Euclidean Spanners Short Thin andLanky Sunil Ary a Gautam Das y David M Moun tz Je rey SSalo ex Mic hiel Euclidean spanners areimp ortan datastructures algorithm design b ecausethey pro vide ID: 113498
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Euclidean Spanners: Short, Thin, andLanky Sunil Ary a Gautam Das y David M. Moun tz Je rey S.Salo ex Mic hiel SmidAbstract Euclidean spanners areimp ortan datastructures algorithm design, b ecausethey pro- vide ameansof approximating thecompleteEu-clideangraphwith onlyO ( n ) shortestpathlengthbeteeneachpairofpointsisbetpoin In importanspannerpossessanberofadditionalproperties: boundedsideredonepropertyortheother. eshowthatitispossibletobuildspannersintimeand)spacethataceoptimalornearoptimaltradeo s beteenallcombinationsofthese Max-Planc k-Institut f ur Informatik, D-66123 Saarbr uc-ken, German y . Email: f arya,michi el g @mp i-s b.m pg .de .Supp orted ythe ESPRIT Basic hActions Program, undercontractNo. 7141 (project ALCOM II). y MathSciencesDept.,TheUniv yofMemphis,Mem- Supported GrantCCR-9306822. E-mail:dasg@next1 .msci.memst.edu .z tofComputerScienceandInstituteforAd-supported fSaarbr umd.edu.x QuesT ech,Inc.,7600ALeesburgPike,FallsChurch,V fSaarbr vl. properties.becauseofanewstructure,calledthe el ltr hprovidesamethodofdecomposingaspannertoaconstantnberoftrees,sothateachofthe )spannerpathsismappedtoapathinoneofthesetrees. In tro duction Let G = (V ; E be G (u; v )bebet � 1 beanyconstan subgraphG0 isa anner if,forallpairsof u;v u;v isasetofpointsinI k isthecompletegraph,thelengthofedgeu;v)isbetpoinEuclideangraphand ant-spanner.F thepurposesofderivingasymptoticbounds,weas-sumethatthedimension andthespannerfactorareconstantsindependentofItiskno wnhowtoconstructaEuclidean-spannerha)edges)time[5, 13, 14 ]. Spanners importansincetheyprovideamechanismforapprothecompleteEuclideangraphinamhmoreeco- course,aspannershouldeasmallnberofedges(ideally)),butformanapplications,itisquite importantthatthespannerbeproperties. thefollo w weigh t: inthespannershouldbeassmallaspossible.bestbehopedttimesoftheningtree, (MST )). Bounded degree: Thenberofedgesincidentoanyvertexshouldbebounded. spanner diameter:Thespannerdiame-ter(orsimplydiameter)isde ned asthesmallestintegerD bettainingatmostorspannersofboundeddegreethebestthatbehopedsomeapplicationsevensmallerdiametersmabedesirable,butthiscomesattheexpenseofincreasingdegree.bebettransportationberlocations.meansthattheamountofconcreteneededtobuildtheroadsisboundedmeansthatnolocationorkhasmorethanaboundedbertermeansthatitispossibletodescribeyspan-nerpathconciselyorkonspannershasfocusedpropert er,atransportationnetorkwhichacsmalldiameterbymassivelyincreasingtotalwisoflittlepracticalvThissuggeststheimpor-tquestionofwhetherthereexistspannersthat esomeoralloftheseproper-Inthispaperwepresentastrongpositiveanswtothisquestion.epresentanberofnewcon-per-spectivperfor-propertiesabotradeo sbetproperties.orex-longedges,whichinturnincreasestotalwt,orberorthisreason,weconsiderpossiblebinationsoftheseproperties.paperberbutonedeservesparticularmenAnimportandecomposition,troducedbyCallahan andKosara ju Thisstruc-turerepresentsthe)pairsofpointsusingonly)pairsofgeometrically ell-separated" ofsubsetsofpoints(de nitionswillbegivenlater).Inthispaper,wepresentanoelmethodoffurtherdecomposingdecompositiontoaconstantnberofhierarc setsofwell-separatedpairs.(Theconstantdependsonthedimensionandtheseparationfactor.) decomposition,bebeingtnberoftrees,whichwecallelltr er,eachofthe)spannerpathsarisesastheuniquepathbeteentoleaesinoneofthesethatthe)spannerbeberisaratherremarkablefactinitself,andsuggestsagreatdealaboutspecialstructureofthesegraphs. importancedecompositionstoavyofgeometricproblems,esuspectthatthisdecompositionmaybeofusetobbellshasappearedbefore[7 ],buttheiruseindecompos-ingspannerpathsisnewtothispaper.paper.describedbeloruninoptimal)timeand)spacefory xeddimension boundedbestestwhichrunsinO(nlogkn)time. spannerconstructionthathasoptimalwbestconstructionforspannersoflowwt,whicasduetoDas andNarasimhan [8 ],andwhicrunsin)time. edgesandpossibleediameter )+ 2withthesamenberofedges,where)istheinerseofAc spectrumbetofdiameter2with)edges,diameter loglognershaeanoptimalnberofedgesfortheendiameter.boundedandhenceprovidesanoptimalsolutiontothisproblemaswTherearenopreviousresultsonthisproblem.tool,ol,]hasw)logasdiameterThiscombinationisop-boundsouslyknoboundedspectbothboundboundswnforthisproblem. boundedgree,w)log),anddiameterNosimultaneousboundswereprevi-ouslyknoInsummary,allofourresultsareoptimalintermsofprovidingthebesttradeo sbeteentheseprop-boundedpossiblysuboptimal)factorinwTherestofthispaperisorganizedasfollo2,webrie\ry thewdecomposition.3,wede nethedumbelltree,andshowthatthereexistsaspannerthatcanbedecomposedintoaconstantnberofsucInSection4,wegiveasimpleoptimalalgo-rithmforconstructingaofbounded5,weshowthatthespannerresultsfromthewell-separatedpairdecompositioncanbeprunedinsuchawythatwegetaspanner nersoflowdiameter.OurresultsofSection3implythatit toaddedgestoaconstantnberofboundedtogetaspanneroflowdiameter.ThisisdonebyusingatecduetoAlonandSchieber[1].InSection7,weshobbelltopologytrees[10 ]inordertogetaspannerofboundedde-greeand)diameter.InSection8,weshopairdecompositionhaew)logloggivesat-spannerofweightO(w(MST)log econsiderpropertiesdegree,tanddiameter Split trees andw ell-separated pairsVirtually allofourspannerconstructionswill airdeofasetofpoints[4,13,14].thissection,wereviewthesedatastructures.splittrisatreethatstemsfromahierarcdecompositionpoin-dimensionalrectanglesofboundedaspectra-Thereareanberofvtsonasplittree.eoutlinethefairsplittree,duetoCallahanandKosaraju[4]. asmallest-possible 0 aboutpoinrootChoosethereforesplittotosmaller Thentheleftsubtreeofisthesplittreefor TheprocessisrepeateduntilasinglepoinInordertosimplifysomeofourarguments,itisform,whic dbox thistree,rectanglesareypercubes,ypercubesofhalfthesidelength.Actualconstructionswillbeoutusingthefairtheideal-izedboxsplittreeprovidesacleanwyofconcep-tualizingthefairsplittreeforpurposesofanalysis.decompo- 0beaconstanopointsets andB wellsepiftheycanbeenclosedin-spheresofradius ,whosedistanceofclosestap- hisatleastwell-sepdppositionisasetofpairsofnonemptysubsetsof ,ff ;B ;B;: ::; ;B,suchthat(1) andBiaredisjoint,(2)foreachpaira; b 2 thereisaunique;BandKosarajuuseasplittocomputeasetof)well-separatedpairsin)time. Kosarajushowthatacanbeoreachpair;Binthewpairdecomposition,choosearbitrarypoints,called ]andSalo The dum bbellOneofthemajordicultiesinestablishingthere-ofthispaperstructureindecompositionsthatarederivfromthem.decompositionsnotpossessyobvioushierarchicalstructure.Oneofthema-jorinnoationsofthispaperistheobservationthatell-separatedpairdecompositions,andhencebedecomposedtoaconstantnberofhierarcThisgreatlysimpli estheanalysisandconstructionofspanners,byreducingproblemsongeneralgraphstomhsimplerproblemsontrees.decompositionberofotherproblemswheresparsegraphsareused.doespermitdecomposition,ell-separatedpair;Bcanbeviewedasage-object,consistingoftorectanglesrespectivshape,ellandtherectangles(orinfact,smallperturba-tionsoftheserectangles)arecalledthe ofthebbell. bbellthedistancebeteenthecentersofitsheads.ofaheadisde nedtobehalfitsdiameter.diameter.introducedbbell.possiblebbellsfromthedecompositiontnberofgroups,sucthatwithinbbellbbellisnestedtheheadofthebbell.wing(proofswillappearinthefullpaper): 1 Consider ells well-sep In ellsinto suchthatwithine ells theydi er byafactoratleells aterthan,andell ells) suchthatadumb-elloflengthhasaheadofsizeatmostells dwithinaadoftheother.Thenestingofdumbbellsprovidesuswithatreeellportanaboutbbelldecomposi-tionisthatspannerscanbederivedfromthewdecompositioncanbeasconsistingoftheunionofaconstantnberoftrees.more,wwthatmappedtirelytoonetree.Ourmainresultissummarizedinthefollowingtheorem: Given ,andgivent�,aforestconsistingofe,havingthefollowingpr oreachtreintheest,thereisa ebentheleavesofthistreandthepointsof Eachinternalnodehasaunique pointhepointsinanyofitsdescendentleu;v oftheforest,sothatthepathformedbywalking enthesedes,isaannerponstantfactorsessingtimeeark= 1)).With additionofanaugment-ingdatastructureofsize,wecancomputeaannerpathbenanytwopointsin +logtime,wheristheerofedgesonthep Spannersof bounded sult,whicbeusedtoconstructintimea-spannerofbounded L dsuchthateachpointhasoutdeeatmost (t0 t))k ,forsomecInordertoproethisresult,weneedthenotionofsingle-sinkbeasetofpointsin,letbeapointof,andlett�Adirectedgraphhavingthepointsofasitsverticesiscalledpoin thereisa-spannerpathfrom be =4and1=(cos sin)t.LetC beconessuchthat(i)eachconehasapexconescoersIoreachpoin 2C betheconepoinbe bepoin fxgthatarecontainedintheconeCx poinmorethantainsmorethan2points,thenwpartitionit(arbitrarily)intotosubsets ,eachofsizeatmostisobtainedas tainsmorethan2points,foreachsubset poinclosestto,andweaddanedgefromforthissubset.Therecursionstopsifasubsethassizeone.Usingexactlythesameanalysisasin[12],itfol-wsthatthegraphisasingle-sinkthateachpointhasoutdeeatmost1andindedby,forsomecwwearereadytogivethetransformationthateTheorembesetofpoininIandlet�t�bea-spannerforandassumethattheedgesofcanbedirectedhthateachpointhasoutdegreeatmostdenotethisdirectedversionof oreachpoin,wedothefollosiderallpointsofthathaeanedgein bepoinspannerfortheset inggrapheclaimthatisa-spannerforpoinhasadegreeboundedyaoproethis,let beanytopointsofThereisa-spannerpath;p;p;:::;pbetConsideranyedge;p w.l.o.g.thatinisdirectedfromThedirectedgraphtainsa-single-sink()-spannerwitha(pathbetconcatenationofallthesepathshaslengthatmost m 1i=0 (t0=t)j jbothpoinbe poinboundgraphcon t 0=t 1)d 1)=O((ct wletbeanypointsuchthattainsanedgefrom(Thereareatmosthpoinoccursina-single-sinkspanner,andithasindegreeboundedb)inthisHence,inthedirectedgraph,poinhasindegreeboundedb((1+)(hpointhasadegreeboundedyaconstanThisproesTheorem3.Itturnsoutthatsevwnspannershaethepropertythattheiredgescanbehthateachpoinhasboundedorexample,forany0=4,the-graph(see[12,2])isa-spannerfor poinboundedcanbeconstructedin)time.positionspropert,theconstructionistoen)setsof\boxpairs."oreachwell-separatedpairofboA;B,chooseanarbitrarypoina;bisdirectedfromiftheparentboxofisnotboinggraphhasboundedoutdegree.ordetails,seeandKosaraju[5tiregraphcanbeconstructedin)time.bepoinbounded ofLemma3.eapplythegiventransformationandobtainthedesiredThispro easetofointsin Inthissection,wegivean)timeconstruc-tionofa-spannerthathaswordertoboundtofthisgraph,weusetheoremfromDas,NarasimhanandSaloSalo].Letc�0beaconstant,letbeasetofedges,andlet beanedgeofw Ifitispossibletoplaceacylinderofradiusandheigh hthattheaxisasubedgenf)= ,thenissaidtobeThesethastheisolationprifalledgesareisolated. ([9]) isolationpr)=mumspanningtrewithrcttotheendpointsofItiseasytoseethatinthede nitionoftheiso-lationpropert,onecanreplacethecylinderwithabox,etc.,withouta ectingtheaboethe-Thelotspannerisconstructedinthefol-wingwbeacone,andlet)bethesetofedgesintheboxwell-separatedpairconstruc-endpoinapexendpoinsurethattheedgedoesnotintersecttheinterioroftheconexhofthepointswithintherespectivboTheseendpointsarechosenfromamongthepointswithmaximumorminimumcoordinatesinaparticulardimension.oreachpoin,markedgeoneendpoinboxancestorofThespannerconsistsoftheunionofthemarkededges.eclaimthatisaspannerandthatitsedgessatisfythepropertfactthatbeardin-proof.propert,weusesomeofthepruningtechniquesofof].Wemayas-sumethatedgesebeentoaberofgroupssothateachedgehaseitherap-ximatelythesamelengthordi ersinlengthbasucientlylargeamount(butboundedbyacon-tfactor).a;bpropertcorrespondstosomewA;Bbobobo about thecenterpointof e rstclaimthatdoesnottainanypoinwthis,becausesomepoin,sa,forwhich(a;b)isashortestpoin,thisouldimplybebebebeinanearbyconeaswell;thisdetailcanbe xedusingsomepruningtecberofedgesintersectaslighersionofSupposeasmallamounNotetherearenopointsinsideof,sonotbeshrunkenbymorethanasmallpercenSuppose;btheidealizedboissucientlyfar,anypointinouldbeinawcorrespondingbesamedirection.theproofisonlyskNotethatthisiswhereeneedtochoosetheboxrepresenesinacare-fulw,consideranedgethathasapprocorrespondtoidealizedboxeswithindistancekingargumentsthatusethefactthat)isrelatedtothewidthofthatthereberbothisarea.Therefore,thereareonlyatmostacon-tnberofedgesthatcaninusingthedecompositiontechniqueofDasetal.[7],onecanpartitionthegroupofedgesintoaconstanberpossessingpropertThefollowingtheoremispro anbandithasOurconstructionactuallyhasboundeddegreeasThisisbecauseanysetofedgespossessingtheisolationpropertyhasboundeddegree(astraighardproof).ethereforehaethefollo 6.1 Inanydimensionandfort� pointsonaline,andtheoutputisagraphwithsmalldiameter.ausefulconstructionbeenhieber[1].thattherespannerwithdiameter)+2.aspectshieberconstruction(theyaretailoredsomewhattoenhanceanalogytoourproblem).Supposetaspannerthatconfewedgesaspossible.AlonandSchieberdivideuppoin ;thisaccountsforspannerpathswithinpathsbeteenthepieces,selectthepointsineacgroupwithsmallestandlargestvhofthepointsinaparticulargroupareconnecteddirectlyesthemselvesareconnectedwithaspannerofbern;dhiebern;d)+`;d2)+n=`;dBychoosingthevaluesof,itispos-tosho2)=3)=loglog4)=),andsoon.isalsopossibletoshowthatthediameteris)+2ifoneallowsonly)edges.case,weusethefactthatthereexistsaspannerwhichcanberepresentedastheunionofberbounded bemodi edboundedsomeadditionalgeometricproperties.Speci callythismodi eddumbbellisatreewhosevpoinimportanpropertisthefolloifapairofpoininawell-separatedpairthatactuallyappearsasabbellbetisa-spannerpath.Theactualconstructionofisdoneinthefol-bbell hildrendumbbells;:::;andseveraliso-poin;p;:::p poinbebbellsof.Thispossiblylargesetofedgeswillbereplacedbyatreewhosedegreeisboundedyaconstant,describedbelobohoosepoinbespectconnectstheserepresenepoinThistreeisaSteineritconsistsoforiginalpoints(theepoints),Steinerpoints(degree-3vypespoinfromreplacinghofthepathsinwithasingleedge.proofthatbellsbbellboberAdetailedproofisomitted.hieber ;x;:::;xbegleinequalit,anypath;x ;x ;:::;x properties.Thedetailsofhowtheseshortcutsareconstructedisomitted. oranyt�,andanydimensioneisaannercdgesandcstructibleintimewithdiameter)+2beHerearesomeofourresults. oranyt�,andanydimensiondgesandconstructibleintimewithloglogdgesandconstructibleintimewith dgesandconstructibleintimewith boundedanbecsectionhaehighOntheotherhand,itisboundbounded-produceandboundedforasetofpointsonahorizontalline.lossofgeneralit,assumethatisapoerof2andthattheyarenbered0through1fromlefttorighanedge(i;i+1),0inTheresult-thesetofevberedpoin;:::+2),Repeatprocess.ofedgespointshaedegreelogaswInordertoreducethedegree,notethatbeenodd-beredpointswerechosenatthesecond\levorifeither2or2+1was\promoted"tothesecondAsimilarstatementcanbemadeattheel(2 through2+1)1canbepromoted).oneiscarefulaboutalternating\promotions,"the boundedgreeskip-list,hasboundeddegreeandlogarithmicproofsions,weneedtoapplythesamestrategytotrees,speci callymodi edbbellideaseemstobeFkson'stopologytrees[10methodpropertiesSupposerootedboundednodehasauniquelabelandthatanyinternalnodebelabeledlabelnode.hoosenodespropagateleaflabels.Anodechoosesoneofthepropagatedlabelspropagatesthelabelataaninternalnode.performalanodesnodeimportanumdegreeboundedateverylevlabelstributedsothatnolabelisusedmorethanacon-tnberoftimes.labelingpoin(correspondinglabels)boundedthelevelingprocessensuresthatthepathhaslink-bethe nalpaper. eusethefollowingspannerconstruction,duetotoStartwithafair-splittree,anddesignatesomenodesasheavyandsomeaslighAnodeisheavyifitcontainsmorepoininitssubtreethanitssibling,anditislightother-bothsubtreesconberpoinisligheusethisdesignationtodetermineboesforthewell-separatedpairs;speci -,aparentboxinheritstherepreseneofitsaetet]showthatiftherepre-esarechoseninthisw,thentheresultingspannerhasdiameterwsho)loglog].ThisimprovestheresultsofLenhofetal.[11],whoproethatthesumofthediametersoftheboxesinaboxsplittreeis)logandthatthelengthofthewell-separatedpairedgesOurtechniquescanbeusedtoshowthat)logfocusfocusbbellpropertertAsetofedgesEhasthegappropertyifforeverypaire1ande2,thedis-tancebetendpoinpropert)=)logbebbell hasthegappropertyand)=( esthat)=)logpropertnfethegappropertthefollowingdirectedforest:iseliminatedbecauserootwillremainin,sowewttoshowthat)=properti�c 1isaconstanromdumbbellproperties,ber ybewithinadistanceofa xedpoinsothenberofchildrenofingroupisatmostendpoinbehosenindependentlyof,sothatcItfol-wsthatthetotalwtofthechildrenofisat w (e ) 1 ,where =2c theoreminthissection. 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