Journal ArticleDOI
An Improved Algorithm for Constructing kth-Order Voronoi Diagrams
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The kth-order Voronoi diagram of a finite set of sites in the Euclidean plane E2 subdivides E2 into maximal regions such that all points within a given region have the same k nearest sites.Abstract:
The kth-order Voronoi diagram of a finite set of sites in the Euclidean plane E2 subdivides E2 into maximal regions such that all points within a given region have the same k nearest sites. Two versions of an algorithm are developed for constructing the kth-order Voronoi diagram of a set of n sites in O(n2 log n + k(n - k) log2 n) time, O(k(n - k)) storage, and in O(n2 + k(n - k) log2 n) time, O(n2) storage, respectively.read more
Citations
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Journal ArticleDOI
Voronoi diagrams—a survey of a fundamental geometric data structure
TL;DR: The Voronoi diagram as discussed by the authors divides the plane according to the nearest-neighbor points in the plane, and then divides the vertices of the plane into vertices, where vertices correspond to vertices in a plane.
Journal ArticleDOI
New applications of random sampling in computational geometry
TL;DR: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry by creating a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer.
Book ChapterDOI
Chapter 5 – Voronoi Diagrams*
TL;DR: In this article, the authors proposed a method to solve the problem of unstructured data in the context of the Deutsche Forschungsgemeinschaft (DFG).
Proceedings ArticleDOI
Iterated nearest neighbors and finding minimal polytypes
David Eppstein,Jeff Erickson +1 more
TL;DR: A new method for finding several types of optimalk-point sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of theO(k) nearest neighbors to each point, which is better in a number of ways than previous algorithms, which were based on high-order Voronoi diagrams.
Book ChapterDOI
Voronoi Diagrams of Moving Points in the Plane
TL;DR: This paper presents a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has a nearly cubic upper bound of O(n2λs(n), where λs,(n) is the maximum length of an (n, s)-Davenport-Schinzel sequence and s is a constant depending on the motions of the point sites.
References
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Computational geometry. an introduction
TL;DR: This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.
Proceedings ArticleDOI
Closest-point problems
Michael Ian Shamos,Dan Hoey +1 more
TL;DR: The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space, and is used to obtain O(N log N) algorithms for most of the problems considered.
Journal ArticleDOI
Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire. Sur quelques propriétés des formes quadratiques positives parfaites.
Proceedings ArticleDOI
A sweepline algorithm for Voronoi diagrams
TL;DR: A transformation is used to obtain simple algorithms for computing the Voronoi diagram of point sites, of line segment sites, and of weighted point sites with sweepline technique.