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Algorithmic aspects of energy efficient transmission in multihop cooperative wireless networks
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Algorithmic aspects of energy efficient transmission in multihop cooperative wireless networks
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ALGORITHMICASPECTSOFENERGYEFFICIENTTRANSMISSIONIN MULTIHOPCOOPERATIVEWIRELESSNETWORKS by MarjanBaghaie ADissertationPresentedtothe FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (ELECTRICALENGINEERING) December2011 Copyright 2011 MarjanBaghaie Dedication To my dear Maman and Baba, whom I love and admire. ii Acknowledgements Iwouldliketoexpressmygratitudetomyadviser,Prof. Bhaskar Krishnamachari, for his guidance, insightful suggestions and generous support throughout these years. The levelofenthusiasmand intellectualcuriosityhebringstohisresearch hasbeen asource ofinspirationformeandissomethingIhopetobeabletoemulateinmyownresearch. IamgratefultoProf.DoritHochbaumandProf. AndreasMolisch for acting as my thesiscommitteemembers. Iamprivilegedtohaveworkedwiththem.Prof.Hocbhaum’s courseoncombinatorialoptimizationisoneofthebestcoursesIhaveevertaken. Ithank herforthelearningopportunityandforherguidanceinformulatingtheresultsinChapter 5.IalsothankProf. Molischforsharingwithmehisinsightsonmutualinformation accumulation,andforhiskindsupportduringmyinternshipinMERL. Ithankmymentors,friendsandcolleaguesattheMitsubishiElectric Research Lab (MERL) and the Qualcomm NJ Research Center (NJRC). In particular, I would like to express my gratitude to Dr. Matthew Brand, Dr. Samel Celebi, Mr. Luca Blessent, Dr. RiteshMadan,Dr. CyrilMeasson,Dr. AleksandarJovicic,andmyflatmateGhazaleh. Ithankmyfriendandlab-mateDr. ScottMoeller;itwasapleasure working with him on the smart grid paper. I acknowledge Dr. Kyuho Son, for our ongoing work on iii thesmart grid project. I also thank all theothercurrent andpast members ofANRG for having made the experience an enjoyable and memorable one. Thanks are also due to VladandTahafor themanyfundiscussionsandgoodtimesovertheseyears. IamgratefultoallmyfriendsatUSCandLAforthegoodmemories and for the pleasure of their company. I am indebted to my good friends Mohammad and Arian, whohavebeenmydefactofamilymemberssincemyfirstdayinLA,extendingahelping hand most often without me even having to ask; and to Layla, Raj, Majid, Hooman and Terryforshowingmetheropesduringtheconfusingearlystages. Specialthanksarealso duetoDianeDemetrasandTimBostonoftheEEdepartmentforalltheirkindsupport. IexpressmydeepestgratitudetomyMomandDadfortheirunbounded support, affection, andtheir manysacrifices throughthe years. Aboveall,Ithankthemforbeing such inspirational role models. I also thank my dear sister, Nilufar, for setting the bar high and paving the way for the rest of us to follow in her footsteps; I thank my baby brother, Rezza, for his uncanny ability to put things in perspective for me whenever the goinggot tough. Iam grateful to my wonderfulin-laws, Farkhondeh, Abbas, and Mina, foralltheirkindnessandsupport. Finally, I express deep love and gratitude to my husband, Amin, who after three yearsstillfeelstoogoodtobetrue! Thanksforopeningmyeyestonewhorizonsandfor makingmyworldsomuchmorecolorful. MyonlyregretisthatthetimeIspentworking onthisdissertation,Icouldhavebeenspendingwithyouintheeastcoast. iv Table of Contents Dedication ii Acknowledgements iii ListofTables vii ListofFigures viii Abstract ix Chapter1: Introduction 1 1.1 ThesisFocus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 MainContributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 ThesisStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 OrganizationoftheThesis . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter2: BackgroundandRelatedWork 9 2.1 ComputationalComplexity . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 CooperativeCommunication . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Chapter3: GeneralizedAlgorithmicFormulation&ComputationalComplex- ity 23 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 SystemModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 GeneralizedFormulation . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 ComputationalComplexity . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4.1 InterpretingtheSet CoverProbleminNetworkingContext . . . 31 3.4.2 InapproximablityofDMECB . . . . . . . . . . . . . . . . . . 32 3.4.3 InapproximablityofDMECM . . . . . . . . . . . . . . . . . . 36 3.4.4 HardnessResultsforDMECU . . . . . . . . . . . . . . . . . . 36 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 v Chapter4: MinimumEnergyDelayConstrainedTransmission 41 4.1 OptimalTransmissionGivenOrdering . . . . . . . . . . . . . . . ... 42 4.1.1 Instantaneousoptimalpowerallocation . . . . . . . . . . ... 42 4.1.2 JointSchedulingandpowerallocation . . . . . . . . . . . . ..43 4.2 OptimalUnicastwithEnergyAccumulation . . . . . . . . . . . .... 45 4.3 ApproximationAlgorithmforBroadcast withEnergyAccumulation . . 48 4.4 PerformanceEvaluaion . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.5 DecentralizedImplementation . . . . . . . . . . . . . . . . . . . . ..60 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Chapter5: TransmissioninPresenceofInterferingFlows 65 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.2 ProblemFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3 SpecialCaseofk=1........................... 75 5.4 InapproximabilityResults . . . . . . . . . . . . . . . . . . . . . . . ..77 5.5 PerformanceBounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.5.1 AnAnalyticalLowerBound . . . . . . . . . . . . . . . . . . . 80 5.5.2 AnAnalyticalUpperBound . . . . . . . . . . . . . . . . . . . 81 5.6 APolynomial-timeHeuristic . . . . . . . . . . . . . . . . . . . . . . . 82 5.7 PerformanceEvaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Chapter6: ConclusionsandFutureDirections 91 References 95 AlphabetizedReferences 103 vi ListofTables 3.1 Summaryofthealgorithmicnegativeresults. . . . . . . . . ...... 40 4.1 Summaryofthealgorithmicpositiveresults. . . . . . . . . ....... 64 vii ListofFigures 2.1 Anodereceivinginformationfrommultiplesources. . . ........ 17 3.1 ConstructionofG ! fora givenGinDMECB. . . . . . . . . . . . . . . 33 3.2 ConstructionofG ! fora givenGinDMECU. . . . . . . . . . . . . . . 37 4.1 AsimplifiedexampleofhowclustersareconstructedinG ! .. . . . . . . 51 4.2 PerformancewithoptimalorderingvsDijkstra’salgorithm-basedheuris- ticordering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3 Effectofcooperationforvaryingnetworksize. . . . . . . ....... 56 4.4 Power-delaytradeoffincooperativevsnon-cooperativecase. . . . . . . 57 4.5 Energyaccumulationvsmutualinformationaccumulation. . . . . . . . 58 4.6 Effectofnetworkdensityonpower-delaytradeoff. . . . ........ 59 4.7 Different η valuesforinformationaccumulation. . . . . . . . . . . . . 60 5.1 Applyingthemulticommodityflowtechniqueforunicastcast . . . . . . 74 5.2 An example ofk> 2,withT =3,wheretheoptimalsolutionisnota singlepath. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.3 ExampleconstructionofG ! ,foragivenG.. . . . . . . . . . . . . . . . 79 5.4 Performanceoftheheuristicagainsttheanalyticalupperandlowerbound. 88 5.5 Effectofchanneldegradationonthetotalenergyconsumed. . . . . . . 89 viii Abstract We consider the problem of energy-efficient transmission incooperativemultihopwire- less networks. Although the performance gains of cooperative approaches are well known,becauseofthecombinatorialnatureoftheseschemes,designingefficientpolynomial- time algorithms to decide which nodes should take part in cooperation, and when and with what power they should transmit, has remained a challenge. We propose to tackle thisprobleminthisdissertation. We provide a generalized algorithmic formulation of the problem that encompasses the two main cooperative approaches, namely: energy accumulation and mutual Infor- mation accumulation. We investigate the similarities and differences of these two ap- proachesunderourgeneralizedformulation,focusinginparticularonthescenariowhere adelayconstraintispresent. Weprovethatthebroadcastandmulticastproblemsare, in general, o(log(n)) inapproximable. We break this NP hard problem into three parts: ordering, scheduling and power control and proposea generalized novel algorithm that, given an ordering, can optimally solve the joint power allocation and scheduling prob- lemssimultaneouslyinpolynomialtime. Wefurthershowempiricallythatthisalgorithm used in conjunction with an ordering derived heuristically using the Dijkstra’s shortest ix path algorithm yields near-optimal performance in typical settings. In the unicast case, weprovethatalthoughtheproblemremainshardwithmutualinformationaccumulation, itcanbesolvedoptimallyandinpolynomialtimewhenenergyaccumulationisused. We useouralgorithmtostudy,numerically,thetrade-offbetweendelayandpower-efficiency incooperativebroadcastandcomparetheperformanceofenergyaccumulationvsmutual informationaccumulationaswellastheperformanceofourcooperativealgorithmwitha smartnon-cooperativealgorithminabroadcastsetting. WealsoprovideanO(T log 2 (n)) approximationalgorithmforthebroadcastcasewhereenergyaccumulationisused. We further formulate the problem of minimum energy cooperative transmission in adelayconstrainedmultiflowmultihopwirelessnetwork,asacombinatorialoptimiza- tionproblem,forageneral settingofk-flows andformallyprovethatthe problemis not only NP-hard but it is o(n 1/7−! ) inapproxmiable. To our knowledge, the results in this dissertation provides the first such inapproxmiablity proofinthecontextofmultiflow cooperative wireless networks. We show that for a special case of k =1,thesolution is a simple path and develop an optimal polynomial time algorithm for joint routing, scheduling and power control. We then use this algorithm to establish analytical upper and lower bounds for the optimal performance for the general case ofk flows. Further- more,weproposeapolynomialtimeheuristicforcalculatingthesolutionforthegeneral case and evaluate the performance of this heuristic under different channel conditions andagainsttheanalyticalupperandlowerbounds. x Chapter1 Introduction Inawirelessnetwork,atransmitsignalintendedforonenodeisreceivednotonlybythat node but also by othernodes. In a traditional point-to-pointsystem,wherethereisonly oneintendedrecipient,thisinnatepropertyofthewirelesspropagationchannelcanbea drawback, as the signal constitutes undesired interferenceinallnodesbuttheintended recipient. However, this effect also implies that a packet can be transmitted to multiple nodes simultaneouslywithoutadditionalenergy expenditure. Exploitingthis“broadcast advantage”, broadcast, multicast and multihopunicast systemscan be designed to work cooperatively and thereby achieve potential performance gains. As such, cooperative transmission in wireless networks has attracted a lot of interest not only from the re- search community in recent years [36,37,39,41–44] but also from industry in the form ofpracticalcooperativemobilead-hocnetworksystems[49]. 1 1.1 ThesisFocus We focus on the problem of cooperative transmission in this work, starting with a case whereasinglenodeissendingapackettoeithertheentirenetwork(broadcast), asingle destination node (unicast) or more than one destination node(multicast),inamultihop wireless network. Other nodes in the network, that are neither the source nor the des- tination, may act as relays to help pass on the message throughmultiplehops. The transmissionis completed when all thedestinationnodes have successfullyreceived the message. We particularly focus on the case where there is a delay constraint, whereby thedestinationnode(s)shouldreceivethemessagewithinthedelayconstraint,however, we also discuss how our results apply to the unconstrained case. We also look at the case where there are morethan one sources, each tryingto sendtheirmessage(possibly throughrelays)totheircorrespondingdestinations. Akeyprobleminsuchcooperativenetworksisroutingandresource allocation, i.e., the question which nodes should participate in the transmission of data, and when, and withhowmuchpower,theyshouldbetransmitting. Thesituationisfurthercomplicated bythefactthattheroutingandresourceallocationdependsonthetypeofcooperationand other details of the transmission/reception strategies of the nodes. We consider a time- slottedsysteminwhichthenodesthathavereceivedanddecodedthepacketareallowed to re-transmitit in futureslots. Duringreception, nodesperceive added up signal power (energy accumulation, EA), or the added up mutual information (mutual information accumulation, MIA) received from multiple sources. EA, which has been discussed in 2 priorwork[37,39,41,42],canbeimplementedbyusingmaximalratiocombining(MRC) oforthogonalsignalsfromsourcenodesthatuseorthogonaltime/frequencychannels,or spreading codes, or distributed space-time codes. MIA can beachievedusingrateless codes[44,45]. Althoughthesetechniquesareoftentreatedseparatelyintheliterature,we shall see how our formulation of the problem encompasses bothapproachesandallows manyoftheresultstobeextendedtoboth. We furthermore assume that the nodes are memoryless, i.e., accumulation at the re- ceiver is restricted to transmissions from multiple nodes inthepresenttimeslot,while signals from previous timeslots are discarded. This assumption is justified by the lim- ited storage capability of nodes in ad-hoc networks, as well as the additional energy consumption nodes have to expand in order to stay in an active reception mode when theyoverhearweak signalsinprecedingtimeslots. Notethatmuchoftheliteraturecited abovehasusedtheassumptionofnodeswithmemory,sothattheirresultsarenotdirectly comparabletoours. Akeytradeoffisbetweenthetotalenergyconsumption 1 andthetotaldelaymeasured intermsofthenumberofslotsneededforalldestinationnodesinthenetworktoreceive the message. At one extreme, if we wish to minimize delay, eachtransmittingnode should transmit at the highest power possibleso that the maximumnumber of receivers can decode the message at each step (indeed, if there is no power constraint, then the source node could transmit at a sufficiently high power to reach all destination nodes in 1 Asweconsiderfixedtimeslotdurations,weusethewordsenergyandpowerinterchangeablythrough- out. 3 the first slot itself). On the other hand, reducing transmit power levels to save energy, mayresultinfewernodesdecodingthesignalateachstep,andthereforeinalongertime to complete the transmission. We therefore formulate the problem of performing this transmissioninsuchawaythatthetotaltransmissionenergyoveralltransmittingnodes isminimized,whilemeetingadesireddelayconstraintonthemaximumnumberofslots thatmaybeusedtocompletethetransmission. Thedesignvariableinthisproblemisto decidewhichnodesshouldtransmit,when,andwithwhatpower. 1.2 MainContributions Thekeycontributionsofthisdissertationareasfollows: • We formulate the problem of minimum energy transmission in cooperative net- works. Althoughthe priorliterature have focusedon either EA ([37,42])orMIA ([44,45])andhavetreatedthemseparately,ourgeneralizedformulationcantreat bothmethodsasvariationsofthesameproblem. • Our formulation of delay-constrained minimum energy broadcast in cooperative networks, goes beyond the prior work in the literature on cooperative broadcast whichhasfocusedeitheronminimizingenergywithoutdelayconstraints[37,42], or on delay analysis without energy minimization [67]. Our extended problem formulationallowsustoexposeandinvestigatetheenergy-delaytradeoffsinherent incooperativenetworking. 4 • We not only prove that the delay constrained minimum energy cooperativebroad- cast(DMECB)andmulticast(DMECM)problemsareNP-complete 2 ,butalsothat they areo(log(n))-inapproximable (i.e., unlessP = NP,itisnotpossibletode- velopa polynomialtime algorithmfor this problem thatcan obtainasolutionthat isstrictlybetterthanalogarithmic-factoroftheoptimuminall cases). Weare not aware of prior work on cooperative broadcast or multicast that shows such inap- proximabilityresults. • Weshowthatthedelayconstrainedminimumenergycooperativeunicast(DMECU) problem is solvablein polynomialtime using EA but is NP-complete using MIA. We are unaware of any hardness results on unicast approachesusingmutualinfor- mationaccumulation. • For the cases where we prove the transmission problem to be hard, we are able to show that for any given ordering of the transmissions (which dictates that a node laterintheorderingmaynottransmitbeforethenodesearlierintheorderinghave decoded successfully), then the problem of joint schedulingandpowerallocation caninfactbesolvedoptimallyinpolynomialtimeusingacombinationofdynamic programming for the scheduling and convex optimization or linear programming forthepowerallocation. 2 Throughout the thesis, the terms NP-complete and NP-hard might be used interchangably when re- ferring to the hardness of the same problem. The reader shouldnotethattheformerisreferringtothe decisionversionoftheproblemandthe lattertothe optimizationversionofthe sameproblem. Chapter2 providesabackgroundontheseconcepts. 5 • For small network instances, we computethe optimalsolutionthroughexhaustive search,andshowempiricallythroughsimulationsthatourproposedjointschedul- ingandpowercontrolmethodworksnear-optimallyintypicalcaseswhenusedin conjunctionwithanorderingprovidedbytheDijkstratree construction. • Wealsoshowthroughsimulationsthedelay-energytradeoffsandminimumenergy performance for larger networks and demonstrate the significant improvements that can be achieved by our solution compared to non-cooperative broadcast. We further comparethe performance ofour proposedbroadcast algorithmunderMIA andEAapproaches. • For DMECB where EA is used, we present a reduction that would allow for a polynomial time algorithm for the joint ordering-scheduling-power control prob- lem that is provably guaranteed to offer a O(n ! ) approximation, for any " > 0. This algorithm is based on the current best-known algorithm for the bounded di- ameter directed Steiner tree problem [54]. Using the same reduction, we can also getan approximationfactorofO(T log 2 (n)) for afixed delayconstraintT.Given that DMECB is o(log(n)) inapproximable for anyT> 2,thisprovidesafairly tightapproximation,especiallyforsmallT. • We further formulate the problem of minimum energy cooperative transmission in a delay constrained multiflow multihop wireless network, as a combinatorial optimizationproblem, for a general settingofk-flows and formally provethat the problem not only NP-hard but it iso(n 1/7−! ) inapproxmiable. To our knowledge, 6 the results in this dissertation provide the first such inapproxmiablityproof in the contextofmultiflowcooperativewirelessnetworks. • We observe that for a special case of k =1,thesolutionisasimplepathand offered an optimal polynomial time algorithm for joint routing, scheduling and power control. We then use this algorithm to establish analytical upper and lower boundsfortheoptimalperformanceforthegeneralcaseofk flows. • Furthermore,we proposea polynomialtimeheuristicschemetoaddresstheprob- lemofminimumenergycooperativetransmissioninadelayconstrainedmultiflow multihop wireless network for the general case and evaluate the performance of this heuristic under different channel conditions and against the analytical upper andlowerbounds. 1.3 ThesisStatement Thethesisstatementcan besummarizedasfollows: Energy efficient transmissionin delay constrainedcooperativewireless net- works is computationallyhard in general,however, careful considerationof the combinatorial structure of the problem can yield (near-)optimal algo- rithmsin typicalsettings. 7 This dissertationmakes several contributionsthat significantly enhance our understand- ing of complexity and algorithm design for cooperative transmission in wireless net- works. The summary of the algorithmic results developed in this thesis for the single- flowproblemarepresentedinTables3.1and4.1andtheresultsforthemultiflowproblem arehighlightedinchapter5. Itisworthnoticingthatalthoughwemaintainedamemory- less assumption throughout, all the negativeresults presented in Table 3.1 extend to the casewhere thereisnomemory. 1.4 OrganizationoftheThesis Therestofthisthesisisorganizedasfollows: Chapter2providesabriefhistoryofcoop- erative communication and places this dissertation in the context of prior related work. It also highlightssome of the key concepts in computational complexity theory that are later used throughout the thesis. References are provided for the interested reader to sourceswithcomprehensivediscussionsofeachtopic. Chapter3providesageneralized formulation of energy-efficient transmission problem in cooperative multihop wireless networks, encompassing both EA and MIA. We further establishhardnessresultsfora variety of settings in that chapter. In chapter 4, we propose approximation algorithms and positive results for the hard problems described in chapter 3, and evaluate the per- formance of these algorithms using simulations. The multiflow problem is discussed in chapter 5, where we investigate the delay constrained minimum energy problem in presence of interfering flows. Concluding comments and directions for future work are discussedinchapter6. 8 Chapter2 BackgroundandRelatedWork In this chapter we provide a brief tutorial on key concepts in computational complexity theory that will be used later on in this thesis. We also provide a background on the conceptofcooperationandbrieflydiscussthehistoryofcooperationinwirelesscommu- nicationandhighlightthecurrentstateofartasitrelatetothetopicofthisdissertation. 2.1 ComputationalComplexity Inthissectionwebrieflyreviewsomeofthemostbasicconceptsofcomputationalcom- plexity, including NP-hardness and NP-completeness that will be used later on in the thesis. Thediscussionsinthischapterarelargelyfrom[3,4],andtheinterestedreaderis referred tothesesourcesandthereferences thereinforamorethoroughdiscussion. Computational complexity theory 1 is a branch of the theory of computation in the- oretical computer science and mathematics that focuses on classifying computational 1 DefinitionadoptedfromWikipedia,thefreeencyclopedia. 9 problems according to their inherent difficulty [3,74]. In this context, a computational problem is understood to be a task that is in principle amenable to being solved by a computer (which basically means that the problem can be stated by a set of mathemat- ical instructions). Informally, a computational problem can be viewed as an infinite collection of instances together with a solution for every instance. The input stringfor acomputationalproblemisreferredtoasaprobleminstance,andshouldnotbecon- fused with the problem itself. In computational complexity theory, a problem refers to the abstract question to be solved. In contrast, an instance of this problem is a rather concrete utterance, which can serve as the input for a decision problem 2 .Forexample, primality testingis the problem ofdeterminingwhether a givennumberis primeor not. Theinstancesofthisproblemarenaturalnumbers,and thesolutiontoan instanceisyes ornobasedonwhetherthenumberisprimeornot. The running time of many of the algorithms we encounter are bounded by some polynomial in the size of the input. These algorithms are efficient algorithms, and the corresponding problems are traceable.Inotherwords,wesayanalgorithmisefficient if its running time is O(P(n)),where P(n) is a polynomial in the size of the input n. Theclassofallproblemsthatcan besolvedby efficientalgorithmsisdenotedbyP (for polynomialtime). 2 Adecisionproblemisaquestioninsomeformalsystemwithonly a yes-or-noanswer,dependingon thevaluesofsomeinputparameters. 10 There are also many problems for which no polynomial time algorithm is known. Some of these problemsmaybe solvedby efficient algorithmsthat are yet to bediscov- ered. Formanysuchproblemshowever,thereisastrongbeliefthattheycannotbesolved efficiently. It is desirable to be able to identify such problems, so one does not have to spendtimesearch fornon-exissenetalgorithms. Onespecialclassofsuchproblemsthat we are interested in is a class of decision problems called NP-complete problems [13]. We can group these problems in one class because they are all equivalent in a strong sense, there exists an efficient algorithm for any on NP-complete problem if and only if there exist efficient algorithms for all NP-complete problems. NP-complete is a sub- setofNP,thesetofalldecisionproblemswhosesolutionscanbeverifiedinpolynomial time. Incomputationalcomplexitytheory,NPisoneofthemostfundamentalcomplexity classes. The abbreviation NP refers to “nondeterministic polynomial time”. The com- plexity class P is also contained in NP, but NP contains many important problems, the hardest ofwhichare called NP-completeproblems,forwhichno polynomial-timealgo- rithms are known. The most important open question in complexity theory, the P = NP problem,askswhethersuchalgorithmsactuallyexistforNP-complete,andbycorollary, allNPproblems. Itiswidelybelievedthatthisisnotthecase[29]. NP-hard, non-deterministic polynomial-time hard, is a class of problems [3,4] that are, informally, at least as hard as the hardest problems in NP.NP-hardproblemsmay beofanytype: decisionproblems,searchproblems,oroptimizationproblems. 11 The above mentioned notions are the basis for an elegant theory that allows us to identify the problems for which no polynomial algorithm is likely to exist. But proving that a given problem is hard does not make it go away, we still need to solve the prob- lem! However, given that a polynomial algorithm is unlikely to exist, we need to make compromises. Themostcommoncompromisesconcerntheoptimality,robustness,guar- anteed efficiencyor the completenessof thesolution. Analgorithmthatmay notleadto theoptimal(precises)result iscalled an approximationalgorithm.Ofparticularinterest are approximation algorithms that can guarantee a bound on the degree of imprecision. We will see an example of such algorithms in chapter 4. 2.2 CooperativeCommunication Inthissectionweprovideabriefbackgroundoncooperationconceptandabriefhistory of cooperative communication, based largely on the materials in [1,2,9,10] and the referencestherein. Theinterestedreaderisreferredtothesesourcesforamorethorough background. Wealsohighlightthestateoftheartrelatedtothethepremissofthisthesis, inparticularcross-layertechniquesforcooperativetransmissioninmultihopnetworks. The word cooperate derives from the Latin words co- and operate (to work), con- noting the idea of working together.Cooperationisthestrategyofagroupofentities workingtogether to achievea commonor individualgoal[1]. Cooperation has been the subject of intensive study in mathematics, artificial intelligence, social and biological sciences. Examples of cooperation can be found in different areas ranging from animal 12 behavior in nature, including population of ants, termites,bees,huntinglions,vampire batstohumaninteractions,toinformationsystemsandsuccessofopensource[5–8]. Wireless networks provide yet another realm in which cooperationamong groupsof entities can be attained, provided that the right framework can be designed and imple- mented. Cooperative communication has become one of the fastest growing areas of research in wireless communicationin recent years. The key ideain user-cooperation is the resource-sharing among multiple nodes in the network., which would often lead to savings of overall network resources. An enormous application space for user coopera- tionstrategiesisMeshnetworks[1,2]. Cooperationispossiblewhenevertherearemorethantwocommunicatingterminals. As such, a three-terminal network, introduced by [11,12], can be thought of as a sim- plest form and a fundamental unit of user cooperation and as such has been the subject of intensestudies [2] . Indeed, a vast portion of the literature, especially in the realm of informationtheory,has been devotedtoa special three-terminalchannel, labeled the re- laychannel. Inhisoriginalwork,vanderMeulendiscoveredupperandlowerboundson thecapacityoftherelaychannel,andmadeseveralobservationsthatledtoimprovement of his results in later years. The capacity of the general relay channel is still unknown, but the mostprominent work on relaying to date is [14], in whichtheauthors developed lower and upper bounds on the channel capacity for specific non-faded relay channel models. Most of the results in this work have still not been superseded [1], however there has been a lot of work in the area which has improved our understanding of the 13 problem [15–20]. In particular, several works have studied the capacity of relay chan- nelsanddevelopedcodingstrategiestoachievetheergodiccapacityofthechannelunder certainscenarios(see[20]andthereferences therein). The terms decode-and-forward and amplify-and-forward are introduced in [21,22], where theauthors proposedifferent cooperativediversityprotocolsand analyzethe per- formance in terms of outage behavior. In the former protocol,relaysreceiveandde- code the signal transmitted by the source, before forwardingthedecodedmessageto the destination. The destination combines the copies in a proper way. The latter proto- col works by the relay amplifying the received signal and forwarding it to the destina- tion. This protocol is clearly simpler, and although it amplifies noise, it can be shown to achieve spatial diversitygain if the messageis transmitted overspatiallyindependent channels. Compress-and-forward is discussed in [14,20]. More information on related distributedsourcecodingtechniques and on alternativecooperativediversitytechniques can be found in [23] and [24,25] respectively. In this work, wefocusondecode-and- forwardprotocol. Themajorityoftheworkdiscussedsofarconsiderscooperationwithveryfewnodes and in a two-hop setting. In a different direction, the authorin[26]haveproposeda new approach towards finding network information carrying capacity, which has led to research on finding scaling laws for wireless networks in a variety of settings. This work shows that an aggregate throughput scaling ofΘ( √ n) is an upper bound for what is achievable by multihop transmission. Their results however are limited, in that in 14 heir model they assume no cooperation is allowed between networks. As such, a signal notintendedtobetransmittedtoanodeistreated asinterference. However,cooperation benefitsfromthebroadcastnatureofthewirelesschannelandutilizesthisinnateproperty ofthechannelinsteadoftreatingtheoverheadsignalasinterference. In [27], the authors improve the throughput compared to traditional schemes by proposing distributed collaborative schemes over multihopnetworks,achievinganag- gregate throughput ofΘ(n 2/3 ).Theauthorsin[28]proposeahierarchicalcooperation scheme, achieving linear scaling in ad hoc networks. This means that as the number of nodes per unit area increases the throughput per node does notdrop. Thisisaninter- esting result which shows that cooperation can overcome the interference limitation in wirelessnetworks. Amorecomprehensivediscussionofthistopiccanbefoundin[1]. Although the topic of cooperation has been discussed extensively in physical layer and informationtheory, the majorityof the work has been focused on singleor two-hop settings [1,2]. Recently there has been an increase interestinacross-layerdesignfor cooperative networks. In [30–32], the authors propose a cooperative MAC protocol to introduce cooperation in 802.11 networks. The proposed protocol is shown to achieve substantialthroughput and delay performance improvementsbyintegratingcooperation at the physical layer with the MAC sublayer. The protocol recruits a single relay on the fly to support the communication of a particular source-destination pair. This work is extended in [35], where the authors propose utilizing multiple cooperative nodes by 15 developinga randomized cooperativeframework [33,34]. However,thefocus isstillon physicalandMAClayeronlywithasingle-sourcesingle-destinationsetup. In order to harvest the cooperative gains predicted by analytical models in multihop settings, one needs to take routing, scheduling and resourceallocationintoaccountas well. While the optimum networking performance strongly depends on the physical- layer technique used, often routing, scheduling and power allocation and physical layer designaretreatedseparately. Inthisthesis,wefocusoncooperativecommunicationina multihopsettingandinvestigatethecomplexityoftheproblemfromanalgorithmicpoint ofviewandthealgorithmicaspectsofaddresstheproblemofdesigningenergyefficient cross-layercooperativealgorithms. In this thesis, we focus on two of the main physical layer techniques used in the literatureconcerningcooperativecommunicationinmultihopnetworks,namely: Energy accumulation(EA) [37,39,41,42]and mutualinformationaccumulation(MIA) [44,45, 80,82]. Another technique is maximum-ratio transmission (virtual beamforming) [86– 89], not considered in this thesis. For cooperation in a non-multihop context see [1,21, 24,93,94]. Figure 2.1 information shows arriving at the receiving noder from a set of senders S.Let p s denote the power used by sender s ∈ S to transmit the message and let h sr denotethemeanchannelgainbetweennodessandr. 16 Figure2.1: Anodereceivinginformationfrommultiplesources. An ideal EA receiver can reliably decode the message so long astheaccumulated energycanexceedsomethresholdτ.Thiscanbeshownas: ! s∈S h sr p s N > τ (2.1) where N represents the noise. Notice that τ can be re-adjusted to absorb noise in this formulation. A Rake receiver, in CDMA, is a good approximation for such an EA re- ceiver the information from the different source nodes arrives with relative delays that arelargerthanthechipduration[60]. Alternatively,space-timecodes[57]couldbeused for transmission. EA can also be achieved by providing orthogonal resources for each channel [67]. Recently a commercially developed cooperative mobile ad hoc network system has been developed which utilizes a pragmatic cooperation method requiring minimal information exchange, based on a combination of phase dithering and turbo 17 codes [48,49,83]. It is shown in [48] that the performance of this pragmatic scheme is closetothatofanidealEAapproachbasedonspace-timecoding. In MIA [44,45], the receiving node accumulates mutual information for a packet frommultipletransmissionsuntilitcanbedecodedsuccessfully. Thedecodingcriterion inthiscasecanbeexpressedas: ! s∈S log " 1+ h sr p s N # > θ (2.2) where θ isthedecodingthreshold. Thiscanbeachievedusingfountaincodesanddecoders [44,45,47],sothatinforma- tionstreamsfromdifferentrelaynodescanbedistinguished,andthemutualinformation of signals transmitted by relay nodes can be accumulated. Note that in this case the CDMA system needs to used different spreading codes for different nodes in order for thedestinationtobeabletoresolvethedifferentstreams. Noticethatalthoughthistech- niqueissimilartothatofcodedcooperation(wheredifferentcodesaretransmittedfrom different nodes and thenodes can help each other, see e.g. [93]), it is different most no- tably in that the underlying source information in MIA is the same for all nodes, and the nodes can start transmissionat different times, making it particularly appealing in a multihopsetting. NoticealsothatinMIA,nodesaredesignedtouseindependentlygen- erated codes for relaying. If the same code were used by each transmitter, the receiver would get multiple looks at each codeword symbol. This would be EA; however by 18 getting looks at different codes (generated from the same information bits) the receiver accumulatesmutualinformationrather thanenergy[80]. It has been shown that one can achieve significant saving in energy and/or transmis- sion time when using an EA, compared to traditional protocols[37,42,67]. Ifenergy accumulationisachievedbytransmittingtheexact samepacket either from differentre- lays or through successivere-transmissions, the scheme is shown to achieve capacity in anasymptoticallywidebandregime[37]. Aspreviouslymentioned,inthisdissertation,weareinterestedinenergy-efficiency 3 inmultihopsettings(broadcast, multicastandunicast),inparticularasitrelatestodelay constraint. Wenowbrieflyreviewthestateoftheartinthescopeofthisthesis. Many network protocols in mobile ad hoc and sensor networks need to operate in broadcast mode to disseminate certain control messages to the entire network (for in- stance, to initiate route requests, or to propagate a query).Thesubjectofbroadcast transmission in multi-hop wireless networks has attractedalotofattentionfromthere- search community in both non-cooperative[50,51,53] and cooperative settings [37,39, 42,43,67]. Fortraditionalnon-cooperativewireless networks,Cagalj et al.[53] showed that theproblem of minimumenergy broadcast is NP-hard. In [39], Mergen et al.show throughacontinuumanalysistheexistenceofaphasetransitioninthebehaviorofcoop- erativebroadcast: ifthedecodingthresholdisbelowacriticalvaluethenthebroadcastis successful,elseonlyafractionofthenetworkisreached. In[42],MergenandScaglione, show that the problem of scheduling and power control for minimum energy broadcast 3 Fora discussionofmultihopthroughputoptimality,see [84,90–92]. 19 istractablysolvableforhighlydense(continuum)networksandshowthegainsobtained with respect to noncooperative broadcast. In [67], we examined the delay performance ofcooperativebroadcastandshowthatcooperationcanresultinextremelyfastmessage propagation, scaling logarithmically with respect to the network diameter, unlike the linear scaling for non-cooperative broadcast. We discuss the hardness of the broadcast problem with both MIA and EA in chapter 3 and provide a polynomial-time algorithm (given ordering) in chapter 4, as well as an O(T log 2 (n)) approximation algorithm for thecasewhenEAisused. Algorithmic aspects of cooperative communication and computational complexity are topics that have remained largely unaddressed in the literature. In [37], Maric and Yates address the computation in the context of cooperative broadcast with EA. Their workissimilartooursinthattheyalsoconsideraminimumenergycooperativesolution, however delay constraint is not addressed in their work. Furthermore, they consider a modelwith memory,where thenodes can savesoft informationfrom all previoustrans- missions throughout time and use it to decode data later on. They prove that the prob- lem is NP-complete in this case. In their setting, because of the memory, it suffices to haveeachtransmittertransmitonlyonce;thereforethereisnodistinctionbetweenorder- ing and scheduling. This is no longer true in our memoryless setting where the energy frompasttransmissionscannotbeaccumulated. Moreover,unicastandmulticastsettings (whicharediscussedinthisthesis)arenotconsideredin[37]. WefurtherconsiderMIA, whichisnotconsideredin[37]. 20 MIA is mostnotablydiscussedin [44,45,82]. However,thediscussionsare focused on unicast routing. The algorithms presented are heuristics, the performance of which are verified via simulations. We are not aware of any hardness results on cooperative MIAoranydiscussionsontheuseofMIAinageneral cooperativebroadcastsetting. There is very limited work addressing the power-delay tradeoff in a cooperativeset- ting which is a focus of this dissertation. One prior work addressing this issue is [60]; however,thefocusofthatworkisonspace-timecodesusedforunicast,anddoesnotdis- cuss broadcast or multicast. The authors in [82] also addresstheproblempower-delay tradeoffinthecontextofunicastwithMIA,byproposingaheuristicthatrunsasequence of LP-based route optimizationsunder increasingly tight energy constraints, revealing a trade-off between energy consumption and delay. These work however do not address theproblemofhardnessinadelayconstrainedsettinganddonotconsiderthebroadcast ormulticastproblem. Inearlierwork[68],wehadconsideredthistradeoffinabroadcast cooperative setting using EA and conjectured that many of theresultswouldextendto MIA but the investigationof that conjecture had remained an open problem. This open problem was addressed in [64]. The results of these works are discussed in chapter 3, andchapter4. Themajorityoftheworkin multihopcooperativecommunicationconsiders asingle flow. In [40],theauthorsconsidertheproblemof broadcastingindependentsourcesina densewirelessnetwork. Theycharacterizethepropagationofthesourceflowsacrossthe network and show that in the limit of an infinitely dense network, the relaying proceeds 21 inlevels. Theproblemofjointlycomputingschedules,routing,andpowerallocationfor multiple flows in cooperative networks has recently been discussed in [61–63]. These papers propose heuristics and conjecture that the problem isingeneralNP-hard. We are not aware of any work considering energy-delay tradeoff in multiflow cooperative networks, or any proof of hardness for such problems in the literature. We will address thisprobleminchapter5. Partofthatchapterhaspreviouslyappearedin[71]. 2.3 Summary In this chapter, we briefly discussed the history of cooperation in wireless communica- tion. Mentioning in particular that the historically, the majority of the literature in this area has been on information theoretic and coding aspects of cooperation in small net- works. Wethenhighlightedthecurrentstateofartasitrelatestothepremissofthisthe- sis, in particular on cross-layer algorithms for energy-efficient cooperative transmission inmultihopnetworks. Wealsoprovidedabrieftutorialonkeyconceptsincomputational complexitythatwillbeusedlateroninthisthesis. 22 Chapter3 GeneralizedAlgorithmicFormulation&Computational Complexity This chapter considers theproblem of energy-efficient transmissionin cooperativemul- tihop wireless networks 1 .Althoughtheperformancegainsofcooperativeapproaches are well known, the combinatorial nature of these schemes makes it difficult to design efficientpolynomial-timealgorithmsfordecidingwhichnodesshouldtakepartincoop- eration, and when and with what power they should transmit. Inthischapter,wetackle this problem in memoryless networks with or without delay constraints, i.e., quality of service guarantee. We analyze a wide class of setups, including unicast, multi-cast, and broadcast,andtwomaincooperativeapproaches,namely: energyaccumulation(EA)and mutualinformationaccumulation(MIA).Weprovideageneralizedalgorithmicformula- tionoftheproblemthatencompassesallthosecases. Weinvestigatethesimilaritiesand differencesofEAandMIAinourgeneralized formulation. Weprovethatthe broadcast 1 The work described in this chapter and the following chapter,wasdoneincollaborationwithB. KrishnamachariandA.F.Molischand,hasappearedpartin[64,68]. 23 and multicast problems are, in general, not only NP hard but also o(log(n)) inapprox- imable. WefurtherprovethattheproblemisNP-hard fortheunicast casewithMIA. 3.1 Introduction Inthischapterwefocusonformulatingtheproblemofcooperativetransmissioninwire- lessnetworks,whereasinglenodeissendingapackettoeithertheentirenetwork(broad- cast), a single destination node (unicast) or more than one destination node (multicast), in a multihop wireless network. Other nodes in the network, that are neither the source northedestination,mayactasrelaystohelppassonthemessagethroughmultiplehops. Thetransmissioniscompletedwhenallthedestinationnodeshavesuccessfullyreceived themessage. Weparticularlyfocusonthecasewherethereisadelayconstraint,whereby thedestinationnode(s)shouldreceivethemessagewithinthedelayconstraint,however, wealsodiscusshowourresultsapplytotheunconstrainedcase. As previouslymentionedakey problemin suchcooperativenetworks isroutingand resource allocation,i.e., thequestionwhich nodesshouldparticipateinthetransmission ofdata,andwhen,andwithhowmuchpower,theyshouldbetransmitting. Thesituation isfurthercomplicatedbythefactthattheroutingandresourceallocationdependsonthe typeofcooperationandotherdetailsofthetransmission/receptionstrategiesofthenodes. We consider in this work a time-slotted system in which the nodes that have received and decoded the packet are allowed to re-transmit it in futureslots. Duringreception, nodes add up the signal power (energy accumulation, EA) or themutualinformation 24 (mutual information accumulation, MIA) received from multiple sources. EA, which has been discussedin priorwork [37,39,41,42],can beimplementedby usingmaximal ratio combining (MRC) of orthogonal signals from source nodes that use orthogonal time/frequency channels, or spreading codes, or distributed space-time codes. MIA can be achieved using rateless codes [44,45]. A brief backgroundonthesetechniquesis highlighted in Chapter 2. Although these techniques are often treated separately in the literature,weshallseehowourformulationoftheproblemencompassesbothapproaches andallowsmanyoftheresultstobeextendedtoboth. We furthermore assume that the nodes are memoryless, i.e., accumulation at the re- ceiver is restricted to transmissions from multiple nodes inthepresenttimeslot,while signals from previous time slots are discarded. This assumption is justified by the lim- ited storage capability of nodes in ad-hoc networks, as well as the additional energy consumption nodes have to expand in order to stay in an active reception mode when theyoverhearweak signalsinprecedingtimeslots. Notethatmuchoftheliteraturecited abovehasusedtheassumptionofnodeswithmemory,sothattheirresultsarenotdirectly comparabletoours. Akeytradeoffisbetweenthetotalenergyconsumption 2 andthetotaldelaymeasured intermsofthenumberofslotsneededforalldestinationnodesinthenetworktoreceive the message. At one extreme, if we wish to minimize delay, eachtransmittingnode should transmit at the highest power possibleso that the maximumnumber of receivers 2 Asweconsiderfixedtimeslotdurations,weusethewordsenergyandpowerinterchangeablythrough- outthisthesis 25 can decode the message at each step (indeed, if there is no power constraint, then the source node could transmit at a sufficiently high power to reach all destination nodes in the first slot itself). On the other hand, reducing transmit power levels to save energy, mayresultinfewernodesdecodingthesignalateachstep,andthereforeinalongertime to complete the transmission. We therefore formulate the problem of performing this transmissioninsuchawaythatthetotaltransmissionenergyoveralltransmittingnodes isminimized,whilemeetingadesireddelayconstraintonthemaximumnumberofslots thatmaybeusedtocompletethetransmission. Thedesignvariableinthisproblemisto decidewhichnodesshouldtransmit,when,andwithwhatpower. Therestofthischapterisorganizedasfollows: Theassumptionsmadeonthesystem model is described in section 3.2. The generalized problem formulation is presented in section 3.3, encompassing both EA and MIA for unicast, multicast and broadcast scenarios. Wediscussthecomputationcomplexityofdifferentvariationsoftheproblem insection3.4. Thechapterissummarizedinsection4.6. 3.2 SystemModel We consider a wireless network withnnodes. Radio propagationismodeledby agiven symmetric n by n static channel matrix, H = {h ij },representingthe(power)gainon the channel between each pair of nodesi andj.Timeisassumedtobediscretizedinto fixed-durationslots;withoutlossofgeneralityweassumeunitslotdurations. Weassume 26 cooperativecommunicationinthereceivers,encompassingtwoscenarios: EAandMIA. Onlyasinglemessageistransmittedthroughthenetwork. In EA, the received power at a given receiver in a specific timeslot is sum of the powersreceivedfromthetransmittersthatareactiveduringthatslot.Asdescribedin[39, 41,42],thiskindofadditivereceivedpowercanbeachievedviamaximalratiocombining underdifferentscenariosincludingtransmissionusingTDMA,FDMAchannels,aswell as with CDMA spreading codes and space-time codes. MIA can be implemented using rateless codes and decoders at receivers, as described in [44–46]. With proper design (e.g., different spreading codes), information streams from different relay nodes can be distinguished, and the mutual information of signals transmitted by different nodes can be accumulated. A brief background on this is highlighted in chapter 2. We consider a per-nodebandwidthconstraintanddynamicpowerallocation. We assume appropriate coding is used so that each receiving node can decode the messagesolongasitsaccumulatedreceivedmutualinformationexceedsagiventhresh- old θ that represents the bandwidth-normalized entropy of the information codeword in nats/Hz. Furthermore, all nodes are assumed to operate in half-duplex mode, i.e. they cannot transmit and receive simultaneously. If used in transmission, the nodes operate based on a decode and forward protocol. Therefore, they are not allowed to take part in transmissionuntiltheyhavefullydecodedtheirmessage. Assumingthenoisepoweristhesameatallreceivers,wecanassumewithoutlossof generalitythenoisepowertobenormalizedtounitysothatthetransmitpowerattenuated 27 by the channel becomes equivalent to the signal to noise ratio(SNR).Aspreviously mentioned, we assume a memory-less model in which nodes do notaccumulateenergy orinformationfromtransmissionsoccurred inprevioustimeslots. 3.3 GeneralizedFormulation Inthissectionweprovideageneralizedformulationforthedelayconstrainedminimum- energy cooperative transmission (DMECT) problem in the setting described in section 3.2. We assume that the transmission begins from a single source node. The aim is to get the messageto all the nodes in a destinationset D,withtheminimumpossibletotal energy, within a time T (which can take on any value from 1 to n− 1). Every node in the network is allowed to cooperate in the transmission, solongastheyhavealready decoded the message. The problem now becomes: which nodes should take part in the cooperation, when and with what power should they transmit toachievethisaimwhile meetingtheconstraintsandincurringminimumtotaltransmissionpower. Recallingthememorylessassumption,theconditionforsuccessfuldecodingatsome receiver noder at timet when a set of nodesS(t) is transmittingpackets, with transmit powerp st ,∀s∈S(t)is: y rt ≥ θ (3.1) 28 with y rt being the mutual information accumulated by node r at time t.Let x it be an indicator binary variable that indicates whether or not node i is allowed to transmit at time t.Inotherwords,wedefine x it to be 1,ifnode i is allowed to transmit at time t (i.e. hasdecoded the messageby the beginningof timeslott as per equation (3.1)), and 0otherwise. Letp it bethetransmitpowerforeachnodeiateachtimet.Withoutlossof generality,thesourcenodeisassignednodeindex 1. The DMECT problem can then beformalized as a combinatorialoptimizationprob- lem: min P total = $ T t=1 $ n i=1 p it (3.2) s.t. 1.p it ≥ 0, ∀i,∀t 2.x iT+1 ≥ 1, ∀i∈ D 3.x it+1 ≤ 1 θ y it +x it , ∀i,∀t 4.x 1t =1, ∀t 5.x i1 =0, ∀i'=1 6.x it ∈ {0,1} where,fortheenergyaccumulation(EA)case 3 : y it = log 1+ ! s∈S(t) p st x st h si (3.3) 3 Notice that because of the monotonicity of the log function, y it ≥ θ in this case is equivalent to $ s∈S(t) p st x st h si ≥e θ −1 29 andformutualinformationaccumulation(MIA)case: y it = ! s∈S(t) log(1+p st x st h si ) (3.4) Constraint2ensuresthateverynodesinthedestinationsetsuccessfullydecodesthemes- sagewithinthetimeconstraintT,constraint3ensuresthatanodecannottransmitunless it has already received the message while simultaneously making sure that a node that has decoded the message in previous time slots will not be prevented from transmitting in futuretimeslots(ifitwants to transmit),constraint 4 assignsthe source node, and all otherconstraintsareself-explanatory. In general, there are three variations of this problem, basedonthesizeofthedesti- nationset: • Delay constrained minimumenergy cooperativeunicast (DMECU): where the set D includesasingledestinationnode. • Delay constrained minimum energy cooperative multicast (DMECM): where the setD includesmorethanonedestinationnode. • Delay constrained minimum energy cooperative broadcast (DMECB): where the setD includesallthenodesapart fromthesourcenode. Thedecisionversionoftheseproblem,canbedefinedcorrespondinglyasfollows: “Given some power bound C,doesthereexistanallocationofpowers, p it ,satisfyingthecon- straints in (3.2) such that P total ≤ C?” An instance of this decision problem is defined 30 by giving the symmetric n× n matrix H,withadesignatedsourcenode(vertex),a destinationsetD,adelayboundT,andapowerboundC. Notice that assigning T ≥ n,intheaboveformulation,resultsintheproblemdef- inition in the case where there is no delay constraint. Note also that a requirement for per-node maximum power can be trivially added to the above formulation as additional constraint; we have left that out for simplicity. Shouldthe maximumpowerbeadded, it shouldbelargeenoughtoensureafeasiblesolutionexistsforthegivenconnectivityand delayconstraint. 3.4 ComputationalComplexity Inthissection,weprovethatfindinganoptimalsolutionforDMECBandDMECMprob- lemsisnotonlyNP-hardbutalsoo(log(n))inapproximable i.e.,findinganypolynomial time algorithm that approximates the optimal solutions within a factor of o(log(n)) is alsoNP-hard. Weshowthisbydemonstratingthatanyinstanceofthesetcoverproblem canbereducedtoaninstanceofDMECB(andbyextensionDMECM).Wefurtherprove NP-completeness for DMECU when MIA is used; note that DMECU with EA will be treatedinsection4.2. 3.4.1 InterpretingtheSetCoverProbleminNetworkingContext The set cover problem is a classical problem in computer science [55]. It is stated as follows: Given a universe U of n elements and a collection of subsets of U, S = 31 S 1 ,S 2 ,...S k ,findaminimumsubcollectionof S that covers all elements of U.This problemisNP-completeandwasshown,in[56],tobeo(log(n))inapproximable. The set cover problem can be thought of as a bipartite graph G(V,E),with |V| = k +n,representingthek setsandnelementsandtheedgesare usedtoconnecteachset toitselements. This isshownin Figure3.1(a), whereweassigna vertexforeach setin the top part of the graph, and assign a vertex for each element in the bottom part of the graph. We connect each set to its elements using an edge. One can think of each vertex in this graph as a node in a network, in which edges exist between any pair of nodes for which h ij > 0,andtheedgesarelabeledwithaweight w ij that corresponds to the transmit power needed at nodei to exceed a threshold of θ at the receiverj,inasingle time slot if i was the only transmitter. Given an instance, G,ofthesetcoverproblem, theoptimalsolutiontothesetcoverproblem,OPT sc ,wouldfindtheminimumsubsetof verticesinthetoppartofthegraph,sothattheirtransmissionofamessagecanbroadcast themessagetoalltheverticesinthebottompartofthegraph. 3.4.2 InapproximablityofDMECB Given an instance, G,ofthesetcoverproblem,with k sets and n elements, let us con- struct a new graph G ! as follows: Assign a root node r,whichisthesourcewiththe messageatthestartingtime,callthislevel0.Includek nodesinlevel1,representingthe k sets in the set cover problem, all connected to the root node, as shown in Figure 3.1 (b). This is followed by the bipartite graph of G,whichmakesuplevel 2 and 3 of G ! . 32 Connecteachoftheknodesinlevel2totheirrepresentativeinlevel1andtoalltheother nodesinlevel 2.Noticethenodesinlevel 2arealsoconnectedtotheirelementsinlevel 3 of the graph, as shown in the Figure. We make all the weight on the edges arbitrarily small(say 1), withthe exceptionof the edges inbetween the nodes inlevel 1 and 2.We makethoseedgestobesufficientlylarge,sayM,tobespecifiedlater. r Level 0 Level 1 Level 2 Level 3 M M M G ! G Figure3.1: ConstructionofG ! foragivenGinDMECB. Assume the the weight on the edges represent the power needed for the message to betransmittedacross that edge. Ifwe wereto run theoptimalDMECBalgorithmonG ! withT =3 thealgorithmwould haveto act as follows,to beable to coverall the nodes inthegiventimeframe: Step 1: Roottransmitswithpower 1,turningonallitsk neighborsonlevel 1. Step 2: The algorithm picks a subset of thek nodes on level 1 to transmit the message. This subset must be chosen to be as small as possible, given thelargeweighttheyhave to endure to pass on the message on to the bipartite graph, and the fact that DMECB 33 is trying to minimize the total weight. Yet it has to be large enough so that when the nodesinlevel2transmit,allthenodesinlevel3wouldreceivethemessage. Theoptimal algorithmmustbeabletofindsuchasubset. Step 3: The nodes that receive the message in level 2 transmit the message in this step, turning on all the nodes in level 3 of the graph, as well as all the nodes in level 2 of the graphthatwerenotselectedfortransmission,thuscoveringthewholegraph. Let uscall thesolution 4 ofthisoptimalalgorithmOPT DMECB .Thenthefollowingtwo lemmaswithrespecttotheaboveconstructionofGandG ! hold: Lemma1. OPT DMECB ≤M.OPT SC +1+OPT SC Proof. Consider an instance of SC (with graph G), whose optimal solution isOPT SC . Construct a graph G ! ,asexplainedandruntheDMECBalgorithmtogetOPT DMECB . Theaboveinequalityholdsbyconstructionofthegraph. Lemma2. OPT SC ≤ OPT DMECB M Proof. ConsideraninstanceofDMECBonG ! anditsoptimalsolutionOPT DMECB for delayT =3.NoticethatifT> 3,weaddadditionalsinglenodes(asvirtualroots)to reducetheproblemtothecasewhereT =3.LookingatG ! ,weobservethattomeetthe delayconstraint,byendofstepi,atleastonenodeinlevelimusthaveheardthemessage -elseitisimpossibletogetthemessagethroughtotherestofthelevelsinthetimeframe left. Let’ssaytherootisonlevel0.Considerthesubsetoflevel1thathascomeonatthe end of time 1, s 1 ,andfromlevel 2 consider the set, s 2 ,thatcameonattheendoftime 4 Minimumenergyneededfortransmission. 34 step 2.Wenowwanttoshowthat s 2 is a feasible solution for set cover. To do so, we makethefollowingtwoclaims: Claim 1:Nodesresponsibleforturningon s 2 mustbea subset of s 1 . Claim 2: s 2 is a feasiblesolution to set cover. Claim 1 holds because only nodes that have received the message by the end of time 1 can transmit the message at time2.Notallofthemmighttransmitthough,sos 2 isasubsetofcorrespondingnodesin s 1 .Claim2istruebecauseifthereexistsanelementinlevel3thatisnotacorresponding node to any node in s 2 ,itcannotdecodebyT =3.Therefore, s 2 ,isafeasiblesolution to set cover. OPT DMECB must spend at leastM for each element of s 2 to come on, so OPT SC ≤ OPT DMECB M . Theorem1. The DMECB problemiso(log(n)) inapproximable,forT ≥ 3. Proof. For an instance of the set cover problem, withk being the total number of sets, lemma 1 can be re-written as OPT DMECB ≤M.OPT SC +1+k.Wealsoknowby lemma2 thatOPT SC ≤ OPT DMECB M .Therefore,forasufficientlylargeM,wecanwrite OPT SC = OPT DMECB M +o(1).Therefore,thereductionusedinconstructionofthegraph G ! preserves the approximation factor. That is, if one can find an α-approximation for DMECB, by extension there must exist an α-approximation for set cover. We know, by [56], that the set cover problem iso(log(n)) inapproximable, thus DMECB must be o(log(n)) inapproximable. In other words, finding a polynomial time approximation algorithmthatapproximatesOPT DMECB withafactorofo(log(n))isNP-hard. The DMECB problem can be solved in polynomial time for cases whenT< 3. The optimal algorithm for T =1 is trivial and an optimal polynomial algorithm for 35 T =2 is discussed in section 4.1 5 .Itisalsotrivialtoverifythefeasibilityofagiven power allocation, and verify whether or not it satisfies the decision version of DMECB given in section 3.3. Therefore, the problem belongs to the class of NP. Notice that the inapproximability result, given by Theorem 1, is stronger than, and implies, the NP- completenessresult. Itisalsoworthnoticingthatwithoutanydelayconstraint(i.e. when T ≥ n), the problem is still NP-complete and the proof can be obtained, using directed Hamiltoneanpath,followingtheapproachin[37]. 3.4.3 InapproximablityofDMECM Theproofofthefollowingtheorem,followsfrom Theorem 1 bynoticingthatbroadcast canbethoughtofasaspecialcaseofmulticast. Theorem2. The DMECM problemiso(log(n)) inapproximable,forT ≥ 3. 3.4.4 HardnessResultsforDMECU In the unicast case, the hardness of the problem depends on whether we are using EA orMIA.In theformercase, DMECU can beshownto bepolynomiallysolvableandthe algorithmfor thatisprovidedinsection4.2. Intheremainderofthissection,wediscuss DMECUwithMIA. Given an instance, G,ofthesetcoverproblem,with k sets and n elements, similar to that in section 3.4.2, let us construct a new graph G ! as follows: Assign a root node 5 DMECT goalgorithm,discussed in section4.1, alongwith an orderingbased onchannelgainsfrom thesource,providesanoptimalpolynomialtimealgorithmforDMECBforthecasewhenT =2. 36 r,whichisthesourcewiththemessageatthestartingtime,call this level 0.Includek nodesinlevel1,representingtheksetsinthesetcoverproblem,allconnectedtotheroot nodewithasmallweight(sayweight1),asshowninFigure3.2. Thisisfollowedbythe bipartitegraphofG,whichmakesuplevel2and3ofG ! .Connecteachofthek nodesin level 2 to their representative in level 1 with edge weights, of say W.Noticethenodes in level 2 are also connected to their elements in level 3 of the graph, as shown in the Figure, with low-weight edges. Add a single destination noded,inlevel 4 and connect allthenodesinlevel3tod.Letthechannelbetweenallnodesonlevel 3anddestination dbeequalandofgainh.Therefore,theedgeweightontheedgesconnectingthelevel3 nodes tod,canbeassignedtobeM,whereM is defined so that the following equality holds: log(1+Mh)= θ. M 1 W 1 r d Level 0 Level 1 Level 2 Level 3 Level 4 Figure3.2: ConstructionofG ! foragivenGinDMECU. Assume that the weight on the edges represent the power neededforthemessageto be transmitted across that edge. If we were to run the optimal DMECU algorithm on G ! withT =4 the algorithm would have to act as follows, to be able to turn noded on 37 withinthegiventimeframe: Step 1: Roottransmitswithpower 1,turningonallitsk neighborsonlevel 1. Step 2: The algorithm picks a subset thek nodes on level 1 to transmit the message to thenodesinlevel 2. Step3:Asubsetofnodesthathavereceivedthemessageinlevel2,transmitthemessage inthisstep,turningonasubsetofnodesinlevel3ofthegraph. Step4:Asubsetofnodesthathavereceivedthemessageinlevel3,transmitthemessage inthisstepwithsufficientpowertoturnond. LetuscallthesolutionofthisoptimalalgorithmOPT DMECU . Theorem3. The DMECU problem,with MIA, isNP-complete forT ≥ 4. Proof. GivenaninstanceofG,weconstructG ! asabove. LetusrunDMECUonG ! and call the optimal solutionOPT DMECU for delay T =4.NoticethatifT> 4,weadd additionalsinglenodes(asvirtualroots)toreducetheproblemtothecasewhereT =4. Defineptosatisfythefollowing:nlog(1+ph) = log(1+Mh),meaningpisthepower required for nodes on level 3 to turn on d,ifallofthemweretransmittingatthesame time. Claim: OPT DMECU needs to use all the nodes in level 3 for transmission. This claim holds by contradiction, as follows: If all the nodes on level 3 are used for trans- mission, each node on that level must transmit with power p.Let’sassumeoneofthe nodesinthatlevelisnotusedfortransmission. Thentheremainingnodesinlevel3need to transmitwith powerp ! ,wherenlog(1 +ph)=(n− 1)log(1+p ! h)= θ.Therefore, the ratio of the sum power needed with one fewer node transmitting to the case where 38 all nodes in level 3 are transmitting can be written as (n−1)p ! np = (n−1)(e θ/n−1 −1) n(e θ/n −1) ,forsuf- ficiently large θ,thisratiocanbecomearbitrarilylarge.Therefore,forsufficiently large θ,theclaimholds.Giventheclaimholds,weknowthatbydefinitionOPT SC provides theoptimalway(minimumenergy)toturnonallthenodesinlevel3withintherequired time frame, therefore, for non-zero edge weightsOPT DMECU needs to optimally solve thesetcoverprobleminstep 2. Itisworthnoticingthatallthehardnessresultspresentedinthissectionextendtothe casewhere thereisnomemory. 3.5 Summary In this chapter we formulated the novel problem of delay constrained minimum energy cooperative transmissionin memoryless wireless networks,encompassingbothEAand MIA. We analyzed a wide class of setups, including unicast, multi-cast, and broadcast, and two main cooperative approaches, EA and MIA. We providedageneralizedalgo- rithmicformulationoftheproblemthatencompassesallthosecases. Weinvestigatedthe similarities and differences of EA and MIA in our generalizedformulation. Weproved that the broadcast and multicast problems are, in general, not only NP hard but also o(log(n)) inapproximable. We further proved that the problem is NP-hard for the uni- cast case with MIA. Table 3.1 provides a summary of the algorithmicresultsprovedin thischapter. 39 NEGATIVE RESULTS EnergyAccumulation DelayConstraint(T) Unconstrained Broadcast o(log(n))inapproximableforT ≥ 3 NP-complete Multicast o(log(n))inapproximableforT ≥ 3 NP-complete Unicast Polynomialtime Polynomialtime NEGATIVE RESULTS MutualInformationAccumulation DelayConstraint(T) Unconstrained Broadcast o(log(n))inapproximableforT ≥ 3 NP-complete Multicast o(log(n))inapproximableforT ≥ 3 NP-complete Unicast NP-completeforT ≥ 4 — Table 3.1: Summary of the algorithmic negative results. 40 Chapter4 MinimumEnergyDelayConstrainedTransmission In chapter 3 we considered the problem of energy-efficient transmission in coopera- tive multihop wireless networks. We proved NP-hardness for several variations of the DMECT problem (namely, the DMECB, DMECM and DMECU (with MIA),withthe former two being o(log(n)) innaproximable). In this chapter we break these problems into three parts: ordering, schedulingand power control, and propose a novel algorithm that, given an ordering, can optimally solve the joint power allocation and scheduling problems simultaneously in polynomial time. We further showempiricallythatthisal- gorithm used in conjunction with an ordering derived heuristically using the Dijkstra’s shortestpath algorithmyieldsnear-optimal performanceintypicalsettings.Forthe uni- cast case,weprovethatalthoughtheproblemremainsNPcompletewithMIA,itcanbe solvedoptimallyandinpolynomialtimewhenEAisused. Wefurtheruseouralgorithm to study numerically the trade-off between delay and power-efficiency in cooperative broadcast and compare the performance of EA vs MIA as well as the performance of 41 our cooperative algorithm with a smart non-cooperative algorithm in a broadcast set- ting. We also briefly discuss how the algorithms discussed could be implemented in a decentralizedfashion. 4.1 OptimalTransmissionGivenOrdering Insection3.4,weprovedNP-hardnessforseveralvariationsoftheDMECTproblem.In this section, we break this NP-hard problem into three subproblems, namely ordering, schedulingandpowerallocation,andweproposeanoptimalpolynomialtimealgorithm for joint scheduling and power allocation when the ordering is given. We evaluate a heuristicfortheorderinginsection4.4. Definition 1. An ordering, for a vector of n nodes, is an array of indices from 1 to n; anynodethathasdecodedthemessagewillonlybeallowedtoretransmitwhenallnodes with smaller index have also decoded the message (and are thusallowedtotakepartin transmission). Given an ordering, what remains to be determined is which nodes should take part intransmission,howmuchpowertheyshouldtransmitwithandatwhattimeslots,such thatminimumenergyisconsumedwhiledelayconstraintsaresatisfied. 4.1.1 Instantaneousoptimalpowerallocation Ifweknowwhichnodesaretransmittingthemessageandwhichnodesarereceivingit,at anysingletime-slot,we can use a convexprogram(CP) todeterminethe optimalpower 42 allocationfor thattimeslot. Consider an ordered vectorofnnodes (1,...,k,...,i,...,n). Letusassumethatbytimeslott,node 1toihavedecodedthemessageandnodesi+1 ton are to decode it during that time slot. At time instancet,theoptimalinstantaneous powerallocationforaset oftransmittingnodes(sayS(t)=(k,...,i))toturnonasetof receivingnodesnodes(sayR(t)=(i+1,...,n))canbecalculatedbythefollowingCP: min $ s∈S(t) p st (4.1) s.t. p st ≥ 0, ∀s y rt ≥ θ, ∀r∈R(t) We use the notation CP([{k...i},{i+1...n}],θ,H) to refer to solution of the above CP. As a notation,CP([{x...y},{z...α}],θ,H)=0,if z ≥ α.Noticethatinthecase where EA is used, this CP simply reduces to a linear program, using the manipulation highlightedinfootnote 2insection3.3. 4.1.2 JointSchedulingandpowerallocation Knowing the instantaneous optimal power allocation given the set of senders and re- ceivers at each time slot, all that remains to be done is to determine these sets at each timeslot,inordertominimizetheoverallpowerwhilemeetingthedelayconstraint. Let C(j,t) be the minimum energy needed to cover up to node j in t steps or less. We can calculate this, using the following algorithm: 43 C(j,t) = min k∈(1,..,j) [C(k,t−1)+CP({1...k},{k+1...j},θ,H)] (4.2) whereC(k,1) =CP(1,{2...k},θ,H),C(1,t)=0 ∀t. Thus, in DMECB, the total minimum cost for covering n nodes by time T can be found by calculating C(n,T).InDMECM,andDMECU(MIA),thesameapproach could be used, except for noden being replaced by the highest order destination in the formerandbythedestinationorderinthelatter. ApseudocodeforthealgorithmispresentedinAlgorithm1.The complexityof the Algorithm 1 Delay constrained minimum energy cooperative transmission, given an ordering(DMECT go) 1: INPUT: an ordered array of nodes of size n (where node i is the ith node in the array),T (delay),d(destination),H (channel), θ (threshold). 2: OUTPUT:C (costmatrix) 3: Begin: 4: fori := 2tondo 5: C(i,1) :=CP([1,{2...i}],θ,H); 6: endfor 7: fort := 1toT do 8: C(1,t) := 0; 9: endfor 10: fort := 2toT do 11: fori := 1toddo 12: fork := 1toido 13: x(k) :=C(k,t−1)+CP([{1...k},{k+1...i}],θ,H); 14: endfor 15: C(i,t) := minx 16: endfor 17: endfor optimalschedulingandpowerallocationcanbeobtainedbyinspectionoftheabovealgo- rithm: itinvokesatmostO(n 2 T)callstotheCPsolver,each ofwhichtakespolynomial 44 time. HencetheDMECT goalgorithmthatdoesjointschedulingandpower-controlisa polynomialtimealgorithm. Note that the delay constraint T can be made sufficiently large (≥ n),orremoved entirely from the formulation, to cover the case of no delay constraints. In that case the two-dimensional dynamic program proposed in (4.2), reduces to a one-dimensional dynamicprogram: C(n) = min k [C(k)+CP({1...k},{k+1...n}),θ,H] (4.3) whereC(n) is the minimumcost of coveringnoden usingour cooperativememoryless approach,startingfromnode 1andC(1) = 0. 4.2 OptimalUnicastwithEnergyAccumulation Inthissectionweproposeanoptimalpolynomialtimealgorithmsforsolvingtheunicast problemwithEA. Theorem 4. In DMECU with EA, there exists a solution consisting of a simple path between source and destination,which isoptimum. Proof. Let us prove by induction: For delay T =1,theclaimistriviallytrue,asthe optimal solution is a direct transmission from the source, s,tothegivendestination, d.ForT> 1,weprovetheclaimbyinduction. Assumethattheclaimistruefor T = k− 1.Pickanynodeinthenetworkasthedesireddestinationd.Ifthemessage 45 can be transmitted from source s to d with minimum energy in a time frame less than k,thenanoptimalsimplepathexistsbytheinductionassumption. So considerthe case when it takes exactly T = k steps to turn on d.Thesystemismemoryless,so d must decodebyaccumulatingtheenergytransmittedfromasetofnodes,v,attimek.Thiscan be represented as log(1+ $ v i ∈v p v i k h dv i )≥ θ.Weobservethattheremustexistanode v o ∈vwhosechanneltodisequalorbetterthanalltheothernodesinv.Therefore,given h dvo ≥h dv i ,∀v i ∈v−{v o }thenlog(1+ $ v i ∈v p v i k h dvo )≥ log(1+ $ v i ∈v p v i k h dv i )≥ θ. In other words, if we add the power from all nodes in v and transmit instead from v o , our solution cannot be worse. v o must have received the message by timek− 1,tobe able to transmit the message to d at timek.Weknowbytheinductionassumptionthat the optimal simple path solution exists from source to any node to deliver the message withink−1 timeframe. Thus,forT =k,thereexistsasimplepathsolutionbetweens andd,whichisoptimum. Notice that the above theorem holds in the case where there is no delay constraint as well. The proof follows an straightforward modification oftheaboveproofandis omittedforbrevity. Corollary 1. The Dijkstra’s shortest path algorithm provides the optimalorderingin the case of minimum energy memoryless cooperative unicast, when there is no delay constraint. Proof. We have already established that an optimal minimum energy solutionexistsbe- tweensourceanddestination,whichisasimplepath. Thewell-knownDijkstra’sshortest 46 path algorithm can find the minimum cost simple path between source and destination. Therefore, Dijkstra’salgorithmprovidestheoptimalordering. Using theorem 4 we know that the optimal unicast solution fromsourcetoanydes- tination in DMECU (with EA) is given by a simple path. The cost paid by the optimal solution can be calculated using the following algorithm: Let C(i,t) be the minimum cost it takes for source nodes to turn oni,possiblyusingrelays,withinatmostt time slots. Thenwecanwrite: C(i,t) = min k∈Nr(i) [C(k,t−1)+w(k→i)] (4.4) withC(s,t)=0,forallt andC(i,1) = w(s→ i),whereNr(i) is theset that contains i and its neighboring nodes that have a non-zero channel to i, w(k → i) represents the power it takes for k to turn on i using direct transmission. Given that, the solution to OPT DMECU (with EA) is given by C(d,T).Computingthislower-boundincursa runningtimeofO(n 3 ). The unicast case (with EA), with no delay constraint, is stillpolynomiallysolvable. GivenTheorem4andCorollary1,theoptimalsolutionissimplytheweightoftheshort- estpathgivenbytheDijkstra’salgorithm. It is worth noticing that the crux of the difference between DMECU with EA and with MIA, that allows the former to be polynomially solvable,whilethelatterisNP- complete,liesintheoptimalityofsingle-nodetransmission. Namely,intheEAcase,the 47 multi-transmitterssingle-receivercase(multi-single)makesnosenseasexplainedabove and instead it is optimal to put the combined power into the best channel. This allows for the overallsolutionto be a simplepath. However,in the MIA case, themany-to-one transmissioncasedoesinfactmakesense. Thatisduetothepropertyofthelogfunction, creatinganeffectsimilartowhatwe observein water-filling,whereitisbesttotransmit from the best channel up until some point, then from the secondbestchannelandso forth. 4.3 ApproximationAlgorithmforBroadcastwithEnergy Accumulation In section 3.4, we proved that DMECB is NP-complete and o(log(n)) inapproximable, therefore it is hard to approximate DMECB to a factor strictly better than log(n).It is of theoretical interest to know how close we can get to the optimal solution, using a polynomial-time algorithm. In this section we show that existing approximation algo- rithmsforthebounded-diameterdirectedSteinertreeproblemcanbeusedtoprovidean O(n ! )approximationforDMECBinthecasewhereEAisused. We dosobyproposing anapproximation-preservingreductiontothedirectedSteinertreeproblem. TheSteinertreeproblemisaclassicproblemincombinatorialoptimization[55]. We focusona variationofthisproblem,namelyboundeddiameterdirectedSteinertree,de- fined as follows. Givena directed weighted graphG(V,E),aspecifiedrootr∈ V,and 48 asetof terminal nodesX⊆ V (|X| = n), the objectiveis to find the minimumcost ar- borescencerootedinrandspanningallverticesinX,subjecttoamaximumdiameterT. Diameterreferstothemaximumnumberofedgesonanypathinthetree. Noticethatthe tree may include vertices not inX as well, these are known as Steiner nodes.Directed Steiner tree problem is known to beNP-complete andO(log(n)) inapproximable[55]. In [54], the authors give the first non-trivial approximationalgorithmsforSteinertree problemsand proposeapproximationalgorithmsthatcan achievean approximationfac- torofO(n ! )foranyfixed "> 0inpolynomialtime. Tothebestofourknowledgethisis currentlythetightestapproximationalgorithmknownfor thisproblem. In order to reduce a given instance of the DMECB to an instance of the Steiner treeproblem,wefirstrestrictDMECBbynotallowingmany-transmitter-to-one-receiver (many-to-many) transmissions. Notice that in the proof of theorem 4, we had estab- lished that many-to-one transmissions can be replaced with one-to-one transmissions withoutloss of optimality. Therefore, by not allowingmany-to-manytransmissions,we are left withone-to-oneand one-to-manytransmissions. We call thisan integralversion ofDMECB,DMECB-int. Theintegralitygapoftheweightedsetcoverproblemisshown tobelog(n)[55];itisstraightforwardtoextendthatresulttoshowthatDMECB-intalso losesafactorof log(n),comparedtooptimalDMECB. Consider an instance of DMECB-int,G(V,E),with(|V| = n)ands∈ V being the source node. To reduce this problem to an instance of directedSteinertreeproblem,let usconstructanewgraphG ! ,consistingofnclusters,x ! ,eachcorrespondingtoeachnode 49 in G.Leteachclusterbeabipartitegraph,with n nodes on the left (marked as “− ”) and n nodes on the right (marked as“+”), as shown in Figure 4.1. The “− ” nodes are intra-connected within a cluster with edges of weight 0.Ineachcluster, x ! ∈ G ! correspondingtonodev∈G,the“+”and “−”nodesoneach level,i,ofthebipartite graph are connected to each other with an edge of weight w i ,representingthepower needed bythecorrespondingnodev∈ G toturn onitsi closestneighbors. Thei + node is then connected, with edges of weight 0,toallthe “− ” nodes in the corresponding neighbor clusters. We further add a single root node,r∈ G ! ,andconnectitviaazero- weight edge to all the “− ” nodes in the cluster corresponding tos, x ! s .Weassignthe rootrandonedesired“−”nodefromeachclusterasterminalnodesandallothernodes inG ! asSteiner nodes. Let us look at an example of this construction, say node v 1 ∈ G,whoseclosest 3 neighbors are (v 2 ,v 4 ,v 6 ). We have an equivalentclusterx ! 1 ∈ G ! corresponding to node v 1 . x ! 1 has 2n nodes, arranged in n levels. The weight between the two nodes in say level 3 is equivalent to the power it takes forv 1 to turn on (v 2 ,v 4 ,v 6 ). Furthermore, the node3 + inclusterx ! 1 isconnectedtothe“−”nodesinclusters(x ! 2 ,x ! 4 ,x ! 6 )withedgesof weight 0.Thisconstructionallowsustofindawaytoallowv 1 totransmitwithdifferent power levels, without knowing what those powers might be in advance. We first add a single root node, r,andconnectitviaazero-weightedgetoallthe “− ” nodes in the clustercorrespondingtos. 50 Figure4.1: Asimplifiedexampleofhowclustersare constructedinG ! . 51 Run the directed Steiner tree algorithm on G ! to obtain a solution. The solution must choose at least one node from each cluster, to meet the mandatory terminal nodes requirement. Recall that each cluster in G ! corresponds to a node in G and that multi- multi was not allowed. To convert the solution of the Steiner tree algorithm on G ! to a solutionofDMECB-intonG,welookattheparentofeachcluster,whichisa“+”node inanothercluster. Let’ssaywewanttoseewhichnodeturnsonv 6 bylookingatG ! .We look at the parent of x ! 6 and see that it’s 3 + ∈ x ! 1 .Soin G,wefigureoutthat v 1 must transmitwithenoughpowerto turnon 3 ofitsclosestneighbor(w 3 ), and itisas a result of this transmission that v 6 comes on. Going through all the clusters and their parents, we can establish an ordering and transmission power for all the nodes that should take partincooperationinG,andthuswehaveasolutionforDMECB-int. Theorem5. ForDMECBproblemwithEA,anO(n ! )approximationratiocanbeachieved inpolynomialtime,forany fixed "> 0. Proof. Asmentioned,thedirectedSteinertreeiso(log(n))inapproximable,andthebest approximation algorithm currently available [54] gives an O(n ! ) approximation on the optimal solution. We had already lost O(log(n)) to convert DMECB to its integral form. The approximation algorithm proposed in [54] can approximate the optimal in- tegral solution within O(n ! ).Therefore,usingtheabovereduction,andapplyingthe directed Steiner tree approximationalgorithm,we can approximatetheoptimalsolution toDMECBwithinO(n ! ×log(n)),whichisequivalenttoO(n ! ). 52 The running timeof the Steiner tree approximationalgorithmisafunctionof ",and the tighter the approximation, the worse the running time. Similarly, using the above mentioned reduction, the following result holds by directlyapplyingtheapproximation algorithms in [54]. Detailed discussions of the algorithms in [54] are beyond the scope ofthisdissertation. Theorem 6. For any fixedT> 0,thereisanalgorithmwhichrunsintime n O(T) and gives anO(T log 2 (n)) approximationof theDMECB with EA. 4.4 PerformanceEvaluaion For the simulations,we focus on the broadcast case. We consider a network ofn nodes uniformlydistributedona 15by 15squaresurface. Thetransmissionstartsfromanode, arbitrarily located at the left center corner of the network (0,7).Thechannelsbetween allnodesarestatic,withindependentandexponentiallydistributedchannelgains(corre- spondingtoRayleighfading),whereh i j denotesthechannelgainbetweennodeiandj. The mean value ofthe channel between two nodes,h ij ,ischosentodecaywiththedis- tance between thenodes, so thath ij = d −η ij ,withd ij being thedistancebetween nodesi andj andη beingthepathlossexponent. Thecorrespondingdistributionforthechannel gainsisthengivenby f h ij (h ij )= 1 h ij exp " h ij (k) h ij # 53 Based on the intuition developed in section 4.2, we use the Dijkstra’s shortest path al- gorithm as our ordering heuristic. Simulations are repeatedmultipletimeswiththe same node locations but different fading realizations and average values are shown in the graphs. Notice that the minimum power calculated by different algorithms, shown on the y-axes of the graphs in this section, are normalized by unit power (rendering it unit-less). Thevalueof θ is,arbitrarily,chosentobe log(2)throughoutthissection. In Figure 4.2, we calculate the optimal ordering by brute-force for a small number of nodes and compare the performance of our algorithm, which uses Dijkstra’s shortest path-based ordering, with the optimal performance. The results for the broadcast case, with EA, is shown in this figure. As can be seen, Dijkstra’s algorithm provides a good heuristic for ordering in this example and will be used throughout this section. Note that, as can be seen in the figure, although the problem was proved to be o(log(n)) in- approximable,itispossibletoachievenear-optimalresultsinpolynomialtimeincertain practical settings where the network does not have any pathological properties. The in- approximablityresultscontainallpossible(includingpathologicalnetworks)scenarios. We next compare the performance of our cooperative algorithmwithasmartlyde- signed non-cooperative algorithm (both using EA). Notice that in our cooperative algo- rithm we make use of the wireless broadcast advantage (WBA),wheretransmissionby one node can be received by multiple nodes and cooperative advantage,whereanode canaccumulatepowerfrommultipletransmitters. 54 4 5 6 7 8 100 150 200 250 300 350 400 450 500 Number of Nodes Normalized Power Heuristic ordering Optimal ordering Figure 4.2: Performance with optimal ordering vs Dijkstra’salgorithm-basedheuristic ordering. 55 IfanalgorithmisusingtheWBA,butnotthecooperativeadvantage,itcanbethought ofasanintegralversionofDMECB.Thismeansthateachnodecanreceivethemessage from one transmitter only (and cannot accumulate from multipletransmitters), however onetransmittercan transmittomultiplereceivers. We had established in section 4.3 that DMECB-int is also NP complete. It is how- ever interesting to note that DMECB-int needs to solve a weighted set cover problem when allocating powers as well; we know that set cover problemiso(log(n)) inapprox- imable[55],so thenon-cooperativecaseiso(log(n))inapproximable,evenwhen order- ingisprovided. Greedyalgorithmsexist[55]thatgiveO(log(n))approximationsforthe weightedsetcoverproblem,andthusprovideatightpolynomialtimeapproximation. 10 20 30 40 50 60 70 80 90 50 100 150 200 Number of nodes Normalized Power Non−Cooprative Cooperative Figure4.3: Effectofcooperationforvaryingnetworksize. 56 2 3 4 5 6 7 8 9 10 50 100 150 200 250 300 350 400 450 500 Delay Constraint (T) Normalized Power Non−Cooperative Cooperative Figure4.4: Power-delaytradeoffincooperativevsnon-cooperativecase. Therefore,tosimulateasmartnon-cooperativealgorithm,weuseDijkstra’salgorithm- basedorderingandtheDMECTalgorithmofsection4.1,withtheexceptionthatinstead ofusinganLPweusethegreedyalgorithmforpowerallocation. The performance comparison between our proposed cooperative algorithm and the smartnon-cooperativealgorithm,fordifferentvaluesofnisshowninFigure4.3andthe power-delay tradeoff for cooperative and non-cooperative algorithms are presented in Figure 4.4. As can be seen, the cooperative algorithm outperforms the non-cooperative algorithm,andtheadvantageismorepronouncedwhenadelayconstraintisimposed. The performance gains obtained by using MIA is shown in Figure4.5,forasample networkof 30nodes. 57 2 3 4 5 6 7 8 9 10 50 100 150 200 250 300 Delay Constraint (T) Normalized Power Energy Accumulation Info Accumulation Figure4.5: Energyaccumulationvsmutualinformationaccumulation. 58 We next study the power-delay tradeoff of the cooperative algorithm for different channel conditions and different values of network density ρ (in nodes/area). Figure 4.6 and Figure 4.7, show results for EA and MIA, respectively.Thesefigureshighlight the sensitivity of the dense networks and those with poor channel conditions to delay constraints and the importance of having smart algorithms tominimizetheenergycon- sumption. 2 3 4 5 6 7 8 9 10 50 100 150 200 250 Delay Constraint (T) Normalized Power ! = 0.3 ! = 0.2 ! = 0.1 Figure4.6: Effectofnetworkdensityonpower-delaytradeoff. 59 3 4 5 6 7 8 9 10 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Delay Constraint (T) Normalized Power " = 4 " = 3 " = 2 Figure4.7: Different η valuesforinformationaccumulation. 4.5 DecentralizedImplementation The algorithms discussed in this chapter are centralized algorithms. In this section, we discussapossibleapproachforadecentralizedimplementationofthesealgorithmsinthe casewhere EAisused. Forsimplicity,westartoffbyconsideringthesinglesourcewithenergyaccumulation without a delay constraint. As previously discussed, the unicast problem in this case is optimallysolvableandconsistsofasinglepath. WeobservedthattheDijkstraalgorithm could provide us with that optimal single path. An alternative algorithm is Bellman’s shortestpathalgorithmdevelopedin 1957.Thedistributedalgorithmbasedonthiswork is often referred to as Bellman-Ford algorithm, named after its inventors. Although, not 60 regardedtobeasfastasDijkstra’salgorithmforgraphswithnon-negativeedges,thedis- tributed Bellman-Ford algorithm seems to provide a viable alternative to Dijkstra’s that wouldallowustohaveadistributedsolutionfortheunicastcase,inenergyaccumulation. Next, let us consider the algorithm discussed in section 4.1,withEA.Forthisal- gorithm, we already established that, given an ordering, an optimal polynomial time solutionexistsandcanbefoundusingdynamicprogramming,forscheduling,andlinear programming,forpowercontrol,jointly. WeproposedusingDijkstra’salgorithmforthe ordering and had verified, using simulations, that this ordering performs near-optimally in typical settings. So, if ordering is given, using this insight we could try to find a distributedapproach. Inspecting the dynamic program closely, for the case with delay constraint, we can see that it works by filling out theelements of ann×T table, withn being the number of nodes in the network and T being the delay. Assuming all nodes have the ordering vector, we can have each node calculate their own corresponding row of the table using the information that has been passed on to them by the node immediately preceding them and then amend this to what they have received from the previous node and pass it on to the node that proceeds them in the ordering. Notice that since the next node in the neighboring is not necessarily their closest neighbor (in terms of channel strength) they might need to pass on this information using some sort of distributed shortest path algorithm. Thefinalnodeintheorderingwillthenbeinthepossessionoftheentiretable 61 andcanpassitontoallthenodesinthenetwork. Oncethenodesgetacopyofthetable, theycanlocallycalculatewhentheyneedtotransmitandwithwhatpower. Assuming channel conditions do not vary too frequently, thisinitialstepneednot happen too frequently. This way, givenordering, we can find adistributedsolutionwith each node only having access to the ordering vector and the channel gains of its neigh- bors. If ordering is not given, the problem becomes more challenging. We have already established that, without ordering, the problem is o(log(n)) inapproximable and have proposed an O(n ! ) approximation algorithm based on the best existing approximation algorithms for directed Steiner tree. A possible decentralized approximation algorithm forthebroadcastversionoftheproblemwithEAisusingtheobservationthatifweareto broadcastthemessagetoallthenodesinthenetwork,wecannotpossiblydobetterthan what it takes to transmit the message in an optimal unicast fashion to each of the nodes in thenetwork. Thus, for a givensource, theminimumunicast optimalsolutionoverall nodes provides a lowerbound on the optimalsolutionfor the broadcast. We can further observe that the optimal solution cannot be worse than n times the maximum unicast optimalsolutionoverallnodes. Usingtheseobservations,wecan thinkofthefollowing distributedalgorithm(inabsenceofordering): WefirstrunthedistributedBellman-Ford algorithm, to establish a shortest path tree in the network - in the unconstrained case and then use the distributed unicast solution developed above with the leaf nodes being thedestinations. Similarly,inthedelayconstrainedcase,weusethedistributedsolution 62 for the delay constrained unicast case. This provides a distributed approximate for the optimalbroadcastproblem. TheapproximationisO(n)inthiscase. 4.6 Summary We considered the DMECT problem in memoryless wireless networks. We had proved in the previous chapter that this problem is o(logn) inapproximable in broadcast and multicast cases and is NP-complete in the unicast case when mutual information accu- mulation is used. In this section we developed a polynomial-time algorithm that can solve this NP-hard problem optimally for a fixed transmissionordering. Ourempirical results suggest that for practical settings, a near-optimalorderingcanbeobtainedby using Dijkstra’s shortest path algorithm. We have further showed that the unicast case can be solved optimally and in polynomial time when EA is used.Wehavestudiedthe energy-delay tradeoffs and the performance gain of MIA usingsimulations,andevalu- ated the performance of our algorithm under varying conditions. For the broadcast case with EA, we presented an O(T log 2 (n)) approximation algorithm. We discussed how some of the algorithms discussed could be implemented in a decentralized fashion and proposeda decentralizedO(n)approximationalgorithmfor the broadcast problemwith EA. The summary of the algorithmic results developed in this chapter are presented in Tables 4.1. 63 POSITIVE RESULTS EnergyAccumulation MutualInformationAccumulation Delay Constraint (T) Unconstrained Delay Constraint (T) Unconstrained Broadcast •O(n ! )for "> 0 •O(T log 2 (n)), forfixedT • Polynomial time given order- ing(DMCT go) • Polynomial time algorithm forT =1,2 Polynomial time givenordering 1Ddynamicpro- gram • Polynomial time given order- ing(DMCT go) • Polynomial time algorithm forT =1,2 Polynomial time givenordering 1Ddynamicpro- gram Multicast Polynomial time given ordering (DMCT go) Polynomial time given ordering (DMCT go) Polynomial time given ordering (DMCT go) Polynomial time given ordering (DMCT go) Unicast Polynomialtime Polynomialtime • Polynomial time given order- ing (DMCT go) T ≥ 4 • Polynomial time algorithm forT =1,2 — Table 4.1: Summary of the algorithmic positive results. 64 Chapter5 Transmissionin Presenceof InterferingFlows In this thesis so far, we have focused on the case where there isnointerferencepresent. In this chapter, we consider the problem of energy-efficient transmission in multi-flow multihop cooperative wireless networks 1 .Aswediscussedinpreviouschapters,the combinatorial nature of these schemes makes it difficult to design efficient polynomial- time algorithms for joint routing, scheduling and power control. This becomes more so when there is more than one flow in the network. It has been conjectured by many authors, in the literature, that the multiflow problem in cooperative networks is an NP- hardproblem. Inthischapter,weformulatetheproblem,asacombinatorialoptimization problem,forageneralsettingofk-flows,andformallyprovethattheproblemisnotonly NP-hardbutitiso(n 1/7−! )inapproxmiable. Toourknowledge,theresultsinthischapter provide the first such inapproxmiablity proof in the context of multiflow cooperative wireless networks. For the special case ofk=1,weprovethatthesolutionisasimple 1 The work described in this chapter, was done in collaborationwithB.KrishnamachariandD.S. Hochbaumand,is presentedinpartin[71] 65 path, and offer a polynomial time algorithm for jointly optimizing routing, scheduling and power control. We then use this algorithm to establish analytical upper and lower bounds for the optimal performance for the general case of k flows. Furthermore, we proposeapolynomialtimeheuristicforcalculatingthesolutionforthegeneral caseand evaluatetheperformance ofthisheuristicunder differentchannelconditionsandagainst theanalyticalupperandlowerbounds. 5.1 Introduction In a wireless network, a transmit signal intended for one nodeisreceivednotonlyby that node but also by other nodes. In a traditional point-to-point system, where there is only one intended recipient, this innate property of the wireless propagation channel can be a drawback, as the signal constitutes undesired interference in all nodes but the intended recipient. However, this effect also implies thatapacket can be transmitted tomultiplenodessimultaneouslywithoutadditionalenergyexpenditure.Exploitingthis broadcastadvantage,broadcast,multicastandmultihopunicastsystemscanbedesigned to work cooperatively and thereby achieve potential performance gains. As such, coop- erativetransmissioninwirelessnetworkshasattractedalotofinterestnotonlyfromthe research community in recent years [36,37,39,42,43,67,68]butalsofromindustryin theformoffirstpracticalcooperativemobilead-hocnetworksystems[49].Themajority of the work in the cooperative literature has so far focused onthesingleflowproblem, 66 thoughrecently there has been an increased interest in considering multiflowsettingsin cooperativenetworks[58,59,61–63]. As previously discussed, networks is routing and resource allocation are key prob- lems in cooperative networks. The situation is further complicated by the fact that the routing and resource allocation depends on the type of cooperation and other details of the transmission/receptionstrategies of thenodes. We consider a time-slottedsystemin which the nodes that have received and decoded the packet are allowed to re-transmit it in future slots. During reception, nodes add up the signal power (EA) received from multiple sources. Details of EA, and possible implementations have been extensively discussedinpriorwork[37,39,42,59]andhavebeenbrieflyhighlightedinChapter 2. We focus on the problem of minimum-energy multiflow cooperativetransmissionin thischapter, wheretherearek source-destinationpairs,withthesourcenodewantingto send a packet to its respective destination nodes, in a multihopwireless network. Other nodesinthenetwork,thatareneitherthesourcenorthedestination,mayactasrelaysto help pass on the message through multiple hops. The transmission is completed when allthedestinationnodeshavesuccessfullyreceivedtheircorrespondingmessages. Ithas beennotedintheliterature([64,68])thatakeytradeoffincooperativesettingsisbetween thetotalenergyconsumptionandthetotaldelaymeasuredintermsofthenumberofslots needed for all destination nodes in the network to receive themessage. Therefore,we take delay into consideration and focus on the case where there is a delay constraint, whereby the destination node(s) should receive the message within some pre-specified 67 delayconstraint. Wethereforeformulatetheproblemofperformingthistransmissionin such a way that the total transmission energy over all transmitting nodes is minimized, while meeting a desired delay constraint on the maximum number of slots that may be usedtocompletethetransmission. Thedesignvariablesinthisproblemdeterminewhich nodesshouldtransmit,when,andwithwhatpower. We furthermore assume that the nodes are memoryless, i.e., accumulation at the re- ceiver is restricted to transmissions from multiple nodes inthepresenttimeslot,while signals from previous time slots are discarded. This assumption is justified ( [64,68]) by the limited storage capability of nodes in ad-hoc networks, as well as the additional energy consumption nodes have to expand in order to stay in an active reception mode whentheyoverhearweaksignalsinprecedingtime-slots. Themaincontributionoftheworkpresentedinthischapterisasfollows:Ithasbeen conjecturedintheliteraturethattheproblemofjointlycomputingschedules,routing,and power allocationfor multipleflows in cooperativenetworksisNP-hard [61–63]. Inthis chapter we formulate the jointproblem of scheduling, routingand powerallocation in a multiflowcooperativenetworksettingandformallyprovethatnotonlyitisNP-hard,but it is alsoo(n 1/7−! ) inapproximable. (i.e., unlessP = NP,itisnotpossibletodevelop apolynomialtimealgorithmforthisproblemthatcanobtainasolutionthatisstrictly better than a logarithmic-factor of the optimum in all cases). We are not aware of prior work on multiflow cooperative networks that shows such inapproximabilityresults. We further prove that for a special case of k =1,thesolutionisasimplepathandoffer 68 an optimal polynomial time algorithm for joint routing, scheduling and power control. We establish analytical upper and lower bounds based on this algorithm and propose a polynomial-timeheuristic,theperformanceofwhichisevaluatedagainstthosebounds. The rest of this chapter is organized as follows: In section 5.2 we provide a mathe- maticalformulationoftheproblem. Insection5.3weconsiderthespecialcaseofk=1 and prove the solution is a simple path and can be found optimally in polynomial time. Theinapproximablityresultsarepresentedinsection5.4usingreductionfromminimum graphcoloringproblem. Weestablishanalyticalupperandlowerboundsforoptimalper- formance in section 5.5. A polynomial-time heuristic is proposed in section 5.6 and its performanceisevaluatedunderdifferentchannelconditionsandagainsttheperformance bounds. Concludingremarksaresummarizedinsection5.8. 5.2 ProblemFormulation Consider a network, G,withatotalof n nodes, I = {1,..,n}.Assumewehave r source nodes, labeled S = {s 1 ,s 2 ,...,s r },andr corresponding destination nodes, D = {d 1 ,d 2 ,...,d r }.Thesource-destinationnodescanbethoughtofaspairs, {(s k ,d k )} r k=1 , allwiththe samedelay constraintT.Thegoalistodeliveraunicastmessagefromeach source to its corresponding destination, possibly using other nodes in the network as relays. The objectiveis to do so usingthe minimumamount of sum transmitpowerand withinthedelayconstraint. 69 WeconsideracooperativewirelesssettingwithEAandconsidersignal-to-intereference- plus-noise(SINR)thresholdmodel,[37,59,69,70]. Thatis,inorderfornodeitobeable todecodemessagek attimet,thefollowinginequalityneedstobesatisfied: $ j∈s k (t) p jt h ji $ u/ ∈s k (t) p ut h ui +N ≥ θ. (5.1) Here s k (t) is the set of nodes transmitting the message k at time t, h ij is a constant between 0 and 1 representing the channel gain between node i and j,and N and θ are constantsrepresentingthenoiseandthedecodingthresholdrespectively. Equation(5.1)canbere-writtenas n ! j=1 h ji p k jt −θ r ! q=1 q$=k n ! u=1 h ui p q ut −θN≥ 0, (5.2) wherep k it isthepowerusedbynodeiattimettotransmitmessagek. Thesystemismemoryless,meaningalthoughweareallowedtoaccumulatethesame message from multiple sources during each time slot, we cannot accumulate over time. The relays are half-duplex, meaning they cannot transmit andreceivesimultaneously. Therelayscannottransmitmorethanonemessageatthesametimeeither. In order to apply ideas driven by the rich literature on multicommodity flows [3] to our problem, we need to somehow introduce the notion of delay constraint into the multicommodity setting. What follows is a transformation ofournetworkgraphthat would allow for the multicommodity flow technique to be applied, while observing the 70 delay constraint: For a delay constraint T,mapthegivennetworktoalayeredgraph with T layers as shown in Figure 5.1. Place a copy of all the nodes in the network on each of the layers. Connect each node, on each layer, to its corresponding copy on its neighboring layers with an edge weight of 0.Alsocreatedirectededgesbetween each node, on each layer k,andthenodesonthenextlayer k+1,withedgeweights representingtheamountofpowerrequiredtotransmitthemessagefromthenodeonthe topleveltothenodeonthebottomlevel,asawhole. Noticethatthereisnoedgebetween the nodes on the same level. Call the new graphG ! .Assignthenodescorrespondingto thesourcenodesofGonlevel1ofG ! assourcenodesinG ! andthedestinationnodeson levelT ofG ! ,correspondingtodestinationnodesinG,asdestinationsin G ! ,asshown in the figure. Similar transformations have been used in the literature in the context of multiflowtransmission[62]. Without loss of generality, we assumeunit length timeslots.Thenodeswhowantto transmitaretodosoatthebeginningofeach timeslot,andthedecoding(bynodeswho receive enough information during that time slot) will happen by the end of that time slot. Letz k it be an indicatorbinary variablethat indicates whether ornotnodei decodes the message k during time slot t,asperinequalityinequation(5.1). Inotherwords, we define z k it to be 1,ifnode i decodes message k during time slot t,and 0 otherwise. Let p k it be the transmit power used by nodei at each timet to transmit message k.We define another binary variable x k it ,thatis 1 if node i is allowed to transmit message k at time t,and 0 otherwise. A node is allowed to transmit during a particular time 71 slot, if it has already decoded that message in previous time slots, and it’s not receiving or transmitting any other messages during that time slot. Notice that being allowed to transmit does not necessarily mean that a transmission actually occurs. To take care of actual transmissions, let us definev k it to be a binary variable that is 1 if nodei transmits messagek attimet,and 0otherwise. Theproblemcanthenbeformalizedasacombinatorialoptimizationproblem: min P total = $ T t=1 $ n i=1 $ r k=1 p k it (5.3) s.t. 1.p k it ≥ 0, ∀i,t,k 2.x k d k T+1 =1, ∀k 3.x k it+1 ≤z k it +x k it , ∀i,t 4. (−M)(1−z k it )≤y k it , ∀i,t 5.p k it ≤Mv k it , ∀i,t 6. $ r k=1 ) v k it +z k it * ≤ 1, ∀i,t 7.v k it ≤x k it , ∀i,t,k 8.x k s k 1 =z k s k 1 =1,∀k 9.x k i1 =z k i1 =0,∀i∈I\{s k } 10.x k it ∈ {0,1} 11.z k it ∈ {0,1} 12.v k it ∈ {0,1}. 72 Herey k it = $ n j=1 h ji p k jt −θ $ r q=1 q$=k $ n u=1 h ui p q ut −θN,M isalargepositiveconstant,and theconstraintshavethefollowinginterpretations: 1. Nonegativepowerisallowed. 2. Every node in the destination set is required to have decoded the data by the end oftimeslotT. 3. If a node has not decoded a message by the end of time slot t,thatnodeisnot allowedtotransmitthatmessageattimet+1. 4. z k ti isforcedtobe 0ifmessagek isnotdecodedintimeslott. 5. p k it isforcedtobe0,ifnodeiisnottransmittingmessagekattimet(i.e. ifv k it =0). 6. A node cannot transmit and receive at the same time and can only transmit or receivea singlemessageateachtimeslot. 7. v k it is forced to be 0,nodei is not allowed to transmit messagek at timet (i.e. if x k it =0). 8. Onlysourceshavethemessageatthebeginning. 9. Nooneelsehasthemessageatthebeginning. 10. x,z andv are binaryvariables. WecallthisoptimizationproblemMCUE,for multiflowcooperativeunicastwithEnergy Accumulation. 73 Level 1 Level 2 Level 3 LevelT s 1 s 2 s r d 1 d 2 d r Figure5.1: Applyingthemulticommodityflowtechniqueforunicastcast 74 5.3 SpecialCaseofk=1 In this section we consider MCUE for the special case ofk=1 and prove the problem can be solved optimallyand in polynomialtime for this special case. We also provide a polynomial-timealgorithmtoachievetheoptimumsolution. Theorem 7. The optimal solution for MCUE is a simple path fork=1,butnotneces- sarilysofork> 2. Proof. The claim can be proved by induction on T:Fordelay T =1,theclaimis trivially true, as the optimal solution is direct transmission from the source, s,tothe given destination, d.Letusassumetheclaimistruefor T = t− 1.Tocompletethe proof, we need to show the claim holds for T = t.Pickanynodeinthenetworkas the desired destination d.Ifthemessagecanbetransmittedfromsource s to d with minimum energy in a time frame less than t,thenanoptimalsimplepathexistsbythe inductionassumption. So considerthecase when it takes exactlyT = t steps to turnon d.Thesystemismemoryless,sodmustdecodebyaccumulatingtheenergy transmitted from a set of nodes, v,attime t.Thiscanberepresentedas $ v i ∈v p v i t h dv i ≥ θ.We observe that there must exist a node v o ∈ v whose channel to d is equal or better than all the other nodes inv.Therefore,givenh dvo ≥ h dv i ,∀v i ∈ v\{v o } then $ v i ∈v p v i t h dvo ≥ $ v i ∈v p v i t h dv i ≥ θ.Inotherwords,ifweaddthepowerfromallnodesin v and transmit instead from v o ,oursolutioncannotbeworse. v o must have received the message by timet− 1,tobeabletotransmitthemessagetod at timet.Weknowbytheinduction 75 assumptionthattheoptimalsimplepathsolutionexistsfromsourcetoanynodetodeliver themessagewithint−1timeframe. Thus,forT =t,thereexistsasimplepathsolution betweensandd,whichisoptimum. Consideringtheabovetheorem,theMCUEproblemformulation(forthespecialcase ofk=1)reducesto: min P total = $ T t=1 $ n i=1 p it (5.4) s.t. 1.p it ≥ 0, ∀i,t 2.x dT+1 =1 3. −M(1−x it+1 )≤ n $ j=1 h ji p jt −θN, ∀i,t 4.p it ≤Mx it , ∀i,t 5.x s1 =1 6.x i1 =0,∀i'=s 7.x it ∈ {0,1} This can be solved optimally in polynomial time using dynamicprogramming. Let C(i,t)betheminimumcostittakesforsourcenodestoturnoni,possiblyusingrelays, withinatmostttimeslots. Thenwecan write: C(i,t) = min j∈Nr(i) [C(j,t−1)+w ji ] (5.5) 76 withC(s,t)=0,forallt andC(i,1) = w si ,whereNr(i) is theset that containsi and itsneighboringnodesthathaveanon-zerochanneltoi,w ji representsthepowerittakes forj toturnoniusingdirecttransmission. Thusthesolutionto(5.4)isgivenbyC(d,T) anditscomputationincursarunningtimeofO(n 3 ). 5.4 InapproximabilityResults For k =1,weprovedinTheorem5.3,thattheoptimalsolutionisasimple path. For k> 2,wecanconsiderthefollowingcounter-exampletoarguethatthesolutionisnot necessarily a single-path. Consider the scenario shown in Figure 5.2, where T =3, wheretheedgeweightsareequalandtheedgesshowningrayshowstronginterference. Therednodescannotbythemselvestransmitthemessagetod 2 ,asitcausesinterference for d 1 and d 3 preventing them from being able to decode the data. However, they can cooperate with each other, by each sending with half power to get the message to d 2 withoutcausingtoomuchinterferencefortheotherdestinations. s 3 s 2 s 1 d 1 d 2 d 3 Figure5.2: Anexampleofk> 2,withT =3,wheretheoptimalsolutionisnotasingle path. 77 To investigate the complexity of MCUE, let us start by lookingatasub-problem. Imagineaonehopsettingofksourcenodesandtheircorrespondingkdestinationnodes, with no relay nodes. Due to interference, not all sources can transmit simultaneously. Thetaskistoschedulethesourcesappropriately,sothateveryonecan gettheirmessage delivered to their corresponding destination within a time delay T.Theproblemisto find the minimum such T.LetuscallthisproblemMOSP,formulti-sourceone-hop scheduling problem 2 .ItisimportanttonotethatMCUEisatleastashardasMOSP. Thus,anyhardnessresultsobtainedforMOSPimplyhardnessofMCUE. In this section, we derive inapproximablity results for MOSPbyshowingthatany instanceofminimumgraphcoloringproblem[3]canbereducedtoaninstanceofMOSP. Lemma3. MOSPiso(n 1/7−! ) inapproximable,forany "> 0. Proof. Given an instance G(V,E), |V| = n,oftheminimumgraphcoloring,wecon- structa bipartitegraphG ! ,withthebi-partitionX andY with|X| = |Y| = n.Foreach node v i ∈ G,weplacetwonodesu i ∈ X and u ! i ∈ Y and connect them with an edge (u i ,u ! i ).AlsoforeveryedgeinG,e ij = {v i ,v j },placetwoedges (u i ,u ! j )and (u j ,u ! i )in G ! .Weassignu i andu ! i to bea sourceand destinationpair respectivelyfor alli.Weset equaledgeweightsforalltheedgesinG ! andset θ > 1togetaninstanceofMOSP. AsimpleexampleisshowninFigure5.3. Noticethatthegrayedges in the figure represent interference, and by setting θ > 1,amessagecanbesuccessfullydecodedif andonlyifthereisnointerferenceatthatnode. 2 This is essentially the problem considered in [66], though noproofofcomplexityisgiveninthat paper. 78 u 3 u 4 v 1 v 2 v 3 v 4 u 2 u 1 u ! 1 u ! 2 u ! 3 u ! 4 G G ! Figure5.3: ExampleconstructionofG ! ,foragivenG. This in turn means two sources in G ! can simultaneously transmit if and only if there is no edge in between them in G.Thus,thesetofnodesthataretransmitting simultaneouslyinG ! correspond to an independent set inG.Consequently,theoptimal solution to MOSP is equal to the minimum graph coloring of G,whichisknowntobe o(n 1/7−! ) inapproximable [65]. The following theorem follows by noticing that MOSP isaspecialcaseofMCUE. Theorem8. MCUE iso(n 1/7−! ) inapproximable,forany "> 0. Notice that the inapproximability result, given by Theorem 8, is stronger than, and implies, the NP-hardness result. In other words, it implies that not only finding the optimalsolutionisNP-hard butfindingapolynomialtimeapproximationalgorithmthat approximatestheoptimalsolutiontoMCUEwithafactorofo(n 1/7−! )isalsoNP-hard. 79 5.5 PerformanceBounds In section 5.4, we proved that MCUE problem is in general inapproximable. However, it was shownin section 5.3 that the problem can be solvedoptimallyand in polynomial time for the special case ofk=1.Inthissection,weusedtheresultsofsection5.3to obtainperformanceboundsforMCUE. 5.5.1 AnAnalyticalLowerBound InthissectionweestablishalowerboundontheoptimumsolutiontoMCUE. Togetabetterintuitionforthislowerbound,letusstartoffbyconsideringtheoptimal solution to MCUE for the case when there is only one flow presentinthenetwork. As before, we haven nodes and a channel H,butthistimethesource s wants to transmit themessagetoaparticulardestinationd,usingtheminimumenergywithinagivendelay constraint T.Thesystemiscooperativeinthatothernodesinthenetwork,maybe utilizedasmemorylessenergy accumulatingrelaystohelpachievetheminimumenergy goal. Based on section 5.3, the solution can be found by calculating C(d,T) where C(d,T)isdefinedasperequation(5.3). TofindalowerboundforMCUEforageneralcaseofrflows,withsource-destination pairs{(s k ,d k )} r k=1 ,allwiththesamedelayconstraintT,wenoticethatthecostpaidby optimalMCUEtocovereach nodecannotbelowerthantheoptimalminimumcostpaid 80 byeach sources k tocoveritscorrespondingdestinationd k intheabsenceofotherinter- fering flows. Based on that observation we derive the following lower-bound, LB(T), fortheOPT MCUE forr flowswhenthedelayconstraintisT: LB(T)= r ! k=1 C(d k ,T) (5.6) whereC(d k ,T)is defined as perequation(5.3). In otherwords,C(d k ,T)calculates the minimumcostofoptimalsingleflowtransmissiontocoveradestinationd k ,startingfrom itscorrespondingsourceunderadelayconstraintT. LB(T)takesthesumofthosecosts and use it as lower-bound - since we knowOPT MCUE has to cover all these flows and cannotdosoanybetterthantheoptimalsolutionforasingleflow. Computingthislower boundincursarunningtimeofO(n 3 ). 5.5.2 AnAnalyticalUpperBound In this section we establish an upper bound on the optimum solution to MCUE, for the generalcaseofr flows,withT ≥r. The upper bound is established by considering the multiplexing solution. At the extreme end ofT = r,wewouldallowonetimeslotforeachofther flow to transmit its message, while the other flows are silent. For a general time T(>r) we break the time intor blocks T =(τ 1 ,τ k ,...τ r ),suchthat $ r k=1 τ k = T.Weassigneachblockto one ofthe flows, whilethe otherflows are silent. We calculateC(d k ,τ k ),definedasper 81 equation (5.3). For a given tuple T,thesummationofthetotalenergyrequiredbyall flowstocompletetheirtransmissioncanbeachievedbycalculating: UB(T)= r ! k=1 C(d k ,τ k ) (5.7) This sum would provide an upper bound forOPT MCUE .Forageneral T ≤ r,wewill have ) T−1 k−1 * possibilitiesfor assigningthetimeslotsto different flows. Theupper bound iscalculatedasfollows: UB(T) = min T UB(T ) (5.8) To compute this upper bound we need to carry on the computationforcalculating asingleflowMCUE,discussedinsection5.3, ) T−1 k−1 * ×r times. Thus the upper bound incursarunningtimeofO(n 3 ). 5.6 APolynomial-timeHeuristic In this section we propose a polynomial time heuristic for MCUE, the performance of whichislaterevaluatedagainstthatoftheboundsestablishedearlierinthischapter. To recap, consider a network, G,withatotalof n nodes, I = {1,..,n}.Assume we have r source nodes, labeled S = {s 1 ,s 2 ,...,s r },and r corresponding destination nodes, D = {d 1 ,d 2 ,...,d r }.Thesource-destinationnodescanbethoughtofaspairs, 82 {(s k ,d k )} r k=1 ,allwiththesamedelayconstraint T.Thegoalistodeliveraunicast message from each source to its corresponding destination, possibly using other nodes in the network as relays. The objective is to do so using the minimum amount of sum transmitpowerandwithinthedelayconstraint. The algorithm works greedily by scheduling flows one by one. Each flow is given moreslotsthanitspreviousflows,toensureafeasiblesolutionalwaysexist. Thatmeans the algorithm works for T ≥ r.Eachflow,withtheexceptionofthefinalflow,uses morepowerthan required to deliverits message. Thisis achieved by assigning a higher threshold to that flow when scheduling the flow. After scheduling, the nodes that will betransmittingat each timeslotand thepowertheyusefortransmissionispassed onto the next flow. Each flow, when scheduling itself, will ensure that its transmission will not disturbthetransmissionofpreviouslyscheduledflows.Alowerthresholdisusedto check for disturbance, than the one used for scheduling the flow itself. Let us now look atthedetailsofthealgorithm. We schedule the r flows greedily, starting from the one that causes the least distur- bance. Without loss of generality, let us assume that we are scheduling the flows in the order 1 to r.Wehaveatotalof T time slot, for flow 1,weassign T 1 time slots for transmission,forflow 2,weassignT 2 andsoforth,suchthat: 1≤T 1 <T 2 <...<T k <..T r =T (5.9) 83 Recall that time-slots are in unit durations, thus can increment in integer units, thus each flow has at least one more time slot at its disposal than itsimmediatepredecessor, ensuringthata feasiblesolutionalwaysexists. In section 5.2, we defined θ to be the decoding threshold as per equation (5.1). For thismulti-flowsetting,eachflowisassigneditsown θ value,such that: θ 1 > θ 2 >...> θ k >...θ r = θ (5.10) Flow 1 is scheduled withT 1 and θ 1 ,asperalgorithminsection5.3.Westorethenodes that are scheduled to transmit in each time slot, and their transmit power and their cor- respondingreceivers ina blacklistB,assuchB(t) givesusthesetofalready scheduled nodes that are transmitting at time t and their corresponding powers, and their corre- spondingreceivingnodes. For thekth flow, we pick a subset of nodes (as potential relays) and useamodified versionof the algorithmdiscussedin section4.1, and assignanorderingtothosenodes. Let us call this ordered subset I k =(1 k ,2 k ,...,j k ,...,n k ),where 1 k corresponds to s k and n k corresponds tod k .Thissetcouldforinstancebeobtainedbypickingthenodes that would have been picked if we were to run the single-flow algorithm of section 5.3 forflowk.Recallthatgivenorderingthealgorithminsection4.1,wouldfindtheoptimal scheduling and power allocation in polynomial-time. In order to use that algorithm, we needtomodifythepowerallocationparttoensurethatwhenassigningpowerstothecur- rent flow we are not disturbing the previouslyscheduled flows.Letuscallthemodified 84 versionofpowerallocationalgorithmLPMF (forlinearprogrammulti-flow).LPMF, to be specified shortly, would calculate the instantaneous optimal power allocation for flowk at timet,giventhesetofinstantaneoussendersandreceiversforflowk and the setB(t) (of senders and receivers of flows 1 tok− 1 and their corresponding powers at thattimeslot). Given this modified power allocation algorithm, all that remains to be done is to determine these sets at each time slot, in order to minimize the overall power while meeting the delay constraint. Let C k (j k ,t) be the minimum energy needed for flow k to cover up to node j k in t steps or less. We can calculate this, using the following deterministicdynamicprogram: C k (j k ,t) = min i k ∈(1,..,j k ) [C k (i k ,t−1)+LPMF({1...i k },{i k +1...j k },θ k ,H,B(t))] (5.11) where C k (i k ,1) = LPMF(1 k ,{2 k ...i k },θ k ,H,B(t)) ∀i k ∈ I k \1 k ,and C k (1 k ,t)= 0 ∀t. Thus, for flow k,thetotalminimumcostforcovering n k nodes by time T k can be foundbycalculatingC k (n k ,T k ). Let us now look at the design of LMPF. LMPF is defined similar to that of power allocation algorithm in (4.2), except we have to take a few new points into ac- count. ConsiderLMPF for flowk,thealgorithmtakesasaninputasetoftransmitters (TX k (t)= {1...i k })andasetofreceivers(RX k (t)= {i k +1...j k }),andthesetofalready 85 scheduled nodesfor thattime-slotand theircorresponding powersB(t),thechannelbe- tween the nodes and the receiving threshold θ k .Forthisflow,thenewsetofrulesto abidebywouldbe: 1. Anodecannottransmitamessageforflowkattimet,ifithasalreadybeensched- uledtoparticipateinanotherflowinthattimeslot. Inanotherwords, ∀q∈TX k (t), ifq∈ B(t)thenp k qt =0. 2. Anodecannotreceiveamessageforflowk attimet,ifithasalreadybeensched- uled to participate in another flow in that time slot. This renders the power allo- cation task infeasible with the given set of transmitting andreceivingnodes. In anotherword, ∀q∈RX k (t), ifq∈ B(t)thenLPMF({1...i k },{i k +1...j k },θ k ,H,B(t)) =∞. Afterhavingtakentheabovetwoconditionsintoaccount,wecanproceedwiththelinear programasfollows: min $ q∈TX k (t) p k qt (5.12) s.t. 1.p k qt ≥ 0, ∀q∈TX k (t) 2. $ q∈TX k (t) h qj p K qt −θ k $ u∈TX f (t) h uj p f ut −θ k N≥ 0, ∀j∈RX k (t) ∀f∈{1,...,k−1} 3. $ v∈TX f (t) h vz p f vt −θ f $ u∈TXg(t) g$=f h uz p g ut −θ f N≥ 0, ∀z∈RX f (t) ∀g∈{1,...,k} ∀f∈{1,...,k−1} 86 Where constraint 1 ensures that there are no negative powers. Constraint 2 ensures that the nodes assigned to receive flowk at timet will in fact accumulate enough energy to decodethemessage,despitetheexistinginterference. Constraint3ensuresthatthepower beingassignedtonodesinflowk,isnotdisturbingthepreviouslyscheduledflows.This algorithm is referred to as LPMF({1...i k },{i k +1...j k },θ k ,H,B(t)).Asanotation, LPMF([{x...y},{z...α}],θ,H,B(t)) = 0,ifz≥ α 5.7 PerformanceEvaluation Inthissectionwecomparetheperformance oftheproposedheuristicagainsttheanalyt- icalboundsforanexamplenetworkwithanarbitrarilychosenthreeflows.Wealsolook attheeffect ofchanneldegradationintheoverallperformance. We consider a network of 100nodes uniformlydistributedon a 20 by 20 squaresur- face. Thechannelsbetween allnodesarestatic,withindependentandexponentiallydis- tributedchannelgains(correspondingtoRayleighfading),whereh ij denotesthechannel gain between node i and j.Themeanvalueofthechannelbetweentwonodes, h ij ,is chosentodecaywiththedistancebetweenthenodes,sothath ij =d −η ij ,withd ij beingthe distance between nodesi and j and η being the path loss exponent. The corresponding distributionforthechannelgainsisthengivenby f h ij (h ij )= 1 h ij exp " h ij (k) h ij # 87 Notice that the minimum power calculated by different algorithms, shown on the y-axesofthegraphsinthissection,arenormalizedbyvalueof θ (renderingitunit-less). Figure5.4showstheperformanceoftheheuristicagainstthatoftheanalyticalbounds. Ascanbeseentheheuristicisperformingclosetothelowerbound. Noticethatthelower boundisanunachievablelowerbound,inthatitassumesnointerferenceispresent. This meansthatitsperformanceisnotachievablebyanyalgorithm. Thisismoreemphasized whenwehavefewertimeslotsavailable,andthusweneedtousemorepowertotransmit themessagecreating alot of interference that is ignored bythelowerbound. As weget moretime-slotsavailableto us,theperformanceoftheheuristicand theboundsseemto converge, which is what we expect as the solution goes to a multiplexingsolutionin all cases. 4 5 6 7 8 9 10 1 10 2 10 3 10 4 Delay Constraint (T) Normalized Power Upperbound Heuristic Lowerbound Figure5.4: Performance oftheheuristicagainsttheanalyticalupperandlowerbound. 88 4 5 6 7 8 9 10 1 10 2 10 3 10 4 10 5 Delay Constraint (T) Normalized Power " = 3 " = 2 Figure5.5: Effectofchanneldegradationonthetotalenergyconsumed. We see the effect of poor channel conditions in Figure 5.5. As expected the perfor- manceisdegradedasthechannelconditionsbecomepoor,thishighlightstheimportance ofhavingsmartalgorithmstominimizetheenergyconsumptioninsuchscenarios. 5.8 Summary Inthischapterwe formulatedtheproblemof minimumenergycooperativetransmission inadelayconstrainedmultiflowmultihopwirelessnetwork,asacombinatorialoptimiza- tionproblem,forageneralsettingofk-flowsandformallyprovedthattheproblemisnot only NP-hard but it is o(n 1/7−! ) inapproxmiable. To our knowledge, the results in this chapter provide the first such inapproxmiablity proof [64] inthecontextofmultiflow cooperativewirelessnetworks. Itisinterestingtonotethatalthoughtheminimumgraph 89 coloring problem is NP-hard, the fractional graph coloring can be solved in polynomial time. That presents an interesting venue for future work and for designing possible ap- proximationalgorithmsforthisproblem. We further proved that for a special case ofk=1,thesolutionisasimplepathand offered an optimal polynomial time algorithm for joint routing, scheduling and power control. We then used this algorithm to establish analyticalupperandlowerboundsfor the optimal performance for the general case of k flows. Furthermore, we proposed a polynomial time heuristic for calculating the solution for the general case and evalu- ated theperformance of thisheuristicunderdifferent channelconditionsand againstthe analyticalupperandlowerbounds. 90 Chapter6 ConclusionsandFutureDirections In this thesis, we formulated the novel problem of delay constrained minimum energy cooperative transmission (DMECT) in wireless networks, encompassing both EA and MIA. We have shown that this problem is o(logn) inapproximable in broadcast and multicast cases and is NP-complete in the unicast case when mutual information accu- mulation is used . For the broadcast case with EA, we have presented an O(T log 2 (n)) approximationalgorithm. Another key algorithmiccontribution has been to show a polynomial algorithm that cansolvethisNP-hardproblemoptimallyforafixedtransmissionordering. Ourempiri- cal resultssuggestthatfor practicalsettings,a near-optimalorderingcan beobtainedby using Dijkstra’s shortest path algorithm. We have further showed that the unicast case can be solved optimally and in polynomial time when EA is used.Wehavestudiedthe energy-delay tradeoffs and the performance gain of MIA usingsimulations,andevalu- ated the performance of our algorithm under varying conditions. The summary of the 91 algorithmic results developed are presented in Tables 3.1 and 4.1.The empty slots are stillopenproblems. We further formulated theproblemof minimumenergy cooperativetransmissionina delayconstrainedmultiflowmultihopwirelessnetwork,as acombinatorialoptimization problem,forageneralsettingofk-flowsandformallyprovedthattheproblemisnotonly NP-hardbutitiso(n 1/7−! )inapproxmiable. Toourknowledge,theresultsinthischapter provide the first such inapproxmiablity proof in the context of multiflow cooperative wirelessnetworks. We observed that for a special case of k =1,thesolutionisasimplepathand offered an optimal polynomial time algorithm for joint routing, scheduling and power control. We then used this algorithm to establish analyticalupperandlowerboundsfor the optimal performance for the general case of k flows. Furthermore, we proposed a polynomial time heuristic for calculating the solution for the general case and evalu- ated theperformance of thisheuristicunderdifferent channelconditionsand againstthe analyticalupperandlowerbounds. There are many interesting open problems and research directions yet to be investi- gatedinthisfield. Inthefollowing,wehighlightanumberofthesedirections,whichare ofparticularinteresttotheauthorandtheresultsofwhichweconsidertobeofsignificant theoreticalandpracticalimpacts. • One area, that is of particular interest, is investigating the approximability gap in thebroadcastcasewithEA.Wehavealreadyprovedthatthisproblemiso(log(n)) 92 inapproximable, however the current best positive approximation result we have forthisproblemgivesanapproximationfactorofO(T log 2 (n)).Itwouldbeinter- estingtoseewhetherornotwecan tightenupthisgap. Noticethatweobtainedourapproximationalgorithmbyproposingamappingbe- tween our problem and the directed Steiner tree problem that preserved the ap- proximation factor. We, thus, argued that the best known approximation results for directed Steiner tree applies directly to our work. Hence, we are so far using theexistingapproximationalgorithmsofdirectedSteinertreeas ablockbox. The conjectureisthatacloserexaminationofhowthoseapproximationalgorithmsfor directed Steiner tree were developed, and applying those techniques [72,73] di- rectlytoourproblem,mightleadtoatighterapproximabilitygap. • Throughout the thesis, we did not make any assumptions on the structure of the network when deriving inapproximability results. We did however note, through simulations, that our proposed algorithms were performing close to optimal for uniform distribution of nodes for example. It would be interesting to investigate achieving better inapproximablityresults by making assumptionson the structure ofthenetwork. • We proved that the multiflow problem is o(n 1/7−! ) inapproxmiable and proved a heuristic that performed good in simulations. It would be interesting to establish approximation algorithms for this problem and see how close we can get to the theoretical limit. We drove the inapproximablity results using minimum graph 93 coloring problem. Good approximation algorithms exists forthisproblem[95– 97], and it is interesting to note that the fractional graph coloring can be solved in polynomial time. That presents an interesting venue for future work and for designingefficientheuristicsforthisproblem. Investigatingmulticommodityflow problem and its approximation algorithms [74,76] in the context of our problem mightalsoleadtointerestingresults. • Applying game theory to cooperative wireless settings is currently a vibrant field ofresearch [98–100]. Anotherinterestingvenuethatmightbepromisingtoinves- tigateisapplyinggametheorytotheabovesettingbyeithertryingtodevelopanew distributedapproach based on theassumptionthatthe nodes are selfish and might notbewillingto,leftontheirown,cooperate;orbyapplyinggametheoryanalysis to the distributed version of our existing algorithms and analyze the performance undertheassumptionthatthenodesareselfish. • More recently there has been a growing interest in the research community in in- vestigatingthe practical aspects of cooperative schemes settinggeared toward de- velopingpracticalschemescapableofharvestingthegainspredictedbyanalytical models [49,81,83]. 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Baghaie, Marjan
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Algorithmic aspects of energy efficient transmission in multihop cooperative wireless networks
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Viterbi School of Engineering
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Doctor of Philosophy
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Electrical Engineering
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11/25/2011
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