Asymptotics of geometrical navigation on a random set of points in the plane
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 con...
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| Veröffentlicht in: | Advances in applied probability 2011-12, Vol.43 (4), p.899-942 |
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| container_title | Advances in applied probability |
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| creator | Bonichon, Nicolas Marckert, Jean-François |
| description | 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. |
| doi_str_mv | 10.1239/aap/1324045692 |
| format | Article |
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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. 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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.</abstract><cop>Cambridge, UK</cop><pub>Cambridge University Press</pub><doi>10.1239/aap/1324045692</doi><tpages>44</tpages><orcidid>https://orcid.org/0000-0003-3773-262X</orcidid><oa>free_for_read</oa></addata></record> |
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| ispartof | Advances in applied probability, 2011-12, Vol.43 (4), p.899-942 |
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| source | Jstor Complete Legacy; JSTOR Mathematics & Statistics; Cambridge University Press Journals Complete |
| subjects | 60B10 60D05 60G55 68W40 Asymptotic methods Asymptotic properties Convergence Cost functions Density Differential equations Geometric planes Geometry Graph theory Graphs greedy routing Integers Mathematical analysis Mathematical functions Mathematics Navigation Nonuniform Planes Poisson distribution Poisson point process Probability proximity graph Random variables Real numbers spatial network Stochastic Geometry and Statistical Applications Studies Trajectories |
| title | Asymptotics of geometrical navigation on a random set of points in the plane |
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