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Journal ArticleDOI

New applications of random sampling in computational geometry

Kenneth L. Clarkson1
01 Jun 1987-Discrete and Computational Geometry (Springer New York)-Vol. 2, Iss: 1, pp 195-222
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.
Abstract: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requiresO(sd+?) expected preprocessing time to build a search structure for an arrangement ofs hyperplanes ind dimensions. The expectation, as with all expected times reported here, is with respect to the random behavior of the algorithm, and holds for any input. Given the data structure, and a query pointp, the cell of the arrangement containingp can be found inO(logs) worst-case time. (The bound holds for any fixed ?>0, with the constant factors dependent ond and ?.) Using point-plane duality, the algorithm may be used for answering halfspace range queries. Another algorithm finds random samples of simplices to determine the separation distance of two polytopes. The algorithm uses expectedO(n[d/2]) time, wheren is the total number of vertices of the two polytopes. This matches previous results [10] for the cased = 3 and extends them. Another algorithm samples points in the plane to determine their orderk Voronoi diagram, and requires expectedO(s1+?k) time fors points. (It is assumed that no four of the points are cocircular.) This sharpens the boundO(sk2 logs) for Lee's algorithm [21], andO(s2 logs+k(s?k) log2s) for Chazelle and Edelsbrunner's algorithm [4]. Finally, random sampling is used to show that any set ofs points inE3 hasO(sk2 log8s/(log logs)6) distinctj-sets withj≤k. (ForS ?Ed, a setS? ?S with |S?| =j is aj-set ofS if there is a half-spaceh+ withS? =S ?h+.) This sharpens with respect tok the previous boundO(sk5) [5]. The proof of the bound given here is an instance of a "probabilistic method" [15].

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Citations
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Journal ArticleDOI
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

Proceedings ArticleDOI
Kenneth L. Clarkson1
06 Jan 1988
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Abstract: Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.

1,163 citations

Proceedings ArticleDOI
01 Jan 1993
TL;DR: The up-tree (vantage point tree) is introduced in several forms, together‘ with &&ciated algorithms, as an improved method for these difficult search problems in general metric spaces.
Abstract: We consider the computational problem of finding nearest neighbors in general metric spaces. Of particular interest are spaces that may not be conveniently embedded or approximated in Euclidian space, or where the dimensionality of a Euclidian representation 1s very high. Also relevant are high-dimensional Euclidian settings in which the distribution of data is in some sense of lower dimension and embedded in the space. The up-tree (vantage point tree) is introduced in several forms, together‘ with &&ciated algorithms, as an improved method for these difficult search nroblems. Tree construcI tion executes in O(nlog(n i ) time, and search is under certain circumstances and in the imit, O(log(n)) expected time. The theoretical basis for this approach is developed and the results of several experiments are reported. In Euclidian cases, kd-tree performance is compared.

1,145 citations


Cites methods from "New applications of random sampling..."

  • ...More recently, the Voronoi digram [21] has provided a useful tool in low- dimensional Euclidian settings { and Figure 1: vp-tree decomposition Figure 2: kd-tree decomposition the overall eld and outlook of Computational Geometry has yielded many interesting results such as those of [22, 23, 24, 25] and earlier [26]....

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Proceedings ArticleDOI
01 Oct 1998
TL;DR: New packet classification schemes are presented that, with a worst-case and traffic-independent performance metric, can classify packets, by checking amongst a few thousand filtering rules, at rates of a million packets per second using range matches on more than 4 packet header fields.
Abstract: The ability to provide differentiated services to users with widely varying requirements is becoming increasingly important, and Internet Service Providers would like to provide these differentiated services using the same shared network infrastructure. The key mechanism, that enables differentiation in a connectionless network, is the packet classification function that parses the headers of the packets, and after determining their context, classifies them based on administrative policies or real-time reservation decisions. Packet classification, however, is a complex operation that can become the bottleneck in routers that try to support gigabit link capacities. Hence, many proposals for differentiated services only require classification at lower speed edge routers and also avoid classification based on multiple fields in the packet header even if it might be advantageous to service providers. In this paper, we present new packet classification schemes that, with a worst-case and traffic-independent performance metric, can classify packets, by checking amongst a few thousand filtering rules, at rates of a million packets per second using range matches on more than 4 packet header fields. For a special case of classification in two dimensions, we present an algorithm that can handle more than 128K rules at these speeds in a traffic independent manner. We emphasize worst-case performance over average case performance because providing differentiated services requires intelligent queueing and scheduling of packets that precludes any significant queueing before the differentiating step (i.e., before packet classification). The presented filtering or classification schemes can be used to classify packets for security policy enforcement, applying resource management decisions, flow identification for RSVP reservations, multicast look-ups, and for source-destination and policy based routing. The scalability and performance of the algorithms have been demonstrated by implementation and testing in a prototype system.

741 citations

Journal ArticleDOI
Kenneth L. Clarkson1
TL;DR: These results are tied together, stronger convergence results are reviewed, and several coreset bounds are generalized or strengthened.
Abstract: The problem of maximizing a concave function f(x) in the unit simplex Δ can be solved approximately by a simple greedy algorithm. For given k, the algorithm can find a point x(k) on a k-dimensional face of Δ, such that f(x(k) ≥ f(xa) − O(1/k). Here f(xa) is the maximum value of f in Δ, and the constant factor depends on f. This algorithm and analysis were known before, and related to problems of statistics and machine learning, such as boosting, regression, and density mixture estimation. In other work, coming from computational geometry, the existence of ϵ-coresets was shown for the minimum enclosing ball problem by means of a simple greedy algorithm. Similar greedy algorithms, which are special cases of the Frank-Wolfe algorithm, were described for other enclosure problems. Here these results are tied together, stronger convergence results are reviewed, and several coreset bounds are generalized or strengthened.

456 citations

References
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Proceedings ArticleDOI
01 Nov 1986
TL;DR: Blumer and Haussler as discussed by the authors extended Valiant's learnability model to learning classes of concepts defined by regions in Euclidean space E. They showed that the essential condition for distribution-free learnability is finiteness of the Vapnik-Chervonenkis dimension, a simple combinatorial parameter of the class of concepts to be learned.
Abstract: We extend Valiant's learnability model to learning classes of concepts defined by regions in Euclidean space E". Our methods lead to a unified treatment of some of Valiant's results, along with previous results of Pearl and Devroye and Wagner on distribution-free convergence of certain pattern recognition algorithms. We show that the essential condition for distribution-free learnability is finiteness of the Vapnik-Chervonenkis dimension, a simple combinatorial parameter of the class of concepts to be learned. Using this parameter, we analyze the complexity and closure properties of learnable classes. Authors A. Blumer and D. Haussler gratefully acknowledge the support of NSF grant IST-8317918, author A. Ehrenfeucht the support of NSF grant MCS-8305245, and author M. Warmuth the support of the Faculty Research Committee of the University of California at Santa Cruz. Part of this work was done while A. Blumer was visiting the University of California at Santa Cruz and M. Warmuth the Univer-

154 citations

Journal ArticleDOI
TL;DR: The k-hull is the set of points p such that for any hyperplane containing p there is a k-Hull of X such that p = 1,2, \cdots.
Abstract: For any set X of points (in any dimension) and any $k = 1,2, \cdots $, we introduce the concept of the k-hull of X. The k-hull is the set of points p such that for any hyperplane containing p there...

153 citations

Proceedings ArticleDOI
Andrew Chi-Chih Yao1, F F Yao
01 Dec 1985
TL;DR: In this paper, it was shown that any bounded region in Ed can be divided into 2d subregions of equal volume in such a way that no hyperplane in Ed cannot intersect all 2d of the sub-regions.
Abstract: It is shown that any bounded region in Ed can be divided into 2d subregions of equal volume in such a way that no hyperplane in Ed can intersect all 2d of the subregions. This theorem provides the basis of a data structure scheme for organizing n points in d dimensions. Under this scheme, a broad class of geometric queries in d dimensions, including many common problems in range search and optimization, can be solved in linear storage space and sublinear time.

142 citations

Proceedings Article
01 Jan 1985
TL;DR: It is shown that any bounded region in E = d can be divided into 2 subregions of equal volume in such a way that no hyperplane in E can intersect all 2 of the subRegions.
Abstract: It is shown that any bounded region in Ed can be divided into 2d subregions of equal volume in such a way that no hyperplane in Ed can intersect all 2d of the subregions. This theorem provides the basis of a data structure scheme for organizing n points in d dimensions. Under this scheme, a broad class of geometric queries in d dimensions, including many common problems in range search and optimization, can be solved in linear storage space and sublinear time.

134 citations


"New applications of random sampling..." refers methods in this paper

  • ...The O(log s) query time of the algorithm given here is much faster that of several algorithms previously known [29, 30, 7]....

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Proceedings ArticleDOI
06 Jan 1988
TL;DR: The concept of an infinitesimal perturbation is introduced and it is shown that the method is consistent relative to such perturbations.
Abstract: In a previous paper, we introduced a generic solution to the problem of data degeneracy in geometric algorithms. The scheme is simple to use: algorithms qualifying under our requirements just have to use a prescribed blackbox for polynomial evaluation in order to achieve a symbolic perturbation of data. In this paper, we introduce the concept of an infinitesimal perturbation and show that our method is consistent relative to such perturbations.

124 citations


"New applications of random sampling..." refers methods in this paper

  • ...Recently systematic methods have been developed to apply such perturbations \formally," that is, to break ties in an arbitrary but consistent way, so as to simulate nondegeneracy with degenerate input [22, 44 ]....

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