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1 StaticFrequencyAssignmentinCellular LataNarayanan Networks August,998 SunilM.Shendey tice.inthestaticfrequencyassignmentproblem,eachvertexofthegraphisa Acellularnetworkisgenerallymodeledasasubgraphofthetriangularlat- Abstract distinctfrequenciespercall.theedgesofthegraphmodelinterferenceconstraintsforfrequenciesassignedtoneighboringstations.thestaticfrequency representsthenumberofcallsthatmustbeservedatthevertexbyassigning basestationinthenetwork,andhasassociatedwithitanintegerweightthat anyouterplanargraph.fortheproblemofmulticoloringanarbitraryconnected describeanecientalgorithmtooptimallymulticoloranyweightedevenorodd lengthcyclerepresentingacellularnetwork.thisresultisfurtherextendedto assignmentproblemcanbeabstractedasagraphmulticoloringproblem.we whichguaranteesthatnomorethan4=3timestheminimumnumberofrequired subgraphofthetriangularlattice,wedemonstrateanapproximationalgorithm weightsatasmallneighborhood. adistributedmanner,whereeachstationneedstohaveknowledgeonlyofthe colorsareused.further,weshowthatthisalgorithmcanbeimplementedin DepartmentofComputerScience,ConcordiaUniversity,Montreal,Quebec,Canada,H3GM8. rithms,graphmulticoloring,distributedalgorithms. Keywords:Frequencyassignment,cellularnetworks,approximationalgo- ydepartmentofcomputerscience,rutgersuniversity,camden,nj0802,usa.

2 Cellulardataandcommunicationnetworkscanbemodeledasgraphswitheachver- texrepresentingabasestation(sometimescalledacell)inthenetwork.cellscan Introduction anygiventime,acertainnumberofactiveconnections(orcallsincellularnetwork terminology)areservicedbytheirnearestbasestation.thisserviceconsistsmainly communicatewiththeirneighborsinthegraphviadirectionalradiotransceivers.at interferencebetweentwodistinctcallsinthenetwork.however,cellularnetworksuse ofassigningafrequencytoeachclientcallinamannerthatminimizesoravoidsradio staticfrequencyassignmentproblem,therefore,consistsofdesigninganinterferencefreefrequencyallocationprotocolforanetworkwherethenumberofcallspercelliitedavailablebandwidthiscriticaltotheviabilityandeciencyofthenetwork.the axedspectrumofradiofrequenciesandtheecientsharedutilizationofthelim- nitetriangulargridembeddedintheplane.verticesrepresentingcellsareplaced knownapriori.thisformsthemotivationfortheproblemsstudiedinthispaper. attheapexesofsimilartriangles,andeachvertexhasatmostsixotherneighbors Inparticular,cellularnetworksareusuallymodeledasniteportionsoftheinsurroundingitinthegrid.Thereasonforadoptingthisparticulargeometrystems fromthefactthatcellsareuniformlydistributedinthegeographicareaofthenetwork,andanindividualcellgenerallyhassixdirectionaltransceivers.hence,the asaregularhexagon.thetriangulartilingrepresentingthenetworkissimplythe planardualoftheresultingvoronoidiagram.weshallrefertotheresultinggraphs Voronoiregionaroundacell(orequivalently,thatcell'scallingarea)canbeidealized ashexagongraphs. hasanassociatedintegerweight,w(v)0.aw-coloring(ormulticoloring)ofg abstractedasfollows.letg=(v;e)denoteanhexagongraph.eachvertexv2v Thefrequencyassignmentproblemincorporatinginterferenceconstraintscanbe w(v)distinctcolorswherebyforeveryedge(u;v)2e,thesetofcolorsassignedto theendpointsuandvaredisjoint.inparticular,weareinterestedinaminimum isanassignmentofsetsofcolorstotheverticessuchthateachvertexvisassigned multicoloringoraw-coloringofgthatusestheleastnumberofcolors. thecellularnetworkliterature.however,inpractice,cellularsystemstendtobemorecomplicated functions;seeforexampleborndorferet.al.[].nevertheless,thehexagongraphmodelcontinues andtherehavebeenrecentstudiesthatattempttomodelmoregeneralinterferencepatternsandcost Wenoteherethatthisisthemostcommonlyusedmodelforfrequencyassignmentproblemsin tobeofsignicancefromthehistoricalstandpointandasanabstractionthatissucientlycloseto realitytoprovideusefulinsights. 2

3 distinctfrequencyanditisassumedthattwocallsmayusethesamefrequencyifand ausefulabstractionoftheessentialinterferenceconstraints:eachcolorrepresentsa Inthecontextoffrequencyassignment,amulticoloringasdenedabove,provides oftheradiospectrum,andfrequencyreuseiscontrolledbyasequenceofnon-negative inpractice,theavailablecellularfrequencyspectrumisacontiguouslinearsubinterval onlyiftheyoriginateindistinctcellsthatarenotneighbors.itshouldbenotedthat incellsthatareadistanceiapartintheunderlyinggraphmustbeassignedfrequencies integers,c0c:::,withc0,calleddistancereuseconstraints.twodistinctcalls thesimplestconstraints,viz.whenc0=c=andci=0,i2.underthis andversionsofthefrequencyassignmentproblem.weformulateourproblemunder thatdierbyciinthefrequencyspectrum.hale[3]discussesmanygeneralizations formulation,theproblemreducestobeingabletocomputeaminimummulticoloring a(vertex)weightedgraphthatisanite,inducedsubgraphoftheinnitetriangular toagivenhexagongraph. grid.thus,thegraphisplanar,andeveryvertexv2vhasdegreeatmostsix Inthesequel,weassumethatG=(V;E;w)denotesahexagongraph,i.e.itis andanassociatedintegerweight,w(v)0.theweightedchromaticnumberofg, ofdetermining(g)isnon-trivial.infact,ithasbeenestablishedonlyrecentlythat forgraphswitharegularstructuresuchasthoseconsideredinthepaper,theproblem denoted(g),istheminimumnumberofcolorsrequiredinaw-coloringofg.even interesttostudyapproximationalgorithmsfortheproblem. apolynomialtimealgorithmforcomputing(g)canbedevised.naturally,itisof thecorrespondingdecisionproblemisnp-complete[7],andhenceitisunlikelythat atverticesinanymaximalcliqueinthegraphisatriviallowerboundon(g).note atanysetofmutuallyadjacentvertices.thusthemaximumoverthesumofweights Itiseasytoseethat(G)mustbegreaterthanthetotalnumberofcolorsrequired upperbounds,whilethereisavastliteratureonalgorithmsforfrequencyassignment thatforhexagongraphs,edgesandtrianglesaremaximalcliques.inthedirectionof arenoprovenboundsontheperformanceoftheproposedalgorithms,intermsofthe numberofcolorsusedinrelationtotheweightedchromaticnumber[2,5,6,8,9].we ongraphs(especiallyhexagongraphs)thatclaimtousefewcolors,generallythere noteheretwoexceptions.awell-knownalgorithm,sometimesreferredtoasfixed usescolorsfromtherstset,andavertexthatbasecolor2or3usescolorsfromthe usesthreexedsetsofcolors,oneforeachbasecolor.avertexthathasbasecolor Allocation,usesthefactthattheunderlyinggraphcanbe3-colored.Thealgorithm manyas3timesthenumberofrequiredcolors.janssenet.al.[4]proposeadierent secondorthirdsetsrespectively.itiseasytoshowthatthisalgorithmcoulduseas 3

4 3=2timestheminimumnumberofcolorsrequired. algorithmcalledfixedpreferenceallocationthatisguaranteedtousenomorethan graphs.insection4,weaddressthequestionofmulticoloringanarbitraryhexagon Section3,wepresentoptimalalgorithmsformulticoloringcyclesandouterplanar Inthenextsection,weformallydenesomebasicterminologyandproblems.In graph.ourmainresultisanecientapproximationalgorithmwithaperformance offuturedirectionsinsection6. guaranteeofwithin4=3oftheoptimal.finally,insection5,weshowhowtoimplementtheabovealgorithminadistributedmanner2.weconcludewithadiscussion denedontheverticesofthegraph,wherew(v)representsthenumberofcallsto 2LetG=(V;E;w)beahexagongraphwithanon-negativeintegerweightvectorw Preliminaries withverticesandedgescontainedintheinnitetriangularlattice(tessellation)ofthe beservedatvertexv.weassumehereafterthatghasaxedplanarembedding plane.thusanyvertexvcanbeconnectedtoatmost6neighbors,andforaxed edgeincidentonv,anyotheredgeincidentonvisatanangleof=3;2=3;;4=3or 5=3fromthatedge.Sincethetriangularlatticeis3-colorableintheordinarysense (i.e.wheneachvertexhasunitweight),theunderlyinggraphcorrespondingtounit weightsatverticesofgisalso3-colorable. thepalettecsuchthat C(thecolorpalette)andafunctionfthatassignstoeachv2Vasubsetf(v)of Aw-coloringormulticoloringofthegraphG=(V;E;w)consistsofasetofcolors 8(u;v)2E;f(u)\f(v)=:twoneighboringverticesgetdisjointsetsofcolors. 8v;jf(v)j=w(v):eachvertexgetsw(v)distinctcolors,and maticnumberofg,denoted(g),isthesmallestnumbermsuchthatthereexists amulticoloringofgofspanm.thusgivenahexagongraphg,ourobjectiveisto ThespanofamulticoloringisthecardinalityofthesetC.Theweightedchro- ndamulticoloringforgwhosespanisascloseto(g)aspossible. inadistributedsetting. dentlyderived,andunlikethemcdiarmid-reedalgorithm,hastheadvantageofbeingimplementable havealsodescribedadierent4=3-approximatealgorithmfortheproblem.ourresultwasindepen- 2WenotethatinadditiontoshowingtheNP-hardnessofthisproblem,McDiarmidandReed[7] 4

5 thattheweightofanymaximalcliqueofgisalowerboundon(g).letd[2] D[3] anedge(triangle)ingtobethesumoftheweightsofitsendpoints(apexes).note TheonlymaximalcliquesinGbeingedgesandtriangles,wedenetheweightof denedg=maxfd[2] GdenotetherespectivemaximaovertheweightsofedgesandtrianglesinG,and G;D[3] Gg.Then,ifthereexistsamulticoloringofGwithspan, Gand itfollowsthat: besuitablyorderedorpartitioned;inparticular,wewilloftenassumethatvertices Wewillassumewithoutlossofgeneralitythatanypaletteofavailablecolorscan (G)DG: areassignedcolorsfromthecircularlyorderedinterval[;m]=f;2;:::;mg,where itmeansthatthevertexiscoloredwiththesetfi;i+;:::;jginacyclicmanner Forinstance,whenavertexisassignedthesubintervalofcolors[i;j]fromthepalette, Misapositiveintegerthatdependsontheparticulargraphunderconsideration. wherecolorisassumedtofollowthecolorm. ConsiderahexagongraphG=(V;E;w)withnverticesintheformofasimple 3cycle,labeledu;u2;:::;uninclockwiseorder.Forsimplicity,letwi;in, OptimalMulticoloringofCycles onwhethern,thenumberofverticesonthecycle,isevenorodd. optimallycoloredwithexactly(g)colors.therearetwocasestoconsiderdepending denotetheweight,w(ui),ofvertexui.weshowthatanysuchhexagongraphcanbe maximalcliquesofgbeingedges,dg=d[2] Supposethatn=2m,i.e.thegraphconsistsofanevenlengthcycle.Thenall colorpalette[;dg].theideaistoassignforim,thecolors[;w2i ]tothe cycle.weobservethataverysimplegreedystrategysucestomulticolorgwiththe Gisthemaximumweightofanedgeinthe vertexu2i.notingthatforim, odd-numberedvertexu2i andthecolors[dg w2i+;dg]totheeven-numbered withsubscriptsinterpretedcyclically,itfollowsthatthegivenmulticoloringisproper. DGw2i +w2i; DGw2i+w2i+ and optimalmulticoloringofg. Byconstruction,(G)=DGandthesimpleparity-basedalgorithmthusprovidesan [8,4],butitwasonlyappliedtonetworksconsistingofsimplepaths(itiseasyto Wenotethataverysimilarideawasalreadyusedinthecellularnetworkliterature 5

6 unweightedodd-lengthcycleneedsatleastthreecolorsinanyordinarycoloring.for parityargumentfailstomulticolorodd-lengthcycles,preciselybecausetheunderlying seethatthisstrategyworksingeneralforanybipartitegraph).unfortunately,the Denition3.LetG=(V;E;w)beanodd-lengthsimplecycle,withverticeslabeled bemulticoloredwithdg=4colors,butneeds5colorsinstead. instance,ifgisa9-cyclewithweight2oneachvertex,itiseasytoseethatgcannot u,u2,:::,u2m+,m,inclockwiseorder.wedene D0G=maxfD[2] G;dP2m+ i=wi m eg Theorem3.2LetG=(V;E;w)beacycleofoddlengthn=2m+3.Then Proof.ItisclearthatD[2] (G)=D0GandGcanbeoptimallymulticoloredwithexactlyD0Gcolors.Further,the multicoloringcanbeobtainedintimeo(n). isatmostm,anysinglecolorcanbeusedonlyatmverticesorfewerinthecycle. andhenced0g,isalowerboundon(g).sincethesizeofanindependentseting Gisalowerboundon(G);weestablishthatd2m+ i=wi Thetotalnumberofcolorsneededatallverticesbeing2m+ m e, (G)d2m+ NextweshowthatGcanbecoloredwithD0Gcolorsusingalineartimealgorithm; i=wi m e.thus,d0gisindeedalowerboundon(g). i=wi,weconcludethat thiscompletestheproofofthetheorem.werstobservethattheremustbeasmallest indexk,km,whichsatisestheinequality 2k+ iswell-denedandcanbefoundeasilyinlineartime. NotethatthispropertyholdstruefortheindexmfromDenition3.,andhencek i=wikd0g:.verticesuthroughu2kareassignedcontiguouscolorsinacyclicmannerfrom Theverticesofthecyclearenowcoloredasfollows: thepalette[;d0g].specically,forj2k,vertexujisassignedthecolors cyclically.byconstruction,thisensuresthatthepathu;u2;:::;u2kisproperly [(+j Xi=wi);jXi=wi]; multicoloredsinced0gd[2] G. 6

7 2.Verticesu2k+throughu2m+arecoloredbasedontheirparity(asintheevencyclealgorithm).Inparticular,for2k+i2m+,thevertexuiisassigned thecolors[;wi]ifiiseven,orthecolors[d0g wi+;d0g]ifiisodd.again, thisensuresthatthepathu2k+;u2k+2;:::;u2m+isproperlymulticolored. theedge(u2k;u2k+)isproperlymulticolored:thisisaconsequenceoftheminimality ofk,forweknowthat2k theedge(u;u2m+)isalsoproperlymulticolored.allthatremainsistoverifythat Sincevertexuhasthecolors[;w]andvertexu2m+thecolors[D0G w2m++;d0g], amongthecolors[d0g w2k++;d0g]assignedtovertexu2k+. Weillustratethelabelingschemeusinga9-cyclewithweight2oneachvertexas i=wi>(k )D0G.Hence,nocolorassignedtou2kcanbe anexample.asd0g=maxf4;5g=5,weusethepalette[:::5].since5i=wi= assigncolorsasinabipartitegraph,fromthetwoendsoftheinterval[;5].finally, alwaystakingthenextfouravailablecolorsinthepalette.forthelastvevertices,we 02D0Gbut3i=wi=6>D0G,wecolortherstfourverticesinacyclicmanner, wenotethatouralgorithmcanactuallymulticoloranycycle,andnotjustcyclesthat arehexagongraphs(i.e.embeddedinthetriangularlattice). everyvertexofgliesontheboundaryoftheexteriorface.itisstraightforwardto graph.agraphissaidtobeouterplanarifitcanbeembeddedintheplanesothat Theorem3.2canbeusedtoderiveanoptimalmulticoloringofanyouterplanar seethattheweightedchromaticnumberofanygraphisthemaximumtakenoverthe chords.abiconnectedouterplanargraphgwithoutchordsisasimplecycleandcan siderabiconnectedouterplanargraph:anysuchgraphisacyclewithnon-intersecting weightedchromaticnumbersofitsbiconnectedcomponents.thusitsucestocon- bemulticoloredoptimallyusingtheconstructionintheproofoftheorem3.2.otherwise,let(u;v)beachordandletgandg2bethetwopartsofgonthesidesof thecolorassignmentofg2sothatthecolorsassignedtouandving2agreewith thischord,eachoneincludingtheedge(u;v).recursivelycolorgandg2.relabel coloredoptimallyusing(g)colors. Corollary3.3LetG=(V;E;w)beanarbitraryouterplanargraph.ThenGcanbe thoseassigneding.thefollowingcorollaryisimmediate: multicoloring.weremarkthatcorollary3.3appliestoanyouterplanargraph,and notjustouterplanarhexagongraphs. Acarefulimplementationofthealgorithmsketchedabove,resultsinalineartime 7

8 4Inthissection,weconsidertheproblemofcomputinganapproximatemulticoloring ofanarbitraryhexagongraph.sinceahexagongraphmaycontainanoddcycleasan Approximatemulticoloringofhexagongraphs inducedsubgraph,itfollowsasaconsequenceoftheorem3.2thatd[3] D[3] weighttakenoveralltriangles,isnotalwaysatightbound.forexample,consider a9-cyclewhereeveryvertexisgiventheweightk,forsomeintegerk2.while G=D[2] G,themaximum choosea9-cyclebecauseitisthesmallestoddcyclethatcanbeaninducedsubgraph ofthetriangularlattice.thisshowsthatanyalgorithmtocolorhexagongraphsmust useatleastd9d[3] G=2kforthegraph,weknowfromTheorem3.2that(G)=d9k=4e.We graphusingatmost4dd[3] demonstrateanecientapproximationalgorithmthatcanmulticoloranyhexagon G=8ecolorsonsomegraphsGwithtriangleboundD[3] G=3ecolors(andhence,atmost4d(G)=3ecolors). G.Infact,we weletm=dd[3] ponentsofgcanbeindependentlycoloredwithoutanycolorconicts.forsimplicity, Withoutlossofgenerality,weassumethatGisconnected,sincedisconnectedcom- withabasecoloringofgsothateveryvertexgetsbasecolorred,blueorgreen.with eachbasecolor,weassociateaclassofmhuesidentiedwiththeinterval[;m]. G=3eandwechoosethefollowingcolorpaletteinouralgorithm.Start withtheinterval[;m].theentirecollectionof4mdistincthuesformsourcolor palette. Inaddition,wehaveatourdisposalaclassofauxiliarypurplehues,againidentied theauxiliarypurpleclass.wedescribethealgorithmasproceedinginvephases; possiblebeforetryingtousehueseitherfromtheremainingtwobaseclassesorfrom Theideaistoleteachvertexvuseasmanyhuesfromitsbasecolorclassas wemaintaintheinvariantthatattheendofeachphase,thegraphispartiallybut completed.wealsoassignanarticialprioritytovertices:redverticesdominateover letgi=(vi;ei;wi)denotetheremaininggraphafterphasei(i4)hasbeen correctlycolored.tofacilitatereasoningaboutthecorrectnessofthealgorithm,we blueoneswhichinturndominateovergreenones.thispriorityschemeisusedin phases2and3toselect,ineachcase,asuitablesubsetofverticesforpartialcoloring. of24colorsequallydividedamongthered,blue,greenandpurplehues. forwhich3m=d[3] WeillustrateouralgorithmwitharunningexampleshowninFigure,agraphG Avertexv2Gisdenedtobelightifw(v)Mandtobeheavyotherwise.This Gcaneasilybeveriedtobe8.Hence,thecolorpaletteconsists distinctioniscriticaltoeachofthethevephasesbelow: Phase:Everyvertexvisassignedtherstw(v)huesfromitsbasecolorclass,in particular,thehues[;minfw(v);mg].allthelightverticesthusgetcompletely 8

9 Figure:Ahexagongraphwithinitialweights 2 0 Red Blue Green vertexvisdecreasedbym,resultinginthegraphg. coloredandaredeletedfromthegraph.theweightofeveryremainingheavy everytriangleingmustcontainatleastonelightvertexthatiseliminatedinthe rstphase.lethdenotethesubgraphofginducedbythedegree3verticesing. ItiseasytoseethatGhasnomaximalcliquesofsizegreaterthan2,because 2=3radiansinorder;furthermorethegeometryimpliesthatallthreeneighborshave embedding)ing,thentheincidentedgestotheneighborsformsuccessiveanglesof Notethatifavertexv2Ghasthreeneighbors(say,inclockwiseorderinthexed thesamebasecolor.itfollowsthateachconnectedcomponentinhcontainsvertices orblueandgreenvertices. consistsofeitheranisolatedvertex,orcontainsonlyredandblue,orredandgreen, thatbelongtoatmosttwobasecolorclasses.thus,everyconnectedcomponentofh amongitsneighbors(ifany)inh(recallthatreddominatesbluewhichdominates green).clearly,thepriorityverticesformanindependentset(infact,adominating Callavertexv2Hapriorityvertexifandonlyifithasthehighestpriority set)inh. Phase2:Withoutlossofgenerality,letvbearedpriorityvertexinHwiththree blueneighborsinh.letg(v)bethemaximumamongtheweightsofthe colortheremainingweightonvsinceallthreeblueneighborsofvareheavy ThenvcanborrowfromamongthelastM g(v)greenhues;thesesuceto threegreenneighborsofv;theseverticesmusthavebeenlightverticesing. verticesandhence,w(v)m g(v).accordingly,visassignedthegreen 9

10 r-6 r-6 r-6 r-6 r-6 r-6 b-6 b-6 b-6 b-6 g-6 g-3 g g g g-3 b-6 g-2 r (a) (b) g x v u y t b-2 g-6 2 b a c b d e Figure2:Colorassignmentduring(a)Phaseand(b)Phase2 hues,[m g(v)+;m],andeliminatedfromfurtherconsideration.notethat thepartialcolorassignmentattheendofphase2hasnocolorconictsamong neighbors,andtheremaininggraphisdesignatedg2. Figure2detailsthepartialcolorassignmentattheendofphaseandphase2 respectively.notethatthesixredhuesaredenotedasr-6andsoforthinthe gure.sincethesubsetofpriorityverticeseliminatedinphase2isadominatingset ofh(thedegree3verticesofg),everyremainingvertexing2nowhasdegreeat most2.equivalently,theconnectedcomponentsofg2consistofisolatedvertices, cyclesandpathsinthetriangulargrid.notealsothatanyedgeofg2hasaresidual weightofatmostm,aconsequenceofthedenitionofheavyvertices.ifthegraph G2containsonlyevencyclesorpaths,thenwecancoloralltheverticesusingtheM purplecolors.howeverg2maycontainisolatedverticesandoddcycles.inthenext phase,weessentiallyeliminateallpotentialcyclesing2. Callavertexv2G2acornervertexifandonlyifithastwoneighborsx;yofthe samebasecolorclassing2suchthattheanglesubtendedatvbytheincidentedges (v;x)and(v;y)isexactly2=3radians.further,acornervertexvisapriorityvertex ing2ifandonlyifvhasthehighestpriorityamongallitsneighbors,ifany,thatare alsocornervertices.itisnotdiculttoseethatthesubsetofpriorityverticesing2 formsanindependentseting2.also,everycornervertexiseitheritselfapriority vertexorisadjacenttoapriorityvertex;hence,thesubsetofpriorityverticesis adominatingsetofthesubgraphinducedbythecornerverticesing2.finally,by denitioneverycycleing2containsatleastonecornervertex.thus,coloringpriority verticesandeliminatingthemalsobreaksallcycles.forexample,infigure2-(b), 0

11 a x p Red t u Figure3:LocalgeometryaroundabluepriorityvertexvinPhase3 r Blue v Green priorityvertices. theverticeslabeledv,v0andxarecornerverticesofg2;amongthem,vandv0are b y q Phase3:Withoutlossofgenerality,letvbeabluepriorityvertexinG2withred neighborsxandying2asshowninfigure3.notethatudenotesthethird blueverticespandqmustbelightvertices.whileuisabsenting2,thereare priorityconsiderationsthatxandymustbenon-cornerverticesandhence,the neighborofvofthesamebasecolorclassasxandy.itisalsoeasytoseefrom (i)uwaseliminatedinphase(i.e.w(u)=0):letg(v)bethemaximum twopossibilities. overtheweightsofv'sgreenneighbors(i.e.verticesa,bandtinfigure3) ing.sinceudidnotparticipateinphase2,vcanborrowfromamongthe lastm g(v)greenhues;thesesucesincetwoofv'sredneighborsareheavy [M w2(v)+;m],andeliminated. verticesandhencew2(v)m g(v).accordingly,visassignedthegreenhues, andrecallthatuwasassignedthelastw(u)greenhues,[m w(u)+;m], greenneighborsofuandving.letwabbethemaximumamongw(a)andw(b) (ii)uwasapriorityvertexinphase2:consideraandb,thecommon(light) thegreenhues[wab+;wab+w2(v)]fromthemiddleofthegreenspectrum. However,wemustensurethatthisassignmentwillnotconictwiththegreen duringphase2.sincew2(v)+w(u)+wabm,itappearsthatvcouldborrow huesassignedtot(seefigure3). Ifw(t)wab,thenclearlythenewassignmentdoesnotconictwithprior nowusestherstwabandthelastw(t) wabgreenhues.thenewassignment colorassignments.however,ifw(t)>wab,thenwecanrecolortasfollows:its originalassignmentoftherstw(t)greenhues(inphase)ischangedsothatt

12 g2 p p-4 g2-5 b4-5 4 p p g3-4 g6-2 Figure4:Colorassignmentduring(a)Phase3and(b)Phase4 8 p p p-6,b5-6 sincew(t)+w2(v)m. totcannotconictwiththegreenhues,[wab+;wab+w2(v)],borrowedbyv (b) otherneighboroft.clearlysuchaconictcouldoccuronlyifsomeotherneighborof talsoborrowsgreenhuesinphases2or3.fromtheobservationsabove,weconclude Itremainstoverifythatrecoloringtdoesnotcauseacascadingconictwithany phase)andp;q(theyarelighting).theonlyremainingpossibility,viz.vertexr,is thatthiswouldbeimpossibleforverticesx;y(theyarenotpriorityverticesineither 3,itwouldhavegreenneighborsinG2andwouldborrowbluehues.Figure4-(a) notaproblemeither:rcouldnothavebeenapriorityvertexinphase2(sinceithas threeconsecutivelightneighborsp;tandq)andifitwereacornervertexinphase lightvertexlabeledtinfigure2-(b)isrecoloredasdescribedabove. depictsthecolorassignmentduringphase3fortherunningexample;notethatthe paths.itiseasytoseethatanyremainingisolatedvertexmayhavearesidualweight furthermore,theremaininggraphg3consistsonlyofisolatedverticesorstraight-line Thus,attheendofphase3,wehaveacorrectpartialassignmentofcolors,and Phase4:Toanyisolatedvertexv2G3,werstassignminfw3(v);Mgpurplehues. betweenand2m,whereastheweightoneveryremainingedgeisatmostm. Ifw3(v)>M,thenwestillneedtond=w3(v) MMadditionalcolorsto nishcoloringv.withoutlossofgenerality,weassumethatvisaredvertex, andobservethatalltheneighborsofvmusthavebeenlightverticesing. Hence,theblue(green)neighborsofvmusthavehadcolorseitherassigned totheminphaseorinphase3(asaresultofrecoloring).weclaimthat 2

13 vcanstillborrowcolorsfromeithertheblueorthegreenpaletteswithout beusingcolorsfromeitherendoftheirbasecolorpalettes.letb3b2bm Fromthedescriptionofphasesand3,observethatthelightneighborsofvmay conictingwithanyneighboringassignment. andg3g2gmbetheweightsofv'sblueandgreenneighborsing thebluepalette.likewise,thereareatleastm (g+g2)greenhuesavailable assignedbluehues,vhasatleastm (b+b2)huesavailableforitsusefrom respectively.clearly,regardlessofthemannerinwhichtheblueneighborsare tog.sincethegreenvertexwithweightgformsatriangleingwithvand that M (b2+g).asimilarargumentshowsthatm (g2+b).itfollows eitheroneoftheblueverticeswithweightborwithweightb2.inanyevent, orinotherwords,thatvcanobtaintheremainingcolorsbyborrowingeither onlybluehuesoronlygreenhueswithoutanycolorconictswithassignments maxfm (b+b2);m (g+g2)g; verticesinourrunningexample.attheendofphase4,theremaininggraphconsists Figure4-(b)demonstratesthecolorsassignedinPhase4totheremainingisolated priortothephase. onlyofstraight-linepaths,i.e.pathsinwhichanytwoconsecutiveedgessubtendan Phase5:Sinceeveryremainingconnectedcomponentisastraight-linepathwitha edgehasaresidualweightofatmostm. angleofdegreesatthecommonvertex.further,asnotedabove,everyremaining weightedchromaticnumberofatmostm,itsucestousethegreedyparitybasedstrategydescribedinsection3tonishcoloringthegraphusingthem sincethepurplehueswereusedonlyinphase4tocolorisolatedverticesinp; purplehues.thiscannotcauseanyconictwithpreviouslycoloredvertices runningexample.thefollowingresultisimmediate: Figure5showstheentirecolorassignmentconstructedbyouralgorithmforthe thelatteraredisconnectedfromanyremainingvertexinthecurrentphase. foranyhexagongraphg=(v;e;w)canbeecientlycomputedinlineartime. Theorem4.Anapproximatemulticoloringthatusesnomorethan4dD[3] G=3ecolors 3

14 r-6,g2 b-6,p b-6,p g r-6,p6 b-2 g-6,p-4 g r-6,g2-5 b-3 g-6,b4-5 b-6,p g r-6,p g-3 r-6,g5-6 g6-2 b-6,g3-4 r-6,p-6,b5-6 5 Adistributedimplementation Figure5:Completecolorassignmentintheexamplegraph b-6,p g-2 r-6,p modelinganetworkofprocessors(basestations),witheachprocessorresponsiblefor implementedinacompletelydistributedmanner.weconsiderthehexagongraphas Thealgorithmgivenintheprevioussectionhastheadditionalpropertythatitcanbe asinglevertexinthehexagongraph.thenetworkhasthesamespatialembeddingas thegraph,andprocessorsatneighboringbasestationsinthenetworkcanexchange localinformationeciently.foreaseofdescription,wewillsometimesidentifythe verticesofthehexagongraphwiththeirprocessors. arbitraryvertextobetheorigin,andthreedirectionalaxesthatintersecttheorigin: onedesignatedthehorizontalaxis,andtheremainingtwoatangles=3and2=3 Inthexedplanarembeddingoftheinnitetriangulargrid,wecanselectan fromthehorizontalaxis.itiseasytoseethatanypathinthegraph,wheretheangles subtendedbyallintermediateedgesinthepathareexactly,isorientedalongone ofthethreedirectionalaxes.foreveryvertex,wewishtoassignaparitywithrespect toeachdirectionalaxis.thiscaneasilybedoneasfollows.theparityofavertexv alongthehorizontalaxisisdenedtobetheparityofthelengthofthepathoriented alongthehorizontalaxisfromvtoavertexonthe=3axisthatintersectstheorigin. 4

15 axisintersectingtheorigin.thusgivenanarbitrarynitehexagongraph,anypath thepathfromvorientedalongthe=3axis(2=3axis)toavertexonthehorizontal Similarly,theparityofvalongthe=3axis(2=3axis)istheparityofthelengthof canbepre-computedaccordingtotheparitiesoftheverticesalongthepath. inthegraphthatisorientedalongoneofthedirectionalaxeshasa2-coloringthat information: Abasecoloringforthegraphisknown:eachprocessorknowswhetheritsvertex Ouralgorithmassumesthateachprocessorinitiallyhasaccesstothefollowing A2-coloringalongeachpathorientedalongadirectionalaxisisknown.This meanseachprocessorknowsthreebitscorrespondingtowhetherithasevenor isred,greenorblue,andalsothecolorofeachofitsneighbors. ThevalueofD[3] classesamongprocessorsisknown.additionally,thisimpliesthateveryvertex oddparityalongeachofthe3directionalaxes. hasaccesstoaknownsetofmpurplehuesifneeded. Gisknown;thisalsoimpliesthatthedivisionofbasecolor insection4.thealgorithmstartswiththreeroundsofinformationgathering,after participateinoneormoreofthevephasesoftheapproximationalgorithmdescribed Thedistributedalgorithmconsistsofeachprocessordeterminingwhetheritshould whichnomorecommunicationotherthaninformingneighborsofthecurrentcolor assignmentisrequired;essentially,processorscancontinueindependentlytocompute Round:psendsitsweighttoeachofitssixneighbors. thehuestoassigntothemselves.wedescribethecommunicationroundsfromthe perspectiveofaxedprocessorp. Round2:Havingreceivedtheweightsofallitsneighbors,pdecidesifitwouldbea degree3vertexafterphase,andsendsthisinformation(asinglebit)toeach Round3:Theinformationreceivedintheprevioustworoundssucesforptodecide ofitsneighbors.thiswouldbethecaseifpisitselfaheavyvertexandhas threeneighborsthatareheavyvertices. isabluevertexwithdegree3afterphaseeitherwithnoneighborsthatwill ifitwillbeapriorityvertexinphase2aswellasifanyofitsneighborswill alsobedegree3verticesafterphase,orwithgreenneighborsofdegree3after bepriorityverticesinphase2.forinstance,pwillbeapriorityvertexifit phase(seesection4). 5

16 Next,pdeterminesifitwillbeacornervertexinphase4andsendsthisinformation(asinglebit)toeachofitsneighbors.pisacornervertexifallthe pisaheavyvertex. phasexactlytwoneighborsofthesamecolorclassthatareheavyvertices pwillnotbeapriorityvertexinphase2. followingconditionsaremet: Round4:TheinformationderivedfromRound3enablesptodetermineifitwillbe apriorityvertexinphase3,asdescribedinsection4.ifpwouldbeapriority butnotpriorityverticesinphase2. neighbor,sayqthatwouldneedtoberecoloredinthatphase.inthiscase,p vertexinphase3,andwouldfallintocase(ii)ofphase3,thenithasalight Round5:AlightvertexthatgotamessagetorecoloritselfinRound4informs sendsamessagetoqwiththemaximumweightofitsremainingtwoneighbors allitsneighborsofhowmanycolorsitwillusefromtheendofitsbasecolor ofthesamecolorasq,sothatpcancoloritselfappropriately. Fromtheweightsofitsneighboringprocessorsanditslimitedglobalknowledge, appropriatelyinphase4. spectrum.thisenablesanisolatedvertexinitsneighborhoodtocoloritself cessorcaneasilycomputethecolorsitwilluseineachofthevephases.without lossofgenerality,consideraprocessorthatcorrespondstoabluevertexv2g.the andtheinformationcollectedinthecommunicationroundsdescribedabove,apro- Phase:Ifw(v)<=M,theprocessorassigns(toitself)theappropriatebluehues. processoremulatesthevephasesofthesequentialalgorithmasfollows: Phase2:IftheprocessorisapriorityvertexinPhase2,itsimulatesphase2and continues. Otherwise,itassignstoitselfallthebluehues,reducesitsweightbyMand wouldborrowfromoneofitsneighboringcolorclassarethelastcolorsfrom stops,orelsecontinuestophase3.recallthatthecolorsthatapriorityvertex Phase3:IftheprocessorisapriorityvertexinPhase3,itcandeterminethecolors atapriorityvertex. thatclass;thusnoconsultationisrequiredwithneighborstocomputethecolors itneedstoborrowfromtheappropriateneighboringcolorclass. 6

17 distributedalgorithmaswell.todoso,itusestheinformationitreceivedin inphase3ofthesequentialalgorithm,thenitcanemulatethisbehaviorinthe If,however,theprocessorisalightvertexthatwouldhaveundergonerecoloring Phase4:Ifaprocessorisanisolatedvertexinthisphase,itassignsitselfanypurple itsneighborinround4,andrecolorsitselfasdescribedinsection4sothatno huesthatitneeds.additionalcolorsthatitmayneedareborrowedfromone conictappearsamongthebluecolors. recoloreditselfinthelastphasesentinformationinround5aboutthenumber ofcolorsitwouldusefromtheendofitsbasecolorspectrum,theprocessorcan oftheneighboringcolorclasses.sinceanyofitslightneighborsthatmayhave Phase5:Anyvertexwithunassignedcolorsatthisstageliesalongsomestraight-line determineallcolorsusedbyitslightneighborsandcanborrowcolorswithout anypossibilityofconict. caneasilycomputetheidentityoftheparticularaxisfromtheinformation neighborsthatliealongexactlyoneofthethreedirectionalaxes.theprocessor pathinthegrid.inparticular,itcandetectitsoneortwoincompletelyassigned thesetofpurplehues,asinthecoloringforbipartitegraphs. alongthisaxis,itassignsitselfthenecessaryhuesfromthebeginningorendof gatheredafterround3.dependingonwhetheritisanevenoroddvertex completelydistributedmanner,afteraninitialconstanttimecommunicationprotocol foranyhexagongraphg=(v;e;w),canbeecientlycomputedinconstanttimeina Theorem5.Anapproximatemulticoloringthatusesnomorethan4dD[3] G=3ecolors whereeachprocessorexchangesvemessageswitheachofitsneighbors. 6Inthispaper,wehavecasttheproblemoffrequencyassignmentincellularnet- worksasamulticoloringproblemforhexagongraphs.forsomeparticularinduced Discussion algorithmsformulticoloringthemusinganoptimalnumberofcolors.insection4, subgraphsofhexagongraphs,i.e.cyclesandouterplanargraphs,weshowecient wedescribeamulticoloringalgorithmthatusesatmost4dd[3] themaximumweightonany3-cliqueing,isatriviallowerboundontheminimum numberofcolorsrequired.weshowedalsoahexagongraphthatrequiresd9d[3] G=3ecolorswhereD[3] G=8e colors.determininganexactboundon(g),theweightedchromaticnumberofan arbitraryhexagongraphisnp-hard;ourresultsdoestablishthatforallhexagon 7

18 graphsg,(g)4dd[3] forhexagongraphswhichalwaysusesatmostd9d[3] G=3e.Whetherornotthereisanapproximationalgorithm distributedmanner.incontrast,thealgorithmofmcdiarmidandreed[7],which openproblem.ausefulfeatureofouralgorithmisthatitcanbeimplementedina G=8ecolorsremainsanintriguing madedistributedinanyobviousway. hasthesameperformanceratioasours,seemsinherentlycentralizedandcannotbe quirednotmerelytobedierent,butalsotobefarenoughapart[3].anotherrecently wherethefrequenciesassignedataparticularvertexoratadjacentverticesarere- Aninterestingavenueforfutureresearchisthegeneralizedversionoftheproblem, proposedmodel[]considersarbitraryinterferencegraphswithpre-denedcostson totalcost,orequivalently,thenetinterferenceofanassignment.finally,thedynamic frequenciesareassignedtoneighboringnodes.theobjectivehereistominimizethe theedges;thesecostsreecttheinterferencepenaltieswhenthesameoradjacent inthissettingandwhatboundscanbeprovedonitsperformance. versionoftheprobleminvolveschangingweightsatvertices.itwouldbeinteresting toseeifthedistributedalgorithmwedescribeinsection5canbeadaptedtowork Acknowledgments WethankJeannetteJanssenforintroducingustotheproblemofchannelassignment, formofcorollary3.3. totheanonymousrefereesfortheirusefulcomments,andforsuggestingthecurrent andforcommentsthatgreatlyimprovedthepresentationofsection3.wearegrateful References []R.Borndorfer,A.Eisenblatter,M.Grotschel,andA.Martin.Frequencyassignmentincellularphonenetworks.Technicalreport,Knorad-ZuseZentrumfur InformationstechnikBerlin,997. [2]D.DimitrijevicandJ.Vucetic.Designandperformanceanalysisofalgorithmsfor [3]W.K.Hale.Frequencyassignment:Theoryandapplications.Proceedingsofthe ogy,42(4):526{534,993. channelallocationincellularnetworks.ieeetransactionsonvehiculartechnol- IEEE,68(2):497{54,980. 8

19 [5]T.KahwaandN.Georganas.Ahybridchannelassignmentschemeinlargescalecellular-structuredmobilecommunicationsystems.IEEETransactionson forcellulartelephonesystems.unpublishedmanuscript,april995. [4]J.Janssen,K.Kilakos,andO.Marcotte.Fixedpreferencefrequencyallocation [6]S.KimandS.L.Kim.Atwo-phasealgorithmforfrequencyassignmentincellular mobilesystems.ieeetransactionsonvehiculartechnology,994. Communications,4:432{438,978. [8]P.Raymond.Performanceanalysisofcellularnetworks.IEEETransactionson [7]C.McDiarmidandB.Reed.Channelassignmentandweightedcoloring.submitted Communications,39(2):787{793,99. forpublication,997. [9]W.WangandC.Rushforth.Anadaptivelocal-searchalgorithmforthechannelassignmentproblem.TechnicalReport,August995. 9