B. Chazelle, H. Edelsbrunner, L. Guibas, J. Hershberger, R. Seidel, M. Sharir
{"title":"Slimming down by adding; selecting heavily covered points","authors":"B. Chazelle, H. Edelsbrunner, L. Guibas, J. Hershberger, R. Seidel, M. Sharir","doi":"10.1145/98524.98551","DOIUrl":null,"url":null,"abstract":"<italic>We show that for any set &Pgr; of n points in three-dimensional space there is a set Q of &Ogr;(n<supscrpt>1/2</supscrpt> log<supscrpt>3</supscrpt> n) points so that the Delaunay triangulation of &Pgr; ∪ Q has at most &Ogr;(n<supscrpt>3/2</supscrpt> log<supscrpt>3</supscrpt> n) edges — even though the Delaunay triangulation of &Pgr; may have &OHgr;(n<supscrpt>2</supscrpt>) edges. The main tool of our construction is the following geometric covering result: For any set &Pgr; of n points in three-dimensional space and any set S of m spheres, where each sphere passes through a distinct point pair in &Pgr;, there exists a point x, not necessarily in &Pgr;, that is enclosed by</italic> &OHgr;(<italic>m</italic><supscrpt>2</supscrpt>/<italic>n</italic><supscrpt>2</supscrpt> log<supscrpt>3</supscrpt> <italic>n</italic><supscrpt>2</supscrpt>/<italic>m</italic>) <italic>of the spheres in S.</italic>\npar><italic>Our results generalize to arbitrary fixed dimensions, to geometric bodies other than spheres, and to geometric structures other than Delaunay triangulations.</italic>","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"53 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"16","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SCG '90","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/98524.98551","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 16
Abstract
We show that for any set &Pgr; of n points in three-dimensional space there is a set Q of &Ogr;(n1/2 log3 n) points so that the Delaunay triangulation of &Pgr; ∪ Q has at most &Ogr;(n3/2 log3 n) edges — even though the Delaunay triangulation of &Pgr; may have &OHgr;(n2) edges. The main tool of our construction is the following geometric covering result: For any set &Pgr; of n points in three-dimensional space and any set S of m spheres, where each sphere passes through a distinct point pair in &Pgr;, there exists a point x, not necessarily in &Pgr;, that is enclosed by &OHgr;(m2/n2 log3n2/m) of the spheres in S.
par>Our results generalize to arbitrary fixed dimensions, to geometric bodies other than spheres, and to geometric structures other than Delaunay triangulations.
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