abcde1221012012Figure1Stabbersfromatoelinewedgedoublewedge2leveltreezigzagAstandardgeometrictoolwhichwillbeusedthroughoutthis ID: 178170
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StabbersoflinesegmentsintheplaneM.ClaverolyD.GarijozC.I.GrimaxA.Marquez{C.SearakAbstractTheproblemofcomputingarepresentationofthestabbinglinesofasetSofnlinesegmentsintheplanewassolvedbyEdelsbrunneretal.withan(nlogn)timeandO(n)spacealgorithm.Wepresentastudyofdi erenttypesofstabberssuchaswedges,double-wedges,2-leveltrees,andzigzags;providingecientalgorithmswhosetimeandspacecomplexitiesdependonthenumberofcombinatoriallydi erentextremelineshSorcriticallinescS,andthenumberkSofdi erentslopesthatappearinS.1IntroductionLetS=fs1;:::;sngbeasetoflinesegments(orsegments)intheplane.Forconvenience,werequirethatifpandqareendpointsofasegment,thenp6=q,andconsequently,lines,rays,andpointsarenotconsideredtobesegments.Inordertoavoidtediouscaseanalysis,weassumethattheendpointsofthesegmentsareingeneralposition.Nevertheless,theresultspresentedinthepapercanbeextendedtoarbitrarilysegmentsets.Alineisatransversalof(orstabs)SifitintersectseachsegmentofS.Edelsbrunneretal.[7]presentedan(nlogn)timeandO(n)spacealgorithmforsolvingtheproblemofconstructingarepresentationofalltraversallinesorstabbinglinesofS.SeeEdelsbrunner[6]forananalysisofthisproblemfrombothacombinatorialandcomputationalpointofview.ThelowerboundfromEdelsbrunneretal.[7]doesnotapplytothedecisionproblem:determiningifthereexistsalinestabberforS.Avisetal.[2]presentedan\n(nlogn)timelowerboundinthe xedorderalgebraicdecisiontreemodeltodeterminetheexistenceofalinestabberforS.Forasetofnverticalsegments,astabbinglinecanbecomputedinO(n)time.Astabbingline`forSclassifytheendpointsofthesegmentsintwoclasses:endpointsabove`,sayredpoints;andendpointsbelow`,saybluepoints.Theendpointon`isclassi edaccordingtotheotherendpoint.Thus,wecanseetheproblemofstabbingSasaproblemofclassifyingtheendpointsofthesegmentsintodisjointmonochromaticredandblueregionsde nedbythestabber,i.e.,asaseparabilityproblem.SincewewantthatthestabbersforSclassifytheendpointsofthesegmentsinthatway,wecanconsidertheconditionthatthereisnosegmentstabbedbymorethanoneelementofthestabber.Wecallthisconditiontheseparabilitycondition.SowelookforstabbersforSsuchthatwecanassignredandbluecolorstotheendpointsofthesegmentsandsplittheplaneintodisjointmonochromaticregions,i.e.,obtainingared/blueclassi cationoftheendpointsofthesegments.Hurtadoetal.[11]classi edredandbluepointsintheplanewithseparatorswhicharesimilartoourstabbers.Followingthislineofresearch,wedealwiththeproblemof ndingdi erentkindsof\simple"stabberswhenthereexistsnostabbinglineforS.Concretely,weshallconsiderthestructuresshowninFigure1.ThegoalistodesignecientalgorithmsforcomputingthesestabbersforSwithorwithouttheseparabilitycondition.SupportedbyprojectsMECMTM2006-01267andDURSI2005SGR00692.yDept.deMatematicaAplicadaIV,UniversitatPolitecnicadeCatalunya,Spain,merce@ma4.upc.eduzDepto.deMatematicaAplicadaI,UniversidaddeSevilla,Sevilla,Spain,dgarijo@us.esxDepto.deMatematicaAplicadaI,UniversidaddeSevilla,Sevilla,Spain,grima@us.es{Depto.deMatematicaAplicadaI,UniversidaddeSevilla,Sevilla,Spain,almar@us.eskDept.deMatematicaAplicadaII,UniversitatPolitecnicadeCatalunya,Spain,carlos.seara@upc.edu (a)(b)(c)(d)(e)``1`2`2`1`0`1`2.........................................`0`1`2Figure1:Stabbersfrom(a)to(e):line,wedge,double-wedge,2-leveltree,zigzag.Astandardgeometrictoolwhichwillbeusedthroughoutthisworkisduality[7]:thegeometrictransformdenotedbyDwhichmapsapointintoanon-verticallineandviceversa.Givenapointp:=(a;b)andaline`:=y=cx+d,wehaveD(p):=y=ax+bandD(`):=( c;d).Asegmentsi2Sisdeterminedbyitsendpoints.TheendpointsaretransformedbyDintotwolines.Ifsiisnotvertical,D(si)isadouble-wedgewhichdoesnotcontainaverticallineinitsinterior.Thus,thedouble-wedgeisformedbytwoupperraysandtwolowerrays.Ifsiisaverticalsegment,D(si)isastrip.ThesetofendpointsofthesegmentsinSistransformedbyDintoanarrangementof2nlinesdenotedbyA(S).ThetransformDsatis esthefollowingproperties:(i)thetransformDmaintainstherelativeposition(above/below)ofpointsandlines;(ii)aline`intersectsasegmentsiifandonlyifthepointD(`)liesinthedouble-wedgeD(si);(iii)thestabbinglinesofSstandinone-to-onecorrespondencewiththeintersectionpointsoftheirdouble-wedges,i.e.,Tsi2SD(si).Relatedworks.Claverol[5]asapartofherPhDthesisinitiatedthestudyheredeveloped.Inthispaperweimprovethecomplexitiessheobtainedandsomeotherstabbingproblemsarealsoconsidered.AtallahandBajaj[1]presentedanO(n (n)logn)algorithmforlinestabbingnsimpleobjectsintheplane,where (n)istheinverseoftheAckerman'sfunction.AsimpleobjectisanobjectwhichhasanO(1)storedescriptionandforwhichcommontangentsandintersectionscanbecomputedinO(1)time.Edelsbrunner,GuibasandSharir[8]showedhowtoconstructarepresentationofthelinestabbersofconvexpolygonswithatotalofnverticesinO(n (n)logn)time.LaterimprovedtoO(nlogn)usinganO(nlogn)timealgorithmfromHershberger[10]for ndingthelowerenvelopeofasegmentsetintheplane.O'Rourke[14]presentedanalgorithmfor nding(ifitexists)astabbinglineofverticallinesegments.GoodrichandSnoeyink[9]presentedanaturalvariantconsideringanothertypeofstabbersdi erentfromthelinesbysolvingtheproblemofcomputingatransversalconvexpolygonforasetofparallelsegmentsinO(nlogn)time.Bhattacharyaetal.[3]workedontheproblemofcomputingtheshortesttransversalsegmentforasetoflinesintheplaneandalsoforasetofconvexpolygons.Lyonsetal.[12]studiedtheproblemofcomputingtheminimumperimeterconvexpolygonwhichstabsasetofisotheticlinesegments.Rappaport[15]consideredtheproblemofcomputingasimplepolygonwithminimumperimeterwhichstabsorcontainsasetoflinesegments.Bhattacharyaetal.[4]andMukhopadhyayetal.[13]consideredtheproblemofcomputingtheminimumareaconvexpolygonwhichstabsasetofparallellinesegments. 2IdeasandtoolsSomeideasandtoolsaresharedbymostofourresults.Inthissectionwepresenttheminauni edcontext.ThelinescontainingthesegmentsofScanhavedi erentslopes.Denotebymitheslopeof(thelinecontaining)thesegmentsi2S.Thecomplexityofmanyofthealgorithmsthatwepresentheredependsonthenumberofdi erentslopesofthelinescontainingthesegmentsofS,writtenaskS.Itisduetothefollowingfact:theendpointsofthesegmentsfallintotwoclassesdeterminedbyastabbingline`,endpointsabove`andendpointsbelow`,sayredandbluerespectively.Theendpointofasegmenton`isclassi edaccordingtoitsotherendpoint.Thus,foragivenslopeof`ourproblemofstabbingthesetScanbeviewedasared-blueseparabilityproblemofclassifyingtheendpointsofthesegmentsintodisjointmonochromaticregionsoftheplanedeterminedbythestabber.Infact,itisnotdiculttoshowthatthereexistasmanydi erentclassi cationsoftheendpointsofSaskS.Arelevantpropertyaboutourstabbersisthatnotallthelinescanbecandidatetode neoneofthesestructures.Forinstance,consideraray`whichispartofawedgethatstabsS,anditsextensiondenotedby`0.Wehavethatallthesegmentsthatarenotintersectedby`mustlieonthesamehalf-planede nedby`0.Thus,wesaythataline`0isanextremelineforSif`0stabsasubsetofsegmentsS1S,S16=;,andtheremainingsegmentsS2=SnS1areinonlyoneoftheopenhalf-planesde nedby`0.Otherwise,theline`0issaidtobeanon-extremelineforS.Thus,aswehavementionedbefore,ourinterestinextremelinescomesfromthefollowingfact:thelinecontaininganyrayofastabbingwedgeforSisextremelineforS.Inthispaper,extremelinesarestudiedfromtwodi erentpointsofview:computationalandcombinatorialview.ThereasonisthatouralgorithmsdependonthecomputationofthesetofextremelinesforS.Thus,wesaythattwonon-verticallines,`1and`2,arecombinatoriallydi erentwithrespecttoSifeither:(1)thesubsetofsegmentsS1Sstabbedby`1isdi erentfromthesubsetofsegmentsS2Sstabbedby`2;or(2)ifS1=S2theneither:(i)thesubsetofendpointsofsegmentsofSabove`1isdi erentfromthesubsetofendpointsofsegmentsofSabove`2,or(ii)thesubsetofendpointsofsegmentsofSbelow`1isdi erentfromthesubsetofendpointsofsegmentsofSbelow`2.IfhSisthenumberofcombinatoriallydi erentextremelinesofS,wecomputearepresentationofthecombinatoriallydi erentextremelinesforSinO(hS+nlogn)timeandO(hS+n)space.Observethatthenumberofcombinatoriallydi erentextremelinesforSisatmostO(n2),butdependingonthepropertiesofthestabbingproblem,wecanconsideronlyasubsetofthemnamedthecriticalextremelineswhichsizecSisatmostlinear.3StabbingwedgesOur rstaimistostudytheproblemofdecidingwhetherthesetScanbestabbedbyawedge,andcomputingthisstructureincaseofexistence.Obviously,itisassumedthatthesetSisnotstabbedbyaline.Wedistinguishtwocases:stabbingwedgesWsatisfyingtheseparabilitycondition(describedinSection1)orthosethatdonotsatisfysuchcondition.Inthe rstcase,weprovideanO(hSkSlogn+nlogn)timeandO(hS+n)spacealgorithm.TherangeforhSisfromO(1)toO(n2),andtherangeforkSisfromO(1)toO(n).Inthesecondcase,wedesignanO(cSkSlogn+nlogn)timeandO(n)spacealgorithm,withrangeforcSinbetweenO(1)andO(n).Wenowintroducesomeusefulnotationforourpurpose.Givenaline`andasegments,wecanclassifytheendpointsofswithrespectto`whenever`andthelinecontainingsarenotparallel.Itsucestodoaparallelsweepwith`untilitcrossess,leavingoneendpointin`+,andtheotheronein` .Theseendpointsaredenotedbye+ande ,respectively.LetWbeastabbingwedgeforS.ThetworaysthatformWaredenotedby`1and`2,andWiswrittenasW=f`1;`2g.Thelinecontaining`ifori=1;2isdenotedby`0.Thehalf-planesde nedby`iarewrittenas`0+and`0i .Supposethat`1stabsasubsetofsegmentsS1(S,S16=;,andthesetS2=SnS1isstabbedby`2. 3.1StabbingwedgessatisfyingtheseparabilityconditionSincebyde nitionweconsiderWasaseparatorstructure,asegmentcannotbestabbedbybothrays.LetS+1(S 1)bethesetofendpointsofthesegmentsofS1classi edase+(e )withrespectto`0.Analogously,S+2(S 2)isthesetofendpointsofthesegmentsofS2classi edase+(e )withrespectto`0.Thus,S 1andS+2arecontainedinsidethewedgeW,andS+1andS 2arelocatedoutsideit.Noticethattheseassignmentsf+; gdependontherelativepositionofthelines`01and`02,i.e.,ontheslopeandtheapertureangleofthestabbingwedge.Weconcentrateinaparticularcasebuttheremainingconstantnumberofcasescanbehandleinasimilarway.Lemma3.1.IfW=f`1;`2gisastabbingwedgeforS,`01and`02areextremelinesforSandatleasthalfofthesegmentsofSarestabbedbyeither`1or`2.Thenextlemmaassumesthefollowingconditions:(i)let`01beanextremelineforSwhereS1(S,S16=;,isthesubsetofsegmentsstabbedby`0.LetS+1andS 1betheclassi cationoftheendpointsofthesegmentsofS1givenby`0,andletS2=SnS1.(ii)Letmbea xedslopeandletS+2andS 2betheclassi cationoftheendpointsofthesegmentsofS2bysweepingalinewithslopem.Lemma3.2.ThereexistsastabbingwedgeW=f`1;`2gforSwith`1containedin`01ifandonlyifS 2islineseparablefromS 1[S+2.ThelocusofapicesofthestabbingwedgesforSrespectingthisclassi cationofendpointsisa(possibleunboundedanddegenerate)convexquadrilateralQde nedbythefollowingfourlines:theinteriorsupportedlinesbetweenCH(S+1)andCH(S 1[S+2)andtheinteriorsupportedlinesbetweenCH(S 1[S+2)andCH(S 2).Lemmas3.1and3.2arethekeytoolstodesignanalgorithmthatprovesthefollowingresult.Theorem3.3.Thesetofcombinatoriallydi erentstabbingwedgesforSwiththeseparabilitycondi-tion,togetherwitharepresentationofthemformedbythelocusofapicesofthestabbingwedgescanbecomputedinO(hSkSlogn+nlogn)timeandO(hS+n)space.3.2StabbingwedgesnotsatisfyingtheseparabilityconditionLetW=f`1;`2gbeastabbingwedgeforSnotverifyingtheseparabilitycondition.Apropertythatonlyholdsforthistypeofwedgesisthefollowing:therealwaysexistsastabbingwedgeWforS(notsatisfyingtheseparabilitycondition)formedbytworaysbothanchoredon xedpointsofS,i.e.,bothraysarecontainedincriticalextremelines.Thispropertyletusprovethefollowingresult.Theorem3.4.Thesetofcombinatoriallydi erentstabbingwedgesforSnotsatisfyingtheseparabilityconditioncanbecomputedinO(cSkSn+nlogn)timeandO(n)space.Nextweshowan\n(nlogn)lowerboundfortheproblemofdecidingifthereexistsastabbingwedgeforasetofarbitrarilysegments.Wereducethedecisionofthestabbingwedgeproblemtotheproblemofdecidingwhetherthereexistsastabbinglineforasegmentset,whichhasan\n(nlogn)timelowerboundinthe xedorderalgebraicdecisiontreemodel[2]).Theorem3.5.Decidingwhetherthereexistsastabbingwedgeforanarbitrarysegmentsetrequires\n(nlogn)timeinthe xedorderalgebraicdecisiontreemodel.4StabbingwedgesforparallelsegmentswithequallengthLetS=fs1;:::;sngbeasetofnparallelsegmentsintheplanewithequallengthwhicharenotstabbedbyaline.WenowconsidertheproblemofcomputingastabbingwedgeWforSsatisfyingtheseparabilitycondition.Uptosymmetrywithrespecttoeitherthex-axisorthey-axis,wedistinguishthreetypesofwedgesaccordingtotherelativepositionoftherays`1and`2ofthepossiblestabbingwedgeW=f`1;`2gforS.Let Wbetheapertureangleorintervaldirectionde nedbytherays`1and`2ofW.Thethreetypesarethefollowing:(a) Wcontainstheverticaldirection;(b) Wcontainsthehorizontaldirection;and(c)bothrays`1and`2ofWhavepositiveslope. ........................................................rbrrrrrrrbbbbbbb(a)(b)(c)Figure2:Thethreetypesofstabbingwedgesforaparallelsegmentset.4.1Type(a)When Wcontainstheverticaldirection,itispossibletodesignthefollowingO(nlogn)timealgorithmforcomputingastabbingwedgeforS.ItisbasedonanO(nlogn)timeandO(n)spacealgorithmfordecidingthewedgeseparabilityofared-bluepointsetintheplane[11].1.InO(n)time,classifytheendpointsofthesegmentsofSasfollows.Foreachsegments,colorredtheendpointofswithbiggery-coordinateandcolorbluetheotherendpoint.LetRandBbethesetsofredandblueendpoints.2.InO(nlogn)time,decidewhetherthereexistsaseparatingwedgeforRandB,andcomputeitincaseofexistence.Inthesametime,wecancomputethelocusofapicesofalltheseparatingwedgesformedbyconvexquadrilaterals.Theorem4.1.GiventhesetS,astabbingwedgeoftype(a)forScanbecomputedinO(nlogn)timeandO(n)space.Theassumptionthatthesegmentshaveequallengthisnotusedinthealgorithm.Iftheslopesoftheraysofthewedgeareknown,thereexistsanO(n)timealgorithmtocomputeastabbingwedgeforSusingthemedianofthex-coordinatesoftheendpointsofthesegments.4.2Type(b)WehavealsoobtainedanO(nlogn)timealgorithmforcomputingastabbingwedgeforSwhen Wcontainsthehorizontaldirection.Foreachsegmentsi2S,consideritsmidpointi.InO(nlogn)time,sortthesemidpointsbydecreasingy-coordinate,andletydenotethisorder.DenotebyS=fs;:::;sgthesetofsegmentsofSsortedbytheyorderoftheirmidpoints.Thekeytoolstodesignouralgorithmaregivenbythefollowinglemma.Lemma4.2.IfthereexistsastabbingwedgeforS,W=f`1;`2g,suchthat Wcontainsthehorizontaldirection,thenthemidpointsofthesegmentsstabbedby`1appearbeforeintheyorderthanthemidpointsofthesegmentsstabbedby`2.Let`0bealinewithpositiveslopethatstabsS1(S,S16=;,andlet`02bealinewithnegativeslopethatstabsS2=SnS1.Considertheclassi cationoftheendpointsofSprovidedby`01and`02.ThereexistsastabbingwedgeW=f`1;`2gforSifandonlyifbothS+1andS 2arelineseparablefromS 1[S+2.Theorem4.3.GiventhesetS,astabbingwedgeoftype(b)forScanbecomputedinO(nlogn)timeandO(n)space.4.3Type(c)Whenbothrays`1and`2ofWhavepositiveslope,themainproblemistoobtainaconsistentclassi cationofthemidpointsofthesegmentsaccordingtothepossiblestabbingwedgeoftype(c).DenotebydthelengthofthesegmentsofS,andrecallthatiisthemidpointofthesegmentsi2S. Assumethatthereexistsastabbingwedgeoftype(c)forS,denotedbyW=f`1;`2g.Supposealsothat Wisknown.Let`0fori=1;2,bethelinecontainingtheray`i.Denoteby`00thelinebelowandparallelto`0,suchthattheverticaldistancebetweenthetwolines`001and`01isexactlyd=2.Similarly,`00isthelineaboveandparallelto`02suchthattheverticaldistancebetween`002and`02isalsod=2.Let`bethebisectoroftheanglede nedby`01and`02oranylinewithslopebetweentheslopesof`0and`02(Figure3).Obviouslytheanglede nedby`001and`002is W.Let`bethebisectorlineof W.Infact,as`wecantakeanylinewithslopewithintheslopeintervalde nedby W.Considerthedouble-wedgeDWformedbythelines`00and`002,andthecorrespondingupperraysandlowerraysofDW.Byde nitionof`00and`002,allthemidpointsofthesegmentsofSstabbedby`1(`2)areabove(below)orovertheupperrays(lowerrays)ofDW.Let`betheorderofthemidpointsofthesegmentsofSaccordingtoasweepingbythebisectorline`of W.Thus,ifweknowthat W ,forsomegiven ,wecancomputeaconstantnumberd2 e=tofslopecandidatesforline`andcheckeachoneinO(nlogn)timeandO(n)space.Forstabbingwedgeswithverysmallapertureangle W,thevaluetcandominaten.......................................................................................`1...........................................................`2`d`00`00Figure3:Bothrays`1and`2ofastabbingwedgehavepositiveslope.Theorem4.4.GiventhesetSandapositivevalue ,astabbingwedgeoftype(c)forSwithapertureangle W canbecomputedinO(ntlogn)timeandO(n)space.5OtherstabbersTable1summarizestheobtainedresultsforthedecisionalproblemsofthestabberswehaveconsid-ered.By(sc)and(nsc)wedenotesatisfyingseparabilityconditionandnotsatisfyingtheseparabilitycondition,respectively.StabberTimeSpaceWedge(sc)O(hSkSlogn+nlogn)O(hS+n)Wedge(nsc)O(cSkSn+nlogn)O(n)Double-wedge(sc)minfO(n4);O(n3kSlogn)gO(n2)Double-wedge(nsc)O(n2kSlogn)O(n2)2-leveltree(nsc)O(n2kSlogn)O(n2)Zigzag(sc)O(n2kSlogn)O(n2)Zigzag(nsc)O(n3kS)O(n2)Table1:Summaryofresultsofdecisionproblem.Acknowledgements.WewanttogivespecialthankstoM.AbellanasandF.Hurtadofortheirhelpinthe rststudyofthestabbingproblemswhichwaspartoftheClaverol'sPhDthesis.E.ArkinandJ.S.B.Mitchellalsodeserveourthanksfortheircomments. 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