Abstract. A data cube is a popular organization for summary data. A cube is simply a multidimensional structure that contains in each cell an aggregate value, i.e., the result of applying an aggregate function to an underlying relation. In practical situat

LOGLINEAR-BASEDQUASICUBES259one(thelaterusedtodemonstratescalability).FinallySection4presentstheconclusionsandfuturework.

2.Ourmethod

Inthissectionwedescribeindetailhowwecompressthedatacubebymeansofconstructingloglinearmodelsfordenseregionsofthecube.Wemodelregionsofthecorecuboid2andemploythesemodelstoestimatethevaluesoftheindividualcells.Thereasontofocusinthecorecuboidissimple:theerrorguaranteesforqueriestothecorecuboidholdforanyothercuboidinthelattice.(Inreality,asweshallsee,theerrorswhenweaggregatecellsofthecorecuboiddecreasedramatically.)

Aswestatedintheintroduction,toavoidincurringinlargeerrorsbytheestimation,weretainallthecellvalueswhoseestimationsarefartherawayfromtherealvaluebymorethanapre-establishedthreshold.Thisthresholdbecomestheguaranteeoftheapproximateanswer.(Aswewillshowlater,manyanswersare,inrealityclosertotherealanswerthanwhatthethresholdpredicts.)Westorethemodelparameters(foreachmodeledregionofthecuboid)alongwiththeretainedcellstoprocessthequeries.

Theissuesinvolvedincompressingthecube,givingapproximateanswerstothequeriesandevaluatingourtechniquecanbesummarizedasfollows:

Selectingchunksofthecorecuboidthatwillbedescribedbymodels(regionsofthecorecuboidthataresuf cientlydense).

Foreachchunktobemodeled,computingthemodelparametersbasedonthedatacon-tainedinthechunkandthen,basedontheestimatedvaluescomputeforeachnon-zerocellinthechunktheestimationerroranddetermineifthecellneedstoberetained.

Organizingthemodelparametersandretainedcellstoef cientlyaccessthemwhenprocessingqueries.

Evaluatingthetradeoffsbetweenspacetakenbythecompressedcube,responsetimeforthequeriesandthequalityoftheapproximateanswersobtained(errors).

Intherestofthissection,wedescribeindetailhowwesolvedeachoftheseissues.

2.1.Dividingthecorecuboid

Inordertoselectthechunksinwhichwedividethecorecuboidweuseadensitybasedapproachcalledhierarchicalclustering(JainandDubes,1988)togethighdensityportionsofthecube.Thisapproachhasbeenpreviouslyutilizedtoidentifyregionsofhighdensityinmultidimensionaldata(Agrawaletal.,1996).(TheaiminAgrawaletal.(1996)istohaveadensityapproachtoclustering,whereaclusterisde nedasaregionwithhigherdensityofpointsthanitssurroundingregions.)Weassumethatthecorecuboidhasbeencomputed(analgorithmsuchastheonepresentedinRossandSrivastava(1997)iswellsuitedforthetask).Givenad-dimensionaldatacubewithdimensionsA={D1,D2,...,Dd},wheredisthenumberofdimensions,weassumeitbeasetofbounded,totallyordereddomainsspace.3Accordingly,S= D1 × D2 ×···× Dd ,isthepossiblenumberofcellsin