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1、FewestRepeaters131FewestRepeatersforaCircularArea:IterativeExtremalOptimizationBasedonVoronoiDiagramsWenqiangWangYueCaoZimoYangUniversityofElectronicScienceandTechnologyChengdu,Sichuan,ChinaAdvisor:TaoZhouAbstractWeproposeatwo-tierednetworkinwhichlower-
2、poweruserscommuni-catewithoneanotherthroughrepeaters,whichamplifysignalsandretrans-mitthem,havelimitedcapacity,andmayinterferewithoneanotheriftheirtransmitterfrequenciesarecloseandtheysharethesameprivate-linetone.Ourobjectiveisthefewestrepeaterssothatei
3、thereveryuseriscoveredbyatleastonerepeaterorelseeveryusercancommunicatewithanyotheruseranywhereintheconsideredarea.Motivatedbycellularnetworks,wegiveana¨ıvesolutionwherethenum-berofrepeatersandtheirpositionscanbeobtainedanalytically.Inacircularareawithr
4、adius40miles,12repeaterscanaccommodate1,000simultaneoususers.WefurtherproposeaniterativerefinementalgorithmconsistingofthreefundamentalmodulesthatdrawtheVoronoidiagram,determinethecenterswww.madio.net数学中国ofthecircumscribedcirclesoftheVoronoiregions,andes
5、capethelocalopti-mumbyusingextremaloptimization.Thealgorithmobtainsasolutionwith11repeaters,whichweprovetobetheabsoluteminimum.For10,000users,ituses104repeaters,betterthanthena¨ıvesolution’s108.Wefurtherdiscusshowtoassignfrequenciesandprivate-linetones(
6、basedonmaximumandminimumspanningtreetechniques),accommodatingsi-multaneoususers,thefluctuationofuserdensityinreality,howthelandscapecanaffectrepeaters’locations,andthestrengthsandweaknessesofthemodelandthealgorithms.TheUMAPJournal32(2)(2011)131–148.cCopy
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8、reditispermitted,butcopyrightsforcomponentsofthisworkownedbyothersthanCOMAPmustbehonored.Tocopyotherwise,torepublish,topostonservers,ortoredistributetolistsrequirespriorpermissionfromCOMAP.132TheUMAPJournal32.2(2011)IntroductionA