Asymptotic of geometrical navigation on a random set of points of the plane

Details Asymptotic of geometrical navigation on a random set of points of the plane Asymptotic of geometrical navigation on a random set of points of the plane

2010-09-23

arxiv.org/abs/1009.4679v2

Mathematics

Probability

Scientific paper

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 non uniform Poisson point process (in a finite domain) with intensity going to $+\infty$. We show the convergence of the traveller path lengths, the number of stages done, and the geometry of the traveller 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 $\theta$-graphs.

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