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Journal ArticleDOI

ź-nets and simplex range queries

01 Dec 1987-Discrete and Computational Geometry (Springer New York)-Vol. 2, Iss: 1, pp 127-151
TL;DR: The concept of an ɛ-net of a set of points for an abstract set of ranges is introduced and sufficient conditions that a random sample is an Â-net with any desired probability are given.
Abstract: We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,dÂ?2, with linear storage andO(nÂ?) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ . These bounds are better than those previously published for alldÂ?2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an Â?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an Â?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.

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Citations
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Proceedings ArticleDOI
01 Apr 1990
TL;DR: It is shown that every graph that is the 1-skeleton of a simplicial complex K in 3-dimensions has a separator of size O(c 2/3 + ~), which gets an O(n 2) time algorithm for solving linear systems that arise from the finite element method.
Abstract: We show that every graph that is the 1-skeleton of a simplicial complex K in 3-dimensions has a separator of size O(c 2/3 + ~), where c is the number of 3-simplexes in K and 0 is the number of 0simplexes on the boundary of K, if every 3-simplex has bounded aspect-ratio. This is natural generalization of the separator results for planar graphs, such as the Lipton and Tarjan planar separator theorem. We also show that a family of separators of size O(c 2/3) exists and is constructible. Using this family of separators we get an O(n 2) time algorithm for solving linear systems that arise from the finite element method. In particular, we solve linear systems in O(n 2) time where the underlying graph is the 1-skeleton of a simplicial complex having bounded aspect-ratio and small boundary. All the constructions work in RNC with a reasonably small number of processors. 1 I n t r o d u c t i o n The Divide-and-Conquer paradigm is fundamental for a large number of both sequential as well as *This work was s u p p o r t e d in p a r t by Na t iona l Science F o u n d a t i o n g r a n t DCR-8713489. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. parallel algorithm design. Divide-and-Conquer can give both fast and efficient algorithms. For graph algorithms and numerical analysis the efficiency of the algorithm is determined by the size and quality of the separator used in the algorithm. D e f i n i t i o n 1.1 A subset of vertices B of a graph with n vertices is said to & s e p a r a t e if the remaining vertices can be partitioned into 2 sets A and C such that there are no edges from A to C, IA[, ICl < ~f. n. The subset B is an f ( n ) s e p a r a t o r if there exist a constant ~f < 1 such that B $-separates and IBI < f (n) . Two of the most important classes of graphs with small separators have been trees and planar graphs. It is well known that a tree has a single vertex separator that 2/3-separates. Another natural class of graphs with smM1 separators are the planar graphs. Lipton and Tarjan showed that any planar graph has x/-8" n-separator that 2/3-separates, [LT79]. They gave a linear time algorithm to find this separator. There have been many extensions of this work, [Mi186, GM87, GM, Dji82, Gaz86]. All these results only consider planar graphs. There have also been several results on finding separators for graphs of a given genus, [GHT82, HM86]. Many applications of these separators exist, [Lei83, FJ86] One of the main applications of these separators is the finite element method, [LRT79, PR85a, PR85b, GT87]. We show that our separators do not generate too much fill-in and, therefore, can also be used for solving these linear systems in O(n 2) sequential time or O(log2nloglogn) time using O(n 2) processors on a PRAM. Thus, we show that n 2 direct © 1990 ACM 089791-361-2/90/0005/0300 $1.50 300 methods exist for the finite element method in the 3-dimensions (possibly the most important dimension). The algorithms we present for finding these family are randomized. But linear system solvers are otherwise deterministic. To motivate our result we view the planar separator theorem as a statement about 2-dimensional simplicial complexes. We first give a few definitions. Def in i t ion 1.2 A k-dimensional simplex (k-s implex) is the convex hull of k + 1 affinely independent points in ~d space. A simplicial complex is a collection of simplexes closed under subsimplex and intersection. A k-complex K is a simplicial complex such that for every kl-simplex in K, k I <_ k. Thus, a 3-complex is a collection of tetrahedra or cells (3-simplexes), triangular patches or faces (2-simplexes), edges (1-simplexes), and vertices (0-simplexes). The k -ske le ton of a simplicial complex K is the k-complex consisting of all k Isimplexes in K for k I _< k. Thus, the 1-skeleton of a 2-complex in the plane can be viewed as a graph that is planar. On the other hand, by F£ry's Theorem we know that every planar graph can be embedded in the plane such that each edge maps to a straight line, [Tho80a]. Thus, if G is a triangulated planar graph then it can be embedded in a 1-skeleton of a 2-complex in the plane. Thus we can view the planar separator theorem as statements about the 1-skeletons in 2-dimensions. The main goal of this paper is to show that under reasonable assumptions small separators exist and can be found for 1-skeletons of 3-complexes embedded 3-dimensions. We next discuss the restriction we place on the 3:omplex. It is not hard to see that any graph can be embedded in 3-dimensions. In particular, one can show that the complete graph can be embedded in ~;he 1-skeleton of a 3-complex in ~3. The 3-complex will have O(n 2) 3-simplexes. We can accomodate r, his example by allowing the size of the separator 1;o be a function of the number of 3-simplexes. This restriction alone is not sufficient to insure ~;he existence of small separators. In Section 6 we exhibit a 3-complex such that its 1-skeleton has only separators for size >_ t . c~ log c, where c is the number of 3-simplexes and t is some fixed constant. The 3-simplexes in the example are long and thin. For many applications we can restrict our attention to those complexes where the simplexes have bounded aspect-ratio. There are many equivalent definitions of the aspect-ratio of a simplex such as: all angles have minimum size, the number of simplexes that can share a point is bounded below, and ratio between the diameter of the circumscribing sphere and diameter of the inscribing sphere is bounded. We have picked the following definition: Def in i t ion 1.3 The d i a m e t e r Dia(S) of a ksimplex ,,q is the maximum distance between any pair for points in S. While the a spec t r a t i o equals

83 citations

01 Jan 2003
TL;DR: In this paper, the authors study a variety of problems in combinatorial and computational geometry, which deal with various aspects of arrangements of geometric objects, in the plane and in higher dimensions.
Abstract: In this thesis we study a variety of problems in combinatorial and computational geometry, which deal with various aspects of arrangements of geometric objects, in the plane and in higher dimensions Some of these problems have algorithmic applications, while others provide combinatorial bounds for various structures in such arrangements The thesis involves two main themes: (i) Counting Crossing Configurations in Geometric Settings and its Applications: Suppose we “draw” a simple undirected graph G = (V, E) in the plane using points to represent vertices, and Jordan arcs connecting them to represent edges Assume that G has n vertices and m edges and that m ≥ 4n Then, using a planarity argument, there must exist two crossing arcs in this drawing This fact can be exploited to show that the number of such crossings is Ω(m/n), no matter how the graph is drawn The proof of this “Crossing Lemma” is due to Leighton [Lei83] and to Ajtai et al [ACNS82] A probabilistic proof of this fact was entitled “A proof from the book” [AZ98] Adapting and extending the proof technique of the Crossing Lemma, we provide improved asymptotic bounds on well-studied geometric combinatorial problems, such as the “k-set” problem (Chapter 4), the complexity of polytopes spanned by sets of points in the plane and in space (Chapter 3), etc In Chapter 2 we provide some sharp asymptotic Ramsey type theorems for intersection patterns of “nice” objects that are spanned by finite point sets: For example, we prove that for any dimension d, there exists a constant c = c(d) such that for any set P of n points in IR and any set S of m > cn distinct balls, each bounded by a sphere passing through a distinct pair of points of P , there exists a subset S ′ of S of size at least Ω(m/n) with nonempty intersection This is asymptotically tight and improves the previously best known bound (see [CEG94]) We extend this result to other families of objects, including pseudo-disks in the plane and axis-parallel boxes in any dimension The proofs rely on the same probabilistic proof technique of the Crossing Lemma, and can be regarded as extensions of that lemma The results of this chapter are joint work with Micha Sharir and appear in [SS03b] In Chapter 3 we prove that the maximum total complexity of k non-overlapping convex polygons in a set of n points in the plane is Θ(n √ k) This bound was already proved in the dual plane by Halperin and Sharir [HS92] However, our proof is much simpler and uses the Crossing Lemma applied to the collection of edges of the given polygons Similar results are obtained for more restricted collections of polygons We then generalize these results to bound the total complexity of k distinct non-overlapping

82 citations

Proceedings ArticleDOI
01 Sep 1991
TL;DR: An optimal algorithm for computing hyperplane cuttings results in a new kind of cutting, which enjoys all the properties of the previous ones and, in addition, can be refined by composition.
Abstract: An optimal algorithm for computing hyperplane cuttings is given. It results in a new kind of cutting, which enjoys all the properties of the previous ones and, in addition, can be refined by composition. An optimal algorithm for computing the convex hull of a finite point set in any fixed dimension is also given. >

82 citations

Proceedings ArticleDOI
01 Jul 1992
TL;DR: This work applies Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improve solutions for them, including an algorithm for computing the diameter of a point set in 3-space, and a very simple solution which bypasses parametric search altogether.
Abstract: We apply Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improve solutions for them. We obtain, for any fixed e > 0, an O(n1+e) algorithm for computing the diameter of a point set in 3-space, an O(n8/5+e) algorithm for computing the closest pair in a set of n lines in space. All these algorithms are deterministic. We also look at the problem of computing the k-th smallest slope formed by the lines joining n points in the plane. In 1989 Cole, Salowe, Steiger, and Szemere´di gave an optimal but very complicated O(n log n) solution based on Megiddo's technique. We follow a different route and give a very simple O(n log2n) solution which bypasses parametric searching altogether.

81 citations

Journal ArticleDOI
TL;DR: It is proved that every set system of bounded VC-dimension has a fractional Helly property and the assumption about bounded dual shatter function applies to families of sets in $\Rd$ definable by a bounded number of polynomial inequalities of bounded degree.
Abstract: We prove that every set system of bounded VC-dimension has a fractional Helly property. More precisely, if the dual shatter function of a set system $\FF$ is bounded by $o(m^k)$, then $\FF$ has fractional Helly number $k$. This means that for every $\alpha>0$ there exists a $\beta>0$ such that if $F_1,F_2,\ldots,F_n\in\FF$ are sets with $\bigcap_{i\in I}F_i eq\emptyset$ for at least $\alpha{n\choose k}$ sets $I\subseteq\{1,2,\ldots,n\}$ of size $k$, then there exists a point common to at least $\beta n$ of the $F_i$. This further implies a $(p,k)$-theorem: for every $\FF$ as above and every $p\geq k$ there exists $T$ such that if $\GG\subseteq\FF$ is a finite subfamily where among every $p$ sets, some $k$ intersect, then $\GG$ has a transversal of size $T$. The assumption about bounded dual shatter function applies, for example, to families of sets in $\Rd$ definable by a bounded number of polynomial inequalities of bounded degree; in this case we obtain fractional Helly number $d{+}1$.

81 citations


Cites background from "ź-nets and simplex range queries"

  • ...well-known theorem of Haussler and Welzl [ 9 ] about the existence of e-nets for systems of bounded VC-dimension....

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References
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Book ChapterDOI
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


"ź-nets and simplex range queries" refers background or methods or result in this paper

  • ...The drawback is that the constants, if deri~,ed from the results in [ 17 ], can be quite large....

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  • ...More generally, we characterize the classes of ranges for which there exists a function f(E) for e S0 such that any finite point set A has an e-net of size f(e), independently of the size of A. These are precisely the classes of ranges with finite Vapnik-Chervonenkis dimension, known as Vapnik-Chervonenkis classes [ 17 ], [9], [19], [1]....

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  • ...The key concepts and proof techniques of this section are based on the pioneering work of Vapnik and Chervonenkis [ 17 ]....

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  • ...Example 5. Let A be a set of n points in E 2. Since the dimension of (E 2, H~-) is 2, the results in [ 17, Theorem 2 ] show that there exists a 0.01-approximation V of A for positive half-planes (and thus for all half-planes) with I VI = 2,525,039....

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  • ...Using the related notion of an e-approxirnation (directly from [ 17 ]), we also point out trivial data structures of constant size that give approximate solutions to the counting problem for halfspaces in constant time (compare [13])....

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Book
01 Jan 1987
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Abstract: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

2,284 citations


"ź-nets and simplex range queries" refers background in this paper

  • ...We conclude this section by examining the relationship between the notion of an e-net and the established notion of a centerpoint [21], [11] in combinatorial geometry....

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  • ..., [11] for a general treatment of arrangements....

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Journal ArticleDOI
TL;DR: This paper will answer the question in the affirmative by determining the exact upper bound of T if T is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that •A| c for each A⩾ N and some constant c.

1,029 citations


"ź-nets and simplex range queries" refers background in this paper

  • ...Now the assertion can be seen as the dual formulation of Caratheodry's theorem (see [ 15 ], Theorem 2.3.5), which states that if a point x is in the convex hull of a set A in E d, then there exists a subset A' of A such that JA'I -< d + 1 and x is in the convex hull of A'. []...

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Journal ArticleDOI
TL;DR: In this article, the convergence of a stochastic process indexed by a Gaussian process to a certain Gaussian processes indexed by the supremum norm was studied in a Donsker class.
Abstract: Let $(X, \mathscr{A}, P)$ be a probability space. Let $X_1, X_2,\cdots,$ be independent $X$-valued random variables with distribution $P$. Let $P_n := n^{-1}(\delta_{X_1} + \cdots + \delta_{X_n})$ be the empirical measure and let $ u_n := n^\frac{1}{2}(P_n - P)$. Given a class $\mathscr{C} \subset \mathscr{a}$, we study the convergence in law of $ u_n$, as a stochastic process indexed by $\mathscr{C}$, to a certain Gaussian process indexed by $\mathscr{C}$. If convergence holds with respect to the supremum norm $\sup_{C \in \mathscr{C}}|f(C)|$, in a suitable (usually nonseparable) function space, we call $\mathscr{C}$ a Donsker class. For measurability, $X$ may be a complete separable metric space, $\mathscr{a} =$ Borel sets, and $\mathscr{C}$ a suitable collection of closed sets or open sets. Then for the Donsker property it suffices that for some $m$, and every set $F \subset X$ with $m$ elements, $\mathscr{C}$ does not cut all subsets of $F$ (Vapnik-Cervonenkis classes). Another sufficient condition is based on metric entropy with inclusion. If $\mathscr{C}$ is a sequence $\{C_m\}$ independent for $P$, then $\mathscr{C}$ is a Donsker class if and only if for some $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.

555 citations

Journal ArticleDOI
TL;DR: A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane.
Abstract: This paper uses a new formulation of the notion of duality that allows the unified treatment of a number of geometric problems. In particular, we are able to apply our approach to solve two long-standing problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane; the other is to produce an optimal algorithm for the half-plane range query problem. This problem is to preprocessn points in the plane, so that given a test half-plane, one can efficiently determine all points lying in the half-plane. We describe an optimalO(k + logn) time algorithm for answering such queries, wherek is the number of points to be reported. The algorithm requiresO(n) space andO(n logn) preprocessing time. Both of these results represent significant improvements over the best methods previously known. In addition, we give a number of new combinatorial results related to the computation of line arrangements.

286 citations


"ź-nets and simplex range queries" refers methods in this paper

  • ...It should be noted that better bounds are possible for reporting in two dimensions (specifically O(log n + t) time, where t is the number of points reported [3]), but these techniques only work for half-planes....

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