Expected Length of the Voronoi Path in a High Dimensional Poisson–Delaunay Triangulation

Castro, Pedro Machado Manhães de; Devillers, Olivier

doi:10.1007/s00454-017-9866-y

Cited by 3 publications

(4 citation statements)

References 4 publications

(3 reference statements)

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“…In a companion paper [5] we prove that the expected length of the Voronoi path between two points at unit distance increases with the dimension from about 3 2 in 3D to Θ…”

mentioning

confidence: 88%

Walking in a Planar Poisson–Delaunay Triangulation: Shortcuts in the Voronoi Path

Devillers

Noizet

2018

Int. J. Comput. Geom. Appl.

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Let [Formula: see text] be a planar Poisson point process of intensity [Formula: see text]. We give a new proof that the expected length of the Voronoi path between [Formula: see text] and [Formula: see text] in the Delaunay triangulation associated with [Formula: see text] is [Formula: see text] when [Formula: see text] goes to infinity; and we also prove that the variance of this length is [Formula: see text]. We investigate the length of possible shortcuts in this path, and define a shortened Voronoi path whose expected length can be expressed as an integral that is numerically evaluated to [Formula: see text]. The shortened Voronoi path has the property to be locally defined; and is shorter than the previously known locally defined paths in Delaunay triangulation such as the upper path whose expected length is [Formula: see text].

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“…In a companion paper [5] we prove that the expected length of the Voronoi path between two points at unit distance increases with the dimension from about 3 2 in 3D to Θ…”

mentioning

confidence: 88%

Walking in a Planar Poisson–Delaunay Triangulation: Shortcuts in the Voronoi Path

Devillers

Noizet

2018

Int. J. Comput. Geom. Appl.

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“…The angle is symmetric, so we can instead consider the integrand in (3) as the projection of a unit p-cube in a random p-plane onto L 0 . Formally, we write St p,d for the Stiefel manifold of orthonormal p-frames in R d , we identify a frame with the unit p-cube it spans, and we integrate using the uniform probability measure of St p,d to arrive at (4).…”

Section: Random Projectionsmentioning

confidence: 99%

“…Considering the Voronoi tessellation of a stationary Poisson point process and a line segment in R 2 , Baccelli et al [2] proves that the expected distortion is 4 π . Extending this work to d > 2 dimensions, de Castro et al [4] proves that the expected distortion is…”

Section: Introductionmentioning

confidence: 99%

Average and Expected Distortion of Voronoi Paths and Scapes

Edelsbrunner,

Nikitenko

2024

Discrete Comput Geom

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The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about $$\tfrac{4}{\pi }$$ 4 π . We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes.

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“…They proved that this Voronoi path has expected length 4/π 1.27, giving an upper bound for the expected stretch factor. This result has been improved by introducing shortcuts [12] and generalizing to higher dimensions [10].…”

Section: Introductionmentioning

confidence: 99%

Stretch factor in a planar Poisson–Delaunay triangulation with a large intensity

Chenavier

Devillers

2018

Adv. Appl. Probab.

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Let X := Xn ∪ {(0, 0), (1, 0)}, where Xn is a planar Poisson point process of intensity n. We provide a first non-trivial lower bound for the distance between the expected length of the shortest path between (0, 0) and (1, 0) in the Delaunay triangulation associated with X when the intensity of Xn goes to infinity. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to 35 3π 2 , giving an upper bound for the expected length of the smallest path.

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Expected Length of the Voronoi Path in a High Dimensional Poisson–Delaunay Triangulation

Cited by 3 publications

References 4 publications

Walking in a Planar Poisson–Delaunay Triangulation: Shortcuts in the Voronoi Path

Walking in a Planar Poisson–Delaunay Triangulation: Shortcuts in the Voronoi Path

Average and Expected Distortion of Voronoi Paths and Scapes

Stretch factor in a planar Poisson–Delaunay triangulation with a large intensity

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