The Size of the Open Sphere of Influence Graph in L ∞ Metric Spaces

Soss, Michael

doi:10.1007/3-540-37623-2_45

Cited by 5 publications

(7 citation statements)

References 1 publication

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“…The size of these graphs is also a basic parameter that has attracted significant attention. In particular, it has been proved that the size of the kth order Relative Neighborhood graph is linear in kn [31] and that the size of the Sphere of Influence graph 1 is at most 15n [29].…”

Section: Introduction and Preliminary Definitionsmentioning

confidence: 99%

On Structural and Graph Theoretic Properties of Higher Order Delaunay Graphs

Abellanas

Bose

García

et al. 2009

Int. J. Comput. Geom. Appl.

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Given a set P of n points in the plane, the order-k Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order-k Delaunay graph. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the order-k Delaunay graph is connected under the flip operation when k ≤ 1 but not necessarily connected for other values of k. If P is in convex position then the order-k Delaunay graph is connected for all k ≥ 0. We show that the order-k Gabriel graph, a subgraph of the order-k Delaunay graph, is Hamiltonian for k ≥ 15. Finally, the order-k Delaunay graph can be used to efficiently solve a coloring problem with applications to frequency assignments in cellular networks.

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Section: Introduction and Preliminary Definitionsmentioning

confidence: 99%

On Structural and Graph Theoretic Properties of Higher Order Delaunay Graphs

Abellanas

Bose

García

et al. 2009

Int. J. Comput. Geom. Appl.

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show abstract

“…In this section we combine the reasoning of Bateman and Erdős [3] with that of Soss [10] to prove that Theorem 2.1 implies Theorem 1.2. Let p ≥ 1 be a constant to be chosen later, and denote q = 1/p.…”

Section: Reducing the Problem To A Bounded Onementioning

confidence: 99%

“…For this selection of p and q, this bound is optimal. Soss [10] used p = 3/2, q = 2/3 to prove an upper bound of of 15 for the total weight of the points in P 1 ∪ P 1/2 . In a private communication, Soss admitted he chose these values because they were "nice."…”

Section: Reducing the Problem To A Bounded Onementioning

confidence: 99%

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An improved upper bound for the size of the sphere of influence graph

Ismailescu¹,

Kim²,

Park³

2020

Preprint

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Let V be a set of n points in the plane. For each x ∈ V , let B x be the closed circular disk centered at x with radius equal to the distance from x to its closest neighbor. The closed sphere of influence graph on V is defined as the undirected graph where x and y are adjacent if and only if the B x and B y have nonempty intersection.It is known that every n-vertex closed sphere of influence graph has at most cn edges, for some absolute positive constant c. The first result was obtained in 1985 by Avis and Horton who provided the value c = 29. Their result was successively improved by several authors: Bateman and Erdős (c=18), Michael and Quint (c=17.5), and Soss (c=15).In this paper we prove that one can take c = 14.5.

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“…The nearest-neighbor circle of point p is a circle centered at p, and the radius is the distance from p to its nearest-neighbor. Soss [18] has proven that the number of edges in SIG is at most 15n.…”

Section: Msbr With Square Labelsmentioning

confidence: 99%

Polynomial Time Algorithms for Label Size Maximization on Rotating Maps

Yokosuka

Imai

2017

Journal of Information Processing

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Map labeling is the problem of placing labels at corresponding graphical features on a map. There are two main optimization problems: the label number maximization problem and the label size maximization problem. In general, both problems are NP-hard for static maps. Recently, the widespread use of several applications, such as personal mapping systems, has increased the importance of dynamic maps and the label number maximization problem for dynamic cases has been studied. In this paper, we consider the label size maximization problem for points on rotating maps. Our model is as follows. For each label, an anchor point is chosen inside the label or on its boundary. Each label is placed such that the anchor point coincides with the corresponding point on the map. Furthermore, while the map fully rotates from 0 to 2π, the labels are placed horizontally according to the angle of the map. Our problem consists of finding the maximum scale factor for the labels such that the labels do not intersect, and determining the placing of the anchor points. We describe an O(n log n)-time and O(n)-space algorithm for the case where each anchor point is inside the label. Moreover, if the anchor points are on the boundaries, we also present an O(n log n)-time and O(n)-space exact and approximation algorithms for several label shapes.

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The Size of the Open Sphere of Influence Graph in L ∞ Metric Spaces

Cited by 5 publications

References 1 publication

On Structural and Graph Theoretic Properties of Higher Order Delaunay Graphs

On Structural and Graph Theoretic Properties of Higher Order Delaunay Graphs

An improved upper bound for the size of the sphere of influence graph

Polynomial Time Algorithms for Label Size Maximization on Rotating Maps

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