On locally delaunay geometric graphs

Rom Pinchasi, Shakhar Smorodinsky

Research output: Contribution to conferencePaperpeer-review

10 Scopus citations

Abstract

A geometric graph is a simple graph G = (V, E) with an embedding of the set V in the plane such that the points that represent V are in general position. A geometric graph is said to be k-locally Delaunay (or a D k-graph) if for each edge (u, v) ∈ E there is a (Euclidean) disc d that contains u and v but no other vertex of G that is within k hops from u or v. The study of these graphs was recently motivated by topology control for wireless networks. We obtain the following results: (i) We prove that if G is a Di-graph on n vertices, then it has O(n3/2) edges. (ii) We show that for any n there exist D1-graphs with n vertices and Ω(n4/3) edges. (iii) We prove that if G is a D 2-graph on n vertices, then it has O(n) edges. This bound is worst-case asymptotically tight. As an application of the first result, we show that: (iv) The maximum size of a family of pairwise non-overlapping lenses in an arrangement of n unit circles in the plane is O(n3/2). The first two results improve the best previously known upper and lower bounds of O(n 5/3) and Ω(n) respectively (see [6]). The third result improves the best previously known upper bound of O(n log n). Finally, our last result improves the best previously known upper bound (for the more general case of not necessarily unit circles) of O(n3/2k(n)) (see [1]), where k(n) = (log n)O(α2(n)) and where α(n) is the extremely slowly growing inverse Ackermann's function.

Original languageEnglish
Pages378-382
Number of pages5
DOIs
StatePublished - 1 Jan 2004
Externally publishedYes
EventProceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04) - Brooklyn, NY, United States
Duration: 9 Jun 200411 Jun 2004

Conference

ConferenceProceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04)
Country/TerritoryUnited States
CityBrooklyn, NY
Period9/06/0411/06/04

Keywords

  • Delaunay
  • Extremal graph theory
  • Geometric graphs
  • Sensor networks

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