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Asymptotics of geometrical navigation on a random set of points in the plane

Part of: Algorithms - Computer Science Probability theory on algebraic and topological structures Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Nicolas Bonichon  and
Jean-François Marckert
Nicolas Bonichon*
Affiliation:
Université de Bordeaux
Jean-François Marckert*
Affiliation:
Université de Bordeaux
*
∗ Postal address: CNRS, LaBRI, Université de Bordeaux, 351 cours de la Libération, 33405 Talence cedex, France.
∗ Postal address: CNRS, LaBRI, Université de Bordeaux, 351 cours de la Libération, 33405 Talence cedex, France.
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Abstract

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A navigation on a set of points S is a rule for choosing which point to move to from the present point in order to progress toward a specified target. We study some navigations in the plane where S is a nonuniform Poisson point process (in a finite domain) with intensity going to +∞. We show the convergence of the traveller's path lengths, and give the number of stages and the geometry of the traveller's trajectories, uniformly for all starting points and targets, for several navigations of geometric nature. Other costs are also considered. This leads to asymptotic results on the stretch factors of random Yao graphs and random θ-graphs.

Keywords

NavigationPoisson point processgreedy routingspatial networkproximity graph

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 68W40: Analysis of algorithms 60B10: Convergence of probability measures 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 43 , Issue 4 , December 2011 , pp. 899 - 942
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

Partially supported by the ANR project ALADDIN.

Partially supported by ANR-08-BLAN-0190-04A3.

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