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On the relation between graph distance and Euclidean distance in random geometric graphs

  • J. Díaz (a1), D. Mitsche (a2), G. Perarnau (a3) and X. Pérez-Giménez (a4)

Abstract

Given any two vertices u, v of a random geometric graph G(n, r), denote by d E (u, v) their Euclidean distance and by d E (u, v) their graph distance. The problem of finding upper bounds on d G (u, v) conditional on d E (u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=ω(√logn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on d E (u, v) conditional on d E (u, v).

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Corresponding author

* Postal address: Department of Computer Science, Universitat Politècnica de Catalunya, Jordi Girona Salgado 1, 08034 Barcelona, Spain. Email address: diaz@lsi.upc.edu
** Postal address: Laboratoire J. A. Dieudonné, Université Nice Sophia Antipolis, Parc Valrose, 06108 Nice cedex 02, France. Email address: dmitsche@unice.fr
*** Postal address: Department de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Jordi Girona Salgado 1, 08034 Barcelona, Spain. Email address: guillem.perarnau@ma4.upc.edu
**** Postal address: Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, N2L 3G1, Canada. Email address: xperez@uwaterloo.ca

References

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