Central limit theorems for the radial spanning tree

Schulte, Matthias; Thäle, Christoph

doi:10.1002/rsa.20651

Cited by 8 publications

(4 citation statements)

References 24 publications

(42 reference statements)

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“…More complex models assign edges between vertices in a randomized fashion (see, e.g., [12,21,51]). Some models introduce an ordering on the sampled points (see, e.g., [1,52,53,61,67]) which results in a directed graph that resembles the RRT tree [35].…”

Section: Random Geometric Graphsmentioning

confidence: 99%

New perspective on sampling-based motion planning via random geometric graphs

Solovey¹,

Salzman²,

Halperin³

Robotics: Science and Systems XII

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Abstract-Roadmaps constructed by many sampling-based motion planners coincide, in the absence of obstacles, with standard models of random geometric graphs (RGGs). Those models have been studied for several decades and by now a rich body of literature exists analyzing various properties and types of RGGs. In their seminal work on optimal motion planning Karaman and Frazzoli [31] conjectured that a sampling-based planner has a certain property if the underlying RGG has this property as well. In this paper we settle this conjecture and leverage it for the development of a general framework for the analysis of sampling-based planners. Our framework, which we call localization-tessellation, allows for easy transfer of arguments on RGGs from the free unit-hypercube to spaces punctured by obstacles, which are geometrically and topologically much more complex. We demonstrate its power by providing alternative and (arguably) simple proofs for probabilistic completeness and asymptotic (near-)optimality of probabilistic roadmaps (PRMs). Furthermore, we introduce two variants of PRMs, analyze them using our framework, and discuss the implications of the analysis.

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Section: Random Geometric Graphsmentioning

confidence: 99%

New perspective on sampling-based motion planning via random geometric graphs

Solovey¹,

Salzman²,

Halperin³

Robotics: Science and Systems XII

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show abstract

“…More complex models assign edges between vertices in a randomized fashion (see, e.g., Broutin et al, 2014; Frieze and Pegden, 2014; Penrose, 2016). Some models introduce an ordering on the sampled points (see, e.g., Bhatt and Roy, 2004; Penrose and Wade, 2010a,b; Schulte and Thaele, 2014; Wade, 2009), which results in a directed graph that resembles the aforementioned RRT.…”

Section: Related Workmentioning

confidence: 99%

New perspective on sampling-based motion planning via random geometric graphs

Solovey

Salzman

Halperin

2018

The International Journal of Robotics Research

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Roadmaps constructed by many sampling-based motion planners coincide, in the absence of obstacles, with standard models of random geometric graphs (RGGs). Those models have been studied for several decades and by now a rich body of literature exists analyzing various properties and types of RGGs. In their seminal work on optimal motion planning, Karaman and Frazzoli conjectured that a sampling-based planner has a certain property if the underlying RGG has this property as well. In this paper, we settle this conjecture and leverage it for the development of a general framework for the analysis of sampling-based planners. Our framework, which we call localization-tessellation, allows for easy transfer of arguments on RGGs from the free unit hypercube to spaces punctured by obstacles, which are geometrically and topologically much more complex. We demonstrate its power by providing alternative and (arguably) simple proofs for probabilistic completeness and asymptotic (near-)optimality of probabilistic roadmaps (PRMs) in Euclidean spaces. Furthermore, we introduce three variants of PRMs, analyze them using our framework, and discuss the implications of the analysis.

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“…Originally introduced in [30] and further developed in many subsequent works it has turned out to be a versatile device with a vast of potential applications. As concrete examples we mention the works [8,20,21,22,23,31,32,37] on various models for geometric random graphs, the paper [7] dealing with geometric random simplicial complexes, the application to the classical Boolean model [17], the works [8,16,25,31] dealing with Poisson hyperplane tessellations in Euclidean and non-Euclidean spaces, the applications in [22,36] to Poisson-Voronoi tessellations, the works on excursion sets of Poisson shot-noise processes [19,21] as well as the papers [4,5,22,40,41,42] considering different models for random polytopes. For an illustrative overview on the Malliavin-Stein method for functionals of Poisson processes we refer to the collection of surveys in [29].…”

Section: Introductionmentioning

confidence: 99%

Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes

Betken¹,

Schulte²,

Thäle³

2022

Electron. J. Probab.

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This paper deals with the union set of a stationary Poisson process of cylinders in R n having an (n − m)-dimensional base and an m-dimensional direction space, where m ∈ {0, 1, . . . , n − 1} and n ≥ 2. The concept simultaneously generalises those of a Boolean model and a Poisson hyperplane or m-flat process. Under very general conditions on the typical cylinder base a Berry-Esseen bound for the volume of the union set within a sequence of growing test sets is derived. Assuming convexity of the cylinder bases and of the window a similar result is shown for a broad class of geometric functionals, including the intrinsic volumes. In this context the asymptotic variance constant is analysed in detail, which in contrast to the Boolean model leads to a new degeneracy phenomenon. A quantitative central limit theory is developed in a multivariate set-up as well.

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Central limit theorems for the radial spanning tree

Cited by 8 publications

References 24 publications

New perspective on sampling-based motion planning via random geometric graphs

New perspective on sampling-based motion planning via random geometric graphs

New perspective on sampling-based motion planning via random geometric graphs

Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes

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