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.
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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02 Jul 2006
TL;DR: New techniques using color, line and texture to visualize k-order diagrams are proposed, which have several interesting characteristics such as providing useful multi-dimensional information when even a small portion of the figure is available.
Abstract: Higher-order Voronoi diagrams provide a useful tool for studying problems where more than one nearest site is of interest. Understanding and visualizing higher-order Voronoi diagrams is more difficult than ordinary Voronoi diagrams. We propose new techniques using color, line and texture to visualize k-order diagrams. Approaches we develop have several interesting characteristics such as providing useful multi-dimensional information when even a small portion of the figure is available.
11 citations
Cites methods from "New applications of random sampling..."
...Efficient construction of order-k Voronoi diagrams has been studied by many researchers including Lee [8], Chazelle and Edelsbrunner [4], Aurenhammer [2], Clarkson [5], and Agarwal et al....
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...Efficient construction of order-k Voronoi diagrams has been studied by many researchers including Lee [8], Chazelle and Edelsbrunner [4], Aurenhammer [2], Clarkson [5], and Agarwal et al. [1]....
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DOI•
01 Jan 1995TL;DR: Although the focus of the thesis is on the convex hull problem, applications of the techniques to many related problems in computational geometry are also explored, including the computation of Voronoi diagrams, extreme points, convex layers, levels in arrangements, and envelopes of line segments.
Abstract: The construction of the convex hull of a nite point set in a low-dimensional Euclidean space is a fundamental problem in computational geometry. This thesis investigates e cient algorithms for the convex hull problem, where complexity is measured as a function of both the size of the input point set and the size of the output polytope. Two new, simple, optimal, output-sensitive algorithms are presented in two dimensions and a simple, optimal, output-sensitive algorithm is presented in three dimensions. In four dimensions, we give the rst output-sensitive algorithm that is within a polylogarithmic factor of optimal. In higher xed dimensions, we obtain an algorithm that is optimal for su ciently small output sizes and is faster than previous methods for sublinear output sizes; this result is further improved in even dimensions. Although the focus of the thesis is on the convex hull problem, applications of our techniques to many related problems in computational geometry are also explored, including the computation of Voronoi diagrams, extreme points, convex layers, levels in arrangements, and envelopes of line segments, as well as problems relating to ray shooting and linear programming. ii Table of
10 citations
TL;DR: A randomized divide-and-conquer algorithm to compute the order-k abstract Voronoi diagram in expected O ( k n 1 + e ) operations and provides basic techniques that can enable the application of well-known random sampling techniques to the construction of Vor onoi diagrams in the abstract setting and for non-point sites.
Abstract: Given a set of n sites in the plane, their order-k Voronoi diagram partitions the plane into regions such that all points within one region have the same k nearest sites. The order-k abstract Voronoi diagram offers a unifying framework that represents a wide range of concrete order-k Voronoi diagrams. It is defined in terms of bisecting curves satisfying some simple combinatorial properties, rather than the geometric notions of sites and distance.In this paper we develop a randomized divide-and-conquer algorithm to compute the order-k abstract Voronoi diagram in expected O ( k n 1 + e ) operations. For solving small sub-instances in the divide-and-conquer process, we also give two auxiliary algorithms with expected O ( k 2 n log ź n ) and O ( n 2 2 α ( n ) log ź n ) time, respectively, where α ( ź ) is the inverse of the Ackermann function. Our approach directly implies an O ( k n 1 + e ) -time algorithm for several concrete order-k instances such as points in any convex distance, disjoint line segments or convex polygons of constant size in the L p norms, and others. It also provides basic techniques that can enable the application of well-known random sampling techniques to the construction of Voronoi diagrams in the abstract setting and for non-point sites.
10 citations
01 Jun 1991
TL;DR: If all arcs have the same radius, the (expected) running time can be improved to $O(n^{3/2+\epsilon})$, for any $\ep silon < 0$.
Abstract: \indent In this paper we present efficient algorithms for counting intersections in a collection of circles or circular arcs. We present a randomized algorithm to count intersections in a collection of $n$ circles whose expected running time is $O(n^{3/2+\epsilon})$, for any $\epsilon < 0.$ We also develop another randomized algorithm to count intersections in a set of $n$ circular arcs whose expected running time is $O(n^{5/3+\epsilon})$, for any $\epsilon < 0.$ If all arcs have the same radius, the (expected) running time can be improved to $O(n^{3/2+\epsilon})$, for any $\epsilon < 0$.
10 citations
Posted Content•
TL;DR: It is shown that, for any $\varepsilon >0$, there exists an $\vARpsilon-net of $P$ for halfspace ranges, of size $O(1/\varpsilon)$, and five proofs of this result are given.
Abstract: Given a set $P$ of $n$ points in $\mathbb{R}^3$, we show that, for any $\varepsilon >0$, there exists an $\varepsilon$-net of $P$ for halfspace ranges, of size $O(1/\varepsilon)$. We give five proofs of this result, which are arguably simpler than previous proofs \cite{msw-hnlls-90, cv-iaags-07, pr-nepen-08}. We also consider several related variants of this result, including the case of points and pseudo-disks in the plane.
10 citations
References
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Book•
01 Jan 1968
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Abstract: A fuel pin hold-down and spacing apparatus for use in nuclear reactors is disclosed. Fuel pins forming a hexagonal array are spaced apart from each other and held-down at their lower end, securely attached at two places along their length to one of a plurality of vertically disposed parallel plates arranged in horizontally spaced rows. These plates are in turn spaced apart from each other and held together by a combination of spacing and fastening means. The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. This apparatus is particularly useful in connection with liquid cooled reactors such as liquid metal cooled fast breeder reactors.
17,939 citations
01 Jan 1985
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.
Abstract: From the reviews: "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...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."
6,525 citations
"New applications of random sampling..." refers background in this paper
...(In fact the mapping γ is not unique in this regard: see [13, 23, 2]....
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Book•
01 Jan 1973
4,243 citations
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Abstract: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady. The paper was first published in Russian as Вапник В. Н. and Червоненкис А. Я. О равномерноЙ сходимости частот появления событиЙ к их вероятностям. Теория вероятностеЙ и ее применения 16(2), 264–279 (1971).
3,939 citations
"New applications of random sampling..." refers background in this paper
...Vapnik and Chervonenkis [27] have derived general conditions under which several probabilities may be uniformly estimated using one random sample....
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Book•
01 Jan 1978TL;DR: In this article, the authors present a coherent treatment of computational geometry in the plane, at the graduate textbook level, and point out the way to the solution of the more challenging problems in dimensions higher than two.
Abstract: From the reviews: "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...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."
3,419 citations