ź-nets and simplex range queries
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.
Citations
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Proceedings Article•
01 Jan 2008
TL;DR: In this paper, a technique for proving the existence of small μ-nets for hypergraphs satisfying certain simple conditions is described. But the technique is not suitable for proving o(1/μ log 1/μ) upper bounds which the standard VC-dimension theory does not allow.
Abstract: We describe a new technique for proving the existence of small μ-nets for hypergraphs satisfying certain simple conditions. The technique is particularly useful for proving o(1/μ log 1/μ) upper bounds which the standard VC-dimension theory does not allow. We apply the technique to several geometric hypergraphs and obtain simple proofs for the existence of O(1/μ) size μ-nets for them. This includes the geometric hypergraph in which the vertex set is a set of points in the plane and the hyperedges are defined by a set of pseudo-disks. This result was not known previously. We also get a very short proof for O(1/μ) size μ-nets for half-spaces in R3.
62 citations
17 Sep 2002
TL;DR: In this article, the running time of the algorithm is O(n log n), with the constant of proportionality depending on k, d, and?, where k is the number of points in the set P of n points in Rd and d is an integer k? 1, where w* denotes the minimum value so that P can be covered by k cylinders of radius at most w *.
Abstract: Given a set P of n points in Rd and an integer k ? 1, let w* denote the minimum value so that P can be covered by k cylinders of radius at most w*. We describe an algorithm that, given P and an ? > 0, computes k cylinders of radius at most (1 + ?)w* that cover P. The running time of the algorithm is O(n log n), with the constant of proportionality depending on k, d, and ?. We first show that there exists a small "certificate" Q ? P, whose size does not depend on n, such that for any k-cylinders that cover Q, an expansion of these cylinders by a factor of (1 + ?) covers P. We then use a well-known scheme based on sampling and iterated re-weighting for computing the cylinders.
61 citations
01 Jul 1993
TL;DR: This work presents a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo, based on expander graphs and is conceptually much simpler and has more explicit geometric flavor.
Abstract: We present a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo. Our new approach is based on expander graphs and is conceptually much simpler and has more explicit geometric flavor. It does not require parallelization or randomization, and it exploits recent range-searching techniques of Matouscek and others. We exemplify the technique on three problems, the slope selection problem, the planar distance selection problem, and the planar two-center problem. For the first problem we develop an O(n log3n)) solution, which, although suboptimal, is very simple. The second and third problems are more typical examples of our approach. Our solutions have, respectively, running time O(n4/3 log3+d n), for any d > 0, and O(n2 log3 n), comparable with the respective solutions of [2, 5].
60 citations
TL;DR: Using ideas from computational geometry and @e-net theory, an O(@D) bound is attained for the maximum interference where @D is the interference of a uniform-radius ad-hoc network.
60 citations
Cites background from "ź-nets and simplex range queries"
...The following theorem is a fundamental theorem in learning theory and computational geometry [10,9]....
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...It is known that the VC dimension of the set of all triangles in the plane is finite [10]....
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Book•
18 Aug 2011
TL;DR: The maximum number of faces boundingm distinct cells in an arrangement ofn planes is O(m2/3n logn +n2); the authors can calculatem such cells specified by a point in each, in worst-case timeO(m3/5−δn4/5+2δ+m+n logm), for any collection of points no three of which are collinear.
Abstract: We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m2/3n logn +n2); we can calculatem such cells specified by a point in each, in worst-case timeO(m2/3n log3n+n2 logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m2/3n logn+n2), but this number is onlyO(m3/5??n4/5+2?+m+n logm), for any?>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m3/4??n3/4+3?+m) log2n+n logn logm) for any?>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m3/4??n3/4+3?+m] log2n+n logn logm) for any?>0. (v) The maximum number of facets (i.e., (d?1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m2/3nd/3 logn+nd?1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.
59 citations
References
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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 1987TL;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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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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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
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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