Finding k points with minimum diameter and related problems
TL;DR: In this article, the authors consider the problem of finding k points of a set S that form a small set under some given measure, and present efficient algorithms for several natural measures including the diameter and variance.
About: This article is published in Journal of Algorithms.The article was published on 1991-01-02. It has received 153 citations till now.
Citations
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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
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TL;DR: In this article, a survey of clustering from a mathematical programming viewpoint is presented, focusing on solution methods, i.e., dynamic programming, graph theoretical algorithms, branch-and-bound, cutting planes, column generation and heuristics.
Abstract: Given a set of entities, Cluster Analysis aims at finding subsets, called clusters, which are homogeneous and/or well separated. As many types of clustering and criteria for homogeneity or separation are of interest, this is a vast field. A survey is given from a mathematical programming viewpoint. Steps of a clustering study, types of clustering and criteria are discussed. Then algorithms for hierarchical, partitioning, sequential, and additive clustering are studied. Emphasis is on solution methods, i.e., dynamic programming, graph theoretical algorithms, branch-and-bound, cutting planes, column generation and heuristics.
600 citations
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TL;DR: A new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations is given that takes expected timeO(nℝgn) and spaceO( n), and is eminently practical to implement.
Abstract: In this paper we give a new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations. The new algorithm is more “on-line” than earlier similar methods, takes expected timeO(nℝgn) and spaceO(n), and is eminently practical to implement. The analysis of the algorithm is also interesting in its own right and can serve as a model for many similar questions in both two and three dimensions. Finally we demonstrate how this approach for constructing Voronoi diagrams obviates the need for building a separate point-location structure for nearest-neighbor queries.
520 citations
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TL;DR: The relation between RCPVs and chance-constrained problems (CCP) is explored, showing that the optimal objective of an RCPV with the generic constraint removal rule provides, with arbitrarily high probability, an upper bound on the optimal objectives of a corresponding CCP.
Abstract: Random convex programs (RCPs) are convex optimization problems subject to a finite number $N$ of random constraints. The optimal objective value $J^*$ of an RCP is thus a random variable. We study the probability with which $J^*$ is no longer optimal if a further random constraint is added to the problem (violation probability, $V^*$). It turns out that this probability rapidly concentrates near zero as $N$ increases. We first develop a theory for RCPs, leading to explicit bounds on the upper tail probability of $V^*$. Then we extend the setup to the case of RCPs with $r$ a posteriori violated constraints (RCPVs): a paradigm that permits us to improve the optimal objective value while maintaining the violation probability under control. Explicit and nonasymptotic bounds are derived also in this case: the upper tail probability of $V^*$ is upper bounded by a multiple of a beta distribution, irrespective of the distribution on the random constraints. All results are derived under no feasibility assumptions on the problem. Further, the relation between RCPVs and chance-constrained problems (CCP) is explored, showing that the optimal objective $J^*$ of an RCPV with the generic constraint removal rule provides, with arbitrarily high probability, an upper bound on the optimal objective of a corresponding CCP. Moreover, whenever an optimal constraint removal rule is used in the RCPVs, then appropriate choices of $N$ and $r$ exist such that $J^*$ approximates arbitrarily well the objective of the CCP.
315 citations
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01 Jan 2000TL;DR: In this article, the authors proposed a method to solve the problem of unstructured data in the context of the Deutsche Forschungsgemeinschaft (DFG).
Abstract: 1 Partially supported by the Deutsche Forschungsgemeinschaft, grant Kl 655 2-2.
296 citations
References
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TL;DR: It is shown that the k-nearest neighbor problem and other seemingly unrelated problems can be solved efficiently with the Voronoi diagram.
Abstract: The notion of Voronoi diagram for a set of N points in the Euclidean plane is generalized to the Voronoi diagram of order k and an iterative algorithm to construct the generalized diagram in 0(k2N log N) time using 0(k2(N − k)) space is presented. It is shown that the k-nearest neighbor problem and other seemingly unrelated problems can be solved efficiently with the diagram.
361 citations
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TL;DR: It turns out that the standard Euclidean Voronoi diagram of point sets inRd along with its order-k generalizations are intimately related to certain arrangements of hyperplanes, and this fact can be used to obtain new Vor onoi diagram algorithms.
Abstract: We propose a uniform and general framework for defining and dealing with Voronoi diagrams. In this framework a Voronoi diagram is a partition of a domainD induced by a finite number of real valued functions onD. Valuable insight can be gained when one considers how these real valued functions partitionD ×R. With this view it turns out that the standard Euclidean Voronoi diagram of point sets inRd along with its order-k generalizations are intimately related to certain arrangements of hyperplanes. This fact can be used to obtain new Voronoi diagram algorithms. We also discuss how the formalism of arrangements can be used to solve certain intersection and union problems.
346 citations
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TL;DR: An algorithm is presented for computing certain kinds of three-dimensional convex hulls in linear time and it is shown that the Voronoi diagram ofn sites in the plane can be computed in Θ(n) time when these sites form the vertices of a convex polygon in counterclockwise order.
Abstract: We present an algorithm for computing certain kinds of three-dimensional convex hulls in linear time. Using this algorithm, we show that the Voronoi diagram ofn sites in the plane can be computed in ?(n) time when these sites form the vertices of a convex polygon in, say, counterclockwise order. This settles an open problem in computational geometry. Our techniques can also be used to obtain linear-time algorithms for computing the furthest-site Voronoi diagram and the medial axis of a convex polygon and for deleting a site from a general planar Voronoi diagram.
337 citations