On a random directed spanning tree

Bhatt, Abhay G.; Roy, Rahul

doi:10.1017/s0001867800012854

Cited by 23 publications

(67 citation statements)

References 5 publications

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“…In this paper we define a new model of rooted tree generated over a point set, the radial spanning tree (RST). This tree holds similarities with the minimal directed spanning tree [6,21,22] and the Poisson forest (as defined in [12,13]). The mathematical analysis of the RST is conducted on a homogeneous Poisson point process (PPP) on the plane.…”

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confidence: 96%

The radial spanning tree of a Poisson point process

Baccelli¹,

Bordenave²

2007

Ann. Appl. Probab.

161

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We analyze a class of spatial random spanning trees built on a realization of a homogeneous Poisson point process of the plane. This tree has a simple radial structure with the origin as its root. We first use stochastic geometry arguments to analyze local functionals of the random tree such as the distribution of the length of the edges or the mean degree of the vertices. Far away from the origin, these local properties are shown to be close to those of a variant of the directed spanning tree introduced by Bhatt and Roy. We then use the theory of continuous state space Markov chains to analyze some nonlocal properties of the tree, such as the shape and structure of its semi-infinite paths or the shape of the set of its vertices less than $k$ generations away from the origin. This class of spanning trees has applications in many fields and, in particular, in communications.Comment: Published at http://dx.doi.org/10.1214/105051606000000826 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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confidence: 96%

The radial spanning tree of a Poisson point process

Baccelli¹,

Bordenave²

2007

Ann. Appl. Probab.

161

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“…where we have written f = f h to emphasize the dependence of f (·) on h(·). Taking supremum over h ∈ H 1,1 first on the right, and then on the left now yields the result upon applying definition (11). Now we will demonstrate a few applications of Theorem 2.1.…”

Section: Dickman Approximation In Number Theorymentioning

confidence: 80%

“…We begin with a simple application of Theorem 1.1 to the minimal directed spanning tree, or MDST, following [11], first pausing to describe the construction of the MDST. For two points (…”

Section: Weighted Bernoulli and Poisson Sumsmentioning

confidence: 99%

Dickman approximation in simulation, summations and perpetuities

Bhattacharjee¹,

Goldstein²

2019

Bernoulli

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The generalized Dickman distribution D θ with parameter θ > 0 is the unique solution to the distributional equality W = d W * , wherewith W non-negative with probability one, U ∼ U [0, 1] independent of W , and = d denoting equality in distribution. Members of this family appear in number theory, stochastic geometry, perpetuities and the study of algorithms. We obtain bounds in Wasserstein type distances between D θ and the distribution of0 MSC 2010 subject classifications: Primary 60F05, 60E99, 91B16 0

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“…-In [6,16] the authors study the property of a so-called "minimal directed spanning tree" (MDST). To each finite subset S of (0, 1] 2 is associated a tree as follows : x ∈ S is connected by a directed edge to the nearest y ∈ S ∪ {(0, 0)}, having both coordinates smaller.…”

Section: Link With Other Random Objectsmentioning

confidence: 99%

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

Marckert

2011

Adv. Appl. Probab.

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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.

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On a random directed spanning tree

Cited by 23 publications

References 5 publications

The radial spanning tree of a Poisson point process

The radial spanning tree of a Poisson point process

Dickman approximation in simulation, summations and perpetuities

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

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