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Algorithmica

, Volume 62, Issue 3–4, pp 713–732 | Cite as

Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs

  • Victor Chepoi
  • Feodor F. DraganEmail author
  • Bertrand Estellon
  • Michel Habib
  • Yann Vaxès
  • Yang Xiang
Article

Abstract

δ-Hyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4-point condition: for any four points u,v,w,x, the two larger of the distance sums d(u,v)+d(w,x),d(u,w)+d(v,x),d(u,x)+d(v,w) differ by at most 2δ. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. In this paper, we study unweighted δ-hyperbolic graphs. Using the Layering Partition technique, we show that every n-vertex δ-hyperbolic graph with δ≥1/2 has an additive O(δlog n)-spanner with at most O(δn) edges and provide a simpler, in our opinion, and faster construction of distance approximating trees of δ-hyperbolic graphs with an additive error O(δlog n). The construction of our tree takes only linear time in the size of the input graph. As a consequence, we show that the family of n-vertex δ-hyperbolic graphs with δ≥1/2 admits a routing labeling scheme with O(δlog 2 n) bit labels, O(δlog n) additive stretch and O(log 2(4δ)) time routing protocol, and a distance labeling scheme with O(log 2 n) bit labels, O(δlog n) additive error and constant time distance decoder.

Keywords

Algorithms Distance and routing labeling schemes Additive spanners δ-Hyperbolic graphs 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Victor Chepoi
    • 1
  • Feodor F. Dragan
    • 2
    Email author
  • Bertrand Estellon
    • 1
  • Michel Habib
    • 3
  • Yann Vaxès
    • 1
  • Yang Xiang
    • 4
  1. 1.Laboratoire d’Informatique Fondamentale, Faculté des Sciences de LuminyUniversité de la Mediterranée, MarseilleMarseille Cedex 9France
  2. 2.Algorithmic Research Laboratory, Department of Computer ScienceKent State UniversityKentUSA
  3. 3.LIAFAUniversity Paris Diderot - Paris 7Paris Cedex 13France
  4. 4.Department of Biomedical InformaticsThe Ohio State UniversityColumbusUSA

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