{"title":"How to net a lot with little: small ε-nets for disks and halfspaces","authors":"J. Matoušek, R. Seidel, E. Welzl","doi":"10.1145/98524.98530","DOIUrl":null,"url":null,"abstract":"It is known that in general range spaces of VC-dimension <italic>d</italic> > 1 require <italic>ε</italic>-nets to be of size at least &OHgr;(<italic>d</italic>/<italic>ε</italic> log 1/<italic>ε</italic>). We investigate the question whether this general lower bound is valid for the special range spaces that typically arise in computational geometry. We show that disks and pseudo-disks in the plane as well as halfspaces in R<supscrpt>3</supscrpt> allow <italic>ε</italic>-nets of size only <italic>&Ogr;</italic>(1/<italic>ε</italic>), which is best possible up to a multiplicative constant. The analogous questions for higher-dimensional spaces remain open.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"17 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"127","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SCG '90","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/98524.98530","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 127
Abstract
It is known that in general range spaces of VC-dimension d > 1 require ε-nets to be of size at least &OHgr;(d/ε log 1/ε). We investigate the question whether this general lower bound is valid for the special range spaces that typically arise in computational geometry. We show that disks and pseudo-disks in the plane as well as halfspaces in R3 allow ε-nets of size only &Ogr;(1/ε), which is best possible up to a multiplicative constant. The analogous questions for higher-dimensional spaces remain open.
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