Two-Dimensional Voronoi Diagrams in the Lp-Metric
TL;DR: Many proximity problems revolving a set of points, such as finding the nearest neighbor of a given point, finding the minimum spamung tree, findmg the smallest circle enclosing the point set, etc., can be solved very efficiently via the Voronoi diagram.
Abstract: The Voronoi diagram, also known as the Thiessen diagram, for a set of N points in the Cartesian plane in which the L,-metnc is the distance measure, where p is a real number between 1 and 0o inclusive, is defined, and an algorithm for constructing the dmgram m O(NlogN) tune is presented This algonthm uses the divide-and-conquer technique. Many proximity problems revolving a set of points, such as finding the nearest neighbor of a given point, finding the minimum spamung tree, findmg the smallest circle (m the Lp-metric) enclosing the point set, etc., can be solved very efficiently via the diagram The running time of the algorithm presented is also shown to be optimal to within a constant factor. KEY WORDS AND PHRASES Voronoi diagram, L,-metric, computational geometry, computational complexity, analysis of algorithm, divide-and-conquer CR CATEGORIES 4 49, 5 25, 5 32 1. Introduction
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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.
Abstract: Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor
4,236 citations
TL;DR: This paper provides a unified discussion of the Delaunay triangulation and two algorithms are presented for constructing the triangulations over a planar set ofN points.
Abstract: This paper provides a unified discussion of the Delaunay triangulation. Its geometric properties are reviewed and several applications are discussed. Two algorithms are presented for constructing the triangulation over a planar set ofN points. The first algorithm uses a divide-and-conquer approach. It runs inO(N logN) time, which is asymptotically optimal. The second algorithm is iterative and requiresO(N
2) time in the worst case. However, its average case performance is comparable to that of the first algorithm.
1,460 citations
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01 Jan 1995TL;DR: A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Abstract: An $(n,s)$ Davenport--Schinzel sequence, for positive integers $n$ and $s$, is a sequence composed of $n$ symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation $a \cdots b \cdots a \cdots b \cdots$ of length $s+2$ between two distinct symbols $a$ and $b$. The close relationship between Davenport--Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive, because a wide variety of geometric problems can be formulated in terms of lower envelopes. A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems. This paper gives a comprehensive survey on the theory of Davenport--Schinzel sequences and their geometric applications.
1,052 citations
TL;DR: The close relationship to convex hulls and arrangements of hyperplanes is investigated and exploited, and efficient algorithms that compute the power diagram and its order-k modifications are obtained.
Abstract: The power pow $(x,s)$ of a point x with respect to a sphere s in Euclidean d-space $E^d $ is given by $d^2 (x,z) - r^2 $, where d denotes the Euclidean distance function, and z and r are the center and the radius of s. The power diagram of a finite set S of spheres in $E^d $ is a cell complex that associates each $s \in S$ with the convex domain $\{ x \in E^d | {\operatorname{pow}} (x,s) < {\operatorname{pow}} (x,t), {\text{ for all }} t \in S - \{ s\} \}$.The close relationship to convex hulls and arrangements of hyperplanes is investigated and exploited. Efficient algorithms that compute the power diagram and its order-k modifications are obtained. Among the applications of these results are algorithms for detecting k-sets, for union and intersection problems for cones and paraboloids, and for constructing weighted Voronoi diagrams and Voronoi diagrams for spheres. Upper space bounds for these geometric problems are derived.
836 citations
TL;DR: This work presents a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage.
Abstract: A planar subdivision is any partition of the plane into (possibly unbounded) polygonal regions. The subdivision search problem is the following: given a subdivision $S$ with $n$ line segments and a query point $p$, determine which region of $S$ contains $p$. We present a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage. Our subdivision search structure can be constructed in linear time from the subdivision representation used in many applications.
810 citations
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13 Oct 1975
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.
Abstract: A number of seemingly unrelated problems involving the proximity of N points in the plane are studied, such as finding a Euclidean minimum spanning tree, the smallest circle enclosing the set, k nearest and farthest neighbors, the two closest points, and a proper straight-line triangulation. For most of the problems considered a lower bound of O(N log N) is shown. For all of them the best currently-known upper bound is O(N2) or worse. 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. The Voronoi diagram is used to obtain O(N log N) algorithms for all of the problems.
1,140 citations
"Two-Dimensional Voronoi Diagrams in..." refers background or methods in this paper
...Shamos and Hoey [ 15 ] present an O(NlogN) algorithm to construct it and show that a number of seemingly unrelated problems can be solved very efficiently once the diagram is available....
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...Since the minimum spanning tree for the set of N points is embedded m the triangulation [14, 15 ], any minimum spanning tree algorithm [1, 17] which runs in less than O(NlogN) time can be applied to find the minimum spanning tree....
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1,121 citations
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TL;DR: The existence of reasonably dense lattice coverings and reasonably economical lattice covers has been studied in this article, where the authors show that simplices cannot be very dense and coverings with spheres cannot have very economical coverings.
Abstract: Introduction 1. Packaging and covering densities 2. The existence of reasonably dense packings 3. The existence of reasonably economical coverings 4. The existence of reasonably dense lattice packings 5. The existence of reasonably economical lattice coverings 6. Packings of simplices cannot be very dense 8. Coverings with spheres cannot be very economical Bibliography Index.
631 citations
TL;DR: This paper studies methods for finding minimum spanning trees in graphs and results include relationships with other problems which might lead general lower bound for the complexity of the minimum spanning tree problem.
Abstract: This paper studies methods for finding minimum spanning trees in graphs. Results include 1. several algorithms with $O(m\log \log n)$ worst-case running times, where n is the number vertices and m is the number of edges in the problem graph; 2. an $O(m)$ worst-case algorithm for dense graphs (those for which m is $\Omega (n^{1 + \varepsilon } )$ for some positive constant $\varepsilon $); 3. an $O(n)$ worst-case algorithm for planar graphs; 4. relationships with other problems which might lead general lower bound for the complexity of the minimum spanning tree problem.
427 citations
TL;DR: The existence of reasonably dense lattice coverings and reasonably economical lattice covers has been studied in this paper, where the authors show that simplices cannot be very dense and coverings with spheres cannot have very economical coverings.
Abstract: Introduction 1. Packaging and covering densities 2. The existence of reasonably dense packings 3. The existence of reasonably economical coverings 4. The existence of reasonably dense lattice packings 5. The existence of reasonably economical lattice coverings 6. Packings of simplices cannot be very dense 8. Coverings with spheres cannot be very economical Bibliography Index.
421 citations