Recommendations for two-way selections using skyline view queries

28 Pages·2012·1.74 MB·English

KnowlInfSyst
DOI10.1007/s10115-012-0489-6
REGULAR PAPER
Recommendations for two-way selections using skyline
view queries
JianChen · JinHuang · BinJiang ·
JianPei · JianYin
Received:28November2010/Revised:3December2011/Accepted:6March2012
©Springer-VerlagLondonLimited2012
Abstract Westudyapracticalandnovelproblemofmakingrecommendationsbetween
twopartiessuchasapplicantsandjobpositions.Wemodelthecompetentchoicesofeach
partyusingskylines.Inordertomakerecommendationsinvariousscenarios,weproposea
seriesofskylineviewqueries.Tomakerecommendations,weoftenneedtoanswerskyline
view queries for many entries in one or two parties in batch, such as for many applicants
versusmanyjobs.However,theexistingskylinecomputationalgorithmsfocusonanswering
asingleskylinequeryatatimeanddonotconsidersharingcomputationwhenanswering
skyline view queries for many members in one party or both parties. To tackle the batch
recommendationproblem,wedevelopseveralefficientalgorithmstoprocessskylineview
queriesinbatch.Theexperimentresultsdemonstratethatouralgorithmssignificantlyout-
performthestate-of-the-artmethods.
Keywords Mutualrecommendation·Skylinequery·Multi-objectiveoptimization·
Stablematching
J.Chen
SouthChinaUniversityofTechnology,Guangzhou,China
B
J.Huang( )
SouthChinaNormalUniversity,Guangzhou510631,China
e-mail:[email protected]
B.Jiang
FacebookInc.,MenloPark,CA,USA
J.Pei
SimonFraserUniversity,Burnaby,Canada
J.Yin
SunYat-senUniversity,Guangzhou,China
123
J.Chenetal.
1 Introduction
Ithasbeenwellrecognizedthatskylinequeriesareimportantinmanymulti-criteriadecision-
makingapplications.Forexample,considerascenariowhereanemployerscreensapplicants
forajobposition.Supposeeachapplicanthastwoattributes:theexperiencelevelandthe
qualificationlevel,thehigherthebetter.Anapplicantadominatesanotherapplicantb,that
is,aisabetterapplicantthanbinthecontextofthisexample,ifaisbetterthanbinatleast
oneaspectandisnotworsethanbintheotheraspect.Amongasetofgivenapplicants,the
ones not dominated by any others in the set form the skyline. In other words, the skyline
applicantsareallpossibletrade-offsbetweenexperienceandqualificationthataresuperior
tootherapplicants.
Therearemanystudiesonefficientskylinecomputation,aswillbebrieflyreviewedin
Sect.3.Existingstudiesfocusoncomputingtheskyline(s)foragivendatasetineitherthe
full space or subspaces. In real applications when selections involve more than one party,
skylineanalysiscanbemuchmorecomplicated,whichmotivatesourstudy.
Consider a headhunter making recommendations between job applicants and job posi-
tions.GivenasetofapplicantsinTable1andasetofpositionsinTable2,supposethehigher
theexperiencelevelandqualificationlevel,thebetteranapplicant,andthehigherthesalary
levelandthebenefitlevel,themoreattractiveajob.InTable1,eachapplicantindicatesthe
jobs interesting to her/him. Symmetrically, each job has some requirement on applicants.
InTable2,eachjobhasalist ofapplicantswho satisfy therequirement ofthejob.Those
applicantsarequalifiedforthejob.
Anaturalquestionfromanapplicantistofindtheskylinejobsthattheapplicantisqualified.
Wecallthisthereciprocalskylineoftheapplicant,whichrepresentsthebestchoicesforthe
applicant.Forexample,AdainTable1isqualifiedforjobsJ1,J2,andJ3.J2dominatesJ3,
sinceJ2isbetterthanJ3inbothsalaryandbenefit.Thus,Ada’sreciprocalskylineconsists
ofJ1andJ2,thoughAdaisnotcurrentlyinterestedinJ1.ThisinformationmayhelpAda
to adjust her interest. Symmetrically, we can also provide a job the reciprocal skyline of
applicantsthatcontainsthebestapplicantswhoareinterestedinthejob.
Anapplicantmayalsowanttoknowwhichjobsregardtheapplicantaskylineapplicant.
We can model such an information need by an inverse skyline query. For example, Cathy
inTable1isinterestedinandalsoqualifiedforjobsJ1,J2,andJ3.Unfortunately,Cathy’s
inverseskylineisemptysinceCathyisnotintheskylineofanyjobs.Knowingherinverse
skyline,Cathymayadjustherexpectationandjobhuntingstrategyproperly,sinceshemay
Table1 Asetofapplicants
Applicant Experience Qualification Interest
Ada 4 3 J2,J3
Bob 2 4 J1,J3
Cathy 3 1 J1,J2,J3
Dan 1 2 J2,J3
Table2 Asetofjobpositions
Job Salary Benefit Interest
J1 80K 1 Ada,Bob,Cathy
J2 50K 3 Ada,Cathy,Dan
J3 40K 2 Ada,Bob,Cathy,Dan
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Recommendationsfortwo-wayselections
not be the first choice of any jobs. Symmetrically, we can also provide a job the inverse
skylineofapplicantsthatcontainstheapplicantswhoviewthejobaskylineone.
Bothreciprocalskylinequeriesandinverseskylinequeriesaretheexamplesofaseriesof
skylineviewqueriestobediscussedinthispaper.Asillustrated,variousskylineviewque-
riesmaybeaskedinthecourseofrecommendationsfortwo-wayselections.Althoughthose
queriescanbeansweredforoneapplicantoronejobindividuallybyadaptingsomeexisting
skylinecomputationalgorithms,aheadhunteroftenhastoanswersuchqueriesinbatchforall
oralargenumberofapplicantsand/orjobs.Forexample,onlinejobsearchengines,suchas
mon\discretionary-ster.com,havemillionsofjobpostsandregisteredapplicants.Evenmore,
thousandsofnewjobpostsandapplicantsareaddedeveryday.Thesearchengineshaveto
makerecommendationsfornewjobsandapplicantsaswellasupdatetherecommendations
forexistingjobsandapplicants.Suchsystemsneedtoprocessmillionsofskylineviewque-
ries periodically. The existing skyline computation algorithms only consider computing a
singleskylineatatimeandthusarenotoptimizedforansweringmultiplequeriesinbatch.
Canweanswerabatchofskylineviewqueriesinamoreefficientandmorescalablewaythan
processingthemonebyone?Itisfarfromtrivialtodevelopefficientcomputationsharing
strategieswhencomputingmanyskylinesatthesametime.
The scenario of matching applicants and jobs is just one example of recommendations
between two parties towards two-way selections. Recommendations between two parties
fortwo-wayselectionsareneededinmanyapplications.Onlinedatingservicesrecommend
appropriate dates for men and women, who may specify requirements on various factors
suchasage,location,andincome.Onlineadvertisementserviceapplications,suchasGoogle
AdSense,makemutualrecommendationsbetweenwebpagesandadvertisers.Skylineview
queriesarehighlyusefulinrecommendationsbetweentwopartiesfortwo-wayselections.
However, to the best of our knowledge, skyline queries for recommendations in two-way
selectionhavenotbeenstudiedsystematically.
In this paper, we identify a series of skyline view queries useful for recommendations
betweentwopartiesanddevelopseveralnovelalgorithmstoanswertheskylineviewqueries
inbatchefficiently.Anexperimentalstudyindicatesthatourbatchalgorithmssignificantly
outperformansweringqueriesindividuallyusingtheexistingmethods.
Therestofthepaperisorganizedasfollows.InSect.2,weformulate8typesofskyline
viewqueries.WereviewtherelatedworkinSect.3.Section4developsefficientalgorithms
foransweringthe8typesofskylineviewqueries.Wereportanextensiveempiricalstudyin
Sect.5.Section6concludesthepaper.
2 Skylineviewqueries
Inthissection,wefirstrecallthepreliminariesofskylines.Then,weformallydefine8skyline
viewqueries.
2.1 Preliminaries
Consider a set of points S in a multidimensional space D = (D ,...,D ). To keep our
1 d
discussionsimple,weassumethatthedomainofeachattribute D (1 ≤ i ≤ d)isnumeric.
i
Ourdiscussioncanbegeneralizedtoattributesofothertypeswherepartialordersaredefined.
Considertwopointsx,y ∈ S.xissaidtodominatey,denotedbyx (cid:4) y,ifforeveryattri-
buteD , x.D ≥ y.D ,andthereexistsoneattributeD (1≤ j ≤d)suchthatx.D > y.D .
i i i j j j
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J.Chenetal.
Table3 Asummaryof
Notion Meaning
frequentlyusednotions
V(A) TheviewofA,{J ∈J|A.P(J)=true}
IV(A) TheinverseviewofA,{J ∈J|J.P(A)=true}
SV(A) TheskylineviewofA, Sky(V(A))
iSky(A) TheinverseskylineofA,{J|A∈SV(J)}
rSky(A) ThereciprocalskylineofA, Sky(IV(A))
MV(A) ThemutualviewofA,{J|J ∈V(A)∧A∈V(J)}
SMV(A) TheskylinemutualviewofA, Sky(MV(A))
SIS(A) TheskylineofinverseskylineofA, Sky(iSky(A))
Apointx ∈ SisaskylinepointinSiftheredoesnotexistanotherpointx(cid:7) ∈ Ssuchthat
x(cid:7) (cid:4)x.TheskylineofS,denotedbySky(S),isthesetofskylinepointsinS.
ApredicatePdefinedinspaceDspecifiesaregionineitherDorasubspaceofD.Apoint
x issaidtosatisfy P,denotedby P(x) = true,ifx fallsintheregionspecifiedby P.We
alsooverloadsymbol P todenotetheregionspecifiedbythepredicateofthesamename.
Inthispaper,wefocusonsimplepredicateswhoseregionsarerectangles.Tobeconcrete,a
predicatespecifiesthemaximumand/orminimumvaluesonsomeattributes.Forexample,
anapplicantmayspecifythepredicateofhisjobinterestsasSalary>50K,andajobposition
mayspecifythepredicateoftheapplicantrequirementsasExperience>3andQualification
∈[2,3].
2.2 Skylineviewqueries
Inthispaper,westudyskylineviewqueriesinvolvingtwoparties.Tomakeourdiscussion
easytofollow,weuse“applicants”and“jobs”asthealiasesofthetwopartiesinthequestion.
Ifoneprefersbeingformal,wecanalwaysreplace“applicants”and“jobs”by“Party1”and
“Party2,”respectively,inthemathematicalstatements.
GivenanapplicantdatasetAandajobdatasetJ,westudytheproblemofrecommen-
dationsbetweenthetwodatasets.
Inpractice,itmaynotbeeasytoobtaintheexplicitandcompleteinterestofanapplicant
orajob.Instead,anapplicantmayspecifytheminimumand/ormaximumrequirementson
attributessuchassalaryandbenefit.Hence,wemodeltheinterestofanapplicant A ∈ A
asapredicate A.P inthespaceofthejobdatasetJ.Symmetrically,thequalificationofa
job J ∈ J isalsomodeledasapredicate J.P inthespaceoftheapplicantsA.Applicant
A isinterestedinjob J if J satisfiesthepredicateof A,thatis, A.P(J) = true.Job J is
interestedinapplicant A,thatis,theapplicantqualifiesjob J,if Asatisfiesthepredicateof
J,thatis, J.P(A)=true.
Intherestofthissection,weonlydefinetheskylineviewqueriesforapplicants.Their
counterpartsforjobscanbeobtainedsymmetrically.Table3summariesthe8typesofskyline
viewqueries.Tohelpunderstandthequeries,Table4showstheresultsofthesequerieson
thedatasetsinTables1and2.
Definition1 (View)Theviewofanapplicant A∈AconsistsofalljobsinJ satisfyingthe
predicateof A,thatis,V(A)={J ∈J|A.P(J)=true}.
Symmetrically, we define the view of a job J with predicate J.P as V(J) = {A ∈
A|J.P(A)=true}.
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Recommendationsfortwo-wayselections
Forexample,inTable1,Ada’sviewincludesJ2andJ3.
Itisalsoimportantforanapplicanttoknowthejobsthatshe/heisqualified.Thiscanbe
modeledbytheinverseviewoftheapplicant.
Definition2 (Inverseview)Theinverseviewofanapplicant AisthesetofjobsinJ that
areinterestedin A.Thatis, IV(A)={J ∈J|J.P(A)=true}.
Ifajob J isintheinverseviewofapplicant Athen Aisintheviewof J,andviceversa.
Therefore, IV(A) = {J ∈ J|A ∈ V(J)}. Similarly, the inverse view of a job J ∈ J is
IV(J)={A∈A|A.P(J)=true}={A∈A|J ∈V(A)}.
Foranapplicant A, V(A)includesalljobsthatsatisfy A’spredicate.Amongthesejobs,
theonesintheskylineofV(A)aremoreattractiveto Asincetheyarenotdominatedbyany
otherjobsinV(A).
Definition3 (Skylineview)Foranapplicant A ∈ A,the skylineview of A istheskyline
of the jobs that A is interested in, that is, the skyline of A’s view denoted by SV(A) =
Sky(V(A)).
Symmetrically,theskylineviewofajob J ∈AisSV(J)= Sky(V(J)).
AnapplicantAmayliketoknowthejobsJwhoseskylineviewSV(J)containsA.Those
jobsaretheones Ahasgoodpromise.
Definition4 (Inverseskyline)Foranapplicant A∈A,theinverseskylineof Aisthesetof
jobswhoseskylineviewscontain A,denotedbyiSky(A)={J|A∈ SV(J)}.
Jobsintheinverseview IV(A)of Ashouldberecommendedto A,since Aisqualified
thosejobs.Moreover,AshouldpaymoreattentiontothejobsintheskylineofIV(A)because
theyarethebestjobs Acanget.Wemodelthissetofjobsasthereciprocalskylineof A.
Definition5 (Reciprocalskyline)Foranapplicant A ∈ A,thereciprocalskyline of A is
theskylineofthesetofjobsthat Aisqualified,denotedbyrSky(A)= Sky({IV(A)}).
An important task for headhunters is to match applicants and jobs. When an applicant
andajobsatisfytherequirementofeachother,theyhavemutualinteresttoeachother.For
an applicant, the set of jobs which the applicant is interested in and is qualified form the
applicant’smutualview.
Definition6 (Mutualview)Foranapplicant A ∈ A,themutualview of A is MV(A) =
{J|J ∈V(A)∧ A∈V(J)}.
Furthermore,Adamaybeinterestedintheskylineinhermutualview,capturedbyher
skylinemutualview,whichconsistsofthebestjobswhichsheisqualifiedandinterestedin.
Definition7 (Skylinemutualview)Foranapplicant A ∈A,theskylinemutualviewof A
isSMV(A)= Sky(MV(A)).
Theinverseskylineofanapplicantcaresaboutthedominancerelationshipsamongappli-
cants.Foreachjobinanapplicant’sinverseskyline,heisagoodcandidate(i.e.,doesnot
dominatedbyanyotherapplicant).However,thejobsinhisinverseskylinemaybedominated
byotherjobs,andthisdominancerelationshiphasnotbeenexploreedininverseskylines.
Therefore, we consider the skyline of an applicant’s inverse skyline and call it theskyline
ofinverseskyline.Theskylineofinverseskylineconsistsofthebestjobsthatconsiderthe
applicant as one of their best candidates. It takes into account the preferences of both the
applicantsandthejobs.
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J.Chenetal.
Definition8 (Skyline of inverse skyline) For an applicant A ∈ A, the skyline of inverse
skylineof AisSIS(A)= Sky(iSky(A)).
Itiseasytoseethataviewqueryisamultidimensionalrangequerywherethepredicate
specifiesthesearchregion.Aninverseviewqueryisastabbingquery.Theinverseviewof
anapplicantconsistsofalljobswhosepredicateregionscontaintheapplicantinthespace
ofapplicants.Theother6typesofqueriesareessentiallyskylinequeriesonasubsetofthe
originaldataset.
Straightforwardly,wecanemployeeanexistingrangequeryalgorithm,astabbingquery
algorithmandaskylinealgorithmtoanswerthosequeries.Intwo-wayselectionrecommen-
dationsystems,recommendationsmaybemadetomanyorallmembersofoneorbothparties
inbatch.Forexample,aheadhuntermaywanttomakerecommendationstoallapplicants
andjobs.Apparently,whenansweringaqueryforallindividuals,muchcomputationmaybe
redundant.Forexample,somejobsarecomparedagainandagainwhencomputingskyline
viewsofdifferentapplicants.Therefore,inSect.4,wedevelopalgorithmstoanswereach
typeofqueriesforallapplicants/jobsinbatchmodeinordertoshareasmuchcomputation
aspossible.
2.3 Properties
Followingwiththedefinitionsoftheskylineviewqueries,wehaveseveralinterestingprop-
ertiesaboutthequeries.
Property1 Forany A ∈ A, SV(A) ⊆ V(A), iSky(A) ⊆ IV(A), SMV(A) ⊆ MV(A),
andSIS(A)⊆iSky(A).
Property2 Forany A∈A, MV(A)=V(A)∩IV(A).
We observe that every job in the inverse skyline view of A is also in the inverse view
of A.
Property3 Foranyapplicant A∈A, iSky(A)⊆ IV(A).
Proof1 According to Definition 4, for any job J ∈ iSky(A), A ∈ Sky(V(J)). Thus,
A∈V(J).Thatis, J ∈ IV(A).Therefore,iSky(A)⊆ IV(A). (cid:10)(cid:11)
Figures1and2showthederivativerelationshipsandthecontainmentrelationshipsamong
the8typesofskylineviewqueriesaccordingtotheaboveproperties.Itiseasytoverifythat
theabovepropertiesholdfortheexampleshowninTable4.
3 Relatedwork
Ourstudyishighlyrelatedtotheexistingworkonskylinecomputationandthestablematch-
ingproblem.
3.1 Skylinecomputationonsingledataset
Since Börzsönyi et al. [1] introduced the skyline operator into the database community,
therehasbeenagreatamountofresearchdedicatedtoskylinequeryprocessingaswellas
extensionsandapplicationsofskylineinpreference-baseddataanalysis.
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Recommendationsfortwo-wayselections
Fig.1 Thederivativerelationshipsofthe8typesofskylineviewqueries
Fig.2 Thecontainment
relationshipsamongthe8types
ofskylineviewqueries
rSky
iSky
Table4 Theresultsofthe8skylineviewqueriesondatasetsinTable1and2
Queryonapplicants Ada Bob Cathy Dan
V J2,J3 J1,J3 J1,J2,J3 J2,J3
IV J1,J2,J3 J1,J3 J1,J2,J3 J2,J3
SV J2 J1,J3 J1,J2 J2
iSky J1,J2,J3 J1,J3 ∅ ∅
MV J2,J3 J1,J3 J1,J2,J3 J2,J3
SMV J2 J1,J3 J1,J2 J2
rSky J1,J2 J1,J3 J1,J2 J2
SIS J1,J2 J1,J3 ∅ ∅
Queryonjobs J1 J2 J3
V Ada,Bob,Cathy Ada,Cathy,Dan Ada,Bob,Cathy,Dan
IV Bob,Cathy Ada,Cathy,Dan Ada,Bob,Cathy,Dan
SV Ada,Bob Ada Ada,Bob
iSky Bob,Cathy Ada,Cathy,Dan Bob
MV Bob,Cathy Ada,Cathy,Dan Ada,Bob,Cathy,Dan
SMV Bob,Cathy Ada Ada,Bob
rSky Bob,Cathy Ada Ada,Bob
SIS Bob,Cathy Ada Bob
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J.Chenetal.
Thealgorithmsofskylinecomputationfallintotwocategories:thenon-indexedalgorithms
and the index-based algorithms. The non-indexed algorithms do not assume any index on
thedata.SuchalgorithmsincludetheDivideandConqueralgorithmandtheBlockNested
LoopsalgorithmbyBörzsönyietal.[1],theSortFilterSkylinealgorithmbyChomickiet
al.[2],andtheLinearEliminationSortforSkylinealgorithmbyGodfreyetal.[3].
Differentfromthenon-indexedalgorithmswhichhavetoscanthewholedatasetatleast
once,index-basedalgorithmstakeadvantageofvariouspre-builtindexesondatasets,sothat
theyonlyaccessaportionofadatasettocomputetheskyline.TheBitmapalgorithmandthe
IndexalgorithmbyTanetal.[4],theNearestNeighboralgorithmbyKossmannetal.[5],
andtheBranchandBoundSkylinealgorithmbyPapadiasetal.[6,7]belongtothiscategory.
Particularly,R-trees[8]havebeenusedextensivelyinindex-basedalgorithms[9].When
computingtheskylineofasetofobjects,theR-treeindexprovidesapossibilitytofilterout
asubsetofobjectswithoutexaminingthemonebyone.Becausethemaximumcornerofthe
MBBofanodeintheR-treedominatesallobjectscontainedintheMBB,ifthemaximum
cornerisdominatedbysomeotherobject,thennoneoftheobjectscontainedintheMBB
canbeintheskyline.
TheBranchandBoundSkylinealgorithm(BBSforshort)[6,7]isanR-treebasedskyline
algorithm and is one of the fastest algorithms for computing skyline in practice. Given a
setofpointsindexedbyanR-tree,BBStraversestheR-treetocomputetheskyline.BBS
maintainsamin-heap H builtagainsttheminimumdistancetotheoriginofthedataspace
mindist ofeveryentry(node).Thealgorithmproceedsiteratively.
Atthebeginning,entriesattherootareinsertedintoH.Ineachiteration,thetopelemente
ofHisprocessed.Ifeisfullydominated(i.e.,theminimumcornerofe.MBBisdominated)
byaknownskylinepoint,theneisdiscarded.Otherwise,ifeisadatapoint,thenitisoutput
asaskylinepoint;ifnot,eisdiscardedandthoseentriesate(thatis,thechildrenofe)which
are not fully dominated by any known skyline point are inserted into H. An in-memory
R-treeonknownskylinepointsismaintainedinordertofacilitateexaminingthedominance
relationship.Thealgorithmterminateswhen H isempty.
BBSensuresthateachoutputpointisintheskyline.Moreover,BBSisI/Ooptimal.
Inthispaper,weassumethateverydatasetisindexedbyanR-tree[8]anduseBBSas
thebaselineinourexperiments.
Recently,thereismuchresearchstudyingskylinesinmultidimensionalsubspaces.Peiet
al.[10,11]andYuanetal.[12]proposedaskylinecubedatastructurethatcompletelypre-
computestheskylinesofallpossiblesubspacesforagivendataset.Xiaetal.[13]addressed
theincrementalmaintenanceofskylinecubes.Taoetal.[14]developedtheSUBSKYalgo-
rithmtoanswersubspaceskylinequeriesefficientlyinanysubspaces.Totackletheproblem
ofskylinesinhighdimensionalspaces,Chanetal.[15]relaxedthedefinitionofdominance
tok-dominanceandproposedk-dominantskylines.
Therearealsonumerousstudiesonskylinecomputationindifferentenvironments,suchas
skylinequeryprocessingoverdatastreams[16–20]andtimeseries[21],computingskylines
in distributed systems [22,23] and on mobile lightweight devices such as MANETs [24],
spatialskylinequeries[25],skylinecomputationinpartiallyordereddomains[26],skyline
queriesinmetricspace[27],skylinecomputationonuncertaindata[28,29],usingskylines
tomineuserpreferencesandmakerecommendations[30,31].
Tothebestofourknowledge,wearethefirsttostudyusingskylinequeriesforrecommen-
dationintwo-wayselections.Althoughtheproposedskylineviewqueriescanbeanswered
byexistingmethodsindividually,thosemethodsonlyconsideransweringasinglequeryata
time.Itisfarfromtrivialtodevelopefficientalgorithmsforansweringskylineviewqueries
inbatch.
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Recommendationsfortwo-wayselections
3.2 Skylinecomputationinvolvingtwodatasets
Mostofthepreviousworkonlyconsiderscomputingskylines(ortheirvariants)onasin-
gle data set, whereas our skyline view queries deal with two distinct data sets. Recently,
Dellis et al. [32] studied the concept of reverse skyline and bichromatic reverse skyline.
Theyconsiderasetofproductsdescribedbymultiplefeaturesandasetofcustomers,each
of whom specifies his/her favorite product features as a data point in the product space.
Given a product q, the reverse skyline consists of the customers whose dynamic skyline
containsq.Thedynamicskylineofacustomerpreference p correspondstoatransformed
dataspacewhere p becomestheoriginandallotherproductsarerepresentedbytheirdis-
tance vectors to p. Lian et al. [29] studied the bichromatic probabilistic reverse skyline
queries over uncertain data. Wu et al. [33] proposed several heuristics to enhancing the
performance.
Ourskylineviewqueriesarefundamentallydifferenttobichromaticskylinesandreverse
skylines.
Fromtheapplicationpointofview,bichromaticskylinesandreverseskylinescareabout
only the preference on one party. Customers have interests and preferences in products,
but products cannot choose customers. However, our skyline views queries consider two-
way mutual selections between two parties and both parties have preferences in each
other.
Giventhisfundamentaldifference,ourproblemsaremodeleddifferentlytobichromatic
skylinesandreverseskylines.Bichromaticandreverseskylinescomputetheskylineofthe
datasetofonepartyinatransformedspaceaccordingtothequerypoint,andtheyconsider
allpointsincomputation.Ontheotherhand,ourskylineviewqueriesessentiallycompute
theskylineofasubsetofonepartyofthedatasetaccordingtotheconstraintsspecifiedby
thequery,andwedonotusespacetransformation.
Asaresult,thequeryansweringtechniquesarequitedifferent.Thealgorithmsforbichro-
maticandreverseskylinesfocusonreducingtheoverheadofdataspacetransformationin
ordertoansweronequery.However,ouralgorithmsaimatminimizingrepetitivecomputation
whenansweringmultiplequeriesinbatch.
Nevertheless,wepointedoutthattheinverseskylines,andthereverseskylinesareequiv-
alenttoeachotherforspecialinputdataasfollows.
– Forinverseskylines,foreachapplicant,choosehis/herpredicatesuchthatalljobsinthe
jobdatasetsatisfythispredicate.
– Forreverseskylines,foreachcustomer,choosehis/herpreferencetobetheoriginpointin
theproductspacesuchthatnospacetransformationisneededtoanswerareverseskyline
query.
Intheabovetwospecialcases,theinverseskylineofajobisthesameasthereverseskyline
ofaproduct.Infact,theinverseskylineandthereverseskylineofajob/productareequalto
thesetofapplicants/customers,ifthisjob/productisintheskylineofthesetofjobs/products.
Otherwise,theinverseskylineandthereverseskylineofthisjob/productareanemptyset.
Inourexperiments,weexperimentallycompareouralgorithmsforinverseskylinesandthe
state-of-the-artalgorithmsforreverseskylines[33].
3.3 Thestablematchingproblem
Making recommendations between two parties is related to the classical stable matching
problem[34](alsoknownasthestablemarriageproblem).Anapplicantandajobcanbe
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J.Chenetal.
matched only if they satisfy the requirements from each other. In a matching between all
(cid:7)
applicantsandalljobs,assumeapplicant Aispairedwithjob J,andapplicant A ispaired
(cid:7) (cid:7) (cid:7)
withjob J .If Aand J satisfytherequirementsfromeachother,and Aprefers J to J and
(cid:7) (cid:7) (cid:7)
J prefers Ato A,thenthepair Aand J iscalledadissatisfiedpair.Amatchingissaidto
bestableifthereisnodissatisfiedpair.
Intheoriginalversionofthestablemarriageproblem[34],theapplicantsetandthejob
sethavethesamecardinality(denotedbyn).Thepreferenceofeachapplicantonjobsisa
completerankedlist(i.e.,includingalljobsinthejobset)withouttie.Thepreferenceofeach
jobonapplicantsisalsosucharankedlist.GaleandShapley[34]showedthattherealways
existsastablematchinganditcanbefoundinO(n2)time.
Numerousvariantsofthestablemarriageproblemarestudied,suchasthecollegeadmis-
sionproblemwhereacollegecanacceptmorethanonestudent[34],theroommateassign-
mentproblemwheretwopartiesarefromthesameset[35],thestablematchingproblemwith
incompletepreferencelistsand/orwithties[36–38],etc.Ingeneral,thepreferencecanbea
partialorder.Inthiscase,theproblemisprovedNP-hardandhardtobeapproximated[38].
Inthispaper,thedominancerelationshipisemployedtomodelthepreference,whichinfact
definesapartialorder.
Althoughthestablemarriageproblemandourproblemsharesimilarmotivations,they
are very different in terms of the goal and the computational issues. The stable marriage
focuses on the global picture and tries to find an assignment such that every one in both
parties is satisfied. Each individual does not have the ability to choose after the assign-
ment is made. However, instead of giving the matching pairs straightforwardly, our paper
addressesaseriesofflexiblequeriesformutualdecision-makingparties.Ourproblemcares
abouttherecommendationsforindividuals.Weprovidetheresultsofskylineviewqueriesas
recommendationswhicharegoodcandidatesbasedoneachindividual’sinterest.Individuals
canthenmakedecisionsandchoicesfromourskylineviewquerieswithoutprocessingthe
originallargedataset.
4 Queryansweringalgorithms
In this section, we develop algorithms to answer the 8 skyline view queries in batch. In
particular,wedesignefficientalgorithmsforreciprocalskylinequeriesandinverseskyline
queriesinSects.4.2and4.3,respectively.Otherqueriescanbereducedtothesetwoqueries
asdiscussedinSect.4.1.Weassumethatboththeapplicantdatasetandthejobdatasetare
indexedbyR-trees.Weonlydescribethealgorithmsforqueriesonapplicantssincethesame
algorithmscanbeappliedtoanswerqueriesforjobssymmetrically.
4.1 Queryreduction
Accordingtothedefinitionofviewqueriesandthatofinverseviewqueries,withlineartime
complexity, we can derive the inverse views of all applicants from the views of all jobs.
Thus, an inverse view query can be answered efficiently using the view query answering
algorithm.Thispropertyalsoholdsbetweenskylineviewqueriesandinverseskylineview
queries.Besides,amutualviewcanbeobtainedbyintersectingthecorrespondingviewand
thecorrespondinginverseview.
Example1 (Query Reduction) Consider the example in Table 4. By scanning the view of
jobsJ1whichis{Ada,Bob,Cathy},wecanassignJ1totheinverseviewsofAda,Bob,and
Cathy.Similarly,weassignJ2totheinverseviewsofAda,Cathy,andDan,andJ3tothe
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