Geometric clustering to minimize the sum of cluster sizes

  • 格式:pdf
  • 大小:204.24 KB
  • 文档页数:12

GeometricClusteringtoMinimize

theSumofClusterSizes

VittorioBil`o1,IoannisCaragiannis2,ChristosKaklamanis2,and

PanagiotisKanellopoulos2

1DipartimentodiMatematica“EnnioDeGiorgi”Universit`adiLecce,ProvincialeLecce-Arnesano,73100Lecce,Italy.2ResearchAcademicComputerTechnologyInstitute&DepartmentofComputerEngineeringandInformaticsUniversityofPatras,26500Rio,Greece

Abstract.Westudygeometricversionsofthemin-sizek-clusteringproblem,aclusteringproblemwhichgeneralizesclusteringtominimizethesumofclusterradiiandhasimportantapplications.Weprovethattheproblemcanbesolvedinpolynomialtimewhenthepointstobeclus-teredarelocatedonaline.ForEuclideanspacesofhigherdimensions,weshowthattheproblemisNP-hardandpresentpolynomialtimeap-proximationschemes.Thelatterresultyieldsanimprovedapproximationalgorithmfortherelatedproblemofk-clusteringtominimizethesumofclusterdiameters.

1Introduction

Clusteringisanareaofcombinatorialproblemswhichisbothalgorithmicallyrich

andpracticallyrelevant.Severalclusteringproblemshavebeenextensivelystud-

iedsincetheyhaveapplicationsinmanyfieldsincludingdatabasesystems,image

processing,datamining,informationretrieval,molecularbiology,andmore.

GivenasetofpointsX,wecallaclusteranynonemptysubsetofX.Aset

ofclustersisaclusteringforXifeachpointofXbelongstosomecluster.A

clusteringiscalledk-clusteringifitconsistsofatmostkclusters.Ingeneral,

clusteringproblemsarestatedasfollows:Aninstanceofsuchaproblemconsists

ofasetXofnpoints,adistancefunctiondist:X×X→Randaninteger

kandtheobjectiveistocomputeak-clusteringofthepointsinXminimiz-

ingf(C1,...,Ck),wherefisafunctiondefinedontheclusters,typicallyusing

thedistancefunctiondist.Dependingonthedefinitionofthefunctionf,many

differentclusteringproblemscanbedefined.Themostlystudiedonesarethe

k-center,k-median,andk-clustering.Theirobjectivesaretoassignthepointsto

atmostkclusterssothatthemaximumdistancefromanypointtoitscluster

center(k-center)orthesumofdistancesfromeachpointtoitsclosestclus-

tercenter(k-median)orthesumofalldistancesbetweenpointsinthesame

cluster(k-clustering)isminimized.TheseproblemsareNP-hardandseveralap-

proximationalgorithmshavebeenproposed[3,5,13]includingpolynomialtime

approximationschemesforgeometricinstancesoftheseproblems[1,2,10,16].Inthispaper,westudyavariationoftheproblemofclusteringasetofpoints

intoaspecificnumberofclusterssoastominimizethesumofclustersizes.The

sizeofaclustermaybeproportionaltotheradius/diameterofthecluster,toits

area,etc.Inparticular,minimizingthesumofclusterradii/diametershasbeen

suggestedasanalternativetothek-centerobjectiveincertainapplicationsso

astoavoidthedissectioneffect[8]:usingthemaximumdiameter/radiusasthe

objectivesometimesresultsinobjectsthatshouldhavebeenplacedinthesame

clustertobeplacedindifferentclusters.

Clusteringtominimizethesumofdiameters/radiihasbeenstudiedforpoints

inmetricspacesin[6]and[8].Anapproximationalgorithmwhichcomputesa

solutionwithatmost10kclustersofcostatmostafactorofO(logn/k)within

theoptimalsolutionforkclusterswaspresentedin[8].Thisresultwasim-

provedbyCharikarandPanigrahyin[6]whereanalgorithmthatcomputesa

constantapproximatesolutionusingatmostkclustersispresented.Inmetric

spaces,ρ-approximationalgorithmsforclusteringtominimizethesumofdi-

ametersgive2ρ-approximationalgorithmsforthecorrespondingradiiproblem

(andviceversa).Negativeresultsincludea2−󰀃inapproximabilityboundfor

minimizingthesumofdiametersinmetricspaces[8]whilethecomplexityofthe

correspondingradiiproblemisopen.Fornon-metrics,noapproximationbound

ispossiblefordiametersinpolynomialtimeunlessP=NPevenfork=3

[8].Whenkisfixed,theoptimalsolutionforradii/diameterscanbefoundin

polynomialtimebyenumeratingtheO(nk)possiblesolutions.Thepapers[12]

and[15]presentfastpolynomialtimealgorithmsforthecasek=2,addressing

theEuclideancaseaswell.Capoyleasetal.[7]studyageneralizedversionofthe

problemforpointsontheEuclideanplaneandshowthat,forfixedkandany

functionoftheclusterdiameters,itcanbesolvedinpolynomialtime.

Inthispaper,weconsidergeometricversionsofthemin-sizek-clustering

problem.Formally,aninstance(X,F,d,α)oftheproblemhasasetXofn

pointswithrationalcoordinatesonthed-dimensionalEuclideanspace,acost

functionFthatassociatesafixednon-negativecostwitheachpoint,anda

constantvalueα.Theobjectiveistocomputeak-clusteringCtogetherwith

centerpointsc∈XineachclusterCsuchthat󰀈

C∈CCOST(C)isminimized,

whereCOST(C)isdefinedas(maxp∈Cdist(p,c))α+Fcanddist(p,c)denotesthe

Euclideandistancebetweenthepointspandc.Thequantitymaxp∈Cdist(p,c)

istheradiusofclusterCwithcenterc.

Besidesitsimportanceforclusteringoptimization,anothermotivationfor

studyingthemin-sizek-clusteringproblemisthefollowingscenario.Assumethat

atelecommunicationagencywishestogivewirelessaccesstousersscatteredin