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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01 Jan 2017
TL;DR: In this paper, the authors review the known results and techniques, including recent developments, for simplex range searching and its variants, and present a survey of the range-searching literature.
Abstract: A central problem in computational geometry, range searching arises in many applications, and numerous geometric problems can be formulated in terms of range searching. A typical range-searching problem has the following form. Let S be a set of n points in \(\mathbb{R}^{d}\), and let \(\mathbb{R}\) be a family of subsets of \(\mathbb{R}^{d}\); elements of \(\mathbb{R}\) are called ranges. Preprocess S into a data structure so that for a query range \(\gamma \in \mathbb{R}\), the points in S ∩γ can be reported or counted efficiently. Notwithstanding extensive work on range searching over the last four decades, it remains an active research area. A series of papers by Jirka Matousek and others in the late 1980s and the early 1990s had a profound impact not only on range searching but also on computational geometry as a whole. This chapter reviews the known results and techniques, including recent developments, for simplex range searching and its variants.
32 citations
Cites background from "New applications of random sampling..."
...Clarkson [40] had described a random-sampling based data structure of size O(nbd/2c+ε) that could anser a point-location query in O(log n) time....
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...Clarkson [40] and Haussler and Welzl [58] were the first to show the existence of a (1/r)-cutting of H of size O(rd logd r)....
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Posted Content•
TL;DR: A polynomial partitioning algorithm for semialgebraic range searching was proposed in this paper, with running time bounds similar to Agarwal, Sharir, and the first author's algorithm.
Abstract: The polynomial partitioning method of Guth and Katz [arXiv:1011.4105] has numerous applications in discrete and computational geometry. It partitions a given $n$-point set $P\subset\mathbb{R}^d$ using the zero set $Z(f)$ of a suitable $d$-variate polynomial $f$. Applications of this result are often complicated by the problem, what should be done with the points of $P$ lying within $Z(f)$? A natural approach is to partition these points with another polynomial and continue further in a similar manner. So far it has been pursued with limited success---several authors managed to construct and apply a second partitioning polynomial, but further progress has been prevented by technical obstacles. We provide a polynomial partitioning method with up to $d$ polynomials in dimension $d$, which allows for a complete decomposition of the given point set. We apply it to obtain a new algorithm for the semialgebraic range searching problem. Our algorithm has running time bounds similar to a recent algorithm by Agarwal, Sharir, and the first author [SIAM~J.~Comput. 42(2013) 2039--2062], but it is simpler both conceptually and technically. While this paper has been in preparation, Basu and Sombra, as well as Fox, Pach, Sheffer, Suk, and Zahl, obtained results concerning polynomial partitions which overlap with ours to some extent.
31 citations
TL;DR: A A (H) Monte Carlo algorithm for this problem is obtained, improving a resuit of Edelsbrunner é tal and has numerous conséquences for the construction offurther randomized algorithms, using the above problems as a subroutine.
Abstract: Let P be a point set in the plane and T a spanning tree on P, whose edges are realized by segments. We define the crossing number of T as the maximum number of edges of T intersected by a single Une. We give a A(n) deterministic algorithm finding a spanning tree with crossing number O(fh) on a given n point set {this crossing number is asymptotically optimal), and a A(«) randomized (Las Vegas) algorithm finding a spanning tree with crossing number O(^Jnlogn) (hère f (n) = A (g (n)) means f(n) = O(g(n)\\o%n) for a constante). This improves results of Welzl and Edelsbrunner et al. We also consider the construction of a family of OQogri) spanning trees, such thatfor every Une X there is a tree in this family such that X crosses only O(fn.\\o%n) ofits edges. We obtain a A (H) Monte Carlo algorithm for this problem, improving a resuit of Edelsbrunner é tal . This resuit has numerous conséquences for the construction offurther randomized algorithms, using the above problems as a subroutine. Résumé. Soit P un ensemble de points du plan et soit T un arbre recouvrant de P, dont les arêtes sont des segments. Le nombre de croisements de T est le nombre maximal d'arêtes de T intersectées par une même droite. Si f et g sont deux fonctions, on pose f(n) = A(g(n)) s'il existe une constante c telle que ƒ (n) = O(g(n)\\off(ri)). Nous donnons un algorithme déterministe en A (AÏ') pour construire un arbre recouvrant dont le nombre de croisement est O(/n), où n est le nombre de points (ce nombre de croisement est asymptotiquement optimal); on donne également un algorithme probabiliste en A(«) pour construire un arbre recouvrant dont le nombre de croisement est O(fn\\ogn). Ceci améliore des résultats de Welzl, Edelsbrunner et al. On considère également la construction d'une famille de O(\\ogn) arbres recouvrants tels que, pour chaque droite X il existe un arbre de la famille tel que X intersecte seulement O(fiï log M) arêtes de l'arbre. On obtient un algorithme probabiliste en A (n) pour ce problème, ce qui améliore un résultat de Edelsbrunner et al. Ce résultat a de nombreuses conséquences pour la construction d'autres algorithmes probabilistes, qui utilisent alors les solutions des problèmes ci-dessus comme sous-programmes.
31 citations
TL;DR: It is proved that the combinatorial complexity ofℒ(ℬ) has an % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYb
Abstract: A line intersecting all polyhedra in a set? is called a "stabber" for the set?. This paper addresses some combinatorial and algorithmic questions about the set?(?) of all lines stabbing?. We prove that the combinatorial complexity of?(?) has an % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGpbGaai% ikaiaad6gadaahaaWcbeqaaiaaiodaaaGccaaIYaWaaWbaaSqabeaa% caWGJbWaaOaaaeaaciGGSbGaai4BaiaacEgacaWGUbaameqaaaaaki% aacMcaaaa!4368! $$O(n^3 2^{c\sqrt {\log n} } )$$ upper bound, wheren is the total number of facets in?, andc is a suitable constant. This bound is almost tight. Within the same time bound it is possible to determine if a stabbing line exists and to find one.
31 citations
17 Jan 2010
TL;DR: In this paper, a streaming algorithm for maintaining a blurred ball cover whose working space is linear in d and independent of n is presented, and lower bounds on the worst-case approximation ratio of any streaming algorithm that uses poly(d) space.
Abstract: We develop (single-pass) streaming algorithms for maintaining extent measures of a stream S of n points in Rd. We focus on designing streaming algorithms whose working space is polynomial in d (poly(d)) and sublinear in n. For the problems of computing diameter, width and minimum enclosing ball of S, we obtain lower bounds on the worst-case approximation ratio of any streaming algorithm that uses poly(d) space. On the positive side, we introduce the notion of blurred ball cover and use it for answering approximate farthest-point queries and maintaining approximate minimum enclosing ball and diameter of S. We describe a streaming algorithm for maintaining a blurred ball cover whose working space is linear in d and independent of n.
30 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