Farthest-Polygon Voronoi Diagrams

Cheong, Otfried; Everett, Hazel; Glisse, Marc; Gudmundsson, Joachim; Hornus, Samuel; Lazard, Sylvain; Lee, Mi-Ra; Na, Hyeon-Suk

doi:10.1007/978-3-540-75520-3_37

Cited by 22 publications

(32 citation statements)

References 11 publications

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“…A natural question is: can one compute the farthest-color Voronoi diagram FCVD(S) much faster, provided that S is well-clustered? We believe that computing FCVD(S) can be done in O(n polylog n) time if S is well-clustered; one possible approach which must be examined is to use the parametric search technique by Cheong et al [14], which was used to merge two farthest-polygon Voronoi diagrams.…”

Section: Discussionmentioning

confidence: 99%

On Linear-Sized Farthest-Color Voronoi Diagrams

Bae

2012

IEICE Trans. Inf. & Syst.

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SUMMARYGiven a collection of k sets consisting of a total of n points in the plane, the distance from any point in the plane to each of the sets is defined to be the minimum among distances to each point in the set. The farthest-color Voronoi diagram is defined as a generalized Voronoi diagram of the k sets with respect to the distance functions for each of the k sets. The combinatorial complexity of the diagram is known to be Θ(kn) in the worst case. This paper initiates a study on farthest-color Voronoi diagrams having O(n) complexity. We introduce a realistic model, which defines a certain class of the diagrams with desirable geometric properties observed. We finally show that the farthest-color Voronoi diagrams under the model have linear complexity. key words: Voronoi diagrams, farthest-color Voronoi diagrams, realistic models, computational geometry

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Section: Discussionmentioning

confidence: 99%

On Linear-Sized Farthest-Color Voronoi Diagrams

Bae

2012

IEICE Trans. Inf. & Syst.

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“…The Farthest-Polygon Voronoi Diagram (FPVD) [6] subdivides the plane in O(n) regions with total complexity O(n). In each region, the same polygon is furthest and the same feature (vertex or edge) of that polygon is closest (see Fig.…”

Section: General Labelsmentioning

confidence: 99%

“…5(a)). Cheong et al [6] show that it can be computed in O(n log 3 n) time. The Hausdorff Voronoi Diagram (HVD) [7,11] subdivides the plane in a set of regions, with total complexity O(n).…”

Section: General Labelsmentioning

confidence: 99%

Circles in the Water: Towards Island Group Labeling

Goethem

Kreveld

Speckmann

2016

Geographic Information Science

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Abstract. Many algorithmic results are known for automated label placement on maps. However, algorithms to compute labels for groups of features, such as island groups, are largely missing. In this paper we address this issue by presenting new, efficient algorithms for island label placement in various settings. We consider straight-line and circular-arc labels that may or may not overlap a given set of islands. We concentrate on computing the line or circle that minimizes the maximum distance to the islands, measured by the closest distance. We experimentally test whether the generated labels are reasonable for various real-world island groups, and compare different options. The results are positive and validate our geometric formalizations.

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“…We first use SPMs to define a mixed vertex on a bisector of two points, which is similar to a mixed (Voronoi) vertex defined in [6,10]. Then we introduce a degenerate mixed vertex to deal with disconnected bisectors.…”

Section: Mixed Vertices and Their Degenerate Versionmentioning

confidence: 99%

Higher-Order Geodesic Voronoi Diagrams in a Polygonal Domain with Holes

Liu¹,

Lee²

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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We investigate the higher-order Voronoi diagrams of n point sites with respect to the geodesic distance in a simple polygon with h > 0 polygonal holes and c corners. Given a set of n point sites, the k thorder Voronoi diagram partitions the plane into several regions such that all points in a region share the same k nearest sites. The nearest-site (first-order) geodesic Voronoi diagram has already been well-studied, and its total complexity is O(n+c). On the other hand, Bae and Chwa [3] recently proved that the total complexity of the farthest-site ((n − 1) st -order) geodesic Voronoi diagram and the number of faces in the diagram are Θ(nc) and Θ(nh), respectively. It is of high interest to know what happens between the first-order and the (n − 1) storder geodesic Voronoi diagrams. In this paper we prove that the total complexity of the k th -order geodesic Voronoi diagram is Θ(k(n − k) + kc), and the number of faces in the diagram is Θ(k(n − k) + kh). Our results successfully explain the variation from the nearest-site to the farthest-site geodesic Voronoi diagrams, i.e., from k = 1 to k = n − 1, and also illustrate the formation of a disconnected Voronoi region, which does not occur in many commonly used distance metrics, such as the Euclidean, L 1 , and city metrics. We show that the k th -order geodesic Voronoi diagram can be computed in O(k 2 (n+c) log(n+c)) time using an iterative algorithm.

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Farthest-Polygon Voronoi Diagrams

Cited by 22 publications

References 11 publications

On Linear-Sized Farthest-Color Voronoi Diagrams

On Linear-Sized Farthest-Color Voronoi Diagrams

Circles in the Water: Towards Island Group Labeling

Higher-Order Geodesic Voronoi Diagrams in a Polygonal Domain with Holes

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