UnconstrainedDelayOptimization Delayisanimportantoptimizationmetricinanycircuitopt imizationproblem.Sinceadelay optimizedcircuithashighertimingyield,thedelayiscons ideredasthemostcriticalmetrictobe optimizedinanygeneralizedframework.Inthepresenceofp rocessvariationsatthenanometerlevel, delayuncertaintycanbereducedbyperformingunconstrain eddelayoptimization.Themathematical programmingmodelforunconstraineddelayoptimizationis givenbyequation(4.1).Thenoisemargin constraintsthatcontrolthemaximumnoiseanetcantolerat ehasbeenderivedinEquation(3.28).The designconstraintsintermsofthenodedelaysinthepathsfo rmtheconstraintsfortheproblem. mint spec (4.1) s : tat i ( p )+ d i at i + 1 ( p ) 8 i 2 n ; 8 p 2 P N i U i 8 i 2 n d i = i 2 p ( a i b i s i + c i j 2 fo i s j ) l i s i u i 8 i 2 n SimilartotheEquation(3.30),the at i ( p ) isthearrivaltimeatthegate i inpath p d i istheinternalgate delayof i ,and at i + 1 ( p ) isthearrivaltimeatthenextgate i + 1inthepath p U i istheupperboundon thenoisemargin,and N i isthenoisemarginofthecurrentnet.Thevalues b i ,and c i aretheuncertain parametercoefcientvalues,and P isthesetofallthepathsinthecircuit.Thelineardelaymod elis adaptedfrom[95]andhasbeendescribedinEquation(3.22).4.2.2PowerOptimizationUnderDelayConstraints ThepowermodelsproposedbyGaoandHayes[51]havebeenadap tedtoidentifytheleakage powerandthedynamicpowerasafunctionofgatesizesinEqua tions(3.26)and(3.25)respectively. 67

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Theseequationscanbeusedtoformulatethemathematicalpr ogrammingmodelforpoweroptimizationunderdelayconstraints. From(3.26)and(3.25),itisidentiedthattheleakagepowe rofagateisdirectlyproportionalto itssize,andthedynamicpowerisproportionaltothesumoft hesizesofitsfan-outgates.Hence, usingthelineardelaymodel,theproblemofleakageanddyna micpoweroptimizationunderdelay constraintscanbegivenby(4.2). min n i = 1 ( q s i + f s i ) (4.2) s : t : d i d i ( max ) 8 i 2 n l i s i u i 8 i 2 n where, q correspondstothenormalizedimpact(weight)ofgatesizeo ntheleakagepower,and f correspondstothenormalizedimpactofthegatesizeonthef an-ingatesofthedesign,andeffectively thedynamicpowerimpact.Here, d i isthedelayofgate i d i ( max ) istheupperboundonthedelay ofgate i ,accordingtothetimingspecicationsinanodebasedmodel n istotalnumberofgatesin thedesign,and l i and u i aretheminimumandmaximumavailablegatesizesinthestand ardlibrary, respectively.4.2.3CrosstalkNoiseOptimizationUnderDelayConstraint s Therelationshipbetweenthesizesofthedrivinggatesofth ecouplednets,derivedinEquation (3.27)canbeincorporatedinformulatingalinearprogramm ingformulationforthecrosstalknoise optimizationunderdelayconstraints.Here,foreverygate s i ,thenoiseonitsfan-outnetisafunction ofthetotalcross-couplingcapacitanceonthenet.So,give naweightvector x asafunctionofthe cross-couplingcapacitanceonthenet,suchthat x i 8 i 2 n ,and x i 0,anequivalentnodebasedlinear programmingmodelforcrosstalknoiseoptimizationbygate sizingcanbeformulatedformaximizing theweightedsumofthegatesizes,underdelayandnoisemarg inconstraints.Thelinearprogramming 68

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formulationforcrosstalknoiseoptimizationcanbeformal lystatedas(4.3). max n i = 1 x i s i (4.3) s : t : d i d max 8 i 2 n N i U i 8 i 2 n l i s i u i 8 i 2 n Theproblemisformulatedasamaximizationproblemtominim izethenoiseoneachgatebymaximizingitssize,weightedbytheimpactofeachgateonthecro sstalkofeachoutputnet.Since,the impactofsizingissymmetriconthecouplednets,theoptima lsizesofthegatesareobtainedthat wouldreducethecouplingeffect.4.3IntegratedFrameworkforVariationAwareGateSizing Inthissection,adetaileddescriptionoftheuniedgatesi zingapproachispresented.First,we discussthevariousaspectsoftheobjectivefunctionmodel ing,inwhichtherelationshipsbetweenthe threemetrics,dynamicpower,leakagepower,andcrosstalk noisearecapturedasafunctionofgate sizes.Theobjectivefunctionshouldberecongurableinth esensethatanymetriccanbeinserted ordeleted,orweightedasrequired.Next,theintegratedfr ameworkisdiscussed,followedbythe mathematicalprogrammingformulationforvariationaware optimization.Finally,wewillbriey discusstheprocessofconvertingapathbasedapproachtoan odebasedapproach,whichimprovesthe runtimeofthealgorithm.4.3.1ObjectiveFunctionModeling Inthecontextofgatesizing,theimpactofthethreedesignp arameters,leakagepower,dynamic power,andcrosstalknoiseasafunctionofgatesizesisinco rporatedintheobjectivefunctionthatis tobeoptimized.Specically,theinter-relationshipofth ethreemetricsisasfollows. AsshowninEquation(3.26),theleakagepowerofagateisdir ectlyproportionaltothesizeof thegate,andhencehasadirectimpact.So,increasingthesi zeofthegatewouldincreasethe 69

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leakagepowerdissipationofthecircuit.Thecomponentsli ketheinputtransitionprobabilities, leakagecurrentandtheinputpatternsalsoimpacttheleaka gepowerdissipationofthecircuit. Therelationshipbetweenthedynamicpowerdissipationand thegatesizesisshowninEquation (3.25).Thedynamicpowerdissipationofagateisprimarily affectedbythetotalsizeofitsfanoutgatesinthecircuit.Thus,foragate i ,thetotalnumberofgatesitsfan-innetsareconnected to,determinetheimpactofthegate i onthedynamicpowerofthegatesthatareinitsfan-in. Hence,theweightforsizingthisgate i isproportionaltothenumberofgatesthatitisconnected tointheirfan-out. Asdiscussedintheprevioussection,thecrosstalknoiseon anetprimarilydependsuponthe sizeofitsdrivergateandthesizesofthedrivergatesofthe couplednets.Hence,crosstalknoise hasaninverserelationshipwiththegatesize.Ifthedrivin ggateofanetissizedup,thesignal strengthonthenetincreasesandhencethecrosstalknoiseo nthenetreduces.However,the up-sizingofthegatehasanadverseeffectonthecouplednet s.Byup-sizingthedrivergatesof thecouplednets,thenoiseonthosenetscanbereduced. Theseperformancemetricscanbemodeledinasingleobjecti vefunction,whichisoptimized inthepresenceofdelayconstraints.Hence,threedimensio nlessnormalizedcoefcients q f ,and x ,referringtotheimpactofgatesizingontheleakagepower, dynamicpower,andcrosstalknoise respectivelyareincorporatedintheobjectivefunction.T hecoefcient q isdirectlyproportionaltothe sizeofthegate,andcoefcient f isafunctionofthenormalizedimpactofthegatesizeontheg ates thatareinitsfan-in.Soifagate i isinthefan-outofalargenumberofgates,theimpactofup-s izing i willbehigherforthecircuit.However, x isinverselyproportionaltothesizeofthegate,andits normalizedvalueisafunctionofthemaximumcouplingcapac itanceofitscorrespondingnetwiththe aggressornets.Thehigherthecross-couplingcapacitance valuehigheristhecoefcientvalue. Toincorporatethecapabilitytoselectamongtheobjective functions,threeconstants a b ,and g aremultipliedtothecoefcients q f ,and x ,controllingtheimpactofthesecoefcientsonthenal objectiveachievement.Forexample,ifallthethreeobject ivesleakagepower,dynamicpower,and crosstalknoiseareequallyweighted,then a = b = g = 0 : 33. 70

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Theobjectivefunctionisgivenbythefollowingequation: min n i = 1 ( a q s i + b f s i g x s i ) (4.4) where, s i isthesizeofthegate i ,and n isthetotalnumberofgatesinthedesign. 4.3.2IntegratedFramework Aftertheobjectivefunctionisidentiedasshownin(4.4), anintegratedframeworkforthemultimetricoptimizationisformulated.Figure4.1showsthetop levelowchartfortheframework.During therststep,alinearprogramisformulated,asshownin(4. 1),withthedelay( t spec )astheobjective function,whichisminimizedtoidentifythebestpossiblec ircuitdelay.Thenominal(best)casedelay coefcientsareusedduringthisoptimization.Thisdeterm inisticoptimizationsolutiongeneratesthe delayspecicationsforallthepathsinthedesign,whichar eusedasconstraintsinthenextsteps. Inthenextstep,anymathematicalprogrammingmethodology canbeimplementedtoperformthe stochasticoptimizationbyincorporatingthedelayconstr aints(3.22)andthenoisemarginconstraints (3.28).Theweightedcoefcients q f ,and x areincorporatedintheobjectivefunction,whichallow toincorporatetheprioritiesforoptimizingthemetricsac cordingtothedesignerrequirements.The mathematicalprogrammingformulationisgivenin(4.5). min n i = 1 ( aq + bf gx ) s i (4.5) s : td p t spec 8 p 2 P N i U i 8 i 2 n l i s i u i 8 i 2 n d i = a i b i s i + c i j 2 fo ( i ) s i Aftertheproblemisformulatedinthemathematicalprogram mingframework,itcanbeconvertedinto astandardoptimizationlanguageformat(AMPLetc.),andca nsolvedusinganylinearprogramming solver. 71

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Figure4.1Gatesizingframeworkformulti-metriccircuito ptimization.Sincedelayistheprimary objectiveinVLSIoptimization,itisoptimizedseparately astherststepintheprocess.Next,theother optimizationmetricsaresimultaneouslyoptimizedbyinco rporatingthemintheobjectivefunction. Theobjectivescanberelativelyprioritized.4.3.3MathematicalProgrammingMethodology Anystochasticmathematicalprogrammingtechniquecanbei ncorporatedintheframeworkto solvethemulti-metricoptimizationproblem,providedtha ttheprobabilitydistributionsfortheinterdieandintra-dievariationsareavailable.However,theev aluationandoptimizationofthedistributions iscomputationallyintensive.Thisisattributedtothefac tthatexhaustiveMonte-Carlosimulationsare requiredtogeneratetheprobabilitydistributionsforall theparametervariations.Inseveralcases,appropriateempiricalinformationisnotavailable,thuslea dingtoinaccurateapproximation.However, 72

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itispossibleforexpertstopredictthepessimisticcorner s,andoptimisticcornersforthedifferentuncertainparameters.Intervalmathematicsbasedtechnique ssuchasfuzzymathematicalprogramming techniquecanusesuchtoplevelinformationtomakebetterd ecisionsinsuchsituations.Also,Buckley[100]hasshownthatfuzzyprogrammingbasedoptimizati onguaranteessolutionsthatarebetter oratleastasgoodastheirstochasticcounterparts,sincet heyidentifythesupremumofallthefeasible solutionsandnottheaverages.Thus,wechoosethefuzzymat hematicalprogrammingtechniqueas thesolutionmethodologytoillustrateourframework. Here,wewillbrieypresentthemethodologyandtheformula tions.AlgorithmAlgorithm4.1 showsthestepsinvolvedintheprocess.Inthefuzzymathema ticalprogrammingmethod[48],the parametricvariationsinthedelayequationaremodeledasf uzzynumbertripletsoftheform( b i b i g i b i + g i )and( c i c i h i c i + h i ).Here, g i and h i correspondtomaximumvariationsforthecoefcients b i and c i respectively.Thecoefcient b i approximatesthevariationineffectivechannellength( L eff ), whereas c i approximatesthevariationsinoxidethickness( t ox ). Algorithm4.1 Multi-metricgatesizingalgorithm Require: ParasiticinformationfromSPEFFile,Designvariablesfro mDEF,CAP,andstructural Verilogles,characterizedvaluesfornominalandworstca sedelaycoefcients a b ,and c Ensure: OptimalGatesizes 1: Evaluatethenominalcasedelay( t spec )bysolvingapathconstrainedlinearprogrammingformulation,incorporatingthenominalcasedelaycoefcientsint helineardelaymodelgiveninEquation (3.22) 2: Formulateandsolvethenominalcasemulti-metricgatesizi ngproblemthroughadeterministicLP formulationwithnoiseandpowerobjectives,andnominalca sedelayandnoisemarginconstraints. Also,incorporatethespatialcorrelationsinthemodeling oftheproblem 3: Storethenominalcaseresultsfornoise( N nc )andgatesizes( S nc ) 4: Formulateandsolvetheworstcasemulti-metricgatesizing problemthroughadeterministicLP formulationwithnoiseandpowerobjectives,andworstcase delayandnoisemarginconstraints. Also,incorporatethespatialcorrelationsinthemodeling oftheproblem 5: Storetheworstcaseresultsfornoise( N wc )andgatesizes( S wc ) 6: Usinggatesizevalues S nc and S wc ,andthenoiseresults N nc and N wc ,formulateacrispfuzzy non-linearprogramtomaximizethevariationparameter l ,underdelayandnoiseconstraints Intheworstcaseoptimizationscenario,themaximumpossib levariationsareassumedandapessimisticapproximationisperformed,andthedelayequatio nisgivenby(4.6). d i = a i ( b i g i ) s i +( c i + h i ) j 2 fo ( i ) s i (4.6) 73

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Inthenominalcaseoptimization,thedelayequationisgive nasfollows: d i = a i b i s i + c i j 2 fo ( i ) s i (4.7) Thenominalcase(step2)andtheworstcase(step4)gatesizi ngformulationsforfuzzyprogramming aregivenby(4.8)and(4.9)respectively.Thespatialcorre lationsareincorporatedinthemodeling bymultiplyingaconstantmultipliertoweightthevariabil ityimpactofafan-outgateonthedelay ofaparticulargate.Thefartherthefan-outgateis,thelow eristheweightandhencetheimpactof variationsonthatgate. min n i = 1 ( aq + bf gx ) s i (4.8) s : td p t spec 8 p 2 P N i U i 8 i 2 n l i s i u i 8 i 2 n d i = a i b i s i + c i j 2 fo ( i ) s i min n i = 1 ( aq + bf gx ) s i (4.9) s : td p t spec 8 p 2 P N i U i 8 i 2 n l i s i u i 8 i 2 n d i = a i ( b i g i ) s i +( c i + h i ) j 2 fo ( i ) s i Afterthedeterministicnominal( S nc N nc )andworstcase( S wc N wc )problemsaresolvedforthedelay, noiseandpowervaluesusingamathematicalprogrammingsol ver,thenoiseandgatesizevaluesare usedforformulatingacrispnon-linearprogrammingmodel. Usinganewvariationparameter l ,the fuzzyoptimizationproblemisformulatedusingthesymmetr icrelaxationmethod[101].Thegate 74

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sizingprobleminthepresenceofprocessvariationsisgive nby(4.10). max l (4.10) l ( S wc S nc ) GS + S wc 0 ; l ( N wc N nc ) GS + N wc 0 ; s : t : D p t spec 8 p 2 P andD p = i 2 p ( a i ( b i g i l ) s i +( c i + h i l ) j 2 fo ( i ) s j ) where,theparameter l isboundedby0and1.However,forthegatesizingproblem,as mallerbounds ofrangebetween0.5and0.75canbegivenfor l .Suchasmallerboundissufcientduetothedual requirementofhighyieldandlowoverheadforthegatesizin goptimizationinpresenceofvariations, andspeedsuptheprocedure2-3times,withoutaffectingthe nalsolution.Physically, l canbe consideredasthevariationresistance(robustness)prope rtyofthecircuit,meaningtheabilitytomeet timingconstrainteveninthepresenceofvariations.Hence ,theLPtriestomaximizethisvariation resistance.Thenoiseandsizingconstraintsensurethatth ecrosstalknoiseandthepowerarebetween theworstcaseandthenominalcasevalues.Thevariationres istancetriestoensurethattheoptimal solutionvaluesareclosertothenominalcasevalues,andth usminimizesthepowerandcrosstalknoise ofthecircuit.Ithasbeenshownintheliterature[102–104] thatthefuzzynon-linearprogramming solutionsproducethemostsatisfyingoptimizationresult sinthepresenceofuncertainty. 4.3.4PathstoNodes Animportantissueintheaforementionedoptimizationprob lemisthatitisintrinsicallyapath basedformulation.Thisissuecanbeaddressedbyconvertin gapathbasedformulationtoanode basedone,witheachnodecorrespondingtoagate[47].Ifwec onsidertwosimplepaths a c and b c ,where a and b arethenodescorrespondingtoprimaryinputsand c asprimaryoutputnode, 75

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thenconsideringadummysinknode s suchthat c s ,thenodebasedformulationcanbegivesas: min 3 i = 1 s i (4.11) s : t : d a t c d b t c t c + d c t spec where, t c isthearrivaltimeat c and d i saregivenbythelineardelaymodel.Thesuboptimalityintr oducedduetothistransformationiscloseto2%forcircuitsw ith20levelsoflogic.Thesub-optimality referstothevalueofdynamicpowerobtainedwhencomparedt othepathbasedformulation.However, thistransformationsignicantlyimprovestheruntimeand thefeasibilityofoptimizinglargecircuits. 4.4ExperimentalResults Inthissection,wepresenttheexperimentsconductedtoeva luatetheperformanceofthismultimetricoptimizationframework.Theframeworkwasrigorous lytestedforoptimizationinvarioussettingslikeequallyweightedmulti-metricoptimization,si nglemetricoptimization,andadaptivemultimetricoptimizationwherethemetricsareoptimizedwithdi fferentprioritiesbyassigningdifferent weightvectorstothemetrics.4.4.1SimulationSetup Themulti-metricoptimizationalgorithmforgatesizingun derprocessvariationswasrigorously testedontheITC'99benchmarkdesigns.Thesimulationsetu phadthreeimportantsteps.Duringthe rststep,theRTLlevelVHDLnetlistsofthebenchmarkcircu itswereextractedforgeneratinggate levelVeriloglesusingtheSynopsysDesignCompilertool. ThesegatelevelVeriloglesandthe TSMC180nmStandardCelllibraries(LEF,TLF,DBFilesetc.) arethenusedtoplaceandroutethe designsandgeneratetheDEFles,celldelayinformationet cusingtheCadenceDesignEncounter toolbox.WesynthesizedthebenchmarkcircuitsusingTSMC1 80nmlibraries. Inthesecondstep,theparasiticresistanceandcapacitanc einformation(SPEFle)wasextracted fromtherouteddesignsusingtheCadenceFireN'IceRCextra ctor.Thisinformationisrequiredfor 76

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extractingthecouplingnoiseoftherouteddesigns.TheSPE Flewasthenusedtoobtainthecoupling capacitanceandresistanceinformationofeachnetwithits aggressornetsusingaPERLscriptthatwas writtentoextractthiscouplinginformationforeachnet,w ithitstopthreeaggressornets.Duringthe thirdstep,thedelaycoefcientsforavailablesizes(1x-6 x)andfan-outsofthestandardcellsin theTSMC180nmstandardcelllibrarywerecharacterizedusi ngtheHSPICEsimulations.Also,the variationswereassumedtobe25%ofthenominalvalues[56], andwereappropriatelytranslatedto thecoefcients a b and c .Withalltheinformationaboutthedelays,couplingnoisee tc.available forthebenchmarkdesigns,thenominalcaseandtheworstcas egatesizingoptimizationproblems wereformulated.TheseproblemswereprogrammedinClangua ge,whichgeneratedtheoptimization modelsinthestandardAMPLformat.Theseextremecaseprobl emswerethensolvedusingaLinear programmingsolvercalledKNITRO.KNITROisarobustnon-li nearprogrammingsolverforboth convexandnon-convexoptimizationproblems.Itisspecic allydesignedtosolveproblemswithlarge dimensionality.KNITROusesinteriorpointandactivesetm ethodsforoptimization,andiscapable ofutilizingmultipleprocessors.Thesolverisavailablea sapartoftheNEOSoptimization[37]suite. Theoptimalsolutionsoftheworstcaseandthenominalcases ettingsarethenutilizedtoformulatea crispnon-linearprogrammingproblem,andsolvedusingKNI TRO.Adetailedowofthesimulation setupisshowninFigure4.2.4.4.2OptimalNoiseMargins Ideally,thenoisemarginisgivenasthedifferencebetween theminimumsizedvictimandthe maximumsizedaggressorgates.Thiscorrespondstothemaxi mumcouplingimpactonthevictimnet. Whenthevictimdrivergateissizedatitsminimumpossibles ize,thesignalstrengthonthevictimnet isverylow.Additionally,iftheaggressornet'sdrivergat eissizeduptothelargestpossiblesize,the aggressornet'ssignalstrengthincreases,therebyinduci ngalargecross-couplingcapacitance,which canaffectthevictimnet'ssignalintegrity.Hence,aconst rainttocontrolthemaximumtolerablenoise onanetisincorporatedinthemodel. Thegatesizesconsideredinthissimulationsetuprangefro m0 : 25 e 6 to1 : 5 e 6 .Therefore,the noisemarginforanetcanbegivenby(4.12). N =( 1 : 5 e 6 ) ( 0 : 25 e 6 ) ( 1 : 25 e 6 ) (4.12) 77

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Figure4.2Flowchartforsimulationsetup.180nmstandardc elllibrarieshavebeenusedtoextractthe requiredlesfortheITC'99benchmarkcircuits.However,theseidealnoisemarginsarenottight,andcouldn otcapturetheimpactofcouplingnoise effectively.Thus,experimentswereperformedtoidentify theoptimalnoisemargins,byevaluating theimpactofdifferentnoisemarginvaluesontheobjective functions.Theexperimentalresultsare showninFigure4.3.Here,ifthenoisemarginisbelow0 : 65 e 6 ,thedynamicpowerisadversely affected,eventhoughleakagepowerandcrosstalknoiseare unaffected.Thus,afteraveragingthe effectsofnoisemarginonthecrosstalk,andthepower,atig hternoisemarginconstraintof0 : 65 e 6 wasidentied.Thistighternoisemargincanefcientlymin imizetheeffectofcouplingnoise,and 78

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generatebettersolutionpoints.Thesenoisemarginsareus edasconstraintinthe(4.9)tocontrolthe crosstalknoisebetweenthenetsduringthegatesizing. Figure4.3Effectofdifferentnoisetolerancevaluesonthe optimalityoftheobjectives.Tightertolerancevaluesforthenoiseconstraintderivedin(4.12)hav ebeenappliedandtheirimpactonthe leakagepower,dynamicpowerandcrosstalknoisehasbeenpl ottedtoobtainatighternoisemargin fortheoptimizationprocess.4.4.3DeterminationofTimingSpecication Intherststepoftheoptimizationprocess,thecircuitdel ayisoptimized,sinceitistheprimary optimizationmetric.Anodeconstrainedlinearprogramisf ormulatedusing(4.1),andsolvedto evaluatethebesttimingspecicationsforeachgateinthed esign.Thenoisemarginconstraintsensure thatthemaximumtolerablenoisemarginsaremaintaineddur ingtheoptimizationprocess.However, thedelayoptimizationresultsinsub-optimalvaluesforle akagepower,dynamicpowerandcrosstalk noise.Theoptimaldelayvaluesidentiedduringthissteps erveastheconstraintsforthenodedelays duringthenextstepsofmulti-metricoptimization.Theopt imaldelayvalues( t spec )fortheITC'99 benchmarkcircuitsareshowninthethirdcolumnofTable4.1 79

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4.4.4LeakagePower,DynamicPower,andCrosstalkNoiseOpt imization Weevaluatedtheoptimizationresultswhenleakagepower,d ynamicpower,andcrosstalknoise aresimultaneouslyoptimizedwithequalpriority( a = b = g = 0 : 33).Thetechnologicalconstraintsof nodedelayswithdelayvaluesandnoisemarginconstraints, alongwiththeboundingconstraintscorrespondingtominimumandmaximumavailablesizesofthegat esareusedtoformulatetheworstcase andthenominalcaseoptimizationproblems.Foreachofthes eproblems,therespectivecharacterized lineardelaycoefcients a b ,and c and a i b i ,and c i areincorporatedin(4.9),whichwassolvedusing theKNITROsolver.Thedeterministicnominalcase( S nc N nc )andworstcase( S wc N wc )powerand noiseresultsarethenutilizedinformulatingthefuzzymat hematicalprogramasshownin(4.10).The solutionofthecrispnon-linearproblemobtainedusingthe KNITROsolvergivestheoptimizationsin theleakagepower,dynamicpowerandcrosstalknoiseforthe circuit. Theoptimizationimprovementsindynamicpower,leakagepo wer,andcrosstalknoiseascomparedtothesub-optimalvaluesobtainedduringthedelayop timizationareshowninTable4.1.As evidentfromthetable,themulti-metricoptimizationresu ltsaresignicantlyimprovedoverthesuboptimalvaluesfromunconstraineddelayoptimization.The incorporationofspatialcorrelationsduring themodelingoftheproblemfurthereliminatethepessimism byreducingtheeffectofvariationsin circuitelementsthatarenotinthesamegridasthecurrente lement. Table4.1Improvementintheoptimizationofmetricsformul ti-metricoptimizationwithequalpriority ( a = b = g = 0 : 33),ascomparedtothevaluesobtainedduringunconstraine ddelayoptimization. ITC'99 Number UnconstrainedDelay †ImprovementinMetrics Execution Benchmark ofgates t spec (ns) LeakagePower DynamicPower CrosstalkNoise Time(secs) b11 385 0.71 12.75% 19.8% 28.1% 2.35 b12 834 0.36 14.18% 20.5% 34.76% 38.65 b13 249 0.26 57.5% 66.2% 59.98% 0.848 b14 4232 2.5 38.0% 17.92% 125.25% 177 b15 4585 3.43 25.63% 42.0% 42.35% 213 b17 21191 2.68 46.38% 57.33% 62.87% 2338 b20 8900 3.59 18.95% 21.57% 72.79% 713 b22 12128 2.63 14.57% 58.69% 59.41% 978 AverageSavings 28.49% 38.0% 60.7% †:Percentageimprovementovertheunconstraineddelayopt imization Anotableaspectofthismethodistheruntimeofthealgorith m.Asshownincolumn7ofthe Table4.1,theruntimeforthealgorithmiscomparativelylo wforeven21000gatesdesigns.Thisis attributedtotheoptimummodelingoftheproblemasanodeba sedapproachascomparedtoapath 80

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basedapproach.Also,selectionoftheoptimizationsolver playsanimportantroleincontrollingthe runtime.KNITROisafastandaccuratesolver,availablefor bothlinearoptimizationandnon-linear optimizationproblems. Next,acomparativestudybetweenthepessimisticworstcas eanalysisandthefuzzyanalysiswas performedtostudytheeffectivenessofthemathematicalpr ogrammingtechniquebeingimplemented asasolutionmethodologyforourframework.AsshowninFigu re4.4,fuzzymathematicalprogrammingidentiedthesolutionpointsthatsignicantlyimpro vedovertheworstcasevalues,andthe valueswereclosertonominalcaseanalysis.Theimprovemen tsintheoptimizationarenotablesince theaveragetotalpowersavingsaremorethan30%,andthecro sstalknoiseimprovementismorethat 40%. Figure4.4Averageimprovementinthemetricsvaluesforsim ultaneousmulti-metricoptimizationas comparedtothedeterministicworstcasepessimisticanaly sis.Themetricsleakagepower,dynamic powerandcrosstalknoiseareweightedas a = b = g = 0 : 33respectively. 4.4.5SingleMetricOptimizationResults Thisframeworkallowsforselectiveoptimizationofthemet rics,dependinguponthedesignrequirements.Themetrictobeoptimizedcanbeprioritizedby assigningahighweightvectortoit.For 81

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example,ifadesignerintendstooptimizeonlytheleakagep ower,thecoefcients a b ,and g will beassignedthevaluesas1,0,0respectively.However,when onlyleakagepowerisoptimized,the crosstalknoisemaybeaffectedadversely.Theimpactofsin glemetricleakagepoweroptimization ascomparedtotheequallyweightedmulti-metricoptimizat ionisshowninFigure4.5.Asshown, whenonlyleakagepowerisoptimized,theoptimizationmayr esultinsub-optimalityintroducedin othermetrics,likecrosstalknoiseinthiscase.Thisisint uitive,sincetheleakagepowerisdirectly proportionaltothegatesizes.However,sincethecrosstal knoisehasaninverserelationshipwiththe gatesizes,thenoisemayincreaseasaresultofoptimizatio n.Sincedynamicpowerisaffectedbythe sizeofthefan-outgates,ifthegatessizedduringtheleaka gepoweroptimizationaresameastheones thataffectthedynamicpower,thendynamicpowerwouldalso reduce.However,intheotherscenario, dynamicpowerwouldbeadverselyaffected. Figure4.5Effectofsinglemetricleakagepoweroptimizati onascomparedtoequallyweightedmultimetricoptimization.Forsinglemetricoptimizationtheme tricsareweightedas a = 1 ; b = 0 ; g = 0, andformulti-metricoptimizationthemetricsareweighted as a = 0 : 33 ; b = 0 : 33 ; g = 0 : 33. Wealsoperformedexperimentsforsinglemetricoptimizati onoftheothertwometrics,dynamic power(withweights a = 0 ; b = 1 ; g = 0),andcrosstalknoise(withweights a = 0 ; b = 0 ; g = 1).The resultsforsinglemetricdynamicpoweroptimizationascom paredtoequallyweightedmulti-metric 82

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optimizationareshowninTable4.2.Thedynamicpowerdissi pationforsinglemetricoptimizationis lowerthanthemulti-metricoptimization.However,aninte restingobservationinthisscenarioisthat ontheaverage,dynamicpowerdissipationisreducedatthec ostofleakagepowerandnotcrosstalk noise.Thistrendoccursduetothefactthatduringthedynam icpowerminimization,fewergatesare resizedfromthesub-optimalsizes(afterdelayoptimizati on)ascomparedtoleakagepoweroptimization.Thisresultsinadecreaseindynamicpower,butthelea kagepowerislargelyunaffected. Whenthesinglemetriccrosstalknoiseminimizationisperf ormed,thegatesizingproblemtranslatesintoamaximizationproblem.Theresultsforcrosstal knoiseminimizationascomparedtothe equallyweightedmulti-metricoptimizationarealsoshown inTable4.2.Anotableimprovement(almost2x)incrosstalknoiseascomparedtothemulti-metrico ptimizationscenarioisidentied.Thisis duetothefactthatthemaximizationproblemsatisesthede layconstraintsmucheasilyascompared totheminimizationproblem.Increasingthegatesizesredu cesthecrosstalknoise,aswellasthegate delays.However,thisincreasesthepowerdissipationofth edesignbymorethan40%. 4.4.6ResultsforPriorityBasedOptimization Theframeworkcanbeutilizedforadaptivemulti-metricopt imizationinsituationswherethedesignrequiresthemetricstobeoptimizedwithdifferentpri orities.Insuchscenarios,thecoefcients a b ,and g areassignedweightscorrespondingtotherelativepriorit ies.Weperformedtheexperimentswithtwosuchscenarios.First,anoptimizationispe rformedwithequalprioritiesassigned toleakagepoweranddynamicpower,whileneglectingtheimp actofcrosstalknoise.Theweights wereassignedas a = 0 : 5 ; b = 0 : 5 ; g = 0.Theresultsforoptimizationwerecomparedwiththeequal ly weightedmulti-metricoptimizationscenariotoidentifyt hepercentageimprovementinthetwometricsleakagepoweranddynamicpower.Sincetheweightswere increasedbyapproximately17%for eachmetric,weevaluatediftheoptimalityofeachmetricfo llowsthesametrend.Theresultsforthe percentageimprovementinmetricsforthebenchmarksaresh owninFigure4.6.Theaveragedynamic powerimprovementwas11.1%andaverageleakagepowerimpro vementwas12.7%.Although,the improvementswerenotofthesameorder,theywerecoherentw iththeexpectations. Finally,tocomparethethreescenariosdiscussedinthisse ction,unconstraineddelayoptimization, singlemetricoptimization,andmulti-metricoptimizatio n,wecomparedtheleakagepowervalues 83

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Table4.2Comparisonofsinglemetricdynamicpowerandcros stalknoiseoptimizationwiththe equallyweightedmulti-metricoptimizationvalues. DynamicPowerOptimization a = 0 ; b = 1 ; g = 0 ITC'99 †ImprovementinMetrics Benchmark DynamicPower LeakagePower CrosstalkNoise b11 13.49% -15.11% -12.02% b12 8.02% -11.73% 2.88% b13 13.62% -34.76% -6.66% b14 5.00% -10.26 0.70% b15 20.57% -10.82 2.30% b20 6.42% -14.75 8.83% b22 14.32% -6.74 25.30% Average 11.64% -14.88% 3.05% CrosstalkNoiseOptimization a = 0 ; b = 0 ; g = 1 ITC'99 †ImprovementinMetrics Benchmark DynamicPower LeakagePower CrosstalkNoise b11 -19.66% -21.34% 143.20% b12 -20.69% -22.38% 197.71% b13 -35.91% -38.40% 106.84% b14 -15.81% -16.11% 97.23% b15 -19.15% -18.88% 183.50% b20 -19.15% -19.81% 146.88% b22 -10.01% -10.19% 111.26% Average -20.05% -21.02% 140.95% †:Percentageimprovementascomparedto equallyweightedmulti-metricoptimization obtainedduringthesinglemetricleakagepoweroptimizati on,andthemulti-metricoptimizationwith leakageweightedas a = 0 : 3withsub-optimalleakagepowervaluesobtainedduringunc onstrained delayoptimization.Theimprovementsinthemetricareshow ninFigure4.7.Theresultsindicate thatthesingleobjectiveoptimizationidentiesmostopti malvaluesforthemetric,followedbythe multi-metricoptimization,whichisintuitive.However,s uchoptimizationsintroducesub-optimality inothermetricslikecrosstalknoiseanddynamicpower.4.5Discussion Inthiswork,anewintegratedframeworkforvariationaware multi-metricoptimizationhasbeen developedforoptimizationofseveralmetricslikedelay,l eakagepower,dynamicpower,andcrosstalk noise.Anymathematicalprogrammingapproachcanbeutiliz edtoimplementthisframework.Inthis 84

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Figure4.6Theimprovementinleakagepoweranddynamicpowe rwhenoptimizedwithpriorities a = 0 : 5, b = 0 : 5,and g = 0ascomparedtothescenariowhere a = 0 : 33, b = 0 : 33,and g = 0 : 33. Figure4.7Comparativestudyofleakagepoweroptimization inthreedifferentscenarios,unconstraineddelayoptimization,singlemetricleakagepowero ptimization,andmulti-metricoptimization. Thecoefcientscorrespondingtothemetricsinmulti-metr icoptimizationareassignedas a = 0 : 3, b = 0 : 45and g = 0 : 25. 85

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work,weidentifytherelationshipsbetweenthedelay,leak agepower,dynamicpower,andcrosstalk noisemetricsasafunctionofgatesizesandmodeltheminaun iedmanner.Additionalmetricslike security,reliabilityetc.intermsofgatesizescanbeinco rporatedintheoptimizationframeworkwithoutanymodications.Theframeworkiscompletelyrecongu rableintermsofdesignrequirements toselectivelyoptimizeoneormoremetricsbyassigningapp ropriateweightstothemetrics. TheexperimentsperformedontheITC'99benchmarkcircuits indicatethattheequallyweighted multi-metricoptimizationachievesgoodresultsintermso foptimizingthevaluesofallthemetrics. Also,theweightsassignedtoeachmetricinthemodelareapp roximatelylinearlycorrelatedwith theaverageimprovementsintheoptimizationvalues,andhe ncecanbeusedtoprioritizethemetrics. Although,singlemetricoptimizationachievesmaximumsav ingsforthecorrespondingmetric,such anoptimizationintroducessignicantsub-optimalityint hevaluesofothermetrics.Experimentsand comparativeanalysisofdifferentoptimizationscenarios advocatetheefcacyofthisframeworkasa generalizedpost-layoutmulti-metricoptimizationtool. 86

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CHAPTER5 AMICROECONOMICAPPROACHTOSPATIALDATACLUSTERING Inanoptimizationproblem,asthesizeofaproblemincrease s,themostefcientwayofsolving itistoformmultiplepartitionsonthebasisofcertaincrit eria,andtosolveeachpartitionseparately. However,clusteringisanoptimizationproblem,sinceclus tersarerequiredtobeidentiedonthe basisofspecicobjectives.Ingeneralspatialpatternclu steringdomain,severaltechniqueshavebeen developedinawidevarietyofscienticdisciplinessuchas biology,patternrecognition,information systemsetc.Whilethesetraditionaldisciplinesfocusond evelopingalgorithmstoperformsingle metricclustering,variousengineeringandmulti-discipl inaryapplicationsinemergencymanagement, computernetworks,VLSI,androboticsentailsimultaneous examinationofmultiplemetricsforspatial patternclustering.Inthiswork,wedevelopanovelmulti-o bjectiveclusteringapproachthatisbased ontheconceptsofmicroeconomictheory.Thealgorithmmode lsamulti-step,normalformgame consistingofrandomlyinitializedclustersasplayerstha tcompetefortheallocationofresources(data objects).ANashequilibriumbasedmethodologyevaluatesa solutionthatissociallyfairforallthe players.Aftereachstepinthegame,theclustersareupdate dusinganymathematicalclustering algorithms.Extensivesimulationswereperformedonsever alrealdatasetsaswellasarticially synthesizeddatasetstoevaluatetheefcacyofthealgorit hm.Theexperimentalresultsindicatethat ouralgorithmyieldssignicantlybetterresultsascompar edtothetraditionalalgorithms.Further, thealgorithmyieldsahighvalueforthe fairnessindex ,whichindicatesthequalityofthesolutionin termsofsimultaneousclusteringonthebasisofmultipleob jectives.Also,thesensitivityofthevarious designparametersontheperformanceofouralgorithmisana lyzedandreported. 5.1SpatialDataClustering Formally,theclusteringproblemcanbedenedasanoptimiz ationproblem[12,90]:Givenaset ofinputpatterns X = f x 1 ; ; x j ; ; x N g ,apositiveinteger K ,adistancemeasure d ,andacriterion 87

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function J ( C ; d ( :;: )) on K -partition C = f C 1 ; ; C K g of X and d ( :;: ) ,where x j =( x j 1 ; x j 2 ; ; x jd ) T 2 d ,andeach x ji isafeatureinthefeaturespace,partition X intodisjointsets C 1 ; ; C K ( K N ) such that J ( C ; d ( :;: )) isoptimized(minimizedormaximized).Thedifferentclust eringcriteria J ,andthe distancefunctions d ( :;: ) denethevariousclusteringobjectives,whichmaybeconi ctinginnature. Asanexample,theobjectiveslikespatialseparationandco nnectednessfollowaninverserelationship. Similarly,thecompactnessoftheclusterisinverselyrela tedtotheequi-partitioningobjective.Hence, mostoftheexistingclusteringmethodologiesattempttoop timizejustoneoftheobjectivesthatare identiedtobethemostappropriateinthatcontext.Thisre sultindiscrepanciesbetweensolutions providedbydifferentalgorithmsandcouldcauseaclusteri ngmethodtofailinthecontextswherethe criterionisinappropriate. Theapplicationssuchasrescuerobotsdeployment,ad-hocn etworks,wirelessandsensornetworks,andmulti-emergencyresourcemanagementhaveneces sitatedtheidenticationofnewclusteringmechanismsthatcouldsimultaneouslyoptimizemult ipleobjectives,whichmaybecompetitive innature.Asanexample,letusconsideraproblemofestabli shmentofanad-hocnetworkofnodes thatcommunicatewitheachotheroverawirelesslink.Altho ugheachnodehasidenticaltransmission andcomputingcapabilities,duetopowerconstraints,clus tersarerequiredtobecreatedtoreduce thecommunicationoverhead.Eachclustershouldhaveaclus terheadthatisresponsibleforinterandintra-clustercommunication.Anoptimalclusteringme chanismneedstoensurethatthenodesdo notdropoutofthenetwork.Hence,clusteringshouldbeperf ormedonthebasisofmultiplecriteria; compactnessforlowpowerintra-clustercommunication,an dequi-partitioningforuniformpowerdistribution.Theseobjectivesarecompetitiveinnatureandn eedstobeoptimizedsimultaneouslyusing a multi-objectiveclusteringtechnique Inthisresearch,weinvestigateanovelmethodologythatid entiesoptimalclustersinapplications withmultipleconictingobjectives.Thismethodologycon sistsofthreecomponents,aniterative hillclimbingbasedpartitioningalgorithm,amulti-stepn ormalformgametheoreticformulation,and aNashequilibriumbasedsolutionmethodology.Specicall y,inthisclusteringmechanism,initial clustersareidentiedusingamathematicalapproach(KMea nsorKMedoids)followedbyagameformulationwiththeseclustersasplayersandresources.Itth enidentiesasolutionusingtheconceptsof Nashequilibrium.Sincetheobjectivesareconvexinnature ,asshownin[13]aNashequilibriumsolu88

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tionalwaysexists,andtriestoachieveglobaloptima.Also ,dependingupontheproblemformulation, thecomplexityoftheNashequilibriumliesbetweenPandNP[ 13]. Abriefreviewtheexistingclusteringmethodologies,thev ariousapplicationdomainsofgame theory,andtheapplicationsofload-sharingarediscussed inSection2.5.Therestofthechapteris structuredasfollows.InSection5.2,wediscussthemotiva tionforidentifyingtheapproachformultiobjectiveclustering.Theclusteringmethodologyisprese ntedindetailinSection5.3.Experimental resultsarepresentedinSection5.4,followedbyadiscussi onontheapplicationsandthepossible futureresearchinSection5.5.5.2WhyGameTheoryforClustering? Traditionally,indataclustering,asingleparameterisop timizedwhileassumingtheotherparametersasconstraints.However,theclusteringrequirem entsofmulti-disciplinaryapplicationshave resultedintheneedfornewmulti-metricclusteringmethod s.Incontrasttotheensemblemethodsthat effectivelyintegratetheresultsofmultiplesingleobjec tiveclusteringmethods,thefundamentalbasis ofgametheoryallowsfortheformulationofproblemsasmult ipleinter-relatedcostmetricscompetingagainstoneanotherforsimultaneousoptimization.Ing ametheory,eachplayer'sdecisionisbased uponthedecisionsofeveryotherplayerinthegame,andheca noptimizehisgainwithrespectto theirgains.Thisresultsinidenticationofglobalgains, andconsequentlyanequilibriumstatefor thesystem.Asanexample,intheprocessofclusteringtheda taobjectswithanobjectiveofmaximizingpartitioncompactness,oftenclustersareformedsu chthatsomepartitionshavefewobjects, whileothershavingmanyobjects,resultinginasituationo fpartition-imbalance.However,aclusteringperformedwithload-sharingorequi-partitioningasob jectivecouldresultinformationofclusters withlargeintra-clusterdistances.Thus,suchsituations areconvexinnature,andcanbesuccessfully modeledinagameframework.Also,asshownin[13],ifthepay offfunctioninagameisconvex,a Nashequilibriumsolutionalwaysexistsandtendstoidenti fygloballyoptimalsolutions[105].This isagoodmotivationformodelingthesysteminagametheoret icframeworkforsimultaneousmultiobjectiveclustering. Auniquepropertyofgametheoryissocialequityorsocialfa irness[13],whichensuresthateach playerinthegameissatisedandtheoverallgoalsarereach ed.Asanexample,forclusteringon 89

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thebasisofthreeparameters,compactness,equi-partitio ning,andconnectedness,theothermethods identifysolutionstargetingtheglobalobjectiveasafunc tionofthesedesignparameters.However, agametheoreticmodelensuresthateachoftheseparameters isoptimizedwithrespecttotheothers. Foranelaboratediscussionofgametheorythereaderscanre ferto[27,28]. 5.3MicroeconomicClusteringAlgorithm Inthissection,adetaileddescriptionofthegametheoreti calgorithmispresented.Initially,one ofthemathematicalclusteringmethods(KMeans,inthiswor k)isbrieyexplained,followedbya thoroughdiscussionofthekeycomponentsofthegametheore ticmodel,andthemodelitself.An alternativeensemblebasedpost-mathematicalpartitioni nggametheoreticmethodisalsopresented. Thesectionconcludeswiththeanalysisofcomplexityofthe model,andtheproofofprogressionof algorithm. Certainassumptionshavebeenmadeduringthemodelingofth eproblemasagametheoretic framework.Mostoftheseassumptionsarenotrestrictivein termsoftheapplicabilityofthemodel, andcanbediscardedwithnoorverylittlechanges.Inthismo del,theobjectivesbeingconsidered arecompactnessandequi-partitioning,butthemethodolog yisapplicabletoanytypeandnumberof objectives,conditionalupontheconvexityoftheproblem. Thenotationsandterminologyusedinthe restofthechapteraregiveninTable5.1.5.3.1MathematicalPartitioning Theinitialsetofclustersisidentiedusingoneofsimples tpartitioningmethodKMeans.This algorithmpartitionsadatasetofsize N into K clustersonthebasisofminimizationofthetotalintraclustervariation(TICV).Thestepsinvolvedintheiterati veKMeansalgorithmareshowninAlgorithm Algorithm5.1. Let f x i ; i = 1 ; ; N g beasetofdatavectorssuchthat x i = f x i 1 ; ; x id g .Deneaboolean w ik for i = 1 ; ; N and k = 1 ; ; K w ik = 8><>: 1if i thvectorbelongsto k thcluster 0otherwise (5.1) 90

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Table5.1Notationsandterminology.Thesenotationsareus edintheequationsandalgorithmsdescribedinrestofthechapter. N Totalnumberofdataobjectsinadataset d Dimensionalityofdataset K Totalnumberofclusters E nk Euclideandistancebetween n and k ,where n 2 N and k 2 K E SumofthesquaredEuclideandistance l k Numberofdataobjectsincluster k ; 8 k 2 K l ideal Numberofdataobjectsperclusterinequi-partitionedstat e; l ideal = j N = K j L Sumofthesquaredloadvalues; L = Kk = 1 ( l k l ideal ) 2 P Totalnumberofplayers; P K p i i th playerinagame; 8 i 2 P p i Setofplayersinthegameotherthantheplayer p i R Totalnumberofresourcecenters; R K r j j th resourcecenterinagame; 8 j 2 R r j Setofalltheresourcecentersnotinthecurrentgame U i Totalnumberofstrategiesofaplayer p i S i Setofallthestrategiesofplayer p i s iu u th strategyoftheplayer p i ; s iu 2 S i and u = 1 ; ; U S Strategysetofallthestrategiesinthegame; S = f S 1 ; S 2 ; ; S P g S i Setofallthestrategycombinationsofallplayersothertha n p i s i v Astrategycombinationconsistingofonestrategyofallpla yersotherthan p i ; s i v 2 S i Deneamatrix W =[ w ik ] suchthat K k = 1 w ik = 1,i.e.,adatavectorcanbelongtoonlyonecluster(hard partitioning).Now,let c k =( c k 1 ; ; c kd ) bethecentroidof k thcluster,where c kj isgivenbyequation 5.2. c kj = N i = 1 w ik x ij N i = 1 w ik (5.2) Then,theintra-clustervariationfor k thclusterandtheTICVbasedupontheEuclideandistancemea sureisgivenbyEquations(5.3)and(5.4)respectively. E ( k ) ( W )= N i = 1 w ik d j = 1 ( x ij c kj ) 2 (5.3) E ( W )= K k = 1 N i = 1 w ik d j = 1 ( x ij c kj ) 2 (5.4) 91

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TheobjectiveoftheKMeansclusteringistoidentifytheclu stersthatminimizethesumofsquared Euclidean(SSE)distancemeasureandhenceisgivenas, E ( W )= min W f E ( W ) g (5.5) AlthoughKMeansisfast,thealgorithmissensitivetothese lectionofinitialcluster-headpositionsand Algorithm5.1 KMeanspartitioning Require: K,datasetofsize N anddimensionality d Ensure: theassignment w nk 8 n 2 N ,where k 2 K 1: randomlyinitialize K locationson d dimensionspacewithcentroids c k ; 8 k 2 K 2: initializeiterationnumber i 0 3: repeat 4: i i + 1 5: for n = 1to N do 6: calculate E nk ; 8 k 2 K 7: nd k 0 ,suchthat E nk 0 = min f E nk g 8: w ink 0 1,and w ink 0 ; 8 k 6 = k 0 9: endfor 10: update c k accordingtoequation5.2, 8 k 2 K 11: until w ink = w i 1 nk ; 8 n 2 N and k 2 K 12: return: w nk w ink ; 8 n 2 N and k 2 K caneasilyconvergetolocaloptimaifthechoiceofinitialp artitionsisimproper.Also,thealgorithm isapplicableonlyforsingleobjectiveclustering.5.3.2Multi-StepNormalFormGameModel TheKMeansidentiesthepartitionsonthebasisofminimiza tionofSSE.However,thisprocess adverselyaffectsthecomplementaryequi-partitioningob jective.Agametheoreticmethodologyisdescribedinthissectiontoperformclusteringofthedatabys imultaneouslyoptimizingalltheconicting objectives.Specically,theprocessinvolvestheidenti cationofinitialclustersusingtheinitialization step.Theseclustersarethencategorizedasplayersandres ourcecenters,andagameisformalized. Theplayersinthegamecompeteforallocationofresourcesf romtheresourcecenters.Theresource centersconsistofadiscretesetofdataobjects.Thestrate gyofaplayerismodeledasatupleconsistingofthenumberofresourceunitsrequestedfromeveryreso urcecenter.Thepayoffcorresponding toeverystrategyisafunctionoftheconictingobjectives .ANashequilibriumsolutiontothegameis 92

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thenevaluated,andtheallocationsareperformedaccordin gly.Afterthereallocation,theclustersare updated.Thiscompleteprocessisrepeateduntilthestoppi ngcriteriaaresatised.Thestepsinvolved inthealgorithmaredescribedinAlgorithmAlgorithm5.2.T hefollowingsub-sectionsdescribethe normalformgametheoreticmodelindetails. Algorithm5.2 Gametheoreticalgorithm Require: K,datasetofsize N anddimensionality d Ensure: Kpartitionsoptimizedonthebasisofobjectives 1: initialize K clustercenterson d dimensionalspace 2: performoneiterationofAlgorithmAlgorithm5.1,steps5-1 0 3: repeat 4: load before getLoad () ; SSE before getSSE () 5: if 9 k 2 K j l k < l ideal then 6: initializeanewgame G 1 7: P f m j l m < l ideal g ; R f n j l n > l ideal g 8: forall r n j n = 1 ; ; R do 9: r n : overhead l n l ideal ; r n : consistent 0 10: endfor 11: forall p m j m = 1 ; ; P do 12: performminimumcostinitialallocationofresourcesfrom R ,suchthat l m l ideal 13: endfor 14: update l n ; 8 n 2 R 15: forall r n j n = 1 ; ; R do 16: if l n > r n : overhead then 17: r n : conflict 1 18: G 1 : createStrategySet () ;%seeAlgorithmAlgorithm5.3% 19: G 1 : createPayoff () ;%seeAlgorithmAlgorithm5.4% 20: G 1 : evaluateNashEquilibrium () ;%seeAlgorithmAlgorithm5.5% 21: performtemporaryreallocationofunitstoplayersaccordi ngtoNashequilibrium 22: endif 23: r n : conflict 0; r n : consistent 1 24: endfor 25: load after getLoad () ; SSE after get S SE () 26: if % D ( load ) > % D ( SSE ) then 27: commitreallocations 28: updateclustercentersaccordingtostep2 29: else 30: break 31: endif 32: endif 33: until FALSE 93

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Figure5.1Identicationofoptimumclustersusinggamethe oreticclustering(GTKMeans)and KMeansmethodologies.(a)Initialclustersidentiedbysi ngleiterationofKMeans,(b)nalclustersafterKMeans,(c)formulationofagamewithplayers p 1 ; p 2 ; p 3 andresources r 1 ; r 2 ,(d)nal clustersafterGTKMeansalgorithm5.3.2.1IdenticationofPlayers Thestepsinvolvedduringthealgorithmcanbedescribedwit hthehelpofanexample 1 givenin Figure5.1.Duringinitialization,theclustercentersare randomlygeneratedforthe d -dimensional dataset.Thisisfollowedbytheidenticationofinitialcl ustersbyperformingasingleiterationofthe KMeans.AsshowninFigure5.1(a),the L andthe SSE valuesoftheinitialclustersisnotoptimal. IftheiterativeKMeansasshowninAlgorithmAlgorithm5.1i simplementedwiththeobjectiveof minimizationof SSE ,thenalvalueofthe SSE is38716(Figure5.1(b)).However,thecorresponding L valueis106.8,signifyingthattheclustersarenotequi-pa rtitioned.Hence,agameisformulatedwith theobjectiveofsimultaneousclusteringofobjectsontheb asisofcompactnessandequi-partitioning. 1 ThedataistakenfromGermanTownData,whichisatwodimensi onaldatasetwith59observations,obtainedfrom [106].TheSSEvalueforKMeansclusteringfor5clustersist hereportedminimumvalueinliterature[85]. 94

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Therststepintheformulationofthegameisdeningthecom ponentsofthegame,i.e.,the players,theresources,thestrategies,thepayofffunctio nsetc.Inthismodel,theclustercenterswith l k < l ideal ; 8 k 2 K areidentiedastheplayersinthegame.Alternatively,the clustercenterswith l k > l ideal ; 8 k 2 K ,areconsideredastheresourcesinthegame.Theobjectiveo faplayeristoreceivethedataobjectsfromtheresourcesinsuchamannertha thiscompactnessobjectiveandthe equi-partitioningobjectivesareoptimizedsimultaneous ly.Inasituationwheremultipleplayersare requestingunitsfromthesameresourcecenter,thereisaco nictamongtheplayers,soeveryplayer competesagainsteveryotherplayerinthegameinordertoma ximizeitsownutility.OnesuchexamplescenarioisdisplayedinFigure5.1(c),wheretheplayer s p 2 and p 3 willcompetetoreceiveunits fromtheresourcecenter r 1 5.3.2.2DenitionofStrategy Thefeasibilityofagametheoreticmodellargelydependsup onthenotionofstrategy,whichis amajorfactorindeterminingthecomputationalcomplexity ofthemodel.Onewayofdeningthe strategyforaplayeristocreateatupleconsistingofthenu mberofunitsthattheplayercanrequest fromeveryresourceavailableinthesystem.Forexample,in Figure5.1(c),theplayer p 3 ,which requires6resourceunitstorealizeequi-partitionedsitu ation,couldhaveastrategy f 1 ; 5 g ,i.e.,receive oneresourceunitfrom r 1 ,and5unitsfrom r 2 .Thestrategysetfortheplayerwouldconsistofall possiblecombinationsofresourceunitsfromtheresourcel ocations,andthestrategyspaceincreases exponentiallywitheveryunitincreaseinthenumberofreso urcecenters.Hence,suchanotionof strategyisapplicableonlyforthegameswithveryfewresou rces,andanalternatenotionofstrategy hastobeidentiedforthismodel. AlgorithmAlgorithm5.3outlinesthestepsinvolvedinthef ormulationofthestrategysetfora player.Essentially,itisatwostepprocess,inwhich,duri ngtherststep,theplayerstrytoreceive resourceunitsfromtheresourcelocationsonthebasisofmi nimumcostallocationmethodology,irrespectiveoftheallocationsmadetotheotherplayers.Due tothis,asituationmayarisewherethe resourcelocationshaveallocatedmoreresourcesthantheo verheadavailablewiththem.Therefore, foreverysuchresourcelocation,agameneedstobeformulat edandsolvedtoensureequi-partitioning. Hence,duringsteptwo,theclustercentersthathavetriedr eceivingresourcesfromtheresourcelo95

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cationinconictareconsideredastheplayersinthegame.T heplayers'strategiesconsistofthe numberofresourceunitstheymayhavetolooseinordertoens urethatthecorrespondingresource location(forwhichthegamehasbeenformulated)isinconsi stentstate,i.e.,theresourcelocation isequi-partitioned.AnexamplescenariodescribedinFigu re5.2wouldbehelpfulinimprovingthe understanding.AsshowninFigure5.2,theplayer p 1 hasrequested1resourceunitsfromlocation r 1 andplayer p 2 hasrequested4units.Duetotherequests, r 1 mayloose5units,whichwouldlead toasituationwhere l r 1 < l ideal .However,theplayersonlyneedtolooseatotalof3unitsand try toreceivethoseunitsfromotherresourcelocationstoensu rethat l r 1 = l ideal .So,agameisplayed betweentheplayers p 1 and p 2 ,withplayer p 1 'sstrategysetas f 0 g ; f 1 g ,andplayer p 2 'sstrategyset as f 0 g ; f 1 g ; f 2 g ,withthenumbersindicatingtheresourceunitstheplayers mayhavetolooseinorder toensurethattheresourcecenterisequi-partitioned.The playerswouldreceiveapayoffforevery strategy,whichwouldbeafunctionoftheadditionalcostin curredforreceivingtheresourcesfrom thecentersthatarefartherfromtheplayer,andthechangei n L valuefortheplayersandtheresource. Modelingofthestrategyintheproposedmannerreducesthes trategyspaceconsiderably.Also,the numberofactualplayerspergameissignicantlylessthant hetotalnumberofplayersinthesystem, sincenotallplayerswouldhaverequestedunitsfromtheres ourcelocationthatisintheconictsituation.Effectively,usingthismethodology,onelargegam eissubdividedintoseveralsmallgames playedinmultiplesteps. Algorithm5.3 Generationofstrategyset Require: resourcelocationinconict( r n ), r n : overhead ,setofallplayers P Ensure: strategyset S j S = f S 1 ; S 2 ; S P g 1: identifythesetofplayers P 0 thatreceivedallocationofresourceunitsfrom r n 2: forall p i j i = 1 ; ; P 0 do 3: numstrategies = min ( r n : overhead ,unitsreceivedby p i from r n ) 4: for j = 0to numstrategies do 5: S i += f j g %numberofunitsaplayermayhavetoloosefrom r n % 6: endfor 7: S += f S i g 8: endfor 9: return S 96

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Figure5.2Anexamplefordenitionofstrategy.Here,r1and r2arethetwoclusterswithmoreunits availablewiththemaftertheinitialclustersareidentie d.p1andp2aretheplayersinthegame, competingwitheachotherinordertoreceivemaximumresour ceunitsfromr1. 5.3.2.3PayoffFunction Theplayersinagameplaytheirstrategiesinordertooptimi zetheequi-partitioningandthecompactionobjectivesoftheresourcecenterforwhichthegame isplayed.Anexpectedutilityisassociated correspondingtoeachstrategycombinationthataplayerin thegamewouldreceive.Thisutilityis mathematicallymodeledasapayofffunction,whichevaluat esthegainorlossaplayerincurswhenit playsitsownstrategy,andtheotherplayersplaytheircorr espondingstrategies.Inthisscenario,the payoffforaplayer p i 'sstrategy s iu andtheplayers p i strategycombination s i v foragameplayedfor resourcecenter r j isaffectedbythefollowingfactors. Everyresourceunitthattheplayerintendstoloosefrom r j isreceivedfromtheotherresource locations r j .ThisincreasestheSSEvaluefortheplayer. Whentheotherplayers p i inthegameplay s i v before p i 'sstrategy s iu ,thecostincurredfor receivingtheresourcesfrom r j furtherincreasesbecausesomeofthecloserresourcelocat ions mighthavealreadyallocatedresourcestotheplayers p i 97

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Theloadvalue l r j valuefor r j improvesastheplayerstrytoreceiveunitsfrom r j .However, asthetotalnumberofunitslostbytheplayersbecomegreate rthan l ideal ,theload l r j starts worsening.Hencetheabsolutevalueofchangein l r j needstobeminimized. Thepayofffunctioncapturestheinter-relationshipofthe abovementionedcriteria,andismodeledas ageometricmeanofthetotallossincurredbytheplayer p i intermsofthedifferencebetweentheSSE beforeandaftertheotherplayers p i playtheirstrategies s i v ,andtheabsolutevalueoftheload l r j correspondingtothestrategy s iu Algorithm5.4 Payoffmatrixgeneration Require: strategyset S ,players P 0 ,conictresource( r n ) Ensure: Payoffmatrices po i ofplayers p i j i = 1 ; ; P 0 1: forall p i j i = 1 ; ; P 0 do 2: rows j S i j 3: columns P 0 b = 1 ; b 6 = i ( j S b j ) 4: createemptypayoffmatrix po i ofsize rows columns 5: for j = 0to columns do 6: for k = 0to rows do 7: rc before cost(asadistancemeasure)incurredto p i forreceiving k resourceunitsfrom resourcelocations r m j m 6 = n ; r m : consistent = 0 8: cc cost changeintheloadvalueofsystemwhenplayers p i playtheirstrategycombinationcorrespondingtocolumn j ,andreceiveresourcesunitsfromlocations r m j m 6 = n ; r m : consistent = 0 9: rc after cost(asadistancemeasure)incurredto p i forreceiving k resourceunitsfrom resourcelocations r m j m 6 = n ; r m : consistent = 0,aftertheotherplayers p i haveplayed theirstrategies 10: rc final rc after rc before 11: cc final j r n : overhead ( cc cost + k ) j 12: po i [ k ][ j ] p rc final cc final 13: endfor 14: endfor 15: endfor 5.3.2.4NashEquilibriumSolution Themulti-objectiveclusteringproblembeingmodeledasag ameissolvedusingtheNashequilibriummethodology.Ascomparedtotheothersolutionconcept savailableintheliterature,onlyNash equilibriummethodidentiesthesocialoptima.Thepayoff matricesevaluatedduringtheprevious stepserveastheinputtothealgorithm,whichgeneratesano utputasaNashequilibriumstrategyset consistingofonestrategychosenforeveryplayerinthegam e.AttheNashequilibriumpoint,no 98

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playerhasincentivetochangeitsstrategyunilaterally.T heNashequilibriumalgorithmisexplained inAlgorithmAlgorithm5.5.Aftertheequilibriumstrategi esareidentied,thereallocationofresourceunitsisperformedaccordingly.Thegameisthenplay edforotherresourcelocationsinconict andtheallocationsareperformedaccordingly.Thecluster meansarethenupdated,andthecomplete processisrepeateduntilthereisnofurtherimprovementin oneoftheobjectiveswithoutworseningof theother. Algorithm5.5 Nashequilibriumalgorithm Require: PayoffMatrices po i ofplayers p i j i = 1 ; ; P 0 Ensure: Nashequilibriumstrategycombination S 1: forall pay i j i = 1 ; ; P 0 do 2: identifyastrategy s i suchthat 3: po i ( s 1 ; ; s i ; ; s P 0 ) po i ( s 1 ; ; s i ; ; s P 0 ) 4: %Nashequilibriumstrategycombinationidentiedontheba sisof[36]% 5: endfor 6: S = f s 1 ; ; s P 0 g 7: return S 5.3.3EnsembleBasedGameTheoreticClustering Asshownintheprevioussub-section,thesimultaneousclus teringonthebasisofmultipleobjectivesisperformedusingmultiplegameiterations,wherean iterationconsistsofmulti-stepgames.The complexityofthismethoddependsuponthenumberofdataobj ectsaswellasthenumberofclusters, andthustheresponsetimeofalgorithmishighforlargedata sets.Hence,anensemblemethodthat performsthecompleteclusteringonthebasisoffastmathem aticalmethodsfollowedbyagametheoreticalgorithmhasbeenpresentedhere.Inthismethod,dur ingtherststep,aKMeansclusteringof thedataobjectsisperformedonthebasisoftheobjectiveof minimizationoftheintra-clusterdistance asexplainedintheAlgorithmAlgorithm5.1.Theclustersob tainedaftertheKMeansalgorithm arenotoptimalontheequi-partitioningparameter,hencea gameisformulatedwiththeplayersasthe clusterswithnumberofdataobjectslessthan l ideal ,andtheresourcesastheclusterswiththenumber ofdataobjectsgreaterthan l ideal .ThegameisthenplayedandaNashequilibriumsolutionpoin tis identied.Areallocationofthedataobjectsisperformedi frelativechangeinthecompactnessand theequi-partitioningvaluesisbelowthethreshold.Since ,thegameisplayedonlyonceinthisscenario,thenotionofthestrategyasdescribedintheparagra ph1ofSection5.3.2.2canbeadopted. 99

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Although,postKMeansgametheoreticmodel,referredas PKGame henceforth,doesnotperformsimultaneousoptimizationofmultipleobjectives,themetho dologyisfast,andtheresultsobtainedfor theexperimentsarepromising.5.3.4AnalysisofGameTheoreticAlgorithm Inthissubsection,themethodologyisanalyzedtoevaluate itspracticability.First,thecomputationalcomplexityofthemethodologyfortheextremecase saswellastheworstcasescenariois identied,thensomeoftheuniqueattributesofNashequili briumalgorithmthatmakesitattractiveas asolutionmethodforthismodelarediscussed.Thediscussi onwillconcludewithabriefdiscussion abouttheprogressionofalgorithm.5.3.4.1ComputationalComplexityAnalysis Inanormalform P -playergamewithanaveragenumberofstrategies S perplayer,theworst casetime-complexityisgivenby O ( P S P ) [36]whenthegameisplayedinsingleshot.However, inthemodeldiscussedinsection5.B,amulti-stepgamehasb eenformulatedandsolved.So,the overallcomputationalcomplexityofplaying R suchgamesis O ( R P S P ) ,where R K ; P K and R + P K K isthetotalnumberofclusters.Among R ; P ,and S ,thecomplexityislargelygoverned bythevalueof S ,whichdependsuponthedenitionofastrategy.Asopposedt othenaturalnotion ofstrategyasacombinationofresourcerequestsfromevery resourcelocation,thestrategyinthis contexthasbeendenedasthenumberofresourcesaplayerma yhavetolooseinordertoensure thattheresourcelocationisinconsistent,equi-partitio nedstate.Thisrestrictsthesizeofstrategy setofaplayer p i as j S i j = b N = K c .Hence,theworstcasetimecomplexityofonegameisgiven as K b N = K c K ,since P K .Now,ifthenumberofclusters K is1,thecomputationalcomplexity wouldbe R ( 1 N 1 )= R N .Similarly,if K = N ,thecomplexitywouldbe N 1 N ,since l ideal = 1. Therefore,fortheextremecases,thecomplexityofthesyst emis O ( N 2 ) O ( R P S P ) .Intheworst casescenario,thenumberofplayersinthegameisequaltoth enumberofresourcesinthegame. Hence, K = N = 2,andthecomplexityofthesystemisgivenbyEquation5.6. ( N = 2 ) ( N = 2 ) b N = ( N = 2 ) c N = 2 = N 2 2 ( N 2 ) = 2 (5.6) 100

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Thecomplexityofthisalgorithmdependsprimarilyonthenu mberofgamesandthenumberofdata objectsinthedataset.Hence,thismethodologyisideallys uitedformulti-objectiveclusteringinsmall tomediumsizeddatasets. TheNashequilibriumsolutionpointspossesscertainattri butesthatmakethemethodologyappropriateforcertainapplications.ANashsolutionpointisso ciallyequitable,whichmeansthatevery playerinthesystemissatisedwithrespecttoeveryotherp layer,andhenceisinequilibrium.Social satisfactionisimportantinthescenarioswhereeveryobje ctiveinamulti-objectiveclusteringhasequal priority.AnotherimportantaspectofNashequilibriumist hat,foramixedstrategynon-cooperative game,aNashequilibriumsolutionpointalwaysexist[36].A lthough,apurestrategygamehasbeen modeledinthiswork,themodelcanbeeasilyextendedasamix ed-strategygamebyassociating probabilitiescorrespondingtothestrategiesofaplayer.5.3.4.2NatureofAlgorithmExecution Thealgorithmconsistsofmultiplegames,oneforeveryreso urcelocationinconict.Theplayer setcorrespondingtoagameconsistsofthesetofclustersth athaverequesteddataobjectsfromthe resourcecenterinconict.Onceagameisplayedforapartic ularresourcelocation,andplayers receivetheexcessallocationfromotherresourcelocation s,thelocationforwhichthegamewasplayed becomesconsistentintermsofequi-partitioning.however ,asituationmayariseatalatertimethat thislocationagainbecomesinconsistentduetoallocation ofunitstootherplayersasaresultofa gameplayedforsomeotherresourcelocation.Inextremecas es,thismayleadtocycling,andthe methodologywouldtakeinnitelylongtimetocomplete.Ino rdertoensure,thatsuchasituationdoes notoccur,a ag isassociatedwitheveryresourceforwhichagameisplayed, andissetto FALSE initially.The ag issetto TRUE afteragameisplayedforthatresource.Alltheresourceswi th ag=TRUE arenotconsideredforreallocation.Thisensuresthatthea lgorithmprogressesinforward directionandnishesinnitenumberofsteps.However,thi smayaffectthequalityofsolution. 5.4ExperimentalResults Severalsingle-objectiveclusteringmethodologieshaveb eendevelopedandemployedforvarious applications.However,inthemulti-objectiveclustering domainveryfewmethodshavebeenpro101

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posed,whichsignicantlylimitsthecomparativestudyoft heperformanceofouralgorithm.The performanceofouralgorithm,referredas GTKMeans henceforthascomparedtotheKMeansalgorithm,andamodiedalgorithmemulatingtheweightedmulti -objectiveoptimizationmethodology hasbeenevaluatedinthissection.Therstsetsofexperime ntswereperformedwithrealdatasets beingusedinthepreviousstudies.Toanalyzethealgorithm morecloselyintermsofefciencyand qualityofthesolution,articialdatasetswerecreatedto simulatetherealworldscenarios,andthe methodwasexhaustivelytestedonthosedatasets.Also,the sensitivityofthismethodintermsofthe variousparameterslikethenumberofclusters,thenumbero fdataobjectsperclustersandthestrategy setsoftheplayershasbeeninvestigatedinthissection.5.4.1SimulationSetup TheGTKMeanswastestedonsomeofthedatasetsthathavebeen widelyusedinliteraturefor theevaluationofgeneralpurposeclusteringapproaches.T hedatasetsarelistedasfollows: BritishTownData(BTD):Adatasetconsistingoffourprinci palsocio-economicdatacomponentscorrespondingto50Britishtowns.Thesetwasobtaine dfrom[107]. GermanTownData(GTD):Atwodimensionaldatasetcontainin gthelocationcoordinatesof 59Germantowns.Thedatasetwasobtainedfrom[106]. IrisData(IRIS):Afourdimensionaldatasetconsistingoft hesepallength,sepalwidth,petal length,andthepetalwidthmeasurementson150samplesofIR ISobtainedfrom[107]. Therealdatasetsavailableintheliteratureoftenhaveani ntrinsicstructurethataspecicclustering methodologyattemptstocomprehendandclusteraccordingl y.Duetothisproperty,theclustering methodsthatarettingforcertaindatasetsmaynotbeappro priateforotherssincetheyoptimizea singleobjective.Hence,tobetterevaluatetheperformanc eofanalgorithm,andanalyzethesensitivity ofvariousattributesofit,awiderrangeofarticialdatas etsneedbeconstructed.Inthiswork, 704normallydistributeddatasetsconsistingofthelocati oncoordinatesofdataobjectsonatwo dimensionalgridofsize12*12werecreated.Thevaluesofme anandvariancewerevariedfrom; s = 2and0 10.Thesizeofdatasetswasvariedfrom50to150dataobjects partitionedinto 3to10clusters.Also,intra-clustersimilaritymeasuresi ntermsofnumberofobjectsperclusterwere 102

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takenintoconsideration.Asanexample,adataset6 8 90wouldhave90dataobjectspartitioned into6clusters,witheachclusterhavingthenumberdataobj ectsrangingfrom b 0 : 8 ( 90 = 6 ) c = 12 to b ( 0 : 2 ( 90 = 6 ))+( 90 = 6 ) c = 15.Foreachexperiment,averagesof200repetitionswerepe rformed withrandomclustercenter(clusterhead)initializations .TheNashequilibriumsolutiontothen-person normalformgamewasidentiedusingthe SimplicalSubdivision algorithm.AmongtheseveralNash equilibriummethodologiesavailableinliterature,thesi mplicalsubdivisionmethodhasbeenidentied toworkconsistentlybetterthanotherexistingmethodolog ies.Thealgorithmisacceptablyfastforthe moderatesizedproblems.Baseduponthesimplexmethod,the algorithmstartswithagivengridsize, andconvergestoanapproximatesolutionpointbyiterative labelingofthesub-simplexes. Gambit [108],anopensourceClibraryofgametheoryanalyzersoftw aretoolkitwasusedforidentication ofNashequilibriumsolution.Gambitincorporatesseveral Nashequilibriumalgorithmsforsolving normalform,extensiveform,andBayesiangames.Allexperi mentswereperformedonaSunblade 1500workstationthathad4GBofRAM.5.4.2ExperimentswithExistingDataSets ToevaluatetheperformanceofGTKMeansalgorithm,wecompa reditwiththeclassicalKMeans algorithmfortheBTD.SincebothKMeansandGTKMeansmethod ologieshavesimilarstartingpoints andboththemethodsidentifysameclustersduringtheiniti alizationphase,theinitialknowledgeof theenvironmentissameforbothmethods.Afterward,theKMe ansalgorithmproceedswithanobjectiveofclustercompaction( SSE ),whereastheGTKMeanssimultaneouslyoptimizesthecompa ction aswellastheequi-partitioningmeasures( L ).Figure5.3displaysacomparativegraphofGTKMeans andKMeansperformancefortheclusteringperformedontheB ritishtowndata[107].Thepercentageimprovementin SSE (Y-axisonleft)and L (Y-axisonright)valuesfromtheinitialclustersfor differentclustersizesisdisplayedinthegraph.Aseviden tfromthegraph,for K = 4 ; ; 10the percentageimprovementinthe L objectiveforGTKMeansismuchhigherthanthatoftheorigin al KMeansalgorithm,whereasthepercentageimprovementin SSE ismoreforKMeansascomparedto GTKMeans.ThisisduetothefactthattheKMeansalgorithmpe rformsasingleobjectiveoptimizationonlyonthebasisofcompaction,whereastheGTKMeansal gorithmidentiesclustersonthebasis ofsimultaneousconsiderationofboththeclusteringobjec tives.Theaverageimprovementin SSE and 103

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L forGTKMeansis87.3%and62.7%respectively.Althoughthei mprovementin SSE measureis 95.8%incaseofKMeans,theequi-partitioningmeasureimpr ovesbyonly30.7%.Overall,theGTKMeansalgorithmshowedameanimprovementof20%higherthan thatoftheKMeansalgorithmfor simultaneouslyoptimizingboththeobjectivefunctions.T oevaluatetheperformanceofthePKGame methodology,experimentswereperformedontheGermanTown Data[106].Theperformanceofthe algorithminoptimizingthetwoobjectivesisshowninFigur e5.4.ThegraphdisplaystherelativeperformanceofthePKGameandtheKMeansalgorithms.ThePKGame methodologyoutperformedthe KMeansmethodintermsoftheaveragepercentageimprovemen tinthe L fortheclusters.Theoutput characteristicsweresimilartothepreviousexperiment,a ndanaverageoverallimprovementof18% wasnoted.Inanattempttoevaluatetheperformanceofthecl usteringmethodinamulti-objective Figure5.3PerformanceofthealgorithmsontheBritishTown dataset.KMeansandGTKMeans algorithmsarecomparedonthebasisoftheirperformancein optimizingthecompaction(SSE)and theequi-partitioning(L)objectives.setting,wemodiedtheoriginalKMeansalgorithmtoincorp oratetheequi-partitioningobjectiveto theoriginalcompactionobjective.InthismodiedKMeans( MKMeans)method,theclusteringwas performedonthebasisofafunctionthatwasaweightedavera geofthe SSE andthe L valuesofcluster.Theweightswerekeptat0.5sothatbothobjectivesaree quallyrepresentedinthesolution.The resultsfromthesetofexperimentsperformedontheIRISdat aset[107]areshowninTable5.2.The tableliststheimprovementsinthe L andthe SSE valuesobtainedafter200iterationsofGTKMeans, 104

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Figure5.4PerformanceofthealgorithmsontheGermanTownd ataset.KMeansandPKGamealgorithmsarecomparedonthebasisoftheirperformanceinop timizingthecompaction(SSE)andthe equi-partitioning(L)objectives.KMeans,MKMeans,andPKGamealgorithms.OnaveragetheGTKM eansmethodoutperformed othermethodsformajorityofexperiments.Theensemblebas edPKGamemethodalsoperformed wellonmostofthedatasets.TheimprovementofPKGameovert heKMeansmethodisattributedto thefactthattheformerisarenementthatisperformedafte rthelatternishes.Theexperimentson theexistingdatasetswerepromising,andshowedthepotent ialapplicabilityofthismethod.Overall, thegametheorybasedmulti-metricclusteringmethodoutpe rformedtheKMeansalgorithminterms ofsimultaneousoptimizationofthemultipleobjectives.A lthough,themethodisslowerthanKMeans methodinidentifyingclusters,itprovidessociallyfairs olutions.However,athoroughanalysisofthis methodrequiredfurtherexperimentation,andhence,arti cialdatasetsweregeneratedtoevaluatethe varioussensitivitymeasuresaswellastheperformancemea suresofthemethod. 5.4.3ExperimentswithArticialDataSets Toevaluatetheperformanceofthetwomicroeconomicsbased methodsthemulti-objectiveclusteringwasperformedonthearticialdatasetsdescribedin thebeginningofthissection.Anaverage oftheoutputsforimprovementsin SSE and L valueswereplottedongraphsasshowninFigure5.5 andFigure5.6respectively.FromFigure5.5,itcanbeident iedthattheKMeansalgorithmperforms 105

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betterthanthegametheoreticmethodsforthecompactionob jective.Also,theperformanceofthe MKMeansmethodfollowstheKMeansclosely.Thisbehavioris intuitiveasthemeansbasedpartitioningmethodologiesoptimizeonlythe SSE attribute.However,fromtheFigure5.6,itisevident thattheperformanceofKMeansforequi-partitioningobjec tiveissignicantlyinferiorascompared totheGTKMeansandPKGamemethods.Thisfollowsfromthefac tthatthetwoobjectivesareoften inverselycorrelated,andtheunilateralimprovementinon eobjectivefunctionadverselyaffectsthe otherobjective.However,sincetheGTKMeansmethodsimult aneouslyoptimizesboththeobjectives, theclusteringperformancewasimprovedbymorethan50perc entforboththeobjectives,asshown inthegraphs.Anotherobservationwasthattheperformance oftheensemblebasedPKGamemethod didnotimprovemuchforthesmallerclusters,i.e. K = 3 ; 4,butforthelargernumberofclusters, theensemblemethodalsoperformedwell.SincetheKMeanswo rksverywellforsmallernumber ofclusters(3-4),thecompactnessvaluesarehigh(alsoevi dentfromgraph),andhence,whengame theoreticmethodisappliedafterKMeans,the L improvesatthecostof SSE ,whichisnotdesired. Figure5.5Averageimprovementinthecompactionobjective fortheexperimentsonarticialdata sets.Theoptimizationinthe SSE metriccomparedtotheworstcasevaluesisevaluatedforKMe ans, GTKMeans,MKMeansandPKGame. 106

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Figure5.6Averageimprovementintheequi-partitioningob jectivefortheexperimentsonarticial dataset.Theoptimizationinthe L metriccomparedtotheworstcasevaluesisevaluatedforKMe ans, GTKMeans,MKMeansandPKGame.5.4.4FairnessofClustering Thestrengthofthegametheoreticclusteringmethodologyl iesinthefairnessofoptimizingeach objectivewithequalpriority.Toappropriatelyevaluatet heperformanceofthealgorithms,aquantitativemeasureofthefairnessofthealgorithmsforoptimizin g SSE and LOAD canbeidentiedusingthe Jain'sFairnessIndex [109],orageometricmeanoftherelativeimprovementinthe clusteringcriteria. AccordingtotheJain'sindex,thefairnessofthemethodolo gyisidentiedusingEquation5.7. fairness = ( n i = 1 x i ) 2 ( n n i = 1 x 2i ) (5.7) Here, x i correspondstheimprovementinthe i th objective.Thefairnessvaluerangesfrom0(worst case)to1(bestcase).Similarly,Thegeometricmeanofthei mprovementsintheclusteringcriteria identiestheaverageperformanceofthemethodology,equa llyweighingallthecriteria.Table5.3 showsthefairnessmetricvaluesfordifferentnumberofclu sters.Asshown,theGTKMeansmethod hasahighJain'sfairnessindexaveraging0.98ascomparedt otheKMeansvalueof0.93.Thissignies 107

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thattheGTKMeansmethodoptimizesboththeobjectiveswith equalpriority.Similarly,thegeometric meanoftheGKT-MeansishigherthantheKMeansbymorethan15 percent.Thefairnessperformance oftheMKMeansmethodandthePKGamemethodisalsoinferiort otheGTKMeansfairness. 5.4.5SensitivityAnalysis Theexperimentalresultsonthearticialdatasets,showni ntheprevioussubsectionsgivehints aboutthesensitivityofthismethodologyfordifferentdes ignattributevalues.Inthissubsection,we willcloselyanalyzethesensitivityoftheGTKMeansmethod .Thenumberofplayers,numberof strategiespergame,responsetimeofthealgorithm,andstr uctureofthedatasetsignicantlyaffect thepracticabilityofthismethod.Inthefollowingsubsect ions,weexperimentallyanalyzetheseparameters.5.4.5.1DataSetSimilarityMeasure Inmanycases,thestructureofthedatasethasasignicanti mpactontheperformanceofan algorithm.Wegeneratedawiderangeofarticialdatasetsi ntermsfornumberofdataobjectsper clusterdenedasthesimilaritymeasure,andradiusofaclu steras s = 2ona10*10grid.Theeffect ofstructureontheexecutiontimeofthealgorithmfordiffe rentsimilaritymeasuresandclustersizesis showninFigure5.7.Asshown,thesimilaritymeasuredoesno tsignicantlyimpacttheperformance ofthealgorithm,i.e.,onaverage,theexecutiontimeofthe GTKMeansalgorithmisindependentof thestructureofthedataset,andhenceitissuitableasagen eralclusteringmethodology.Theaverage performanceintermsoffairnessofallocationisshowninTa ble5.3.Thegeometricmeanfairnessis inrange60-80percent,whichisagoodmeasureoffairness.H ence,thestructureofadatasetdoes notadverselyaffecttheperformanceofthismethodology.5.4.5.2NumberofPlayersandStrategies Animportantconsiderationduringthemodelingofaproblem inagametheoreticframeworkis theimpactofthesizeofgame.Thesizedeterminesthecomple xity,andconsequentlytheperformance ofthesystem.Thus,weevaluatedtheaveragesizeofthegame intermsofthenumberofplayersand thestrategiesfordifferentclusters.ThegraphshowninFi gure5.9displaystherangeofplayersand 108

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Figure5.7Effectofdatasetsimilaritymeasureontheexecu tiontimeoftheGTKMeansalgorithm. Theinitialdatasetsimilaritymeasureisgivenasthedegre eofsimilarityinthesizesoftheinitial clusters.Ahigherdegreeofsimilarityresultsininitialc lusterswithalmostequalnumberofdataunits percluster.consequentlythestrategiesfordifferentclusters.Animp ortantobservationisthatalthoughtheaverage numberofplayersincreasesastheclustersizeincreases,t hetotalnumberofplayersissignicantly lessthanhalftheclustersize,whichistheworstcasescena rio.Forexample,onaveragethereareat most3.5playersforthedatasetswith9clusters.Itisalsoi mportanttonotethattheaveragestrategy sizedoesnotincreasesexponentiallyasafunctionofthenu mberofplayers,whichistheintuitive notioninagametheoreticsetup.Thisbehaviorisattribute dtothealternativedenitionofaplayerand strategyforourmodelasdiscussedinsection5.3.2.2.Them odelingcontrolledthecomplexityofthe systemsignicantly.However,thesurgeinthenumberofstr ategiesfordatasetswithlargenumber ofclustersindicatethattheGTKMeansisbettersuitedform ulti-objectiveclusteringofmediumsized datasetswithalessnumberofclustersperdataset. 109

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Figure5.8Relationshipbetweentheexecutiontimeandthen umberofclusters.ThealgorithmexecutiontimeofKMeansandGTKMeansarecomparedinthissetofex periments.Additionallytheworst caseandtheaveragecaseexecutiontimesareplottedandcom pared. Figure5.9Averagenumberofplayersandstrategiesfordiff erentclustersizes.Fordifferentclustersizes,theaveragenumberofplayersandstrategiesperg ameindicatethesizeofthegameand consequentlytheexecutiontimeandthefeasibilityofthec lusteringalgorithm. 110

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5.4.5.3ExecutionTime Themulti-objectiveclusteringmethodologypresentedint hisworkisslowerthantheKMeans methodbymultipleordersofmagnitude.Similaristhecasew ithotherheuristicsbasedmethodologies. Inordertoquantifytheeffectofnumberclustersontheexec utiontimeofthealgorithm,andanalyze theperformanceextremes,weplottedaverageexecutiontim eandthemaximumexecutiontimefor differentnumberofclusters.AsshowninFigure5.8,forsma llernumberofclusters,i.e., K = 3 ; ; 8, theGTKMeansperformswellandidentiestheoptimumcluste rswithin10seconds.Also,theworst caseperformancefollowssimilartrendandiswithin100sec onds.However,forlargernumberof clusters,theperformancedecaysexponentially.Thisisdu etothefactthatasthenumberofclusters increase,thepotentialnumberofclustersandcorrespondi nglythestrategiesincreasesignicantly,and thegamebecomeslarge.ThetimecomplexityoftheNashequil ibriumalgorithmisexponential,which resultsinslowerexecutiontimeforsuchcases.5.5Discussion Anovelmicroeconomicsbasedalgorithmformulti-objectiv eclusteringproblemhasbeendevelopedinthisresearch.Inthisalgorithm,anon-cooperative multi-playernormalformmulti-stepgame isformulatedwiththesubsetsofinitiallyidentiedclust ersasplayers.Anymathematicalpartitioning methodcanbeemployedtoidentifytheinitialclustersandt oupdatetheclustersafteraniterationof thegame.ANashequilibriumbasedmethodisusedtosolvethe gametheoreticformulation.Thisalgorithmisindependentofthetypeandthenumberofobjectiv esthatcanbesimultaneouslyoptimized. Also,anensemblebasedgametheoreticoptimizationalgori thmhasbeendevelopedinthiswork.In theensemblebasedmethod,theKMeanspartitioningisperfo rmedrst,followedbyagametheoretic formulationbaseduponthesizesoftheclusters.Theexperi mentalstudyontheexistingandarticialdatasetsprovidesimportantinsightsforthegametheo reticclusteringalgorithm.Ascompared totheKMeans,thisalgorithmperformssignicantlybetter intermsofthefairnesstowardimprovingtheclusteringcriteria.Also,thecomplexityofthealg orithmintermsofplayersandstrategies ismuchlowerascomparedtotheclassicalnormalformgameth eoreticmodeling.Thisisattributed anoveldenitionofstrategy.Thisalgorithmisnotsensiti vethestructureofthedataset.However, thealgorithmdoesnotscaleverywellwiththesizeofthedat asetsincaseswherethenumberof 111

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clustersincrease.Overall,thismodeliswellsuitedforam ulti-objectiveclusteringproblemwherethe objectivefunctionsarecomplementaryandneedtobeoptimi zedsimultaneously. Thedomainofmulti-objectiveclusteringisreceivingsign icantattentionasthenewermultidisciplinaryresearchareasareemerging.Thisrstattemptin propoundingagametheoreticsolutionis attractive.Theapplicationsofthisalgorithmmayrequire severalobjectivestobeconsideredsimultaneously,dependingupontheapplicationarea.Also,analte rnatemodelingofthepayofffunctionmay improvethecostfunctionintermsofcapturingtheessenceo fcompetitiveobjectives,andthusneed furtherinvestigationandrenement.Alogicalnextstepin researchistomodelthisgametheoretic clusteringapproachfordynamicallychangingscenarios.S imilarly,techniquesforpruningthestrategy setswouldalsoresultinmulti-foldimprovementintheperf ormanceandcomplexityofthemodel. 112

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Table5.2PerformanceofthealgorithmsonIrisdataset.The clusteringalgorithmsKMeans,MKMeans,GTKMeans,andPKGa mearecomparedfor theirperformanceontwooptimizationmetrics,compaction ( SSE )andequi-partitioning( L ). Total AverageImprovementinSSE†Value(in%) AverageImprovementinL‡Value(in%) Clusters KMeans MKMeans§ GTKMeans PKGame†† KMeans MKMeans GTKMeans PKGame 2 94.5 93.9 84.3 94.5 45.8 40.3 57.1 45.8 3 93.7 97.4 76.8 93.8 35.9 75.9 57.6 35.9 4 93.6 98.2 72.4 86.1 35.6 64.5 94.2 99.9 5 93.6 98.3 75.4 90.4 33.6 55.8 82.0 97.9 6 93.2 98.7 67.2 91.4 30.1 55.2 61.7 91.4 7 93.4 98.8 63.9 93.6 29.5 44.7 75.2 90.8 8 93.9 98.9 51.9 94.9 30.8 51.7 59.4 82.9 9 93.8 99.0 59.8 95.2 29.6 45.7 60.2 83.4 10 94.2 99.0 72.2 95.0 29.0 43.1 69.9 86.1 11 94.7 99.1 66.3 95.7 31.8 43.0 66.2 85.1 †:SumofSquaredEuclideanDistance(SSE)correspondstoth ecompactionobjective ‡:Load(L)correspondstotheequi-partitioningobjective §:ModiedKMeans(MKMeans)algorithm :GameTheoreticKMeans(GTKMeans)algorithmdevelopedin thisresearch ††:PostKMeansGameTheoretic(PKGame)ensemblebasedalgo rithmdevelopedinthisresearch 113

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Table5.3Fairnessoftheclusteringalgorithms.TheKMeans ,MKMeans,GTKMeansandPKGamealgorithmsarecomparedonth ebasisofthe quantitativemeasureofthefairnessoftheclustering.The twofairnessindexesusedforthecomparisonareGeometricm eanfairnessindexandJain's fairnessindex. Clustering GeometricMeanFairnessIndex Jain'sFairnessIndex Algorithm 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10 KMeans 57.1 60.5 61.2 59.4 60.1 60.9 56.9 62.0 0.89 0.94 0.95 0.93 0.93 0.95 0.93 0.96 MKMeans 64.0 65.9 65.2 64.9 63.7 62.2 62.0 62.7 0.92 0.94 0.94 0.94 0.94 0.92 0.92 0.92 GTKMeans 78.0 73.8 72.7 71.2 77.1 73.8 74.3 76.2 0.90 0.98 0.98 0.97 0.97 0.98 0.97 0.99 PKGame 57.1 66.5 56.7 46.3 36.5 41.3 41.8 44.9 0.90 0.88 0.80 0.72 0.66 0.72 0.74 0.78 114

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CHAPTER6 GAMETHEORETICAPPROACHTOROBOTTEAMFORMATION Theaggregationofrobotsintoteamsisnecessitatedduetot helimitedpowerandcommunication capabilitiesinemergencyenvironments.Theformationoft eamsofrobotssignicantlyenhances theperformanceandefciencyofsearchandrescuemissions insuchenvironments.Asopposed totheclassicalpartitioningapplicationdomains,therob otaggregationrequiresmultipleconicting objectivestobeoptimized.Wepresentanewmethodforsimul taneousmulti-objectivepartitioning ofrobotsintoteams,whichisbasedontheconceptsofmicroe conomics.Themethodutilizesthe strengthsofKMeansalgorithm,gametheoreticmodeling,an dNashequilibriummethodologyfor fastandsociallyfairpartitioning.Inthiswork,partitio nsarecreatedtoidentifydecentralizedteams ofrobotsinsuchamannerthateachrobotinateamclosesttoi tscommunicationgateway,aswell aseachteamisequallyrepresentedintermsofitsstrength( batterypower).Rigoroussimulations wereperformedtoevaluatetheperformanceofthemethod,an dtheresultsindicatethatourmethod performssignicantlybetterthantheKMeansmethodology, andidentiesgoodsolutionpoints. 6.1ProblemDescription Intherecentyears,searchandrescueroboticshasemergeda sanimportantemergencyresponse function.Mobilerobotshavebeenshowntobeavaluablereso urceduringtheexplorationmissions intheeventofsuchemergencies[110].Theserobotsareinvo lvedincollectingandintegratingthe information,andtransmittingittothebasestationforfur therdeliberation.Inacentralizedsystem, thisrequireseachrobottomaintainawirelessconnectionw iththebasestationandconstantlysend theinformationpackets.However,thiscommunicationissi gnicantlylimitedbythestrictconstraints ofbatterypower,lowradiorange,andconstantlychanginge nvironmentforeveryrobot. Foradetailedexplanationofthestepsinvolvedinthemulti -emergencyrobotdeploymentandthe issuesfacedintheprocess,pleaserefertoFigures6.1–6.5 115

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Figure6.1showsascenariowheremultipleemergencysituat ionshaveemergedinalocality inatime-overlappedmanner.Ofteninsuchsituations,thed eploymentofemergencyresponse personnelisnotfeasible,andhenceroboticunitsplayanim portantroleinthesearchandrescue missions. Therobotsdeployedintheeldrequirestwotypesofcommuni cation.Eachunitneedstocommunicatewiththebasestationtoreceivethecommandandcon trol.Thefeedbackfromthe emergencylocationiscontinuouslytransmittedtothebase station.Also,therobotscommunicatewitheachotherinordertocoordinatethecoveragearea amongthem.Thisensuresthatthe completeterrainiscovered.Thedeploymentofrobotsandth einterconnectionnetworkbeing establishedinsuchscenarioisshowninFigure6.2. However,apoint-to-pointgridbasednetworkingschemewhe reeachnodecommunicateswith everyothernodeandthebasestationisnotfeasibleinthese scenarios.ThisisshowninFigure 6.3.Duetothelimitedbatterypowerandhighcommunication overhead,afewroboticunitsmay dropoutofthesystemasthetimeprogresses,resultinginas ituationwherethecommunication withsomeoftheemergencylocationwouldbelost. Thus,apartitioningmechanismmaybeusedtoformteamsofro botunits,suchthateachpartitionhasasetofrobotsthatareclosetoeachotherandhenc edissipatinglesspowerinintraclustercommunication.Also,apartitionheadisdecidedam ongthenodesofthepartition (possiblytheonewithmaximumavailablebatterypower)whi chisresponsiblefortheinterclusteraswellastheclustertobasestationcommunication .Inthismanner,thecommunication overheadisreducedandtherobotsmaysustainintheeldfor longerduration.Ifaclassical clusteringschemelikeKMeansisusedforpartitioning,the teamsareformedasshowninthe Figure6.4. However,thepartitioningrequirementsinamulti-emergen cyrescueandresponsearedifferent fromotherenvironments,inthesensethatthepartitioning isrequiredtobeperformedonthe basisofmultiplecriteria.Inthisparticularcase,thetea msofrobotsbeingformedshouldpossess twoimportantproperties;theintra-clustercommunicatio nshouldbeminimizedtoreducethe powerdissipationincommunication,andeachteamshouldbe equallyrepresentedintermsof 116

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itscapabilities.Thesetofcapabilitiescouldbeanything rangingfromtheequaldistributionof theroboticunitsintermsoftheirrescuecapabilities,the distributionofroboticunitsthatare specializedtoperformcertainjobs,orasimpleequaldistr ibutionofthetotalbatterypowerin eachclustertoensurethateachemergencylocationisexami nedwithequalcapabilities.The classicalpartitioningalgorithmsarelargelysinglemetr icoptimizationmethods,andthuscan notbeusedforpartitioning.Theyoftenresultinformation ofpartitionsthatareeithertoolarge ortoosmall.OnesuchpartitioningresultisshowninFigure 6.5,whereacoupleofpartitions aretoolargeandacoupleofthemaretoosmall.Ifthepartiti onistoosmall,therobotsinthat partitionwillhavetoperformallthework,aswellascommun ication,andwillsoondropoutof thenetworkduetorapidpowerdissipation. Figure6.1Exampleofamulti-emergencysituationinasubur banarea.Severalemergencymanagementresourcesareallocatedtotheemergencylocationsfor search,rescue,responseandrecovery process. Although,theissueofworkdistributiontotherobotswithi nateam[111,112]hasreceivedsignicantattention,thedevelopmentofspecializedalgorit hmsforoptimalaggregationofrobotsinto teamshasnotbeenexplored.Unlikeclassicalapplicationd omainslikedatamining,bio-informatics, computervisionandpatternrecognition,computerandcomm unicationnetworks,andinformation systems[11,12],whereobjectanddataclusteringareperfo rmedonthebasisofsingleobjective,multidisciplinaryapplicationslikerobotteamformationrequi remultiplecriteria(thatmaybeconicting 117

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Figure6.2Anexampleofsearchandrescuerobotdeploymenti nmulti-emergencyscenario.The roboticunitsaredeployedinsituationswheresearchandre scuesituationsarecomplexandinaccessibletohumans.Aprimitiveinter-connectionnetworkises tablishedtomonitorprogressinrealtime andsharetheinformationamongtherobots,andbetweenrobo tsandthebasestation. Figure6.3Effectofhighcommunicationoverheadonthesear chandrescueprocess.Asaresultof highcommunicationbandwidthandthelimitedbatterypower ofeachrobot,theroboticunitsmaydie, therebydisruptingtheresponsefromsomeemergencylocati ons. 118

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Figure6.4Partitioningofrobotssuchthattheintra-clust ercommunicationisminimized,andeach partitionhasaheadnoderesponsibleforinter-clustercom munication. Figure6.5Partitioningresultsforrobotteamformationus ingKMeansalgorithm.SinceKMeans performsthepartitioningonthebasisofasingleobjective ofclustercompaction,theteamsidentied usingthisalgorithmaresuchthatsomeoftheteamsareveryl argeinsize,whereassomeoftheteams areverysmallinsize. 119

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innature)tobeoptimized.Someofthepartitioningcriteri ainthisdomainincludecompaction,equipartitioningofrobotsonthebasisofcapabilitiesperpart ition,numberofunitsperpartition,availabilityofaveragebatterypowerperpartition,orequi-distrib utionofworkloadperpartition,etc.Hence,a techniqueforsimultaneousoptimizationofconictingobj ectivesneedstobedeveloped.Inthiswork, wehavedevelopedanovelmethodologythatperformssimulta neousmulti-objectivepartitioningof robotsintoteams.Themethodologyconsistsofthreeimport antcomponents: Aniterativehillclimbingpartitioningalgorithm Amulti-stepnormalformgametheoreticmodel ANashequilibrium(NE)basedsolutionmethodology 6.2WhyMicroeconomicsforRobotTeamFormation? Inthecontextofrescuerobots,duetothepowerandcommunic ationconstraints,compactnessand uniformpowerdistributionhavebeenconsideredastheobje ctivestobeoptimized.Sincethesetwo objectivesareconicting,andthusconvexinnature,thesy stemcanbenaturallymodeledasagame. Also,asshownin[13],ifthepayofffunctionisconvex,aNas hequilibriumsolutionalwaysexistsand tendstoevaluategloballyoptimalsolutions. Inmulti-emergencyenvironments,itisdesirabletoensure thatallemergenciesreceiveresources inafairmanner.Fairnesshasseveralconnotations,butint hiscaseitcorrespondstoasituationwhere eachemergencyreceivesitsfairshareofrescuerobots.Gam etheoryexhibitsauniquepropertysocial equityorsocialfairness[13],whichensuresthateachplay erinthegameissatisedandtheoverall goalsarereached.Thegametheoreticsocialequilibriumin herentlyensurestheoptimumvaluesof eachobjectivewithrespecttootherobjectives,whichisde sirableinthesescenarios. 6.3Background Therealtimeapplicabilityofmobilerobotsforurbansearc handrescue(USAR)wasrstrecognizedduringtheWorldTradeCenterdisaster[113].There searchinthedomainofUSAR,and human-robotintegration(HRI)[114]hasidentiedexplici tcommunicationamongtherobotsasa bigobstacle.Thisisattributedtothelimitednetworkband width,limitedbatterypowerofrobots,and 120

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noisycommunicationchannels.Aneffectivesolutiontothi sproblemistoclustertherobotsintoteams toensurerobustnessandreliability.In[115],theauthors havediscussedtheperformanceofseveral rescuerobotsattheRoboCupRescueRealRobotLeaguecompet itions,andhaveidentiedthatthe multiplerobotcooperation,andteamsofrobotscouldmaxim izethesearchregions,andutilizeand enhancetheabilitiesofrobotsinsearchanddetectionmiss ions.However,muchresearchconcentrates ontheidenticationandoptimizationoftaskdistribution ,andcooperationamongtherobotswithina team[111]. In[112],theuseofstochasticgametheorytomodelcooperat ionamongtherobotteamonthe basisofobservationhistoryhasbeendemonstrated.Simila rly,in[116],theauthorsproposeahybrid roboticcommunicationmechanismthatusesrobotvisionand radiosignalsforimprovedcommunication.In[117]themulti-robotexplorationproblemhasbe enaddressedfromadifferentperspective bysuggestingaKMeansbasedclusteringoftheunknownsearc hspaceandallocatingthespacetothe robotsforexploration. Theproblemofobjectpartitioninghasbeeninvestigatedin thecontextofawiderangeofapplications,andreportedinliterature.Detailedsurveysofthes eworkscanbefoundin[11,69].Anelaborate discussionofthesemethodsonthebasisofpartitioningcri terialikecompaction,equi-partitioning, connectedness,andspatialseparationcanbefoundin[12,7 8].TheKMeans[74]isthesimplestand mostwidelyusedmathematicalalgorithmforpartitioningo nthebasisofcompaction.Itisusedfor creatinginitialpartitionsinourapproachdiscussedlate r.Additionally,someheuristicsbasedtechniques[83,88]andhybridapproaches[85]havebeenpropose dinliterature.However,allofthese methodologiesarelimitedtosingleobjectiveoptimizatio n.Intherealmofmulti-objectiveoptimization,theproposedmodelsprimarilyconsistofensemblemet hods[89]thatperformsingleobjective optimizationusingdifferentmethodsfordifferentobject ives,andintegratetheresults aposteriori Thesemethodsdonotexploittherealstrengthofsimultaneo usmulti-objectiveoptimization. Microeconomicoptimization methodsarecapableofnaturallymodelingthesituationsof conictand cooperationinagametheoreticsettingasdiscussedinthep revioussection.Itmodelsoptimization problemsinaframeworkconsistingofplayerswithconicti ngobjectivescompetingtooptimizetheir individualaswellasthesystemwideutilities[27,28].The gameissolvedusingNashequilibrium basedmethodologythatidentiesasociallyfairsolution[ 36]. 121

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Inthiswork,weidentifytherobotteamformationproblemas amulti-stepnormalformnoncooperativegame.Asubsetofinitialpartitionsidentied byKMeansalgorithmaremodeledasplayers,andtheremainingasresources,differentcombination sofrobotrequestsbyplayersfromdifferent resourcecentersasstrategies,andafunctionofcompeting objectivescompactnessanduniformpower distributionasthepayoff.Thepartitionsareupdatediter ativelyonthebasisofNEsolutionsuntilthe stoppingcriterionissatised.6.4MicroeconomicModeling Inthissection,wedescribethepartitioningalgorithmfor multi-robotteamformation.Sincethis isanapplicationofthemulti-objectiveclusteringapproa chbeingpresentedinChapter5,themethodologyfollowsthesamestepsformostpart.Inthissection,w ewillbrieydiscussthestepsinvolved inthealgorithm.PleaserefertoSection5.3fordetailedde scriptionofthesesteps. ThealgorithmidentiestheinitialpartitionsusingtheKM eansclusteringmethod,andiftheinitial partitionsarenotoptimal,agameisformulatedwiththepar titionsasplayersandresources.ANash equilibriumsolutionofthegameidentiestheoptimalreal locationofrobotstothepartitions.The notationsandterminologybeingusedintherestofthepaper aregiveninTable6.1. 6.4.1KMeansPartitioning ThismethodologyrequirestheKMeansalgorithmtoidentify initialaswellastheupdatedteams ofrobots.TheKMeansalgorithmpartitionsthetotalnumber ofrobots N intopartitions( K )dependinguponthenumberofemergencylocationsinaregion.Eacht eamwouldperformthesearchand rescueoperationsatthecorrespondingemergencylocation .Thestepsinvolvedinthemathematical partitioningprocessare: Initializetherandompartitionheadsatthecoordinateloc ationsneartheemergencylocations. Calculatethedistance(Euclideaninthiscase)ofeachrobo tfromeachofthepartitionheads, givenas: E ( k ) ( W )= N i = 1 w ik d j = 1 ( x ij c kj ) 2 (6.1) 122

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Table6.1Notationsforrobotpartitioning.Thenotationsa reusedindevelopingthealgorithmfor robotteamsformationonthebasisofcompactionforlowpowe rdissipationincommunication,and equi-partitioningforuniformpowerdistribution. N Totalnumberofrobotsinthesystem d Totalnumberofattributesofarobot(coordinates) K Totalnumberofpartitions E nk Euclideandistancebetween n and k ,where n 2 N and k 2 K E SumofthesquaredEuclideandistance l k Numberofrobotsinpartition k ; 8 k 2 K l ideal Numberofrobotsperpartitioninauniformpowerdistributi onsituation; l ideal = j N = K j L UniformPowerDistributionMeasure; L = Kk = 1 ( l k l ideal ) 2 P Totalnumberofplayers; P K p i i th playerinagame; 8 i 2 P p i Thesetofalltheplayersinthegameotherthantheplayer p i R Totalnumberofresourcecenters; R K r j j th resourcecenterinagame; 8 j 2 R r j Setofalltheresourcecentersnotinthecurrentgame U i Totalnumberofstrategiesofaplayer p i S i Setofallthestrategiesofplayer p i s iu u th strategyoftheplayer p i ; s iu 2 S i and u = 1 ; ; U S Strategysetconsistingofallthestrategiesinthegame; S = f S 1 ; S 2 ; ; S P g S i Setofallthestrategycombinationsofalltheplayersother than p i s i v Astrategycombinationconsistingofonestrategyofallthe playersotherthan p i ; s i v 2 S i E ( W )= K k = 1 N i = 1 w ik d j = 1 ( x ij c kj ) 2 (6.2) Here,Equation(6.1)correspondstothedistancemeasurefo rthe k thpartitionandEquation(6.2) correspondstothetotalintra-partitionvariation. Assigntherobottothepartitionaccordingtothesumofsqua redEuclideandistance(SSE) measure,asgivenbyEquation(6.3). E ( W )= min W f E ( W ) g (6.3) 6.4.2GameTheoreticPartitioningofRobots Theprocessofidentifyingpartitionswiththeobjectiveof minimizationofSSEmeasureadversely affectsthecomplementarypowerdistributionobjective(d enotedby L ).Hence,agameisrequired tobeformulatedtosimultaneouslyoptimizealltheconict ingobjectives.Specically,theprocess involvestheidenticationofinitialpartitionsusingthe initializationstepofKMeansalgorithm.These 123

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partitionsarethencategorizedasplayersandresourcecen ters,andagameisformalized.Theplayers inthegamecompeteforallocationofresources(robotunits )fromtheresourcecenters.Thestrategy ofaplayerismodeledasatupleconsistingofthenumberofro botsrequestedfromeveryresource center.Thepayoffcorrespondingtothevariousstrategies representsconictingobjectives.Afterthe formulationofthegame,aNashequilibriumsolutionpointi sevaluatedandtheallocationsareperformedaccordingly.Afterthereallocationofrobotstothe partitionsaccordingtothegametheoretic solution,thepartitionsareupdatedusingtheKMeansalgor ithm.Thiscompleteprocessisrepeated untiloptimumpartitionsareidentied.Thestepsinvolved inthealgorithmaredescribedinAlgorithm Algorithm6.1. Algorithm6.1 Microeconomicrobotteamclusteringalgorithm Require: Locationsofrobots,initialnumberofpartitions,initial allocationofrobotstothepartitions afterKMeans Ensure: OptimalPartitioningofrobotsintoteamsonthebasisofpow erdistributionandcompaction objectives 1: if foreachpartition k then 2: Thecondition l k = l ideal issatised,thenreportthesolutionasoptimal,andexit. Here, l ideal =numberofunitsperclusterattheuniformpowerdistributi onstate 3: else 4: Classifytheunequalpartitionsasplayersandresources: 5: Players:all k ,suchthat l k l ideal ; 6: Resources:all k ,suchthat l k l ideal ; 7: endif 8: for Foreachresourcelocation do 9: Playersformulateagamewiththeirstrategies,toreceiveu nitsfromtheresource,sothatthe overheadsaredistributedamongtheplayersandtheresourc eachievesaconsistentstate( l ideal ). 10: Thegameisthensolvedforanequilibriumsolutionpointusi ngNashequilibriumalgorithm 11: endfor 12: Afterthereallocation,thenewpartitioncentersareident ied,andtheprocessisrepeateduntil convergence Thegenerationofthestrategysetinvolvestheplayerstryi ngtoreceiveunitsfromtheresource locationsonthebasisofminimumcostallocationmethodolo gy,irrespectiveoftheallocationsmade totheotherplayers.Howeverinthisprocess,asituationma yarisewherelocationhasallocatedmore unitsthanitsoverhead.Therefore,agameisformulatedand solvedforthatresourcelocation,and subsetofpartitionsthathavetriedreceivingunitsfromit playthegame.Thestrategiesoftheplayers consistofthenumberofunitstheymayneedtoloseinorderto ensurethattheresourcelocationisin consistentstate,i.e.,ithasauniformpowerdistribution .Duetothisalternativedenitionofstrategy, 124

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asinglestepgamewithanexponentialstrategysetandlarge numberofplayersisreducedtomultiple gamesofwithsignicantlysmallerstrategysetsandplayer s.Thedetailsofthestepsfollowedfor deningthestrategysetforthegametheoreticformulation aregiveninSection5.3.2.2. Eachstrategycombinationinagamehasanexpectedutilityt hataplayerwouldreceive.The utilityismathematicallymodeledasapayofffunctioneval uatingthegainorlossaplayerwouldincur whenitplaysitsownstrategyandtheotherplayersplaythei rcorrespondingstrategies.Thepayoff functioninthismodelcapturestheinter-relationshipoft heoptimizationcriteria,andismodeledasa geometricmeanofthetotallossincurredbyaplayerwhenitp laysaparticularstrategy.Specically, aplayerwouldhavetoreceiveunitsfromadistantresourcel ocationiftheotherplayersrequestfor alltheunitsavailablewiththecurrentresourcelocation. Thepayoff,afunctionofpowerdistribution metricandcompactionmetricisthelosstotheplayerwhensu chasituationoccurs.Thealgorithmfor payofffunctionisgiveninAlgorithmAlgorithm5.4insecti on5.3.2.3. Thepayoffmatricesevaluatedduringthepreviousstepareg ivenasinputtotheNashequilibrium (NE)algorithm,whichgeneratesanoutputasaNEstrategyse tconsistingofonestrategychosen foreveryplayerinthegame.AttheNashequilibriumpoint,n oplayerhasincentivetochangeits strategyunilaterally.Mathematically,theNEpointisgiv enbyEquation(6.4).Aftertheequilibrium strategiesareidentied,thereallocationofunitsisperf ormedaccordingly.Thegameisthenplayed forotherlocationsinconictandtheallocationsareperfo rmedaccordingly.Thepartitionmedoids arethenupdated,andthecompleteprocessisrepeateduntil therelativeimprovementinthepower distributionsdoesnotsupersedethresholddecreaseinthe valueofSSE. po i ( s 1 ; ; s i ; ; s P 0 ) po i ( s 1 ; ; s i ; ; s P 0 ) (6.4) 6.5ExperimentalResults Inthissection,wepresenttheexperimentsthatwerecarrie douttoevaluatetheefcacyofthe methodologyforrobotteamformation.Sincetherearenoben chmarksavailableformulti-objective robotpartitioning,severalarticialdatasetswerecreat edtosimulatetherealworldscenarios.The performanceofthemicroeconomicmodelwascomparedwithth eclassicalKMeansalgorithm. 125

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6.5.1SimulationToolsandSetup Tosimulatethelocationsofrobotsonaterrainthefollowin gsetupwasformulated: Atwodimensionalgridofsize12*12wascreated,andnormall ydistributeddatasetsconsisting ofthex-ycoordinatesoftherobotlocationsonthegridwere generated. Thevaluesofmeanandvariancewerevariedfrom0 10and s = 2respectivelyforeach dataset. Thedatasetswith35nodesweregenerated,with3to7cluster sperdataset. Theintra-partitionsimilaritymeasuresintermsofnumber ofrobotsperpartitionweretaken intoconsideration.Forexample,adataset5 7wouldhave5partitions,eachpartitionhaving thenumberofrobotsrangingfrom b 0 : 7 ( 35 = 5 ) c = 4to b ( 0 : 3 ( 35 = 5 ))+( 35 = 5 ) c = 9. Eachexperimentalresultwasanaverageof200repetitionsw ithrandomgatewaylocationinitializations. TheNashequilibrium(NE)solutiontothen-personnormalfo rmgameisidentiedusingthe SimplicalSubdivisionalgorithm,whichhasbeenidentied toworkconsistentlybetterthan otherexistingNEmethodologiesavailableinliterature.B aseduponthesimplexmethod,thealgorithmstartswithagivengridsize,andconvergestoanapp roximatesolutionpointbyiterative labelingofthesub-simplexes. Gambit[108],anopensourceClibraryofgametheoryanalyze rsoftwaretoolkitforidenticationofNEsolutionwasusedasasolutionmethodology. 6.5.2Analysis Experimentswereconductedtostudytheperformanceofthis methodinsimultaneouslyoptimizingtheobjectivefunctions;thecompactionmeasure( SSE )andtheuniformpowerdistributionmeasure ( L ).AsshowninFigure6.6,foradataset6 7thatconsistsof6gateways,and35robotsdistributed amongthegateways,ourmethodologyidentiesthepartitio nswiththe SSE of58.86,andthe L being 10.83,whichcontributestowardanimprovementof90.3%and 90.2%respectivelyfromtheinitial values.However,fortheKMeansalgorithm,althoughtheimp rovementincompactionis1.3%higher 126

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thangametheoreticmethod,thepowerdistributionobjecti veis45.1%worse.Overall,thegametheory basedmulti-metricoptimizationmethodoutperformstheKM eansalgorithmintermsofsimultaneous optimizationofthemultipleobjectives. Figure6.6Identicationofoptimumsizesoftheclustersan dthelocationsoftheclustercentersusing gametheoreticalgorithm,andKMeansalgorithm.Totalnumb erofrobots=35,totalnumberof gateways=6,andnameofdataset=6 7.txt. Theaverageperformanceofthenewmethodwasalsocomparedw iththeKMeansalgorithm.All 35datasetswereexecutedandaverageoftheoutputsforimpr ovementsin SSE and L wereplottedon agraphasshowninFigure6.7.Asshown,theimprovementinth ecompactionobjectiveishigherfor theKMeansalgorithm.ThisisintuitivesinceKMeansperfor msthepartitioningonlyonthebasisof optimizationofcompactionobjective.However,thisadver selyaffectstheuniformpowerdistribution objective,andisevidentfromthegraph. Incontrast,ouralgorithmsimultaneouslyoptimizesbotht heobjectives.Itisimportanttonote thatboththeobjectivesareoptimizedwithanaverageimpro vementofmorethan50%intermsof results.Aninterestingobservationisthatasthenumberof partitionsincrease,theperformanceofthis methodimprovesandaftercertainlimititdegrades.Thisis duetotheincreasingdimensionalityofthe problem.Ifthenumberofpartitionsaretoofew,theinitial partitionsidentiedbytheinitialiteration 127

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Figure6.7Averageperformanceofalgorithmsonarticiald atasets.TheKMeansandthegame theoreticalgorithmarecomparedfortheirperformanceont heclustercompaction( SSE )anduniform powerdistribution( L )metrics. ofKMeansoptimallypartitionsthedata,andagameisnotfor mulatedoften.Asthepartitionsincrease, theKMeansinitializationisunabletoidentifyoptimumclu stersresultinginmultipleiterationsofgame formulationandhencesimultaneousoptimizationofobject ives.However,asthepartitionsincrease beyondacertainlimit,thenumberofstrategiespergameinc reaseandthegametheoreticmodelin itscurrentformprunesthestrategysettocontrolthedimen sionalityoftheproblem.Duetothis, occasionallythestrategiesthatarenotlocallyoptimalbu thaveaglobaleffectmaygetprunedthereby affectingtheperformance. Theresponsetimeofamicroeconomicmodellargelydetermin esitspracticabilityinanapplicationdomain.Theparametersthatlargelygoverntherespons etimeforgametheoreticmodelinthis contextincludethenumberofplayers,thenumberofpartiti ons,andthetotalnumberofstrategiesof players.Table6.2showstheaveragevaluesoftheseparamet ersfordifferentnumberofgateways.For smallernumberofpartitions,theinitialKMeansclusterin gisoftenoptimalandagameisnotrequired tobeplayed,andhenceaveragenumberofiterationsofgamei slessthanone.Theresultsonthe simulateddatasetsarepromisingbecauseasthenumberofpa rtitionsincrease,thenumberofstrategiesdonotincreaseexponentially,whichisaconcernwithm ostoftheproblemsmodeledinagame theoreticframework.Thelinearrelationshipbetweenthes izeofstrategysetandnumberofpartitions isattributedtothenoveldenitionofthestrategyandthem odelingofthegameinthiscontextofthis 128

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work.Thestrategysetforaplayerinthismodeldoesnotdepe ndoncombinationsofthenumberof resourcelocationsthatareavailingtheresources,butont henumberofunitsaplayermayhavetolose forkeepingtheresourceinaconsistentstate.Table6.2Parametersaffectingthegametheoreticmodel.Th einter-relationshipbetweentheimportant attributesofagamesuchasthenumberofplayers,numberofs trategies,numberofclusters,total numberofgameiterations,andtheexecutiontimeofthealgo rithmisidentied. Partitions 3 4 5 6 7 Avg.IterationsofGame 0.23 0.80 1.52 1.96 2.02 Avg.NumberofPlayers 0.22 0.60 1.39 1.60 2.13 Avg.NumberofStrategies 0.32 2.08 4.71 5.27 6.77 ResponseTime(sec.) 0.0003 0.0627 0.1447 0.1615 0.1968 6.6Discussion Anovelmicroeconomicapproachformulti-objectiverobott eamformationproblemhasbeendevelopedinthisresearch.Itmodelstheproblemasahybridap proachinvolvingKmeansandnoncooperativemulti-playernormalformgamewithNashequili briumbasedsolution.Theobjective functionsbeingconsideredinthemodelarecompactness,an duniformpowerdistribution.Thesimulationshavebeenconductedusingnormallydistributedar ticialdatasets.Theperformanceofthis methodascomparedtotheKMeansalgorithmconformstothecl aimthatourmodelisbettersuited forrobotaggregationthantheexistingpartitioningmetho ds.Theaveragecomplexityofthesystem isnon-exponential.Thisistherstsuccessfulattemptint hedirectionofrobotteamformationon thebasisofmultipleobjectives.Currently,themodelissi mplistic,andoptimizesonlytwoobjectives simultaneously.However,thepracticalimplementationof themodelmayrequiremoreobjectives,like improvedradiocommunication,minimuminter-teamcommuni cation,etc.tobeconsidered.Insuch scenarios,thepayoffmodelingwouldneedfurtherinvestig ationandrenement.Also,inpractice,the capabilitiesofeachrobotaredifferent,andsuchconsider ationsmustbereectedinthemodeling.It isrequiredtodeployrobotsinseveralrealworldtestscena riostoefcientlyandaccuratelyevaluate theperformanceofthealgorithm. 129

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CHAPTER7 CONCLUSIONSANDFUTUREDIRECTIONS Successfulpackingtwobilliontransistorsonasinglechip [10]givesaclearideaaboutthelevel ofminiaturization,anddensityofthenextgenerationVLSI circuits.Thisincreaseintheintegration uncoversnumerousissuesthathavetobeaddressedbythedes ignersinordertorealizehighperformance,lowpowerdissipating,andreliablecircuits.Someo ftheseconcernsincludetheimpactof processvariationsatnanometerlevel,theeffectofvariou sperformancemetricsoneachother,andthe efciencyofthecircuitoptimizationmethods.Itisachall engingtasktoaddressalltheseissuesina singlemodel.Thefocusofthisdissertationistoaddressal ltheseconcernsintheVLSIdomain,and todevelopaframeworkthatiscapableofsolvingthecurrent aswellasnextgenerationVLSIcircuit optimizationproblems. Thesizeofanoptimizationprobleminanyengineeringdisci plineencouragestheuseofclustering mechanismstopartitionalargeproblemintosmallerproble ms,andsolvethemseparately.However, itisdifculttoadapttheknowledgeandintelligencefromc lassicalclusteringdisciplinestosolvethis problem.Specically,insituationswheretheclusteringn eedstobeperformedonthebasisofmultiple objectivesthatmaybecompetitiveinnature,singleobject iveclusteringalgorithmscannotgenerate goodclusters.Thus,thedevelopmentofageneralizedclust eringmechanismforsuchproblemsis imperative. Inthisdissertation,wehavedevelopedmulti-metricoptim izationframeworkstosolvetheVLSICADcircuitoptimizationproblemsandspatialpatternclus teringproblems,usingutilitarianmethods. Thespecicproblemsbeingsolvedinthisdissertationarea sfollows: Apostlayoutgatesizingalgorithmformulti-metricoptimi zationofdelay,leakagepower,dynamicpower,andcrosstalknoiseinthepresenceofprocessv ariations[118].Thealgorithm generatesadeterministicequivalentoftheinherentlysto chasticoptimizationproblem,while ensuringhighutilitylevels.Itisindependentoftheproce ssvariationdistributionsandcanin130

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corporatetheimpactofvariationsduetogatesizesaswella sinterconnects.Thealgorithmis alsocapableofincorporatingrandomnessintheobjectivef unctions. Developmentofauniedmathematicalprogrammingbasedfra meworkformulti-metricoptimizationofdelay,leakagepower,dynamicpower,andcrosst alknoiseinthepresenceofprocess variation.Theframeworkcanbeimplementedusinganymathe maticalprogrammingtechnique, andiscompletelyrecongurableintermsofprioritizingor selectingthemetricstobeoptimized. Developmentofasimpleyeteffectivecross-talknoisemode landidenticationofrelationships betweenthedifferentperformancemetricsintermsofgates izes. Developmentofanovelgametheoreticclusteringapproachf orsimultaneousmulti-metricclusteringofspatialdataobjects.Ageneralframeworkisdevel opedthatcanincorporateanynumber ofconictingclusteringobjectives. Thegametheoreticclusteringapproachisappliedtosolvet hemulti-objectiverobotteampartitioningprobleminmulti-emergencysearchandrescuemis sions[119].Thepartitioningis performedonthebasisofclustercompactionanduniformpow erdistribution. Theutilitarianmethodsbeingappliedinthisdissertation possesscertainuniqueattributesthathave madetheirapplicationsuitabletosolvetheseproblems,an dtheidenticationofthesemethodsis animportantcontributionofthisdissertation.Theexpect edutilitybasedapproacheschangetheperspectiveofsolvingthestochasticgatesizingproblemwith randomconstraintstoadeterministicrisk minimizationproblemwithanobjectiveofmaximizationofe xpectedutilityofthesatisfactionofthe constraints.Thistransformationsignicantlyreducesth etimecomplexityofthealgorithm,while maintainingahighyield.Thisisaprimecontributionofthi sdissertation.Themodelingofaclusteringprobleminagametheoreticframeworkisnovel.Thenewde nitionofstrategiesfortheplayers hascontributedtowardsignicantreductioninthetimecom plexityofthealgorithm.Anoveldenitionofpayoffsasafunctionofequi-partitioningandcompa ctionisunique.Theapplicationofspatial clusteringalgorithmfortherobotteamformationisapract icalproblem,andthisisoneofthevery fewworksthathaveaddressedtheproblemwiththisperspect ive. 131

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Theapproachespresentedinthisdissertationarenoveland havewideapplicabilityinthevarious areasofresearch.Someofthefuturedirectionstoimproveo verthisdissertationwork,andother interestingresearchideasarelistedasfollows. Themulti-metricoptimizationmodelforVLSIcircuitoptim izationpresentedinthisdissertationincorporatesfourmetricsthathavebeenoptimized.Ad ditionalmetricslikesecurityand reliabilityetc.canbeincorporatedeasilyoncetherelati onshipbetweenthemetricsintermsof gatesizesisidentied. Theexpectedutilitybasedmethodscanbeutilizedforvario uscircuitoptimizationtechniques likebufferinsertionorrepeaterinsertionandwiresizing .Thesemethodsalsondapplications insolvingthemulti-metricoptimizationproblemsusingga tesizingatthelogiclevelorRTL level. Theexpectedutilitybasedoptimizationpresentedinthisd issertationassumesthatscarceinformationintermsofonlymeanandvarianceoftheprocessvaria tionsisavailable.However,if moreinformationintermsofcoefcientofcorrelationsisa lsoavailable,themodelcanbefurtherextendedtoincorporatesuchinformationandformulat ealinearprogrammingequivalent modelwithquadraticconstraints[94]. TheVLSImulti-metricoptimizationproblemcontainsanobj ectivefunctionthatisdeterministic innature.However,theexpectedutilitybasedmethodiscap ableofsolvingtheproblemswith randomobjectivesalso.Thisisaninterestingfutureworkf ormulti-metricoptimizationwith differentlevelsofrandomnessintheindividualmetrics.S uchasolutionwillgiveafrontierof solutionpoints. Thegametheoreticspatialclusteringalgorithminitscurr entformiscapableofclustering mediumsizeddatasets.Thisisattributedtothenon-linear increaseinthenumberofstrategies asthenumberofplayersincrease.However,ifbettertechni quesareincorporatedtoaggressively prunethestrategyset,thealgorithmwouldbeabletocluste rlargerdatasets. Analternativenotionofapplyingthegametheoreticmethod totheclusteringproblemistoconsidertheobjectivesastheplayers.Itwouldbeinteresting toseethechangesintheoptimization 132

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performance,sincethenumberofplayersinthatscenariowo uldbeconstant,butthestrategyset maybelarger. Thegametheoreticclusteringapproachhasseveralusefula pplications.Onesuchapplicationis inthedomainofad-hocandsensornetworks.Thead-hocnetwo rksneedclusterstobeformed withtheobjectiveofminimizinginter-aswellasintra-clu stercommunication.Tosatisfythese requirements,eachclusterdesignatesoneofthenodesasag atewayforinter-clustercommunicationandonenodeasaclusterheadforintra-clustercommu nication.Gametheoreticclustering approachcanbeutilizedtosolvethisclusteringproblem. 133

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REFERENCES [1]A.J.Strojwas.ConqueringProcessVariability:AKeyEn ablerforProtableManufacturing inAdvancedTechnologyNodes. InternationalSymposiumonSemiconductorManufacturing 2006. [2]ITRS.InternationalTechnologyRoadmapforSemiconduc tors. http://www.itrs.net/Links/2007ITRS/ExecSum2007.pdf ,2007. [3]K.T.TangandE.G.Friedman.InterconnectCouplingNois einCMOSVLSICircuits. ProceedingsoftheInternationalSymposiumonPhysicalDesign ,pages48–53,1999. [4]S.Borkar.DesignChallengesofTechnologyScaling. IEEEMICRO ,pages23–29,1999. [5]S.Borkar,T.Karnik,S.Narendra,J.Tschanz,A.Keshava rzi,andV.De.ParameterVariations andImpactonCircuitsandMicroarchitecture. ProceedingsofDesignAutomationConference pages338–342,2003. [6]H.ChangandS.S.Sapatnekar.StatisticalTimingAnalys isunderSpatialCorrelations. IEEE TransactionsonComputerAidedDesign ,24(9):1467–1482,2005. [7]C.Visweswariah,K.Ravindran,K.Kalafala,S.G.Walker ,andS.Narayan.First-OrderIncrementalBlock-BasedStatisticalTimingAnalysis. ProceedingsoftheDesignAutomation Conference ,pages331–336,2004. [8]H.Chang,V.Zolotov,S.Narayan,andC.Visweswariah.Pa rameterizedBlock-BasedStatistical TimingAnalysiswithNon-GaussianParameters,Non-Linear DelayFunctions. IEEE/ACM DesignAutomationConference ,pages71–76,2005. [9]L.Cheng,J.Xiong,andL.He.Non-LinearStatisticalSta ticTimingAnalysisforNon-Gaussian VariationSources. Proceedingsofthe44thAnnualConferenceonDesignAutomat ion ,pages 250–255,2007. [10]IntelCorporation.World'sFirstTwoBillionTransist orMicroprocessor. http://www.intel.com/technology/architecture-silico n/2billion.htm ,2008. [11]A.K.Jain,M.N.Murty,andP.J.Flynn.Dataclustering: areview. ACMComputingSurveys 31(3):264–323,1999. [12]R.XuandD.Wunsch.SurveyofClusteringAlgorithms. IEEETransactionsonNeuralNetworks ,16(3):645–678,2005. [13]A.Vetta.NashEquilibriainCompetitiveSocieties,wi thApplicationstoFacilityLocation, TrafcRoutingandAuctions. IEEESymposiumonFoundationsofComputerScience ,pages 416–425,2002. 134

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[14]O.H.Kwon.PerspectiveoftheFutureSemiconductorInd ustry:ChallengesandSolutions. ProceedingsoftheDesignAutomationConference ,2007. [15]J.W.Friedman.GameTheorywithApplicationstoEconom ics. OxfordUniversityPress ,1986. [16]U.GuptaandN.Ranganathan.MultieventCrisisManagem entusingNoncooperativeMultistep Games. IEEETransactionsonComputers ,56(5):577,2007. [17]N.Ranganathan,U.Gupta,R.Shetty,andA.K.Murugavel .AnAutomatedDecisionSupport SystemBasedonGameTheoreticOptimizationforEmergencyM anagementinUrbanEnvironments. JournalofHomelandSecurityandEmergencyManagement ,4(2),2007. [18]U.GuptaandN.Ranganathan.SocialFairnessinMulti-E mergencyResourceManagement. IEEEInternationalSymposiumonTechnologyandSociety ,pages1–9,2006. [19]U.GuptaandN.Ranganathan.FIRM:AGameTheoryBasedMu lti-CrisisManagementSystem forUrbanEnvironments. ProceedingsoftheInternationalConferenceonSharingSol utionsfor EmergenciesandHazardousEnvironments,AmericanNuclear Society ,pages595–602,2006. [20]A.LazarandN.Semret.AResourceAllocationGamewithA pplicationtoWirelessSpectrum. Technicalreport,ColumbiaUniversity ,1996. [21]Z.Liu,V.Misra,andL.Wynter.DynamicOfoadinginaMu lti-ProviderEnvironment:aBehavioralFrameworkforUseinInuencingPeering. ProceedingsofIEEEInternationalSymposiumonClusterComputingandtheGrid ,pages449–458,2004. [22]Y.K.Kwok,S.Song,andK.Hwang.SelshGridComputing: Game-TheoreticModelingand NASPerformanceResults. ProceedingsofIEEEInternationalSymposiumonClusterCom putingandtheGrid ,2005. [23]D.C.GrosuandA.T.M.Y.Leung.LoadBalancinginDistri butedSystems:anApproachusing CooperativeGames. IEEEInternationalParallelandDistributedProcessingSy mposium ,pages 52–61,2002. [24]N.HanchateandN.Ranganathan.SimultaneousIntercon nectDelayandCrosstalkNoise OptimizationThroughGateSizingusingGameTheory. IEEETransactionsonComputers 55(8):1011–1023,2006. [25]A.K.MurugavelandN.Ranganathan.AGameTheoreticApp roachforPowerOptimization DuringBehavioralSynthesis. IEEETransactionsonVeryLargeScaleIntegrationSystems 11(6):1031–1043,2003. [26]A.K.MurugavelandN.Ranganathan.GateSizingandBuff erInsertionusingEconomicModels forPowerOptimization. InternationalConferenceonVLSIDesign ,pages195–200,2004. [27]F.Forgo,J.Szep,andF.Szidarovszky. IntroductiontotheTheoryofGames:Concepts,Methods,Applications .KluwerAcademicPublishers,1999. [28]E.Rasmusen. GamesandInformation:AnIntroductiontoGameTheory .BlackwellPublishers, 2001. [29]GameTheory.net.AResourceforStudentsandEducators ofGameTheory. http://www.gametheory.net/Dictionary 135

PAGE 149

[30]W.F.Lucas.SomeRecentDevelopmentsinn-PersonGameT heory. SIAMReview ,13(4):491– 523,Oct.1971. [31]D.Dutta,A.Goel,andJ.Heidermann.ObliviousAQMandN ashEquilibria. ACMSIGCOMM ComputerCommunicationsReview ,32(3),July2002. [32]J.VonNeumannandO.Morgenstern.TheoryofGamesandEc onomicBehavior. Wiley,New York ,1944. [33]S.J.RussellandP.Norvig.ArticialIntelligence:AM odernApproach. Prentice-HallSeries inArticialIntelligence ,page932,1995. [34]M.FriedmanandL.J.Savage.TheUtilityAnalysisofCho icesInvolvingRisk. TheJournalof PoliticalEconomy ,56(4):279–304,1948. [35]J.VonNeumann.ZurTheoriederGesellschaftsspiele. MathematischeAnnalen ,100:295–320, 1928. [36]J.F.NashJr.EquilibriumPointsinN-PersonGames. ProceedingsofNationalAcademyof ScienceoftheUnitedStatesofAmerica ,36(1):48–49,1950. [37]E.D.Dolan,R.Fourer,J.J.More,andT.S.Munson.TheNE OSServerforOptimization:Version4andBeyond. MathematicsandComputerScienceDivision,ArgonneNation alLaboratory ,2002. [38]C.Visweswariah.OptimizationTechniquesforHigh-Pe rformanceDigitalCircuits. IEEE/ACM InternationalConferenceonComputer-AidedDesign ,pages198–207,1997. [39]J.Cong,L.He,C.K.Koh,andP.H.Madden.PerformanceOp timizationofVLSIInterconnect Layout. VLSIJournalonIntegration ,21(1-2):1–94,1996. [40]J.M.Rabaey.DigitalIntegratedCircuits:ADesignPer spective. Prentice-HallElectronicsAnd VLSISeries ,page702,1996. [41]B.C.Paul,A.Agarwal,andK.Roy.Low-PowerDesignTech niquesforScaledTechnologies. Integration,theVLSIJournal ,39(2):64–89,2006. [42]T.XiaoandM.Marek-Sadowska.GateSizingtoEliminate CrosstalkInducedTimingViolation. ProceedingsoftheInternationalConferenceonComputerDe sign ,pages186–191,2001. [43]T.XiaoandM.Marek-Sadowska.CrosstalkReductionbyT ransistorSizing. Proceedingsof theAsiaandSouthPacicDesignAutomationConference ,pages137–140,1999. [44]S.Bhardwaj,Y.Cao,andS.Vrudhula.StatisticalLeaka geMinimizationThroughJointSelectionofGateSizes,GateLengthsandThresholdVoltage. ProceedingsoftheAsiaandSouth PacicDesignAutomationConference ,pages953–958,2006. [45]A.Srivastava,D.Sylvester,andD.Blaauw.Statistica lOptimizationofLeakagePowerConsideringProcessPariationsusingDual-VthandSizing. Proceedingsofthe41stAnnualConference onDesignAutomation-Volume00 ,pages773–778,2004. [46]R.Rao,A.Srivastava,D.Blaauw,andD.Sylvester.Stat isticalAnalysisofSubthresholdLeakageCurrentforVLSICircuits. IEEETransactionsonVLSISystems ,12(2):131–139,2004. 136

PAGE 150

[47]M.ManiandM.Orshansky.ANewStatisticalOptimizatio nAlgorithmforGateSizing. ProceedingsoftheIEEEInternationalConferenceonComputerD esign ,pages272–277,2004. [48]V.Mahalingam,N.Ranganathan,andJ.E.HarlowIII.AFu zzyOptimizationApproachfor VariationAwarePowerMinimizationduringGateSizing. IEEETransactionsonVLSISystems 2008,toappear. [49]M.Mani,A.Devgan,andM.Orshansky.AnEfcientAlgori thmForStatisticalMinimizationOfTotalPowerUnderTimingYieldConstraints. ProceedingsoftheDesignAutomation Conference ,pages309–314,2005. [50]R.W.Brodersen,M.A.Horowitz,D.Markovic,B.Nikolic ,andV.Stojanovic.MethodsforTrue PowerMinimization. ProceedingsoftheInternationalConferenceonComputerDe sign ,pages 35–42,2002. [51]F.GaoandJ.P.Hayes.TotalPowerReductioninCMOSCirc uitsviaGateSizingandMultiple ThresholdVoltages. ProceedingsoftheDesignAutomationConference ,pages31–36,2005. [52]D.Nguyen,A.Davare,M.Orshansky,D.Chinnery,B.Thom pson,andK.Keutzer.MinimizationofDynamicandStaticPowerThroughJointAssignmentof ThresholdVoltagesandSizing Optimization. ProceedingsofISLPED ,pages158–163,2003. [53]D.SinhaandH.Zhou.YieldDrivenGateSizingforCoupli ng-NoiseReductionUnderUncertainty. ProceedingsoftheAsiaandSouthPacicDesignAutomationC onference ,pages 192–197,2005. [54]D.SinhaandH.Zhou.GateSizingforCrosstalkReductio nUnderTimingConstraintsby LagrangianRelaxation. ProceedingsoftheIEEE/ACMConferenceonComputerAidedDe sign pages14–19,2004. [55]N.HanchateandN.Ranganathan.StatisticalGateSizin gforYieldEnhancementatPostLayout Level. IEEE/ACMInternationalSymposiumonVLSIDesign ,pages245–252,2007. [56]V.Mehrotra,S.L.Sam,D.Boning,A.Chandrakasan,R.Va llishayee,andS.Nassif.AMethodologyforModelingtheEffectsofSystematicWithin-DieInt erconnectandDeviceVariationon CircuitPerformance. ProceedingsoftheDesignAutomationConference ,2000. [57]E.JacobsandM.Berkelaar.GateSizingUsingaStatisti calDelayModel. Proceedingsofthe DesignAutomationandTestinEurope ,pages283–291,2000. [58]S.Nakagawa,D.Sylvester,J.G.McBride,andS.Y.Oh.On -ChipCrossTalkNoiseModelfor Deep-SubmicrometerULSIInterconnect. Hewlett-PackardJ ,pages39–45,1998. [59]A.DavoodiandA.Srivastava.VariabilityDrivenGateS izingforBinningYieldOptimization. Proceedingsofthe43rdAnnualConferenceonDesignAutomat ion ,pages959–964,2006. [60]S.NeiroukhandX.Song.ImprovingtheProcess-Variati onToleranceofDigitalCircuitsusing GateSizingandStatisticalTechniques. ProceedingsoftheDesign,AutomationandTestin Europe ,pages294–299,2005. [61]X.Bai,C.Visweswariah,N.Strenski,H.Philip,andJ.D avid.Uncertainty-AwareCircuit Optimization. ProceedingsoftheDesignAutomationConference ,pages58–63,2002. 137

PAGE 151

[62]A.Agarwal,D.Blaauw,V.Zolotov,S.Sundareswaran,M. Zhao,K.Gala,andR.Panda.PathBasedStatisticalTimingAnalysisConsideringInter-andI ntra-DieCorrelations. Proceedings ofACM/IEEEWorkshoponTimingIssuesintheSpecicationan dSynthesisofDigitalSystems pages16–21,2002. [63]M.HashimotoandH.Onodera.APerformanceOptimizatio nMethodByGateSizingusingStatisticalStaticTimingAnalysis. Proceedingsofthe2000InternationalSymposiumonPhysica l Design ,pages111–116,2000. [64]M.R.Guthaus,N.Venkateswarant,C.Visweswariah,and V.Zolotov.GateSizingusingIncrementalParameterizedStatisticalTimingAnalysis. ProceedingsoftheIEEE/ACMInternational ConferenceonComputer-AidedDesign ,pages1029–1036,2005. [65]S.Raj,S.Vrudhula,andJ.Wang.AMethodologytoImprov eTimingYieldinthePresenceof ProcessVariations. ProceedingsoftheDesignAutomationConference ,pages448–453,2004. [66]J.Singh,V.Nookala,Z.Q.Luo,andS.Sapatnekar.Robus tGateSizingbyGeometricProgramming. ProceedingsoftheDesignAutomationConference ,pages315–320,2005. [67]K.Agarwal,D.Sylvester,D.Blaauw,F.Liu,S.Nassif,a ndS.Vrudhula.VariationalDelay MetricsforInterconnectTimingAnalysis. ProceedingsoftheDesignAutomationConference 1,2004. [68]Y.Liu,S.R.Nassif,L.T.Pileggi,andA.J.Strojwas.Im pactofInterconnectVariationsonthe ClockSkewofaGigahertzMicroprocessor. Proceedingsofthe37thConferenceonDesign Automation ,pages168–171,2000. [69]P.Scheunders.AComparisonofClusteringAlgorithmsA ppliedtoColorImageQuantization. PatternRecognitionLetters ,18(11-13):1379–1384,1997. [70]E.Kolatch.ClusteringAlgorithmsforSpatialDatabas es:ASurvey. OnlineAvailable, http://citeseer.nj.nec.com/436843.html ,2001. [71]P.Berkhin.SurveyofClusteringDataMiningTechnique s. AccrueSoftware ,10:92–1460,2002. [72]F.Murtagh.ASurveyofRecentAdvancesinHierarchical ClusteringAlgorithms. TheComputerJournal ,26(4):354–359,1983. [73]A.BaraldiandP.Blonda.ASurveyofFuzzyClusteringAl gorithmsforPatternRecognition. II. IEEETransactionsonSystems,ManandCybernetics,PartB ,29(6):786–801,1999. [74]J.MacQueen.SomeMethodsforClassicationandAnalys isofMultivariateObservations. ProceedingsoftheFifthBerkeleySymposiumonMathematicalSt atisticsandProbability ,1:281– 297,1967. [75]E.Vorhees.TheEffectivenessandEfciencyofAgglome rativeHierarchicalClusteringinDocumentRetrieval. PhDthesisDepartmentofComputerScience,CornellUnivers ity ,1985. [76]G.J.McLachlanandT.Krishnan. TheEMAlgorithmandExtensions .Wiley,1997. [77]M.Ester,H.P.Kriegel,J.Sander,andX.Xu.ADensity-B asedAlgorithmforDiscoveringClustersinLargeSpatialDatabaseswithNoise. Proceedingsofthe2ndInternationalConfference onKnowledgeDiscoveryandDataMining ,pages226–231,1996. 138

PAGE 152

[78]W.Gale,S.Das,andC.T.Yu.ImprovementstoanAlgorith mforEquipartitioning. IEEE TransactionsonComputers ,39(5):706–710,1990. [79]Y.T.WangandR.J.TMorris.LoadSharinginDistributed Systems. IEEETransactionson Computers ,34:204–217,1985. [80]D.GrosuandA.T.Chronopoulos.AlgorithmicMechanism DesignforLoadBalancinginDistributedSystems. IEEETransactionsonSystems,ManandCybernetics,PartB ,34(1):77–84, 2004. [81]M.Demirbas,A.Arora,V.Mittal,andV.Kulathumani.AF ault-LocalSelf-StabilizingClusteringServiceforWirelessAdHocNetworks. IEEETransactionsonParallelandDistributed Systems ,17(9):912–922,2006. [82]A.D.AmisandR.Prakash.Load-BalancingClustersinWi relessAdHocNetworks. Proceedings3rdIEEESymposiumonApplication-SpecicSystemsand SoftwareEngineeringTechnology ,pages25–32,2000. [83]S.Kirkpatrick,C.D.GelattJr,andM.P.Vecchi.Optimi zationbySimulatedAnnealing. Science 220(4598):671,1983. [84]K.KrishnaandM.N.Murty.GeneticK-MeansAlgorithm. IEEETransactionsonSystems,Man andCybernetics,PartB ,29(3):433–439,1999. [85]M.LaszloandS.Mukherjee.AGeneticAlgorithmusingHy per-QuadtreesforLowDimensionalK-MeansClustering. IEEETransactionsonPatternAnalysisandMachineIntelligence ,28(4):533–543,2006. [86]J.HandlandJ.Knowles.EvolutionaryMultiobjectiveC lustering. ProceedingsoftheEighth InternationalConferenceonParallelProblemSolvingfrom Nature ,pages1081–1091. [87]F.Glover.FuturePathsforIntegerProgrammingandArt icialIntelligence. Computers& OperationsResearch ,13:533–549,1986. [88]M.Dorigo,G.D.Caro,andL.M.Gambardella.AntAlgorit hmsforDiscreteOptimization. ArticialLife ,5(2):137–172,1999. [89]A.Topchy,A.K.Jain,andW.Punch.ClusteringEnsemble s:ModelsofConsensusandWeak Partitions. IEEETransactionsonPatternAnalysisandMachineIntellig ence ,27(12):1866– 1881,2005. [90]J.HandlandJ.Knowles.MultiobjectiveClusteringand ClusterValidation,Chapter1. Studies inComputationalIntelligence,Springer-Verlag,Berlin, Germany ,pages1–24,2005. [91]D.GrosuandA.T.Chronopoulos.AGame-TheoreticModel andAlgorithmforLoadBalancing inDistributedSystems. IEEEInternationalParallelandDistributedProcessingSy mposium pages146–153,2002. [92]R.R.Schaller.Moore'sLaw:Past,PresentandFuture. IEEESpectrum ,34(6):52–59,1997. [93]H.Chang,V.Zolotov,S.Narayan,andC.Visweswariah.P arameterizedBlock-BasedStatistical TimingAnalysiswithNon-GaussianParameters,Non-Linear DelayFunctions. Proceedingsof theDesignAutomationConference ,pages71–76,2005. 139

PAGE 153

[94]E.Ballestero.StochasticLinearProgrammingwithSca rceInformation:AnApproachfrom ExpectedUtilityandBoundedRationalityAppliedtotheTex tileIndustry. EngineeringOptimization ,38(4):425–440,2006. [95]M.BerkelaarandJ.Jess.GateSizinginMOSDigitalCirc uitswithLinearProgramming. ProceedingsoftheDesign,AutomationandTestinEurope ,pages217–221,1990. [96]R.H.Byrd,J.Nocedal,andR.A.Waltz.KNITRO:AnIntegr atedPackageforNonlinearOptimization. Large-ScaleNonlinearOptimization ,pages35–59. [97]N.Ranganathan,U.Gupta,andV.Mahalingam.Simultane ousOptimizationofTotalPower, CrosstalkNoise,andDelayUnderUncertainty. ProceedingsofACMGreatLakesSymposium onVLSI ,2008. [98]N.Ranganathan,J.E.HarlowIII,V.Mahalingam,andU.G upta.StatisticalGateSizingfor Multi-MetricOptimizationofDelay,PowerandCrosstalkNo iseatPostLayoutLevel. SemiconductorResearchCorporationTechnicalReport,1596.00 1 ,2007. [99]P.Gargini,J.Glaze,andO.Williams.TheSIA's1997Nat ionalTechnologyRoadmapfor Semiconductors:SIAroadmappreview. SolidStateTechnology ,41(1):73–76,1998. [100]J.J.Buckley.StochasticVersusPossibilisticProgr amming. FuzzySetsandSystems ,34(2):173– 177,1990. [101]R.E.BellmanandL.A.Zadeh.DecisionMakinginFuzzyE nvironment. ManagementScience pages141–164,1970. [102]MasatoshiSakawa. GeneticAlgorithmsandFuzzyMultiobjectiveOptimization .KluwerAcademicPublishers,2002. [103]N.R.GasimovandK.Yenilmez. FuzzyLinearProgrammingProblemswithFuzzyMembership Functions .2000. [104]J.G.KlirandB.Yuan. FuzzySetsandFuzzyLogic .PrenticeHall,1995. [105]C.Papadimitriou.Algorithms,Games,andtheInterne t. ProceedingsoftheThirty-ThirdAnnual ACMSymposiumonTheoryofComputing ,pages749–753,2001. [106]H.Spath. ClusterAnalysisAlgorithmsforDataReductionandClassi cationofObjects .Ellis Horwood,1980. [107]Y.Chien. InteractivePatternRecognition .M.DekkerNewYork,1978. [108]R.McKelvey,A.McLennan,andT.Turocy.Gambit:Softw areToolsforGameTheory. The GambitProject ,2002. [109]R.Jain,D.M.Chiu,andW.Hawe.AQuantitativeMeasure ofFairnessandDiscriminationfor ResourceAllocationinSharedComputerSystem. DigitalEquipmentCorp.EasternResearch Lab,DEC-TR-301 ,1984. [110]L.M.BranscombandR.D.Klausner.MakingtheNationSa fer:TheRoleofScienceandTechnologyinCounteringTerrorism. CommitteeonScienceandTechnologyforCounteringTerrorism,NationalResearchCouncil ,2002. 140

PAGE 154

[111]B.P.GerkeyandM.J.Mataric.Sold!:AuctionMethodsf orMultirobotCoordination. IEEE TransactionsonRoboticsandAutomation ,18(5):758–768,2002. [112]R.Emery-Montemerlo,G.Gordon,J.Schneider,andS.T hrun.GameTheoreticControlfor RobotTeams. Proceedingsofthe2005IEEEInternationalConferenceonRo boticsandAutomation ,pages1163–1169,2005. [113]J.CasperandR.R.Murphy.Human–RobotInteractionsD uringtheRobot-AssistedUrban SearchandRescueResponseattheWorldTradeCenter. IEEETransactionsonSystems,Man andCybernetics,PartB ,33(3):367–385,2003. [114]R.R.Murphy.Human–RobotInteractioninRescueRobot ics. IEEETransactionsonSystems, Man,andCybernetics,PartC ,34(2),2004. [115]N.Sato,F.Matsuno,T.Yamasaki,T.Kamegawa,N.Shiro ma,andH.Igarashi.Cooperative TaskExecutionbyaMultipleRobotTeamanditsOperatorsinS earchandRescueOperations. ProceedingsofIEEE/RSJInternationalConferenceonIntel ligentRobotsandSystems ,pages 1083–1088,2004. [116]Y.Meng,J.V.Nickerson,andJ.Gan.Multi–RobotAggre gationStrategieswithLimitedCommunication. IEEE/RSJInternationalConferenceonIntelligentRobotsa ndSystems ,pages 2691–2696,2006. [117]A.SolanasandM.A.Garcia.CoordinatedMulti-RobotE xplorationThroughUnsupervised ClusteringofUnknownSpace. ProceedingsofIEEE/RSJInternationalConferenceonIntel ligentRobotsandSystems ,1,2004. [118]U.GuptaandN.Ranganathan.AnExpected-UtilityBase dApproachtoVariationAwareVLSI OptimizationUnderScarceInformation. IEEEInternationalSymposiumonLowPowerElectronicsandDesign ,2008,toappear. [119]U.GuptaandN.Ranganathan.AMicroeconomicApproach toMulti-RobotTeamFormation. Proceedingsofthe2007IEEE/RSJInternationalConference onIntelligentRobotsandSystems pages3019–3024,2007. 141

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LISTOFPUBLICATIONS UpavanGuptaandNagarajanRanganathan,“AnExpected-Util ityBasedApproachtoVariation AwareVLSIOptimizationUnderScarceInformation”, IEEEInternationalSymposiumonLow PowerElectronicsandDesign(ISLPED'08) ,2008(toappear).(chapter3) NagarajanRanganathan,UpavanGupta,andVenkatramanMaha lingam,“SimultaneousOptimizationofTotalPower,CrosstalkNoiseandDelayUnderUnc ertainty”, ACMGreatLakes SymposiumonVLSI(GLSVLSI'08) ,pages171–176,2008.(chapter4) UpavanGuptaandNagarajanRanganathan,“AMicroeconomicA pproachtoMulti-Objective SpatialClustering”, IEEEInternationalConferenceonPatternRecognition(ICP R'08) ,2008 (toappear).(chapters5,6) UpavanGuptaandNagarajanRanganathan,“AMicroeconomicA pproachtoMulti-RobotTeam Formation”, IEEE/RSJInternationalConferenceonIntelligentRobotsa ndSystems(IROS'07) pages3019–3024,2007.(chapters5,6) UpavanGuptaandNagarajanRanganathan,“MultieventCrisi sManagementUsingNoncooperativeMultistepGames”, IEEETransactionsonComputers ,56(5):577–589,2007. NagarajanRanganathan,UpavanGupta,RashmiShetty,andAs hokMurugavel,“AnAutomated DecisionSupportSystemBasedonGameTheoreticOptimizati onforEmergencyManagement inUrbanEnvironments”, JournalofHomelandSecurityandEmergencyManagement,Ber keley ElectronicPress ,4(2),Article1,2007. UpavanGuptaandNagarajanRanganathan,“SocialFairnessi nMulti-EmergencyResource Management”, IEEEInternationalSymposiumonTechnologyandSociety,IS TAS`06 ,2006. UpavanGuptaandNagarajanRanganathan,“FIRM:AGameTheor yBasedMulti-CrisisManagementSystemforUrbanEnvironments”, ProceedingsoftheInternationalConferenceon SharingSolutionsforEmergenciesandHazardousEnvironme nts,AmericanNuclearSociety, SaltLakeCity,Utah ,pages595–602,2006. 142

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ABOUTTHEAUTHOR UpavanGuptareceivedtheBachelorofComputerApplication s(Honors)degreefromtheInternationalInstituteofProfessionalStudies,Indore,India ,in2002,andtheM.S.degreeinComputer SciencefromtheUniversityofSouthFlorida,Tampa,in2004 .HeiscurrentlypursuingthePh.D. degreeintheDepartmentofComputerScienceandEngineerin g,attheUniversityofSouthFlorida. Hisresearchinterestsincludethedevelopmentofmulti-me tricoptimizationmethodologiesandimplementingthoseindifferentdomainsofcomputersciencea ndengineering.Heisarecipientofthe IEEEComputerSocietyR.E.MerwinScholarship.Heisastude ntmemberoftheIEEEandtheIEEE ComputerSociety.


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ABSTRACT: In the field of VLSI circuit optimization, the scaling of semiconductor devices has led to the miniaturization of the feature sizes resulting in a significant increase in the integration density and size of the circuits. At the nanometer level, due to the effects of manufacturing process variations, the design optimization process has transitioned from the deterministic domain to the stochastic domain, and the inter-relationships among the specification parameters like delay, power, reliability, noise and area have become more intricate. New methods are required to examine these metrics in a unified manner, thus necessitating the need for multi-metric optimization. The optimization algorithms need to be accurate and efficient enough to handle large circuits.As the size of an optimization problem increases significantly, the ability to cluster the design metrics or the parameters of the problem for computational efficiency as well as better analysis of possible trade-offs becomes critical. In this dissertation research, several utilitarian methods are investigated for variation aware multi-metric optimization in VLSI circuit design and spatial pattern clustering. A novel algorithm based on the concepts of utility theory and risk minimization is developed for variation aware multi-metric optimization of delay, power and crosstalk noise, through gate sizing. The algorithm can model device and interconnect variations independent of the underlying distributions and works by identifying a deterministic linear equivalent model from a fundamentally stochastic optimization problem.Furthermore, a multi-metric gate sizing optimization framework is developed that is independent of the optimization methodology, and can be implemented using any mathematical programming approach. It is generalized and reconfigurable such that the metrics can be selected, removed, or prioritized for relative importance depending upon the design requirements. In multi-objective optimization, the existence of multiple conflicting objectives makes the clustering problem challenging. Since game theory provides a natural framework for examining conflicting situations, a game theoretic algorithm for multi-objective clustering is introduced in this dissertation research. The problem of multi-metric clustering is formulated as a normal form multi-step game and solved using Nash equilibrium theory.This algorithm has useful applications in several engineering and multi-disciplinary domains which is illustrated by its mapping to the problem of robot team formation in the field in multi-emergency search and rescue. The various algorithms developed in this dissertation achieve significantly better optimization and run times as compared to other methods, ensure high utility levels, are deterministic in nature and hence can be applied to very large designs. The algorithms have been rigorously tested on the appropriate benchmarks and data sets to establish their efficacy as feasible solution methods. Various quantitative sensitivity analysis have been performed to identify the inter-relationships between the various design parameters.
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