On the Structure of Higher Order Voronoi Cells

Martínez-Legaz, Juan Enrique; Roshchina, Vera; Todorov, Maxim I.

doi:10.1007/s10957-019-01555-2

Cited by 7 publications

(9 citation statements)

References 13 publications

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“…The lower bounds in Property 18 were already given in [18]. It was also proved in [21] with a different method that V 2 (S) does not contain triangles. As for the upper bounds in Property 18, there exist point sets S such that some face of V k (S) has exactly n − k vertices of type I and k vertices of type II, see Figure 17 Note that Property 18 implies that every bounded face of V 2 (S) contains exactly two vertices of type II, and every bounded face of V n−2 (S) contains exactly two vertices of type I.…”

Section: Properties Of V K (S)mentioning

confidence: 90%

“…The most studied Voronoi diagrams of point sets are V 1 (S), the classic Voronoi diagram, and V n−1 (S), the furthest point Voronoi diagram, which only has unbounded faces. Many properties of V k (S) were obtained by Lee [18], we also mention [7,10,11,13,19,21,22,24] among the sources on the structure of V k (S). In this work we review several of these structural results with new proofs, and we also present new results on V k (S).…”

Section: Introductionmentioning

confidence: 98%

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The edge labeling of higher order Voronoi diagrams

Aguas¹,

Parrilla²,

Huemer³

et al. 2021

Preprint

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We present an edge labeling of order-k Voronoi diagrams, V k (S), of point sets S in the plane, and study properties of the regions defined by them. Among them, we show that V k (S) has a small orientable cycle and path double cover, and we identify configurations that cannot appear in V k (S) for small values of k. This paper also contains a systematic study of well-known and new properties of V k (S), all whose proofs only rely on elementary geometric arguments in the plane. The maybe most comprehensive study of structural properties of V k (S) was done by D.T. Lee (On k-nearest neighbor Voronoi diagrams in the plane) in 1982. Our work reviews and extends the list of properties of higher order Voronoi diagrams.

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Section: Properties Of V K (S)mentioning

confidence: 90%

Section: Introductionmentioning

confidence: 98%

The edge labeling of higher order Voronoi diagrams

Aguas¹,

Parrilla²,

Huemer³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The motivation for this paper comes from the construction of regular tessellations of the Euclidean space via Voronoi diagrams on lattices on the one hand [15], and from several puzzling examples and negative results related to Voronoi diagrams on the plane obtained in [14], on the other. We have succeeded in explaining the lack one-dimensional cells in R 2 observed in [14] proving that there can be no higher-order cells of dimension n − 1 (see Theorem 3.1), however the question of generalising other negative results from that study remains open. For instance, it was shown in [14] that in the case when |T | = 4 and S T , |S| ≥ 2, the higher-order cell V T (S) can not be a triangle or a quadrilateral.…”

Section: Discussionmentioning

confidence: 99%

“…In particular, this setting motivates the generalisation of the notion of boundedly exposed points used by the authors to obtain a characterisation of nonempty farthest Voronoi cell. Our work is in a similar spirit: we are focussed on structural properties of higher-order Voronoi cells, motivated by the puzzling observations on low-dimensional cells presented in [14] and by the work [15] focussed on generating regular tessellations from higher-order cells.…”

Section: Introductionmentioning

confidence: 99%

Dimension and structure of higher-order Voronoi cells on discrete sites

McKewen,

Roshchina

2019

Preprint

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We study the structure of higher-order Voronoi cells on a discrete set of sites in R n , focussing on the relations between cells of different order, and paying special attention to the ill-posed case when a large number of points lie on a sphere. In particular, we prove that higher order cells of dimension n − 1 do not exist, even though high-order Voronoi cells may have empty interior. We also present a number of open questions.

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“…These cells are in general nor convex, nor connected, making analysis of k-th order Voronoi cells complex [12]. Several results on fast algorithms to construct higher-order Voronoi diagrams exist [2,9,29], as well as results on the complexity of its cells [8] as well as the cell shapes [12,18]. However, to our knowledge, no results on the cell sizes of higherorder Voronoi cells or k-th order Voronoi cells exist under any type of underlying point process.…”

Section: Introductionmentioning

confidence: 99%

Degree distributions in AB random geometric graphs

Stegehuis,

Weedage

2021

Preprint

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In this paper, we provide degree distributions for AB random geometric graphs, in which points of type A connect to the closest k points of type B.The motivating example to derive such degree distributions is in 5G wireless networks with multi-connectivity, where users connect to their closest k base stations. In this setting, it is important to know how many users a particular base station serves, which gives the degree of that base station. To obtain these degree distributions, we investigate the distribution of area sizes of the k−th order Voronoi cells of B-points. Assuming that the A-points are Poisson distributed, we investigate the amount of users connected to a certain B-point, which is equal to the degree of this point. In the simple case where the Bpoints are placed in an hexagonal grid, we show that all k-th order Voronoi areas are equal and thus all degrees follow a Poisson distribution. However, this observation does not hold for Poisson distributed B-points, for which we show that the degree distribution follows a compound Poisson-Erlang distribution in the 1-dimensional case. We then approximate the degree distribution in the 2-dimensional case with a compound Poisson-Gamma degree distribution and show that this one-parameter fit performs well for different values of k. Moreover, we show that for increasing k, these degree distributions become more concentrated around the mean. This means that k-connected AB random graphs balance the loads of B-type nodes more evenly as k increases. Finally, we provide a case study on real data of base stations. We show that with little shadowing in the distances between users and base stations, the Poisson distribution does not capture the degree distribution of these data,

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On the Structure of Higher Order Voronoi Cells

Cited by 7 publications

References 13 publications

The edge labeling of higher order Voronoi diagrams

The edge labeling of higher order Voronoi diagrams

Dimension and structure of higher-order Voronoi cells on discrete sites

Degree distributions in AB random geometric graphs

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