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NearestNeighbourSearchForXMLTreesLszlKovcs,TiborRpsi,ErikaBaksa-Vargakovacs@iit.uni-miskolc.hu,repasi@iit.uni-miskolc.hu,iitev@uni-miskolc.huAbstract:

Thereisasignificantresearcheffortonefficientcomputingofsimilaritiesbetweenobjectsofnontraditionaldatatypesasstrings,documents,soundtracksorpictures.ItisreasonabletousetheresultsoftheseeffortsintheproblemofXMLtreematching,too.AsanXMLdocumenthasatreestructureandthetreescanbetransformedintoalinearstructure,atreecanberegardedasaspecialkindofstring.Inourapproach,theresultsofstringcomparisonwillbeextendedtomeasurethesimilaritybetweenXMLtrees.Keywords:

Treeeditdistance,Nearestneighboursearching1IntroductiontoXMLandXPathTheExtensibleMarkupLanguage（XML）isasimple,veryflexibletextformatderivedfromSGML（StandardGeneralizedMarkupLanguage）[1].XMLprovidesawayofdescribingdatainarich,flexibleandefficientwaybymarkingupdatawithdescriptivetags.However,XMLdoesnotprovideawaytolocatespecificpiecesofstructureddatawithinagivendocument.Instead,theXMLPathLanguage（XPath）providesasyntaxforlocatingspecificpartsofanXMLdocumenteffectivelyandefficiently.ItgetsitsnamefromitsuseofapathnotationasinURLsfornavigatingthroughthehierarchicalstructureofanXMLdocument.InXPath,anXMLdocumentisviewedconceptuallyasatreeinwhicheachpartofthedocumentisrepresentedasanode[2].ThestandardXPathmodelassumesrandomaccesstoXMLdataandalsotheabilitytoqueryXMLdataatanytime.XPathisnotastructurallanguagelikeXML;rather,itisastring-basedlanguageofexpressions.Anexpressionisevaluatedtoyieldanobject,whichhasoneofthefollowingfourbasictypes:

node-set（anunorderedcollectionofnodeswithoutduplicates）;boolean;numberorstring.XPathexpressionsoftenoccurinXMLattributes.Expressionevaluationoccurswithrespecttoacontext.Thecontextconsistsof:

anode（thecontextnode）,apairofnon-zeropositiveintegers（thecontextpositionandthecontextsize）,asetofvariablebindings,afunctionlibraryandthesetofnamespacedeclarationsinscopefortheexpression[3].Animportantkindofexpressionisalocationpath.Therearetwokindsoflocationpaths:

relativelocationpathsandabsolutelocationpaths.

（1）Arelativelocationpathconsistsofasequenceofoneormorelocationstepsseparatedby/,andeachstepinturnselectsasetofnodesrelativetothecontextnode.Alocationstephasthreeparts:

anaxis,whichspecifiesthetreerelationshipbetweenthenodesselectedbythelocationstepandthecontextnode;anodetest,whichspecifiesthenodetypeandexpanded-nameofthenodesselectedbythelocationstep;andzeroormorepredicates,whichusearbitraryexpressionstofurtherrefinethesetofnodesselectedbythelocationstep.Thenode-setselectedbythelocationstepisthenode-setthatresultsfromgeneratinganinitialnode-setfromtheaxisandnode-test,andthenfilteringthatnode-setbyeachofthepredicatesinturn.

（2）Anabsolutelocationpathconsistsof/optionallyfollowedbyarelativelocationpath.A/byitselfselectstherootnodeofthedocumentcontainingthecontextnode.Ifitisfollowedbyarelativelocationpath,thenthelocationpathselectsthesetofnodesthatwouldbeselectedbytherelativelocationpathrelativetotherootnodeofthedocumentcontainingthecontextnode[3].Thusalocationpathselectsasetofnodes.Theresultofevaluatinganexpressionthatisalocationpathisthenode-setcontainingthenodesselectedbythelocationpath.Theinitialnode-setconsistsofthenodeshavingtherelationshiptothecontextnodespecifiedbytheaxis,andhavingthenodetypeandexpanded-namespecifiedbythenodetest.Theinitialnode-setisfilteredbythefirstpredicatetogenerateanewnode-set;thisnewnode-setisthenfilteredusingthesecondpredicate,andsoon.Thefinalnode-setisthenode-setselectedbythelocationstep.Theaxisaffectshowtheexpressionineachpredicateisevaluatedandsothesemanticsofapredicateisdefinedwithrespecttoanaxis.Everyaxishasaprincipalnodetype.Fortheattributeaxis,theprincipalnodetypeisattribute;forthenamespaceaxis,theprincipalnodetypeisnamespace;forotheraxes,theprincipalnodetypeiselement[3].ThedisadvantageofXPathisthatitisbasedonpartialmatching.Itmeansthatsomepartsofthetreemustmatchthegivenpatternexactly,whileothersaretotallyignored.Theresultedmeasuredoesnottakeintoaccountthedeviationfromthepattern.However,thisdeviationvaluewouldcarryusefulinformationforusers.Ouraimistofindanefficientalgorithmthatmeasuresthedifferencebetweenthepatternandtheselectedsubtrees.ThereforeallowingsearchingfornearestneighbourapproximationsofthepatternXMLdocument.2ThedistancebetweentwoXMLtreesTheproblemofobjectmatchingisaninterdisciplinarareaasitcollectsmethodsfromthreecurrentdisciplines:

computerscience,statisticsandoperationsresearch.Thereisasignificantresearcheffortonefficientcomputingofsimilaritiesbetweenobjectsofnontraditionaldatatypesasstrings,documents,soundtracksorpictures.ItisreasonabletousetheresultsoftheseeffortsintheproblemofXMLtreematching,too.AsanXMLdocumenthasatreestructureandthetreescanbetransformedintoalinearstructure,atreecanberegardedasaspecialkindofstring.Inourapproach,theresultsofstringcomparisonwillbeextendedtomeasurethesimilaritybetweenXMLtrees.ThestringsandXMLtreescanbeconsideredasobjectslocatedinmetricspace.AspaceXismetricifthereisarealnon-negativefunctionoftwoobjectsd（A,B）defined.Thefunctionisknownasthedistancebetweenthetwopoints.Itischaracterizedbythefollowingproperties.ForAX,BX,andCX1.d（A,B）=0ifandonlyifA=B（thedistanceis0ifandonlyifthepointscoincide）;2.d（A,B）=d（B,A）（thedistancefromAtoBisthesameasthedistancefromBtoA）;3.d（A,B）+d（B,C）d（A,C）（thesumoftwosidesofatriangleisneverlessthanthethirdside）.ThebestknownstringmatchingalgorithmsincomputersciencearetheRusselSoundexcodingandtheLevenshteineditdistancemethods.TheSoundexmethodisusedinrecordlinkagemethodsmainlyforblockingthemasterrecordset.TheLevenshteineditdistanceisdefinedasthesmallestnumberofinsertions,deletions,andsubstitutionsrequiredtochangeonestringortreeintoanother.TheefficiencyofthealgorithmtocomputethedistancebetweenstringsisO（mn）,wheremandnarethelengthsofthestrings.TheLevenshteindistanceisdefinedforstringsofarbitrarylength.Itcountsthedifferencesbetweentwostrings,wherewewouldcountadifferencenotonlywhenstringshavedifferentcharactersbutalsowhenonehasacharacterwhereastheotherdoesnot.Ingeneral,thetransformationofstringscanbeperformedbetweenstringsofdifferentcharactersets.Thus,theinputparameterforstringdistanceproblemisatriple（A,B,c）,whereAisthecharactersetoftheinputstring,Bisthecharactersetoftheoutputstring,andcisthecostfunction.Thetransformationmayusuallyconsistsofthefollowingelementaryoperations:

insertion,deletionandsubstitution.Thereareseveralvariantsforelementarycostfunctions,amongwhichthesimplestoneistheunit-costfunction.Inthiscasetheinsertion,thedeletionofasinglecharacterhasthevalueof1,andthesubstitutionoftwodifferentcharactersisequalto1,too.Forastrings,lets（i）standforitsithcharacter.Fortwocharactersaandb,definec（0,a）=1（insertion）c（a,0）=1（deletion）c（a,b）=0ifa=b;1,otherwise（substitution）Assumewearegiventwostringssandtoflengthnandm,respectively.Thelengthvalueisdenotedbyanupperindex:

sn,tm..Thedistance,i.e.theminimalcostoftransformationfromstotisusuallycalculatedwithusingadynamicprogrammingalgorithm.Thedistancevalueisdefinedwitharecursiveformula:

d（sn,tm）=min{c（sn,tm）+d（sn-1,tm-1）c（sn,0）+d（sn-1,tm）c（0,tm）+d（sn,tm-1）wheresidenotesthecharateratthepositioniinstrings.Basically,editdistancecanbecomputedinO（nm）orO（nm/log（n））timeifthecostweightsarerational[7].Inalotofapplications,amoreprecisedistancevaluecanbeachievedwiththenormalizededitdistancemeasure.Thismeasuretakesnotonlythenumberofelementarytransformationsintoaccount,butalsothelengthoftheinputandoutputstrings.Thenormalizededitdistanceistheratioofthetotalweighttothelengthofthesequence.ThecomplexityofthenormalizededitdistanceisO（nmlog（n））[7].Thedistancevaluefortreescanbedefinedsimilarlytothedistanceofstrings.Animportantassumptioninourinvestigationisthatthelabelsofthetreesareordered.ThisassumptioncorrespondstotheXMLstandard.Usually,likein[4],thefollowingelementaryerationsaredefinedfortreeobjects:

-relabel:

assignsanewnodenametotherootofthetree-insert:

insertinganewnodeintothechildrenoftherootnode-delete:

deletinganodefromthechildrenoftherootnode-inserttree:

insertingatreeundertherootnode-deletingtree:

deletingatreefromthechildrenofthenodeInourapproach,thetreeinsertionortreedeletionoperatorsarereplacedwithalistofsinglenodeoperations.Accordingto[7],thetreedistancevaluecanbecalculatedusingthefollowingrecursiveformula:

d（0,0）=0d（F,0）=d（F-v,0）+c（v,0）d（0,F）=d（0,F-v）+c（0,v）d（F1,F2）=min{d（F1-v,F2）+c（T（v）,0）d（F1,F2-v）+c（0,T（v））d（F1-T