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Design, Implementation, and Database Integration of a Data Model for Vague Spatial Data as a Foundation of the Next Generation of Spatial Databases and Geographical Information Systems

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Design, Implementation, and Database Integration of a Data Model for Vague Spatial Data as a Foundation of the Next Generation of Spatial Databases and Geographical Information Systems
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PAULY, ALEJANDRO ( Author, Primary )
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2008

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Algebra ( jstor )
Approximation ( jstor )
Bytes ( jstor )
Data types ( jstor )
Database management systems ( jstor )
Databases ( jstor )
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Spatial data ( jstor )
Spatial models ( jstor )
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University of Florida
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Copyright Alejandro Pauly. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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ThisworkwouldnothavebeenpossiblewithouttheguidanceandsupportofDr.MarkusSchneider.Iamthankfulnotonlyforhisacademicsupport,butalsoforhisprofessionalandpersonalguidance.Iamalsoverygratefultothemembersofmycommittee.ThankstoDr.JoachimHammerforhisinvaluablesupport;toDr.AlinDobraforallhisprofessionalsuggestionsandhiswillingnesstohelp.IgreatlyappreciatethetimeandeortthatDr.ChrisJermaineandDr.SabineGrunwaldprovidedtohelpmyworkgothrough.IamforevergratefultoDr.ManuelBermudezandhisfamilyforalltheguidanceandsupportthatatalllevelshashelpedmethroughmyyearsinschool.MarkMcKenneyandReaseyPraingaregreatworkmatesandfriendsandhelpedmethroughsomeofthosenot-so-bearabletimes.Ithankallmyfriendswhohavecomebythroughtheyears,andinonewayortheotherhelpedshapethepathItooktogethere.Fromfaraway,myparentsAlbertoandJudy,mybrotherMauricio,andmylittlesisterDaniela,havebeencompletelysupportive.Iamherebecauseofthem.Iamverythankfultomyin-laws,Bob,Dee,andCarliwhowecouldalwayscountonwhentheschedulesimplydidnottthegoal.EventhoughmysonsEthanandLucaprobablydonotknowityet,theyhavebeenmysinglemostpowerfulsourceofmotivationandhappiness.Finally,likethisdissertation,alltheworkthatbearsmynameimplicitlyincludesthenameofmywifeCamille.Shehasworkedmorethanmesothatwecangethere,andIfeeltheluckiestbyknowingIcanalwayscountonher.Foreverandforeverything. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 2RELATEDWORK .................................. 16 2.1SpatialDatabases ................................ 16 2.1.1HistoryandMotivation ......................... 16 2.1.2SpatialDataTypes ........................... 17 2.1.3SpatialOperations ........................... 19 2.1.4TopologicalPredicates ......................... 20 2.2ImprecisioninSpatialData .......................... 23 2.2.1ATaxonomyofApproachestoSpatialImprecision .......... 24 2.2.2CurrentApproachestoHandlingImprecisioninSpatialData .... 25 2.2.2.1Exactmodelbasedapproaches ............... 25 2.2.2.2Spatialdatamodelingwithroughsets ........... 27 2.2.2.3Fuzzysettheoreticapproachestospatialdatahandling .. 28 2.2.2.4Probabilisticmethods .................... 28 2.2.2.5Comparisonofapproaches .................. 29 2.3UncertaintyinDatabases ............................ 31 3VASA:CONCEPTUALFRAMEWORK ...................... 35 3.1OverviewofVASA ............................... 35 3.2VagueSpatialDataTypes ........................... 37 3.2.1WhatareVagueSpatialObjects? ................... 37 3.2.2DenitionofVagueSpatialDataTypes ................ 39 3.3VagueSpatialOperations ........................... 42 3.3.1WhatareVagueSpatialOperations? ................. 43 3.3.2AGenericDenitionofVagueSpatialSetOperations ........ 45 3.3.3OtherGenericVagueSpatialOperationsandPredicates ....... 50 3.3.4Type-DependentVagueSpatialOperations .............. 52 3.3.5VagueNumericOperations ....................... 55 3.4VagueTopologicalPredicates ......................... 60 3.4.1WhatareVagueTopologicalPredicates? ............... 61 3.4.2DeterminationofVagueTopologicalPredicates ............ 62 3.4.3RepresentingVagueTopologicalPredicates .............. 64 5

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............. 67 3.4.4.1Preliminaries ......................... 68 3.4.4.2Problemsolution ....................... 72 3.4.5InterpretationofCharacterizationPredicatesforDeterminingVagueTopologicalPredicates ......................... 76 3.5QueryingVagueSpatialObjects ....................... 81 3.5.1CrispQueriesofVagueSpatialData .................. 81 3.5.2AVagueQueryLanguageExtensiontoEnableVagueQueriesonVagueSpatialData ........................... 84 4VASA:IMPLEMENTATION ............................ 87 4.1RATIOforNumericalRobustness ....................... 87 4.1.1WhatisNumericalNonrobustness? .................. 87 4.1.2RATIOImplementation ......................... 89 4.2StructuredLargeObjectsforDBMSExtensibility .............. 92 4.2.1WhatareStructuredLargeObjects? ................. 94 4.2.2WhatareMulti-StructuredLargeObjects? .............. 96 4.2.3ThemSLOBFramework ........................ 98 4.2.3.1Structuralrepresentation ................... 99 4.2.3.2Supportingecientupdates ................. 101 4.2.3.3ThemSLOBinterface .................... 102 4.2.4ImplementationofmSLOB ....................... 105 4.2.4.1Updatesequenceimplementation .............. 105 4.2.4.2HandlingstructureinmSLOBs ............... 108 4.2.4.3ThemSLOBimplementation ................ 108 4.3SPAL2DastheUnderlyingModel ....................... 111 4.3.1SPAL2DSystemArchitecture ..................... 112 4.3.2TechnicalSpecication ......................... 113 4.3.2.1Robustgeometricprimitives ................. 113 4.3.2.2Spatialdatatypes ...................... 116 4.3.3ImplementationofSPAL2DwithmSLOB ............... 118 4.4VASAImplementation ............................. 121 5CONCLUSIONS ................................... 124 6FUTUREWORK ................................... 126 REFERENCES ....................................... 128 BIOGRAPHICALSKETCH ................................ 134 6

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Table page 2-1The8topologicalrelationshipsbetweensimpleregionsdenedbytheRCCtheoryandthe9-Intersectionmodel. ............................ 20 2-2Numberoftopologicalpredicatesbetweenalldatatypecombinationsofspatialobjects.Simplespatialdatatypesareshownontheleftandcomplexspatialdatatypesontheright. ............................... 22 3-1Componentsresultingfromtheintersectionofkernelparts,conjectureparts,andoutsidepartsoftwovaguespatialobjectswitheachotherfor(a)union,(b)intersection,(c)dierence,and(d)complement. .................. 45 3-2DenitionofcharacterizationpredicatesonthebasisofthenonemptinessandemptinessofthekernelandconjecturepartsoftwovaguespatialobjectsAandBandthecrisptopologicalpredicatesp;q;r;s2T;. ............... 67 3-3Thecompositiontablefortopologicalrelationshipsbetweencomplexpoints. .. 70 3-4Numberofcharacterizationpredicatesbetweenallcombinationsofvaguespatialdatatypes. ...................................... 76 7

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Figure page 2-1Examplesofaspatialobjects:(a)simplepoint,(b)simpleline,(c)simpleregion,(d)complexpoint,(e)complexline,(f)complexregion. .............. 18 2-2Exampleofspatialintersectionrequiringregularization.(a)Theoriginalobjects,onedarkerandwithdottedboundary,theotherlighterandwithcontinuousboundaryareshownwiththeirsharedfeaturesboldened.(b)Theunregularizedresultwithanobviousorphanboundary.(c)Theresultoftheregularizationstep. 19 2-3Hierarchyofconceptsforimperfectionandvagueness ............... 24 2-4Examplesofspatialobjectswithindeterminateboundariesrepresentedinapproachesbasedonexactmodels:(a)broadboundaryandegg-yolkregion,(a,b)twovagueregions. ........................................ 26 3-1Theextentofalakedependingonthedegreeofevaporationandontheamountofprecipitation .................................... 38 3-2Examplesofa(complex)vagueobjects:(a)vaguepoint,(b)vagueline,(c)vagueregion.Theterm'complex'indicatesthateachcollectionofcomponentsformsasinglevaguespatialobject. ............................. 39 3-3Examplescenarioswithafocusontheintersectionbetweenvagueregions:(a)intersectionoftwokernelparts,(b)intersectionofakernelpartandaconjecturepart,(c)intersectionoftwoconjectureparts. .................... 44 3-4Examplesforthevaguespatialoperations:(a)k-boundary,and(b)c-interior. . 53 3-5Examplesoftheconvexhullofvaguepoints:(a)vaguepointp,(b)k-convex hull(p),(c)c-convex hull(p). ................................. 54 3-6Examplesshowingtheimpossibilitytodetermineanupperboundforthenumberofcomponentsof(a)avagueline,and(b)avagueregion. ............ 58 3-7Examplevagueregionobjectwherethekernelpartconsistsofthreecomponentsbuttheminimalnumberofcomponentsis1 .................... 59 3-8Exampleshowingimpossibilitytodeterminethe(a)upperbound,or(b)thelowerboundoftheperimeterofavagueregion. .................. 59 3-9Twoexamplescenariosillustratingvaguetopologicalpredicates.(a)Vagueregionsrepresentinganoilspillandacoralreef.Theblackareaindicatestheintersectionoftheconjecturepartofthespillwiththekernelpartofthereef.(b)VaguelinesasroutesofsuspectedterroristswheretheconjecturepartoftherouteofXintersectswiththekernelpartoftherouteofY. ................ 62 3-10Thegeneralmethodfordeterminingvaguetopologicalpredicates. ........ 63 8

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........................... 71 3-12BinarySpatialConstraintNetworkrepresentationsofn-tuples .......... 73 3-13Thevaguespatialobjectrepresentationofananimal'sroamingareas,migrationroutes,anddrinkingspots. .............................. 83 4-1Anexampleregionobjectwithtwofacesandoneholewitheachsegmentidentiedbyadierentnumber. ................................ 93 4-2ExampletypesofelementsinanmSLOB:(a)abaseelement,(b)astructuralelementconsistingofastructuralelementandabaseelement.Unshadedrectanglesdenotestructuralelementswhereasshadedrectanglesdenotebaseelements. .. 94 4-3DiagramdepictionsofmSLOBcomponents:(a)astructuredobject,(b)itshierarchicaltreerepresentation. .................................. 94 4-4Illustrationofstructureandviewsofamulti-structuredlargeobject:(a)objectwiththreedierentstructuresandreferencelinksdenotedbyshadedcirclesandarrows,and(b)thecorrespondingdirectedgraphrepresentationwithreferencesasdashedarrows. ................................... 97 4-5AnexampleofanmSLOB:(a)attheconceptuallevel,and(b)attheimplementationlevel.ThemSLOBhasasinglestructuralelementcontainingabaseelementandareferenceelement,whicharemarkedbyS,B,andRintheirinformationnodesrespectively. .................................. 100 4-6ExamplecompositionofanmSLOBwiththeinitialunfragmenteddataandtheupdatesequence. ................................... 101 4-7Exampleofinsertingblock[j:::l]atpositionkinanupdatesequence. ..... 102 4-8Illustratingthedeletionofblock[m:::n]inanupdatesequence. ......... 102 4-9Illustrationofthereplacementofblock[o:::p]byblock[l:::q]inanupdatesequence. ....................................... 102 4-10ThemSLOBinterface ................................ 103 4-11TheLocatorinterfaceforaccesstomSLOBfunctionality. ............ 104 4-12Anout-of-ordersetofdatablocksandtheircorrespondingsequenceindex. ... 106 4-13Theinterfaceformanipulatingtheupdatesequence. ................ 107 4-14AsamplemSLOBobjectrepresentingtheregioninFigure 4-1 .Informationbytesarestriped,endingbytesareshadeddark,addressesandcountsareshadedlight.Objectsareleftunshaded.Thelabelsundertheboxesindicatetheaddressofthatspeciclocation. ................................. 109 9

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.......... 112 4-16Illustratinginside-aboveagofhalf-segments.Thehalf-segmentwiththebigcircledendpointasitsdominatingpointwillhaveitsinside-aboveagsettotruebecausetheinterioroftheregionisaboveorleftofthathalf-segment.Forthehalf-segmentwhosedominatingpointisdenotedwiththesmallercircle,itsinside-aboveagissettofalse. ........................... 115 4-17Codesamplethatimplementstheoperationtocomputethelengthofacomplexregion,asimplementedinSPAL2DonthebasisofmSLOBs. ........... 120 4-18CodesamplethatillustratestheimplementationofVASAandhowitinteractswithSPAL2Dtoperformtheunionoperationbetweenvagueregions. ...... 123 10

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73 ]suchaspoint,line,andregion.Wespeakofspatialobjectsasoccurrencesofspatialdatatypes.Sofar,themappingofspatialphenomenaoftherealworldleadsalmostexclusivelytopreciselydenedspatialobjects.Butformanygeometricapplicationsthisisaninsucientabstractionprocess,sinceoftenthefeatureofspatialvaguenessorspatialindeterminacyisinherenttomanygeographicdata[ 12 ].CurrentGISandspatialdatabasesystemsarenotcapableofsupportingapplicationsbasedonvaguegeometricdata.Sofar,oftencontrarytoreality,spatialdatamodelingimplicitlyassumesthatthepositionsofpoints,thelocationsandroutesoflines,andtheextentandhencetheboundaryofregionsarepreciselydeterminedanduniversallyrecognized.Lineslinkaseriesofexactlyknowncoordinates(points),andregionsareboundedbyexactlydenedlineboundaries.Thisleadssolelytoexactobjectmodels.Thepropertiesofthespaceatpoints,alonglines,andwithinregionsaregivenbyattributeswhosevaluesareassumedtobeconstantoverthewholeobjects.Examplesareespeciallyman-madespatialobjectsrepresentingengineeredartifacts(likemonuments,highways,buildings,bridges)andpredominantlyimmaterialspatialobjectsexertingsocialcontrol(likecountries,districts,andlandparcelswiththeirpolitical,administrative,andcadastralboundaries).Wedenotethiskindofentitiesascrispordeterminatespatialobjects.Increasingly,researchersarebeginningtorealizethattherearemanygeometricapplicationsinrealityinwhichpositionsofpointsarenotexactlyknown,thelocationsandroutesoflinesareunclear,andregionsdonothavesharpboundaries,ortheirboundariescannotbepreciselydetermined.Examplesarenaturalphenomena(like 13

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2 surveysallthenecessarybackgroundconceptsrelevanttothecontentsofthisdocument.Itcoversbasicspatialdatabaseconceptsaswellasotherapproachesforhandlingspatialvagueness.InChapter 3 weprovideallformaldenitionscomprisingVASA.Chapter 4 describestheprototypeimplementationofVASA.ConclusionsarepresentedinChapter 5 ,andfuturedirectionsoftheworkpresentedherearepresentedinChapter 6 . 15

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2.1.1 bydescribingthehistoryandmotivationbehindtherecentsurgeinpopularityofspatialdatabases.Section 2.1.2 describesspatialdatatypesandtheirformaldenitions.TheoperationsonthesedatatypesaredescribedinSection 2.1.3 .Section 2.1.4 surveystheexistingworkontheimportantconceptoftopologicalrelationships. 40 , 77 ].Thegoalofspatialdatabasesystems(SDS)istoprovidesupportformanipulatingspatialdatainsuchnon-traditionaldatabaseapplications.Therecentwidespreaduseofgeographicalinformationsystems(GIS)hascausedanincreaseinthepopularityofSDS,whicharenowconsideredtobethefoundationforthenextgenerationofGIS.Originally,GISweredevelopedasproprietary,self-containedsystemsthathandletheirownstorageandensuredataconsistencyeventhoughtheseandotherrelatedtasksarealreadysuccessfullyimplementedandcurrentlyavailableinDBMS[ 6 ].TheproprietarydesignofGISimplementationsandtheirinabilitytoeectivelytakeadvantageofbasicdatabasetheoryandstateoftheartcommercialDBMStechnologiesattesttotheneedforanewformalandrobustfoundationforGIS.Researcheortsintheareaofspatialdatabasesresultinthedenitionofavarietyofspatialalgebrasthatformalizethenecessaryspatialdatatypesandoperationslaterimplementedtosupportspatialdataindatabases.Inadditiontospecicresearcheorts,organizationssuchastheopengeospatialconsortium(OGC)seektoprovideconsensusabstractspecications[ 22 ]forenablingspatialandgeographicdatacontentindomainssuchastheinternet,wirelessandpervasivecomputinganddatabases.Suchspecications 16

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39 ]wherethedenitionofspatialdatatypesPOINT,LINEandREGIONisinuencedbyalreadydevelopeddatatypesforgeometry-baseddatabasesystems.POINTobjectsarerepresentedbyapairofcoordinatesintheEuclideanplane.ALINEobjectisdenedbythesetofPOINTobjectsrepresentingsegmentend-pointsinachainofsegments.REGIONobjectsaredescribedaspolygons,equivalenttoaclosedchainofsegments(i.e.,therstandlastsegmentsinthechainshareanend-point).Similardenitionsarefoundin[ 60 ]whereaspartofdevelopmentsofthePROBEproject,spatialmodelingandqueryprocessingareenabledinimagedatabaseapplications.In[ 71 ]thedenitionsofspatialdatatypesareprovidedonthecontextofapictorialquerylanguage(PSQL).The 17

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30 ]wherethedenitionsareprovidedinthecontextofimplementingtheSpatialSQLquerylanguage.Theseapproachesalldealwithwhatareconsideredtherstgenerationofspatialdatatypesorsimplespatialdatatypes.Increasedapplicationrequirementsandthelackofclosurepropertiesofsimplespatialdatatypeshaveresultedinthenextgenerationcomplexspatialdatatypes.Figure 2-1 illustratessimpleandcomplexspatialobjects.Acurrentandcompletedenitionofcomplexspatialdatatypescanbefoundin[ 76 ]wherethedenitionsareprovidedinthecontextofidentifyingthetopologicalrelationshipsbetweensuchcomplexobjects.Acomplexpointobjectincludesanitenumberofsinglepoints,acomplexlineobjectisassembledfromanitenumberofcurvesstructuredintoconnectedcomponentsdenominatedasblocks,andacomplexregionobjectisstructuredasanitenumberofdisjointfaceswhereeachfacecanpossiblycontainanitenumberofdisjointholes.Fromhereon,werefertothedatatypespoint2D;line2D;andregion2Dasthosecomprisingallcomplexpoint,complexline,andcomplexregionobjectsrespectively.Theneedforcomplexspatialdatatypesisinitiallyaddressedin[ 41 , 42 ]wheretheauthorsalsointroducetheconceptofrealmsinaneorttosolvethesometimesignoredproblemofnumericalnon-robustnessingeometricalcomputations(seeSection 4.1 ).In[ 87 ]theformaldenitionsofcomplexspatialobjectsarelimitedtotheconceptofgenericareaanddenitionsforlinesandpointsarenotprovided.Thedatatypescomplexarea,complexlineandcomplexpointaredenedin[ 16 ].Incontrastwith[ 41 , 42 ],thedenitionsof[ 16 ]allowforselfintersectingsegmentsinlineobjects. Figure2-1. Examplesofaspatialobjects:(a)simplepoint,(b)simpleline,(c)simpleregion,(d)complexpoint,(e)complexline,(f)complexregion. 18

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2.1.4 ).Thebasicspatial(orgeometric)setoperationsofunion,intersection,dierence,andcomplementaredenotedwiththeoperators,,,andrespectively.Theseoperationsaredenedasregularizedversionsofthewellknowngeneraldenitionsofunion,intersection,dierence,andcomplementofsets.Theseoperationsareformallydenedin[ 41 , 73 ]onthebasisoftheregularizationconceptsintroducedin[ 82 ].Concisely,theregularizedsetgeometricoperationsavoidpunctures,cuts,andotherirregularitiessuchasorphanpointsinregionsandlines.Wedenethesyntaxfortheseoperationsasf;;g:!,and:!where2fpoint2D;line2D;region2Dg.Figure 2-2 illustratesthetheintersectionoftworegions. Figure2-2. Exampleofspatialintersectionrequiringregularization.(a)Theoriginalobjects,onedarkerandwithdottedboundary,theotherlighterandwithcontinuousboundaryareshownwiththeirsharedfeaturesboldened.(b)Theunregularizedresultwithanobviousorphanboundary.(c)Theresultoftheregularizationstep. Wellknownnumericspatialoperationsincludetheminimumdistanceoperationwhichresultsintheminimumdistancebetweentwogivenspatialobjects.Othernumericoperationsincludethosecommonlyusedtoderiveattributesofspatialobjects,suchasthe 19

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24 , 70 ]thatemploysspatiallogictoderiveasetoftopologicalrelationshipsandthe9-Intersectionmodel[ 31 ](previously4-Intersection[ 29 ]),thatemployspoint-settheoryandpoint-settopology.Basedonthe9-Intersectionmodel,theworkin[ 31 ]identiesthetopologicalrelationshipsforallcombinationsofsimplespatialdatatypes.Theworkin[ 76 ]extendsthe9-Intersectionforallcombinationsofcomplexspatialdatatypes. sTable2-1. The8topologicalrelationshipsbetweensimpleregionsdenedbytheRCCtheoryandthe9-Intersectionmodel. TheRCCtheoryisbasedonspatiallogicandemploysconceptsofconnectivitytodeneasetofaxiomsthatresultineighttopologicalrelationshipsbetweensimpleregionobjects.TheeightrelationshipsdenedbytheRCCtheoryareillustratedinFigure 2-1 andarelabeleddisconnected(DC),partiallyoverlaps(PO),externallyconnected(EC),tangentialproperpart(TPPanditsinverseTPP1),nontangentialproperpart(NTPP,NTPP1),andequal(=).Thegurealsopresentsthecorrespondencebetween 20

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29 ]forsimpleregions,the9-Intersectionmodelgeneratesasetofunique9-intersectionbooleanmatriceseachofwhichrepresentsatopologicalrelationship.Thematrixvaluesaredeterminedbyevaluatingtheemptiness(ornon-emptiness)thatresultsfromtheintersectionofallcombinationsoftheboundary(@),interior()andexterior()fromeachspatialobjectinvolved.The9-intersectiontopologicalmatrixthatrepresentsrelationshipRbetweenspatialobjectsAandBisdenedas:R(A;B)=0BBBB@A\B6=?A\@B6=?A\B6=?@A\B6=?@A\@B6=?@A\B6=?A\B6=?A\@B6=?A\B6=?1CCCCAAtotalof512uniquematricescanbegenerated,thequestionfollowsaboutwhicharespatiallypossible(valid).Asetoflemmasdenetherequirementsthatidentifyamatrixasvalid.Applyingthelemmaseliminatesallinvalidmatricesandreducesthenumberofvalidmatrices,eachofwhichrepresentsauniquetopologicalrelationship.Asimilarbutmorecompleteapproachisfollowedin[ 76 ]todenethesetoftopologicalrelationshipsbetweencomplexspatialobjects.Table 2-2 showsthenumberoftopologicalrelationshipsidentiedbyeachmodelandbetweeneachtypecombination.Thenumberofrelationshipsbetweencomplexspatialobjectsisratherlarge.Forthisreason,aclusteringmechanismisdetailedin[ 76 ]bywhichthesetoftopologicalrelationshipsisreducedtoaconcisesetofclusteredtopologicalpredicates.Itisimportanttonotethatallthemodelsmentionedgeneratesetsofmutuallyexclusivetopologicalrelationships,ensuringthatbetweeneachpairofobjectsoneandonlyonesuchrelationshipapplies.Topologicalreasoningprovidesimportantconceptsusefulindealingwiththetopologicalrelationshipsdescribedabove.Acommonmethodfortopologicalreasoninginvolvesthederivationofthecompositionoftwo(ormore)topologicalrelationships.A 21

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Numberoftopologicalpredicatesbetweenalldatatypecombinationsofspatialobjects.Simplespatialdatatypesareshownontheleftandcomplexspatialdatatypesontheright. pointlineregion simplepoint 233simpleline 33319simpleregion 3198 complexcomplexcomplex pointlineregion complexpoint 5147complexline 148243complexregion 74333 1 , 28 ]isformallydenedforgeneralrelationsin[ 81 ].Givena(topological)relationshipP1between(spatial)objectsAandBanda(topological)relationshipP2between(spatial)objectsBandC,thecompositionP1;P2(atleastpartially)providesthe(topological)relationshipP3thatholdsbetweenAandC.ItissaidthatP3isonlypartiallydenedifitisimpossibletoderivecompleteinformationfromthegivenpairofrelationships.Insuchcases,P3isconsideredadisjunctionofrelationshipswhichintheworstcase(whennothingofP3canbeinferred)coversthecompletesetofrelationshipsdenedbetweentheobjecttypesofAandC.Boththe9-IntersectionandtheRCCtheoryprovidemechanismsforderivingthecompositionoftopologicalrelationships.In[ 28 ]asetofrulesbasedonthetransitivityofthecontainmentofpoint-setsareusedtoderivethecompositionof9-intersectionmatrices.Asimilarbutmoregeneralapproachisfollowedin[ 1 ].Giventhatthederivationin[ 28 ]isbasedonthe9-intersectionmodelandtherulesdenedrepresentsimplepropertiesofpoint-sets,thesameformalismcanbeappliedtothecomplexspatialdatatypesandtheirtopologicalrelationshipswhicharedenedin[ 76 ].TheRCCtheorynaturallyextendsitslogictoinferencesthatcanresultincompositiontables,withsimilarresultstothosein[ 28 ].InSection 3.4 theseitemsoftopologicalreasoningwillproveusefulinidentifyingtopologicalrelationshipsbetweenvaguespatialobjects.Severalextensionstotopologicalpredicatesexistinliterature.Theimportanceandfactibilityofemployingdimensionastheinvariantdening9-intersectionmatrices(insteadofemptinessofintersection)isinitiallysuggestedin[ 15 ].Alsocoveredin 22

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18 ],thisapproachisbasedontheso-calledDimensionExtendedMethod(DEM)andresultsinthedenitionsof52dimensionawaretopologicalrelationshipsbetweensimpleregions.In[ 54 ]dimensionisalsoconsideredastheinvariantfordening9-intersectionmatrices.Thematricesinthiscaserepresentso-calledDimension-RenedTopologicalPredicates(DRTP).ThisapproachisdierenttotheDEMinitsmethodologyanditsconsiderationofallcombinationsofcomplexspatialobjects,notonlyregions.Theconsiderationofdierentinvariantssuchasdimension,allowsfortherenementoftopologicalrelationships,givingwaytomorenegrainedandmeaningfulqueries.Theextensionoftopologicalpredicatesbetweensimpleobjects,specicallyregions,intopredicatesbetweencomplexregionshasproventonotbeatrivialexercise.Instead,thevarietyofsolutionsmentionedaboveallsuerfromasimilarproblem:thelossoflocalityintherelationships.Thisproblem,asidentiedin[ 55 ]appearsduetothedominanceofsometopologicalrelationshipsoverothers.Asanexample,assumethattherearetwocomplexregions,therstwithtwofaces,thesecondwithoneface.Ifoneofthefacesoftherstmeetsthefaceofthesecondregion,andtheotherfaceoftherstoverlapsthefaceofthesecondregion,overall(globally)wesaythatbothcomplexregionsoverlap.Thus,theinformationaboutthe(local)meetingsituationbetweentwoofthefacesislost.Theauthorsin[ 55 ]proposeaformalmodeloftopologicalrelationshipsthathastheabilitytopreservelocaltopologicalrelationships,whileatthesametimedistinguishingglobalscenarios. 62 , 86 ]aswellastheworkpresentedin[ 23 ].AtaxonomyofconceptsrelatedtoimprecisionisdiscussedinSection 2.2.1 .Asummaryof 23

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2.2.2 . 2-3 ).Therstrootsatthetermimperfection,whereasthesecondrootsattheconceptofvagueness.Althoughsemanticallyunrelated,thesetwoconceptsarerelatedinpracticeduetothefactthatimperfectioncan(andoftendoes)leadtovagueness. Figure2-3. Hierarchyofconceptsforimperfectionandvagueness Theconceptofimperfectioncanappearwithtwodierentfaces:inaccuracy,andimprecision.Inaccuracyisrelatedtoerrorsatthetimeofdatacapture.Forexample,thestatement\theUSAhasasmallerareathanCostaRica"isinaccurate.Imprecisiondealswithdatathatisnotnecessarilyinaccuratebutratheritisnotdeterminate,oftenduetotheniterepresentationsavailableforspatialobjects.Forexample,thestatement\CostaRicaisasmallcountry"isimpreciseasitdescribesarelativeattributeandthereisnoinherentexactmeasurementofthesmallconcept.Imprecisionissometimesconsideredasatypeofvaguenessnotinherenttotheobjectsthataredescribedbutratherderivedfromthedatathatisusedtorepresentthem.Uncertaintydescribesatypeofvaguenessorindeterminacythatisassociatedtoacrispconceptthatisnotknownorcannotbemeasuredprecisely[ 74 ].Ontheotherhand,fuzzinessdescribesvaguenessthatisinherenttoobjects(i.e.,theirboundary,location,orextensionisnotpreciselydened).Exampleofvagueobjectsincludemountains,oceans,forests,andevencitiesforwhichadeterminateboundarycannotbeidentied. 24

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19 ],theegg-yolkapproach[ 21 ],andthevagueregionsconcept[ 33 ].Thesemodelsextendthecommonassumptionthatboundariesofregionsdividetheplaneintotwosets(thesetthatbelongstotheregion,andthesetthatdoesnot)withthenotionofanintermediatesetthatisnotknowntocertainlybelongornottotheregion.ThuswesaythatthesemodelseectivelyextendcrispmodelsthatoperateontheBooleanlogic(true,false)intomodelsthathandleuncertaintywithathree-valuedlogic(true,false,maybe).Theregionswithbroadboundariesapproach[ 17 ]leveragescomplexregionstoenablehandlingofspatialuncertainty.Aregionwithabroadboundaryisinitiallydenedasacomplexregionwhosecomponentsareenclosedinazoneconsidereditsbroadboundary.Thiszoneisboundedbytwosharplineboundaries,wheretheinnerlineexpressestheminimalextensionofthecomponent,andtheouterlineexpressesitsmaximalextension.Thespecialcasewherebothboundariesareequal,representsacrispcomponent.Similartothebroadboundaryapproach(seeFigure 2-4 (a)),theegg-yolkmodelintroducedin[ 21 ]leveragestheregionconnectioncalculus(RCC)methodfortopologicalrelationshipsbetweensimpleregions.Theegg-yolkmodeleectivelymodelsanuncertainregionwheretheyolkcertainlybelongstotheregion,andthezonesurroundingtheyolk 25

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72 ]enablesbroadboundarytyperegionswithholestoberepresentedonthebasisoftheROSE[ 42 ]algebra. Figure2-4. Examplesofspatialobjectswithindeterminateboundariesrepresentedinapproachesbasedonexactmodels:(a)broadboundaryandegg-yolkregion,(a,b)twovagueregions. Boththeegg-yolkandbroadboundaryapproachesfocusonestablishingthetopologicalrelationshipsbetweenregionswithuncertainty.Thebroadboundaryapproachrecognizes44topologicalrelationshipsbetweenbroadboundaryregionsbasedonsimpleregions,and56betweenthosebasedoncomplexregions.Theegg-yolkrecognizes46topologicalrelationshipsbetweenregionswithuncertainboundaries.Finally,thevagueregionsapproach[ 33 ]leveragessimpleregions.Avagueregionisrepresentedbytwosimpleregions,onedenotedasthekernel,theotherastheboundary.Thekernelrepresentstheareathatcertainlybelongstheregion.Theboundaryrepresentstheareathatmayormaynotbelongtotheregion(seeFigure 2-4 (b)).Vagueregionsareclosedunderwelldenedspatialsetoperations(i.e.,geometricintersection,union,dierence,complement).Thisisnotthecaseintheprevioustwoapproaches.ThevagueregionsapproachistheprecursortoVASA,whichwepresentinthroughoutthisdocument. 26

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67 , 68 ].Aroughsetisdenedonthebasisofalowerapproximationandanupperapproximation,whicharebothcrispsets.Thelowerapproximationindicatestheelementsthatcertainlybelongtotheset,whereastheupperapproximationalsoincludesthoseelementsthatmayormaynotbelongtotheset.Atthecoreofroughsettheory,indiscernibilityrelationsbetweenattributesofelementsinthesetareusedtodenetheupperandlowerapproximations.Theserelationscanbeusedtomanipulatethegranularityonwhichtheapproximationsareestablished.Worboysin[ 85 , 86 ]wasoneofthersttoproposearoughsetbasedalternativefordealingwithspatialvagueness.Hisproposalisgearedtowardsprovidingabasisforintegratingandreasoningaboutmulti-resolutionspatialdata,thatisdatathatwascapturedatdierentresolutionsbutneedstobehandledtogether.Thisapproachisextendedandformalizedin[ 27 ].Theauthorsin[ 7 , 8 ]useroughsetsasthemathematicalfoundationtomodeluncertaintyintopologicalrelationshipsbetweenegg-yolkregions.IncomparingtheexpressivenessofusingtheRCCmodelversusroughsettheory,somerelationshipsbecomeindistinguishableusingroughsets,whileotherscaninfactbespeciedwithhigherprecision.TheauthorsconcludethatroughsetsprovideavaluablefoundationformodelinguncertaintyinspatialdataandcanleverageexistingtheoriessuchastheRCCtheoryfortopologicalrelationships.Furthermore,roughsetscanprovidehighergranularitythanmodelssuchastheegg-yolkapproach.Thisisduetotheuseoftheindiscernibilityrelationship.Incontrast,exactmodelbasedapproachesdonothandlesuchameasurefordeningthepartitionbetweenwhatiscertainandwhatisnot,insteadthisisassumedtobeafeatureoftheexistingdata.Anothermethodforemployingroughsetsinhandlinguncertaintyisproposedin[ 2 ]wheretheauthorspursuethederivationofqualitymeasuresfortheuncertaintyorimprecisionintheirdata.Roughclassicationisusedtoassigndata(areas)toclasses 27

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4 ]assignmembershiptosomepropertyforeverycoordinatepointwithinthe(fuzzy)region.Aformaldenitionoffuzzypoints,fuzzylinesandfuzzyregionsisincludedin[ 74 ].Arecenteortforthedenitionofavaguespatialalgebrabasedonfuzzysetsispresentedin[ 26 ].In[ 3 ],theauthorsextendtheirroughclassicationconceptfrom[ 2 ]toaclassicationoffuzzyregions.Thatis,thelowerapproximationandtheupperapproximationofaregionareeachrepresentedbyafuzzyclassication.Accordingtotheauthors,thisenablestreatmentofindiscernibilityorimprecisionbytheroughsetclassicationandfuzzinessbythefuzzyclassication.Approximatespatialreasoningandfuzzyquerylanguagesarediscussedin[ 48 , 83 ]respectively.Anadaptationoffuzzyobjectstotherelationalmodelisdescribedin[ 84 ]. 9 , 11 , 34 , 78 ],modelspatialobjectsonthebasisoftheprobabilityofmembershipofanentity(i.e.,point,area,object)inaset.Thisresultsinanexpectedmembershipwhichisbasedonthesubjectivelydenedprobabilityfunction.Thiscanbecontrastedtothemembershipvaluesoffuzzysetsthatareobjectiveinthesensethattheycanbecomputedformallyordeterminedempirically. 28

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8 ].Whencomparingthetwotypesofapproaches,itisimportanttoconsidertheelementsthatmakethemseeminglydierent.Therstistheconceptofindiscernibilityrelationsthatexistsinroughsets.Tothisconceptthereisanimplicitfunctionthatdenesthegranularitywithwhichthelowerandupperapproximationaredened.Thefunctionprovidesasimilarroletothewellknownconceptofmembershipfunctionavailableinfuzzysets.Thus,theseparationoflowerandupperapproximations,orcertainanduncertainpartsofaspatialobjectisnotafeatureoftheobject(orthedatarepresentingtheobject)asitiswhenusingexactmodelbasedapproaches,butinsteadavaluecomputeddependentonapossiblysubjectiveanddomainspecicfunction.Thisrepresentsanadvantageinthecaseswheresuchafunctionisformallyknown,andcanevenbeadjustedifneeded.Itcanbeconsideredadisadvantagebecauseofthelackof 29

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2 ],isbasedonaroughsetextensionofacrispspatialalgebra.Thismeans,thattheimplementationoftheoriginalcrispalgebrahadtobemodiedtohandleroughsets.Onaseparatecategory,fuzzysetbasedapproachesandprobabilisticapproachesareplacedtogethernotsomuchforthepurposeofcomparisonbutmoreforthepurposeofusingone'ssolutionforthebenetoftheother.Attheheartoffuzzysetbasedapproaches,amembershipfunctionisinchargeofdecidingtowhichdegreeeachpointorsetofpointsbelongstoaspatialobject.Themembershipvaluedoesnotdescribetheuncertaintyofthepointbelongingtotheobject,butratheranactualdegreeofhowmuchthepointbelongstotheobject.Thus,fuzzysetbasedapproachesareappropriatefordealingwithfuzzinessasaninherentpropertyoftheobjects.Incontrast,thefunctionattheheartofprobabilisticmethodsisusedtodescribetheprobabilitythatapointorsetofpointsbelongstoanobject.Thus,describingtheadegreeofhowuncertainitisthatthepointactuallybelongstotheobject,whichmakesprobabilisticapproachesappropriateformodelingimprecisionanduncertainty.Fuzzysettheoryhasbeenrecentlyconsideredthepreferredoptionforhandlingspatialvagueness.Mostapproaches,likesomeofthosementionedinSection 2.2.2.3 ,focusonprovidingdomainspecicsolutions.Fromanimplementationstandpoint,thisprovidesbetterviabilityofanactualimplementation,butrestrictsthemodelstothedomaintheyweredesignedfor.Themodelin[ 26 ]isimplementedasanextensionofGRASS,an 30

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59 ],Motrodistinguishesbetweenthetwooptionsthatwehaveformaintainingtheintegrityofadatabasewhenlessthanperfectdataisavailable.Therstoptionconsistsofrestrictingthedatamodeltotheportionofthedataforwhichperfectinformationisavailable.Takeforexamplethecommonrelationalmodelwhereinmostcasestheinformationmusteitherbeperfect,orvaluesaremodiedtobeassumedperfect.Thismeansthatinanemployeerelation,weonlyhavethoseemployeesforwhomwehaveperfectandpreciseinformation.Thesecondoptionistousedatamodelsthatenabletreatmentofimperfectinformation.Thenatureanddiversityofthetypesofimperfectionthatcanbepresent,haveproventobemajorhurdlesinthedevelopmentofsuchdatamodels.Nevertheless,currentDBMStypicallyincludebasicfeatures,suchasNullvaluesinordertoenablesometreatmentofimperfectinformation.Inthecontextofthewell-knownrelationalmodel,Codd[ 20 ]discussesthetreatmentofmissinginformationwithinrelationaldatabases.Thequestionisraisedonwhetherthetreatmentofmissingwholetuplesissimilartothetreatmentofmissingsingularatomicvalueswithinatuple.Thesetwocasescanbedistinguishedbyaskingthequestions:doesthistuplebelongtothisrelation?fortheformercase,anddoesthisvaluebelong 31

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13 ].Thepurposeofthistheoryistopresentaprobabilisticmodelthatisdenedasageneralizationoftherelationalmodel.Thispurposepresupposesnoeectsonthetreatmentofpreciseandperfectdatathathasbeensofarsuccessfullymanagedonthebasisoftherelationalmodel.Insteaditintendsonaddingexpressivepowerandappropriatehandlingofperhapsnotsopreciseandperfectdatatorelationaldatabases.Thistheoryappliesprobabilitiesatthetuplelevel,byspecifyingsomeprobabilityfunctionthatcanbeappliedtoeachtupleinordertodetermineitsdegreeofcertaintyinbelongingtothatrelation.Fuhr[ 35 ]introducesaprobabilisticframeworkforhandlingimpreciseinformationfromtheperspectiveofthequerylanguage.Thatis,itsfocusisonhandlingvaguequeriesovereitherpreciseorimprecisedata.TheimportanceofqueriescannotbeminimizedduetotheirroleasinterfacebetweentheuserandtheDBMS.TheusercanonlygetasmuchusefulinformationfromtheDBMSasthequerylanguageallowstheusertoexpress.Fuhrintroducestheconceptofanextendedqueryasacommon(Boolean)queryincombinationwithavaguequery.Avaguequeryallowsforvagueinterpretationofsuchoperatorsasthoseforbinarycomparison(<;>;=),andfortheuseofunaryprobabilisticqualierssuchaslow,high,etc.,andfuzzymodierssuchasmany,andseveral.Basedonawellspeciedquerysyntax,theindexingmechanismisbasedonprobabilisticanalysisusingpredenedsemanticsoftheoperators.Thisframeworkforvaguequeriesislater 32

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36 ].Thealgebrarepresentssomeoftheseminalworkonextendingtherelationalmodelwithprobabilisticnotionsforhandlingimprecisedata.SimilartotheideasofCavalloandPitarelli[ 13 ],thisalgebraallowsforannotationsoftheprobabilityforagiventupletobelongtoagivenrelation.Theprobabilisticrelationalalgebraprovidesthenecessarydenitionsforallthewell-knownrelationaloperators(i.e.,projection,selection,etc.).AmodelsimilarinpurposeispresentedbyLee[ 52 ]inwhichanextendedrelationalmodelispresentedforhandlinguncertainandimprecisedata.ThismodelisbasedonDempster-Shafertheory[ 25 ]forreasoningandcalculationoftheprobabilitiesoftuplesbelongingtorelations.ThemodelbyBarbaraetal.[ 5 ],introducesanewprobabilisticdatamodelthatemploysthenecessaryconceptstoallowhandlingofimpreciseattributes,asopposedtotupleswhichisthecaseofthepreviouslyperusedapproaches.Thatis,functionsofprobabilityareassociatedwithindividualattributeswithintuples,thatreecttheprobabilityforthatvaluetobelongtothattuple.Fromthiscanbederivedthatasingleattributecanhavemorethanonevalue,andtheprobabilitiesofallpossiblevaluesforanattributewithinatupleaddto1.Descriptionofrelationalalgebraoperatorsforsuchamodelasanextensiontotherelationalmodelarepresented.TheissueofperformanceisthefocusoftheapproachpresentedbyCheng,etal.[ 14 ].Thisapproachincorporatesnotionsofobjectsmoving(orvalueschanging)onthebasisofpredenedconceptssuchthatitispossibletocomputewithcertainqualityguarantees,theircurrentvaluesbasedonapreviouslyknownvalue.Thisapproachincludesthenecessaryconceptstoecientlyandeectivelyevaluatequeriesoverthesepossiblyimprecisedata.Theimportanttopicofaggregatequeriesisalsoaddressed.Ingeneral,probabilisticapproachesareofgreatusewhenthenecessaryvaluescanbederivedfromwithintherelations,otherwiseanexplicitprobabilityneedstobeassociatedwitheachvalue.Thisinpracticebecomesproblematicbecauseinmanycasessuchmeasuresarenotavailableandtendtobesubjectiveatbest.Aprobabilisticindexingtechniquecalled 33

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80 ].U-Treesareecientinprobabilisticrangesearchesonthebasisofdistancefunctionsdeningtheuncertaintyofgivenvalues.Theincorporationoffuzzysettheoryindatabasesystemshasbeenalongstandingideafordealingwithuncertaintyandimprecision.AgeneralizationoftherelationalalgebraforenablingthetreatmentofincompleteanduncertaindataispresentedbyPradeandTestemale[ 69 ].Theirapproachenablespossibilitydistributions(addingto1)tobeassociatedwiththeattributevalueswithinatuple.Crisprelationaldatabasesthathandleperfectinformationarethenconsideredspeciccasesofsuchageneralization,whereallthepossibilitiesareequalto1.TheconceptoffuzzyrelationaldatabaseswasoriginallycoinedbyBucklesandPetry[ 10 ].Afuzzydatabaseisorganizedinaccordancetotherelationalmodel(relations,tuples,etc.),buttheattributeandtupledomainsareassociatewithafuzzysimilarityrelationships.Suchaddedinformationenablesthedenitionofrelationalalgebraoperatorstoeectivelymanagecrispandvaguequeriesovercrispandvaguedata.Finally,Motro[ 57 , 58 ]introducestheVAGUEsystemasaninterfacethatallowsvaguequeriesovercrispandvaguedata.ThissystemisdevelopedonthebasisoffuzzysettheoryandexploitsthecurrentcapabilitiesofrelationalDBMSstoimplementasucientlyeectivevaguequerysystem.VAGUEemployssystemrelationstomaintainfuzzyclassicationsfordierentdomains.Thesetablesarereferencedbyqueriesthatmustreturnvaluesonthebasisofvaguemodiers(e.g.,small,big)oroperators(i.e.,rangeoperatorssuchasthesimilar-to[ 45 ]).Itisworthnotingothereortsthatprovideimportantideasforhandlingimpreciseanduncertaindataindatabases.TheworkofImielinksiandLipski[ 46 ]providessomeofthemostvaluableearlyworkbymodifyingtherelationalalgebraoperatorsinordertoaccountfordefaultnullvalues.BasicworkbyGessert[ 37 ]introducesafour-valuedlogicbasedapproachtodealingwithincompletenessandimprecision.Thefourvaluesreferringtoinapplicable(i.e.,missing),andfalse,maybe,andtrue. 34

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32 ].Anotherclassicationdistinguishesbetweenadescriptivealgebraandanexecutablealgebra[ 73 ].Adescriptivealgebraisanalgebrainthesensedescribedabove;itoerstypesandoperationsataconceptuallevelwhichcanbeusedtoformulatequeriesinadatabasesystem.Suchanalgebraabstractsfromrepresentationsaspects.Designgoalsareconceptualclarity,simplicity,andexpressiveness.Eciencyplaysaminorrole.Anexecutablealgebrapresentstheactualrepresentationsandalgorithms.Thatis,insuchanalgebrathereisadatastructureassociatedwitheachsortandanalgorithmassociatedwitheachoperator.Designgoalsarehereeciencyandsimplicity.Inadatabasesystem,itisthetaskoftheoptimizertotranslateanexpressionofthedescriptivealgebraintoanequivalent,ecientlyevaluableexpressionoftheexecutablealgebra.Asinglefunctionofthedescriptivealgebramay,ingeneral,betranslatedtoseveraldierentoperators,oreven 35

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4 forafurtherdiscussion). 36

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2.2.2 ,ourmodelbelongstothecategoryofexactobjectmodels.Section 3.2.1 motivatesourconceptbyseveralapplicationscenarios.InSection 3.2.2 weintroducevaguespatialdatatypesasourformalconceptandgivetheirdenitions. 2.1.2 .Weassumethatthesedatatypesareclosedunder(appropriatelydened)geometricunion,intersection,anddierenceoperations.Ourconceptmainlydealswiththeaspectofspatialvaguenessbutalsoincludessomeaspectsofspatialfuzziness.Frequently,thereisuncertaintyabouttheprecisepathsorthespatialextentofphenomenainspace;thatis,objectscanshrinkandextendandhencehaveaminimalandmaximalextent.Anexampleisalakewhosewaterleveldependsonthedegreeofevaporationandontheamountofprecipitation.Highevaporationimpliesdryperiodsandthusaminimalwaterlevel.Highprecipitationentailsrainyperiodsandthusamaximalwaterlevel.Islandsinthelakearelessoodedbywaterindryperiodsandmoreoodedinrainyperiods.Ifanislandcanneverbecompletelyoodedbywater,itformsa\hole"inthelake.Butifanislandlikeasandbankcanbeoodedcompletely,itbelongstothevaguepartofthelake.Hence,wehavecondentinformationabouttheminimalandmaximalextentofalake.Buttheactualextentofalake,whichissomewherebetweenthesetwoextremelimits,isvague.Figure 3-1 illustratesthisspatialconstellation.Dark-grayshadingshowsareasthatdenitelybelongtothelake.Light-grayshadingindicatesareas 37

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Theextentofalakedependingonthedegreeofevaporationandontheamountofprecipitation thatperhapsbelongtothelake.Whitecolorindicatesareasthatdodenitelynotbelongtothelake.Anotherexampleisamapofnaturalresourceslikeironore.Forsomeareasexpertsknowtheexistenceofironorewithcertaintybecauseofsoilsamplesandboreholes.Forotherareasexpertsarenotsureandonlyassumetheincidenceofthismineral.Thesearethekindsofvaguespatialobjects(inthiscase:vagueregions)weareespeciallyinterestedin.Ontheotherhand,ourconceptisalsoabletomodeltheaspectoffuzzinessthatarealobjectshaveanextentbutcannotbeboundedbyapreciseborderlikethetransitionbetweenamountainandavalley.Continuouschangesoffeatures(likeairpollutioncontinuouslydecreasingfromcitycenterstoruralareas)cannotbemodeledbythisconcept.Thisisapredestinedapplicationoffuzzyobjectmodels(seeSection 2.2.2.3 ).Afurtherexampleisoilspills.Forenvironmentalauthoritiesitisveryimportanttoobtaininformationaboutthespreadofoilslicksinordertobeabletotakemeasuresfortheirremoval,assesstheconsequencesforthemarinaoraandfauna,andimplementrescuemeasures.Duetoradarandhelicopterobservationsitispossibletodeterminetheminimaldistributionofoilslicks.Mathematicalmodelsfedbyparameterslikewindvelocityandcurrentenablethepredictionofthemovementandthepossibleextentoftheoilpollution.Asanalillustratingexample,whichwealsousetointroduceourterminologydeployedinVASA,weconsiderahomelandsecurityscenario.Secretservices(should) 38

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Examplesofa(complex)vagueobjects:(a)vaguepoint,(b)vagueline,(c)vagueregion.Theterm'complex'indicatesthateachcollectionofcomponentsformsasinglevaguespatialobject. haveknowledgeofthewhereaboutsofterrorists.Foreachterrorist,someoftheirrefugesarepreciselyknown;someothersareassumedandthusonlyconjectures.Wecanmodelalltheselocationsasasinglevaguepointobjectwherethepreciselyknownlocationsarecalledthekernelpointobjectandtheassumedlocationsaredenotedastheconjecturepointobject.Secretservicesarealsointerestedintheroutesaterroristtakestomovefromonerefugetoanother.Theseroutescanbemodeledasasinglevaguelineobject.Someroutescollectedinakernellineobjecthavebeenidentiedwithcertainty.Otherroutescanonlybeassumedtobetakenbyaterrorist;theyaregatheredinaconjecturelineobject.Knowledgeaboutareasofterroristicactivitiesisalsoimportantforsecretservices.Fromsomeareasitiswellknownthataterroristoperatesinthem;wesummarizetheminakernelregionobject.Fromotherareaswecanonlyassumethattheyarethetargetofterroristicactivity;wedenotethemasaconjectureregionobject.Figure 3-2 givessomeexamples.Dark-grayshadedareas,straightcurves,anddark-graypointsindicatekernelparts.Areaswithlight-grayinteriors,dashedlines,andlight-graypointsrefertoconjectureparts.Whiteareasdescribeexteriorparts.Inthissense,manyapplicationscenarioscanbefoundthatcouldleverageourconceptofvaguespatialobjects. 39

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2.1.2 ),whicharedenedasspecialpointsets(Section 2.1.2 ,[ 76 ])andclosedunderthegeometricsetoperations(union),(intersection),(dierence),and(complement).Givenatype2fpoint2D;line2D;region2Dg,thesignaturesoftheoperationsare;;:!and:!.EachtypetogetherwiththeoperationsandformsaBooleanalgebra.Wedenotetheidentityofby1,whichcorrespondstoIR2.Werepresenttheidentityofby0,whichcorrespondstotheemptyspatialobject(emptypointset?).Further,weleveragethethreepointsettopologicalnotionsofboundary(@A),interior(A),andexterior(A)ofaspatialobjectA(seeSection 2.1.4 ).Theuseofanexactmodelforconstructingvaguespatialdatatypesleadstothebenetthatexistingdenitions,techniques,datastructures,andalgorithmsneednotberedevelopedbutcansimplybeusedorintheworstcaseslightlymodiedorextendedasnecessary.Thisleadstothefollowinggenericdenitionofvaguespatialdatatypes: (i)wk\wc=?(ii)wkpoints(w)wkwc(iii)points(w)2Wecallw2v()a(two-dimensional)vaguespatialobjectwithkernelpartwkandconjecturepartwc.Further,wecallwo:=(wkwc)theoutsidepartofw.If=point2Dholds,anelementofv(point2D)=point2Dpoint2D=:vpoint2Discalledavaguepointobject.Correspondingly,anelementofv(line2D)=:vline2Discalledavague 40

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1 (i).Thedisjointednessishereexpressedonthebasisofpointsettopologicalnotions.Wecanalsospecifythisconstraintbymeansoftopologicalclusterpredicatesoncomplexcrispspatialobjects[ 76 ]. 1 (i)intermsoftopologicalclusterpredicatesis: 41

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76 ].Second,thesepredicatenamesareusedindependentlyofaparticulartypecombination.Finally,wegivethedenitionofthecharacteristicfunctionofavaguespatialobject.Thisfunctiondecidesaboutexistenceornon-existenceofapointinavaguespatialobject. 3.3.1 motivatesthembyafewapplicationscenarios.InSection 3.3.2 wegiveaformaldenitionofthevaguespatialsetoperationsunion,intersection,anddierence.Aninterestingaspectisthatwedenethemgenerically,thatis,independentlyoftheunderlyingdatatypes.Section 3.3.3 introducesothergenericvaguespatialoperationsandpredicates.Type-dependentvaguespatialoperationsarediscussedinSection 3.3.4 .Finally,Section 3.3.5 dealswithvaguenumericoperations. 42

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3.4 .Wenowbrieypresenttworeallifeapplicationsandmotivatetheuseofvaguespatialoperations.Therstexampleistakenfromtheanimalkingdom.Weviewthelivingspacesofdierentanimalspeciesanddistinguishkernelpartswheretheymainlyliveandconjecturepartslikeperipheralareasorcorridorswheretheyinparticularhuntforfoodorwhichtheycrossinordertomigratefromonekernelparttoanotherone.Weconsiderthefollowingexamplequeriesregardingtheirlivingspaces:(1)Findtheanimalspeciesthat(partially)sharetheirlivingspaces.(2)Determinehuntersthatpenetrateintothelivingspaceofotheranimals.(3)Ascertaintheareaswheretwospeciescanonlymeetbyaccident.Fortwoanimalspeciesuandv,theinterestingsituationsforthequeriesareillustratedinFigure 3-3 .Thecommontaskofallthreequeriesistocomputethecommonlivingspacesoftwoanimalspecies,thatis,tocalculatetheintersectionoftwovagueregions.Butthenatureoftheintersectionisdierentinallthreecases.Therstqueryasksforanintersectionbetweentwokernelparts(Figure 3-3 (a)).The(non-empty)resultisdenitelyakernelpart.Thesecondqueryamountstoanintersectionbetweenakernelpartandaconjecturepartbutnotbetweentwokernelparts(Figure 3-3 (b)).Sinceaconjecturepartisinvolvedthatdescribesavaguepart,wecannotmakeadenitestatementwhetherthisintersectionbelongstothekernelpart.Itonlyremainstoregard 43

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3-3 (c));itisdenitelyaconjecturepart.Intotal,weseethatthe\strengthofintersection"decreasesfromlefttorightinFigure 3-3 .Weshowlaterthatforanytwovaguespatialobjectsweareabletoexpressboththegeneral,overallintersectionoperationaswellasthespecialintersectionoperationsaddressedintheabovethreequeries.Lastbutnotleast,wementionthattheintersectionofanexteriorpartwithanythingelseisanexteriorpart. Figure3-3. Examplescenarioswithafocusontheintersectionbetweenvagueregions:(a)intersectionoftwokernelparts,(b)intersectionofakernelpartandaconjecturepart,(c)intersectionoftwoconjectureparts. ThesecondexamplereferstoourhomelandsecurityscenariofromSection 3.2.1 .Takingintoaccountspatialvagueness,weareabletoposeinterestingqueries.Wecanaskforthelocationswhereanytwoterroristshavedenitely,perhaps,and/ornevertakenthesamerefuge.Wecandeterminethoseterroriststhatdenitely,perhaps,and/orneveroperatedinthesamearea.Wecancomputethelocationswhereroutestakenbydierentterroristsdenitely,perhaps,and/ornevercrossedeachother.Wecanndoutthesphereofanumberofterroristsonthebasisoftheirlocationsasavagueconvexhull.Manyfurtherapplicationscenariosandqueriesareconceivable.Vagueconceptsoeragreaterexibilityandmorenuancesformodelingandcomputingpropertiesofspatialphenomenaintherealworldthandeterminate\blackorwhite"conceptsdo.Still,vagueconceptscomprisethemodelingpowerofdeterminateconceptsasaspecialcase. 44

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, ,or denotesthekernelpart,conjecturepart,oroutsidepartofuandwrespectively.Eachentryofthetabledenotesapossiblecombination,i.e.,intersection,ofkernelparts,conjectureparts,andoutsidepartsofbothobjects,andthelabelineachentryspecieswhetherthecorrespondingintersectionbelongstothekernelpart,conjecturepart,oroutsidepartoftheoperation'sresultobject. Table3-1. Componentsresultingfromtheintersectionofkernelparts,conjectureparts,andoutsidepartsoftwovaguespatialobjectswitheachotherfor(a)union,(b)intersection,(c)dierence,and(d)complement.union Theunion(Table 3-1 (a))ofakernelpartwithanyotherpartisakernelpartsincetheunionoftwovaguespatialobjectsasksformembershipineitherobjectandsincemembershipisalreadyassuredbythegivenkernelpart.Likewise,theunionoftwoconjecturepartsortheunionofaconjecturepartwiththeoutsideshouldbeaconjecturepart,andonlythepartswhichbelongtotheoutsideofbothobjectscontributetotheoutsideoftheunion.Theoutsideoftheintersection(Table 3-1 (b))isgivenbyeitherregion'soutsidebecauseintersectionrequiresmembershipinbothregions.Thekernel 45

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3-1 (d))ofthekernelpartshouldbetheoutside,andviceversa.Withrespecttotheconjecturepart,anythinginsidethevaguepartofanobjectmightormightnotbelongtotheobject.Hence,wecannotdenitelysaythatthecomplementofthevaguepartistheoutside.Neithercanwesaythatthecomplementbelongstothekernelpart.Thus,theonlyreasonableconclusionistodenethecomplementoftheconjectureparttobetheconjecturepartitself.Thedenitionofdierence(Table 3-1 (c))betweenuandwcanbederivedfromthedenitionofcomplementsinceitisequaltotheintersectionofuwiththecomplementofv.Thatis,removingakernelpartmeansintersectionwiththeoutsidewhichalwaysleadstooutside,andremovinganythingfromtheoutsideleavestheoutsidepartunaected.Similarly,removingaconjecturepartmeansintersectionwiththeconjecturepartandthusresultsinaconjecturepartforkernelpartsandconjectureparts,andremovingtheoutsideofw(i.e.,nothing)doesnotaectanypartofu.Motivatedbythejustinformallydescribed,intendedsemanticsforthefouroperations,wenowdenethemformally.Aninterestingaspectisthatthesedenitionscanbebasedsolelyonalreadyknowncrispgeometricsetoperationsonwell-understoodexactspatialobjects.Hence,weareabletogiveexecutablespecicationsforthevaguegeometricsetoperations.Thismeans,ifwehavetheimplementationofacrispspatialalgebraavailable,wecandirectlyexecutethevaguegeometricsetoperationswithoutbeingforcedtodesignandimplementnewalgorithmsforthem(seeSection 4.4 ). 46

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4 (ii)theintersectionoftwovaguelineobjectsyieldsagainavaguelineobject.Thisisincontrasttomostdenitionsofitscrispcounterpartwhichisusuallyspeciedas:line2Dline2D!point2D[ 73 ].However,ourspecicationstressestheaspectsofgenericdenition,maintenanceofclosureproperties,andconsistencyamongthethreeinstancesoftheintersectionoperation.Anoperationnamedcommon pointsthatcomputesthesharedpointsoftwovaguelinesisintroducedinSection 3.3.3 .ThefollowinglemmashowsthatthespecicationsofDenition 4 ttothespecicationsinTable 3-1 . 4 realizethebehaviorofandareconsistenttothespecicationsinTable 3-1 .Proof.AccordingtoTable 3-1 (a),forz=uunionwinDenition 4 (i),wehavetoshowthethreeidentities (1)zk=ukwkukwcukwoucwkuowk(2)zc=ucwcucwouowc(3)zo=uowoTheprooffor(1)leveragesthatisidempotent.Wecanthereforeduplicatethersttermukwk.Then,usingthefactthatdistributesover,wecanfactorizebothukandwkandobtain: 47

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4 (i)as: (ucwcuc(wc)(uc)wc)(ukwkuk(wk)(uk)wk)Nextweevaluateallcomplementsbyusingthatwc=wkwoandwk=wcwo.Thisleadsto (ucwcuc(wkwo)(ukuo)wc)(ukwkuk(wcwo)(ucuo)wk)Applyingdistributivityofweobtain: (ucwcucwkucwoukwcuowc)(ukwkukwcukwoucwkuowk)Inthisterm,onlyucwkandukwcappearinbothpartsofthedierence;allotherintersectionstobesubtractedhavenoeectatallsinceallintersectionsarepairwisedisjointduetothedenitionofvaguespatialobjects.Weobtain: ((ukuc))((wkwc))whichisbythedenitionofcomplementequaltouowo,theconditionrequiredforzo. 48

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3-1 (b),forz=uintersectionwinDenition 4 (ii),wehavetoshowthethreeidentities (1)zk=ukwk(2)zc=ucwkucwcukwc(3)zo=uowkuowcuowoukwoucwoTherightsidesoftheequations(1)and(2)corresponddirectlytothespecicationsofthekernelpartandtheconjecturepartofintersection.Forprovingequation(3)weduplicatethetermuowo,factorizebothuoandwo,andsimplifytheequationto 3-1 (c),forz=udierencewinDenition 4 (iii),wehavetoshowthethreeidentities (1)zk=ukwo(2)zc=ukwcucwcucwo(3)zo=ukwkucwkuowkuowcuowoWithwo=(wkwc),therightsidesoftheequations(1)and(2)corresponddirectlytothespecicationsofthekernelpartandtheconjecturepartofdierence.Forprovingequation(3)weduplicatethetermuowk,factorizebothwkanduo,andsimplifytheequationto 49

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3-1 (d),forz=complementuinDenition 4 (iv),wehavetoshowthethreeidentities (1)zk=uo(2)zc=uc(3)zo=ukSinceuo=(ukuc)holds,therightsideofequation(1)correspondstothespecicationofthekernelpartofcomplement.Therightsideofequation(2)isequaltotheconjecturepartofcomplement.Forprovingequation(3)wehavebydenitionthat 3.3.2 ,wehavedenedtheoperationintersectionfortwovaguespatialobjectsofthesametype.Weextendthisdenitionnowtoallmixedtypecombinationssuchthattwovaguespatialobjectshaveadierentdimension.Thisallowsus,e.g.,todeterminethecomponentsofavaguelinethatintersect(i.e.,liein)avagueregion,orthecomponentsofavaguepointlocatedonavagueline.Theresultisalwaysanobjectofthelowerdimension.Let\
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4 (ii)forintersectioncanthenalsobeappliedincaseofthemixedtypecombinations.Sometimesitishelpfultobeabletoexplicitlydealwiththekernelpartortheconjecturepartofavaguespatialobject,ortoswapitskernelpartandconjecturepart. (i)kernel(u):=(uk;0)(ii)conjecture(u):=(0;uc)(iii)invert(u):=(uc;uk)Allthreeoperationshavethesignaturev()!v()andproduceaspecialvaguespatialobject.Thekerneloperationsuppressestheconjecturepartofavaguespatialobjectuandfacilitatescomputationsexclusivelywiththeexactpart.Vaguespatialoperations,appliedtovaguespatialobjectswithanemptyconjecturepart,behaveexactlylikethecorrespondingcrispspatialoperations.Thiscanbeeasilyseenfromthedenitionsandisintended.Consequently,crispspatialobjectsareaspecialcaseoftheircorrespondingvaguecounterparts.Theconjectureoperationallowsonetofocusontheindeterminatepartofu.Theinvertoperationchangestheroleofkernelpartandconjecturepartofu.Thefollowingdenitiongivesaccesstothecomponentsofvaguespatialobjectsandenablesstructuralcomparisons. (i)k-proj(u):=uk(ii)c-proj(u):=uc(iii)u=w:=(uk=wk^uc=wc)(iv)u6=w:=(uk6=wk_uc6=wc)Theprojectionfunctionsk-projandc-projhaveboththesignaturev()!andmakeitpossibletoconnectvagueandcrispspatialdatatypesandtomapthekernelpartortheconjecturepartofavaguespatialobjecttothecorrespondingcrispspatialobject. 51

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2.1.4 . 7 (i)to(iv))orproduceanobjectofhigherdimension(Denition 7 (v)to(viii)).Tokeepthebalancebetweenkernelpartandconjecturepart,eachoperationisavailableintwoversionsinwhicheitherthekernelpartortheconjecturepartoftheargumentobjectdominates.Thestructureofthedenitionsisrathersimilarforboththe\kernelversions"andthe\conjectureversions"ofalloperations.Theprex\k-"usedforthekernelversionsbelowcanbeomittedforreasonsofconvenience. (i)k-vertices(m):=(vertices(lk);vertices(lc)vertices(lk))(ii)c-vertices(m):=(vertices(lk)vertices(lc);vertices(lc))(iii)k-boundary(r):=(boundary(rk);boundary(rc)boundary(rk)))(iv)c-boundary(r):=(boundary(rk)boundary(rc));boundary(rc))(v)k-interior(l):=(interior(lk);interior(lc)interior(lk))(vi)c-interior(l):=(interior(lk)interior(lc);interior(lc))(vii)k-convex hull(p):=(convex hull(pk);convex hull(pkpc)convex hull(pk))(viii)c-convex hull(p):=(convex hull(pkpc)convex hull(pc);convex hull(pc))Theoperationsk-verticesandc-verticeswiththesignaturesvline2D!vpoint2Dandvregion2D!vpoint2Dmakeuseoftherepresentationofvaguelinesandvagueregions 52

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Examplesforthevaguespatialoperations:(a)k-boundary,and(b)c-interior. aslinearapproximations;theycollecttheendpointsofthesegmentsofavaguelineandtheendpointsofthesegmentsoftheboundaryofavagueregionrespectively.Theirdenitionisbasedonthecrispoperationverticeswiththesignaturesline2D!point2Dandregion2D!point2D.Sincesegmentsofthekernelpartandtheconjecturepartofavaguelinemayshareboundaryendpoints,wehavetodecidewhetherthesecommonpointsaresupposedtobelongtotheinteriorofthekernelpart(operationk-vertices)ortheinterioroftheconjecturepart(operationc-vertices)oftheresulting,newvaguepointobject.Thisdecisionhastobemadesince,accordingtoDenition 1 (i),theintersectionoftheinteriorsofthekernelpartandtheconjecturepartofthesamevaguespatialobjectmustbeempty.Theoperationsk-boundaryandc-boundarywiththesignaturevregion2D!vline2Dallowustoextractthevagueboundaryofavagueregionrasavaguelinelnewwithexclusivelyclosedlinecomponents.Theirdenitionrequiresthecrispoperationboundary:region2D!line2Dwhichdeterminestheboundaryofacrispregionobjectandrepresentsitasaclosedcrisplineobject.Forthesamereasonasbefore,sincethekernelpartandtheconjecturepartofavagueregionmaysharecommonboundarylineswhichnowbecomepartoftheinterioroflnew,wemusttakethedecisionwhetherthesecommonlinesaresupposedtobelongtotheinteriorofthekernelpart(operationk-boundary(Figure 3-4 (a)))ortheinterioroftheconjecturepart(operationc-boundary)oflnew.Cycleslocatedwithinothercyclesaremaintained.Viceversa,theoperationsk-interiorandc-interiorwiththesignaturevline2D!region2Didentifycyclesinavaguelineobjectandtransformthemintoavagueregion. 53

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3-4 (b)))ofthenewvagueregion.Thismayalsoresultinholes.Theoperationsk-convex hullandc-convex hullwiththesignaturevpoint2D!vregion2Dcomputeaconvexhullofavaguepointobject.AsubsetSoftheplaneiscalledconvexif,andonlyif,foranypairofpointsp;q2SthelinesegmentbetweenpandqiscompletelycontainedinS.Thewellknowncrispoperationconvex hull:point2D!region2Dcomputesthesmallestconvexregionthatcontainsallpointsofagivenpointobject.Theoperationk-convex hull(c-convex hull)computesthesmallestconvexhullofavaguepointobjectp,asitisgivenbytheconvexhullofitskernelpart(conjecturepart).Ifallpointsoftheconjecturepart(kernelpart)ofplieinsideorontheboundaryoftheconvexhullofitskernelpart(conjecturepart),theconjecturepart(kernelpart)oftheresultingvagueregionis0,i.e.,theemptyregionobject.Otherwise,theconvexhullinvolvingallpointsbothfromthekernelpartandtheconjecturepartwillbelargerthantheconvexhullofthekernelpart(conjecturepart).Ingeneral,k-convex hull(p)6=c-convex hull(p)holds. Figure3-5. Examplesoftheconvexhullofvaguepoints:(a)vaguepointp,(b)k-convex hull(p),(c)c-convex hull(p). 54

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8 ).Webeginwiththeoperationcommon pointswiththesignaturevline2Dvline2D!vline2D.Itcomputesthepointintersectionsbetweentwovaguelineobjects(seethediscussioninSection 3.3.2 )andleveragesthecrispoperationcommon points:line2Dline2D!point2Dthatdeterminestheintersectionpointsoftwocrisplineobjects.Theoperationcommon bordercomplementstheoperationintersectionappliedtotwovaguelineobjectsandcomputesthesharedboundaryoftwoextendedvaguespatialobjectswhereatleastoneoperandisavagueregionobject;theresultisavaguelineobject.Itspossiblesignaturesarevline2Dvregion2D!vline2D,vregion2Dvregion2D!vline2D,andvregion2Dvregion2D!vline2D. points(l;m):=(common points(lk;mk);common points(lc;mc)common points(lk;mc)common points(lc;mk))(ii)common border(l;r):=intersection(l;k-boundary(r))(iii)common border(r;l):=common border(l;r)(iv)common border(r;s):=intersection(k-boundary(r);k-boundary(s)) 73 ].However,wewillseeinthissubsectionthatvaguenumericoperationsturnouttobemorecomplicatedthanperhapsexpected.Thereasonisthatthevaguecharacterofthespatialargumentobjectsleadstovaguenumericalresults.Forexample,theareaofavagueregionisatleastequaltotheareaofthekernelpartandatmostequaltotheareaofthekernelpartandtheconjecturepart,thatis,itissomewhereinbetween.Wecannotexpressthevagueareavaluebyasinglescalarnumberonly.Butanappropriaterepresentationcouldbeanintervalgivenbytheminimumandmaximumareavalues.However,thisleadstothe 55

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56 ]asaseparatealgebraintoDBMSs.Further,intervalarithmeticcomesatapricesinceitcanbeslowandoftengivesoverlypessimisticresultsforreal-worldcomputations.InourcaseofVASA,theintroductionofintervalarithmeticwouldleadtoalargeoverhead.Wethusdecidetokeepthingssimpleandinsteaddenetwooperationsforcomputingthelowerboundandtheupperboundofsuchintervals.Thisenablesustokeepordinarynumericoperations.Werstdiscussseveralunaryvaguespatialoperatorsyieldingnumericalvalues.TheoperationsinDenition 9 existbothina\lowerbound"versionandan\upperbound"version.Theoperationpairmin-length/max-lengthwiththesignaturevline2D!realcomputestheminimal/maximallengthofavaguelineobject.Theminimallengthisgivenbythekernelpartsincethislengthisguaranteed.Themaximallengthisboundedbythelengthofthekernelpartandtheconjecturepart.Thedenitionmakesuseofthecrispoperationlength:line2D!realwhichcalculatesthelengthofacrisplineobject.Similarly,theoperationpairmin-area/max-areawiththesignaturevregion2D!realcomputestheminimal/maximalareaofavagueregionobject.Againthekernelpartandtheunionofkernelpartandconjecturepartdeterminethelowerandupperbounds.Thedenitionleveragesthecrispoperationarea:region2D!realwhichcomputestheareaofacrispregionobject.Inthesameway,theoperationpairmin-diameter/max-diameterwiththesignature!real,2fvpoint2D;vline2D;vregion2Dg,determinesthediameterofavaguespatialobject.Thediameterofaspatialobjectisthelargestdistancewithintheobject'sextent.Thedenitionusesthecrispoperationdiameter:!real,2fpoint2D;line2D;region2Dg,whichperformsthesametaskforcrispspatialobjects.Theoperationpairmin-no of comp/max-no of compwiththesignaturevpoint2D!realyieldstheminimalandmaximalnumberofcomponentsofavaguepointobject.The 56

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of comp:point2D!realreturningthenumberofpointsofacrisppointobject. (i)min-length(l):=length(lk)(ii)max-length(l):=length(lklc)=length(lk)+length(lclk)(iii)min-area(r):=area(rk)(iv)max-area(r):=area(rkrc)=area(rk)+area(rc)(v)min-diameter(t):=diameter(tk)(vi)max-diameter(t):=diameter(tktc)(vii)min-no of comp(p):=no of comp(pk)(viii)max-no of comp(p):=no of comp(pk)+no of comp(pc)Thereareotherusefuloperationsoncrispspatialobjects,however,forwhichageneralizationtothevaguecaseissurprisinglynotquitesosimpleorevenimpossible.Weillustratethisforthecomputationofthenumberofconnectedcomponentsforvaguelinesandvagueregionsaswellasforthecalculationoftheperimeterofavagueregion.Inallthesecases,wecanneitherdenealowerboundversionnoranupperboundversionofsuchanoperation.Anoperationreturningthenumberofconnectedcomponentsofavaguelineobjectoravagueregionobjectdependsheavilyonthesemanticsoftheconjecturepart.Sincelineandregionobjectsneednotbeconnected,theirconjectureparts,whichexpressa\possiblyline"and\possiblyregion"-semanticsrespectively,mightwellallowseveralunconnectedcomponents(comparetoDenition 1 ),whichareusuallycalledblocksandfacesrespectively.Figure 3-6 showsapossibleinstanceofavagueline(a)andavagueregion(b)anddemonstratestheunpredictabilityofthenumberofcomponents.Hence,wecannotgiveanupperboundonthenumberofcomponents.But,ingeneral,wecannotevengiveanon-triviallowerboundeither(e.g.,thenumberofkernel 57

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3-7 theminimalnumberofcomponentsis1althoughtherearethreekernelpartfaces.Ifweconsiderthecalculationoftheperimeterofavagueregion,inarstapproachonecouldbeattemptedtodenelowerboundandupperboundversionssimilartothedenitionsofareaorlength.This,however,mightleadtowrongresults.Wehaveindicatedthattheconjecturepartcanbethoughtofasdescribingpossiblelocationsoftheregion'scontour(andthustheregionitself).Butthenwecannotgiveanyupperboundonthelengthofsuchaboundarycontour.Inparticular,theactualcontourmightbemuchlongerthantheperimeteroftheconjectureregion.Figure 3-8 (a)illustratesthissituation.Moreover,wecannotsimplytaketheperimeterofthekernelpartastheminimalperimeter.Figure 3-8 (b)showsthis.Usually,holescontributetotheperimeterofaregion.If,e.g.,akernelpartofavagueregionvcontainsaholewhichisequaltotheconjecturepartofv,theperimeteroftheholeisnotcountedfortheperimeterofthemaximalpossibleregion.InFigure 3-8 (b),theminimalpossibleregionhasthemaximalpossibleperimeterlength(C)+length(c)whereasthemaximalpossibleregionhastheminimalpossibleperimeteroflength(C).Next,wediscussseveralbinaryvaguespatialoperatorsyieldingnumericalvalues.Forthesamereasonsasdiscussedabove,theseoperatorsexistbothinalowerboundversionandanupperboundversion(Denition 10 ).Theyallrepresentdistanceoperationsandhavethecommonsignaturev()v()!realwith;2fpoint2D;line2D;region2Dg,thatis,theycanhaveanycombinationofvaguespatialdatatypesasoperandtypes.The Figure3-6. Examplesshowingtheimpossibilitytodetermineanupperboundforthenumberofcomponentsof(a)avagueline,and(b)avagueregion. 58

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Examplevagueregionobjectwherethekernelpartconsistsofthreecomponentsbuttheminimalnumberofcomponentsis1 Figure3-8. Exampleshowingimpossibilitytodeterminethe(a)upperbound,or(b)thelowerboundoftheperimeterofavagueregion. operationsmakeuseofthewellknownandhighlyoverloadedcrispdistancefunctionsmindist;maxdist:!real.Theminimum(maximum)distancefunctionconceptuallydeterminesthosetwopointsoftwogivenspatialobjectsthatarenearestto(farthestawayfrom)eachotherandcomputestheirdistance.Ifweconsidertheaspectofminimumdistancerst,anupperboundisobtainedbythedistancebetweenthekernelpartsoftwovaguespatialobjects,sayuandv.Thatis,wearesurethatthedistanceisatmostthedistancebetweenthekernels(operationmax-min-dist).Thedistancemightbesmaller,butitisatleastaslargeasthedistancebetweenthemaximalextensionsofuandv.Inotherwords,theminimumdistanceisgivenbythedistancetakingkernelpartsandconjecturepartsofuandvintoaccount(operationmin-min-dist).Ifmax-min-dist(u;v)=min-min-dist(u;v)holds,wecanconcludethattheminimumdistanceisexactsincetheconjecturepartshavenoeectonthedistancecomputation.Sincetheminimumdistancecomputationistheusualinterpretationofadistancefunctionbetweenspatialobjects,wespecifytheshortcutsmin-distformin-min-distandmax-distformax-min-dist. 59

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(i)min-min-dist(u;v):=mindist(ukuc;vkvc)(ii)max-min-dist(u;v):=mindist(uk;vk)(iii)min-max-dist(u;v):=maxdist(uk;vk)(iv)max-max-dist(u;v):=maxdist(ukuc;vkvc) 3.4.1 givesamotivationforvaguetopologicalpredicatesandillustratessomeoftheirproperties.Fortheformalderivationanddenitionofthesepredicates,weemployathree-stepmethod.InSection 3.4.2 ,werstgiveanoverviewofthismethod.Theremainingsubsectionsdetailitsthreesteps.TherststepleveragesaconceptforrepresentingvaguetopologicalpredicatesbymeansofcrisptopologicalpredicatesandisdescribedinSection 3.4.3 .ThesecondstepdelineatedinSection 3.4.4 explicitlyidentiesallpossiblerepresentationsforvaguetopologicalpredicates.ThethirdandlaststeppresentedinSection 3.4.5 givessemanticstothepossiblerepresentationsandresultsinasetofvaguetopologicalpredicates. 60

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2.1.4 wehaveseenthatatopologicalrelationshipcharacterizestherelativepositionoftwospatialobjects.Iftheseobjectsarecrisp,topologicalpredicatesoperatingonthemcanpreciselyanswerquestionslike\DostatesAandBshareacommonborder?"or\Dotworoadscross?"bymeansofabinarydecision.Clearlyansweringsuchquestionsturnsouttobemorechallengingwhendealingwithvaguespatialobjects.Forillustrationpurposes,wepresenttwoexamplesbasedonsituationsdescribedinSection 3.2.1 .Afteranoilspill,authoritiesmayneedtomakedecisionsonhowtoactdependingonwhetherthepredictedareaoftheoilspillcertainlyoverlapswithoriscertainlydisjointfromaknowncoralreef.Letusagainassumethattheoilspillareaandthereefaremodeledasvagueregionsandfurtherthatthereefhasanemptyconjecturepart.Theauthoritiesmayneedananswertothequestion\Istheoilspilldisjointfromthecoralreef?"Theanswertosuchaquestionmightnotbeclearduetotheexistenceofaconjecturepartintheoilspillrepresentation.Ifthekernelpartofthespillregionisdisjointfromthereefbuttheconjecturepartisnot(asillustratedinFigure 3-9 (a)),wecansaythatmaybethetwovagueregionsaredisjoint,maybetheyarenot.Incaseofmappingterroristroutesusingvaguelines,authoritiesmightneedananswertothequestion:\HasterroristXmetterroristY?"Asimplelookattheirroutesmightgiveusaclearanswer,oritmightnot.IftheroutesofXandYmeetorintersectatapointthatbelongstothekernelofbothroutes,thenXandYsurelymet.Iftherearenointersectionpointsbetweentheroutes,thentheysurelydidnotmeet.Butiftheirroutesintersectatapointthatisaconjecturepointofeitherorbothroutes(asillustratedinFigure 3-9 (b)),thenmaybetheymet;inthiscasethereisnoclearanswertothequestion.Theexamplesprovideddemonstratethatatwo-valued,Booleanlogicisinsucientfordeterminingthetopologicalrelationshipsbetweenvaguespatialobjects.Weemployathree-valuedlogicthat,inadditiontothetruthvaluestrueandfalse,includesavaluemaybefortakingintoaccounttheaspectofvagueness.Avaguetopologicalpredicate 61

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Twoexamplescenariosillustratingvaguetopologicalpredicates.(a)Vagueregionsrepresentinganoilspillandacoralreef.Theblackareaindicatestheintersectionoftheconjecturepartofthespillwiththekernelpartofthereef.(b)VaguelinesasroutesofsuspectedterroristswheretheconjecturepartoftherouteofXintersectswiththekernelpartoftherouteofY. returnsthenoneofthesethreetruthvalues.Itreturnstrueiftherelationshipdenitelyholds,falseiftherelationshipdoesdenitelynothold,andmaybeifthereisnotenoughinformationtomakeacleardecision.Asanexample,weassumethattwovaguetopologicalpredicatesbetweenvagueregionsarenamedoverlapandequal.Appliedtotheoilspillandreefexample,weobtainoverlap(oil spill;reef)=maybeandequal(oil spill;reef)=false.Itisclearthatthetwoobjectsoil spillandreefwillneverbetopologicallyequalsincetheirkernelpartsaredisjoint.Itisnotclearthoughwhetherornottheobjectsoverlap,asthisdependsontheactualextensionofthesofarunknownoilspill. 3-10 givesanoverviewofourgeneralmethodofdeterminingtopologicalpredicatesonvaguespatialobjects.Asarstinputparameter,themethodrequiresanycombinationofvaguespatialdatatypesv()andv(),whicharebuiltfromthecrispspatialdatatypesandbythetypeconstructorv(Section 3.2.2 ).ThesecondinputparameteristhecompleteandmutuallyexclusivecollectionT;oftopologicalrelationshipsbetweenthetypesand[ 76 ]. 62

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Thegeneralmethodfordeterminingvaguetopologicalpredicates. Ourmethodincorporatestwomainsteps.Therststepisdividedintotwotasks.Initially,wespecifyacharacterizationofvaguetopologicalrelationshipsthatmakesexclusiveuseoftheunderlyingcrispspatialdatatypes,theirtopologicalpredicates,andgeometricsetoperations.Theideaistocharacterizethevaguetopologicalrelationshipthatholdsbetweentwovaguespatialobjectsbymeansofthecrisptopologicalrelationshipsthatholdbetweentheirkernelandconjectureparts.WedescribethischaracterizationtechniqueindetailinSection 3.4.3 .Unfortunately,notallcharacterizationsthatcanbesyntacticallyformedaresemanticallyvalid.Therefore,wesubsequentlyproceedwiththeidenticationofthecompletecollectionofuniquecharacterizationsthatarevalid,crisprepresentationsofthetopologicalrelationshipsbetweentwovaguespatialobjects.Wedenotethemembersofthiscollectionascharacterizationpredicates.TheidenticationprocessisdiscussedinSection 3.4.4 .Thecharacterizationandidenticationstepiscrucialasitprovidesthestructuresfromwhichinformationcanbeextractedtoderivethetopologicalrelationshipbetweentwovaguespatialobjects.Iftherepresentationisnotcompleteenough,thenmanydierentspatialcongurationswillhaveequivalentrepresentationsandremainindistinguishablefromeachother.Ontheotherhand,iftherepresentationinvolvestoomuchinformation,redundancymightaddunnecessarycomplexitytothedenitionandlatertotheimplementationofvaguetopologicalpredicates.Thesecondstepofourmethodistheinterpretationstep,whichwedetailinSection 3.4.5 .Duringthisstep,thesemanticsofthekernelandconjecturepartsaretaken 63

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3-10 .Thissectiongivesananswertothequestionwhichtopologicalrelationshipsbetweentheindividualkernelandconjecturepartsoftwovaguespatialobjectswemusttakeintoaccounttoadequatelyrepresentthetopologicalrelationshipbetweenthevaguespatialobjectsthemselves.WehaveseeninFigure 3-9 thatwecannotappropriatelyrepresentthetopologicalrelationshipsbetweenvaguespatialobjectsbyonlytakingintoaccounttheirkernelpartsastheirlowerapproximations.Instead,wealsohavetoconsidertheupperapproximationofeachobject,thatis,thekernelpartandtheconjectureparttogether.Asaresult,wecharacterizethetopologicalrelationshipoftwovaguespatialobjectsbydeterminingthetopologicalrelationshipsofthelowerandupperapproximationsofonevagueobjectwiththelowerandupperapproximationsoftheothervagueobjectrespectively.ForthescenarioinFigure 3-9 (a),thecharacterizationoftherelativepositionoftheoilspillandthereefresultsinthefollowingconjunctionofcrisptopologicalpredicates:disjoint(oil spillk;reefk)^overlap(oil spillkoil spillc;reefk).EachelementintheconjunctionisatopologicalpredicatefromthesetT;ofpredicatesthatoperateonobjectsofthecrispspatialdatatypesand.Wedenotesuchaconjunctionofcrisptopologicalpredicatesasacharacterizationpredicate.Asashortcut,werepresentitasthen-tupleofncrisptopologicalpredicatescontainedinitsconjunction.Intheexampleabove,itisthe2-tuple(disjoint;overlap). 64

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3-2 listsall16possiblecombinationsofnonemptyandemptykernelandconjecturepartsfromtwovaguespatialobjectsanddeterminesforeachcombinationavalidconjunctionofcrisptopologicalpredicatesthatoperateonthenonemptypartsoftheiroperandobjects.Thisconjunctionservesasthedenitionofthecharacterizationpredicateforsuchapairofvaguespatialobjects.Row1ofthetableformsthebasisfortheconstructionofallcharacterizationpredicates.Itrepresentsthestandardcaseinwhichallkernelandconjecturepartsarenonempty.Hence,thecharacterizationpredicateincludesthecrisptopologicalpredicatesbetweenthelowerapproximations(kernelparts),betweentheupperapproximations(kernelpartsandconjecturepartstogether),andbetweenthecombinationsoflowerandupperapproximationsofbothobjects.Theresultingcharacterizationpredicateisrepresentedbythe4-tuple(p;q;r;s).Allothercharacterizationpredicatesarereductionsofthisconjunctionoffourtopologicalpredicates.AreductionisperformedinthesensethattopologicalpredicatesthatwouldyieldfalseaccordingtoDenition 11 areremovedfromthisconjunction.Forexample,thedierencebetweenrow9androw1isthatAk=?holdsinrow9.Forthe4-tupleofrow1thismeansthatp(Ak;Bk)=p(?;Bk)=falseandr(Ak;BkBc)=r(?;BkBc)=falsehold.Hence,thepredicatespandrareremoved,andweobtainthecharacterizationpredicateq(Ac;Bk)^s(Ac;BkBc). 65

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3-9 wherethereefhasanemptyconjecturepart).Theircharacterizationpredicatesaredenotedbythepairs(p;q)and(p;r)respectively.Row6representsascenarioinwhichbothobjectshavenonemptykernelpartsandemptyconjectureparts.Thiseectivelyrenderstheobjectscrisp.Thus,wedenotethecharacterizationpredicateasthecrisptopologicalpredicatepthatholdsbetweentheirkernelparts.Rows3and9describescenariosinwhichexactlyoneoftheobjectshasanemptykernelpart.Insuchacase,thecharacterizationpredicatesincludethecrisptopologicalpredicatesbetweentheupperapproximationoftherstobject,whichishereitsconjecturepartonly,andthelowerandupperapproximationsoftheotherobject.Thecharacterizationpredicatesarerepresentedbythepairs(q;s)and(r;s).Rows7and10handlethesituationswhereoneobjecthasonlyanonemptykernelpartandtheotherobjecthasonlyanonemptyconjecturepart.Theresultingconjunctioncontainsasingletopologicalpredicatebetweenthelowerapproximationofoneobjectandtheupperapproximationoftheotherobject,whichishereitsconjecturepartonly.Thecharacterizationpredicatesarerepresentedbythetopologicalpredicatesrandqrespectively.Row11representsascenarioinwhichbothobjectshaveemptykernelpartsandnonemptyconjectureparts.Thecharacterizationpredicateofsuchasituationisdenedasthesingleelementtuplesthatdescribesthetopologicalrelationshipsbetweentheupperapproximations(hereconjecturepartsonly)ofbothobjectsinvolved.Finally,rows4,8,and12to16dealwiththosesituationsforwhichthereisnovalidconjunctionofcrisptopologicalpredicatesthatcanbeusedtodenethecharacterizationpredicates.ThisisduetothefactthatinallthesecaseseitherAorBistheemptyvaguespatialobject(0,0).Thelowerandupperapproximationsofanemptyvaguespatialobjectarethereforetheemptycrispspatialobject0,andaccordingtoDenition 11 there 66

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DenitionofcharacterizationpredicatesonthebasisofthenonemptinessandemptinessofthekernelandconjecturepartsoftwovaguespatialobjectsAandBandthecrisptopologicalpredicatesp;q;r;s2T;.Row 1 =?6=?6=?6=? =?6=?6=?=? =?6=?=?6=? =?6=?=?=? =?=?6=?6=? =?=?6=?=? =?=?=?6=? =?=?=?=? 3-2 representsavalidcharacterizationpredicate.Thisidenticationprocessisnecessaryduetopossiblesemanticalcontradictionsbetweendierenttopologicalpredicatesinthesameconjunction.Forexample,forrow5inTable 3-2 letussetp=overlapandr=disjoint.Weobtainthetermoverlap(Ak;Bk)^disjoint(Ak;BkBc).Thisisobviouslyacontradictionsincedisjoint(Ak;BkBc))disjoint(Ak;Bk)6=overlap(Ak;Bk)holds. 67

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3.4.3 ,thetuplesandpairsdenedascandidatecharacterizationpredicatesrepresentconjunctionsofbooleanexpressions.Eachelementoftheconjunctionisatopologicalpredicate.Inorderforanyofthesecandidatestorepresentavalidcharacterizationpredicate,theremustexisttwovaguespatialobjectsforwhichtheconjunctionholds.Ifnosuchpairofvaguespatialobjectsexist,theconjunctionisnotsatisableanddoesnotrepresentavalidcharacterizationpredicate.TheproblemofshowingsatisabilityofsuchaconjunctivebooleanexpressioncanbemodeledasabinaryConstraintSatisfactionProblem(CSP)oraBinaryConstraintProblem(BCP)forshort. 50 ].BCPsareoftenrepresentedandsolvedbyusingbinaryconstraintnetworks(BCN).ABCNisalabeleddigraphwherethenodesrepresentthevariablesoftheBCPandtheedgesrepresenttheconstraints(directedrelations)betweenthevariables.AmethodforshowingwhetheraBCPissolvableornotusesconstraintpropagationinordertoenforcepathconsistencyoftheBCN.Constraintpropagationisusedtochangeaproblem(i.e.,simplifyingit)withoutchangingitssolutionbyemployinginformationfromcurrentconstraintstostrengthenotherexistingconstraintsorinfernewconstraints.Constraintpropagationalsoworksbyreducingthesetofvalueseachvariablecantake,thusembeddingconstraintsinthevariables'domains[ 53 ].PathconsistencyisapropertyofapairofvariablesAandCofaBCNthataresaidtobepathconsistentwithathirdvariableBifforeverypairofvaluesa2D(A) 68

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81 ].IfFisarelationdenedoverxandy,andGisarelationdenedoveryandz,thecompositionofFandG,denotedF;G(orFG)denestherelationHoverxandz.WecanalsosaythatF(x;y);G(y;z)=H(x;z).Noticethattheoperationdescribestherelationbetweenxandzrelativetothecommonvariableintheoperands,inthiscasey.Fortwosetsofrelations,I;(setofrelationsbetweenelementsinthedomainsand)andJ;,theircompositiontableisdenedasatablewithjI;jrows(oneforeachelementi2I;)andjJ;jcolumns(oneforeachelementj2J;).Theelementinthetablecorrespondingtothecolumnforsomeiandtherowforsomejcontainstheresultofi;jwhichrepresentstherelationovertwoelementsfromthedomainsand.Inessence,acompositiontableisusedtorepresenttheresultofcomputingthecompositionofallpairsofelementsfromtworelatedsetsofrelations.Forthesolutionofourproblem,weareinterestedspecicallyincomputingthecompositionoperationontopologicalrelationsbetweencomplexcrispspatialobjects.Wecomputethisoperationonthebasisofasetofinferencerulesthatoperateonthe9-intersectionmatricesthatrepresenttherelationships.Giventwomatrices,theserules 69

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28 ]tooperateontopologicalrelationshipsbetweensimpleregions.However,becausetherulesaredenedasgeneralinferencesoverpointsets,theycanbeappliedtocomputethecompositionoftopologicalrelationshipsbetweenalltypecombinationsofcomplexspatialobjects.Table 3-3 showsthecompositiontableforthetopologicalrelationshipsbetweencomplexpoints.Fromthetable,wecanobservethatthecompositionofsomepairsoftopologicalrelationshipsbetweencomplexpointsresultsinmorethanonerelation(e.g.,disjoint;overlap=fdisjoint;inside;overlapg).Wesaythatforthesecases,thecompositioncanbeonlypartiallyinferredfromtheoperandrelations.Theresultofthecompositionisasetofrelationsthatisreadasadisjunctionoftherelationsthatareincludedintheset.Therefore,wecansaythatforthreecomplexpointsA,BandC:disjoint(A;B);overlap(B;C)=disjoint(A;C)_inside(A;C)_overlap(A;C). Table3-3. Thecompositiontablefortopologicalrelationshipsbetweencomplexpoints. eq=equal in=inside ct=contains op=overlap di fdig fdi;in;opg fdig fdi;in;opg feqg fing fctg fopg fing fing fdi;eq;in;ct;opg fdi;in;opg fctg feq;in;ct;op fct;opg fopg fin;opg fdi;ct;opg fdi;eq;in;ct;op 50 ],whereMijreferstotheedge(setofconstraintsorrelations)betweenvariablesiandj.WeillustratepathconsistencyofabasecaseBCNM(seeFigure 3-11 )whichisatriangleofthreevariables(A;B;C)andthesetsofrelationsoneachpair(p0(A;B);:::;pt(A;B);q0(B;C);:::;qu(B;C);r0(A;C);:::;rv(A;C)).Theconstraintthatmustholdbetweentwovariablesisdenedbythedisjunctionofalltheelementsofthesetofrelationsusedtolabelanedgeinthedigraph.Inotherwords,forourexamplep0(A;B)_

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Figure3-11. AbasecaseBCNof3variables. TheconceptsofBCNandpathconsistencyaretranslatedin[ 79 ]toBinarySpatialConstraintNetworks(BSCN)asamechanismtoreasonaboutspatialrelationships,specicallytopologicalrelationships.ABSCNisconstructedfromagivensetofspatialobjects(thevariables)andsomeknowntopologicalrelationshipsbetweentheobjects(theconstraints).Ifthereareanyunknownrelationships(emptyedges),theycanbeinitiallylabeledwiththeuniversal(U)relationship(adisjunctionofallpossiblerelationshipsbetweenapairofspatialobjects).ReasoningabouttherelationshipsisperformedbyapplyingconstraintpropagationtoreachpathconsistencyandpossiblyeliminatemanyoftheinconsistentrelationshipsincludedthroughtheuseofU.ItispossiblethatthereductionofUmightresultintheinferenceofunknownrelationshipsbetweensomeofthepairsofobjects.

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3-11 representsaBSCNforsimpleregionsA,B,andCforwhichweknowthatMAB=overlap(A;B)andMBC=inside(B;C).BecauseMACisunknownweinitiallyassumeitholdstheuniversalrelationshipMAC=U,inthiscase:U=foverlap;disjoint;meet;contains;inside;equal;covers;coveredBygFollowingthedenitionofpathconsistency,wecomputeMAC:=MAC\overlap(A;B);inside(B;C)whichresultsinMAC=foverlap;inside;coveredByg.Weiterativelyperformthesamecomputationfortherestofedgesnoticingthatatthispointthetriangleisstable.Inthiscase,pathconsistencyallowstoreduceMACfromUandlearnthattherelationshipbetweenAandCcanberepresented(atbest)bythedisjunctionoverlap(A;C)_inside(A;C)_coveredBy(A;C).JustasingeneralBCN,aBSCNisconsideredinconsistentandunsatisableifatanypointMij\Mik;Mkj=?(i.e.thereisnopossiblerelationshipthatcanholdbetweeniandjthatmakestheBSCNpath-consistent). 3-2 byaBSCNasillustratedinFigure 3-12 .Inthegure,theBSCN'snodesrepresenttheobjectsinvolvedinthetuple(Ak,Bk,Ac,Bc,andAkAcandBkBclabeledasAandBrespectively).Theedgesbetweenthenodesrepresenttheirtopologicalrelationships.Theedgeslabeledp;q;r;srepresenttherelationshipsgivenbytherowdenition.Foranyrelationshipzwedenoteitsinverserelationshipas 72

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(a)BSCNforrow1 (b)BSCNforrow2(c)BSCNforrow3 (d)BSCNforrow5(e)BSCNforrow9 Figure3-12. BinarySpatialConstraintNetworkrepresentationsofn-tuples 73

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3-2 istestedinAlgorithm 1 wherethecompletesetofvalid4-tuplesareidentied.Weomitasimpliedversionofthisalgorithmthatisusedtodeterminethevalidpairsfromrows2,3,5,and9inTable 3-2 . checkpathconsistencyofM continueuntilalltrianglesarestableorBSCNisinconsistent 1. Thevaguespatialdatatypes(v();v())thatrepresentthetypecombinationforwhichthevaguetopologicalpredicateswillbeidentied. 2. ThesetsT;,T;,T;ofcrisptopologicalpredicates.T;andT;areusedtodeterminethesetsin(T;),I(T;),in(T;),andI(T;). 74

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3-2 .Inline1thealgorithmpre-computesthecompositionofallpairsoftopologicalpredicatesinT;,T;,andT;.Theresultsaregeneratedbyderivingthecompositionbasedontheinferencerulespresentedin[ 28 ]andstoredincompositiontables.Threecompositiontablesareneeded,oneforthecompositionoftopologicalpredicatesbetweenthetwoobjectsofthepossibledistincttypesand,oneforthecompositionoftopologicalpredicatesbetweentwoobjectsoftypeandoneforthecompositionoftopologicalpredicatesfortwoobjectsoftype.Afterinitializingtheresultsettoempty(line2ofthealgorithm),theloopinline3takeschargeofiteratingovereachandeverycandidate4-tuple.Foreach4-tuple,therststepistogenerateaBSCNthatisbasedonFigure 3-12 (a)andcorrespondstothatspecictuple(line4).PathconsistencyoftheBSCNinFigure 3-12 ischeckedbytheloop(line6ofAlgorithm 1 )thatwillcontinuetobeexecutedifthesetofrelationsrepresentedinanyedgeoftheBSCNischanged(i.e.,ifanon-stabletriangleisfound).ThepathconsistencycheckofaBSCNendsifeitheroneofthefollowingistrue:(1)alltrianglesarestable,inwhichcasethe4-tupleisfoundtobeavalidvaguetopologicalpredicate,or(2)thesetofrelationsatanyedgebecomesempty,signalinganinvalid4-tuple.Todeterminethatalltrianglesarestable,thenoneiterationmusthavecompletedwithoutanychangestotheBSCN.Ifachangeoccurs,itisdetectedinlines9to11.Whileperformingtheoperationinline10,thelabelofanedgecanbecometheemptyset,inwhichcasetheBSCNisdeterminedtonotbepath-consistent(line14).Iftheiterationsendduetocondition(1)fromabove,inline19thecurrent4-tupleisaddedtotheresultset.WeapplyAlgorithm 1 inordertoidentifythe4-tuplesforalltypecombinationsofv(point2D),v(line2D),andv(region2D).Wealsoapplyamodiedversionofthealgorithminordertoidentifyallpairs.Thenumberofvalid(pairs+4-tuples)foreachtypecombinationisshowninTable 3-4 . 75

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Numberofcharacterizationpredicatesbetweenallcombinationsofvaguespatialdatatypes. 12+51 155+974 46+166 155+974 2077+471650 1327+74916 46+166 1327+74916 497+55880 3-9 (a)wemustconsidertheoverlapoftheconjecturepartoftheoilspillwiththekernelpartofthereefdierentlytohowwewouldconsideritifitwasanoverlapbetweenbothkernelparts.Aspecicinterpretationofacharacterizationpredicateresultsinitscorrespondingvaguetopologicalpredicate.Theinterpretationofcharacterizationpredicatesisdonethroughinterpretationrules.Theserulesaredenedtotasetofdeterminedsemanticsthatcorrespondtoadesiredsetofvaguetopologicalpredicates.Forexample,wechoosetodeneinterpretationrulesfortheeighttopologicalpredicatesoriginallydenedbetweensimpleregions.InthecaseofFigure 3-9 (a),therulesmustinterpretthemeaningoftheexistenceofadisjointbetweenthekernelpartsofbothobjectsandanoverlapbetweentheconjecturepartofthespillandkernelpartofthereef..Thefactthatitcannotbeassertedwhetherthisoverlapoccursornotisapropertyintroducedbythesemanticsdenedfortheconjecturepart.Todealwiththisproperty,vaguetopologicalpredicatesaredenedasthree-valuedpredicatesallowingamayberesultforthecasesinwhichtheinterpretationofacharacterizationpredicateidentiestheimpossibilitytocertainlysaywhetherapredicateholdsornot.Eachvaguetopologicalpredicateisdenebythreeinterpretationrules;onethatspecies 76

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Disjoint(A;B)=8><>:trueifdisjoint(AkAc;BkBc)falseif:disjoint(Ak;Bk)maybeotherwiseAtthebottomlevel,interpretationrulesdenevaguetopologicalpredicatesonthebasisofthe9-intersectionmatrixentriesforthetopologicalpredicatesthatmakeupthecharacterizationpredicate.Interpretationrulesatthislevelcanbeconsideredmoregeneralbecausethatarenottiedtoaspecicsetoftopologicalpredicatesbetweenagiventype-combination(i.e.,intheexampleabovedisjoint(AkAc;BkBc)specicallyreferstothecrisptopologicalpredicatedisjointdenedbetweencomplexpoints).Instead,rulesspeciedatthebottomlevelcanevaluatethecharacterizationpredicatesindistinctfromthecrispdatatypesinvolved.Forexample,wedenethevaguetopologicalpredicatefordisjointednessatthislowerlevelsothatitcanoperatenotonlybetweenvpoint2Dobjects,butgenerallybetweentwovaguespatialobjectsofanyofthetypesvpoint2D,vline2Dand,vregion2D.Inthiscasewechoosetodenetoobjectsascertainlydisjointifthe 77

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Disjoint(A;B)=8>>>>>>>>>>><>>>>>>>>>>>:trueif((AkAc)\(BkBc)=?)^((AkAc)\@(BkBc)=?)^(@(AkAc)\(BkBc)=?)^(@(AkAc)\@(BkBc)=?)falseif(Ak\Bk6=?)_(Ak\@Bk6=?)_(@Ak\Bk6=?)_(@Ak\@Bk6=?)maybeotherwiseFortherestofthissectionwecompletethesetoftype-combinationindependentinterpretationrulesthearedenedatthebottomlevelofabstraction,asdonewiththeDisjointabove,andlooselybasedonthesemanticsoftheoriginaleighttopologicalpredicatesbetweensimpleregions.TheMeetvaguetopologicalpredicatebetweenvaguespatialobjectscertainlyresultsintruewhentheboundariesoftwoobjectscertainlyintersect,buttheirinteriorscertainlydonot.Becausethispredicatedirectlyinvolvestheevaluationofanon-emptyintersectionofboundaries,itwillalwaysresultinfalsewhenatleastoneoftheoperandsisoftypevpoint2D.Forotheroperandtypes,itwillcertainlyresultinfalseiftheinteriorsofthekernelpartsoftheobjectsintersectortheobjectsarecertainlyDisjoint.IfanyothercongurationexistsitisnotpossibletocertainlysaywhethertheobjectsMeetornot,i.e., Meet(A;B)=8>>>>>>>>>>>>><>>>>>>>>>>>>>:trueif(@Ak\@Bk6=?_@Ak\Bk6=?)^((AkAc)\(BkBc)=?)falseif(Ak\Bk6=?)_(Ak\(BkBc)=?^(AkAc)\Bk=?^(AkAc)\(BkBc)6=?^@(AkAc)\(BkBc)=?^(AkAc)\@(BkBc)=?)maybeotherwise 78

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Inside(A;B)=8>>>>>><>>>>>>:trueif((AkAc)\Bk6=?)^((AkAc)\Bk=?)^(@(AkAc)\Bk=?)^((AkAc)\@Bk=?)^(@(AkAc)\@Bk=?)falseif(Ak\(BkBc)6=?)_(@Ak\@(BkBc)6=?)maybeotherwiseWedenethevaguetopologicalpredicateContainsastheinverseofInside,i.e.,Contains(A;B)=Inside(B;A).AvaguespatialobjectisCoveredByanotherobjectifitsinterioriscompletelycontainedintheinterioroftheotherobjectanditsboundaryisapropersubsetoftheboundaryoftheotherobject.Thisimpliesthatsomeoftheboundaryfromtherstobjectalsointersectstheinteriorofthesecondobject.WecandistinguishavaguespatialobjecttocertainlynotbeCoveredByanotherobjectifitsinteriorisnotcontainedwithintheinterioroftheotherobjectortheobjectisInside.Likethepreviousdenitions,thepredicateresultsinmaybeifitisnotpossibletosurelydeterminewhetherornottheobjectisCoveredBytheother,i.e., CoveredBy(A;B)=8>>>>>><>>>>>>:trueif((AkAc)\Bk6=?)^(@(AkAc)\Bk6=?)^((AkAc)\Bk=?)^(@(AkAc)\Bk=?)^((AkAc)\@Bk=?)^(@Ak\@Bk6=?)falseif(Ak\(BkBc)6=?)_(Inside(A;B)=true)maybeotherwiseCoversisdenedastheinverseofCoveredBy,i.e.,Covers(A;B)=CoveredBy(B;A).WeconsidertwovaguespatialobjectscertainlyEqualonlyiftheirkernelpartsareequalandtheirconjecturepartsareempty.TwoobjectarecertainlynotEqualif 79

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Equal(A;B)=8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:trueif(Ak\@Bk=?)^(Ak\Bk=?)^(@Ak\Bk=?)^(@Ak\Bk=?)^(Ak\@Bk=?)^(Ak\Bk=?)^(Ak\@(BkBc)=?)^(Ak\(BkBc)=?)^(@Ak\(BkBc)=?)^(@Ak\(BkBc)=?)^(Ak\@(BkBc)=?)^(Ak\(BkBc)=?)^((AkAc)\@Bk=?)^((AkAc)\Bk=?)^(@(AkAc)\Bk=?)^(@(AkAc)\Bk=?)^((AkAc)\@Bk=?)^((AkAc)\Bk=?)^((AkAc)\@(BkBc)=?)^((AkAc)\(BkBc)=?)^(@(AkAc)\(BkBc)=?)^(@(AkAc)\(BkBc)=?)^((AkAc)\@(BkBc)=?)^((AkAc)\(BkBc)=?)falseif(Ak\(BkBc)6=?)_(Ak\(BkBc)6=?)maybeotherwiseFinally,wedeterminetwovaguespatialobjectstocertainlyOverlapiftheinteriorofthekernelpartsofbothobjectsintersecteachotherandtheexteriorofthewholeotherobject.TheobjectscertainlydonotOverlapiftheirinteriorsdonotintersect,ortheirinteriorsdonotintersecttheexterioroftheotherobject.InthemajorityofthecasesitsturnsoutthatitisnotpossibletoassertiftheobjectsOverlap,i.e., Overlap(A;B)=8>>>>>>>>>>><>>>>>>>>>>>:trueif(Ak\Bk6=?)^(Ak\(BkBc)6=?)^((AkAc)\Bk6=?)falseif((AkAc)\(BkBc)=?)_((AkAc)\(BkBc)=?)((AkAc)\(BkBc)=?)maybeotherwise 80

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3.5.1 worksbyadaptingVASAtopartiallyworkwithSQL,currentlythemostpopulardatabasequerylanguage.ThesecondproposalpresentedinSection 3.5.2 extendsSQLsothatithasthenecessaryoperatorstohandlevaguequeries. 81

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True P(A;B)=true)P(A;B)=trueTrue P(A;B)=false)P(A;B)=maybe_P(A;B)=falseMaybe P(A;B)=true)P(A;B)=maybeMaybe P(A;B)=false)P(A;B)=true_P(A;B)=falseFalse P(A;B)=true)P(A;B)=falseFalse P(A;B)=false)P(A;B)=true_P(A;B)=maybeWiththistransformationinplace,queriesoperatingonvaguespatialobjectscanincludereferencestovaguetopologicalpredicatesandvaguespatialoperations.Forexample,basedonFigure 3-9 (a)wecanposeanSQLquerytoretrieveallcoralreefsthatareinanydangerofcontaminationfromanoilspill.WemustndallreefsthatarenotcertainlyDisjointfromtheExxon-Valdezoilspill. 3-9 (b)wewanttoretrievetheminimumlengthoftheintersectionsofallpairsofintersectingroutesofterrorists.Todoso,wechoosetocomputetheintersectionofonlythosepairsthatarecertainlynotDisjointandneglectthecomputationoftheintersectionofthosepairsthathavebeendeterminedtonotcertainlyintersect. 82

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3-13 whereanimalroamingareasarestoredasvagueregions,theirmigrationroutesarestoredasvaguelinesandtheirdrinkingspotsarestoredasvaguepoints. Figure3-13. Thevaguespatialobjectrepresentationofananimal'sroamingareas,migrationroutes,anddrinkingspots. Forexample,wewishtoretrieveallspeciesofanimalswhoseaverageweightisunder40lbs.,theirlastcountwasunder100andmayhaveroamingareascompletelycontainedwithintheroamingareasofcarnivoreanimalswhoseaverageweightisabove80lbs.Thisinformationmightbeusefultorecognizeanimalspecieswithlowcountsthatcouldbeextinctduetolivingamongstlargerpredators.Theextinctionofthesmallerspeciecanbecatastrophicevenforthelargerspeciethatdependsonthesmallerfornutrition.Thisretrievalusesdataelementsthatarebothspatialandnon-spatial: 83

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51 ]vaguenessdoesnotnecessarilyonlyappearinthedatabeingqueried,butcanalsobepartofthequeryitself.Namely,onecanposeaquerytoretrieveall\smallcars."Whilethecardataiscrispandthereisnovaguenessonthedata,thequeryitselfisvagueduetothesemanticsoftheterm"small"whichisanobviousexampleofavagueconcept.InthecontextofVASA,wearenottryingtosolvetheproblemofdealingwithvaguequeries,butweneedtoqueryvaguedata.Workthatfocusesonqueryingvaguedatahaslargelysidedtodealingwithfuzzyrepresentationsofdata.Inaspatialcontext,[ 75 ]proposesclassicationsofmembershipvaluesinordertogroupsetsofvaluestogether.Forexample,aclassicationcouldassigntheterm"mostly"tohighmembershipvalues(e.g.,0.95-0.98).Basedonthisclassicationqueriescanaskfor"mostly"disjointobjectswhenusingfuzzytopologicalpredicatesthusgivingadegreeofdisjointnessonthebasisofthemembershipvalues.Mostfuzzyorientedqueryingpropositionshavetodealwithclassifyingthemembershipvalues,thustheyproveineectiveforourpurposes.Inthecontextofdatabasesingeneral,theapproachesin[ 45 , 49 , 57 , 61 ]allproposeextensionstoquerylanguagesonthebasisofanoperatorthatenablesvagueresultsunderdierentcircumstances.Forexample,in[ 45 ]theoperatorsimilar-toforQBE(Query-by-Example)isproposedalongsiderelationalextensionssothatrelatedresultscanbeprovidedintheeventwherenoexactresultsmatchaquery.In[ 57 ]theoperatorisusedinasimilarwaytothesimilar-tooperator.Alltheseapproachesrequireadditional 84

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4.1 .InordertoextendcurrentDBMSswithVASA,wedevelopthemSLOBmechanismwhichispresentedinSection 4.2 .AcrispspatialalgebrawhichwedenominateSPAL2DisusedtheimplementationbasisforVASA.TheprototypeimplementationofSPAL2DisdescribedinSection 4.3 .Finally,inSection 4.4 wedescribehowallthesecomponentscometogethertoworkinVASA'sprototypicalimplementation. 4.1.1 ),andthenprovidethedetailsoftheRATIOimplementation(Section 4.1.2 ). 87

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44 , 89 ],geometriccomputationsarespeciallysensitivetonumericalnonrobustness.Eventheoatingpointnumberimplementationwhichhasbeenthehighprecisionoptionformostcomputations,isproven[ 88 ]tonotassurerobustnessingeometriccomputations.Thus,theneedarisesfordevelopingnewcomputationparadigmsspecicallygearedtowardsachievingrobustnessingeometriccomputation.Therststepinachievingrobustnessingeometriccomputationsistoidentifywhenacomputationcanbeconsideredgeometric.In[ 89 ],theauthorsdeneacomputationasgeometricwhenbesidesthenumericalpartofthecomputation,thereisanidentiablecombinatorialpart.Thecombinatorialpartcanbesometimesdiscreteortopologicalwhichinessenceaddssomesetofdiscreterelationsbetweentheobjectsinvolvedandwhichhavetoremainconsistent.Followingthiscomputationstructure,theExactGeometricComputation(EGC)paradigmdevelopedin[ 88 , 90 ],providesanadvancedmethodthatworksundertwopremises:(1)allnumericalvaluesarerepresentedexactly,and(2)allbranchingdecisionsduringcomputationareerror-free.Inordertoimplementthisparadigmanalternativetothewellknownxedsizedoatingpointnumbersisused.Oneofmanyalternativesistousesomeformofmultiprecisionnumberlibraryforwhichtheprecisionisonlyboundedbythephysicallimitsofthecomputer.ExamplesaretheGNUMultiplePrecisionArithmeticLibrary(GMP)[ 38 ]andClassLibraryforNumbers(CLN)[ 43 ].Althoughbothareverygoodimplementations,theydonotmeettherequirements(asdenedinthenextsection)thatwehaveestablishedfortheimplementationsofourgeometricalgorithmsinadatabasecontext.InthenextsectionwepresentRATIOwhichisanimplementationofamultipleprecisionrationalnumberlibrarydevelopedforthespecicpurposeofhandlingandstoringsuchbignumberindatabases.Althoughexpertsnote(see[ 90 ])thatusingonlybignumberswillnotgetridofnonrobustness,webelievethatusingRATIOisgoodforourpurposesasitprovideshighprecisionwhilenotaectingperformanceseverely. 88

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38 ]).Eachintegerisinternallyrepresentedasabase216(65536)number,meaningthesetofvaliddigitsfromwhichtocomposeeachintegerisf[0::65535]g.ThebasewaschosenonthegroundsoftheC++(informal)standarddeningeachintvariabletoaccept232dierentvalues.Theworstcase(appearinginthemultiplicationalgorithm)requiresthestorageofanumber(digit)thatisatmostthesquaredvalueofthebiggestnumber(digit)allowedbytheintegerrepresentation'sbase(65535inourcase).Anintegermadeupofndigitsisstoredinanarrayofsizen+1wherethelastpositionisreservedforthesign. 47 ].Theaddition,subtractionandbasicmultiplicationalgorithmsverymuchfollowtheprocessthatisusedformanual(pencilandpaper)executionoftheseoperations.Theadditionandsubtractionbothruninlineartime(O(n))dependentonthesizeoftheinputwhichdependsonthebiggestofthetwointegersinvolved.Themultiplicationontheotherhandrunsinsquaretime(O(n2)).Thedivisionalgorithmisthemostcomplexofallfourandeventhoughithassimilarstructuretothemanualcalculationofanintegerdivisionitrequiresextrafeaturesthatallowfortheguessingstepthatisnormallydonebyahumanduringthecomputation.ThecomplexityofdivisionisalsoO(n2)duetothefactthatitdependsonthemultiplication.Besidesthefourbasicarithmeticoperators,theimplementationincludesthemodulo(%)andexponentiation(^)operatorsaswellasselfincrementanddecrementoperators(++,)forcompatibilitywithC++.Italsoincludestheassignmentoperatorthatallowsoneintegertotakeonthevalueofanotherandnallywealsoincludeall 90

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2 . rat rat newx=x;1 91

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2.1.2 areexamplesofcomplexstructuredobjects.Asthenameimplies,theseobjectshaveacomplexinternalstructureandinimplementationgenerallyhaveavariablerepresentationlength.Thesetypesofobjectsarenotuniquetospatialdatabases,butinsteadappearinotheremergingdatabaseapplicationssuchasgenomicdatabasesandmultimediadatabases.Infactbesiderequiringhandlingofhierarchicallystructuredcomplexobject,manyapplicationsincludingthoseinthespatialdomainhavetheadditionalrequirementofsupportingmultipleviewsofdata.Forinstance,somealgorithmsrequirethatregions,likethatillustratedinFigure 4-1 berepresentedasasequenceofsegmentswhoseorderdependsonthepolygonstructureforregionobjectsthatwasdescribedinSection 2.1.2 .Otheralgorithmsrequirethatthesegmentsbeorderedinadierentorder.Thismeansthatalthoughbothtypesofalgorithmsoperateonthesamesetofsegments,eachalgorithmrequiresadierentviewofthesesegments.Inordertotakeadvantageofbothtypesofalgorithms,agenerictypeofcomplexobjectmanagementisrequiredthatcanrepresentmultiplestructuralviews.Manyotheremergingdatabaseapplicationdomainspresentsimilarcomplexobjectmanagementrequirements.Forexample,proteinanalysisandmultiplesequencealignmentapplicationsoftenrequireprimary,secondary,andtertiaryproteinstructureinformationtoberepresented.Phylogeneticanalysissometimesrequiresboththehierarchicaltreeoflifeviewsofdataaswellasecientaccesstochronologicalinformationofwhenspecieswerediscoveredorbecameextinct.Whilesomescienticleformatscanbeusedtorepresentthesetypesofobjects,thecorrespondingapplicationsaretraditionallytiedtodatabasetechnology.Forexample,spatialdatabasesareinwidespreaduseduetothe 92

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Figure4-1. Anexampleregionobjectwithtwofacesandoneholewitheachsegmentidentiedbyadierentnumber. Inthissection,weintroduceaconceptualframeworkforthestorage,retrieval,andmanagementofhierarchicallystructuredcomplexobjects,whichwecallStructuredLargeOBjects(SLOBs).Theexamplesaboveshowthatsupportingonlyhierarchicallystructureddataisnotsucientforsomeapplications;thus,weextendtheframeworktomodelmoregeneraldirectedgraphrepresentationsofdata.WecallthisextendedframeworkMulti-StructuredLargeOBjects(mSLOB)becausetheyallowthestorageofstructuredlargeobjectswithmultipleviewsofdata.Thegoalofourapproachistoprovideecientquerysupport,andportabilitybetweendatabasesystems,notcomputersystemsingeneral.Thus,wefocusonprovidingrandomaccesstoandupdateofcomponentsandstructuralsupportofcomplexapplicationobjects.WespecifythemSLOBconceptthatprovidestheneededfunctionalityandadatabaseindependent,prototypeimplementationthatcanbepluggedintoanyDBMSthatataminimumprovidesunstructuredobjectsupport,suchasLOBsordatabaseles.Section 4.2.1 describesstructuredlargeobjects,whileSection 4.2.2 doesthesamefortheirmultipleviewcounterparts.InSection 4.2.3 wedescribethemSLOBframeworkofimplementationandinSection 4.2.4 wedescribetheimplementationofmSLOBsontopoftheOracleDBMS. 93

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ExampletypesofelementsinanmSLOB:(a)abaseelement,(b)astructuralelementconsistingofastructuralelementandabaseelement.Unshadedrectanglesdenotestructuralelementswhereasshadedrectanglesdenotebaseelements. Figure4-3. DiagramdepictionsofmSLOBcomponents:(a)astructuredobject,(b)itshierarchicaltreerepresentation. 4-2 showsanexamplerepresentationofeachofthetwotypesofelements.Figure 4-3 (a)depictsastructuredobjectwhichisastructuralelementconsistingoftwobaseelementsandtwostructuralelementseachofwhichcontainsthreebaseelements.Figure 4-3 (b)showsahierarchicaltreerepresentationofthestructuredobject. 94

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4-3 isrepresentedbythestring((bbb)bb(bbb)).Itisclearthatnotallstringscomposedofthesesymbolsrepresentavalidstructuredobject.Forexample,ifthenumbersofleftandrightparenthesisarenotequal,thenatleastonestructuralelementisnotrepresentedcorrectly.Therefore,wemustidentifythesetofvalidstructuredobjectrepresentationstrings.Weachievethisbydeningagrammarthatcansuccessfullyparseanyvalidrepresentationstring.Becausewerepresentabaseelementasasinglecharacter,therequirementthatbaseobjectshavenosubelementsisimplicitlyenforced.Therefore,thegrammarmustbeabletoverifyanywellparenthesizedstringcontainingonlyparenthesisandthebsymbol.Notethateverystructuredobjectimplicitlyhasastructureelementasitsroot,soeveryvalidstructuredobjectrepresentationstringwillbeenclosedinparenthesis.Thefollowinggrammarenforcestheseconstraintsanddenesthesetofallvalidstructuredobjectrepresentations(terminalsymbolsareinlowercaseandSEstandsfor\structuralelement"):SLOB:=(COMPONENT)COMPONENT:=SECOMPONENTjbCOMPONENTjemptySE:=(COMPONENT) 95

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4-4 (a)illustratesamulti-structuredobjectwithonestructurerepresentingasequentialviewofitsbaseelementsandtwootherstructuresrepresentingtwodierenthierarchicalviewsofthesamebaseelements.Figure 4-4 (b)showsadirectedgraphrepresentationofthesamemulti-structuredobject.InordertoformallydenemSLOBs,weuseasimilarnotationandstringrepresentationaswasusedtodenestructuredobjects.Becausestructuralelementsimposeahierarchicalstructureontheobject,weareabletorepresentthemasparenthesisthathierarchically 96

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Illustrationofstructureandviewsofamulti-structuredlargeobject:(a)objectwiththreedierentstructuresandreferencelinksdenotedbyshadedcirclesandarrows,and(b)thecorrespondingdirectedgraphrepresentationwithreferencesasdashedarrows. subdivideastringofbaseelementsandreferenceelements.Abaseelementisagainrepresentedasasinglecharacterb.Areferenceelementissimilartoabaseelementinthatitcontainsdata(inthiscasethedataisinformationthatpointstoanotherelementintheobject)andcannotcontainsubelements.Therefore,wesimplyrepresentareferenceelementasasinglecharacterr.Forexample,thestructureofthemSLOBinFigure 4-4 canberepresentedas(((rrr)(rr))(bbbbrr)((r(rr))r(bb))).WedeneavalidmSLOBtobeanymSLOBwhosestructuralstringrepresentationcanbesuccessfullyparsedbythefollowinggrammar(terminalsymbolsareinlowercaseandSEstandsfor\structuralelement"): 97

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4.2.3.3 .Whiletheconceptualdiscussionoutlinestherequirementsneededtomanagestructure,weimposesomeadditionalrequirementsonourimplementationthataecttheperformanceandgeneralityofourimplementation.First,asignicantproblemwithnewdatabasetechnologiesisthattheyareoftenimplementedinresearchorprototypeDBMSsthattendtohaveveryspecializedfeatures,whichareuniquetoeachsystem.Ifanewtechnologyusesoneofthesespecializedfeatures,itisverydiculttotransferittoanothersystem.Therefore,werequirethatourmSLOBimplementationbegeneralinthesensethatitcanbeusedinanyDBMSthatprovidesaminimalsetofstoragerequirements.Specically,wechoosetoimplementourprototypeontopoflargeobjects(LOBs)providedbyDBMSs.LOBsareavailableinmostsystems,andprovidearelativelystandardinterface;thus,byusingLOBsasanunderlyingstoragemodel,weachievethedesiredgenerality.Furthermore,LOBsprovideasimple,linearstoragesimilartoles.A 98

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4.2.3.1 forrepresentingobjectstructures.EcientdatamodicationimpliesthattheinsertionofdataintoanmSLOBshouldnotrequiredatatobephysicallyshiftedondisk.Forexample,considertheinsertionofanelementintoanarray.Unlesstheelementattheinsertionpointisoverwritten,allelementsaftertheinsertionpointmustbeshiftedonepositiontoaccommodatethenewdata.LOBsinmostcommercialDBMSsprovidefunctionsnamedinsert,butthesefunctionsactuallyhaveoverwritesemantics,meaningthattoactuallyinsertdatasothatnodataisoverwritten,thedataaftertheinsertionpointmustbemanuallyshiftedbytheuser.Thisproblem,whichwecalltheupdateproblem,isageneralproblemencounteredwhenstoringstructureddatainunstructuredstorage.Traditionally,theproblemissimplyignored,anddatashiftingisperformed.WeintroducetheconceptofanupdatesequenceinSection 4.2.3.2 ;itmaintainsthelogicalorderofdatawhileallowinginsertionstoLOBsandpreventingtheneedfordatashifting. 99

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AnexampleofanmSLOB:(a)attheconceptuallevel,and(b)attheimplementationlevel.ThemSLOBhasasinglestructuralelementcontainingabaseelementandareferenceelement,whicharemarkedbyS,B,andRintheirinformationnodesrespectively. ofthemSLOBimplementation.Inourimplementationwechoose4byteaddressingwhichallows232bytes=4GBofcapacityandmaintainsarelativelylowoverhead.Giventhesizeofaddresses,wearenowabletodescribethevariouselements.Eachelementbeginswithaninformationbyte,whichidentiesitselementtype,andendswithanendmarkerimplementedasasinglebyte.Structuralelementsarethemostcomplexofthethreeelementsbecauseeachstructuralelementcancontainanycombinationoftheotherelementtypes.Astructuralelementisrepresentedbyaninformationbyte,afourbytecountidentifyinghowmanyelementsthisstructuralelementcontains,asequenceoffourbyteaddresseswhichidentifytheendmarkersofeachrespectivesub-element,asequenceofsub-elements,andanendmarker.Abaseelementbeginswithaninformationbyte,followedbyafourbytesizeeld,followedbytheactualbaseobjectdata,andnallyanendmarker.Areferenceelementcontainsaninformationbyte,theaddressesoftheinformationbyteandtheendmarkerofthereferencedelement,andtheaddressinthestructuralelementcontainingthereferencedelementofthereferencedelement'saddress(neededforelementremoval).Figure 4-5 showsanexamplewhereeachtypeofelementisdepicted.Usingstructuralelementsandbaseelements,anytypeofhierarchicalcongurationcanbecreated.Theuseofreferencesallowsnon-hierarchicalrelationshipsbetweenobjectstobeeectivelyrepresented.Therefore,ausercanconstructobjectsofarbitrarilycomplexstructurethroughthearrangementofthesethreeelements. 100

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Figure4-6. ExamplecompositionofanmSLOBwiththeinitialunfragmenteddataandtheupdatesequence. WeproposetheconceptofanupdatesequencewhichprovidesthenecessarytoolstoenablerandomupdatesinLOBsandinpersistent,unstructuredstoragesystemsingeneral.TheupdatesequenceconceptisbasedonalazyupdateapproachofphysicallystoringnewdataattheendofaLOBandprovidingamechanismthatlogicallymaintainsthecorrectorderofthedata.Itisanorderedlistofphysicaladdressintervalsthatkeepstrackoftheorderinwhichdatamustberead.Inotherwords,theupdatesequenceprovidesamappingsfromlogicaladdresses(L)tophysicaladdresses(P):s:L!P.Ifthelogicalorderingofbytesprovidedbytheupdatesequenceisthesameasthephysicalordering(i.e.,8l2L:s(l)=p^l=p),wesaythattheobjectissolid(nounusedspace)andfullysequenced.Initially,objectsshouldbesolidandfullysequencedasillustratedinFigure 4-6 .Supposethatatthispoint,theusermakesaninsertionatpositionkinthemiddleoftheobject(seeFigure 4-7 ).Insteadofshiftingdataafterpositionkwithin 101

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4-7 . Figure4-7. Exampleofinsertingblock[j:::l]atpositionkinanupdatesequence. Figure 4-8 reectsthebehavioroftheupdatesequencewhendataisdeleted.BecausetheLOBdoesnotactuallyallowfordeletionofdata,theupdatesequenceismodiedtoeliminateaccesstothedeletedblock[m:::n]ofdata.Thus,deletionsresultininternalfragmentation. Figure4-8. Illustratingthedeletionofblock[m:::n]inanupdatesequence. Thehandlingofanupdatewherethevaluesofablockofdata[o:::p]asaportionofblock[j:::l]arereplacedwithvaluesfromanewblock[l:::q]isshowninFigure 4-9 . Figure4-9. Illustrationofthereplacementofblock[o:::p]byblock[l:::q]inanupdatesequence. 102

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Figure4-10. ThemSLOBinterface Locatorobjectshavexedandsmallsizeasopposedtothevariableandlargesizeoftheobjectstheyrepresent.Thus,itismuchmoreecienttouselocatorsthantousetheactual,frequentlylarge,objectsinanyoperationorasfunctionparameters.Forthisreason,everyobjectstoredinanmSLOB,regardlessofitstype,isrepresentedbyalocator.Asaresult,mostofthemSLOBfunctionalityisaccessedthroughfunctionsdenedonlocators.mSLOBsarecreatedandinitializedthroughthefunctionsshowninFigure 4-10 .OnceanmSLOBisinitializedorreferencedwiththesefunctions,theremainingfunctionalityisaccessedthroughthelocatorfunctionslistedinFigure 4-11 .BesidesthemSLOBandLocatordatatypes,theseguresincludetheprimitivetypeInteger(orInt),theDBMSorsystemspecictypeStorageasastoragestructurehandletype(i.e.,blobhandle,ledescriptor,etc.),atypeStreamwhoseobjectsrepresentanoutputchannelforreadingbyteblocksofarbitrarysizefromanobjectrepresentedwithaLocator,anddataasarepresentationofrawdata. 4{1 )constructsanemptymSLOBobject.Thesecondcreate(s)operation( 4{2 )constructsanmSLOBobjectfromaspecicstoragehandlessuchasaLOBobjecthandleoraledescriptor.Thethirdcreate(oldmSlob)operation( 4{3 )buildsanewmSLOBobjectfromanexistingmSLOBobjectoldmSlob. 4{4 ).Oncethislocatorisretrieved,accesstotherest 103

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Figure4-11. TheLocatorinterfaceforaccesstomSLOBfunctionality. ofthehierarchycanbeachievedwiththelocate(l;i)operator( 4{6 )wherelrepresentstheLocatoranditheindexofthesubobjectoflthatistobelocated. 4{7 )toconsecutivelyreaddataofarbitrarysizefromanyobjectreferencedbylocatorl.Thestreamobtainedfromthisoperatorbehavessimilarlytoacommonleoutputstream.Otherthanreadingdata,theinterfaceallowstheinsertionofabaseobjectdofaspeciedsizesatindexiofanobjectreferencedbylocatorlthroughtheinsert(l;d;s;i)operator( 4{8 ).Italsoallowscreationofareferencetoanexistingobjectrepresentedbylocatorl1throughtheinsert(l;l1;i)operator( 4{10 ).Anemptysubobjectcanalsobeinsertedthroughtheinsert(l;i)operator( 4{9 ).Inallthreecases,thenewsubobjectisinsertedintoindexioftheobjectrepresentedbylocatorl.Toremoveasubobject,theremove(l;i)operator( 4{11 )removesthesubobjectatindexioftheobjectreferencedbylocatorl.Thisoperatoralsotakescareofremovingallreferences(notbysettingthesereferencestonullbutactuallyremovethereferenceelements)totheobjectremoved.Ifthesubobjectisareference,thentheobjectitreferencesremainsuntouched,andonlythereferenceisremoved. 104

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4{12 )whilethenumberofsubobjectsofthisobjectisprovidedbythecount(l)operator( 4{13 ).Finally,theresequence(mslob)operator( 4{5 )rewritestheunderlyingbytesofmslobsuchthattheyarephysicallysequential.Afteraresequenceoperation,theupdatesequencewillalwaysconsistofasinglerange. 4.2.4.1 ,wedescribeourupdatesequenceimplementationforhandlingecientupdates.Then,inSection 4.2.4.2 ,wepresentthedetailsofhowwehandlethestructureofmSLOBs. 105

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4-12 .Assumewehavereadblock1andarenowatpositionj.Toretrievethenextlogicalblock(inthiscaseblock2),wemustndthephysicaladdressofthelogicalbytepositionimmediatelyfollowingblock1(positionj+(1logicalbyteposition)).Notethatinthisparticularexample,thisphysicaladdressisl.Theupdatesequenceprovidesthismappingthroughthelocationoperationsdescribedintheinterfacebelow. Figure4-12. Anout-of-ordersetofdatablocksandtheircorrespondingsequenceindex. WhenanupdateoperationisperformedonamSLOBobject,theupdatesequenceismodiedtorepresentalogicalshiftinthedatasequence.Todeleteablockofdata,weremoveanyreferenceswithintheupdatesequencetophysicaladdressesintheblock.Notethatremovingdatafromtheupdatesequencedoesnotnecessarilyimplyaphysicaldeletionofthereferenceddata.Toreecttheinsertionofnewdataintheupdatesequence,referencestothenewblockofdatamustbeinsertedintotheupdatesequence.Eitheranewrangeisinsertedintotheupdatesequence,oranexistingrangeisextendedtoincludethenewblock.Replacingadatablockwithanewdatablockamountstoadeletionfollowedbyaninsertion.Figure 4-13 presentstheinterfaceoeredbytheupdatesequence.LetSeqIdxbethetypeofallupdatesequence.Theupdatesequenceoperationsspanthefollowingcategories:Construction:Aupdatesequencecanbeconstructedwithcreate()( 4{14 )whichgeneratesanemptyupdatesequencewithnoranges.Whenconstructedwithcreate(d)( 4{15 ),anupdatesequenceisloadedwiththeaddressrangesprovidedinthearrayd.Location:Dataislocatedbytraversingtheupdatesequenceinthreeways.AcalltofromBack(si;b)( 4{16 )ndsthephysicaladdressthatislocatedb(logical)bytesfromtheendofthebytesequencegivenbytheupdatesequencesi.Similarly, 106

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Figure4-13. Theinterfaceformanipulatingtheupdatesequence. fromFront(si;f)( 4{17 )ndstheaddresslocatedf(logical)bytesfromthebeginningoftheupdatesequencesi.FunctionfromPos(si;p)( 4{18 )worksthesamewayasfunction( 4{17 )butinsteadofstartingfromthebeginning,itstartsfromagivenlocationp.Insertion:Operatorinsert(si;l;p;s)( 4{19 )modiesthelogicalsequencebyinsertingsbytesofdatastoredatphysicallocationpatthepositionwhereaddresslisfoundintheupdatesequencesi.Theoperationappend(si;p;s)( 4{20 )appendssbytesofdatastoredatphysicallocationptotheendofthelogicalsequence.Removal:Dataislogicallyremovedbyusingtheupdatesequenceinoneoftwoways.TheoperatorrmBySize(si;s;l)( 4{21 )allowstheremovaloflbytesfromstartaddresssinupdatesequencesi.Thisremovalislogicalandcanspanseveralrangesoftheupdatesequence.TheoperatorrmByEnd(si;s;e)( 4{22 )removesallbytesbetweenthestartaddresssandendaddresseinupdatesequencesi.Bothaddressesrefertophysicallocations,thoughtheremovalmaynotbesequentialtotheLOBaslogicallytheremaypossiblybedierentphysicallocationsinbetweenthestartandendarguments.Replacement:Operatorreplace(si;l;p;s)( 4{23 )allowsthereplacementofsbytesatlocationlwithsbytesstoredatlocationp. 107

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4{24 )providesthenumberofbytesaccessiblebytheupdatesequencebetweensandewheresandecanspanmultiplebyteranges. 4-14 illustratesthelocationofthecountandelementboundarybytesfordierentelementsinanmSLOBrepresentingtheregionobjectillustratedinFigure 4-1 . 108

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Figure4-14. AsamplemSLOBobjectrepresentingtheregioninFigure 4-1 .Informationbytesarestriped,endingbytesareshadeddark,addressesandcountsareshadedlight.Objectsareleftunshaded.Thelabelsundertheboxesindicatetheaddressofthatspeciclocation. RecallthatpointersusedtorefertothemSLOBstructureallrefertophysicalbytepositionsinsteadoflogicalbytepositions.Theimplementationisthiswaybecauseifpointersweretostorelogicaladdresses,everydataupdateinthemSLOBwouldcauseallpointerslogicallyforwardoftheupdatetobecomeobsoleteduetothelogicalshiftofdata.Inthecaseofphysicalpointers,thisproblemonlyappearsifthelastsub-elementofanobjectisupdatedcausingthephysicalendoftheobjecttochangewhichwouldrender 109

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4.2.3.3 areimplemented.WeprovidethedetailsoftheimplementationofeachofthemSLOBinterfacefunctionsasfollows:Construction:TocreateanmSLOBbasedonaLOBhandlewithcreate( 4{2 ),weloadtheupdatesequencewhichislocatedattheendoftheLOB.ThisispossiblebecausewestorethesizeoftheupdatesequenceasthelastpieceofdataofthewholeLOB.InternalReference:TogeneratealocatorforthemSLOBinlocateGlobal( 4{4 ),werstlocateitsendaddressbymovingqbytesfromtheendofthemSLOB,whereqisequaltothesumofthesizeofthemSLOB'scountandthesizeofthedummybyte.ThisisdonewiththeupdatesequencefunctionfromBack( 4{16 ).Similarly,itsstartaddressisretrievedwithacalltotheupdatesequenceoperationfromFront(0)( 4{17 ).Thelocatorofthesub-elementatslotk,givenalocatorltoitsparentobject,isgeneratedusingtheoperationlocate( 4{6 ).Tolocatetheendoftheobjectwemustnditsaddresspointerwhichislocatedkspntfromthestartoftheparent(wherespntrepresentsthenumberofbytesusedbyeachpointer).Tondthestartofthesub-elementwhenk=0(i.e.,itistherstsub-element)wecansimplycalculatecountspntwherecountisthenumberofsub-elementsoftheparent.Forallothersub-elements,theirstartcanbecalculatedbyndingthepointerfortheprevioussub-element(k1).EachelementinanmSLOBisactuallylocatedonlyonce.Afterthersttime,thelocatorinformationisstoredinthelocatortree.Thelocatortreeprovidesauniquedatasourcewhereallupdatesonobjectsarecentralized,thusviewablebyrelatedobjectswhennecessary.Write:First,thedatacorrespondingtothenewsub-elementisappendedtotheendoftheLOB.Thesedataincludethecount,thedummybyte,andtheaddresspointer 110

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4{5 )entailsmodifyingthephysicallocationsoftheobjects.Asaconsequence,theresequencingprocessmustgeneratenewaddresspointers.Byrecursivelyresequencingeachhierarchicallevel,theresequenceoperationisabletorewritethemSLOBwithalldatain-orderandwithoutinternalfragmentation. 2.1.2 .SPAL2DiscurrentlydesignedtoincludeatleastthoseoperationsrequiredbyVASA.Thisincludestheformalconceptsforthecrispspatialsetoperationsofunion,intersection,anddierence,minimaldistancerelationships,andoperationstodetermineobjectattributessuchaslengthandarea(see 2.1.3 ).Italsoimplementstopologicalpredicatesbetweencomplexspatialobjects(see 2.1.4 ).ThearchitectureofthesystemencompassingSPAL2DisdetailedinSection 4.3.1 .TheinternalspecicationofSPAL2DdatatypesandoperationsisprovidedinSection 4.3.2 . 111

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4.3.3 . 4.2 .mSLOBisintendedtofunctionasaninterfacebetweenSPAL2DandtheDBMS,thuswemustimplementSPAL2DtakingintoaccountmSLOBfunctionalitywithoutbeingconcernedofanyspecicDBMS.Figure 4-15 illustratestheinteractionofallelementsinvolvedinthesystemthatincludesSPAL2D. Figure4-15. ThearchitectureofthesystemencompassingSPAL2DandVASA.AnarrowfrommoduleAtoBindicatesthatmoduleAusesmoduleB. AsshowninFigure 4-15 ,SPAL2DusesRATIOasitsnumbersystem,mSLOBasitsstorageinterface,andtheunderlyingoperatingsystemforexecution.TheDBMSneedstobeawareofSPAL2Dtounderstanditsdatatypes,thusthereisaDBMSdependentfeaturethatrequiresthatdatatypesassociatedwithSPAL2DberegisteredwiththeDBMS.Onceregistered,theDBMSwillbeabletorouteSPAL2DspeciccallstotheappropriateSPAL2Dsoftwarelibrary.ClientapplicationscancommunicatedirectlywithSPAL2DbyinstantiatingobjectswithelementsretrievedthroughtheDBMS.Theseapplicationscanalsocommunicatewith 112

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4.3.2.1 ,andSDTsandtheirimplementationaredescribedinSection 4.3.2.2 . 113

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4-16 illustratesthesevalues.Thismeansthateachsegmentusedtorepresentaregion,isactuallyrepresentedbytwohalf-segmentsinimplementation,onewithpasthedominatingpoint,andanotherwithqasthedominatingpoint.Alloperationsandpredicatesoperatingonseg2Dobjects,alsooperateonhseg2Dobjects. Figure4-16. Illustratinginside-aboveagofhalf-segments.Thehalf-segmentwiththebigcircledendpointasitsdominatingpointwillhaveitsinside-aboveagsettotruebecausetheinterioroftheregionisaboveorleftofthathalf-segment.Forthehalf-segmentwhosedominatingpointisdenotedwiththesmallercircle,itsinside-aboveagissettofalse. 115

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2.1.2 .Thelengthofaline2Dinstanceiscomputedwiththefunctionlength:line2D!rat.Thedistance:line2Dline2D!ratoperationreturns,giventwolines,theminimaldistancebetweenanytwopointsfromeachline.Wealsoimplementthespatialunion,intersection,anddierence(f:line2Dline2D!line2D).AlltheseasimplementationsoftheconceptsfromSection 2.1.3 . 117

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2.1.2 .Accordingtothisstructure,thehalf-segmentsintheprimaryviewareclusteredintosetseachrepresentingadierentpolygonoftypepoly2D.Thepolygonsthemselvesarethengroupedtocomposefaces,eachoftypeface2D.Foreachface,thereisexactlyoneouterpolygonrepresentingtheoutsideboundary,andzeroormoreinnerpolygonsrepresentingholesintheface.Wedenethefunctionslength,andareaforcomputingtheperimeterandarearespectivelyofaregion2Dinstance(p:region2D!rat,p2flength;areag).Wealsodenethedistancetoreturntheminimaldistancebetweenanytwo(boundary)pointsfromtworegions.Finally,theunion,intersection,anddierenceoperationsareimplementedinconformancewiththespecicationreferencedinSection 2.1.3 . 118

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4.2.4.3 ,alocatoractsasahandletothemSLOBrepresentationoftheobjectandcanprovideelementbasedaccesstoallsub-elements,inthiscaseeachoftypepoi2DbutviewedbymSLOBasrawdata.Theimplementationoftheline2Dexempliesthesimplestuseofthemulti-structuredlargeobjectconcept.Eachline2Dobjectisrepresentedbytwolocators;onefortheprimaryviewandhavingallsub-elementsoftypeseg2D,andthesecondforthesecondaryviewandhavingblock2Dobjectsassub-elements.Eachblock2Dismadeupofasequenceofsegments,andrepresentedinthemSLOBasreferencestothesegmentsactuallystoredintheprimaryviewoftheline.Finally,theregion2DdatatypeisimplementedasillustratedinFigure 4-14 .Eachcomplexregionhastwoviews,eachrepresentedbyaseparatelocator,bothsub-elementsofthesamemSLOB.Theprimaryviewlocatorhashseg2Dobjectsassub-elements,whilethesecondaryviewlocatorrepresentsathreelevelhierarchythathasface2Dobjectsasitstopsub-elements,withpoly2Dobjectsunderneath,andhseg2Dobjectsrepresentedasreferencestothoseintheprimaryview,inthebottomlevel.Representationofeachofthesestructuresaslocatorsallowsfortraversalofeachlevelofthehierarchyinthesamewayasitwouldbedoneinanarray.Forsimplicity,weimplementaniteratorconceptforeachspatialobject,thusallowingtheiterationovereachelementofitssequence.Forillustrationpurposes,inFigure 4-17 weprovidethecodesegmentthatcomputestheperimeterofaregion2Dobject.ThecodeinFigure 4-17 providesabriefinsightintotheactualemploymentofmSLOBintheimplementationofSPAL2D.Line8demonstratesthecallrequiredforlocatinganelementonthebasisofitsindex.Lines9-14demonstratehowtheobjectis 119

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Codesamplethatimplementstheoperationtocomputethelengthofacomplexregion,asimplementedinSPAL2DonthebasisofmSLOBs. retrievedviaaninputstreamreaderspecicallywrittenformSLOB.Line15illustratesthecalltothede-serializingconstructorimplementedforhseg2Dobjectsandthatisabletogivemeaningtothestringofbytesreadonthebuer.AsfarasSPAL2Dconcerns,thedetailsonthestorageofitsobjectstructureareirrelevant.Thusatthebasedofthesystem,mSLOBcoulduseanytypeofstorageorevenacombinationofmorethanonetoachievetheresultsdesireddependingonaspecicdomainandapplication.TheperceivedadvantageforSPAL2Disportability,andsuch 120

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4-15 ,VASAinteractswithfourcomponentsofthesystemarchitecture.OneofthoseistheoperatingsystemwithwhichVASA,justlikeanyothersoftwarelibrarymustinteracttoexecutecode.Thus,weconsiderunnecessarytofurtherdescribethisinteractionandinsteadconcentrateonprovidingthefollowingdetailsoftheotherthreeinteractions: 1. ThemaininteractionofVASAisdescribedbyitsuseofSPAL2D.EachofVASA'sthreedatatypes,vpoint2D,vline2D,andvregion2Daredenedasapairoftheircrispcounterparts.Thatis,theimplementationofthevpoint2Dobjectspeciestwopoint2Dinstancesasitsmemberelds.Similarlyforvline2D,andvregion2Dwheretheformeriscomposedoftwoline2Dobjects,andthelatteriscomposedoftworegion2Dobjects.AllthefunctionsimplementedwithineachoftheclassesdeningdatatypesinVASA,workbycallingmethodsimplementedfortheirtwounderlyingcrispmemberobjects.Forexample,theunionoperationisgenericallyimplementedonthebasisofitsconceptualdenitionfromSection 3.3 .ThecodeinFigure 4-18 illustratestheimplementationoftheoperationbetweentwovregion2Dobjects.Lines3-5inFigure 4-18 areinchargeofexactlyfollowingtheexecutablespecicationofoperationsovercrispregion2Dobjects,thatwillresultintheunionoftwovagueregions. 2. TheonlydirectinteractionofVASAwiththeDBMSisaonewayinteractionwheretheDBMScanmakecallstoelementsinVASA,butVASAhasnoknowledgeoftheDBMS.ThisispossiblethroughthemechanismsofdatatypeextensionscurrentlyavailableinmostDBMS.Theseextensionmechanisms(e.g.,InformixDataBlades,OracleDataCartridges,etc.)allowforuserdeneddatatypestobespeciedor 121

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3. ThelastsignicantinteractionofVASAiswiththeclientapplication.SimilarinthewaythatVASAmakesuseofSPAL2D,aclientapplicationcandenevaguespatialdatatypecolumnsinthedatabaseandthenperformupdatesandretrievalsofthedatastoredinthiscolumns.Aswecansee,theimplementationofVASAis(notsurprisingly)simple.Perhapsitsonlycomplexityderivesfromitscommunicationwiththerestofcomponents.Butinitself,theVASAlibraryisabletoleveragetheSPAL2Dimplementationsuccessfully,thusachievingthegoalsestablishedoriginallyforthedesignandimplementationofVASA. 122

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CodesamplethatillustratestheimplementationofVASAandhowitinteractswithSPAL2Dtoperformtheunionoperationbetweenvagueregions. 123

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2.2.2 ,ourprototypeimplementationofVASAfollowsthesamelineofthedenitionofitsconceptsandit 124

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55 ].Finally,webelievethattheVASAconceptandgenerallytheeortsforhandlingspatialvaguenesscanbenetfromstudyingthepossibilityofextendingVASAconcepts 126

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AlejandroPaulywasborninSanJose,CostaRicawhereheobtainedhisBSdegreeininformationsystemsengineering.Immediatelyaftergraduation,hemovedtoGainesvilletopursueagraduatedegreeincomputerscienceattheUniversityofFlorida.Hisresearchinterestsincludespatialdatabases,uncertaintyinspatialdata,databaseextensibilityandmorerecentlybiodiversityandecologicaldatamanagement.Heplanstopursueacareerineitherresearchorindustrywherehecanpositivelyaectpeople'slives. 134