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

Part of: Graph theory Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  19 September 2016

J. Díaz ,
D. Mitsche ,
G. Perarnau  and
X. Pérez-Giménez
J. Díaz*
Affiliation:
Universitat Politècnica de Catalunya and BGSMath
D. Mitsche*
Affiliation:
Université Nice Sophia Antipolis
G. Perarnau*
Affiliation:
Universitat Politècnica de Catalunya
X. Pérez-Giménez*
Affiliation:
University of Waterloo
*
* Postal address: Department of Computer Science, Universitat Politècnica de Catalunya, Jordi Girona Salgado 1, 08034 Barcelona, Spain. Email address: [email protected]
** Postal address: Laboratoire J. A. Dieudonné, Université Nice Sophia Antipolis, Parc Valrose, 06108 Nice cedex 02, France. Email address: [email protected]
*** Postal address: Department de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Jordi Girona Salgado 1, 08034 Barcelona, Spain. Email address: [email protected]
**** Postal address: Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, N2L 3G1, Canada. Email address: [email protected]
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Abstract

Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(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 dE(u, v) conditional on dE(u, v).

Keywords

Random geometric graphgraph distanceEuclidean distancediameter

MSC classification

Primary: 05C80: Random graphs
Secondary: 68R10: Graph theory (including graph drawing)
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 848 - 864
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Bradonjić, M. et al. (2010).Efficient broadcast on random geometric graphs. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms,SIAM,Philadelphia, PA, pp. 1412–1421.Google Scholar
[2] Ellis, R. B.,Martin, J. L. and Yan, C. (2007).Random geometric graph diameter in the unit ball.Algorithmica 47,421–438.Google Scholar
[3] Gilbert, E. N. (1961) Random plane networks.J. Soc. Indust. Appl. Math. 9,533–543.CrossRefGoogle Scholar
[4] Friedrich, T.,Sauerwald, T. and Stauffer, A. (2011).Diameter and broadcast time of random geometric graphs in arbitrary dimensions. In Algorithms and Computation,Springer,Heidelberg, pp. 190–199.Google Scholar
[5] Goel, A.,Rai, S. and Krishnamachari, B. (2004).Sharp thresholds for monotone properties in random geometric graphs. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing,ACM,New York, pp. 580–586.Google Scholar
[6] Mehrabian, A. and Wormald, N. (2013).On the stretch factor of randomly embedded random graphs.Discrete Comput. Geom. 49,647–658.Google Scholar
[7] Muthukrishnan, S. and Pandurangan, G. (2005).The bin-covering technique for thresholding random geometric graph properties. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms,ACM,New York, pp. 989–998.Google Scholar
[8] Penrose, D. M. (1997).The longest edge of the random minimal spanning tree.Ann. Appl. Prob. 7,340–361.CrossRefGoogle Scholar
[9] Penrose, M. (2003).Random Geometric Graphs.Oxford University Press.Google Scholar
[10] Walters, M. (2011).Random geometric graphs. In Surveys in Combinatorics, 2011,Cambridge University Press, pp. 365–401.CrossRefGoogle Scholar