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A game theoretic framework for interconnect optimization in deep submicron and nanometer design

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A game theoretic framework for interconnect optimization in deep submicron and nanometer design
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Hanchate, Narender
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Game theory
Crosstalk noise
Interconnect delay
Process variations
Delay uncertainty
Transmission line models
Wire sizing
Gate sizing
Dissertations, Academic -- Computer Science and Engineering -- Doctoral -- USF
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ABSTRACT: The continuous scaling of interconnect wires in deep submicron (DSM)circuits result in increased interconnect delay, power and crosstalk noise. In this dissertation, we address the problem of multi-metric optimization at post layout level in the design of deep submicron designs and develop a game theoretic framework for its solution. Traditional approaches in the literature can only perform single metric optimization and cannot handle multiple metrics. However, in interconnect optimization, the simultaneous optimization of multiple parameters such as delay, crosstalk noise and power is necessary and critical. Thus, the work described in this dissertation research addressing multi-metric optimization is an important contribution.Specifically, we address the problems of simultaneous optimization of interconnect delay and crosstalk noise during (i) wire sizing (ii) gate sizing (iii) integrated gate and wire sizing, and (iv) gate sizing considering process variations. Game the ory provides a natural framework for handling conflicting situations and allows optimization of multiple parameters. This property is exploited in modeling the simultaneous optimization of various design parameters such as interconnect delay, crosstalk noise and power, which are conflicting in nature. The problem of multi-metric optimization is formulated as a normal form game model and solved using Nash equilibrium theory. In wire sizing formulations, the net segments within a channel are modeled as the players and the range of possible wire sizes forms the set of strategies. The payoff function is modeled as (i) the geometric mean of interconnect delay andcrosstalk noise and (ii) the weighted-sum of interconnect delay, power and crosstalk noise, in order to study the impact of different costfunctions with two and three metrics respectively. In gate sizing formulations, the range of possible gate sizes is modeled as the set of strategies and the payoff function is modeled as the geome tric mean of interconnect delay and crosstalk noise. The gates are modeled as the players while performing gate sizing, whereas, the interconnect delay and crosstalk noise are modeled as players for integrated wire and gate sizing framework as well as for statistical gate sizing under the impact of process variations.The various algorithms proposed in this dissertation (i) perform multi-metric optimization (ii) achieve significantly better optimization and run times than other methods such as simulated annealing, genetic search, and Lagrangian relaxation (iii) have linear time and space complexities, and hence can be applied to very large SOC designs, and (iv) do not require rerouting or incur any area overhead. Thecomputational complexity analysis of the proposed algorithms as well as their software implementations are described, and experimental results are provided that establish the efficacy of the proposed algorithms.
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Dissertation (Ph.D.)--University of South Florida, 2006.
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by Narender Hanchte.
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AGameTheoreticFrameworkforInterconnectOptimizationi nDeepSubmicronand NanometerDesign by NarenderHanchate Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofComputerScienceandEngineering CollegeofEngineering UniversityofSouthFlorida MajorProfessor:N.Ranganathan,Ph.D. JustinE.HarlowIII,M.S. SoontaeKim,Ph.D. MichaelKovac,Ph.D. NatashaJonoska,Ph.D. DateofApproval: March24,2006 Keywords:Gametheory,Crosstalknoise,Interconnectdela y,Processvariations,Delay uncertainty,Transmissionlinemodels,Wiresizing,Gates izingCopyright2006,NarenderHanchate

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DEDICATION Tomywonderfulparents, GaneshRaoandAnnapurnaBaiHanchate, withallmyloveandrespect

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ACKNOWLEDGEMENTS Mysincerethankstomyadvisor,ProfessorN.Ranganathanfo rgivingmeanopportunitytoworkinthisveryinterestingarea.Iammostgratef ultohimforhiscontinuous encouragementandvaluablesuggestionsfromhisvastexper ience.Iconsidermyselfextremelyfortunateforhavingworkedwithsuchaprominentle aderintheeldofVLSI CADandAlgorithms.Histeaching,leadershipandguidanceh avebeeninstrumentalin myacademicandprofessionaldevelopment.Iwouldalsolike tothankProfessorsJustin E.HarlowIII,Dr.SoontaeKim,Dr.MichaelKovac,andDr.Nat ashaJonoskaforserving onmycommittee. AspecialthanksandacknowledgementtoMr.RobertTufts,As sistantDirectorof NNRCandawonderfulpersonIhaveknown,forhispersonalsup portandvaluablesuggestionsduringmycourseasgraduateassistantinNNRC.Iwo uldliketoacknowledgethe eortsofallthecontributorswhohavedevelopedASICcores andmadethemavailable onlinefordownloadthroughOpencores.Acknowledgmentsto Cadencedesignsystems, SynopsysInc.,developersofCreteprogramfortheirtoolsa nd180nmstandardcelllibrary, developersof\GALib-aC++geneticalgorithmlibrary"andt oAndrewConn,Nick GouldandPhilippeTointforproviding\LANCELOT:apackage forlargescalenon-linear optimization"foropenuse. Iamverygratefulfortheinvaluablesupportandmotivation thatIrecievedfrommy familywithoutwhichthisworkwouldnothavebeenpossible. Theyhavetakenallthepains onthemselvesandenabledmetoworkwithoutdistractions.I wouldalsoliketothankmy friendsViswa,Bheem,Ninnu,Satish,Sandeep,SridharandM oulifortheirsupportduring mystayatUSF.

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TABLEOFCONTENTS LISTOFTABLES iii LISTOFFIGURES iv ABSTRACT v CHAPTER1INTRODUCTION 1 1.1Motivation:Multi-metricOptimization51.2ContributionsofthisDissertation 7 1.3OutlineofthisDissertation 8 CHAPTER2BACKGROUNDANDRELATEDWORK10 2.1WireSizing 11 2.2GateSizing 13 2.3ProcessVariations 14 2.4GameTheoryandStochasticGames 16 2.4.1GameTheory 17 2.4.2StochasticGames 20 2.5InterconnectModels 23 2.5.1InterconnectDelayModels 24 2.5.2DelayUncertaintyModels 26 2.5.3CrosstalkNoiseModels 26 CHAPTER3WIRESIZING 29 3.1ProblemDenition 29 3.2MotivationforWireSizingProblem 30 3.3SimultaneousOptimizationofDelayandNoise323.4SimultaneousOptimizationofDelay,PowerandNoise373.5TimeandSpaceComplexityofProposedWireSizingAlgori thms38 3.6ProofofExistenceofNashEquilibriumSolutionfortheW ire SizingFormulation 40 3.7Discussion 41 3.8DesignFlow 42 3.9ExperimentalResults 43 3.10Conclusions 48 i

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CHAPTER4NEWINTERCONNECTMODELS49 4.1FastTransmissionLineModels 49 4.2ExperimentalResults 52 CHAPTER5GATESIZING 54 5.1ProblemDenition 54 5.2MotivationforGateSizingUsingGameTheoryModel555.3GameTheoreticGateSizingforMulti-metricOptimizati on56 5.3.1Approach1:GatesOrderedBasedonNoiseCriticality5 7 5.3.2Approach2:GatesOrderedBasedonDelayCriticality6 1 5.4TimeandSpaceComplexityofProposedGateSizingAlgori thms62 5.5ProofofExistenceofNashEquilibriumfortheProposedG ate SizingFormulation 63 5.6DesignFlow 65 5.7ExperimentalResults 66 5.8Conclusions 73 CHAPTER6INTEGRATEDGATEANDWIRESIZING74 6.1ProblemDenition 75 6.2MotivationforIntegratedGateandWireSizing766.3ANewApproachtoIntegratedGateandWireSizing76 6.3.1Approach1:GatesOrderedBasedonNoiseCriticality7 7 6.3.2Approach2:GatesOrderedBasedonDelayCriticality8 2 6.4TimeandSpaceComplexity 83 6.5ProofofExistenceofNashEquilibriumfortheProposedI ntegratedGateandWireSizingFormulation84 6.6DesignFlow 85 6.7ExperimentalResults 87 6.8Conclusions 92 CHAPTER7STATISTICALGATESIZINGUNDERPROCESSVARIATIONS 93 7.1ProblemDenition 94 7.2MotivationforStatisticalGateSizingProblem947.3ProposedStatisticalGateSizing 96 7.3.1Approach1:GatesOrderedBasedonNoiseCriticality9 6 7.3.2Approach2:GatesOrderedBasedonDelayCriticality1 01 7.4ExperimentalResults 102 7.5Conclusions 106 CHAPTER8CONCLUSIONSANDFUTUREWORK107REFERENCES 111 ABOUTTHEAUTHOR EndPage ii

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LISTOFTABLES Table2.1.RecentStatisticalGateSizingWorksFoundinLit erature15 Table2.2.NotationsandTerminology 23 Table3.1.ExperimentalResultsforSimultaneousOptimiza tionofInterconnectDelayandCrosstalkNoiseDuringWireSizingUsingComplexDelayModels 45 Table3.2.ExperimentalResultsforSimultaneousOptimiza tionofDelay,PowerandNoiseDuringWireSizingUsingComplexDe-layModels 46 Table4.1.ExperimentalResultsforGameTheoreticApproac hUsingthe DevelopedFastModelsDuringWireSizing53 Table5.1.AverageSavingsforSimultaneousOptimizationo fInterconnectDelayandCrosstalkNoiseDuringGateSizing69 Table5.2.CriticalPathSavingsforSimultaneousOptimiza tionofInterconnectDelayandCrosstalkNoiseDuringGateSizing70 Table5.3.CrosstalkNoiseOptimizationUnderDelayConstr aintsDuringGateSizing 72 Table6.1.AverageSavingsforSimultaneousOptimizationo fInterconnectDelayandCrosstalkNoiseDuringIntegratedGateandWireSizing 88 Table6.2.CriticalPathSavingsforSimultaneousOptimiza tionofInterconnectDelayandCrosstalkNoiseDuringIntegratedGateandWireSizing 89 Table6.3.CrosstalkNoiseOptimizationUnderDelayConstr aintsDuringIntegratedGateandWireSizing91 Table7.1.ComparisonofStochasticGameTheoreticApproac hwithDeterministicandGeometricProgrammingApproach104 iii

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LISTOFFIGURES Figure2.1.TaxonomyofVariousPioneeringWorksonWireSiz ing12 Figure3.1.AnExampleScenario 36 Figure3.2.IntegrationofProposedWireSizingAlgorithmi ntheDesignFlow42 Figure4.1.InterconnectModel(a)ModeledasaTransmissio nLine(b) UncoupledEquivalentofOneSectionoftheInterconnect50 Figure5.1.IntegrationofProposedGateSizingAlgorithmi ntheDesignFlow65 Figure6.1.IntegrationofProposedGateandWireSizingAlg orithmin theDesignFlow 86 iv

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AGAMETHEORETICFRAMEWORKFORINTERCONNECT OPTIMIZATIONINDEEPSUBMICRONANDNANOMETERDESIGN NarenderHanchate ABSTRACT Thecontinuousscalingofinterconnectwiresindeepsubmic ron(DSM)circuitsresultin increasedinterconnectdelay,powerandcrosstalknoise.I nthisdissertation,weaddressthe problemofmulti-metricoptimizationatpostlayoutleveli nthedesignofdeepsubmicron designsanddevelopagametheoreticframeworkforitssolut ion.Traditionalapproaches intheliteraturecanonlyperformsinglemetricoptimizati onandcannothandlemultiple metrics.However,ininterconnectoptimization,thesimul taneousoptimizationofmultiple parameterssuchasdelay,crosstalknoiseandpowerisneces saryandcritical.Thus,the workdescribedinthisdissertationresearchaddressingmu lti-metricoptimizationisan importantcontribution. Specically,weaddresstheproblemsofsimultaneousoptim izationofinterconnectdelayandcrosstalknoiseduring(i)wiresizing(ii)gatesizi ng(iii)integratedgateandwire sizing,and(iv)gatesizingconsideringprocessvariation s.Gametheoryprovidesanatural frameworkforhandlingconrictingsituationsandallowsop timizationofmultipleparameters.Thispropertyisexploitedinmodelingthesimultane ousoptimizationofvarious designparameterssuchasinterconnectdelay,crosstalkno iseandpower,whichareconrictinginnature.Theproblemofmulti-metricoptimizatio nisformulatedasanormal formgamemodelandsolvedusingNashequilibriumtheory.In wiresizingformulations, thenetsegmentswithinachannelaremodeledastheplayersa ndtherangeofpossiblewire sizesformsthesetofstrategies.Thepayofunctionismode ledas(i)thegeometricmean ofinterconnectdelayandcrosstalknoiseand(ii)theweigh ted-sumofinterconnectdelay, v

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powerandcrosstalknoise,inordertostudytheimpactofdi erentcostfunctionswith twoandthreemetricsrespectively.Ingatesizingformulat ions,therangeofpossiblegate sizesismodeledasthesetofstrategiesandthepayofuncti onismodeledasthegeometric meanofinterconnectdelayandcrosstalknoise.Thegatesar emodeledastheplayerswhile performinggatesizing,whereas,theinterconnectdelayan dcrosstalknoisearemodeledas playersforintegratedwireandgatesizingframeworkaswel lasforstatisticalgatesizing undertheimpactofprocessvariations. Thevariousalgorithmsproposedinthisdissertation(i)pe rformmulti-metricoptimization(ii)achievesignicantlybetteroptimizationandrun timesthanothermethodssuch assimulatedannealing,geneticsearch,andLagrangianrel axation(iii)havelineartime andspacecomplexities,andhencecanbeappliedtoverylarg eSOCdesigns,and(iv)do notrequirereroutingorincuranyareaoverhead.Thecomput ationalcomplexityanalysis oftheproposedalgorithmsaswellastheirsoftwareimpleme ntationsaredescribed,and experimentalresultsareprovidedthatestablishtheecac yoftheproposedalgorithms. vi

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CHAPTER1 INTRODUCTION Indeepsubmicron(DSM)circuits,theinterconnectshavebe comeadominantfactor indeterminingtheoverallcircuitperformance,reliabili ty,andcost.Theincreaseinthe integrationdensityandthechiparearesultsintheincreas eoftotalwirelengthperunitarea anddecreaseininterconnectpitch.AccordingtotheIntern ationalTechnologyRoadmap forSemiconductors(ITRS)releasedin2004,thefeaturesiz ewillcontinuetoscaledownat therateof0.7 pergenerationtoreach22nmby2008[1].Thisreductionrate enforcesan increaseinimpurityconcentrationandthescalingdownofs upplyandthresholdvoltagesto maintaintheelectriceldsinthedevice.Asthesupplyvolt ageisscaled,theinterconnect dimensionsmustalsobereducedtotakeadvantageofthefeat uresizescaling[2].Hence,the combinedeectofchipsizegrowthandscalingresultsinrap idincreaseofcapacitanceand resistanceofinterconnectwires.Thisincreasesthepropa gationdelaythroughinterconnects byafactorof S 2 S 2 C ,whereSisthescalingfactorand S C isthechipsizeincreasefactor whichaccountsfortheincreaseinchipsizefromonegenerat ionofICstothenextgeneration [3]. TheinterconnecteectsliketherisingRCdelayofon-chipw iring,noiseconsiderationssuchascrosstalkanddelayunpredictability,uncert aintyduetoprocessvariations, reliabilityconcernsduetorisingcurrentdensitiesandox ideelectricelds,andincreasing powerdissipationarebecomingincreasinglyprominentind eepsubmicronandnanometerdesigns[4,5].Thelogiccelldelayshavereachedpicose condrangeandcontinueto reduceduetothescalingoftheminimumfeaturesize.Howeve r,theinterconnectdelays areincreasingandarecapableofconsumingthemajorityoft heclockcycletimeinDSM designs[6].ThemainreasonforincreasedwiredelaysinDSM istheincreaseinresistance, 1

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whichisinverselyproportionaltoitscross-sectionalare a.Interconnectcapacitancesare alsoincreasingduetohigherwiringdensitiesneededtorou teDSMchips.Thenoisein digitalcircuitsisdenedasanydeviationofasignalfromi tsstablevalueinthoseintervals whereitshouldhavebeenstableotherwise.Indeepsubmicro ndesigns,themainsources fornoiseareinterconnectcross-capacitanceduetocoupli ng,powersupplyructuations, mutualinductanceandthermalnoiseduetoself-heatingcau sedbycurrentrow[7]. Theinterconnectcapacitancehasthreecomponents:areaor groundcapacitance,fringingeldcapacitanceandcouplingcapacitance.Thehighasp ectratioofwiresresultin morewiretowirecapacitanceamongtheneighboringwiresin thesamelayerthanthe areacapacitancebetweenupper/lowerwiringlayers[7].In addition,wirespacescalingdue toincreasedwiringdensitiesalsoincreasescouplingcapa citance.Theinterconnectcrosscapacitancenoiseisduetothechargeinjectedinquiet/sil entnetsbecauseofswitching inneighboringnetsthroughthecouplingcapacitancebetwe enthem.Thechargeinjected increasesprominentlyinthedeepsubmicronregimeduetoth eincreasedcouplingcapacitancebetweenadjacentnetscausingreliabilityissue s[8].Thenoiseduetocoupling capacitanceisthedominantcomponentamongthenoisesourc esandisamajorconcern indeepsubmicrondesign[9].Powersupplynoiseisthespuri oussignalappearingatthe receiverduetothedierencebetweenthelocalreferencevo ltagelevelsatthedriverand thereceiver.Therearetwocomponentsofpowersupplynoise :lowfrequencyandhigh frequency.ThelowfrequencycomponentisknownasIRdropan disduetothevariationin theDCpowersupplyandgroundlevels.Thesimultaneousswit chingofvarioussub-circuit modulesproducethehighfrequencycomponentofpowersuppl ynoisegenerallycalledas deltanoise.Mutualinductancenoiseiscausedbythechange inthemagneticelddue tothetransientcurrentrowthroughtheloopformedbythesi gnalwireandthecurrent returnpath.Thermalnoiseisduetotheselfheatingcausedb ythecurrentrowinthe interconnects,limitingthemaximumallowedaveragecurre ntdensity. ThenoisecausesdelayandfunctionalfailuresduetoMiller eectsandsignaldeviations, andincreasesthepowerconsumptionduetoglitches.Astati cwirecalledthevictimis 2

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perturbedbytheswitchingactivityontheneighboringwire scalledaggressors.When theaggressorsswitchinsamedirectionsimultaneously,an undesirablevoltagespikeis coupledontothevictimduetocapacitivecouplingcausingf alseswitchingorvoltage overshootaectingsignalintegrity[10].Hence,itisimpo rtanttodesignnoiseimmuneDSM circuitsconsideringthecontinuoustrendofscalingofint erconnectdimensions.Modeling thenoiseofacircuitwillneedcompleteinformationofnets (itsneighboringnets,the lengthofoverlap,spacingbetweenthenets,etc)toanalyze thecouplingeects,andhence, istypicallyperformedafterthenalroutingofthedesign[ 11].Thestandardmethods practicedtoreducetheDSMeectsduetointerconnectsared riversizing,buersizing, buerinsertion,wiresizing,wirespacing,netorderingan dwireshielding. TheaggressivescalingofVLSItechnologyhasgivenrisetoi ncreasedimpactofprocess variationsontheperformance,reliabilityandpowerofthe fabricatedcircuits.Limitations duetothemanufacturingprocessesandenvironmentalnoise degradethequalityofsignals andaectthepropagationdelayofthecircuit[4].Theseee ctsforcethepropagation delaytodeviatefromitstypicalornominalvalue,resultin gindelayunpredictability.This deviationofthepropagationdelayduetodelayunpredictab ilityisdenedasdelayuncertainty.Theexamplesoffactorswhichcausedelayuncertain tyarenon-uniformityofgate oxidethickness;imperfectionsinpolysiliconetching,ph otolithography,planarizationand metaletchingprocesses;andenvironmentalnoiseduetocha ngesinambienttemperature andexternalradiation[12].Uncertaintyinpropagationde layofsignalscancauseviolationsinset-upandholdtimingconstraints,resultinginti mingfailureofthedesign.To eradicatethesetimingviolations,thedesignerhastorela xthetimingconstraintsorhasto reducethedelayuncertainty.Relaxingtimingconstraints increasestheclockperiodand hencedegradesthecircuitperformance.Thus,thedesigner hastoreducethedelayuncertaintyinordertomeetthetighttimingconstraintsofthede signwithoutcompromising theperformanceofthedesign. Thedeviceandinterconnectscalingtrendsmakethephysica lrealizationofdevicesand interconnectsunpredictableduringfront-enddesign,hen cechangingtheparadigmofthe 3

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designproblemfromdeterministictoprobabilisticdomain [5].Inaddition,theprocess tolerancesdonotscaleproportionally,therebycausingth erelativeimpactoftheprocess variationstoincreasedramaticallywitheverynewtechnol ogygeneration.Theperformance ofanintegratedcircuitisimpactedbytwodistinctsources ofvariation[13]:(i)environmentalfactorswhichincludevariationsinpowersupplyvoltag eandtemperature,(ii)physical factorswhichincludevariationsintheelectricalandphys icalparameterscharacterizing thebehaviorofactiveandpassivedevices,causedbyproces singandmaskimperfections. Thephysicalfactorscanbefurtherclassiedas:(i)die-to -diephysicalvariationswhichare largelyindependentofthedesignimplementationandareus uallymodeledusingworst-case corners,(ii)within-diephysicalvariationssuchasvaria tionsingatedimensioninruenced bylayoutdesignimplementations,andforwhichworst-case cornermodelingisinsucient [5,14].Theworst-casecornermodelingmaximizesasingled eviceparameter(e.g.delay, noiseorpower)anddoesnotusuallytakethespatialcorrela tionsintoaccount,thereby resultingintoopessimisticanapproach.Asaresult,someo fthevaliddesignsmaybe rejectedorhavetobeadjustedtomeetarticialandinaccur ateworstcaselimits.Thiscan leadtounnecessarilylargechipareaandpowerconsumption aswellasincreasingdesign eortsandcosts. Thephysicalparameterssuchasthewidth,thethicknessoft heinterconnectsand theeectivelengthoftheMOSdevicesvarysignicantlybet weentheintra-dieandinterdiecomponents.Thesephysicalvariationsleadtosubstant ialvariationsintheelectrical parameterssuchasconductance,capacitance,inductance, thresholdvoltages,andleakage currentsoftheCMOSdevicesandinterconnects.Theintra-d ievariationsexhibitspatial correlations,wheredevicesthatareclosetoeachotherhav eahigherprobabilityofhaving similardevicepropertiesthanthosewhichareplacedfarap art.Also,whencoupledwith processvariations,noiseeectscanproduceworst-casede signcornercombinationsforthe designthatrepresentsextremeoperatingconditions,caus ingprimereliabilityconcerns. Hence,itisessentialforthedesigntoolstoaccountforthe seuncertaintiesanddesign robustcircuitsthatareinsensitivetotheprocessvariati ons. 4

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1.1Motivation:Multi-metricOptimization Tosummarizetheabovediscussion,thecontinuousdevicean dinterconnectscaling trendsindeepsubmicronandnanometerdesignshaveresulte dinprominenteectslike increasedinterconnectdelayduetorisingparasiticresis tanceandcapacitanceofon-chip wiring,noiseandreliabilityconcernsduetocrosstalkand couplingcapacitance,delay uncertaintyduetoprocessvariations,andincreasedinter connectpowerdissipation.In addition,thephysicalrealizationofdevicesandintercon nectsareunpredictableduetothe randomnatureofprocessvariations.Thus,the majorchallenge isinachievingreliable, low-power,andhigh-performancesystemimplementationsf romthemicro-architecturelevel downtothelayoutlevel,consideringunpredictablebehavi orduetoprocessvariations.In ordertorealizesuchasystemimplementation,thetraditio nalmethodofdesignoptimizationfornumerousyears-singlemetricoptimizationwi thotherdesignparameters asconstraints,isnolongereectiveorsucient.Onthecon trary,inDSMcircuits,itis signicantlyimportanttosimultaneouslyoptimizevariou sdesignparameters(interconnect delay,crosstalknoise,delayuncertainty,interconnectp ower).Hence,thereisaneedfor newmethodsandalgorithmscapableofperformingmulti-met ricoptimization. Multi-metricoptimizationinconrictingenvironmentsisa dicultproblemsincethe normaldenitionofanoptimalvaluenolongerappliesorval id.Forexample,anoptimal gatesizeforonemetricmaynotbeoptimalforanothermetric .Theoptimalpolicyat anygiveninstancedependsonthepolicesforothermetrics, keepingthebestinterest oftheentiresysteminview.Whilemostoptimizationmethod ssuchasILP,simulated annealing,andforcedirectedmethodslendthemselveswell tosinglemetricoptimization, thesemethodsareinadequateformulti-metricoptimizatio n.Hence,inthisdissertation, weinvestigatetheapplicationofgametheory,amulti-agen toptimizationframework,to theproblemsofVLSICAD.Wehaveusedgametheoreticmodelst osolvetheproblemsof postlayoutwireandgatesizingformulti-metricoptimizat ionwithoutconsideringprocess variations.Theconsiderationofprocessvariationsdurin gdesignoptimizationrequires probabilisticanalysisduetotheuncertaintyelementintr oducedbyprocessvariations.In 5

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ordertocapturethenondeterministicbehaviorofsystempa rametersduetouncertainty,we haveusedstochasticgamemodelingforsolvingpostlayoutg atesizingproblemconsidering processvariations. Thefundamentalbasisandstructureofgametheoryandstoch asticgamesallowthe formulationofoptimizationproblemsinwhichmultipleint er-relatedcostmetricscompete againstoneanotherfortheirsimultaneousoptimization.T hereexistseveralapproaches suchastheNashequilibriumand -equilibriumforachievingequilibriumstatesolutions inwhicheachmetricisoptimizedwithrespecttotheoptimal ityofothers.Further,the stochasticgamemodelsinherentlycapturethenondetermin isticbehaviorofthesystem parametersduetoprocessvariations.Thesefactorsmakega metheoryapowerfultoolto modeloptimizationproblemsinVLSIdesignautomation.Gam etheorysupportsthefollowingfourfeatures:rationality,coalitionformation,c ompetitionandequilibrium.Game theoreticreasoningtakesintoaccounttheattemptsmadeby themultipleagentstowards theoptimizationoftheirobjectivesforeverydecision.Ea chagentoraplayer'sdecisionis basedonthedecisionofeveryotherplayerinthegameandhen ceitispossibleforeach playertooptimizehisgainwithrespecttotheothers'gains inthegame.Thus,game theoreticmodelstrytoachieveglobalgainamongthesetofg ivenplayers.Intermsof gametheory,asolutionissaidtoreachitsglobalvaluefort hegivenconditionswhenthe equilibriumconditionismet. Thesolutionstogametheoreticmodelsexhibittheproperty ofsocialequilibrium[15], whichenforcesthattheoptimizationofindividualdecisio nshavetotakeintoaccountthe optimizationofotherplayers'decisions.Inotherwords,w hilemakingdecisionstowards optimizingonemetricmaynotbeabletooptimizeothermetri cs.Hence,gametheoryisa naturalframeworkwhichinherentlyconsiderssocialequil ibriumwithrespecttotheindividualdecisionsaswellastheglobalobjectivetoensureth efairnessobjective.Further,if thepayofunctionisconvex,i.e.,theparametersbeingopt imizedcorrespondtoconricting objectives(suchasdelayandcrosstalknoiseininterconne cts),gametheoreticoptimization, 6

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infact,performssignicantlybetterthanmethodssuchass imulatedannealing,genetic searchandLagrangianrelaxationforproblemswithconrict ingobjectives[16,17,18]. 1.2ContributionsofthisDissertation Thisdissertationaddressestheproblemofmulti-metricop timizationinthedesignof deepsubmicronCMOSVLSIcircuits.Thefollowingarethemaj orcontributionsofthis dissertation. (i)Wiresizing:Wedevelopedamulti-metricoptimizationf rameworkforperforming wiresizingatpostlayoutlevel.Theframeworkisbasedonga metheorywhichcan simultaneouslyoptimize(a)interconnectdelayandcrosst alknoise(b)Interconnect delay,powerandcrosstalknoise.Wehaveshownthatwiresiz ingisapowerfuland eectivetechniqueinmakinguseoftheunusedroutingresou rceswhileoptimizing designparametersatpost-routestage.Theworkreportedin theliteratureonwire sizingperformonlydelayoptimization,anddonotconsider routingcongestion.They resultinunconstrainedwiresizeswhichcannotbeappliedd irectlyforsizingthe netsofarouteddesign.Hence,wedevelopnewalgorithmsfor wiresizingatpost layoutlevelwhichneitherneedreroutingofnetsnorareaov erhead.Thedeveloped algorithmshavelinearcomplexityintermsofbothtimeands pace. (ii)Fastinterconnectmodels:Newtransmissionlinebased interconnectmodelshavebeen developedforanarrangementofthreeparallelinterconnec twiresegments.These modelsaresimple,fastandaccurate,andhence,canbeusedi nframeworkswhich needmultiplerepetitivecalculationbasedontheanalytic almodels. (iii)Gatesizing:Wedevelopedanewpost-layoutgatesizin galgorithmforsimultaneous optimizationofinterconnectdelayandcrosstalknoise.Th egatesizingproblem ismodeledusinggametheoryandsolvedusingtheNashequili brium.Wehave proposedtwodierentapproachesinwhichthegamesareorde redaccordingtothe 7

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noisecriticalityorthedelaycriticalityofthenets.Thed evelopedalgorithmsare linearintermsoftimeandspaceanddonotrequirererouting orareaoverhead. (iv)Gateandwiresizing:Wedevelopedanintegratedgatean dwiresizingframeworkfor simultaneousoptimizationofinterconnectdelayandcross talknoise.Theproblemis modeledusinggametheoryandsolvedusingtheNashequilibr ium.Theintegrated frameworkperformsthescalingofgateandwiresizesmoree ectivelythanthesequentialapproachofwiresizingfollowedbygatesizingorg atesizingfollowedbywire sizing.Wehaveproposedtwodierentapproachesinwhichth egamesareordered accordingtothenoisecriticalityorthedelaycriticality ofthenets.Thedeveloped algorithmsarelinearintermsoftimeandspaceanddonotreq uirereroutingorarea overhead. (v)StatisticalGatesizingconsideringprocessvariation s:Wedevelopedamulti-metric optimizationframeworkforminimizingdelayuncertaintya ndcrosstalknoiseundertheimpactofprocessvariations.Wehaveusedstochasti cgamestosolvethe statisticalgatesizingproblem.Theformulationandthede velopedframeworkis completelystochastic.Theprocessparameterdistributio nsaremodeledusingthe stochasticfunctionbycontrollingthestatetransitionan dthepayostotheplayers. Theapproachisindependentofprobabilitydistributionsu sedtomodelprocessvariations.Inotherwords,Itcanworkforanystatisticaldistr ibutionslikeGaussianor Log-Normal,andhence,canbeappliedto65nmdesignsorbelo w.Thedeveloped algorithmsdonotrequirereroutingorareaoverhead. 1.3OutlineofthisDissertation Theremainderofthisdissertationisorganizedasfollows: Chapter2describesthe priorworkbasedonwireandgatesizingwithandwithoutthec onsiderationsofprocess variations,someoftherelevantconceptsofgametheoryand stochasticgamesalongwith theirmodeling,andtheinterconnectmodelsusedinmostpar tofthisdissertation.Chapter 8

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3denestheproblemofpostlayoutwiresizingandprovidest wodierentgametheoretic formulationsfor(i)simultaneousoptimizationofinterco nnectdelayandcrosstalknoiseand (ii)simultaneousoptimizationofinterconnectdelay,pow erandcrosstalknoise.InChapter 4,wehavederivedfastinterconnectmodelsbasedontransmi ssionlinesandveriedthemin thecontextofwiresizingproblem.Thesemodelsarefoundto befasterthanthosegivenin Chapter2.5withsamelevelofaccuracy.Chapter5denesthe problemofpostlayoutgate sizingandprovidesagametheoreticsolutionwithtwodier entstrategiesinwhichgames areorderedaccordingto(i)thenoisecriticalityand(ii)t hedelaycriticalityofthenets. InChapter6,theproblemofintegratedgateandwiresizingf ormulti-metricoptimization isaddressed.Thegametheoreticmodelingdevelopedinthis Chapterisatwo-player gamemodelwhichiscompletelydierentwhencomparedtothe respectiveframeworks forgateandwiresizingproblems.Chapter7denestheprobl emofstatisticalgatesizing consideringprocessvariationsanddevelopsastochasticg amebasedNashequilibrium solutionforminimizationofdelayuncertaintyandcrossta lknoise.Theconcludingremarks andfutureworkrelatedtothedissertationaregiveninChap ter8. 9

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CHAPTER2 BACKGROUNDANDRELATEDWORK Thevariousissuesrelatedtothedesignofinterconnectsar ediscussedinChapter1. Therehasbeenasignicantamountofresearchinestimating andoptimizingtheeectsdue tointerconnects.Inthischapter,wehighlighttheresearc hthatisfocusedonoptimizingthe interconnecteectsandinparticular,theworkwhichisrel atedtotheproblemsaddressed inthisdissertation. Someofthestandardprocedurespracticedtoreducetheeec tsofinterconnectsinDSM circuitsare(i)driver/gatesizing(ii)wiresizing(iii)b uerinsertion(iv)netreordering (v)wirespacing(vi)wireshielding.Driversizingisthepr ocessofappropriatelysizingthe drivergatessoastoreducetheinterconnecteects.Ifthed rivergatesaresized-up,the drivingstrengthofthegateincreasesandhence,theamount ofcurrentdriventhrough theinterconnectwireconnectedattheoutputofthegateinc reases.Thisdecreasesthe timerequiredtochargetheoutputcapacitanceattheothere ndoftheinterconnectand therebyreducesinterconnectdelay.Ontheotherhand,thec hargecoupledtotheadjacent interconnectsthroughthecouplingcapacitancealsoincre asesduetotheincreasedcurrent ontheinterconnectdrivenbythesized-upgate.Thisincrea sesthemagnitudeofthe crosstalknoiseinducedontheadjacentnets.Thussizingth edrivergatesappropriately canstrikeabalancebetweentheinterconnectdelayandcros stalknoise. Buerinsertionisthephenomenonofintroducinghighstren gthbuersonlonginterconnectwires.Onlonginterconnectwires,thesignalgetsw eakerasittravelsthroughits lengthbecauseoftheparasiticresistanceandcapacitance ofthewire.Insertingbuersat intervalsthroughoutthelengthoflongwireshelpinrestor ingtheweaksignalsandthereby movingthesignalfaster.Theinsertionofbuersshouldbed onejudiciouslysincetheir 10

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introductionresultsinareaoverheadandalsoincreasesth ecrosstalknoiseinducedonthe adjacentwires.Itiscostlytoperformbuerinsertionatpo stlayoutlevelbecausethe designwillneedrerouting.Wiresizingistheprocessofinc reasingordecreasingthesize ofthewiresinordertoreducetheinterconnecteects.Incr easingthesizeofawirewill reducetheparasiticresistanceandhence,helpsinmakingt hesignalmovefasterwithout loosingitsstrength.Hence,theeectontheinterconnectd elayandcrosstalknoisedueto sizingupawireisequivalenttosizingupthedrivergates. Wirereorderingisanothertechniquetoreducetheintercon necteectsinDSMcircuits. Reorderingistheprocessofshuingthewires,therebychan gingtheiradjacencieswith respecttootherwiresandmakingthemlesssusceptibletocr osstalkeects.Iteectively changesthelengthofoverlapandspacingofawirewithrespe cttoitsneighbors.Wire shieldingisoneofthesimplesttechniquestominimizecros stalk.Itusesshieldwiresin betweenthesignalwireswhicharehighlysusceptibletocro sstalknoise.Theshieldwires arethenewwiresinsertedinthedesignwhicharekeptatzero potential,inordertoact asaneectivebarrierbetweenthehighlycoupledvictimand aggressornets.However, thedisadvantageofthistechniqueisthatitincreasesthea reaoverheadduetothelarge numberofshieldwireswhichmayberequiredbetweenvictims andaggressorsinlarge designs.Intheliterature,driverorgatesizingandbueri nsertionarethepreferred techniquesforminimizinginterconnecteects,butwiresi zingisrelativelylessinvestigated. Inthisdissertation,wefocusonthewiresizingandgatesiz ingtechniqueswiththegoal ofminimizingtheinterconnecteects.Theproblemsarers tmodeledasdeterministic formulationswithoutconsideringprocessvariations,fol lowedbystochasticformulations whichconsiderprocessvariations.2.1WireSizing Inthissection,wepresentabriefoverviewofpriorworkfoc ussedonwiresizing.Figure 2.1.showsthevariouspioneeringworkfoundintheliteratu refortheproblemofwiresizing. Earlyworkonwiresizing[19,20]considertheinterconnect wiresbydividingeachwireinto 11

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smallersegmentsandassumeeachwiresegmenttobeofunifor mwidth.Incontinuouswire sizing,avariationofuniformwiresizing,eachwireisdivi dedintoinnitelymanysegments [21].Theseapproachestendtousemanydiscreteoraninnit enumberofwirewidthsand hence,makeinterconnectplanningdicult[22].In[23],th eauthorshaveprovidedclosed formsolutionstosimultaneousbuerinsertion/sizingand wiresizing.Theyhavedeveloped mathematicalformulaeusingElmoredelaymodelsforwiresi zingbyconsideringthegiven wireas n netsegments.Theaboveworksdonotexplicitlyattempttoco nsiderthecoupling capacitancebetweentheinterconnectwires,whichisamajo rconcerninDSMcircuits. Also,theseworkshaveminimizationofdelayorarea-delayp roductastheirobjectives. based on power and delay Wire sizing formulations based on delay optimizationunder no constraintsbased on delay optimizationunder area constraintsbased on area optimizationunder noise, delay, powerconstraintsbased on simultaneousoptimization of delay andnoise with no area overhead Cong et. al '94C-Ping et. al '97Alpert et. al '99Chu et. al '01Chen et. al '04Sapatnekar '96He et. al '98Cong et. al '02H-Ru et. al '00This work trade-off Figure2.1.TaxonomyofVariousPioneeringWorksonWireSiz ing Theworksreportedin[24,22]attempttoconsiderthecoupli ngcapacitanceforoptimizingthewiregeometry.In[24],inputprobabilitiesareprop agatedtoobtaintheswitching conditionsofnets.In[22],twosimplewiresizingschemesc alledsingle-widthsizingand two-widthsizingareanalyzed.Thesetwoworksresultinsom einaccuracyastheydonot 12

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considertheinterconnectpositionsinthenallayout,ani mportantfactoraectingcouplingbetweenthegivennets.Inaddition,theworksconside rthewiresasindependentsets toperformwiresizingwithouttakingintoaccounttheeect softheirsurroundingsandthe nallayoutconditionssuchasroutingcongestion,theseto fneighbors,etc.Theydonot developdesignmethodologiestogeneratelargefeasiblede signs.In[25],simultaneouswire sizingandwirespacingisperformedatpostlayoutwithdela yminimizationastheobjective.TheabovementionedworksuseElmoredelaymodelstomo deltheinterconnectsand hencelackaccuracy[26].Theexistingworksonwiresizingd onotconsidertheproblemof simultaneousoptimizationofinterconnectdelayandcross talknoisewithaccuratemodels. InChapter3,wedevelopacompletedesignframeworkcapable ofperformingsimultaneous optimizationofinterconnectdelay,powerandcrosstalkno isethroughwiresizingatpostroutelevel.InChapter4,Wedevelopnewfastmodelsforinte rconnectswhichconsider thetransmissionlinebehaviorofwirestorepresenttheint erconnectdelayandcrosstalk noiseofindividualnets.2.2GateSizing Inthissection,wepresentabriefoverviewofpriorworksfo cussedongatesizing.In oneofthepioneeringworks,Conget.al.[19]developedasim ultaneousgateandwiresizing algorithmforpoweroptimizationunderdelayconstraints. In[24],H-Ruet.al.proposeda Lagrangianrelaxationbasedinterconnectoptimizationun dertheconsiderationofcrosstalk noise.However,theydonotconsidertheexactphysicalloca tionofthenetsinthedesign. Chuet.al.[23]developedaclosedformexpressionforbuer andwiresizingfordelay optimizationwithoutconsideringthecrosstalknoise,for useatearlystagesofdesign.In [27],Xiaoet.al.haveusedacrosstalkawarestatictiminga nalyzertoeliminatethetiming violations.In[28],Alpertet.al.haveusedadelaypenalty estimationtechniquetoachieve timingclosure.In[29],Albrechtet.al.havedevelopedali nearprogrammingbasedbuer andwiresizingalgorithmforroorplanareaminimizationun derdelayconstraints.In[30], Hashimotoet.al.havedevelopedagatesizingalgorithmfor crosstalknoiseoptimization 13

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underdelayconstraintswhichisonlycapableofdown-sizin gtheaggressorgates.In[10], agreedygatesizingapproachisproposedforminimizingthe crosstalknoisebycreatinga couplinggraphwiththehelpofClarinetnoiseanalyzer.The gatesareiterativelysized-up orsized-downtosatisfythenoisecriterion.In[31],aLagr angianrelaxationbasedgate sizingapproachisproposedforreducingthecrosstalknois eunderthedelayconstraints. Thealgorithmisalsoiterativeandusesacouplinggraphext ractedbasedonthecoupling capacitances. In[32],theauthorshavedevelopedagametheoreticalgorit hmforgatesizingandbuer insertionatthelogiclevelforpowerminimizationunderde layconstraints.Thedeveloped algorithmispath-basedandusesauctiontheorytoimplemen tthedelayconstraintsasa divisibleresource.InChapter5,weaddresstheproblemofs imultaneousoptimizationof interconnectdelayandcrosstalknoiseusinggatesizing.W eusegametheoryasanoptimizationtooltondtheoptimalgatesizesforthesimultane ousreductionofinterconnect delayandcrosstalknoise.Inthegamemodel,wehavemodeled thegatesastheplayers, itspossiblegatesizesasthestrategysetandthegeometric meanofinterconnectdelay andcrosstalknoiseasthepayofunction.InChapter6,wein vestigatedtheproblemof simultaneousoptimizationofinterconnectdelayandcross talknoiseusinganintegrated approachofgateandwiresizing.Wehavecreatedatwoplayer normalformgamewith delayandnoiseastheplayers.2.3ProcessVariations Inthissection,wepresentabriefoverviewofpriorworksfo cusedongatesizingwith theconsiderationofprocessvariations.Theimpactofproc essvariations,theirsourceand theirvariationtrendshavebeendiscussedinthepioneerin gworksofNassif[13,4,38]and Borkaretal[5]ingreatdetail.In[39],astatisticaldesig napproachispresentedtostudythe impactofinterconnectprocessvariationsonmemorydesign andperformanceusingMonte Carloandsensitivityanalysis.MonteCarloanalysisshows thatthethresholdvoltage,the eectivegatelength,theeectivegatewidthandthesupply voltagearethekeyparameters 14

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Table2.1.RecentStatisticalGateSizingWorksFoundinLit erature Recent Objective Methodology Assumptions Remarks Works ofWork onProcess Parameters Jacobs Minimize Non-linear Gaussian Statisticaldelaymodel,uses etal2000 delay programming distribution SSTA,doesnotconsiderspatial [33] uncertainty correlations Raj Minimize Non-linear Gaussian denesutilityfunctions,generates etal2004 delay programming distribution statisticaldelaymodel,pathbased, [34] uncertainty cannotbeappliedtolargecircuits Chopra Minimize Gradient Gaussian lumpsGaussiandistributionof etal2005 leakagepower computation distribution twoormorerandomvariables [35] anddelay intoone-anapproximation Sinha Minimize Heuristic Gaussian Asetofheuristicswith etal2005 delay approach distribution perturbation,buildsstatistical [36] violations SSTA delaymodelsforgatesinlibrary Singh tradeo-area Geometric Any PosynomialElmoredelaymodel, etal2005 anddelay programming distribution usesSTAwithstatisticaldelay [37] violations constraints,computationallyfast Ourwork Minimizedelay Stochastic Any Purelystochasticapproach, andnoise games distribution multi-metricoptimization, violations noareaoverhead thatinruencetheinterconnectdelay,thetotalaveragepow erandthecrosstalknoise. Studiesonprocessvariationshavebeenmainlyfocusedonva riabilitymodeling[14,34,40] andstatisticalstatictiminganalysis(SSTA)[33,41,42,4 3,36].In[33],thestatistical gatesizeoptimizationproblemissolvedasanon-linearpro blemwithdelaysmodeledas Gaussianfunctions,ignoringthespatialcorrelationsdue tointra-dievariations.In[14], thegatesizingproblemismodeledasadeterministicnon-li nearoptimizationproblemwith thehelpofapenaltyfunctiontointentionallyimprovethet imingslacksonnon-critical pathsofthecircuit.In[34],aheuristicapproachforstati sticalgatesizingisproposedfor improvingthetimingyieldusingtheconceptofstatistical lyundominatedpaths.However, thisapproachispath-basedandcannotbeappliedtolargeci rcuits,sinceasthepathsgets larger,thenumberofgatesincreasesbecomingcomputation allyintensive. 15

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Thedelayuncertaintyinducedduetoprocessvariationsonb uer-driveninterconnect linesisanalyzedin[44]andabuersizingmethodologyisde velopedtoreducedelay uncertainty.In[41],apruningstrategybasedonperturbat ionboundsisdevelopedtosolve thestatisticalgatesizingproblem.In[42],thegatesizin gproblemissolvedbyidentifying theworstnegativestatisticalslackpathswiththeobjecti veofreducingtheperformance varianceofatechnology-mappedcircuit.In[43],anincrem entalandparametricSSTAis proposedtoperformgatesizingwithpre-targetedyieldopt imization.In[35],aheuristic approachisprovidedtocomputethegradientofyieldwithre specttogatesizesandnonlinearoptimizationisperformedtomaximizetheyield.Apr obabilisticmethodologyis developedin[45]forbuerinsertionproblemusingabottom -uprecursiveapproachto calculatethejointprobabilitydensityfunctiontocorrel atebetweenarrivaltimesanddownstreamcapacitance. In[36],statisticalmodelsforthegatesinthestandardcel llibraryaredevelopedusing SSTAbycharacterizingatdierentpointsintheparameters pace.Aspointedoutin [37],themethodsbasedonSSTAneedtomake(i)theassumptio nssuchasthesignal arrivaltimeandtheslopehavenormaldistributions,and(i i)theapproximationssuch astheresultantoftwoormorenormaldistributionsisalsoa normaldistribution,which maybeinaccurate.In[37],theuncertaintyduetoprocessva riationsisincorporatedin delayconstraintsusingaposynomialdelaymodelandsolved fortradeosbetweenthearea androbustnessusinggeometricprogramming.Thisapproach iscomputationallyecient, howeverdoesnotconsidercrosstalkviolations.Table2.1. givesasummaryoftherecent workonstatisticalgatesizingfoundintheliterature.2.4GameTheoryandStochasticGames Inthissection,webrierydiscusssomeoftherelevantconce ptsofgametheoryand stochasticgames.Fordetailedtreatmentoftheseconcepts alongwithvariousotherconcepts,pleasereferto[46,16]forgametheoryandto[47,48] forstochasticgames. 16

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2.4.1GameTheory Gametheory,inabroadsense,canbedenedasacollectionof mathematicalmodelsformulatedtoanalyzetheinteractionofdecisionmaker sinsituationsofconrictand cooperation.Theobjectiveistondasetofbestactionsfor eachdecisionmakerand recognizethecorrespondingstableoutcomes.Agameconsis tsofplayerswhochoosefrom alistofalternativecoursesofaction,resultinginoutcom esoverwhichtheplayersmay havedierentpreferences.Gametheoryisaguidewhichimpl ementsrationalbehaviorof individualplayersandpredictstheiroutcomes. Gametheorywasformulatedasageneraltheoryofrationalbe haviorbyvonNeumann. Thebasicbuildingblocksofgametheoryarebasedontheorie sproposedbyvonNeumann in1928[49]andNashin1950[50].Theessentialelementsofa gameareplayers,actions, payos,andinformation,whicharecollectivelyknownasth erulesofthegame.Players ofthegamearethesetofrationaldecisionmakers.Thegoalo feachdecisionmakeristo maximizehisownutilitybyasetofactionsinthepresenceof otherdecisionmakers.An actionoramovebyaplayer i ,denotedby a i ,isachoice.Thestrategy s i ofaplayer i isaruletochooseanactionateachinstantofthegame.These tofstrategies S i = f s i g availabletoplayer i isdenotedashisstrategysetorstrategyspace.Astrategyc ombination s =( s 1 ;:::;s N )isanorderedsetconsistingofonestrategyforeachofthe N playersin thegame. Inan N -playergame,thepayoofplayer i ,denotedby P i ( s 1 ;:::;s N ),istheutility obtainedaftertheplayershadchosentheirstrategiesandt hegameisplayedout.Itcanalso bedenedastheexpectedutilityreceivedbyplayer i asafunctionofstrategieschosenby eachplayer.Theinformationsetofaplayer i istheknowledgeofactions,previouslychosen bytheplayersatagivencourseofthegame.Theinformations etoftheplayerschangesas thegameprogresses.Anequilibriumisastrategycombinati onconsistingofabeststrategy foreachofthe N playersinthegame.Theequilibriumstrategiesarethestra tegieschosen byplayerstomaximizetheirindividualpayos,amongthepo ssiblestrategycombinations 17

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obtainedbyarbitrarilychoosingonestrategyperplayer.T heequilibriumoutcomeisthe setofpayovaluesoftheplayerscorrespondingtotheirequ ilibriumstrategies. Gamescanbebroadlyclassiedintotwodistinctcategories :non-cooperativeandcooperativegames.Innon-cooperativegames,playerschoose theirstrategiesindependently andtherulesofthegamedonotallowbindingcommitmentsamo ngtheplayers.Inother words,non-cooperativegamesareplayedwithfullyrationa lplayers.Itfocusesonthe strategieschosenbyeachplayerandtheirrespectivepayo s.Incooperativegames,playersformcoalitionamongasubsetofplayersandplaytheirjo intactionsaccordingtothe agreementsmadeduringtheirbindingcommitments.Itfocus esoncoalitionformationand distributionofthebenetsgainedthroughcooperation.Ag eneralgametheoreticmodel canbeclassiedintothreecategoriesbasedonmathematica lformulations-thenormalor strategicform,theextensiveformandthecharacteristicf unctionform.Thecharacteristic functionformisapplicableonlytocooperativegames. Anon-cooperativegameisrepresentedinoneofthetwogener almathematicalformulationsbasedonthetypesofmovesemployedbytheplayers.T herstformulationisthe normalformgame,inwhichtheplayersmovesimultaneouslyt ochoosetheirstrategies. Inthisgame,thestrategiesaresameastheactionsinranked coordination.Thenormal formshowswhatpayoresultsfromeachpossiblestrategyco mbination,whiletheoutcomematrixshowswhatoutcomeresultsfromeachactioncomb ination.Astheplayers maketheirmovessimultaneously,theydonothaveachanceto learneachother'sprivate informationbyobservingeachother.Thus,innormalformga mes,theinformationsetof eachplayerabouttheotherplayersiszero.Therefore,anor malformgameisrepresented bythelistofplayers,theirstrategysetandthepayofunct ions.Thesecondformulation isextensiveformgameswhicharealsocalledsequentialmov egames.Theplayersofthis gamemovesequentiallyandchoosetheirstrategiesaccordi ngtoanorder.Theorderofthe playisimportantandaectsthenaloutcomeofthegame.The extensiveformgameis representedbyalistofplayers,theirsetofactions,infor mationsetandpayofunctions. Thestrategiesoftheplayersareaseriesofactionmoves.Th einformationsetsrepresents 18

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thevariousstateseachplayercantakeatanygivenpointint hegame.Theimportant dierencebetweensimultaneousmovegamesandsequentialm ovegamesisthatinsequentialmovegamesthesecondplayeracquirestheinformationo nhowtherstplayermoved beforehemakeshisowndecision. Anon-cooperativegameiscalledanitegameifthestrategy sets S 1 ;:::;S N arenite. Finitegamesaregivenbylistingthepayosforeachplayeri ntabularform.Anitegame canbegeneralizedtoconsistof N playerswhochoosefromasetofstrategies S i where, i =1 ;:::;N ,andasetofpayofunctions P i where, i =1 ;:::;n : S 1 ::: S n !< ,where, < isthesetofallrealnumbersandapayovalueisassignedtoe achpairofstrategies chosenbytheplayers.Therationalityortheequilibriumpo intisasetofstrategiesthat maximizesorminimizesthepayooftheplayerassumingthat allotherplayersstrategies areheldxed.Thegameisplayeduntileachplayer'sstrateg yisoptimalwithrespectto thestrategiesofothers.Stackelburg'sequilibrium[51]a ndtheNashequilibrium[52]are someofthetechniqueswhichcanbeusedtoreachagame'sequi librium.Inthiswork,we focusonnon-cooperativenitegames. TheNashequilibriumdenedhereisintermsofnormalformga mes,whichcanbeeasily extendedtoextensiveformgames.Let G = f S 1 ;:::;S N ; P 1 ;:::;P N g beanon-cooperative nitegameinnormalformwith N players.Theset S i containsallthestrategiesandthe set P i containsthecorrespondingpayovaluesforaplayer i .The N -tupleofstrategies s =( s 1 ;:::;s N ),where s 1 2 S 1 ;:::;s N 2 S N ,isdenedtobeNashequilibriumpointof G if P i ( s 1 ;:::;s i ;:::;s N ) P i ( s 1 ;:::;s i 1 ;s i ;s i +1 ;:::;s N ) holds 8 s i 2 S i and i =1 ;:::;N .Itcanbestatedinsimplewordsasoncebeinginthe staterepresentedbythestrategychoices s ,theplayer'spayodoesnotgetbetterif heunilaterallydeviatesfromtheNashequilibriumstrateg y.Here,thewordunilaterally meansthattheotherplayerswillsticktotheirequilibrium strategies.TheNashequilibrium point NE ,denesthepayovaluesforalltheplayersinthegame.Qual itatively,theNash 19

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equilibriumforan N -playernitegameisa N -tuplesetofstrategies s =( s 1 ;:::;s N ), givenby N inequalitiessuchthat,nosingleplayercangainbychangin gonlyhisstrategy. 2.4.2StochasticGames Stochasticgamescanbedenedasthenaturalandhybridexte nsionoftheMarkov DecisionProcesses(MDPs)andmatrixgames.Markovdecisio nprocessesareasingle agentandmultiplestateframework.Matrixgamesareamulti pleagentandsinglestate framework.Onthecontrary,stochasticgamesareamultiple agentandmultiplestate framework,whichcanbevisualizedasthemergingofMDPsand matrixgames.Ina multi-agentsetting,stochasticgamesallowthestatetran sitiontodependjointlyonallthe agentactions,andhavingtheimmediaterewardsateachstat edeterminedbyamulti-agent general-summatrixgameassociatedwiththatstate.A N -playernonzero-sumstochastic gameisdenedasatuple( N;S;A i ;P;R i ),where N isthenumberofplayers, S istheset ofstatesforthegame, A i isthesetofactionsavailabletoplayer i P isthetransition probabilityfunction S A S [1 ; 0],and R i : S A !< isthepayoorreward functionforthe i th player.If s isastateatsomestageofthegameandtheplayersselect an a 2 A ( s ),then p ( : j s;a )istheprobabilitydistributionofthenextstateofthegam e. Thetransitionprobability p hasadensityfunction z withrespecttoaxedprobability measure on S ,satisfyingthefollowingcontinuitycondition:Foranyse quenceofjoint actiontuples a n a 0 Z S j z ( s;t;a n ) z ( s;t;a 0 ) j ( dt ) 0 asn !1 : Inatwo-playermatrixgamescenario,letthematrixpair( M 1 ;M 2 )specifythepayos fortheplayer1(rowplayer)andplayer2(columnplayer),wh erethematrices M 1 and M 2 are n by n withtheirindicesrangingfrom1to n .Iftherowplayerchoosestheindex i andthecolumnplayerchoosestheindex j ,thentheplayer1receivesapayoof M 1 ( i;j ) andplayer2receives M 2 ( i;j ).Theindices i and j arecalledpurestrategiesofplayers1 20

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and2respectively.If and aredistributions(calledmixedstrategies)overtherowan d columnindices,theexpectedpayotoplayer k 2f 1 ; 2 g is M k ( ; ) : = E i 2 ;j 2 [ M k ( i;j )]. Themixedstrategypair( ; )issaidtobeaNashequilibriumforthegame( M 1 ;M 2 )if (i)foranymixedstrategy 0 M 1 ( 0 ; ) M 1 ( ; ),and(ii)foranymixedstrategy 0 M 2 ( ; 0 ) M 2 ( ; ).Astrategypair( ; )isdenedasthe -Nash,anapproximate Nashequilibriumfor( M 1 ;M 2 )if(i)foranymixedstrategy 0 M 1 ( 0 ; ) M 1 ( ; )+ and(ii)foranymixedstrategy 0 M 2 ( ; 0 ) M 2 ( ; )+ Atwo-playerstochasticgame G overastatespace S consistsofadesignatedstartstate s 0 2 S ,amatrixgame( M 1 [ s ] ;M 2 [ s ])foreverystate s 2 S ,andtransitionprobabilities P ( s 0 j s;i;j )forevery s;s 0 2 S ,everypurerowstrategy i ,andeverypurecolumnstrategy j .Thestochasticgameproceedsasfollows:Ifthegameiscurr entlyinstate s andthetwo playersplaymixedstrategies and ,thenpurestrategies i and j arechosenaccording to and respectively,andtheplayersreceiveanimmediatepayoso f M 1 [ s ]( i;j )and M 2 [ s ]( i;j ).Thegamethenmovestothenextstate s 0 accordingtothetransitionprobabilities P ( : j s;i;j ).Thus,theimmediatepayostotheplayersandthestatetra nsitiondepend ontheactionsofboththeplayers.Therearetwodierenttyp esofstochasticgamesbased ontheoveralltotalreturnsorpayosreceivedbytheplayer s.Inrsttypecalledinnitehorizondiscountedstochasticgames,theplaybeginsatsta te s 0 andproceedsforever.If aplayerreceivespayosof r 0 ;r 1 ;r 2 ;::: asthegameprogressesthroughthestages,the expectedpayoobtainedbytheplayerforthegameisgivenby r 0 + rr 1 + r 2 r 2 + ::: ,where 0 r< 1isthediscountfactor.Inthesecondtype,callednite-ho rizonundiscounted stochasticgame,theplaybeginsatinitialstate s 0 andproceedsforexactly T steps.If aplayerreceivespayosof r 0 ;r 1 ;r 2 ;:::;r T 1 ,thetotalpayofortheplayerisgivenby (1 =T )( r 0 + r 1 + ::: + r T 1 ).Inthiswork,wehaveusednite-horizonundiscountedsto chasticgamessincewewantthegametostopafter T steps.Thegoalofeachplayerina stochasticgameistomaximize/minimizetheirexpectedtot alpayosfromthedesignated startingstate. 21

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Thepolicyofaplayerinstochasticgameisdenedasthemapp ing ( s )fromstate s 2 S tothemixedstrategiesplayedbytheplayeratthematrixgam eduringthestate s ,spanningoverallthestatesofthegame.Atime-dependentp olicy ( s;t )allowsthe mixedstrategychosenbytheplayertodependonthenumberof remainingsteps t ina T -stepgame.Intime-dependentpolicies,theplayersgainno advantagebyconsidering thehistoryoftheplay.If 1 and 2 arethepoliciesinamatrixgame G withdesignated startstate s 0 G k ( T;s 0 ; 1 ; 2 ), k 2f 1 ; 2 g denotestheexpected T -stepaveragereturn. =( 1 ; 2 )representsthestrategyproleofboththeplayersandthee xpected T stage payotoplayer k k 2f 1 ; 2 g ,isgivenby Tk ( )( s )= E s P Tn =1 r k ( s; k ) ,where, E s is theexpectationoperatorwithrespecttothetransitionpro bability P s ofthestrategiesof theplayers.Theaveragepayoperunittimeforplayer k isdenedas k ( )( s )= limsup 1 T Tk ( )( s ) : Astrategyprole =( 1 ;:::; N )iscalledaNashequilibriumfortheaveragepayo stochasticgameifnounilateraldeviationsfromitarepro table.Mathematically,itcan berepresentedas:foreach s 2 S k ( )( s ) k ( k ; k )( s ) ; 8 k; k where,( k ; k )denotesthestrategyproleobtainedfrom byreplacing k with k .A matrixgameatanygivenstateinastochasticgamemayhavema nyNashequilibria,and hence,therewillbeexponentiallymanyNashequilibriaint hepolicyspaceofthestochastic game[53].Ithasbeenshownin[54]thatthereexistsnopolyn omialtimealgorithmto computeanexactNashequilibriumina2-playernonzerosums tochasticgames.Hence,we resorttoapproximatemethodofndingtheNashequilibrium .ANashselectionfunction isgenerallyusedtoconvertthelocaldecisionsateachstat eintoaglobalNashoranearNashpolicy.Foranymatrixgame( M 1 ;M 2 ),aNashselectionfunction f returnsapair ofmixedstrategies f ( M 1 ;M 2 )=( ; )thatisaNashpairfor( M 1 ;M 2 ).Inthiscase,the 22

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Table2.2.NotationsandTerminology R i resistanceperunitlengthoftheinterconnectline i R a 1 resistanceofrstaggressorgate a 1 withminsize R a 2 resistanceofsecondaggressorgate a 2 withminsize R vd resistanceofminsizedvictimdriver U i selfinductanceperunitlengthofinterconnectline i C i selfcapacitanceperunitlengthofinterconnectline i C il mutualcapacitanceperunitlengthofoverlapbetweeninter connect line i anditsimmediateleftneighbor C ir mutualcapacitanceperunitlengthofoverlapbetweeninter connect line i anditsimmediaterightneighbor C l selfcapacitanceofleftneighborlineperunitlength C r selfcapacitanceofrightneighborlineperunitlength C vd outputcapacitanceofminimumsizedvictimdriver C a 1 outputcapacitanceofminimumsizedrstaggressorgate a 1 C a 2 outputcapacitanceofminimumsizedsecondaggressorgate a 2 C mi 1 mutualcapacitanceofthenet i withitsrstaggressornet C mi 2 mutualcapacitanceofthenet i withitssecondaggressornet L lengthofthegiveninterconnectline i W i widthofthegiveninterconnectline i W l widthoftheleftneighboringinterconnectline W r widthoftherightneighboringinterconnectline S l spacingbetweenthegivennet i anditsimmediateleftneighbor S r spacingbetweenthegivennet i anditsimmediaterightneighbor g gatesizeofthegivendriverwithrespecttoitsminsizeddri ver Z L loadimpedanceofthegiveninterconnectline i C L loadcapacitanceofthegiveninterconnectline i T thicknessofthegiveninterconnectline i H heightofthegiveninterconnectline i fromthedielectric V i propagationvelocityofthegiveninterconnectline i payototheplayer1isgivenby v 1 f ( M 1 ;M 2 ) : = M 1 ( f ( M 1 ;M 2 ))andthepayotoplayer2 isgivenby v 2 f ( M 1 ;M 2 ) : = M 2 ( f ( M 1 ;M 2 )).Inotherwords,aNashselectionfunctionisan arbitraryfunctionusedtomakechoicesofhowtobehaveinan isolatedmatrixgame. 2.5InterconnectModels Inthissection,wediscusstheinterconnectdelay,delayun certaintyandcrosstalknoise modelsusedinmostpartofthisresearch.Sincetransmissio nlinemodelsaremoreaccurate thanlumpedmodelsinmodelinginterconnectwiresindeepsu bmicrondesigns(aspointed outin[26,55]),theyareadaptedinthiswork.Thenotations andterminologyusedinthis dissertationaregiveninTable2.2.. 23

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2.5.1InterconnectDelayModels Ananalyticalequationforinterconnectdelayofanetisder ivedbasedontransmission lineanalysisin[26].Thepropagationdelay D i ( g i 1 g i )denotesthedelayfromgate g i 1 togate g i .Theinterconnectdelayexpressionisreproducedbelowint ermsofthenotations usedinthisdissertation. D i = L V i + i R d + p U i W i p C i C L (2.1) where, i = ln2 e i +2 i e i 1 2 i = RL p C i 2 p U i ;V i = 1 p U i C i Equation2.1givesthepropagationdelayforasingleinterc onnectwireandhence, doesnotconsiderthecouplingeectsduetoneighboringwir es.Wehaveextendedthis analyticalmodeltoincorporatethecouplingeectsduetoc rosscapacitance,byreplacing selfcapacitance C i withtotalcapacitance C tot i .Whenperformingwiresizing,thecoupling eectsduetotheimmediateleftandrightneighborshavetob econsideredforthereasons indicatedinSection3.3.Referringtothemodeldevelopedi n[26],theleftandright mutualcapacitancesactinparallelwithselfcapacitance. Hence,whileperformingwire sizing,thetotalcapacitanceisgivenas C tot i = C i + C il + C ir .Whenperforminggatesizing, thecouplingeectsduetothestrongestandthesecondstron gestaggressorshavetobe consideredforthereasonsindicatedinSection5.3.1.Refe rringtothemodeldevelopedin [26],themutualcapacitancesduetotheaggressorsactinpa rallelwithselfcapacitancefor thegiveninterconnectwire.Hence,whileperforminggates izing,thetotalcapacitanceof theinterconnectwireisgivenas C tot i = C i + C mi 1 + C mi 2 .Combiningbothwireandgate sizingscenarios,thetotalcapacitance C tot i canberepresentedasgiveninEquation2.2. Hence,theextendedinterconnectdelayequationwhichcons iderstheeectsofcoupling capacitancesisgivenbyEquation2.3. 24

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C tot i = 8>><>>: C i + C il + C ir forwiresizing, C i + C mi 1 + C mi 2 forgatesizing. (2.2) D i = L V i + i R d + p U i W i p C tot i C L (2.3) where, i = ln2 e i +2 i e i 1 2 ; i = R i p C tot i 2 p U i ; V i = 1 p U i C tot i C tot i isgivenbyEquation2.2 Whenperformingwiresizing,theeectsofparasiticcapaci tancesandresistanceshave tobecapturedintermsofthewiresizesandspacingsofthene ighboringnetsegments. Theanalyticalexpressionsfortheselfcapacitanceandmut ualcapacitancesarederivedin termsofitswirewidthsandspacingsin[56]and[57]respect ively.Theseequationsare reproducedbelowintermsofourmodelparameters.Theselfc apacitance C i isgivenby theEquation2.4.Themutualcapacitancebetweenthegivenn etanditsimmediateleft neighborisgivenbyEquation2.5.Themutualcapacitancebe tweenthegivennetandits immediaterightneighborcanbeobtainedbyreplacingtheva luesofwidthandspacingin Equation2.5withthecorrespondingvaluesofrightneighbo r. C i = r 10 : 166 W i H +24 : 752 T H 0 : 222 pF=m (2.4) C il = 55 : 6 r ln h 2 S 2 l 1 W l + T 1 W i + T i pF=m (2.5) Whenperforminggatesizing,thevaluesofparasiticsaretr eatedasconstantssince theydependonlyonwiresizesandspacings,andnotongatesi zes.Hence,thevalues ofthewireresistance,areaandcouplingcapacitances,and inductanceareextractedfrom 25

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theStandardParasiticExchangeFormat(SPEF)netlistgene ratedafterdetailedrouting ofthedesign.ThesevaluesareuseddirectlyintheEquation s2.3and2.2. 2.5.2DelayUncertaintyModels Delayuncertaintyisdenedasthedeviationsortherateofc hangeofpropagationdelay duetogatesizechangesasanimpactofprocessvariations.H ence,theanalyticalequation fordelayuncertaintycanbeobtainedbydierentiatingthe propagationdelaygivenby Equation2.3withrespecttothegatesizes[44].Thedelayun certainty DU i ( gate i ;gate i +1 ) fromthegate, gate i tothenextgate, gate i +1 ,duetoprocessvariationsisgivenbyequation 2.6.Thevalueof C tot i isgivenaccordingtotheEquation2.2. DU i = i R vd ( g i g i +1 ) g 2 i + p U i W i p C tot i C L (2.6) 2.5.3CrosstalkNoiseModels Whenperformingwiresizing,thecrosstalknoisehastobeca lculatedintermsofthe couplingeects(expressedasafunctionofwiresizes)onth egivennetduetoitsimmediate leftandrightneighborsseparately.Aftercalculatingthe sevaluesseparately,thecrosstalk noiseonagivennetisgivenbythesuperpositiontheoremint ermsofthecouplingeects duetoitsleftandrightneighbors.Fordeterminingthecros stalkvoltageonanetdueto itsleftneighbor,thegivennetisconnectedtogroundatits sourceendandterminated withaloadcapacitanceof C L atitsterminalend.Theleftneighborisconsideredtobe drivenbyaunitstepvoltageatitssourceendandterminated withaloadcapacitanceof C Ll atitsterminalend.Hence,thecrosstalkvoltageduetoleft neighborcanbedened asthevoltage V l ( t )inducedacrosstheload C L ofthenetunderconsideration.Ithasbeen shownin[55]thattheamplitudeofcrosstalkvoltageattime t isgivenbytheEquation 2.7. V l ( t )= 1 2 exp t 1 exp t 2 (2.7) 26

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where, 1 = R ( C i + C L ) ;and 2 = R (2 C il + C i + C L ) Usingthetheoryofmaximaandminimaofdierentialcalculu s,itcanbeshownthat themaximumvalueofthecrosstalkvoltageisgivenbyEquati on2.8. V max l = 1 2 h exp h N cl 1 2 N cl ln 1+ N cl 1 N cl ii 1 2 h exp h N cl +1 2 N cl ln 1+ N cl 1 N cl ii (2.8) where,thecapacitancecouplingcoecient N cl isgivenby N cl = C il = ( C i + C il + C L ). Similarly,themaximumcrosstalknoise V max r inducedacrosstheload C L ofthenetunder considerationduetoitsrightneighborisgivenbyEquation 2.9. V max r = 1 2 h exp h N cr 1 2 N cr ln 1+ N cr 1 N cr ii 1 2 h exp h N cr +1 2 N cr ln 1+ N cr 1 N cr ii (2.9) where,thecapacitancecouplingcoecient N cr isgivenby N cr = C ir = ( C i + C ir + C L ).The totalcrosstalknoiseonthegiveninterconnectiscalculat edbyapplyingthesuperposition theoremforvoltages V max l and V max r ,denedinEquations2.8and2.9respectively. Whenperforminggatesizing,thecrosstalknoisehastobeex pressedintermsofthegate sizesandinputresistancesofthevictimandaggressordriv ers.Ananalyticalexpression formaximumcrosstalknoisewithvictimdrivermodeledasan eectiveresistanceand aggressordrivermodeledasavoltagesourceconnectedtoit sgateresistanceisderivedin [58].Thisequationisreproducedhereintermsofournotati ons. V max x isthepeakvalue ofcrosstalknoisebetweenavictimnetanditstwoaggressor s. V max x = R v C c V dd 0 t r 1 Y 1 Y 1 Y 2 2 1 2 2 Y 2 Y 1 Y 2 1 1 2 (2.10) 27

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where, Y 1 =exp t r 1 1 ;Y 2 =exp t r 2 1 ; 0 = f [ R a ( C a + C c )+ R v ( C v + C c )] 2 4 R v R a ( C v C c + C v C a + C a C c ) g 1 2 1 = 2 R v R a ( C v C c + C v C a + C a C c ) R a ( C a + C c )+ R v ( C v + C c )+ 0 ; 2 = 2 R v R a ( C v C c + C v C a + C a C c ) R a ( C a + C c )+ R v ( C v + C c ) 0 ; R a = R a 1 R a 2 g a 2 R a 1 + g a 1 R a 2 ;R v = R i + g vd R vd C a = g a 1 C a 1 + g a 2 C a 2 ;C c = C mi 1 + C mi 2 28

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CHAPTER3 WIRESIZING Inthischapter,wefocusontheproblemofpostlayoutwiresi zingtominimizetheDSM eectsofinterconnects.Specically,wedevelopagamethe oreticframeworkandmultimetricoptimizationalgorithmsforthesimultaneousoptim izationof(i)interconnectdelay andcrosstalknoise,and(ii)interconnectdelay,powerand crosstalknoise,duringwire sizing.Weformulatethewiresizingoptimizationproblema sanormalformgamemodel andsolveitusingNashequilibriumtheory.Thenetsconnect ingthedrivingcellandthe drivencellaredividedintonetsegments.Thenetsegmentsw ithinachannelaremodeled asplayersandtherangeofpossiblewiresizesformstheseto fstrategies.Thepayo functionismodeled(i)asthegeometricmeanofinterconnec tdelayandcrosstalknoisein thecaseofrstformulation,and(ii)astheweighted-sumof interconnectdelay,powerand crosstalknoiseinthesecondformulation.Thenetsegments areoptimizedfromtheones closesttothedrivencelltowardstheonesatthedrivingcel l.Thecompleteinformation aboutthecouplingeectsamongthenetsisextractedaftert hedetailedroutingphase. Thetimeandspacecomplexitiesoftheproposedwiresizingf ormulationsarelinearin termsofthenumberofnetsegments.Wealsoprovideamathema ticalproofofexistence forNashequilibriumsolutionfortheproposedwiresizingf ormulation. 3.1ProblemDenition Theproblemofpostlayoutwiresizingcanbedenedasnding theoptimalwire widthssuchthatinterconnecteectsareminimizedunderth egivenareaconstraintsand withouttheneedforreroutinganyofthenetsinthedesign.T heparasiticresistanceand capacitanceofinterconnectwiresarehighlydependentont hewirewidths.Thecoupling 29

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capacitanceisresponsibleforthemajorityofthedeepsubm icroneects.Hence,itis importanttoextractthecouplingcapacitanceofnetswithh ighaccuracy.Thecoupling capacitanceofanetdependsonitswiresize,thelengthofov erlapandspacingbetween adjacentnets.Thisinformationcanbeecientlyextracted atpost-routingphase.Inthis work,wehavemodeledtheproblemofwiresizingsuchthatitd oesnotrequirere-routing anddoesnotincurareaoverhead.3.2MotivationforWireSizingProblem Theproblemofwiresizinghasbeenaddressedatlogiclevelo rprelayoutlevelbymany researchersintherecentpast.But,thisproblemhasnotbee naddressedatpostlayout levelbefore.Afterthedesignisrouted,thelocationsando rientationsofthetransistorsand interconnectwiresinthedesignarexed.Theapplicationo foptimizationmethodslike buerinsertion,wireshieldingatpost-routestagewouldr esultinareaoverheadandcan leadtoreroutingofthedesign.Re-routingofthedesignist ime-consumingandcostlyto beperformedrepeatedly.Typically,whenadesignisrouted ,thechannelshave\unused" trackswhichremainaswhitespacesandgothroughthefabric ationprocessaswasted resource.Wiresizingcaneectivelymakeuseofthese\unus ed"tracksavailablethroughout thedesign,inoptimizingthedesignparameters.Ifthewire sizingproblemismodeled properly,itispossibletoachieveoptimizationwithoutth eneedforre-routingoradditional areaoverhead.Weshowthatwiresizingcouldbepowerfuland eectiveinmakinguseof theunusedroutingresourcestooptimizedesignparameters atpost-routestage. In[59],ithasbeenshownthatwiretaperingisnotrequireda nduniformwiresizingis sucienttogainthebenetsofdelayreductionduetowiresi zing.Also,itispointedout thatwiresizeoptimizationisnotwidelyusedduetothelack ofintegratedwiresizingdesign framework.Followingthis,wedividethenetsintosegments accordingtochannelsand performuniformwiresizingforeachnetsegment.Theworkso nwiresizingreportedin[19, 21,23,22]useanalyticalexpressionsandtheworksin[24,2 0]usenon-linearformulations whiletargetingfordelayoptimization.Thesemodelsdonot considertheroutingcongestion 30

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andthenetpositions,andhence,resultinunconstrainedwi resizes,whichcannotbeapplied directlyforsizingthenetsofarouteddesign.Theuseofthe seapproachesatpost-route levelwillresultinDRCviolations,andrequiresrerouting toxthem.Hence,thereis aneedtodevelopanewmethodologywhichintegratesinthecu rrentdesignrowand determinesthewiresizeswithinthelimitsofDRCrules,the rebyavoidingtheneedfor re-routing.Inthiswork,wedevelopacompletedesignframe workcapableofperforming simultaneousoptimizationofinterconnectdelay,poweran dcrosstalknoisethroughwire sizingatpost-routelevel,satisfyingtheaboverequireme nts. Thewiresizeofanetwillaectthesizesoftheneighboringn etsresultinginconricting objectives.Asthewiresizeofanetincreases,theintercon nectdelaydecreasesandthe couplingcapacitance,hencecrosstalknoiseandinterconn ectpowerincreases(convexpayo function).Thissuitsthemodelingoftheproblemusinggame theorywiththepossiblewire sizesasstrategiesandthenetsegmentsasplayerswhocolle ctivelyworktowardstheglobal objectiveofoptimizingtheinterconnectdelay,powerandc rosstalkmodeledasthepayo function.Traditionally,thisproblemismodeledusingcro sstalknoiseastheobjective function,whilemaintaininginterconnectdelayasaconstr aintorviceversa.However,game theoreticformulationandNashequilibriumsolutionallow thesimultaneousoptimizationof multiplemetricswithconrictingobjectives.Since,inter connectdelay,powerandcrosstalk noisewithinacircuitareconrictinginnature,thepropose dapproachisbenecial.Also,in gamemodels,itispossibletohaveindividualpayofunctio nswhichcanbettercapturethe couplingeectsofindividualnetsegments.Theperformanc eoftheproposedalgorithm iscomparedwiththatofsimulatedannealingandgeneticsea rchinordertoillustrate theeectivenessofgametheoreticsolutionsforproblemsw ithconrictingobjectives.It isshowninSection3.9thattheproposedapproachyieldsbet terresultsthansimulated annealingandgeneticsearchundertheassumptionsofsamem odels,setup,parameters andobjectivefunctionfortheirimplementation. 31

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3.3SimultaneousOptimizationofDelayandNoise Weuseaplaceandroutetooltoperformtheplacementandrout ingofthegivengatelevelnetlist.Theglobalgridsoftherouterareusedtopart itionthecompleterouting areaintodistinctrectangularsectionscalledchannels.T hechannelboundariesareusedin dividingthenetsintovariousnetsegments.Thefollowingi nformationisextractedfromthe routeddesigntocalculatethewireresistance,wirecapaci tanceandcouplingcapacitances accurately: (i)Netsegmentsbelongingtoeachnet (ii)Channelnumberscorrespondingtothenetsegments (iii)Tracknumbersinthechannel (iv)Wirelengthofthenetsegmentsinachannel (v)Startingpositionofthenetsegmentsinachannel (vi)Metallayerstowhichthenetsegmentsbelong (vii)Directionofthenetsegment Theminimumwiresizeofanynetsegmentisxedbasedonthemi nimumwiresize designrulerequirementoftheprocesstechnology.Themaxi mumwiresizeforanet segmentisdeterminedfromthetrackdistancebetweenitsim mediateadjacentnetsandthe minimumedge-to-edgespacingrequirements.Therangebetw eenminimumandmaximum wiresizesforeachnetsegmentcanbetreatedasitspossible wiresizeswithoutviolating theprocessdesignrules.Therangeforeachnetsegmentisdi videdintoadiscretesetof valueswithequalstepsizes.Thenumberofentriesinthedis cretesetaredierentforeach netsegmentasitdependsonitslocationanditsimmediatene ighbors.Thediscretesetof allowablewiresizesforanetsegmentismodeledasitsstrat egysetwithoutviolatingthe designrules. 32

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Algorithm3.1.WireSizingAlgorithmforSimultaneousOpti mizationofInterconnectDelay andCrosstalkNoise Input: Placedandrouteddesign Output: Optimizedwiresizes Algorithm:extractthenetinformationorganizethenetsintochannelsandtracksidentifyterminalnetsegmentsforalllayersdo forallchannelsdo initializeloads();initializescores();determinestrategies();markthechannelasun-played endfor endforselectachannel i withlowestscorevalue whilethereexistsanun-playedchanneldo calculatemutual-capacitance();calculatewire-capacitance();calculatewire-resistance();forallnetsegments j 2 channel i do createa3-playergamewith j anditsleftandrightneighbors cost-matrix payo(threeplayers,strategies) %forpayofunction,seeAlgorithm3.2. optimized-width nash-solution(threeplayers,payos) %forNashequilibriumsolution,seeAlgorithm3.3. endforupdateloads();updatescores();markthechannelasplayedselecttheanewchannelwithlowestscorevalue endwhilereturn: optimizedwidthsofallnetsegments Agameismodeledforeachindividualchannel.Thechannelsl ocatedondierentlayers areconsideredseparatelyastheyconsistofdierentnetse gments.Foragivenchannel,its netsegmentsaremodeledastheplayersofthegame.Thecoupl ingeectonanetsegment dependsonallthenetsegmentsadjacenttoit.Asthedistanc ebetweenthenetsegments increases,thecouplingcapacitancebetweenthemdecrease srapidlytherebyreducingthe couplingeectsduetoeachother.Aspointedoutin[60],int hecontextofwiresizing,it 33

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issucienttoconsiderthecouplingeectsduetoitsimmedi ateneighborsfortworeasons: (i)Thecouplingeectsofotherneighboringnetsareminima lwhencomparedtoimmediate neighborsduetotheirincreaseddistancefromthegivennet .(ii)Theimmediateneighbors actsasshieldstothegivennetfromtheotherneighboringne ts.Hence,inthiswork,we considerthecouplingeectsduetoitsimmediateleftandri ghtneighborsforagivennet segment. Algorithm3.2.AlgorithmforPayoMatrixCalculationforS imultaneousOptimizationof InterconnectDelayandCrosstalkNoise Input: NumberofPlayers N ,Strategyset S Output: Payomatrix Algorithm:forallplayers i 2 1to N do forallstrategycombinations S j = f s j1 ;:::;s jN g ,where( s j1 2 S 1 ) ;:::; ( s jN 2 S N )do calculatethedelayusingEquation2.3normalizethedelayw.r.trststrategycombinationcalculatethecrosstalknoiseusingEquations2.8and2.9normalizethenoisew.r.trststrategycombinationP [ i S j ] Geometricmeanofnormalizednoiseanddelay endfor endforreturn: payomatrix P 8 strategycombinations Thepayofunctiontriestocapturetheinteractionbetween theneighboringnetsegments(modeledastheplayersofthegame)inthechannel.For eachnetsegmentina givenchannel,itsdelayDandmaximumcrosstalknoiseNarec alculatedbyusingthe Equations2.3,2.8,and2.9.Thesevaluesarecalculatedfor allstrategiesofthegivennet segmentbyconsideringthestrategiesofitsimmediateleft andrightneighbors.Thedelay andcrosstalknoisevaluesobtainedforanetsegmentarethe nnormalizedwithrespectto thecorrespondingrststrategy.Thenormalizationisperf ormedtotransformthedelay andcrosstalknoisevaluesintodimension-lessquantities sothattheybeeasilycorrelated witheachother.Thepayofunctionismodeledasthegeometr icmeanofnormalizeddelay andnormalizedcrosstalknoisevaluesforeachstrategyoft henetsegment.Wehavechosen geometricmeansoastogiveequalweightstobothcrosstalkn oiseandinterconnectdelay componentsduringtheiroptimization. 34

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Algorithm3.3.AlgorithmforNashEquilibriumSolution Input: Numberofplayers N ,Payomatrix P Output: Nashsolution Algorithm:forallplayers i do forallpayosofplayer i do nd s i suchthat, P i ( s 1 ;:::;s i ;:::;s N ) P i ( s 1 ;:::;s i ;:::;s N ) s i istheNashstrategyforplayer i endfor endforNash-solution S = f s 1 ;:::;s N g %setofoptimizedstrategiesforall N players return: Nashsolution S Wehaveusednormalformformulationtomathematicallyrepr esentandsolvethegame. Normalformgamerepresentationsuitsformulationswellbe causeitemphasizesmainlythe competitionbetweentheplayersparticipatinginthegame. Inaddition,normalform gamescanbeeasilymodeledandimplementedastheyrequireo nlyapayomatrixand thecorrespondingstrategysettoreachthegame'sNashequi librium.Mathematically,a normalformgameconsistsofasetof N playerslabeled1 ; 2 ;:::;N ,suchthateachplayer i has (i)achoiceset S i calledstrategysetofplayer i ;itselementsarecalledstrategies. (ii)apayofunction P i : S 1 S 2 :::: S N !< ,assignedtoeachstrategychosenby theplayer i withrespecttootherplayers. Inanormalformgame,alltheplayersplaysimultaneouslywi thoutanyknowledgeabout otherplayers'play.Inotherwords,theplayerssimultaneo uslychooseastrategy s i 2 S i suchthattheirrespectivepayoismaximizedorminimizedw ithrespecttothepayosof theotherplayers.Theequilibriumofthegameiscomputedby usingtheNashequilibriumcondition.ConsiderachannelconsistingofNnetsegme nts.Thewiresizeofany netsegmentinthechannelisinruencedonlybyitsimmediate leftandrightneighbors. Therefore,thepayofunctionofanynetsegmentinthechann eldependsonlyontwoother players(leftandrightimmediateneighbors)ratherthanon alltheN-1players.Thus,for 35

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eachchannel,insteadofhavingasinglegamewithN-players ,wedividethegameintoN sub-gameswitheachsub-gameinvolving3-players-thegive nnetsegment,itsleftneighbor anditsrightneighbor. Segment iSegment k Segment j Cell 1 (Driver Cell) Cell 2 (Driven Cell) Figure3.1.AnExampleScenario Theinterconnectdelaycalculationsforeachnetsegmentre quireitsvaluesofwire andloadcapacitance,inadditiontootherparametersdene dinEquation2.3.Asan example,consideranetconnectedbetweencell1andcell2wi thcell1drivingthenet andcell2receiving,asshowninFigure3.1..Inthisexample ,thenetisdividedintothree segmentsjustasanexampleforillustration.Theloadcapac itanceofsegment k isthe inputcapacitanceofthecell2,whichisknown.Thecell2and segment k actasloads forsegment j .Thewirecapacitanceofsegment k dependsonitswirewidth.Hence,in ordertocalculatetheloadcapacitanceofsegment j ,thewirewidthofsegment k hasto beoptimized,requiringsegment k toplaythegamebeforesegment j .Ingeneral,theload capacitanceofanetsegmentcanbecalculatedonlywhenitsd own-streamwiresegments areoptimized.Henceanorderingforchannelstoplayhastob edenedwhichsatises theloadcapacitancedependency.Ascore,denedasthedie rencebetweenthetotalnet segmentsandthenumberofterminalnetsbelongingtoachann el,isusedfororderingthe channels.Theorderingofchannelsaidsinconsideringthee ectsofwiresizesofdownstreamnetsegments.Eventhoughthegameisplayedforasegm ent,theloadcapacitance takesintoaccounttheeectsofitscompletenet.Thusthere sultingsolutionisnotalocal solutionwhichisconnedtoasegmentofthenet. Achannelwithlowestscoreisselectedtoplaythegamewithi tsnon-terminalnet segmentsassignedwithadefaultloadcapacitance.Nashequ ilibriumisevaluatedforthe 36

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lowestscorechannelanditsNashwidthsareusedtoupdateth eloadcapacitancesofthenet segmentsbelongingtoitsadjacentchannels.Thescoresofo nlytheneighboringchannels havetobeupdatedtorerectthenetsegmentswithknownloadv aluesasterminalnet segments.Hence,whenachannelisplayedout,theloadandsc orevaluesofamaximumof sixadjacentchannelshavetobeupdatedtorerecttheoptima lwidthsforthenetsegments resultedfromtheplayedchannel.Again,achannelwithlowe stscoreisselectedtoplaythe nextgameandthisprocessisrepeateduntilallthechannels areplayedout.Algorithm3.1. showsthepseudo-codeofthecompletewiresizingalgorithm forsimultaneousoptimization ofdelayandcrosstalknoise.3.4SimultaneousOptimizationofDelay,PowerandNoise Insection3.3,wedescribedawiresizingmethodologyforop timizinginterconnectdelay andcrosstalknoise.Inthissection,wedevelopawiresizin gmethodologyformulti-metric optimizationofinterconnectdelay,powerandcrosstalkno ise.Again,theplacedandrouted designisusedasthestartingpoint.Thenetextractionphas e,thestrategygenerationand thegamemodelingforthisformulationaresimilartotheone describedinsection3.3. Themaindierencebetweenthetwoformulationsisthemodel ingofthepayofunction. Theobjectivefunctionorthepayofunctionisacombinatio noftheinterconnectdelay, powerandcrosstalkdueanetsegment.Theinterconnectpowe rcanbemodeledasthe powerdissipatedduetothecharginganddischargingofthec apacitanceexhibitedbythe interconnectwires.Mathematically,interconnectpowerc anberepresentedas P inter = 1 2 V 2 dd f clk C inter where, istheswitchingactivityoftheinterconnectwire, V dd isthesupplyvoltageand f clk istheclockfrequency.Theinterconnectcapacitance, C inter ,isthesumofthewire andcouplingcapacitances.Hence,thepowerdissipateddue tointerconnectsisgivenby equation3.1. 37

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P inter = 1 2 V 2 dd f clk ( C i + C il + C ir )(3.1) Thepayofunctionortheobjectivefunctionismodeledasth eweightedsumofnormalizeddelay,powerandcrosstalknoisevaluesforeachnetseg ment.Foreachnetsegmentin thechannel,thedelayDiscalculatedusingequation2.3,th ecrosstalknoiseNiscalculated asthesumofvaluesgivenbyequations2.8and2.9,andtheint erconnectpowerPisgiven byequation3.1.Thesevaluesarecalculatedforallstrateg iesofthegivennetsegment byconsideringthestrategiesofitsimmediateleftandrigh tneighbors.Thedelay,noise andpowervaluesobtainedforeachnetsegmentarenormalize dwithrespecttotheirrst strategy.TheweightsfordelayD,powerPandcrosstalknois eNcanbeadjustedbythe designeraccordingtotheneed.Inthiswork,wehavechosen0 .33astheweightforboth crosstalknoiseandinterconnectpower,and0.34astheweig htforinterconnectdelay.The algorithmforcalculatingthepayofunctionisgiveninAlg orithm3.4.. Algorithm3.4.AlgorithmforPayoMatrixCalculationforS imultaneousOptimizationof InterconnectDelay,PowerandCrosstalkNoise Input: NumberofPlayers N ,Strategyset S Output: Payomatrix Algorithm:forallplayers i 2 1to N do forallstrategycombinations S j = f s j1 ;:::;s jN g ,where( s j1 2 S 1 ) ;:::; ( s jN 2 S N )do calculatedelay D usingEquation2.3 calculatecrosstalknoise N usingEquations2.8and2.9 calculatepower P usingEquation3.1 P [ i S j ] a D +b P +c N %a,b,caretheweightsofDelay,PowerandNoiserespectivel y endfor endforreturn: payomatrix P 8 strategycombinations 3.5TimeandSpaceComplexityofProposedWireSizingAlgori thms TheworstcasetimecomplexityofevaluatingNashequilibri umforageneralM-player gamewith S strategiesforeachplayerisgivenas O ( M S M )[52].ReferringtoSection3, 38

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wehavemodeledtheproblemofwiresizingasasetof3-player gamesforeachnetsegment. Theincrementalstepsizebetweentwoconsecutivewiresize sforanynetsegmentiskept constant.Hence,thenumberofstrategiesforeachplayerde pendsonitsrangeofpossible wiresizes,whichisdierentfromplayertoplayer.Inthisw ork,wehavechosenthestep sizesuchthatthenumberofstrategiesforanynetsegmentis lessthanve.Considera channelwith N netsegments.Eachnetsegmentinthechannelwillforma3-pl ayergame withitsleftandrightneighbors.Hence,thecomplexityofc alculatingNashequilibriumfor agivenchannelwith N netsegmentsisgivenas O ( N 5 3 ) O ( N ).TheNashequilibrium choosesoptimalwiresizesfortheplayersconsideringeach gameindividually.But,a playerparticipatesinthreedierentgamesformedforitse lf,itsrightneighboranditsleft neighbor.Wenoticedfromourexperimentsthatthewidthsre sultingfromthethreegames areequalforaround70%ofnetsegments.Incaseofdierentw idths,maximallikelihood Nashwidthisassignedtothenetsegment.Consideringallth echannelsandthelayersin agivendesign,theoveralltimecomplexityofproposedalgo rithmisgivenas O L C X i =1 N i = O 0@ X 8 i 2 nets n i 1A where L isthetotalnumberoflayers, C isthenumberofchannelsinalayer, N i isthe numberofnetsegmentsbelongingtochannel i and n i isthenumberofnetsegmentsin whichanet i isdivided.Hence,thetimecomplexityoftheproposedalgor ithmis linear in termsoftotalnetsegmentsinthedesign. Thespacecomplexityoftheproposedalgorithmisdependent entirelyonthenumber ofnetsegmentsinthedesignandthepayomatrix.Thespacec omplexityofthepayo matrixdependsonthenumberofstrategiesforeachplayerin thegame.Asthegames areplayedsequentially,thetotalspacerequiredbyallgam esputtogetherisequaltothe spacecomplexityofagameinvolvingplayerswithmaximumnu mberofstrategies.Hence, mathematically,thespacecomplexityisgivenas O ( S 1 S 2 S 3 ) O (5 5 5),where S 1 ;S 2 ,and S 3 arethestrategysetsof3-playergameinvolvingtheplayers withmaximum 39

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strategies.Hence,thespacecomplexityoftheproposedalg orithmisgivenas O 0@ X 8 i 2 nets n i +5 5 5 1A O 0@ X 8 i 2 nets n i 1A 3.6ProofofExistenceofNashEquilibriumSolutionfortheW ireSizingFormulation Inthissection,weprovidetheproofofexistenceofNashequ ilibriuminthecaseofwire sizingproblemforsimultaneousoptimizationofinterconn ectdelayandcrosstalknoise. Asthewiresizeofanetincreases,theinterconnectdelayde creasesandthecoupling capacitanceincreasesresultinginaconvexpayofunction .Let G = f S 1 ;:::;S n ; f 1 ;:::;f n g beagamewitheachplayer i 2 N havingastrategyset S i containingitspossiblewiresizes anditspayogivenby f i .Wehavemodeledthestrategyset S i foreachplayerasanonempty,compactsetofanitedimensionalEuclideanspace.B ecauseoftheconvexnature oftheinterconnectdelayandcrosstalknoise,themodeledp ayofunction f i becomesupper semicontinuouson S = Ni =1 S i andforanyxed u i 2 S i ,thefunction f i ( u i ;: )isalower semicontinuouson S ( i ) [16].Forany u 2 S ,thebestreplyortheexpectedpayo B i ( u ) isalsoconvex.AccordingtoKakutani'sxedpointtheorem[ 61],thegameGhasatleast oneNashequilibriumpointifthegraph G B = f ( x;y ): x 2 S;y 2 B ( x ) g isclosed. Letsassumethatitsnotclosed.Then, 9 ( x 0 ;y 0 ) = 2 G B ,suchthateveryneighborhood (in S S )of( x 0 ;y 0 )containsapointof G B x 0 isawiresize,ithastobeoneofthosefromthesetofpossible wiresizesforthe givenplayerinordertosatisfytheDRCrulesoftheusedproc esstechnology. ) x 0 2 S ) y 0 = 2 B ( x 0 ) 40

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Inotherwords,foratleastonenetsegmentplayingthegame( saysegment1),thereis an y 1 1 2 S 1 suchthat f 1 ( y 1 1 ;x 02 ;:::x 0n ) f 1 ( y 0 1 ;x 02 ;:::x 0n )(3.2) Let F beafunctionsuchthat F : S 2 !< andgivenas F ( x;y )= f 1 ( y 1 1 ;x 2 ;:::x n ) f 1 ( y 1 ;x 2 ;:::x n ) Since f i isuppersemicontinuouson S and f i ( u i ;: )islowersemicontinuouson S i ,Fis lowersemicontinuousand C = f ( x;y ) 2 S 2 : F ( x;y ) 0 g isclosed.Hence,forany ( x; y ) 2 G B ;F ( x; y ) 0.But,byEquation3.2, F ( x 0 ;y 0 ) 0,contradictingtheclosedness ofC.Thus,thereisapoint s 2 S suchthat s 2 B ( s ),whichisaNashequilibriumpoint. 3.7Discussion Inthissection,weexplaintherationalebehindtheoptimiz ationofentirenetsofthe designratherthanonlythecriticalnets.Wiresizingtechn iquecanecientlyutilizethe \unused"routingresourcestominimizethedesignparamete rsofarouteddesignandhence, itisadvantageoustobeappliedatpost-routephaseofthede sign.Inthecontextofwire sizingatpost-routephase,themaximumsizewithwhichanet canbesizedisxed.The sizingofanethastobeperformedwithinthisfeasiblerange orelseaconsiderablenumber ofnetshavetobere-routed.There-routingofadesignusual lyrequirestremendous amountoftimeandeort.Thisisvalidforcriticalnetsaswe llandhence,havetobe sizedwithintheroutingresourcesavailabletoit.Withthe availableroutingresources,the gametheoreticformulationallowsabetterallocationfort hecriticalnetswhencompared toitsneighbors.Thisisbecausethepayovaluesforcritic alnetsdominatesthatofits neighborsandhence,Nashequilibriumgivesmoreweighttot hecriticalnetsandresultsina solutionwhichisinthebestinterestofbothcriticalnetsa ndtheirneighbors.Therouting resourcesavailableatotherlocationscanbebetterusedto optimizethecorresponding nets,ratherthanleavingthemunused.Hence,wehaveplanne dtooptimizeallthenets 41

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inthedesign.Also,optimizingallthenetsindesignwillha veanadvantageofenforcing thetimingclosure,thesignalintegrityforallnetsandhen ce,aidsinotherpost-layout optimizationtechniques.Theexperimentalresultsvalida teourclaimsdepictingbetter criticalnetsavingsforourapproachwhencomparedtothesi mulatedannealingorgenetic search.3.8DesignFlowCrosstalk Noise Estimator ModelingWith calculated wire widthsScriptWire-Sized Delay and Crosstalk Noise Optimized CircuitPayoff Function DEF Update (StarRCXT, SignalStrom, Celtic)Verilog Design Cadence First Encounter Script Net Extractor and Crosstalk Noise Delay Models Game Theoretic Based Wire Size SolverLEF Library, Cell Timing Library, Cell Noise ModelsPlaced and Routed Design (DEF Format)Payoff function Optimized Wire Sizes Wire Size Optimized DEF This WorkDelay Calculator Extracted Net InformationRC ExtractorFigure3.2.IntegrationofProposedWireSizingAlgorithmi ntheDesignFlow Thedesignrowforobtaininganoptimallywiresizedcircuit fromaverilog/VHDL descriptionisshowninFigure3.2..Thebehavioralverilog /VHDLdescriptionissynthesized ontoalibraryofstandardcellsandgivenasinputtothedesi gnrow.Thestandardcells areplacedandroutedinaccordancewiththesynthesizedcod eusinganystandardcellplace 42

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androutetool.WehaveusedtheFirstEncounterRTL-to-GDSIItoolfromCadenceDesignSystemstoperformtheplacementandroutingofgatelevelRTLdesign.Thenet informationrequiredforcalculationsofinterconnectdel ayandcrosstalknoiseisextracted fromtherouteddesign.Agawkscriptisdevelopedwhichextr actsthisinformationfrom theexportedDEFleoftherouteddesign.Thepayofunction isusedbythegame theoreticbasedwiresizesolverdescribedinAlgorithm3.1 .tominimizetheinterconnect delayandcrosstalknoiseofeveryindividualnetofthedesi gn.Theoptimizedwiresizes resultedfromthegametheoreticwiresizesolverareusedto updatetherouteddesign. Wehavedevelopedanothergawkscriptwhichupdatesthewire sizesofallthenetsinthe originalDEFrouteddesignwiththeircorrespondingoptimi zedwiresizes.Itshouldbe notedherethattheresultingoptimizeddesigndoesnotrequ irere-routingasallthesized netssatisfythedesignrulesofthegivenprocesstechnolog y. 3.9ExperimentalResults WehaveimplementedtheproposedalgorithminCandexecuted onaUltraSPARC-IIe 650MHz,512MBSunBlade150systemoperatingonSolaris2.8a ndtestedwiththeASIC designsfromOpencores[62].A180nm,6-Metalstandardlibr aryisobtainedfromCrete [63],aneducationaluniversitycampusprogramdevelopeda ndmaintainedbyCadence designsystems.Thestandardcelllibrarycontainsabout40 logiccellsandover100I/O cellswiththecorrespondingcelltimingandtransistormod els.ASICdesigns,writtenin behavioralVHDL/VerilogareconvertedtostructuralVHDL/ Verilogusingthestandard cellsinthelibrarywiththehelpofBuildGates,anRTLsynth esistoolofCadencedesign systems.WehavemodiedtheASICdesignssuchthatallthebl ocksinthedesignare rattenedtostandardcellsinthelibrarywithoutmaintaini ngthehierarchy.Theon-chip memorymodulesarerealizedasD-ripropregisterarrays.Th estructuralVHDL/Verilog designisusedasinputbyCadenceFirstEncountertodevelop theroorplan.Wehaveset theoptionofrowutilizationto95%forallthedesignssoast ohaveacompactroorplan. ThedesignisthenplacedandroutedusingAmoebaplaceandNa norouterespectively, 43

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whicharepartofCadenceFirstEncountertool.Thenalplac edandrouteddesignisthen exportedinDEFformat.Thenetinformationisextractedfro mtheDEFleandprovided asinputtothegametheoreticwiresizesolver.Thecalculat edwiresizeforeachnetisused toupdatethewiresintheoriginalDEFletogenerateanopti mizedDEFle.Itcanbe notedthattheoptimizedDEFiscreatedwiththehelpofgawks criptsandisnotre-routed. Theparasiticresistancesandcapacitancesfrombothorigi nalandoptimizedDEFlesare extractedusingStarRCXTfromSynopsysInc.Theinterconne ctdelayandcrosstalknoise areestimatedusingCadenceSignalstormandCelticICtools respectively. Severalofthepioneeringworksreportedintheliteraturef ortheproblemofwiresize optimization[19,21,23,22,25,26],onlypresentresultsf orarbitrarynetsanddonot considerroutingcongestion,roorplancompaction,etc,of thespecicdesignorbenchmark circuits. Thus,itisnotpossibletoprovideadirectcomparisonofour resultswiththose works .Tocompareourresults,wehaveimplementedsimulatedanne alingandgenetic searchbasedalgorithmsandexecutedonthesameSolarismac hinewithsamesetofinputs andconstraints.Theannealingprocessofsimulatedanneal ingapproachisdeterminedby experimentingsignicantlytogetthebestresultsandthem aximumoptimization.The netsaredividedintonetsegmentsandthesetofpossiblewir esizesforeachnetsegmentis calculatedasindicatedinSection3.3.Ineachmoveofthean nealingprocess,anetsegment israndomlyselectedanditssizeisassignedfromthesetofi tspossiblewiresizes.The costfunctionisdenedasthegeometricmeanoftheintercon nectdelayandthecrosstalk noisesummedoverallthenetsegments.Theinitialtemperat ureisdeterminedbynding theaveragechangeinthecostforasetofrandommovesfromth estartingconguration andselectingthetemperaturewhichleadstoanacceptproba bilityof0.95.Thenumber ofmovespertemperatureforeachdesignissetto20timesthe numberofnetsegments inthedesignsoastoallowanaverageofatleast10to15moves foreachnetsegment beforesettlingforitssolution.Theup-hillmovesareacce ptedwithaprobabilityof e C T where C isthechangeinthecostandTisthecurrenttemperatureofth eiteration.The temperatureiscooledattherateof0.95. 44

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Table3.1.ExperimentalResultsforSimultaneousOptimiza tionofInterconnectDelayandCrosstalkNoiseDuringWireS izing UsingComplexDelayModels OpencoreDesign TotalNets DieArea( mm 2 ) GeneticSearch SimulatedAnnealingApproach GameTheoreticApproach Runtime(mins) %Delay %Noise Runtime(mins) %Delay %Noise Runtime(mins) %Delay %Noise Savings Savings Savings Savings Savings Savings Avg. Crit. Avg. Crit. Avg. Crit. Avg. Crit. Avg. Crit. Avg. Crit. Mult 854 0.199 34.16 2.45 5.32 3.10 4.32 14.32 6.15 8.91 5.12 13.26 1.89 9.87 11.79 12.13 17.23 PCIbus 19520 0.434 183.23 12.79 20.16 13.78 19.96 47.35 10.31 21.22 12.36 23.39 5.23 19.32 34.21 21.42 37.38 SerialATA 43563 1.624 418.31 17.94 31.02 11.53 25.82 124.83 18.91 28.65 12.41 26.37 11.86 29.87 39.95 20.14 42.15 RISC 61468 2.102 729.47 11.91 17.37 9.19 15.46 188.33 20.39 39.89 16.31 24.25 16.86 25.22 35.21 22.31 29.73 AVR P 78770 11.103 972.51 10.18 11.25 13.67 22.13 232.67 17.57 21.35 27.67 40.12 21.32 22.45 37.63 31.34 40.31 P16C55 C 102021 19.984 1301.43 11.64 15.48 25.91 29.18 288.36 17.98 29.47 31.67 39.45 28.98 19.86 27.45 43.29 57.98 T80 C 157850 30.388 1689.24 13.76 14.11 15.23 18.10 353.25 14.86 15.97 23.39 29.74 39.48 23.78 34.87 33.12 39.89 Average 11.52 16.39 13.63 19.28 15.17 23.64 18.42 28.08 21.48 31.26 26.25 37.81 Noareaoverheadforallthreeapproaches.Thepercentageva luesindicatedarew.r.tplacedandrouteddesignwithoutwi resizing. 1 TableLegend:Avg:Averagesavingsofallthenetsintheenti redesign; Crit:Savingsonthecriticalpathnetofthedesign;Runtime:indicatestherunningtimeofeachalgorithm.45

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Table3.2.ExperimentalResultsforSimultaneousOptimiza tionofDelay,PowerandNoise DuringWireSizingUsingComplexDelayModels OpencoreDesign TotalNets DieArea( mm 2 ) GameTheoreticApproach Runtime(mins) %Delay %Power %Noise Savings Savings Savings Avg. Crit. Avg. Crit. Avg. Crit. Mult 854 0.199 2.34 4.31 7.49 5.16 7.44 8.15 13.56 PCIbus 19520 0.434 8.12 10.69 15.40 14.23 17.05 19.52 25.81 SerialATA 43563 1.624 14.03 17.90 21.45 17.86 23.61 15.85 24.40 RISC 61468 2.102 23.80 19.92 22.31 15.52 19.03 17.64 24.74 AVR P 78770 11.103 30.05 13.54 19.40 11.74 16.89 18.22 23.01 P16C55 C 102021 19.984 37.58 14.11 18.58 19.35 27.20 21.65 30.94 T80 C 157850 30.388 46.35 18.25 25.05 20.05 28.02 23.94 32.69 Average 14.10 14.84 17.85 Theareaoverheadincurrediszero.Thepercentagevaluesin dicatedarew.r.tplaced androuteddesignwithoutwiresizing. Fortablelegends,pleaserefertoTable3.1. Thewiresizingproblemforsimultaneousinterconnectdela yandcrosstalknoiseoptimizationisalsomodeledasageneticsearchmechanismand solvedusingGALib[64]. Theinitialpopulationcontainsthenetsegmentswiththeir correspondingwiresizesas usedintheoriginalunsizeddesign.Eachindividualinthep opulationcalledchromosome isrepresentedasasetofthreeintegersindicatingthenetn umber,thesegmentnumber andthewiresizeassignedtothesegment.Thechromosomesev olvethroughsuccessive iterationscalledgenerations.Duringeachgeneration,th echromosomesareevaluatedfor theirtnesstest.Wehavedenedthetnesscriterionasthe deviationofthecrosstalk noiseandinterconnectdelayofeachnetsegmentfromitswor st-casevalues.Thechromosomeswithlowervaluesofcrosstalknoiseandinterconnect delayaregivenhighertness values.Wehaveusedsteady-stategeneticalgorithmavaila bleasapartofGALiblibrary togenerateoverlappingpopulationswhichretainsits30%o fttestchromosomesinits newgenerations.Themutationprocessforachromosomeisde nedtorandomlyselecta wiresizefromitssetofpossiblewiresizes.Thenewchromos omesarecreatedusingsingle pointcrossoverandarevalidatedagainsttheirsetofpossi blewiresizes.Theselection 46

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processofchromosomeisadoptedbytheroulettewheelselec tionapproach.Wehaveset theconvergence-of-populationasthestoppingmeasurefor theevolutionofgenerations. Table3.1.showstheexperimentalresultsforthecaseofsim ultaneousoptimizationof interconnectdelayandcrosstalknoise.Firstcolumnindic atesthenameofthedesignand thesecondcolumnindicatesitscorrespondingnumberofnet s.Theareaindicatedinthird columnischipareaoccupiedbythecorewithoutconsidering itsI/Opins.Thefourth, ninthandfourteenthcolumnsindicatetheruntimeofgeneti csearch,simulatedannealing andgametheoreticwiresizesolversrespectively.Columns veandsevenindicatethe averagedelayandnoisesavingsforallthenetsofthedesign obtainedbythegeneticsearch mechanism.Columnstenandtwelveindicatethesameforsimu latedannealingapproach, whileColumnsfteenandseventeenrepresentsthegametheo reticapproach.Columnssix andeightindicatethecriticalnetdelayandnoisesavingso btainedbythegeneticsearch mechanism.Columnselevenandthirteenindicatethesamefo rthesimulatedannealing approach,andColumnssixteenandeighteenindicatethegam etheoreticapproach.Table 3.2.showstheexperimentalresultsforthecaseofsimultan eousoptimizationofinterconnect delay,interconnectpowerandcrosstalknoise. Theexperimentswereconductedsuchthattheareaoverheadi szeroinallthreeapproaches.Thesavingsobtainedintermsofinterconnectdel ay,powerandcrosstalknoise dependonthefactorslikeroorplancompaction,routingcon gestion.Thisisbecausethe routingcongestiondecidesthewiresizescalingofthenets routedthroughthatregion.It canbenoticedthatthegametheoreticapproachyieldsbette rsavingsthangeneticsearch andsimulatedannealingforallthetestcasedesigns.Inadd ition,ouralgorithmhassignificantlysmallerruntimesthangeneticsearchorsimulateda nnealingforfairlylarge-scale designs.Henceourapproachisscalableandfavorabletohan dlethecomplexityoflarge SOCdesigns. 47

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3.10Conclusions Gametheoryallowsthesimultaneousoptimizationofmultip lemetricsinthecontext ofconrictingobjectivesleadingtoaconvexobjectivefunc tionintheproblemformulation. Thisessentiallymakesitpossibletousegametheoryforsim ultaneousoptimizationof interconnectdelayandcrosstalknoise.Optimizingbothin terconnectdelayandcrosstalk noiseisextremelycriticalindeepsubmicronandnanoregim ecircuits.Theuseofgame theoryandNashequilibriumfortheproblemofwiresizingto optimizeinterconnectdelay andcrosstalknoiseisbeingattemptedforthersttime.The proposedmethodresultsin alineartimealgorithmwithsignicantlybetterresultsth ansimulatedannealing,making thisworkanimportantcontribution. Ourintentioninthisworkwastoshowthatwiresizingcanbeu sedtoachievesimultaneousoptimizationofinterconnectdelayandcrosstalkn oiseatpost-routestage.We observedthatthepreviousalgorithmsforwiresizingtarge tforsinglemetricoptimization withotherparametersasconstraintsandhavenotbeenteste dwithallthedesignconstraintssuchasroutingcongestion,roorplancompaction, positionofnets,etc.Further, priorworkshavenotindicatedaviabledesignrowtoinclude wiresizing[59,25].Ithas beenpointedoutin[59]thatwiretaperingfortheentirenet canyield5%moresavings indelaywhencomparedtouniformwiresizing.However,perf orminguniformwiresizing withinanetsegmentforallthesegmentsofanetcanyieldsig nicantsavingsintermsof crosstalknoiseandinterconnectdelayatpost-routestage 48

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CHAPTER4 NEWINTERCONNECTMODELS InthegamemodeldevelopedinChapter3,wedividedthenetsi ntonetsegmentsand modeledthenetsegmentsastheplayersofthegame.Eventhou gh,aplayerismodeled asanetsegment,theeectsoftheentirenetonitsnetsegmen tisconsideredbyordering thechannels,whichaidinaccountingforthedownstreamee cts(pleaserefertoSection 3.3).Hence,theinterconnectdelayhastobecalculatedfor anetsegmentratherthan theentirenet.Thelengthsofthenetsegmentsissuciently smallanddonotresultin treetopologies.Theequation2.3givestheinterconnectde layfortheentirenetandis accurate,butcomplextobecomputedrepeatedlyforthenets egments.Hence,wedevelop simple,fastbutsucientlyaccurateinterconnectmodelst hatcanbeusedforcalculating interconnectdelaysrepeatedlyinanarrangementofthreep arallelnetsegments. 4.1FastTransmissionLineModels Inthissection,wederivenew,simpleandfastmodelsbasedo ntransmissionlinetheory. Thenetsegmentismodeledasatransmissionlinedrivenbyav oltagesourceandterminated byaload Z L ,asshowninFigure4.1.(a).Wehaveconsideredthecoupling eectsduetothe immediateleftandrightnetsegmentswhiledevelopingthem odelforinterconnectdelay throughasegmentofthegivennet.Theseriesresistanceoft heneighboringnetsegments isnotconsideredsinceitdoesnotaectthepropagationcha racteristicsalongthegivennet [65].Figure4.1.(b)showstheequivalentmodelwithuncoup ledcapacitancesforonesection ofinterconnectlinearrangementgiveninFigure4.1.(a).T heelementsshowninFigure 4.1.aredenedperunitlengthoftheinterconnectline.Ref erringtoFigure4.1.(b),the seriesimpedanceperunitlength Z s andparallelimpedanceperunitlength Z p aregiven 49

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byEquations4.1and4.2respectively.Theimpedances Z l and Z r representthecoupling eectsduetoleftandrightneighborsconsideringthepropa gationalongthenetsegment. li i l ri r sC R C C C C V (a)Interconnectmodeledastransmis-sionline L r i l C Z Z Z R/2 R/2 V s (b)Uncoupledequivalentofonesectionofinterconnect Figure4.1.InterconnectModel(a)ModeledasaTransmissio nLine(b)UncoupledEquivalentofOneSectionoftheInterconnect Z s = R (4.1) 1 Z p = 1 Z l + 1 Z r + sC i =( Y l + Y r + C i ) s (4.2) where, Z l = C il + C l sC il C l = 1 sY l ;Z r = C ir + C r sC ir C r = 1 sY r Thepropagationconstant r andthecharacteristicimpedance Z 0 ofthetransmission linemodelaregivenbytheEquations4.3and4.4respectivel y. r = s Z s Z p = q Rs ( Y l + Y r + C i )(4.3) 50

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Z 0 = q Z s Z p = s R s ( Y l + Y r + C i ) (4.4) InLaplacetransformdomain(s-domain),thevoltageandcur rentdistributionsalong aninterconnectlength(denotedbythecoordinatez)satisf ythetransmissionlineequations givenby4.5 d 2 V dz 2 = r 2 V;and d 2 I dz 2 = r 2 I (4.5) Thegeneralsolutionsforthevoltageandcurrentsatisfyin gthesetofdierentialequations4.5aregivenby: v z = Ae rz + Be rz ;andi z = Ae rz Be rz Z 0 (4.6) where,theconstantsAandBcanbedeterminedbyusingthetwo knownboundaryconditions:(i)theinterconnectisdrivenbyvoltagesource V s and(ii)theinterconnectis terminatedbyaload Z L .Quantitatively,theycanberepresentedas: atsource,z=0and v z = V s atload,z=Land v z =i z = Z L ,where Z L =1 =sC L substitutingtheaboveboundaryconditionsinEquations4. 6,andsolvingforAandB,we have, A = V s ( Z L + Z 0 ) e rL e rL ( Z L Z 0 )+ e rL ( Z L + Z 0 ) B = V s ( Z L Z 0 ) e rL e rL ( Z L Z 0 )+ e rL ( Z L + Z 0 ) Therefore,thevoltageattheloadend( z = L )isgivenbyEquation4.7. V L = 2 V s Z L e rL ( Z L Z 0 )+ e rL ( Z L + Z 0 ) (4.7) 51

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Thesteadystatevoltageattheloadoftheinterconnectline hastoreachanidealvalue of V s ,thesourcevoltage.ThepropagationdelayinLaplacedomai nisdenedasthetime takenforthevoltageattheloadterminaltoreach50%ofitss teadystatevalue.Atthis point,thes-parameterisdenedass=2 =T delay [65].Hence,wehave, V Steady State L = V s 2 = 2 V s Z L e rL ( Z L Z 0 )+ e rL ( Z L + Z 0 ) transformingtheaboveequation,wehave e rL + e rL Z 0 Z L e rL e rL =4(4.8) Itshouldbenotedintheaboveequationthat Z L Z 0 and r arefunctionsof T delay in termsof s .Equation4.8canbesolvedfor T delay byusingtheMaclaurin'sseriesexpansion.Wehaveapproximatedtheexpansionseriestosecondor dertermssoastoobtaina quadraticequationintermsof T delay ,whichcanbesolvedeasily.Higherordertermsof theMaclaurin'sseriescanbeincludedifmoreaccuracyisne eded.Theresultingmodel issimplebecauseitignoreswireinductanceandlimitsthen umberoftermsinMaclaurin series.4.2ExperimentalResults Wehaveperformedexperimentsbyusingthedelaymodelsdeve lopedinthischapter toanalyzetheiraccuracywhencomparedtotheresultsprovi dedinTable3.1.usingthe complexmodels.Theexperimentswereperformedusingthewi resizingmethodologyfor simultaneousoptimizationofinterconnectdelayandcross talknoisedevelopedinChapter 3.TheresultsincorporatingthenewmodelsaregiveninTabl e4.1.. Table4.1.showstheexperimentalresultsforgametheoreti capproachusingthedelay modelsdenedinEquation4.8.TherstthreecolumnsofTabl e4.1.representsthesame valuesindicatedinthecorrespondingcolumnsofTable3.1. .Fourthcolumnindicatesthe runtimeofthegametheoreticwiresizesolver.Columnsvea ndsevenindicatetheaverage 52

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Table4.1.ExperimentalResultsforGameTheoreticApproac hUsingtheDevelopedFast ModelsDuringWireSizing OpencoreDesign TotalNets DieArea( mm 2 ) Runtime(mins) %Delay %Noise Savings Savings Avg. Max. Avg. Max. Mult 854 0.199 < 1 8.06 13.32 11.31 16.86 PCIbus 19520 0.434 3.42 17.23 35.68 19.24 36.12 SerialATA 43563 1.624 8.62 26.58 42.12 16.83 45.33 RISC 61468 2.102 10.68 22.12 44.36 20.46 31.16 AVR P 78770 11.103 14.23 24.45 38.78 33.26 47.38 P16C55 C 102021 19.984 19.34 17.68 36.57 40.12 51.46 T80 C 157850 30.388 26.54 22.45 39.18 32.48 47.19 Average 19.8 24.81 Theresultsindicatedforgametheoreticapproachhasnoare aoverhead. Thepercentagevaluesindicatedarewithrespecttoplaceda ndrouted designwithoutwiresizing. 1 TableLegend:Avg:Averagesavingsofallthenetsintheenti redesign; Crit:Savingsonthecriticalpathnetofthedesign;Runtime:indicatestherunningtimeofeachalgorithm. interconnectdelayandcrosstalknoisesavingsrespective lyforallthenetsinthedesign. Columnssixandeightindicatethesavingsofthenetyieldin gmaximumgainintermsof interconnectdelayandcrosstalknoiserespectively.Comp aringthegametheoreticresults fromTables3.1.and4.1.,itcanbenoticedthatthetransmis sionlinemodelsdeveloped inthischapterforinterconnectdelayhavesucientaccura cyandimprovedruntimes, andhencecanbeusedinoptimizationproblemsrequiringext ensiveinterconnectdelay computationsonnetsegments. 53

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CHAPTER5 GATESIZING Inthischapter,wedevelopaframeworkformulti-metricopt imizationwhichiscapable ofoptimizingvariousconrictingdesignparameters.Wemod elthepost-routegatesizing forsimultaneousinterconnectdelayandcrosstalknoiseop timizationasagametheoretic optimizationproblemandsolveitusingNashequilibriumth eory.Thecrosstalknoise inducedonanetdependsonthesizeofitsdrivergateandthes izeofthegatesdriving itscouplednets.Increasingthegatesizeofthedriverincr easesthenoiseinducedbythe netonitscouplednets,whereasincreasingthesizeofthedr iversofcouplednetsincreases thenoiseinducedonthenetitself,resultinginacyclicord erdependencyleadingtoa conrictingsituation.Gametheoryinherentlymodelstheco mpetitionandiswellsuited forconrictingsituations.Thegatesofthedesignaremodel edastheplayers,thepossible setofgatesizesforeachgateismodeledasthestrategyset, andthenormalizedgeometric meanofinterconnectdelayandcrosstalknoiseismodeledas thepayofunctionofthe normalformgame.Wehaveimplementedtwodierentstrategi esinwhichgamesare orderedaccordingto(i)thenoisecriticality,and(ii)del aycriticalityofnets.Thetimeand spacecomplexitiesoftheproposedgatesizingalgorithmar elinearintermsofthenumber ofgatesinthedesign.Also,wehaveprovidedamathematical proofofexistenceforNash equilibriumsolutionfortheproposedgatesizingformulat ion. 5.1ProblemDenition Theproblemofgatesizingcanbedenedasndingtheoptimal sizesforallgatesinthe circuitsuchthattheoverallinterconnecteects(delayan dcrosstalknoiseinthischapter) areminimizedwithoutneedforreroutingorincreaseinarea overhead.Thecoupling 54

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capacitanceisresponsibleforthemajorityofthedeepsubm icroneects.Hence,itis importanttoextractthecouplingcapacitanceofnetswithh ighaccuracy.Thecoupling capacitanceofanetdependsthelengthofoverlapandspacin gbetweenadjacentnets.This informationcanbeecientlyextractedatpost-routingpha se.Thecouplingnoiseinduced onanetdependsonthesizeofitsdriver,drivenandaggresso rgates.Also,theinterconnect delayisafunctionofthegatesizes,andtheinputandloadca pacitances.Equations2.3 and2.10emphasizethatthegatesizesdirectlycontrolthei nterconnectdelayandcrosstalk noiseintermsofthedriverresistances,gateandcouplingc apacitances.Hence,calculating theoptimalgatesizescaneectivelyreducethecrosstalkn oiseandinterconnectdelayin deepsubmicrondesigns.Gatesizingcanbeperformedatpost -routelevelbyutilizingthe existingll-space.Inourapproach,weincrementallyscal ethegatesizestoutilizethe availablell-spacesuchthattheroutedresourcesinadjac entregionsarenotdisturbed. Hence,ourapproachwillneitherresultinareaoverheadnor needre-routingofthedesign. 5.2MotivationforGateSizingUsingGameTheoryModel Thecouplingnoiseinducedonanetdependsonthesizeofthev ictimandtheaggressor gates.Whenthesizeofthevictimgateisincreased,thecros stalknoiseonthevictimnet decreases,butincreasesthenoiseinducedbyitontheaggre ssornets.Hence,theaggressor gatesneedtobesized-upinordertoreducetheeectofsized -upvictimdriver.Increasing thesizeofaggressorswillincreasethenoiseinducedonthe victimnet,resultingina cyclicorderdependencyleadingtoaconrictingsituation. Itispointedoutin[10]that solvingthepost-routegatesizingproblemforcrosstalkno iseoptimizationisdicultdue tothisconrictingnatureoftheproblem.Itispossibletode velopaframeworkbasedon gametheorywhichlendsitselfwelltomodelingsuchconrict ingsituations.Inagame theoreticmodelinvolvingconvexpayofunctions,ithasbe enshownin[16]thattheNash equilibriumsolutionalwaysexistsandtendstoyieldgloba llyoptimalsolutions[17]. Asthesizeofagateincreases,theinterconnectdelaythrou ghthedrivennetdecreases andthecrosstalknoiseinducedontheadjacentnetsincreas es(aconvexfunction).Tra55

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ditionally,thisproblemismodeledusingcrosstalknoisea stheobjectivefunction,while maintaininginterconnectdelayasaconstraintorvicevers a.Throughuseofthegame theoreticformulationandtheNashequilibriumfunction,i tispossibletoachievesimultaneousoptimizationofmultiplemetricswithconrictingo bjectives.Since,interconnect delayandcrosstalknoisewithinacircuitareconrictingin nature,leadingtoaconvex objectivefunction,andaconvexobjectivefunctionisareq uirementfortheNashequilibriumfunctiontoyieldgoodresults,thereisagoodmotivati on.Theperformanceofthe proposedalgorithmiscomparedwiththatofsimulatedannea lingandgeneticsearchin ordertoillustratetheeleganceofgametheoreticsolution sforproblemswithconricting objectives.ItisshowninSection5.7thattheproposedappr oachyieldsbetterresultsthan simulatedannealing,geneticsearch,andLagrangianrelax ationundertheassumptionsof thesamemodels,setup,parametersandtheobjectivefuncti on. 5.3GameTheoreticGateSizingforMulti-metricOptimizati on Inthissection,weformulateanddevelopamethodologyfors imultaneousoptimization ofinterconnectdelayandcrosstalknoiseusinggatesizing .Theproblemofsimultaneously optimizinginterconnectdelayandcrosstalknoiseisbeing attemptedforthersttime.This problemiscomplextosolvebecauseoftheconrictingnature oftheinterconnectdelayand crosstalkinanygivencircuit.Givenaplacedandrouteddes ign,wemodelaone-shotgame foreachinterconnectnetandsolveittosizeaparticularnu mberofgatesassociatedwith thegame.Theorderinwhichthenetsarechosentocreatetheo ne-shotgamesiscriticalin decidingthepercentageoptimizationachievedintermsofi nterconnectdelayandcrosstalk noise.Theinterconnectnetscaneitherbeorderedaccordin gtoitsnoisecriticalityorthe delaycriticalitywithrespecttotheothernetsinthedesig n.Wehaveinvestigatedthese twotypesoforderingsanddevelopedagametheoreticframew orkasgivenbelow. 56

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5.3.1Approach1:GatesOrderedBasedonNoiseCriticality Wedevelopanoptimizationtaskwhichcanbeperformedafter theplaceandroutephase ofthegivengate-levelnetlist.Theinterconnectresistan ce,capacitance,inductance,and thesetofaggressordriversalongwiththeircouplingcapac itanceisextractedforeachnet fromtheSPEFnetlistoftherouteddesign.Weextractthelen gthofinterconnectwires, thelengthofoverlapsofeachnetwithitssetofaggressorne tsandtheirwirespacings fromtherouteddesignexportedinDEFformat.Themulti-ter minalnetisconsidered asdierentnetswithsamedriveranddierentreceivers.Th egatesofthedesignare orderedaccordingtothenoisecriticalityofthedrivennet s.Recentworksongatesizing forcrosstalknoiseoptimization[10,31],employacrossta lknoiseestimatorintheirnoise optimizationenginesforidentifyingthenoisecriticalne tsofthegivendesign.Thisisa timeconsumingprocess.Inthiswork,ratherthanestimatin gthenoiseinducedoneach net,werankthenetsrelatively,toindicatewhetheranetis morenoisecriticalthanan othernetorviceversa.Thecouplingcapacitancebetweenan ytwonetsisproportionalto thelengthoftheiroverlapandinverselyproportionaltoth esquareofthedistanceoftheir separation[57].Hence,foreachnet,wedeneascoreas X 8 aggressors (lengthoftheiroverlap) (spacing) 2 Thenetswithhighscorevaluesarerankedhighertoindicate thattheyaremorenoise critical.Thenetsaresortedinalistaccordingtotheirsco revalues.Themostcritical netwillformtheheadofthelistwhiletheleastcriticalnet willformitstail.Thegates areconsideredfortheirsizeoptimizationintheorderofth eranksoftheirdrivennets. ReferringtotheEquation2.3,theinterconnectdelayofane tdependsonthesizeofits driverandreceivergates.Thecrosstalknoiseinducedonan etdependsonthesizeof allaggressorgatesanditsvictimdriver.Foranygivennet, therecanbemanypotential aggressorgates.Itisindicatedin[10]thatitisvirtually notpossibletoconsiderthenoise eectsofalltheaggressorgatesonagivennet.Hence,wecon sidertheeectsoftwo 57

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mostaectingaggressorgateswhilesizingthegatesrelate dtothegivennet.However,the algorithmdevelopedinthisworkisnotlimitedtotwoaggres sorgatesandcanbeeasily extendedtoconsidertheeectsofmorethantwoaggressorga tesforeachnet.Inaddition, weshowfromexperimentalresultsthatconsiderationoftwo mostaectingaggressorgates issucienttotakecouplingeectsintoaccount. Agameismodeledforeachnetintheorderoftheirsortedlist .Thegamecreatedisa4playergamewhoseplayersarethedriver,thereceiverandth etwomostaectingaggressor gatesofthechosennet.Thetwomostaectingaggressorgate sarechosenamongitspool ofaggressorsbasedonthefractionoftheircontributionto thescoreofthegivennet.The twoaggressorgateswhichcontributetothemajorityofthes corevalueareselected.We haveusednormalformformulationtomathematicallyrepres entandsolvethegame.A normalformgameconsistsofasetof N playerslabeled1 ; 2 ;:::;N ,suchthateachplayer i has:(i)achoiceset S i calledstrategysetofplayer i ;itselementsarecalledstrategies, and(ii)apayofunction P i : S 1 S 2 :::: S N !< ,assignedtoeachstrategychosen bytheplayer i withrespecttootherplayers.Thestrategysetandthepayo matrixofall theindividualplayersaresucienttosolvethenormalform game.Alltheplayersplay simultaneouslywithoutanyknowledgeaboutothertheplaye rs'actions.Inotherwords, eachplayersimultaneouslychoosesastrategy s i 2 S i suchthatthecorrespondingpayo isminimizedwithrespecttothepayosoftheotherplayers. Theequilibriumsolutionof thegameiscomputedusingtheNashequilibriumtheory. Thestrategysetforeachgateismodeledasthesetofvarious possiblegatesizeswith whichitcanbescaled.Thescalablegatesizesforeachgatea rechosensuchthatits replacementinthedesigndoesnotresultinre-routing.The maximumscalablegatesize dependsontwofactors:(i)availablefreespacesurroundin gthegateinthedesign,and(ii) drivestrengthsavailableforagatetypeinthestandardcel llibrary.Foragateatspecic locationinthedesign,allthegatesizessupportedbythest andardcelllibrarycannotbe usedasitsstrategies.Someofthegatesizescannotbetted withintheavailablefree spacewithoutdisturbingtheroutednetssurroundingit.He nce,thenumberofstrategies 58

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Algorithm5.1.GateSizingAlgorithmforInterconnectDela yandCrosstalkNoiseOptimization Input: Placedandrouteddesign Output: Optimizedgatesizes Algorithm:extractthenetparasiticsfromSPEFleforallgatesdo determineaggressors(); %extractaggressorgatesfromSPEFle markthegateasun-played endforforallnetsdo extracttheoverlappinglengthsandspacingbetweentheadj acentnetsfromDEFle calculatescores();sortscores(); endforwhilethereexistsanunsizedgatedo selectanun-playednoisecriticalnet i fromthesortedlist identifytwomainaggressorgatesfornet i intheorderoftheircouplingeects createa4-playergamewithdriver,receiver,andtwomainag gressorgatesofnet i forgate g k amongthefourplayersdo if g k ismarkedassizedthen strategysetof g k calculatedNashsize; else strategysetof g k determinestrategies(); endif endforcost-matrix payo(fourplayers,strategies) %forpayofunction,seeAlgorithm5.2. optimized-size nash-solution(fourplayers,payos) %forNashequilibriumsolution,seeAlgorithm5.3. markthefourplayedgatesassizedmarkthenet i asplayed endwhilereturn: optimizedNashsizesofgates availableforeachgateisalwayslessthanthenumberofdriv estrengthsavailableforthat gatetypeinthestandardlibrary.Theminimumsizeofthegat esissettominimumdrive strengthavailableinthestandardcelllibrary.Ifagatein volvedinthecurrentgameis markedassizeddueitsparticipationinearlierplayedgame s,itsstrategysetismodeledas asingletonsetconsistingofitscalculatedNashwidth.The strategysetforotherplayers 59

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involvedinthegameismodeledastheavailablegatesizesin thestandardcelllibrary betweenitsminimumandmaximumgatesizes. Thepayofunctiontriestocapturetheinteractionbetween thefourgatesidentiedas theplayersofthegame.Forthechosennet,itsdelayDandthe maximumcrosstalknoise NarecalculatedusingtheEquations2.3,and2.7respective ly.Thesevaluesarecalculated forallstrategiesofagateconsideringthestrategiesofot herplayersofthegame.The delayandcrosstalknoisevaluesobtainedforeachstrategy ofaplayerarenormalizedwith respecttoaparticularstrategy.Thenormalizationisperf ormedtotransformthedelayand crosstalknoisevaluesintodimensionlessquantitiessoth attheycanbeeasilycorrelated witheachother.Thepayofunctionismodeledasthegeometr icmeanofnormalizeddelay andnoisevaluesoftheplayerssoastogiveequalweighttobo thinterconnectdelayand crosstalknoise. Algorithm5.2.AlgorithmforPayoMatrixCalculation Input: NumberofPlayers N ,Strategyset S Output: Payomatrix forallplayers i 2 1to N do forallstrategycombinations S j = f s j1 ;:::;s jN g ,where( s j1 2 S 1 ) ;:::; ( s jN 2 S N )do calculatethedelayusingEquation2.3normalizethedelayw.r.trststrategycombinationcalculatethecrosstalknoiseusingEquation2.7normalizethenoisew.r.trststrategycombinationP [ i S j ] Geometricmeanofnormalizednoiseanddelay endfor endforreturn: payomatrix P 8 strategycombinations TheNashequilibriumisevaluatedforthechosennetandtheg ameisplayedout.The fourgatesthatparticipatedintheplayedgameareraggedas \sized"andtheirsizesare setequaltothecalculatedNashsizes.Thechosennet,forwh ichthegameisplayedout, istaggedasplayednetandisremovedfromthesortedlistofo rderednets.Anewnet locatedattheheadofthesortedlistisselectedtoplaythen extgame.Thefourplayers correspondingtotheselectednetareidentied.Thestrate gysetofthegateswhichare markedas\sized"aredenedasasingletonconsistingofonl yitscalculatedNashwidth. 60

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Thestrategysetsforotherplayersofthegameareidentied asdescribedaboveand theNashequilibriumofthegameisevaluated.Thisprocesso fcreatingandplayingthe sequentialgamesisrepeateduntilallthegatesofthedesig naremarkedassized.Itcan benotedthatthenumberofgamesplayedisalwayslessthanth etotalnumberofgatesin thedesign.Thepseudo-codeofthecompletegatesizingalgo rithmdevelopedinthiswork isshowninAlgorithm5.1.. Algorithm5.3.AlgorithmforNashEquilibriumSolution Input: Numberofplayers N ,Payomatrix P Output: Nashsolution forallplayers i do forallpayosofplayer i do nd s i suchthat, P i ( s 1 ;:::;s i ;:::;s N ) P i ( s 1 ;:::;s i ;:::;s N ) s i istheNashstrategyforplayer i endfor endforNash-solution S = f s 1 ;:::;s N g %setofoptimizedstrategiesforall N players return: Nashsolution S 5.3.2Approach2:GatesOrderedBasedonDelayCriticality Theorderingofnetsinthesortedlistdictatestheorderinw hichthegatesareconsidered fortheirsizeoptimization.Insection5.3.1,theintercon nectwiresaresortedinalistbased onthenoisecriticalityofthenets.Hence,theapproachout linedinsection5.3.1,yields slightlybetteroptimizationofcrosstalknoisethaninter connectdelay,whilesimultaneously optimizingbothdelayandnoise.Inthissection,weinvesti gateastrategywhereindelay isconsideredashighercriticalitythannoise,whilesimul taneouslyoptimizingdelayand noise.Itisinterestingtonotethatbothmethodsyieldsign icantlybetteroptimizationof bothdelayandnoisecomparedtoothermethods.Thedesigner canchooseeitherofthe strategiesbasedontheneed.Thedierencebetweenthetwos trategiesisthewayinwhich thesortedlistiscreated.Afterthedesignisplacedandrou ted,thepathdelaysofallthe pathsinthedesignareestimated,andaresortedintoalistb asedontheirdelaycriticality. 61

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Themostdelaycriticalpathischosentocreategamesforgat esizeoptimization.The gamesarecreatedforeachnetinthechosenpathintheorderf romitsprimaryoutputto primaryinputs.Asanexampleforillustration,considerth echosenpathtoconsistoffour gates: A;B;C and D ,insuccessivetransitionconnectedwithnets:1 ; 2and3,respectively. Thegate A isdrivenbyprimaryinputsandgate D drivesaprimaryoutput.Inorderto considerthedown-streamloadcapacitance,thenet3connec tinggates C and D should beoptimizedbeforethenets1and2.Thus,thegamesareplaye dintheorderofnet3 followedbynet2followedbynet1.Thegameformulatedforea chnetinvolvesitsdriver, receiveranditstwomostaectingaggressorgatesasitspla yers.Thetwomostaecting aggressorgatesforthenetanditsstrategiesareidentied asindicatedinsection5.3.1. Afterthegamesareplayedforallthenetsofthechosencriti calpathinitsdirectionof primaryoutputtoprimaryinputs,thenextcriticalpathint hesortedlistisselectedto playgames.Thisprocessofcreatinggamesisrepeateduntil thesortedlistisempty. 5.4TimeandSpaceComplexityofProposedGateSizingAlgori thms TheworstcasetimecomplexityofevaluatingNashequilibri umforageneralM-player gamewith S strategiesforeachplayerisgivenas O ( M S M )[52].ReferringtoSection 5.3.1,wehavemodeledtheproblemofgatesizingforsimulta neousinterconnectdelayand crosstalknoisereductionasagamewithfourplayers.Forea chgatetype,thenumberof dierentdrivestrengthsavailableforitinthestandardce lllibraryactasitsmaximum numberofstrategies.Wehaveusedastandardcelllibraryco ntaininggateswithfour dierentdrivestrengths,builtonTSMC180nmdesignrules. Eventhough,therearefour dierentdrivestrengthsavailableinthelibrary,thescal ablesizesforeachgatedepends onthelocationandthefreespacesurroundingitinthedesig n.Hence,thenumberof strategiesavailableforeachgateislessthanorequaltofo ur.Thus,thecomplexityof calculatingtheNashequilibriumforasinglegameplayedis givenby O (4 4 4 ).Thegames areplayedrepeatedlyuntilallthegatesaresized.Inanyga me,ifagateismarkedassized, thenitsstrategysetismodeledtohavesinglestrategycons istingofitscalculatedNash 62

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size.Whenagameisplayed,theoutcomeisndingthebestgat esizesforalltheplayers whicharenotsizedbefore.Thenumberofgamesplayedisless thanthetotalnumberof gatesinthedesignbecauseaftereachgameisplayedout,atl eastonenewgatewillbe sized.Hence,theoveralltimecomplexityofallthegamespl ayedcanbemathematically givenas O N gates 4 4 4 O ( N gates ) where N gates isthetotalnumberofgatesinthegivendesign.Itcanbenoti cedthatthe timecomplexityoftheproposedalgorithmis linear andisproportionaltothetotalnumber ofgatesinthedesign. Thespacecomplexityoftheproposedalgorithmisdependent entirelyonthenumber ofgatesinthedesignandthespacecomplexityofthepayoma trix.Thespacecomplexity ofapayomatrixdependsonthenumberofstrategiesforeach playerplayingthegame. Asthegamesareplayedsequentially,thetotalspacerequir edbyallthegamesisequalto thespacecomplexityofagameinvolvingplayerswiththemax imumnumberofstrategies. Mathematically,thespacecomplexityrequiredbyallthepa yomatricesisgivenas O ( S 1 S 2 S 3 S 4 ) O (4 4 4 4),where S 1 ;S 2 S 3 ,and S 4 arestrategysetsof4-player gameinvolvingtheplayerswithmaximumstrategies.Hence, thespacecomplexityofthe proposedalgorithmisgivenas O ( N gates +4 4 4 4) O ( N gates ) 5.5ProofofExistenceofNashEquilibriumfortheProposedG ateSizing Formulation Inthissection,weprovidetheproofofexistenceofNashequ ilibriuminthecaseofgate sizingproblemforsimultaneousoptimizationofinterconn ectdelayandcrosstalknoise. Asthegatesizeofagateincreases,theinterconnectdelayd ecreasesandthecoupling capacitanceincreasesresultinginaconvexpayofunction .Let G = f S 1 ;:::;S n ; f 1 ;:::;f n g 63

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beagamewitheachplayer i 2 N havingastrategyset S i containingitspossiblegatesizes anditspayogivenby f i .Wehavemodeledthestrategyset S i foreachplayerasanonempty,compactsetofanitedimensionalEuclideanspace.B ecauseoftheconvexnature oftheinterconnectdelayandcrosstalknoise,themodeledp ayofunction f i becomesupper semicontinuouson S = Ni =1 S i andforanyxed u i 2 S i ,thefunction f i ( u i ;: )isalower semicontinuouson S ( i ) [16].Forany u 2 S ,thebestreplyortheexpectedpayo B i ( u ) isalsoconvex.AccordingtoKakutani'sxedpointtheorem[ 61],thegameGhasatleast oneNashequilibriumpointifthegraph G B = f ( x;y ): x 2 S;y 2 B ( x ) g isclosed. Letsassumethatitisnotclosed.Then, 9 ( x 0 ;y 0 ) = 2 G B ,suchthateveryneighborhood (in S S )of( x 0 ;y 0 )containsapointof G B x 0 isagatesize,ithastobeoneofthosefromthesetofpossible gatesizesforthe givenplayerinordertosatisfytheDRCrulesoftheusedproc esstechnology. ) x 0 2 S ) y 0 = 2 B ( x 0 ) Inotherwords,foratleastonegateplayingthegame(saygat e1),thereisan y 1 1 2 S 1 suchthat f 1 ( y 1 1 ;x 02 ;:::x 0n ) f 1 ( y 0 1 ;x 02 ;:::x 0n )(5.1) Let F beafunctionsuchthat F : S 2 !< andgivenas F ( x;y )= f 1 ( y 1 1 ;x 2 ;:::x n ) f 1 ( y 1 ;x 2 ;:::x n ) Since f i isuppersemicontinuouson S and f i ( u i ;: )islowersemicontinuouson S i ,Fis lowersemicontinuousand C = f ( x;y ) 2 S 2 : F ( x;y ) 0 g isclosed.Hence,forany ( x; y ) 2 G B ;F ( x; y ) 0.But,byEquation5.1, F ( x 0 ;y 0 ) 0,contradictingtheclosedness ofC.Thus,thereisapoint s 2 S suchthat s 2 B ( s ),whichisaNashequilibriumpoint. 64

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Crosstalk Noise Estimator ModelingWith calculated gate sizesScript Payoff Function DEF Update (StarRCXT, SignalStrom, Celtic)Verilog Design Cadence First Encounter Script Net Extractor and Crosstalk Noise Delay Models Game Theoretic Based Gate Size SolverPlaced and Routed Design (DEF Format)Payoff function Optimized Gate Sizes Gate Size Optimized DEF This WorkGate-Sized Delay and Crosstalk Noise Optimized Circuit Standard Cell LEF Library with differnet strengths, Cell Timing Library, Cell Noise ModelsDelay Calculator Extracted Net InformationRC ExtractorFigure5.1.IntegrationofProposedGateSizingAlgorithmi ntheDesignFlow 5.6DesignFlow Thedesignrowforobtaininganoptimallygatesizedcircuit fromaverilog/VHDL descriptionisshowninFigure5.1..Thebehavioralverilog /VHDLdescriptionissynthesized ontoalibraryofstandardcellsandgivenasinputtothedesi gnrow.Thestandardcells areplacedandroutedinaccordancewiththesynthesizedcod eusinganystandardcellplace androutetool.WehaveusedtheFirstEncounterRTL-to-GDSIItoolfromCadenceDesignSystemstoperformtheplacementandroutingofgateleveldesign.Theparasitics fromtherouteddesignareexportedinSPEFformatwiththehe lpofStarRCXTfrom SynopsysInc.AlexandyaccscriptisdevelopedtoreadtheSP EFnetlistandextractthe valuesofinterconnectresistance,capacitance,inductan ceandcouplingcapacitancesalong withtheirsetofaggressorgatesforallthenetsofthedesig n.Also,agawkscriptiswritten 65

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toextracttheinformationaboutthelengthofoverlapsbetw eentwogivennetsalongwith theirdistanceofseparationfromtheDEFnetlist.Themodel susedforInterconnectdelay andcrosstalknoise,andthemodelingofthepayofunctiona redescribedinSections2.5 and5.3.1respectively.Thepayofunctionisusedbythegam etheoreticbasedgatesize solverdescribedinAlgorithm5.1.tominimizetheintercon nectdelayandcrosstalknoise ofindividualnetsintheorderoftheirnoisecriticality.T heoptimizedgatesizesresulted fromoursolverareusedtoupdatetherouteddesign.Wehaved evelopedanothergawk scriptwhichscalesthegatesintheoriginalDEFrouteddesi gnaccordingtotheircalculated optimizedgatesizes.Itshouldbenotedherethattheresult ingoptimizeddesigndoesnot requirere-routingsincethepossiblegatesizesdraftedfo reachgatearewithinitslimitsto satisfythedesignrulesoftheusedprocesstechnology.5.7ExperimentalResults ThegametheoreticgatesizesolverdescribedinAlgorithm5 .1.wasimplementedinC andexecutedonaUltraSPARC-IIe650MHz,512MBSunBlade150 systemrunningSolaris 2.8.TheASICdesignsonwhichwetestedouralgorithmwereob tainedfromOpencores [62].Astandardcelllibrarycontaining10logiccellswith 4dierentdrivestrengthsbased ona6-Metallayer,180nmtechnologyhasbeendevelopedandu sed.ASICdesigns,written inbehavioralVHDL/VerilogareconvertedtostructuralVHD L/Verilogusingthestandard cellsinthelibrarywiththehelpofBuildGates,anRTLsynth esistoolofCadencedesign systems.WehavemodiedtheASICdesignssuchthatallthebl ocksinthedesignare rattenedtostandardcellsinthelibrarywithoutmaintaini ngthehierarchy.Theon-chip memorymodulesarerealizedasregisterarrayswithD-ripro pasbasicbuildingunits.The structuralVHDL/VerilogdesignisusedasinputbyCadenceF irstEncountertodevelop theroorplan.Wehavesettheoptionofrowutilizationto70% forallthedesignssoasto allowsomegatesizescaling.Thedesignisthenplacedandro utedusingAmoebaplaceand Nanorouterespectively,whicharepartofCadenceFirstEnc ountertool.Thenalplaced androuteddesignisthenexportedinDEFformat. 66

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Theparasiticinformationfromtherouteddesignisextract edinSPEFformatusing SynopsysStarRCXT.Theinterconnectresistance,intercon nectcapacitance,interconnect inductance,couplingcapacitancesalongwiththeiraggres sordriversisextractedfromthe SPEFleandisgivenasinputtothegametheoreticgatesizes olver.Thelengthofoverlap withtheaggressornetsandtheirspacingisextractedfromt heDEFleandisalsogiven asinput.Thecalculatedgatesizeforeachgateisusedtoupd atetheoriginalDEFleto generateanoptimizedDEFle.Itcanbenotedthattheoptimi zedDEFleiscreated withthehelpofagawkscriptandveriedforDRCrules.Thede signisnotreroutedto generatetheoptimizedDEFle.Theinterconnectdelayandc rosstalknoiseareestimated usingCadenceSignalstormandCelticICtoolsrespectively withtheirrobustmodels,and notusingtheanalyticalmodelsusedinthedissertation. Theworksreportedinliteraturesolvetheproblemofgatesi zingforcrosstalknoise optimizationunderdelayconstraints.Inthiswork,wehave solvedtheproblemofgate sizingforsimultaneousoptimizationofcrosstalknoisean dinterconnectdelay.Hence,in ordertocompareourresults,wehaveimplementedsimulated annealingandgeneticsearch forsimultaneousoptimizationofcrosstalknoiseandinter connectdelay,andexecutediton sameSolarismachinewithsamesetofinputsandparameters. Thesimulatedannealing algorithmwasimplementedandweexperimentedtoobtainthe bestresultsintermsof optimizationofinterconnectdelayandcrosstalknoise.Th esetofpossiblegatesizesis calculatedasindicatedinSection5.3.1.Ineachmoveofthe simulatedannealingprocess, agateisrandomlyselectedanditssizeisrandomlyassigned fromthesetofitspossible gatesizes.Thecostfunctionisdenedasthegeometricmean ofinterconnectdelayand crosstalknoisesummedoverallthenets.Theinitialtemper atureisdeterminedbynding theaveragechangeinthecostforasetofrandommovesfromth estartingconguration andselectingthetemperaturewhichleadstoanacceptproba bilityof0.95.Thenumber ofmovespertemperatureforeachdesignissetto20timesthe totalnumberofgatesin thedesign.Thisisdonesoastoallowatleast10to15moveson theaverageforeachgate beforesettlingforitssolution.Theup-hillmovesareacce ptedwithaprobabilityof e C T 67

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where C isthechangeinthecostandTisthecurrenttemperatureofth eiteration.The temperatureiscooledattherateof0.95. Thegatesizingproblemforsimultaneousoptimizationofin terconnectdelayandcrosstalk noiseismodeledasageneticsearchmechanismandsolvedusi ngGALib[64].Theinitial populationcontainsthegatesrepresentedbytheircorresp ondinggatesizesasusedinthe originalunsizeddesign.Eachindividualinthepopulation ,calledachromosome,isrepresentedasasetoftwointegersindicatingthegatenumbera ndthegatesizeassigned toit.Thechromosomesevolvethroughsuccessiveiteration scalledgenerations.During eachgeneration,thechromosomesareevaluatedfortheirt nesstest.Wehavedenedthe tnesscriterionasthedeviationofthecrosstalknoiseand interconnectdelayofeachgate fromtheirworst-casevalues.Thechromosomeswithlowerva luesofcrosstalknoiseand interconnectdelayaregivenhighertnessvalues.Wehaveu sedthesteady-stategenetic algorithmavailableasapartofGALiblibrarytogenerateov erlappingpopulationswhich retainsits30%ofttestchromosomesinitsnewgenerations .Themutationprocessfor achromosomeisdenedasrandomlyselectingagatesizefrom itssetofpossiblegate sizes.Thenewchromosomesarecreatedusingsinglepointcr ossoverandarevalidated againsttheirsetofpossiblegatesizes.Theselectionproc essofachromosomeisadopted bytheroulettewheelselectionapproach.Wehavesetthecon vergence-of-populationas thestoppingmeasurefortheevolutionofgenerations. ExperimentalresultsareprovidedinTables5.1.and5.2..T able5.1.indicatetheaverage interconnectdelayandcrosstalknoisesavingsofallnetso fthedesign,whereastheTable 5.2.indicatesthecriticalpathsavingsintermsofinterco nnectdelayandcrosstalknoise. ReferringtoTable5.1.,therstcolumnindicatesthenameo ftheopencoredesignandthe secondcolumnindicatesthecorrespondingnumberofgatesi nthedesign.Columnsthree, six,nine,andtwelveindicatetherunningtimesofgenetics earch,simulatedannealing, gametheoreticapproachbasedonnoisecriticality(GT-NCa pproach)andgametheoretic approachbasedondelaycriticality(GT-DCapproach)respe ctively. 68

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Table5.1.AverageSavingsforSimultaneousOptimizationo fInterconnectDelayandCrosstalkNoiseDuring GateSizing OpenCoreDesign[62] TotalGates GSApproach 1 SAApproach 2 GT-NCApproach 3 GT-DCApproach 4 RunTime(mins) %Average RunTime(mins) %Average RunTime(mins) %Average RunTime(mins) %Average Savings Savings Savings Savings Delay Noise Delay Noise Delay Noise Delay Noise Mult 428 28.40 3.06 3.21 10.28 4.83 3.74 1.13 4.18 5.34 1.06 6.32 3.98 PCI 7882 158.41 5.29 4.64 38.83 8.14 10.68 4.29 10.39 16.49 4.16 12.11 13.89 ATA 21781 352.92 5.78 7.30 64.15 14.12 16.81 10.49 20.31 23.91 10.67 21.83 20.79 RISC 34172 587.12 4.72 7.91 79.61 12.92 13.95 13.31 15.28 14.05 13.46 16.72 13.48 AVR P 41274 716.72 8.31 8.92 112.48 15.67 16.78 15.79 21.41 22.96 15.13 24.41 18.63 P16C55 52128 1089.52 5.94 6.11 159.76 13.35 17.23 19.98 16.96 21.91 20.13 19.40 20.07 T80 C 69973 1426.32 6.43 7.86 220.57 17.42 21.63 27.67 19.86 25.24 27.14 20.74 24.83 Average 5.65 6.56 12.35 14.40 15.48 18.56 17.36 16.52 Noareaoverheadforallfourapproaches.Thepercentageval uesindicatedarew.r.tplacedandrouteddesignwithout gatesizing. 1 GSApproach:Geneticsearchbasedgatesizingforsimultane ousoptimizationofinterconnectdelayandcrosstalknoise 2 SAApproach:Simulatedannealingbasedgatesizingforsimu ltaneousoptimizationofinterconnectdelayandcrosstalk noise 3 GT-NCApproach:Gametheoreticgatesizingwithgatesorder edbasedonnoisecriticalityforsimultaneousoptimizatio n ofinterconnectdelayandcrosstalknoise 4 GT-DCApproach:Gametheoreticgatesizingwithgatesorder edbasedondelaycriticalityforsimultaneousoptimizatio n ofinterconnectdelayandcrosstalknoiseTableLegend : AverageSavings :Averagesavingsofallthenetsintheentiredesign; Runtime :runningtimeofeach algorithm.69

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Table5.2.CriticalPathSavingsforSimultaneousOptimiza tionofInterconnectDelayandCrosstalkNoiseDuring GateSizing OpenCoreDesign[62] TotalGates GSApproach 1 SAApproach 2 GT-NCApproach 3 GT-DCApproach 4 RunTime(mins) %Crit.Path RunTime(mins) %Crit.Path RunTime(mins) %Crit.Path RunTime(mins) %Crit.Path Savings Savings Savings Savings Delay Noise Delay Noise Delay Noise Delay Noise Mult 428 28.40 4.98 4.16 10.28 5.71 6.68 1.13 5.59 7.86 1.06 8.41 7.01 PCI 7882 158.41 7.84 6.02 38.83 14.29 17.51 4.29 17.11 25.29 4.16 20.30 24.12 ATA 21781 352.92 9.47 6.51 64.15 15.91 18.43 10.49 24.78 27.12 10.67 25.99 24.39 RISC 34172 587.12 4.39 8.10 79.61 19.41 17.50 13.31 21.16 19.96 13.46 23.09 17.71 AVR P 41274 716.72 7.49 9.91 112.48 20.69 22.54 15.79 27.19 28.95 15.13 35.52 22.31 P16C55 52128 1089.52 7.03 5.67 159.76 16.75 17.49 19.98 21.54 23.16 20.13 25.11 21.69 T80 C 69973 1426.32 3.95 10.64 220.57 20.78 25.41 27.67 24.21 28.92 27.14 26.81 26.22 Average 6.45 7.29 16.22 17.94 20.23 23.04 23.60 20.49 Fortablefootnotes,pleaserefertofootnotesgivenunderT able5.1.. Crit.PathSavings :Savingsonthecriticalnetof thedesign70

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ThecolumnsfourandveofTable5.1.indicatetheaveragein terconnectdelayand crosstalknoisesavingsforallthenetsofthedesignobtain edbygeneticsearchapproach. Columnssevenandeightrepresentthesevaluesforsimulate dannealingapproach,while columnstenandelevenindicateforGT-NCapproachandcolum nsthirteenandfourteen indicateforGT-DCapproachrespectively.ThecolumnsofTa ble5.2.representthesame correspondingvaluesobtainedforcriticalpathoftheresp ectivedesign.Theexperiments wereconductedsuchthattheareaoverheadiszeroinallfour approaches.ReferringtoTable5.1.,geneticsearchshows5.65%and6.56%ofaverageint erconnectdelayandcrosstalk noiseimprovements.Simulatedannealingshows12.35%and1 4.40%ofaverageinterconnectdelaycrosstalknoiseimprovements.Incomparison,GT -NCapproachshows15.48% and18.56%ofaverageinterconnectdelayandcrosstalknois eimprovements,whileGT-DC approachshowsimprovementsof17.36%and16.52%respectiv ely.ReferringtoTable5.2., geneticsearchresultsin6.45%and7.29%andsimulatedanne alingresultsin16.22%and 17.94%intermsofcriticalpathinterconnectdelayandcros stalknoisesavingsrespectively. Incomparison,GT-NCapproachyields20.23%and23.04%impr ovementsoncriticalnets intermsofinterconnectdelayandcrosstalknoise,whileGT -DCapproachyields23.60% and20.49%improvementsrespectively.Also,wehaveobserv edthatinterconnectpower consumptionfollowsthesametrendasthatofcrosstalknois e.Thedecreaseincoupling capacitanceresultsinasmallerswitchedcapacitanceandt herebywouldresultinlesser powerdissipation.Thegametheoreticgatesizesolver,ina dditiontooutperforminggeneticsearchandsimulatingannealingintermsofinterconn ectdelayandcrosstalknoise savings,hassignicantlysmallerruntimes.Hence,ourapp roachisscalableandfavorable tohandlethecomplexityoflargeSOCdesigns. Toenableadirectcomparisonofourworkwiththerecentwork reportedin[31],wehave modiedourgametheoreticapproach1,giveninSection5.3. 1tominimizethecrosstalk noiseunderdelayconstraints.Theworkdevelopedin[31]is aLagrangianrelaxationbased gatesizingapproachforreducingthecrosstalknoiseunder delayconstraints.Thedeveloped algorithmisiterativeandmakesuseofacouplinggraphdeve lopedbasedonthecoupling 71

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Table5.3.CrosstalkNoiseOptimizationUnderDelayConstr aints DuringGateSizing OpenCoreDesign[62] TotalGates NumberofNoiseViolations NoiseThreshold=0.15 V dd LagrangianBased[31] GameBased[ThisWork] Mult 428 7 5 PCI 7882 23 11 ATA 21781 97 26 RISC 34172 148 53 AVR P 41274 181 59 P16C55 C 52128 239 67 T80 C 69973 289 83 Here,wehaveusedthenumberofnoiseviolationsasthemetri c,sinceit isusedin[31]astheiralgorithmevaluationcriteria. capacitances.ForbothgametheoreticandLagrangianrelax ation[31]approaches,wehave usedthedelayvaluesobtainedfromsimultaneousoptimizat ionofinterconnectdelayand crosstalknoiseasthesetofdelayconstraints,soastoensu reatighterconstraintset forbothapproaches.TheLagrangianrelaxationproblemfor post-layoutcrosstalknoise reductionisformulatedasdevelopedin[31]andsolvedusin gLANCELOT[66].Thegame theoreticapproach1,giveninSection5.3.1,ismodiedsuc hthatthestrategieswhichdo notsatisfythedelayconstraintsareprunedoutfromtheirr espectivestrategysets.The payofunctionisthenmodiedtoaccountsolelyforcrossta lknoiseinducedonthenet underconsideration.Table5.3.showsthecomparisonofgam etheoreticandLagrangian relaxationbasedapproachesindicatedintermsofnumberof noiseviolationsforeachdesign. Noiseviolationsareexpressedasthenumberofnetswhichha veaninducednoiseexceeding athresholdnoisesetto0.15 V dd .Itcanseenthatourapproachresultsinsignicantlyfewer faultswhencomparedtotheLagrangianrelaxation. 72

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5.8Conclusions Gametheoryallowsthesimultaneousoptimizationofmultip lemetricsinthecontext ofconrictingobjectivesleadingtoaconvexobjectivefunc tionintheproblemformulation. Thisessentiallymakesitpossibletousegametheoryforsim ultaneousoptimizationof interconnectdelayandcrosstalknoise.Optimizingbothin terconnectdelayandcrosstalk noiseisextremelycriticalindeepsubmicronandnanoregim ecircuits.Theproposed methodresultsinalineartimealgorithmwithsignicantly betterresultsthangenetic search,simulatedannealingandLagrangianrelaxation,ma kingthisworkanimportant contribution. 73

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CHAPTER6 INTEGRATEDGATEANDWIRESIZING Inthischapter,wedevelopanewpost-layoutintegratedgat eandwiresizingalgorithm forsimultaneousoptimizationofinterconnectdelayandcr osstalknoise.Theproblemof post-layoutgateandwiresizingismodeledasanormalformg ameandsolvedusingNash equilibrium.Thecrosstalknoiseinducedonanetdependson itswiresize,itsdriversize andthesizesofgatesdrivingitscouplednets.Itisreporte din[10]thatsolvingtheproblem ofcrosstalknoiseoptimizationatpost-routelevelisdic ultduetothecyclicdependency, resultinginaconrictingsituation.Gametheoryprovidesa naturalframeworkforhandlingconrictingobjectivesandallowssimultaneousoptim izationofmultipleparameters. Theformulationofaconvexobjectivefunctionisarequirem entinordertoobtainbetteroptimizationinagametheoreticframework.Thisproper tyisexploitedtosolvethe cyclicdependencyofcrosstalknoiseonitsgateandwiresiz es,whilemodelingtheproblem ofsimultaneousoptimizationofinterconnectdelayandcro sstalknoise,whichagainare conrictinginnature.Agameismodeledwiththecrosstalkno iseandinterconnectdelay ofthechosennetastheplayers,thepossiblegateandwiresi zesasthestrategysetand theanalyticalexpressionsforcrosstalknoiseandinterco nnectdelayastheirrespective expectedpayos.Wehaveimplementedtwodierentstrategi esinwhichthegamesare orderedaccordingto(i)thenoisecriticality,and(ii)the delaycriticalityofnets.Thetime andspacecomplexityoftheproposedintegratedsizingalgo rithmislinearintermsofthe numberofgatesandwiresinthedesign. InChapters3and5,wehaveindependentlysolvedtheproblem ofsimultaneousoptimizationofinterconnectdelayandcrosstalknoiseusingwi resizingandgatesizingrespectively.InChapter3,wehavemodeledtheproblemofpostlayo utwiresizingasanumber 74

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of3-playergameswithwiresegmentsastheplayers,possibl ewiresizesasthestrategy setandnormalizedgeometricmeanofinterconnectdelayand crosstalknoiseasthepayo function.InChapter5,wehavemodeledtheproblemofpostla youtgatesizingasanumberof4-playergameswithgatesastheplayer,possiblegate sizesasthestrategysetand normalizedgeometricmeanofinterconnectdelayandcrosst alknoiseasthepayofunction. Oncontrary,inthischapter,weaddresstheproblemofinteg ratedgateandwiresizing forsimultaneousoptimizationofinterconnectdelayandcr osstalknoise.Themodelingof gamesfortheintegratedproblemiscompletelydierentfro mthatgiveninChapters3and 5.Here,wehavemodeledtheinterconnectdelayandcrosstal knoiseastheplayersofthe gameratherthanthegatesorthewiresegments.Hence,every gamecreatedisa2-player non-zerosumgame.Thismodelinghelpsinimprovingtherunt imesignicantly. 6.1ProblemDenition Theproblemofpost-layoutgateandwiresizingcanbedened asfollows:ndthe optimalgateandwiresizessuchthattheinterconnecteect s(interconnectdelayand crosstalknoiseinthiswork)areminimizedunderthegivena reaconstraintsandwithout theneedforreroutinganyofthenetsinthedesign.Theparas iticresistanceandcapacitance ofinterconnectwiresarehighlydependentonthewirewidth sandgatesizes.Thecoupling capacitanceisresponsibleforthemajorityofthedeepsubm icroneects.Thecoupling noiseinducedonanetdependsonitswiresize,andthegatesi zesofthedriver,receiver andaggressorgates.Theinterconnectdelayisalsoafuncti onofgateandwiresizes(see Equation2.3).Hence,calculatingtheoptimalgateandwire sizescaneectivelyreduce bothcrosstalknoiseandinterconnectdelayindeepsubmicr ondesigns.Gateandwire sizingcanbeperformedatpost-routelevelbyutilizingthe existingll-space.Inthiswork, weincrementallyscalethesizesofgatesandwirestoutiliz etheavailablell-space,such thattheroutedresourcesinadjacentregionsarenotdistur bed.Hence,ourapproachwill neitherresultinareaoverheadnorneedreroutingofthedes ign. 75

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6.2MotivationforIntegratedGateandWireSizing Thecouplingnoiseinducedonanetdependsonthewiresizeof thenetandthesizesof thevictimandaggressorgates.Whenthesizeofthevictimga teisincreased,thecrosstalk noiseonthevictimnetdecreases,butincreasesthenoisein ducedbyitontheaggressor nets.Hence,theaggressorgatesneedstobesized-uptoredu cetheeectsofsized-up victimdriver.Increasingthesizeofaggressorswillincre asethenoiseinducedonthe victimnet,hence,resultinginacyclicorderdependencyle adingtoaconrictingsituation. Similarly,changingthewiresizeofanetwillaectthewire sizesoftheneighboringnets, resultinginconrictingbehavioreveninthecaseofwiresiz ing.Itisreportedin[10]that solvingthecrosstalknoiseoptimizationproblematpost-r outelevelisdicultduetothis conrictingnatureoftheproblem.Gametheoryprovidesanat uralframeworkforhandling suchconrictingsituations.Asthesizeofagateandwireinc reases,theinterconnectdelay throughthedrivennetdecreasesandthecrosstalknoiseind ucedontheadjacentnets increases(convexpayofunction).Inagameinvolvingconv expayofunctions,thegame theoryworksbetter[16]andNashequilibriumsolutionalwa ysexistsandtendstoachieve globaloptimalsolutions[17].Ithasbeenshownin[17]that thecomplexityofdetermining theNashequilibriumliesbetweenPandNPdependingonthepr oblemformulation.It isshowninSection6.7thattheproposedapproachyieldbett erresultsthansimulated annealingandLagrangianrelaxationundertheassumptions ofthesamemodels,setup, parametersandtheobjectivefunction.6.3ANewApproachtoIntegratedGateandWireSizing Inthissection,weformulateanddevelopamethodologyfors imultaneousoptimization ofinterconnectdelayandcrosstalknoiseusinggateandwir esizing.Theproblemofsimultaneousoptimizationofinterconnectdelayandcrosstalkn oisecanbesolvedindependently usinggatesizingfollowedbywiresizingorwiresizingfoll owedbygatesizingwiththehelp ofmethodologiesdevelopedinChapters5and3.However,itw ouldbemoreadvantageous 76

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tosolvethegateandwiresizingproblemsinanintegratedfr ameworkratherthanina sequentialframework.Insequentialframework,sincethea lgorithmsforgateandwire sizingareappliedinsequence,oneaftertheother,thefull optimizationpotentialofthe individualalgorithmscannotbeachieved.Thealgorithmap pliedrstwouldhavemore resourcesavailableforittooptimizethanthefollowingal gorithmsandhence,wouldresult insub-optimalsolutionsforalgorithmsappliedfollowing therst.Givenaplacedand routeddesign,wemodela2-playerone-shotgameforeachint erconnectnetandsolveitto sizeaparticularnumberofgatesassociatedwiththegame.T heorderinwhichthenets arechosentocreatetheone-shotgamesiscriticalindecidi ngthepercentageoptimization achievedintermsofinterconnectdelayandcrosstalknoise .Theinterconnectnetscaneitherbeorderedaccordingtoitsnoisecriticalityorthedel aycriticalitywithrespecttothe othernetsinthedesign.Wehaveinvestigatedthesetwotype oforderingsanddeveloped agametheoreticframeworkasgivenbelow.6.3.1Approach1:GatesOrderedBasedonNoiseCriticality Aftertheplaceandroutephaseofthedesign,theinterconne ctresistance,capacitance, inductance,andthesetofaggressordriversareextractedf oreachnetfromtheStandard ParasiticExchangeFormat(SPEF)netlist.Weextractthein terconnectwirelengths,wire widths,thelengthofoverlapsofeachnetwithitssetofaggr essornetsandtheirwirespacingsfromtherouteddesignexportedinDEFformat.Theseval uesareusedforcalculating thecouplingcapacitancesbetweenthegivennetsandtheira ggressornets.Amulti-terminal netisconsideredasdierentnetswithsamedriveranddier entreceivers.Recentworks oncrosstalknoiseoptimization[10,31]useanoiseestimat orintheiroptimizationengines, foridentifyingthenoisecriticalnetsofthedesign.Thisi satimeconsumingprocess.The couplingcapacitancebetweenanytwonetsisproportionalt othelengthoftheiroverlap andinverselyproportionaltothesquareofthedistanceoft heirseparation[57].Hence, ratherthanestimatingthenoiseinducedoneachnet,werank thenetsrelatively,toindicatewhetheranetismorenoisecriticalthananothernetorv iceversa.Wedeneascore 77

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foreachnetasgivenby X 8 aggressors (lengthoftheiroverlap) (spacing) 2 Thenetswithhighscorevaluesarerankedhighertoindicate thattheyaremorenoise critical.Thenetsaresortedinalistaccordingtotheirsco revalues.Thegatesandwires relatedtothenetareconsideredfortheirsizeoptimizatio ndependingontheposition ofthenetinthesortedlist,startingfromthehead.Referri ngtotheEquation2.3,the interconnectdelayofanetdependsonthewiresizeofthenet ,andthegatesizesofthe driverandthereceiver.Thecrosstalknoiseinducedonanet dependsonthewiresizesof thenetanditsaggressornets,andthegatesizesoftheaggre ssorsandthevictimdriver. Foranygivennet,therecanbepotentiallymanyaggressorne ts.Itisindicatedin[10]that itisvirtuallynotpossibletoconsiderthenoiseeectsofa lltheaggressorsonthegivennet. Also,thecrosstalknoisemodelbasedontransmissionlines usedinthisworkcanhandle onlytwoaggressors.Hence,weconsidertheeectsoftwomos taectingaggressorswhile sizingthegatesandwiresrelatedtothegivennet.However, thealgorithmdevelopedin thisworkisnotlimitedtotwoaggressornetsandgates,andc anbeeasilyextendedto considertheeectsofmorethantwoaggressorsforeachnet. Inaddition,weshowfrom experimentalresultsthatconsiderationoftwomostaecti ngaggressornetsandgatesis sucienttotakecouplingeectsintoaccount.Here,wewoul dlikethereaderstonote thatthemeasuredvaluesofcrosstalknoiseandinterconnec tdelayinexperimentalresults arebasedonthecommercialmodelsofcadencetoolsandnotus ingtheanalyticalmodels usedinthiswork.Hence,thesavingsreportedinsection6.7 arenotbasedonthecoupling eectsduetotheconsiderationoftwoaggressornets. A2-playernonzero-sumgameismodeledforeachnet, i ,intheorderoftheirsorted list.Theinterconnectdelayandcrosstalknoiseofthenet i actasthetwoplayersofthe game.Wehaveusednormalformformulationtomathematicall yrepresentandsolvethe game.Anormalformgameconsistsofasetof N playerslabeled1 ; 2 ;:::;N ,suchthat 78

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Algorithm6.1.AnIntegratedGateandWiresizingAlgorithm forInterconnectDelayand CrosstalkNoiseOptimization Input: Placedandrouteddesign Output: Optimizedgateandwiresizes Algorithm:-ExtractthenetparasiticsfromSPEFle-Extracttheoverlappinglengthsandspacingbetweenthead jacentnetsfromDEFle -Markallthenetsas\unsized"forallGatesdo Determineaggressors();-Markthegateas\unsized" endforCalculatescores(); %Calculatesscoreforallnetsofthedesign Sortscores(); %Sortsallthenetintoalistaccordingtotheirscores whileThereexistsan\un-played"netinthelistdo -Selectthemostnoisecriticalnet i fromthesortedlist -Createa2-playergamewithinterconnectdelayandcrossta lknoiseasplayers. -Identifythetwomainaggressornets, a 1 and a 2 ,intheorderoftheircouplingeects onnet i -Tagthenets i;a 1 and a 2 asthestrategydeterminers(atotalofthreewiresforwire sizing)-Tagthedriverandreceivergatesofnets i;a 1 and a 2 asthestrategydeterminers(a totalofsixgatesforgatesizing)strategyset ; ; forall SD k 2 strategydeterminersdo if SD k ismarkedas\sized"then strategyset strategyset S calculatedNashsizeof SD k ; else strategyset strategyset S setofpossiblesizesof SD k ; endif endforcost-matrix payo(twoplayers,strategyset); %forpayofunction,seeAlgorithm6.2. optimized-size nash-solution(twoplayers,payos); %forNashequilibriumsolution,seeAlgorithm6.3. -Markthesixgatesinvolvedinthegameas\sized"-Markthenets i;a 1 and a 2 as\sized"andremoveitfromthesortedlist endwhilereturn: OptimizedNashsizesforgatesandnetsegments eachplayer p has:(i)achoiceset S p calledstrategysetofplayer p ;itselementsarecalled strategies,and(ii)apayofunction P p : S 1 S 2 :::: S N !< ,assignedtoeachstrategy 79

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chosenbytheplayer p withrespecttootherplayers.Thestrategysetandthepayo matrix ofalltheindividualplayersaresucienttosolvethenorma lformgame.Alltheplayers playsimultaneouslywithoutanyknowledgeaboutotherplay ers'play.Inotherwords, theplayerssimultaneouslychoosesastrategy s p 2 S p suchthattheirrespectivepayois maximizedwithrespecttothepayosoftheotherplayers.Th eequilibriumofthegame iscomputedbyusingtheNashequilibriumtheory. Algorithm6.2.AlgorithmforPayoMatrixCalculation Input: NumberofPlayers N ,Strategyset S Output: Payomatrix forallplayers i 2 1to N do forallstrategycombinations S j = f s j1 ;:::;s jN g ,where( s j1 2 S 1 ) ;:::; ( s jN 2 S N )do calculatethedelayusingEquation2.3normalizethedelayw.r.trststrategycombinationcalculatethecrosstalknoiseusingEquation2.7normalizethenoisew.r.trststrategycombinationP [ i S j ] Geometricmeanofnormalizednoiseanddelay endfor endforreturn: payomatrix P 8 strategycombinations Thetwomostaectingaggressornetsarechosenamongitspoo lofaggressorsbased onthefractionofitscontributiontothescoreofthegivenn et.Thetwoaggressornets, a 1 and a 2 ,whichcontributetothemajorityofthescorevaluearesele cted.Thewires i;a 1 and a 2 areaddedtothesetofstrategydeterminersandrepresentth esetofscalable wiresinthegame.Thedriverandreceivergatesofwires i;a 1 and a 2 areaddedtotheset ofstrategydeterminersasthesetofscalablegates(atotal ofsixgates).Theminimum wiresizeofanetisxedbasedontheminimumwiresizedesign rulerequirementofthe processtechnology.Themaximumwiresizeforanetisdeterm inedbasedonthefree spaceavailablearoundthenet.Therangebetweenmaximuman dminimumwiresizes foreachnetistreatedasitspossiblewiresizeswithoutvio latingtheprocessdesignrules. Thisrangeisdividedintoadiscretesetofvalueswithequal stepsizesandrepresentedas thesetofscalablewiresizesforthecorrespondingwires.T hescalablegatesizesforeach gatearechosensuchthatitsreplacementinthedesigndonot resultinrerouting.The 80

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maximumscalablegatesizedependsontwofactors:(i)avail ablefreespacesurrounding thegateinthedesign,and(ii)drivestrengthsavailablefo ragatetypeinthestandard celllibrary.Foragateatspeciclocationinthedesign,al lthegatesizessupportedby thestandardcelllibrarycannotbeusedasitsscalablesize s.Someofthegatesizescannot bettedwithintheavailablefreespacewithoutdisturbing theroutednetssurroundingit. Hence,thenumberofscalablesizesavailableforeachgatei salwayslessthanorequalto thenumberofdrivestrengthsavailableinthestandardcell library.Thecollectionofthe scalablewiresizesofthethreewiresandthescalablegates izesofthesixgatesaremodeled asthestrategysetofeachplayer.Ifadesignerwantstogive weighttoaparticulardesign parameter(saycrosstalknoise),thenthestrategysetcanb eprunedtohavescalablesizes whicharefavorabletothatparameter.Thepayosfortwopla yerarecalculatedbyusing theEquations2.3and2.7respectively,accordingtotheact ionschosenbytheplayers. Algorithm6.3.AlgorithmforNashEquilibriumSolution Input: Numberofplayers N ,Payomatrix P Output: Nashsolution forallplayers i do forallpayosofplayer i do nd s i suchthat, P i ( s 1 ;:::;s i ;:::;s N ) P i ( s 1 ;:::;s i ;:::;s N ) s i istheNashstrategyforplayer i endfor endforNash-solution S = f s 1 ;:::;s N g %setofoptimizedstrategiesforall N players return: Nashsolution S TheNashequilibriumisevaluatedforthechosennet, i ,andthegameisplayedout. Thesixgatesandthreewiresparticipatedintheplayedgame areraggedas\sized"and theirsizesaresetequaltothecalculatedNashsizes.Thene ts i;a 1 and a 2 aretagged asplayedandareremovedfromthesortedlistoforderednets .Anewnetlocatedat theheadofthesortedlistisselectedtoplaythenextgame.T hesixgatesandthree wirescorrespondingtotheselectednetareidentiedtopar ticipateinthenewlyformed 2-playergame.Thescalablesizesofthegateswhicharemark edassizedareassignedto 81

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asingletonconsistingofonlyitscalculatedNashwidth.Th escalablesizesforunmarked elementsinthesetofstrategydeterminersareidentiedas describedaboveandtheNash equilibriumofthegameisevaluated.Thisprocessofcreati ngandplayingthesequential gamesisrepeateduntilthesortedlistofwiresisempty.Itc anbenotedthatthreewires areremovedfromthesortedlistforeverygameplayedandhen ce,thetotalnumberof gamesplayedisequalto N wires = 3,where N wires isthetotalnumberofwiresinthedesign. TheAlgorithm6.1.showsthepseudo-codeoftheintegratedg ateandwiresizingalgorithm developedinthiswork.6.3.2Approach2:GatesOrderedBasedonDelayCriticality Theorderingofnetsinthesortedlistdictatestheorderinw hichthegatesareconsidered fortheirsizeoptimization.Insection6.3.1,theintercon nectwiresaresortedinalistbased onthenoisecriticalityofthenets.Hence,theapproachout linedinsection6.3.1,yields slightlybetteroptimizationofcrosstalknoisethaninter connectdelay,whilesimultaneously optimizingbothdelayandnoise.Inthissection,weinvesti gateastrategywhereindelay isconsideredashighercriticalitythannoise,whilesimul taneouslyoptimizingdelayand noise.Itisinterestingtonotethatbothmethodsyieldsign icantlybetteroptimization ofbothdelayandnoisecomparedtoothermethods.Thedesign ercanchooseeitherof thestrategiesbasedontheneed.Thedierencebetweenthet wostrategiesisthewayin whichthesortedlistiscreated.Afterthedesignisplaceda ndrouted,thepathdelaysof allthepathsinthedesignareestimated,andaresortedinto alistbasedontheirdelay criticality.Themostdelaycriticalpathischosentocreat egamesforthegateandwiresize optimization.Thegamesarecreatedforeachnetinthechose npathintheorderfromits primaryoutputtoprimaryinputs.Asanexampleforillustra tion,considerthechosenpath toconsistoffourgates: A;B;C and D ,insuccessivetransitionconnectedwithnets:1 ; 2 and3,respectively.Thegate A isdrivenbyprimaryinputsandgate D drivesaprimary output.Inordertoconsiderthedown-streamloadcapacitan ce,thenet3connectinggates C and D shouldbeoptimizedbeforethenets1and2.Thus,thegamesar eplayedinthe 82

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orderofnet3followedbynet2followedbynet1.Thegameform ulatedforeachnetisa twoplayergamewiththeinterconnectdelayandcrosstalkno iseasplayers.Thetwomost aggressornetsandthecorrespondingstrategyvaluesareid entiedasindicatedinsection 6.3.1.Afterthegamesareplayedforallthenetsofthechose ncriticalpathinitsdirection ofprimaryoutputtoprimaryinputs,thenextcriticalpathi nthesortedlistisselectedto playthegames.Thisprocessofcreatinggamesisrepeatedun tilthesortedlistisempty. 6.4TimeandSpaceComplexity TheworstcasetimecomplexityofevaluatingNashequilibri umforageneralM-player gamewith S strategiesforeachplayerisgivenas O ( M S M )[52].ReferringtoSection 6.3.1,wehavemodeledtheproblemofgateandwiresizingfor simultaneousinterconnect delayandcrosstalknoisereductionasagamewithtwoplayer s.Foreachgatetype,the numberofdierentdrivestrengthsavailableforitinthest andardcelllibraryactasits maximumpossiblegatesizes.Wehaveusedastandardcelllib rarycontaininggateswith fourdierentdrivestrengths,buildonTSMC180nmdesignru les.Wehavedividedthe availablespacebetweenwiressuchthatthewiresinvolvedi nthegamehasamaximumof vedierentpossiblewiresizes.Theactualscalablesizes foreachgateandwiredependson itslocationandthefreespacesurroundingitinthedesign. Hence,thenumberofscalable sizesavailableforeachgateandwireisalwayslessthanore qualitsmaximumpossible sizes.Eachgameinvolvessixgatesandthreewires.Hence,t henumberofstrategiesfor eachplayerisgivenby S p (6 4+3 5).Thus,theworstcasecomplexityofcalculating theNashequilibriumforasinglegameplayedisgivenby O (2 S 2 p ).Thegamesareplayed repeatedlyuntilthesortedlistisempty.Initially,theso rtedlistconsistsofallthewires ofthedesign.Afteragameisplayed,threewiresaretaggeda s\sized"andremovedthe list.Hence,thetotalnumberofgamesplayedisgivenby N wires = 3.Thus,theoverallworst casetimecomplexityoftheproposedintegratedalgorithmc anbegivenmathematicallyas O N wires 3 2 S 2 p + O ( N gates ) O ( N wires )+ O ( N gates ) 83

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where N wires isthetotalnumberofwiresand N gates isthetotalnumberofgatesinthegiven design.Itcanbenoticedthatthetimecomplexityoftheprop osedalgorithmis linear and isproportionaltothetotalnumberofgatesandwiresofthed esign.Thespacecomplexity oftheproposedalgorithmisdependententirelyonthenumbe rofwiresinthedesignand thespacecomplexityofpayomatrix.Thespacecomplexityo fapayomatrixdepends onthenumberofstrategiesforeachplayerplayingthegame. Asthegamesareplayed sequentially,thetotalspacerequiredbyallthegamesiseq ualtothespacecomplexityof agameinvolvingplayerswithmaximumnumberofstrategies. Mathematically,thespace complexityrequiredbyallthepayomatricesisgivenas O (6 4+3 5).Hence,thespace complexityoftheproposedalgorithmisgivenas O ( N wires +6 4+3 5) O ( N wires ) 6.5ProofofExistenceofNashEquilibriumfortheProposedI ntegratedGate andWireSizingFormulation Inthissection,weprovidetheproofofexistenceofNashequ ilibriuminthecaseofgate andwiresizingproblemforsimultaneousoptimizationofin terconnectdelayandcrosstalk noise.Asthegateandwiresizesincreases,theinterconnec tdelayofthenetdecreasesand thecouplingcapacitanceincreasesresultinginaconvexpa yofunction.Inthiswork,we havemodeledtheinterconnectdelayandcrosstalknoiseoft hechosennetastheplayers ofthegame.Let G = f S 1 ;:::;S n ; f 1 ;:::;f n g beagamewitheachplayer i 2 N having astrategyset S i containingthescalablegateandwiresizes,anditspayogi venby f i Wehavemodeledthestrategyset S i foreachplayerasanon-empty,compactsetof anitedimensionalEuclideanspace.Becauseoftheconvexn atureoftheinterconnect delayandcrosstalknoise,themodeledpayofunction f i becomesuppersemicontinuous on S = Ni =1 S i andforanyxed u i 2 S i ,thefunction f i ( u i ;: )isalowersemicontinuous on S ( i ) [16].Forany u 2 S ,thebestreplyortheexpectedpayo B i ( u )isalsoconvex. AccordingtoKakutani'sxedpointtheorem[61],thegameGh asatleastoneNash 84

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equilibriumpointifthegraph G B = f ( x;y ): x 2 S;y 2 B ( x ) g isclosed.Letsassumethatitsnotclosed.Then, 9 ( x 0 ;y 0 ) = 2 G B ,suchthateveryneighborhood(in S S )of( x 0 ;y 0 )containsapointof G B x 0 isagateorwiresize,ithastobeoneofthosefromthesetofpo ssiblegateandwire sizesforthegivenplayerinordertosatisfytheDRCrulesof theusedprocesstechnology. ) x 0 2 S ) y 0 = 2 B ( x 0 ) Inotherwords,foratleastoneplayerplayingthegame(sayc rosstalknoise),thereis an y 1 1 2 S 1 suchthat f 1 ( y 1 1 ;x 02 ;:::x 0n ) f 1 ( y 0 1 ;x 02 ;:::x 0n )(6.1) Let F beafunctionsuchthat F : S 2 !< andgivenas F ( x;y )= f 1 ( y 1 1 ;x 2 ;:::x n ) f 1 ( y 1 ;x 2 ;:::x n ) Since f i isuppersemicontinuouson S and f i ( u i ;: )islowersemicontinuouson S i ,Fis lowersemicontinuousand C = f ( x;y ) 2 S 2 : F ( x;y ) 0 g isclosed.Hence,forany ( x; y ) 2 G B ;F ( x; y ) 0.But,byEquation6.1, F ( x 0 ;y 0 ) 0,contradictingtheclosedness ofC.Thus,thereisapoint s 2 S suchthat s 2 B ( s ),whichisaNashequilibriumpoint. 6.6DesignFlow Thedesignrowforobtaininganoptimallygateandwiresized circuitfromaverilog/VHDLdescriptionisshowninFigure6.1..Thebehavior alverilog/VHDLdescription issynthesizedontoalibraryofstandardcellsandgivenasi nputtothedesignrow.The standardcellsareplacedandroutedincoherencewiththesy nthesizedcodeusingany standardcellplaceandroutetool.WehaveusedtheFirstEnc ounterRTL-to-GDSII toolfromCadenceDesignSystemstoperformtheplacementandroutingofgatelevel 85

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Crosstalk Noise Estimator ModelingWith calculated Nash sizesScript Payoff Function DEF Update (StarRCXT, SignalStrom, Celtic)Verilog Design Cadence First Encounter Script Net Extractor and Crosstalk Noise Delay Models Game Theoretic BasedPlaced and Routed Design (DEF Format)Payoff functionGate and Wire Size Optimized DEF This WorkStandard Cell LEF Library with differnet strengths, Cell Timing Library, Cell Noise ModelsOptimized Gate and Wire SizesGate and Wire Sized Delay and Crosstalk Noise Optimized CircuitIntegrated Size Solver Delay Calculator Extracted Net InformationRC ExtractorFigure6.1.IntegrationofProposedGateandWireSizingAlg orithmintheDesignFlow RTLdesign.Theparasiticsfromtherouteddesignareexport edinSPEFformatwith thehelpofStarRCXTfromSynopsysInc.Alexandyaccscripti sdevelopedtoreadthe SPEFnetlistandextractthevaluesofinterconnectresista nce,capacitance,inductance, alongwiththeirsetofaggressorgatesforallthenetsofthe design.Also,agawkscript iswrittentoextractstheinformationaboutthelengthofov erlapsbetweentwogivennets alongwiththeirdistanceofseparationfromtheDEFnetlist .Thepayofunctionisused bytheproposedintegratedsizesolverdescribedinAlgorit hm6.1.tominimizetheinterconnectdelayandcrosstalknoiseofindividualnetsintheo rderoftheirnoisecriticality ordelaycriticality.Theoptimizedgateandwiresizesresu ltedfromoursolverareusedto updatetherouteddesign.Wehavedevelopedanothergawkscr iptwhichscalesthegates andwiresintheoriginalDEFrouteddesignaccordingtothei rcalculatedoptimizedsizes. 86

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Itshouldbenotedherethattheresultingoptimizeddesignd oesnotrequirereroutingsince thepossiblegateandwiresizesdraftedarewithinitslimit stosatisfythedesignrulesof theusedprocesstechnology.6.7ExperimentalResults ThegametheoreticgateandwiresizesolverdescribedinAlg orithm6.1.wasimplementedinCandexecutedonaUltraSPARC-IIe650MHz,512MBSu nBlade150system runningSolaris2.8onit.TheASICdesignsonwhichwetested ouralgorithmwereobtainedfromOpencores[62].Astandardcelllibrarycontain ing10logiccellswith4dierent drivestrengthsbasedona6-Metallayer,180nmtechnologyh asbeendevelopedandused. ASICdesignsaremappedtothestandardcellsusingBuildGat es,anRTLsynthesistool ofCadencedesignsystems.Wehavemodiedthedesignssucht hattheblocksarerattenedtostandardcellswithoutmaintainingthehierarchy. Theon-chipmemorymodules arerealizedasregisterarrayswithD-ripropasbasicbuild ingunits.ThestructuraldesignisinputedtoCadenceFirstEncountertoperformthepla cementandrouting.The informationneededforthegametheoreticsolverisextract edfromDEFandSPEFles usingalexandyaccscript.Thecalculatedgateandwiresize sfromthegametheoretic solverisusedtoupdatetheoriginalDEFleinordertogener ateanoptimizedDEFle. ItcanbenotedthattheoptimizedDEFleiscreatedwiththeh elpofgawkscriptand veriedfortheDRCrules.Thedesignisnotreroutedtogener atetheoptimizedDEFle. Theinterconnectdelayandcrosstalknoiseareestimatedus ingCadenceSignalstormand CelticICrespectivelywiththeirrobustmodels,andnotusi ngtheanalyticalmodelsused duringoptimization. 87

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Table6.1.AverageSavingsforSimultaneousOptimizationo fInterconnectDelayandCrosstalkNoiseDuringIntegrated Gate andWireSizing OpenCoreDe-sign[62] TotalGates TotalNets WS-GT 1 GS-GT 2 Integrated-SA 3 Integrated-GT-NC 4 IntegratedGT-DC 5 RunTime(mins) %Average RunTime(mins) %Average RunTime(mins) %Average RunTime(mins) %Average RunTime(mins) %Average Savings Savings Savings Savings Savings Delay Noise Delay Noise Delay Noise Delay Noise Delay Noise Mult 428 854 1.89 9.87 12.13 1.13 4.18 5.34 21.07 7.48 4.91 4.05 14.80 18.02 3.79 16.76 14.51 PCI 7882 19520 5.23 19.32 21.42 4.29 10.39 16.49 64.91 14.20 15.82 8.13 26.71 31.95 8.36 34.12 28.59 ATA 21781 43563 11.86 29.87 20.14 10.49 20.31 23.91 192.53 26.91 19.25 13.92 38.91 36.12 14.33 43.57 35.18 RISC 34172 61468 16.86 25.22 22.31 13.31 15.28 14.05 260.16 23.97 16.01 18.34 30.04 29.56 17.58 34.68 26.91 AVR P 41274 78770 21.32 22.45 31.34 15.79 21.41 22.96 428.91 19.78 25.83 24.87 29.74 42.67 25.84 35.16 36.78 P16C55 52128 102021 28.98 19.86 43.29 19.98 16.96 21.91 514.04 14.19 34.89 32.19 24.07 44.26 31.92 27.48 43.02 T80 C 69973 157850 39.48 23.78 33.12 27.67 19.86 25.24 774.38 20.23 29.37 43.16 28.38 35.49 45.27 31.83 31.97 Average 18.11 20.87 27.52 34.03 31.94 30.99 Noareaoverheadincurredforanyoftheapproaches.Theperc entagevaluesindicatedarew.r.tplacedandrouteddesignw ithout gateandwiresizing. 1 WS-GT:Gametheoreticwiresizingapproachforsimultaneou soptimizationofinterconnectdelayandcrosstalknoise 2 GS-GT:Gametheoreticgatesizingapproachforsimultaneou soptimizationofinterconnectdelayandcrosstalknoise 3 Integrated-SA:Integratedsimulatedannealinggateandwi resizingapproachforsimultaneousoptimizationofinterc onnectdelayand crosstalknoise 4 Integrated-GT-NC:Integratedgametheoreticgateandwire sizingapproachforsimultaneousoptimizationofintercon nectdelayand crosstalknoisewithnetsorderedbasedonnoisecriticalit y 5 Integrated-GT-DC:Integratedgametheoreticgateandwire sizingapproachforsimultaneousoptimizationofintercon nectdelayand crosstalknoisewithgatesorderedbasedondelaycriticali ty TableLegend : Avg :Averagesavingsofallthenetsintheentiredesign; Crit :Savingsonthecriticalnetofthedesign; Runtime : runningtimeofeachalgorithm.88

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Table6.2.CriticalPathSavingsforSimultaneousOptimiza tionofInterconnectDelayandCrosstalkNoiseDuringInteg rated GateandWireSizing OpenCoreDe-sign[62] TotalGates TotalNets WS-GT GS-GT Integrated-SA Integrated-GT-NC IntegratedGT-DC RunTime(mins) %Crit.Path RunTime(mins) %Crit.Path RunTime(mins) %Crit.Path RunTime(mins) %Crit.Path RunTime(mins) %Crit.Path Savings Savings Savings Savings Savings Delay Noise Delay Noise Delay Noise Delay Noise Delay Noise Mult 428 854 1.89 11.79 17.23 1.13 5.59 7.86 21.07 8.56 9.16 4.05 17.19 17.99 3.79 20.11 15.27 PCI 7882 19520 5.23 34.21 37.38 4.29 17.11 25.29 64.91 19.96 20.63 8.13 40.03 45.08 8.36 41.64 39.35 ATA 21781 43563 11.86 39.95 42.15 10.49 24.78 27.12 192.53 23.18 20.37 13.92 41.61 41.16 14.33 48.49 38.92 RISC 34172 61468 16.86 35.21 29.73 13.31 21.16 19.96 260.16 42.16 22.07 18.34 47.94 37.41 17.58 51.42 32.13 AVR P 41274 78770 21.32 37.63 40.31 15.79 27.19 28.95 428.91 28.92 36.10 24.87 39.82 53.58 25.84 43.51 40.10 P16C55 52128 102021 28.98 27.45 57.98 19.98 21.54 23.16 514.04 21.61 35.92 32.19 26.88 57.74 31.92 29.15 53.58 T80 C 69973 157850 39.48 34.87 39.89 27.67 24.21 28.92 774.38 22.84 34.57 43.16 37.33 44.23 45.27 43.08 39.76 Average 23.89 25.55 35.83 42.46 39.63 37.02 ThenotationsusedinthistablearesameasinTable6.1.. Crit.PathSavings :Savingsonthecriticalnetofthedesign89

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Theworksreportedinliteraturesolvetheproblemofgatean dwiresizingforcrosstalk noiseoptimizationunderdelayconstraints.Inthiswork,w ehavesolvedtheproblemof integratedgateandwiresizingforsimultaneousoptimizat ionofcrosstalknoiseandinterconnectdelay.Hence,tocompareourresults,wehaveimplem entedsimulatedannealing forsimultaneousoptimizationofcrosstalknoiseandinter connectdelay,andexecutediton sameSolarismachinewithsamesetofinputsandparameters. Theimplementedsimulated annealingalgorithmisexperimentedtoobtainthebestresu ltsintermsofoptimizationof interconnectdelayandcrosstalknoise.Thesetofpossible gateandwiresizesarecalculated asindicatedinSection6.3.1.Ineachmoveofsimulatedanne alingprocess,agateorawire israndomlyselectedanditssizeisrandomlyassignedfromt hesetofitspossiblesizes. Thecostfunctionisdenedasthegeometricmeanofintercon nectdelayandcrosstalk noisesummedoverallthenets.Theinitialtemperatureofsi mulatedannealingprocessis determinedbyndingtheaveragechangeinthecostforaseto frandommovesfromthe startingcongurationandselectingthetemperaturewhich leadstoanacceptprobability of0.95.Thenumberofmovespertemperatureforeachdesigni ssetto20timesthetotal numberofnetsandgatesinthedesign.Thisisdonesoastoall owatleast10to15moves onanaverageforeachgateornetbeforesettlingforitssolu tion.Theup-hillmovesare acceptedwithaprobabilityof e C T ,where C isthechangeinthecostandTisthecurrent temperatureoftheiteration.Thetemperatureiscooledatt herateof0.95. Theexperimentswereconductedsuchthattheareaoverheadi szeroinbothsimulated annealingandgametheoreticapproaches.ReferringtoTabl es6.1.and6.2.,simulated annealingshowsanaverageimprovementsof18.11%and20.87 %intermsofinterconnect delayandcrosstalknoiserespectively.Whenthegamesareo rderedaccordingtothenoise criticalityofthenets,gametheoreticapproachshowsanav erageimprovementsof27.52% and34.03%intermsofinterconnectdelayandcrosstalknois erespectively.Whereas,when thegamesareorderedaccordingtothedelaycriticalityoft hepaths,gametheoreticapproachshowsanaverageimprovementsof31.94%and30.99%in termsofinterconnectdelay andcrosstalknoiserespectively.Hence,gamesorderedacc ordingtonoisecriticalityresults 90

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inbetternoiseoptimization,whilegamesorderedaccordin gtodelaycriticalityresultsin betterdelayoptimization.Thus,thedesignercanchooseei theroftheproposedstrategies basedontheneedofthedesign.Gametheoreticintegratedsi zesolverbasedonbothnoise anddelaycriticalstrategies,inadditiontooutperformin gsimulatingannealingintermsof interconnectdelayandcrosstalknoisesavings,hassigni cantlysmallerruntimes. Hence, ourapproachisscalableandcanhandlethecomplexityoflar geSOCdesigns Table6.3.CrosstalkNoiseOptimizationUnderDelayConstr aintsDuringIntegratedGate andWireSizing OpenCoreDesign[62] NumberofNoiseViolations NoiseThreshold=0.15 V dd Lagrangian[31] Game[ThisWork] Mult 7 5 PCI 18 7 ATA 63 16 RISC 121 33 AVR P 167 47 P16C55 C 204 58 T80 C 257 71 Toenableadirectcomparisonofourworkwithoneoftherecen tworksdevelopedin [31],wehavemodiedourgametheoreticapproachtominimiz ethecrosstalknoiseunder delayconstraints.Theworkdevelopedin[31]isaLagrangia nrelaxationbasedgatesizing approachforreducingthecrosstalknoiseunderthedelayco nstraints.Wehaveextended theLagrangiangatesizingapproachreportedin[31]basedo ntheworkof[24],which isalsoLagrangianbased,inordertoincludethewiresizing alongwithgatesizing.For bothgametheoreticandLagrangianrelaxationapproaches, wehaveusedthedelayvalues obtainedfromsimultaneousoptimizationofinterconnectd elayandcrosstalknoiseasthe setofdelayconstraints,soastoensureatighterconstrain tsetforbothapproaches.The Lagrangianrelaxationproblemforpost-layoutcrosstalkn oiseoptimizationissolvedusing LANCELOT[66].Ingametheoreticapproach,thestrategiesw hichdonotsatisfythedelay constraintsareprunedoutfromtheirrespectivestrategys ets.Thegameisplayedwiththe 91

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threenets,namely,thechosencriticalnetanditstwomosta ggressornets,astheplayers ofthegame.Thepayofunctionisdeterminedbythecrosstal knoiseinducedonthe chosennet.Table6.3.showsthecomparisonofgametheoreti candLagrangianrelaxation basedapproachesindicatedintermsofnumberofnoiseviola tionsforeachdesign.Noise violationsareexpressedasthenumberofnetswhichhaveani nducednoiseexceedinga thresholdnoisesetto0.15 V dd .Itcanseenthatourapproachresultsinsignicantlyless numberoffaultswhencomparedtotheLagrangianrelaxation 6.8Conclusions Gametheoryallowsthesimultaneousoptimizationofmultip lemetricsinthecontext ofconrictingobjectivesleadingtoaconvexobjectivefunc tionintheproblemformulation. Thisessentiallymakesitpossibletousegametheoryforsim ultaneousoptimizationofinterconnectdelayandcrosstalknoise.Optimizingbothinte rconnectdelayandcrosstalk noiseisextremelycriticalindeepsubmicronandnanoregim ecircuits.Wehavedevelopedanintegratedframeworkforperformingbothgateandwi resizingsimultaneouslyon anygivendesign.Thedevelopedintegratedframeworkperfo rmssignicantlybetterthan sequentialapplicationofgatesizingfollowedbywiresizi ng.Also,theproposedmethod resultsinalineartimealgorithmwithsignicantlybetter resultsthansimulatedannealing andLagrangianrelaxation,makingthisworkanimportantco ntribution. 92

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CHAPTER7 STATISTICALGATESIZINGUNDERPROCESSVARIATIONS Inthischapter,wedevelopanewpost-layoutgatesizingalg orithmforsimultaneous reductionofdelayuncertaintyandcrosstalknoiseunderth eimpactofprocessvariations. Theproblemofpost-layoutstatisticalgatesizingismodel edasa2-playerstochasticgame andsolvedusingNashequilibriumtheory.Thedelayuncerta intyinducedonanetdepends onthesizeofitsdriverandreceiver.Thecrosstalknoisein ducedonanetdependsonthe sizeofitsdriverandthesizesofthegatesdrivingitscoupl ednets.Increasingthesizeof agatewillaectthesizesofotherrelatedgatesandresulti naconrictingsituation,thus, makingthegatesizingachallengingproblem.Inaddition,d uetoprocessvariations,the gatesizesarenolongerdeterministic,butratherbehaveas aprobabilisticdistributionover arange.Stochasticgamesallowthemodelingofprobabilist icdistributionofgatesizespace andalsoeectivelycapturetheconrictingnatureofthepro blem.Wehaveimplemented twodierentstrategiesinwhichthegamesareorderedaccor dingto(i)thenoisecriticality, and(ii)thedelaycriticalityofnets. InChapter5,wehadsolvedthegatesizingproblemforsimult aneousoptimizationof interconnectdelayandcrosstalknoiseusingfour-playerg ames,withoutconsideringprocess variations.Thedevelopedalgorithmisshowntobesignica ntlybetterthannon-linear optimizationusingLagrangianmultipliers,simulatedann ealingandgeneticsearch,interms ofoptimizationandruntimes.Inthischapter,wehaveformu latedthegatesizingproblem asatwo-playerstochasticgameforsimultaneousminimizat ionofdelayuncertaintyand crosstalknoiseundertheimpactofprocessvariations.Weh avealsoconsideredthespatial correlationsduetoprocessvariations.Themodelingofgam esinthisworkiscompletely 93

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dierentfromthatgiveninChapter5.Here,wehavemodeledt hedelayuncertaintyand crosstalknoiseastheplayersofthe2-playerstochasticga me. 7.1ProblemDenition Theproblemofpost-layoutgatesizingunderprocessvariat ionscanbedenedas follows:ndtheoptimalgatesizesundertheimpactofproce ssvariationssuchthatthe delayuncertaintyandcrosstalknoiseoftheoverallcircui tisminimizedunderthegivenarea constraintsandwithouttheneedforreroutinganyofthenet sinthedesign.Thecoupling capacitanceofanetdependsonitswiresize,thelengthofov erlapandthespacingbetween adjacentnets.Thisinformationcanbeecientlyextracted atpost-routingphase.The couplingnoiseinducedonanetdependsonthesizeofitsdriv er,receiverandaggressor gates.Also,thedelayuncertaintyisafunctionofthegates izes,andtheinputandload capacitances.Equations2.6and2.7emphasizethatthegate sizesdirectlycontrolthe delayuncertaintyandcrosstalknoiseintermsofthedriver resistances,gateandcoupling capacitances.Hence,calculatingtheoptimalgatesizesca neectivelyreducethecrosstalk noiseanddelayuncertaintyindeepsubmicrondesigns.Thed eterministicmethodoptimizes onlythecriticalandthenearcriticalnetsofthedesignand doesnotimprovethenoncriticalpaths.Duetotheintra-dievariability,non-crit icalpathsbecomecriticalcausing thestatisticalcircuitdelayandnoisetodeteriorate[67] .Hence,wehaveconsideredall thenetsofthedesignforminimizationoftheirdelayuncert aintyandcrosstalknoiseusing astochasticframework.Gatesizingcanbeperformedatpost -routelevelbyutilizingthe existingll-space.Inthiswork,weincrementallyscaleth egatesizestoutilizetheavailable ll-spacesuchthattheroutedresourcesinadjacentregion sarenotdisturbed.Hence,our approachwillneitherresultinareaoverheadnorneedre-ro utingofthedesign. 7.2MotivationforStatisticalGateSizingProblem Traditionally,inVLSIdesign,asingleparameterisoptimi zedassumingotherparametersasconstraints.Indeepsubmicronandnanometercircui ts,thesimultaneousoptimiza94

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tionofpower,delayandcrosstalknoiseisbecomingimporta nt.Further,theoptimization methodsneedtotakeintoaccounttheimpactofprocessvaria tionsarisingfromthefabricationprocess.Theoptimizationinamulti-metricandconric tingenvironmentsisadicult problemsincethenormaldenitionofanoptimalvaluenolon gerappliesandvalid.For example,anoptimalgatesizeforonemetricmaynotbeoptima lforanothermetric.The optimalpolicyatanygiveninstancedependsonthepolicesf orothermetricskeepingthe bestinterestoftheentiresysteminview.Whilemostoptimi zationmethodssuchasILP, simulatedannealing,andforcedirectedmethodslendthems elveswellforsinglemetric optimization,thesemethodsareinadequateformulti-metr icoptimization.Thus,thereis aneedfornewmethodsandalgorithmstobedevelopedwhichap pliesthemulti-agent optimizationtheoriestotheproblemsofVLSICAD.Theconsi derationofprocessvariationsduringthedesignoptimizationrequiresprobabilist icanalysisduetotheuncertainty elementintroducedbyprocessvariations.Mostoftheprevi ousworksreportedinliterature employsadeterministicframeworktoperformgatesizingby usingeitherstatisticalstatic timinganalysisorbydevelopingastatisticalgatedelaymo del(pleaserefertoTable2.1.). In[68],itispointedoutthatastochasticframeworkisneed edtoanalyzethesystemswith statisticalparameterssincethesystemitselfisanoutcom eofastochasticprocess.This motivatesustodevelopastochasticframeworkformulti-me tricoptimizationinorderto performgatesizingundertheimpactofprocessvariations. Thefundamentalbasisandstructureofgametheoryandstoch asticgamesallowsthe formulationofoptimizationproblemsinwhichmultipleint er-relatedcostmetricscompete againstoneanotherfortheirsimultaneousoptimization.F urther,stochasticgamesinherentlycapturesthenondeterministicbehaviorofthesystem parameters.Gametheoretic reasoningtakesintoaccounttheattemptsmadebythemultip leagentstowardstheoptimizationoftheirobjectivesforeverydecision.Eachagent oraplayer'sdecisionisbased onthedecisionofeveryotherplayerinthegameandhenceiti spossibleforeachplayerto optimizehisgainwithrespecttotheothers'gainsinthegam e.Intermsofgametheory,a solutionissaidtoreachitsglobalvalueforthegivencondi tionswhenitreachesitsequi95

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librium.Stochasticgamesareplayedinstages,wherethepa yovaluesforeachplayerand thestatetransitionfromonestagetoanotherarecontrolle dbyastochasticfunction.This enablesthedesignertoecientlycapturetheuncertaintyd uetoprocessvariationswith thehelpofstochasticfunctionandcontroltheoptimizatio nateachstageofthegameusing itsprobabilitydistribution.Thus,gametheoryideallysu itsformulti-metricoptimization withconrictingobjectivesandthestochasticfunctionoft hestochasticgamescanbeused tomodeltheuncertaintyindesignparametersduetoprocess variations.Further,ifthe payofunctionisconvex,i.e.,theparametersbeingoptimi zedcorrespondtoconricting objectives(suchasdelayuncertaintyandcrosstalknoisei nthiswork),gametheoretic optimization,infact,performssignicantlybetter[18].7.3ProposedStatisticalGateSizing Inthissection,wepresentthestochasticgametheoreticme thodologyforminimizing thedelayuncertaintyandcrosstalknoise.Again,wehavede velopedtwodierentstrategies basedontheorderingofthegames.Thetwoorderingsarebase donthenoisecriticality orthedelaycriticalityofthegatesandeachstrategyisdis cussedindetailbelow. 7.3.1Approach1:GatesOrderedBasedonNoiseCriticality Inthiswork,weperformtheoptimizationtaskafterthedesi gnisplacedandrouted. Theinterconnectresistance,capacitance,inductance,an dcouplingcapacitancesalongwith theiraggressordriversareextractedforeachnetfromtheS PEFnetlistoftherouted design.Thelengthofinterconnectwiresandthelengthofov erlapofeachnetwithitsset ofaggressornetsandtheirspacingareextractedfromthero uteddesignexportedinDEF format.Amulti-terminalnetisassumedasdierentnetswit hsamedriveranddierent receivers.Thegatesofthedesignareorderedinaccordance tothenoisecriticalityofits drivennets.Theuseofastatisticalstatictimingandcross talknoiseanalyzertoestimate andidentifythenoisecriticalnetsistimeconsuming.Inad dition,statisticalstaticanalysis makesassumptionsofnormaldistributionsforthesignalar rivaltimeandtheslope,and 96

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approximatestheresultantoftwoormorenormaldistributi onsasnormaldistribution, thereby,leadingtoinaccurateanalysis[37].Hence,inthi swork,weproposeaheuristicto rankthenetsrelativetoeachothertoindicatewhetheranet ismorenoisecriticalthanother netsorviceversa,ratherthanestimatingthenoiseoneachn et.Thecouplingcapacitance betweentwonetsisproportionaltothelengthofoverlapand inverselyproportionaltothe squareoftheirspacing[57].Hence,wedenetherankforeac hnetas Rankofanet = X 8 aggressors (Lengthofoverlap) (Spacing) 2 : Thenetsaresortedandarrangedinalistaccordingtotheirr ank.Themostcritical netwillformtheheadofthelistandtheleastcriticalnetwi llformthetail.Thegates inruencingthecriticalnoisenetsareconsideredrstfort heiroptimization.Foranygiven net,therecanbemanypotentialaggressornets.Itisindica tedin[10]thatitisvirtually notpossibletoconsideralltheaggressorgatesinthecoupl edsetofanet.Inourwork, wehaveusedcrosstalknoisemodelswhichconsiderstheeec tsoftwoaggressornetson thevictimnet.Thersttwoaggressornetswhichcontribute tothemajorfractionof therankofgivennetaremarkedasitsmostaectingaggresso rnets,andareusedwhile sizingthevictimnet.Thelimitationtotwoaggressornetsi snotduetothemodeling usedinthiswork,butduetothecrosstalknoisemodelsused. Thecrosstalknoisemodels havetoreplacedwithamoreaccuratehigherordermodelsino rdertoconsiderthreeor aboveaggressornetswhilesizinganyvictimnet.Foreachga teinthedesign,wehave chosenitsgatesizessuchthattheirreplacementinthedesi gndoesnotresultinre-routing. Thesetofvariouspossiblegatesizeswithwhichitcanbesca ledwithoutviolatingDRC rulesarestoredasitsscalablesizeset.Themaximumgatesi zewithwhichagatecanbe sizeddependson(i)theavailablegatesizesforitsgatetyp einthestandardcelllibrary and(ii)theexistingfree-spacearoundthegivengateinthe design.Theexistingfreespaceispartitionedamongthegatessuchthatitcanberepla cedwithoutdisturbingthe 97

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Algorithm7.1.Post-RouteGateSizingAlgorithmforSimult aneousOptimizationofDelay UncertaintyandCrosstalkNoiseunderProcessVariations Input: Placedandrouteddesign Output: Optimizedgatesizes -extractthenetparasiticsfromSPEFleforallgatesdo determine aggressors(); %extractaggressorgatesfromSPEFle -markthegateasun-played endforforallnetsdo -extracttheoverlappinglengthsandspacingbetweenthead jacentnetsfromDEFle calculate scores(); sort scores(); endforwhilethereexistsanunsizedgatedo -selectanun-playednoisecriticalnet i fromthesortedlist -identifytwomainaggressorgatesfornet i intheorderoftheircouplingeects -createa2-playergamefordelayuncertaintyandcrosstalk noise -theparticipatingelementsfordeterminingstrategyseto fthetwoplayersarethe driver,thereceiver,andthetwomainaggressorgatesofnet i forgate g k amongthefouridentiedgatesdo if g k ismarkedassizedthen scalable size setof g k calculated best gate size; else scalable size setof g k determine strategies(); endif endfor-representthescalable size setoffourgatesasthestrategysetofbothplayers,representedas( s ) Nash sizes Sparse Game( s;T ); %forSparsegamealgorithm,pleaserefertoAlgorithm7.2. -markthesefourgatesassized-markthenet i asplayed endwhilereturn: optimizedNashsizesofgates neighboringroutesandsatisesDRCrules.Theminimumgate sizeforeachgateischosen asthesizeofminimumstrengthgateavailableinthestandar dcelllibraryforitsgatetype. Atwo-playernonzero-sumstochasticgameisformedforthem ostcriticalnetavailable attheheadofthesortedlist.Thedelayuncertaintyandcros stalknoiseactasthetwo playersofthegame.Thedriver,thereceiverandthetwomost aectingaggressorgatesof 98

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theselectednoisecriticalnetareidentiedtoparticipat einthegame.Thescalablesizeset ofthesefourgatesputtogetherformthestrategysetsofthe twoplayers.Ifagateismarked assized,itsscalablesizeisasingletoncontainingitsbes tsizecalculatedfrompreviously played-outstochasticgame.Thestochasticgameisplayedi nstages,witheachstage representedasaone-shotnormalformgame.Wehaveusednit e-horizonundiscounted stochasticgamesinourmodelingsincethesegamesstopafte raxednumber( T )ofstages. Ateachstage,theplayerschooseanstrategy(gatesizesfor thefourparticipatinggates) whichismorebenecialtohim. Duetotheprocessvariations,thegatesizesarenolongerde terministicquantities,but behaveasrandomvariablesaroundtheirnominalvalues.Asp redictedin[4],wehave usedavariationof25%aroundthenominalvalueingatesizes duetoprocessvariations. Thisvariationinthegatesizeforagate g k iscapturedusingaprobabilitydistribution P ( g k ).Theproposedgatesizingmethodologydoesnotmakeanyass umptionsaboutthe distributionofparametervariations.Thewidelyusedpara meterdistributionsareGaussian orNormalandLog-Normaldistributions.Thestrategyofapl ayerisavectorcontaininga sizeassignmenttoeachofthefourgatesparticipatinginth egame.Forexample,ifscalable sizesetof g 1 = f g 11 ;g 12 ;g 13 g ; g 2 = f g 21 ;g 22 g ; g 3 = f g 31 ;g 32 g ; g 4 = f g 41 ;g 42 ;g 43 ;g 44 g ,then apossiblestrategyforaplayercanbe s 0 = f g 11 ;g 21 ;g 31 ;g 41 g ,anotherpossiblestrategy s 00 = f g 12 ;g 21 ;g 32 ;g 44 g andsoon.Hence,therewillbealargesetofstrategiesforea ch player,resultinginalargestatespace.The\sparsesampli ngalgorithm"developedby Kearnsetal[69]ismodiedtosolvetheformulated2-player stochasticgame(pleaserefer toAlgorithm7.2.).Themodicationsaremadeinordertocon siderthespatialcorrelations duetoprocessvariationsandtoincludedelayuncertaintya ndcrosstalknoiseastheplayers. Theinterestingaspectofthisalgorithmisthatitstimecom plexityisindependentofthe sizeofthestatespace,therebymakingourmethodologytrac tableinspiteofhavingalarge statespace.Thealgorithmisarecursivealgorithmwhichta kesanystate s andtime T astheinputs.ItassumesaccesstoanarbitraryxedNashsel ectionfunction f ,generates immediatepayomatrices M k [ s ],andsamplestheprobabilitydistribution P ( g k ).Wehave 99

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Algorithm7.2.SparseGameAlgorithmforComputingApproxi mateNashEquilibriumin StochasticGames FunctionName: Sparse Game( s;T ) Input: Strategysetofbothplayers,Numberofsteps T Output: Nashequilibriumpolicy:( ; ) if T =0then M 1 [ s ] calculatecrosstalknoiseusingEquation2.7 M 2 [ s ] calculatedelayuncertaintyusingEquation2.6 % s representsthesetcontainingthenominalgatesizevalues f 1 ( M 1 [ s ] ;M 2 [ s ]) f 2 ( M 1 [ s ] ;M 2 [ s ]) % f 1 ;f 2 arearbitraryNashselectionfunctions for k 2f 1 ; 2 g do Q k M k [ s ]( ; ) endforreturn: ( ;;Q 1 ;Q 2 ) endifforallstrategypair( i;j ) 2 s do -selectagate g k atrandom -select m randomsamples s 01 ;:::;s 0m from P ( : j g k ;i;j ) % P ( g k )representstheprobabilitydistributionofthegate sizesarounditsnominalvaluesduetoprocessvariations. forl=1 ;:::;m do -transitthegatesizesofremainingthreegatesinthesampl e s l accordingtothe spatialcorrelationsingatesizevariations( 0 ; 0 ;Q 1 [ s 0l ;T 1] ;Q 2 [ s 0l ;T 1]) Sparse Game ( s 0l ;T 1); endforfor k 2f 1 ; 2 g do Q k [ s;T ]( i;j ) M k [ s ]+(1 =m ) P ml =1 Q k [ s 0l ;T 1] endfor endfor f 1 ( Q 1 [ s;T ] ;Q 2 [ s;T ]) f 2 ( Q 1 [ s;T ] ;Q 2 [ s;T ]) for k 2f 1 ; 2 g do Q k Q k [ s;T ]( ; ) endforreturn: ( ;;Q 1 ;Q 2 ) usedtheNashselectionfunctionastoreturntheminimumpay ofortheplayerunderthe givenstrategyoftheotherplayer. Thespatialcorrelationsingatesizesduetoprocessvariat ionsaremodeledexactlyas givenin[37].Thegatesizesofdeviceslocatedinsamegrida reassignedperfectcorrelations, 100

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gatesizesinneighboringgridsareassignedhighcorrelati onsandzerocorrelationsare assignedbetweenothergates.Ateachstageofthestochasti cgame, m samplesaretaken fromtheprobabilitydistributionofgatesizespace P ( g k )ofarandomlyselectedgate g k Thegatesizesfortheremaininggatesofthe m selectedsamplesarescaledaccordingto theirspatialcorrelationwithrespecttogate g k .Thepayomatricesareupdatedaccording totheselected m samplesandhence,thegamemovestothenextstagedepending onthe probabilitydistributionoftheprocessvariations.Ithas beenprovedin[69]thatthe numberofsamples m requiredateachstageforobtaininganear-Nashsolutionis givenby m> ( T 3 = 2 )log( T= )+ T log( n= ),where > 0and n isthenumberofstrategiesavailable tobothplayersatanystage. Inourwork,wehaveused T as10and m asgivenbytheaboveinequality.Thegame makestransitionstonewstagesuntil T stepsarereached.Thisisthestoppingcriterion forthetwoplayernonzero-sumstochasticgame.Thetimecom plexityofndingtheNash equilibriumgiveninAlgorithm7.2.isproportionalto m T ,duetotherecursionfor T stages. Afterthe2-playerstochasticgameisplayedout,theinvolv edfourgatesaremarkedassized andtheselectednoisecriticalnetisremovedfromthesorte dlist.Thenextnoisecritical netavailableattheheadofthesortedlistisselectedtocre ateanew2-playerstochastic gameasillustratedaboveandthegameisplayedout.Thispro cessofformationof2-player stochasticgamescontinuesuntilallthegatesofthedesign aremarkedassized.Algorithm 7.1.representsthepseudo-codeoftheproposedgatesizing algorithmforsimultaneous optimizationofdelayuncertaintyandcrosstalknoiseunde rprocessvariations. 7.3.2Approach2:GatesOrderedBasedonDelayCriticality Theorderingofnetsinthesortedlistdictatestheorderinw hichthegatesareconsideredfortheirsizeoptimization.Insection7.3.1,thei nterconnectwiresaresortedin alistbasedonthenoisecriticalityofthenets.Hence,thea pproachoutlinedinsection 7.3.1,yieldsslightlybetteroptimizationofcrosstalkno isethanfordelayuncertainty,while simultaneouslyoptimizingbothdelayandnoise.Inthissec tion,weinvestigateastrategy, 101

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wherein,delayisconsideredasofhighercriticalitythann oise,whilesimultaneouslyoptimizingdelayandnoise.Itisinterestingtonotethatbothme thodsyieldsignicantlybetter optimizationofdelayandnoisecomparedtoothermethods.T hedesignercanchooseeitherofthestrategiesbasedontheneed.Thedierencebetwe enthetwostrategiesisthe wayinwhichthesortedlistiscreated.Afterthedesignispl acedandrouted,thepath delaysofallthepathsinthedesignareestimated,andareso rtedintoalistbasedon theirdelaycriticality.Themostcriticalpathintermsofd elayischosentocreategames forgatesizeoptimization.Thegamesarecreatedforeachne tinthechosenpathinthe orderfromitsprimaryoutputtoprimaryinputs.Asanexampl eforillustration,consider thechosenpathtoconsistoffourgates: A;B;C and D ,insuccessivetransitionconnected withnets:1 ; 2and3,respectively.Thegate A isdrivenbyprimaryinputsandgate D drivesaprimaryoutput.Inordertoconsiderthedown-strea mloadcapacitance,thenet3 connectinggates C and D shouldbeoptimizedbeforethenets1and2.Thus,thegames areplayedintheorderofnet3followedbynet2followedbyne t1.Thegameformulated foreachnetinvolvesitsdriver,receiveranditstwomostag gressorgates.Thetwomost aggressorgatesforthenetanditsstrategiesareidentied asindicatedinsection7.3.1. Afterthegamesareplayedforallthenetsofthechosencriti calpathinitsdirectionof primaryoutputtoprimaryinputs,thenextcriticalpathint hesortedlistisselectedto playgames.Thisprocessofcreatinggamesisrepeateduntil thesortedlistisempty. 7.4ExperimentalResults Thestochasticgametheoreticgatesizesolver(SGGS)descr ibedinAlgorithm7.1. wasimplementedinCandexecutedonaUltraSPARC-IIe650MHz ,512MBSunBlade150 systemrunningSolaris2.8onit.Wehavetestedourgatesizi ngapproachonseveralmedium andlargeIPcoresobtainedfromOpencores[62].Astandardc elllibrarycontaining10 logiccellswithupto8dierentdrivestrengthsbasedona6Metallayer,180nmtechnology hasbeendevelopedandused.ASICdesigns,writteninbehavi oralVHDL/Verilogare convertedtostructuralVHDL/Verilogusingthestandardce llsinthelibrarywiththehelp 102

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ofBuildGates,anRTLsynthesistoolofCadencedesignsyste ms.Wehavemodiedthe ASICdesignssuchthatalltheblocksinthedesignareratten edtostandardcellsinthe librarywithoutmaintainingthehierarchy.Theon-chipmem orymodulesarerealizedas D-ripropregisterarrays.ThestructuralVHDL/Verilogdes ignisusedasinputbyCadence FirstEncountertodeveloptheroorplan.Wehavesettheopti onofrowutilizationto70% forallthedesignssoastoallowgatesizescaling.Thedesig nisthenplacedandrouted usingAmoebaplaceandNanorouterespectively,whicharepa rtofCadenceFirstEncounter tool.ThenalplacedandrouteddesignisthenexportedinDE Fformat. Theparasiticinformationfromtherouteddesignisextract edinSPEFformatusing SynopsysStarRCXT.Theinterconnectresistance,intercon nectcapacitance,interconnect inductance,couplingcapacitancesalongwiththeiraggres sordriversareextractedfromthe SPEFleandisgivenasinputtoourSGGS.Thelengthofoverla pwiththeaggressor netsandtheirspacingisextractedfromtheDEFleandisals ogivenasinput.The calculatedgatesizeforeachgateisusedtoupdatetheorigi nalDEFletogeneratean optimizedDEFle.ItcanbenotedthattheoptimizedDEFlei screatedwiththehelp ofgawkscriptandveriedforDRCrules.Thedesignisnotreroutedtogeneratethe optimizedDEFle.Thedelayandcrosstalknoiseviolations aremeasuredusingCadence FirstEncounter'stimingclosureandCelticICrespectivel ywiththeirrobustmodels,and notusingtheanalyticalmodelsusedduringoptimization. 103

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Table7.1.ComparisonofStochasticGameTheoreticApproac hwithDeterministicandGeometricProgrammingApproach OpenCoreDesign[62] TotalGates DeterministicApproach GPApproach[37] SGT-NCApproach 1 SGT-DCApproach 2 RunTime(mins) %Yield RunTime(mins) %Yield RunTime(mins) %Yield RunTime(mins) %Yield Improvement Improvement Improvement Improvement Timing Noise Timing Noise Timing Noise Timing Noise PCI 7882 4.29 47.48 39.07 24.37 88.19 128.12 95.34 96.92 135.98 98.38 92.76 ATA 21781 10.49 53.94 42.47 73.52 90.41 376.04 89.75 94.72 391.68 95.29 91.46 RISC 34172 13.31 32.46 48.32 111.32 84.96 504.78 91.35 96.57 487.83 98.06 90.66 AVR P 41274 15.79 46.18 48.02 135.49 93.36 621.29 97.56 98.04 633.15 97.81 95.69 P16C55 52128 19.98 28.95 31.45 171.55 83.59 783.10 93.75 97.11 768.85 98.72 95.86 T80 C 69973 27.67 30.27 22.74 278.92 79.21 947.54 95.08 98.69 928.61 98.17 96.33 Noareaoverheadforallfourapproaches.Alltheapproaches areimplementedatpostlayoutlevel 1 SGT-NCApproach:SGGSalgorithmwithgatesorderedbasedon noisecriticalityforsimultaneousminimizationofdelay uncertaintyandcrosstalknoise 2 SGT-DCApproach:SGGSalgorithmwithgatesorderedbasedon delaycriticalityforsimultaneousminimizationofdelay uncertaintyandcrosstalknoiseRunTime :runningtimeofeachalgorithm; Yield :fractionoftestinstanceswithouttimingornoiseviolati ons; TimingViolations : setupandholdtimeviolations; NoiseViolations :netswithnoisevaluesgreaterthan0.15 V dd104

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Inordertointroducetheuncertaintyduetoprocessvariati onsinbenchmarkcircuits,we havemodeledtheprobabilitydistributionastheGaussian G ( ; ),with asthenominal gatesizeand3 asthe25%ofthenominalgatesize.Wehavegenerated10,000r andom samplesforthegatesizesobtainedfromdierentgatesizin gapproachesusingaGaussian distributionfortheprocessvariations.Thiscreates10,0 00instancesofthesamedesign withdierentgatesizesarounditsnominalvaluesobtained bythegatesizingapproaches. Thetimingyieldofadesignismeasuredasthefractionoftes tinstanceswithoutset-up andholdtimeviolations,whilenoiseyieldismeasuredasth efractionoftestinstances whosenetshaveannoiseinducedofexceedingathresholdval uesetasto0.15 V dd Theexistingworks[34,41,42,43,35,36,37]onstatistical gatesizingintheliterature comparetheirresultswithadeterministicapproach,butdo notprovideancomparisonwith otherstatisticalgatesizingworks.Wehavecomparedourre sultswithadeterministicgame theoreticgatesizingproposedinChapter5andgeometricpr ogrammingbasedstatistical gatesizingproposedin[37].Thegatesizingalgorithmgive ninChapter5isimplemented asisandGaussiandistributionisusedtocreate10,000inst ancesofthedesignwithgate sizesaroundtheresultednominalvaluesgeneratedbytheal gorithm.In[37]theauthors haveprovidedageometricprogrammingapproachfortradeo betweenareaandrobustness. Inordertoprovideafaircomparison,wehaveimplementedth egeometricprogramming approachasillustratedin[37]forzeroareaoverhead. TheexperimentalresultsarepresentedinTable7.1..Colum nsoneandtwogivethe nameofthebenchmarkandthetotalnumberofgatesinthedesi gnrespectively.The columnsthree,six,nineandtwelvegivetheruntimeofdeter ministic(seeChapter5),geometricprogramming[37],stochasticgameswithgamesorde redaccordingtonoise(refer Section7.3.1)andstochasticgameswithgamesorderedacco rdingtodelay(referSection 7.3.2)respectively.Thecolumnsfour,seven,tenandthirt eengivethetimingyieldresultsforthefourapproachesintheirrespectiveorder,whi lecolumnsve,eight,eleven andfourteengivethenoiseyieldresultsrespectively.Itc anbeobservedthattheSGGS 105

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approachhasbettertimingandnoiseyieldswhencomparedto geometricprogrammingat theexpenseofincreasedruntimes.7.5Conclusions Optimizingbothdelayuncertaintyandcrosstalknoiseisex tremelycriticalindeep submicronandnanoregimecircuits.Thetransitionofstate sandthepayostoplayers ineachstateofstochasticgamesarebasedontheprobabilit ydistribution,whichenables thedesignertocapturethevariationsinprocessparameter s.Hence,theuseofstochastic gamesessentiallymakesitpossibleformulti-agentoptimi zationundertheimpactofprocess variations.ThetimecomplexityofndingtheNashequilibr iumofeachstochasticgame isproportionalto m T ,where m and T areparameterswithsparesamplingalgorithm basedsolutionandarenotcontrolledbythecircuitsize.Th evaluesusedfor m and T provideatradeobetweenthequalityofNashsolutionandth eruntime.Thedesignercan choosethesevalueslogisticallywhichmatchestohisneeds .Theproposedstochasticgame theoreticapproachachievesbettertimingandnoiseyields whencomparedtodeterministic andgeometricprogramming. 106

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CHAPTER8 CONCLUSIONSANDFUTUREWORK ThecurrenttrendsofdeviceandinterconnectscalinginCMO Scircuitsincontinuing eortstosatisfyMoore'slawhavebroughttolightanumerou scomplexproblemsfor bothchipdesignersandmanufacturers.Withsubmicrondevi cedimensionsandmore thanhundredmilliontransistorsintegratedonasinglechi p,theon-chipinterconnectsare playingamajorroleindeterminingtheperformance,thepow erconsumption,thesize andthereliabilityofdigitalsystems.Someoftheimportan tandcomplexproblemsin deepsubmicronandnanometerdesignsare(i)increasedinte rconnectdelayduetorising RCparasiticsofon-chipwiring,(ii)crosstalknoisedueto increasedcouplingcapacitance betweenadjacentwires(iii)increasedpowerdissipationd uetointerconnects(iv)delay andnoiseuncertaintyduetoprocessvariations.Severalat temptshavebeenmadebythe researcherstominimizetheeectsofoneoftheaboveconcer nsonDSMdesignswithothers asthedesignconstraints.But,itisinterestingtonotetha ttheabovementionedproblems areinconrict-minimizingtheeectsofoneparameterwilld rasticallydeterioratethe eectsofothersparametersonthedigitalsystem.Hence,si multaneouslyoptimizingtwo ormoreparametersofthedigitalsystemsbecomesachalleng ingtask.Inthisdissertation, themainfocusisonsimultaneouslyoptimizingmultiplepar ameterssoastocollectively reducetheireectsonVLSIcircuits.Themaincontribution sareintermsofdevelopingnew methodsandalgorithmsformulti-metricoptimizationinde epsubmicronandnanometer designs. Inthisdissertation,wehavedevelopedmulti-metricoptim izationframeworkforperforminggatesizingandwiresizingatpostlayoutlevel.Ins pecic,wehavesolvedthe followingproblems: 107

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(i)Awiresizingframeworkforsimultaneousoptimizationo finterconnectdelayand crosstalknoise-atwometricoptimizationframework (ii)Awiresizingframeworkforsimultaneousoptimization ofinterconnectdelay,power andcrosstalknoise-athreemetricoptimizationframework (iii)Agatesizingframeworkforsimultaneousoptimizatio nofinterconnectdelayand crosstalknoise-atwometricoptimizationframework (iv)Agateandwiresizingframeworkforsimultaneousoptim izationofinterconnectdelay andcrosstalknoise-anintegratedframework (v)Astatisticalgatesizingframeworkforsimultaneousop timizationofdelayuncertainty andcrosstalknoise-astochasticframework (vi)Newinterconnectmodelsbasedontransmissionlines TheuseofgametheorymodelsandNashequilibriumtosolveth epostlayoutinterconnectproblemsforsimultaneousoptimizationofmultipl eparametersisuniquetothis dissertation.Thedevelopmentofastochasticframeworkus ingstochasticgamestosolve thestatisticalgatesizingproblemundertheimpactofproc essvariationsisnovel.Game theoreticandauctionmodelsarepreviouslyusedforpowero ptimizationatbehavioraland logiclevelsofdesignabstraction[70].However,thealgor ithmsdevelopedhaveanexponentialtimecomplexity.Oncontrary,inthisdissertation,we havedevelopedgametheoretic modelsandalgorithmsforpostlayoutmulti-metricoptimiz ationwith lineartimeandspace complexities .Thedevelopmentofalgorithmswithlinearcomplexitiesis anotherprimeand novelcontributionofthisdissertation,especiallyconsi deringtodaysDSMdesignswitha typicaltransistorcountofhundredmillionormore.Thepub licationsresultedfromthis dissertationaregiveninreferences[71,72,73,74]. Basedontheresultspresentedinthisdissertation,wearee ncouragedtodevelopgame theoreticandstochasticgamemodelsformulti-metricopti mizationframeworkwiththe followingfuturedirections: 108

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(i)Thegametheoreticmodelsdevelopedinthisdissertatio ncanbeusedtosolvethe multi-metricoptimizationproblemusinggatesizingatlog iclevelorRTLlevel. (ii)Buerinsertionorrepeaterinsertionisanothertechn iquewhichiswidelyusedto minimizetheinterconnecteectsonlonginterconnectwire s.Thistechniquewould beeectiveifusedatlayoutlevelwhileperformingroorpla nningoratlogiclevel.A gametheoreticframeworkcanbedevelopedtosolvethebuer insertionproblemfor multi-metricoptimization. (iii)Wehavedevelopedastochasticframeworkforsolvingt hestatisticalgatesizingproblemconsideringtheuncertaintyinprocessparametersduet oprocessvariations.This canbeextendedtosolvethestatisticalbuerinsertionpro blembothduringroorplanningandlogicsynthesis. (iv)InChapter6,wehavedevelopedanintegratedframework whichcansimultaneously performbothgateandwiresizing.IncurrentDSMdesigns,th ereisaprimenecessity fordevelopingsuchintegratedframeworkswhichcansimult aneouslyapplytwoor moredesigntechniques.Hence,agametheoreticframeworkw hichiscapableof performingbuerinsertionandgatesizingorbuerinserti onandwiresizingorall threetogethercanbeapossibilityafterdevelopinggameth eoreticalgorithmsfor buerinsertion. (v)Inthisdissertation,wehavedevelopedanobjectivefun ctionwhichcaneectively handletwodesignparameters.Wehavealsoattemptedtoopti mizethreedesign parametersinSection3.4.But,theobjectivefunctionused wasinecientforhandlingthreeparameters,asseenfromtheexperimentalresul ts.Hence,thereisaneed toformulateaneworimprovetheexistingobjectivefunctio nwhichcaneectively handlethreeormoredesignparameters. (vi)InChapter7,thetimecomplexityofthedevelopedstoch asticgamebasedgatesizing algorithmisproportionalto m T ,where m and T areconstants.But,stillthisconstant 109

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isexponential.Thereasonforthisexponentialconstantis duetotheuseof Sparse Sampling algorithmtosolvetheNashequilibriumofthestochasticga mes.Toimprove thetimecomplexityoftheproposedstochasticframework,n ewalgorithmsforsolving theNashequilibriumin2-playerstochasticgameshavetobe developed. 110

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[59]C.J.Alpert,A.Devgan,andS.T.Quay.Iswiretaperingw orthwhile?In Proc.of ICCAD ,pages430{435,1999. [60]S.P.Khatri,R.K.Brayton,A.L.Sangiovanni-Vincente lli. Cross-TalkNoiseImmune VLSIDesignusingRegularLayoutFabrics .KluwerAcademicPublishers,2001. [61]S.Kakutani.AGeneralizationofBrouwer'sFixedPoint Theorem. DukeJournalof Mathematics ,8:457{459,1941. [62]OpenCores.FreeOpenSourceIPCoresandChipDesign.ht tp://www.opencores.org. [63]CRETE.CadenceRepositoryforElectronicTechnicalEd ucation. http://crete.cadence.com. [64]GALib.GAlib-AC++LibraryofGeneticAlgorithmCompon ents. http://lancet.mit.edu/ga/. [65]L.N.Dworsky. ModernTransmissionLineTheoryandApplications .JohnWileyand Sons. [66]LANCELOT.APackageforLarge-ScaleNonlinearOptimiz ation. http://www.numerical.rl.ac.uk/lancelot/blurb.html. [67]H.Hashimoto,andH.Onodera.Increaseindelayuncerta intybyperformanceoptimization.In Proc.ofISCAS ,pages379{382,2001. [68]P.Ghanta,S.Vrudhula,R.Panda,andJ.Wang.Stochasti cPowerGridAnalysis ConsideringProcessVariations.In Proc.oftheDATE ,2005. [69]M.Kearns,Y.Mansour,andS.Singh.FastPlanninginSto chasticGames.In Proc. ofthe16thConf.onUncertaintyinArticialIntelligence ,pages309{316,2000. [70]A.Murugavel.Newmethodsfordynamicpowerestimation andoptimizationinVLSI circuits.Ph.D.Thesis,USFLIBRARY,TampaCirculatingCol lection-TK7885.Z9 M872003,UniversityofSouthFlorida,Tampa,FL,2003. [71]N.HanchateandN.Ranganathan.SimultaneousIntercon nectDelayandCrosstalk NoiseOptimizationthroughGateSizingUsingGameTheory. IEEETrans.onComputers(ToAppear) ,2006. [72]N.HanchateandN.Ranganathan.AGame-TheoreticFrame workforMulti-Metric OptimizationofInterconnectDelay,PowerandCrosstalkNo iseDuringWireSizing. ACMTODAES(ToAppear) ,2006. [73]N.HanchateandN.Ranganathan.ALinearTimeAlgorithm forWireSizingwith SimultaneousOptimizationofInterconnectDelayandCross talkNoise.In Proc.of Intl.Conf.onVLSIDesign ,pages283{290,2006. [74]N.HanchateandN.Ranganathan.Post-LayoutGateSizin gforInterconnectDelay andCrosstalkNoiseOptimization.In Proc.ofISQED ,2006. 115

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ABOUTTHEAUTHOR NarenderHanchatereceivedtheB.E.degreeinElectronicsa ndCommunicationEngineeringfromOsmaniaUniversity,Hyderabad,India,in20 00,andtheM.S.degreein ComputerScienceandEngineeringfromtheUniversityofSou thFlorida,Tampa,in2003. HeiscurrentlyworkingtowardshisPh.D.degreeinComputer ScienceandEngineering atUniversityofSouthFlorida,Tampa,FL.Heisastudentmem berofIEEEandIEEE ComputerSociety.From2000to2001,heworkedasaDesignEng ineerintheeldofVLSI andEmbeddedSystemsatHCLTechnologiesLtd,Chennai,Indi a.Hisresearchinterests includethedevelopmentofmethodologiesforlow-powersyn thesis,multi-metricoptimization,developmentofmethodologiesandtoolsforinterconn ectoptimization,andprocess variationsindeepsubmicrondesigns.


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A game theoretic framework for interconnect optimization in deep submicron and nanometer design
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ABSTRACT: The continuous scaling of interconnect wires in deep submicron (DSM)circuits result in increased interconnect delay, power and crosstalk noise. In this dissertation, we address the problem of multi-metric optimization at post layout level in the design of deep submicron designs and develop a game theoretic framework for its solution. Traditional approaches in the literature can only perform single metric optimization and cannot handle multiple metrics. However, in interconnect optimization, the simultaneous optimization of multiple parameters such as delay, crosstalk noise and power is necessary and critical. Thus, the work described in this dissertation research addressing multi-metric optimization is an important contribution.Specifically, we address the problems of simultaneous optimization of interconnect delay and crosstalk noise during (i) wire sizing (ii) gate sizing (iii) integrated gate and wire sizing, and (iv) gate sizing considering process variations. Game the ory provides a natural framework for handling conflicting situations and allows optimization of multiple parameters. This property is exploited in modeling the simultaneous optimization of various design parameters such as interconnect delay, crosstalk noise and power, which are conflicting in nature. The problem of multi-metric optimization is formulated as a normal form game model and solved using Nash equilibrium theory. In wire sizing formulations, the net segments within a channel are modeled as the players and the range of possible wire sizes forms the set of strategies. The payoff function is modeled as (i) the geometric mean of interconnect delay andcrosstalk noise and (ii) the weighted-sum of interconnect delay, power and crosstalk noise, in order to study the impact of different costfunctions with two and three metrics respectively. In gate sizing formulations, the range of possible gate sizes is modeled as the set of strategies and the payoff function is modeled as the geome tric mean of interconnect delay and crosstalk noise. The gates are modeled as the players while performing gate sizing, whereas, the interconnect delay and crosstalk noise are modeled as players for integrated wire and gate sizing framework as well as for statistical gate sizing under the impact of process variations.The various algorithms proposed in this dissertation (i) perform multi-metric optimization (ii) achieve significantly better optimization and run times than other methods such as simulated annealing, genetic search, and Lagrangian relaxation (iii) have linear time and space complexities, and hence can be applied to very large SOC designs, and (iv) do not require rerouting or incur any area overhead. Thecomputational complexity analysis of the proposed algorithms as well as their software implementations are described, and experimental results are provided that establish the efficacy of the proposed algorithms.
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Adviser: Nagarajan Ranganathan, Ph. D.
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Game theory.
Crosstalk noise.
Interconnect delay.
Process variations.
Delay uncertainty.
Transmission line models.
Wire sizing.
Gate sizing.
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