LATIN 2002: Theoretical Informatics
Volume 2286 of the series Lecture Notes in Computer Science pp 479493
On the Spanning Ratio of Gabriel Graphs and βskeletons
 Prosenjit BoseAffiliated withSchool of Computer Science, Carleton University
 , Luc DevroyeAffiliated withSchool of Computer Science, McGill University
 , William EvansAffiliated withDepartment of Computer Science, University of British Columbia
 , David KirkpatrickAffiliated withDepartment of Computer Science, University of British Columbia
Abstract
The spanning ratio of a graph defined on n points in the Euclidean plane is the maximal ratio over all pairs of data points (u, v), of the minimum graph distance between u and v, over the Euclidean distance between u and v. A connected graph is said to be a kspanner if the spanning ratio does not exceed k. For example, for any k, there exists a point set whose minimum spanning tree is not a kspanner. At the other end of the spectrum, a Delaunay triangulation is guaranteed to be a 2.42 spanner[11]. For proximity graphs inbetween these two extremes, such as Gabriel graphs[8], relative neighborhood graphs[16] and βskeletons[12] with β ∈ [0, 2] some interesting questions arise. We show that the spanning ratio for Gabriel graphs (which are βskeletons with β = 1) is Θ(√n) in the worst case. For all βskeletons with β ∈ [0, 1], we prove that the spanning ratio is at most O(n λ) where λ = (1  log_{2}(1 +√1  β^{2}))/2. For all βskeletons with β ∈ [1, 2), we prove that there exist point sets whose spanning ratio is at least (^{1} _{2}  o(1))√n. For relative neighborhood graphs[16] (skeletons with β = 2), we show that there exist point sets where the spanning ratio is ω(n). For points drawn independently from the uniform distribution on the unit square, we show that the spanning ratio of the (random) Gabriel graph and all βskeletons with β ∈ [1, 2] tends to ∞ in probability as √log n/ log log n.
 Title
 On the Spanning Ratio of Gabriel Graphs and βskeletons
 Book Title
 LATIN 2002: Theoretical Informatics
 Book Subtitle
 5th Latin American Symposium Cancun, Mexico, April 3–6, 2002 Proceedings
 Pages
 pp 479493
 Copyright
 2002
 DOI
 10.1007/3540459952_42
 Print ISBN
 9783540434009
 Online ISBN
 9783540459958
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 2286
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 Additional Links
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 Editors

 Sergio Rajsbaum ^{(4)}
 Editor Affiliations

 4. Compaq Cambridge Research Laboratory, One Cambridge Center
 Authors

 Prosenjit Bose ^{(5)}
 Luc Devroye ^{(6)}
 William Evans ^{(7)}
 David Kirkpatrick ^{(7)}
 Author Affiliations

 5. School of Computer Science, Carleton University, Ottawa, Ontario, Canada
 6. School of Computer Science, McGill University, Montreal, Canada
 7. Department of Computer Science, University of British Columbia, Vancouver, Canada
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