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.
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01 Mar 2014
TL;DR: This paper studies Macbeath's problem in a more general setting, and not only for the Lebesgue measure as is the case in the classical theorem, and proves near-optimal generalizations for several basic geometric set systems.
Abstract: The existence of Macbeath regions is a classical theorem in convex geometry ("A Theorem on non-homogeneous lattices", Annals of Math, 1952). We refer the reader to the survey of I. Barany for several applications. Recently there have been some striking applications of Macbeath regions in discrete and computational geometry.
In this paper, we study Macbeath's problem in a more general setting, and not only for the Lebesgue measure as is the case in the classical theorem. We prove near-optimal generalizations for several basic geometric set systems. The problems and techniques used are closely linked to the study of espilon-nets for geometric set systems.
15 citations
01 Jan 2000
TL;DR: Very general methods for designing efficient parallel algorithms for problems in computational geometry, including the PRAM, are described, providing strong evidence that these techniques yield equally efficient algorithms in more concrete computing models like Butterfly networks.
Abstract: We describe very general methods for designing efficient parallel algorithms for problems in computational geometry. Although our main focus is the PRAM, we provide strong evidence that these techniques yield equally efficient algorithms in more concrete computing models like Butterfly networks. The algorithms exploit random sampling and randomized techniques for solving a wide class of fundamental problems from computational geometry like convex hulls, Voronoi diagrams, triangulation, point-location and arrangements. In addition, the algorithms on the Butterfly network rely critically on an efficient randomized multisearching algorithm. Our description emphasizes the algorithmic techniques rather than a detailed treatment of the individual problems.
15 citations
TL;DR: This work presents an algorithm that performs a point location query with O(d^2\log n) linear comparisons, improving the previous best result by about a factor of d and has currently the best performance for arbitrary hyperplanes.
Abstract: We consider the point location problem in an arrangement of n arbitrary hyperplanes in any dimension d, in the linear decision tree model, in which we only count linear comparisons involving the query point, and all other operations do not explicitly access the query and are for free. We mainly consider the simpler variant (which arises in many applications) where we only want to determine whether the query point lies on some input hyperplane. We present an algorithm that performs a point location query with $$O(d^2\log n)$$
linear comparisons, improving the previous best result by about a factor of d. Our approach is a variant of Meiser’s technique for point location (Inf Comput 106(2):286–303, 1993) (see also Cardinal et al. in: Proceedings of the 24th European symposium on algorithms, 2016), and its improved performance is due to the use of vertical decompositions in an arrangement of hyperplanes in high dimensions, rather than bottom-vertex triangulation used in the earlier approaches. The properties of such a decomposition, both combinatorial and algorithmic (in the standard real RAM model), are developed in a companion paper (Ezra et al. arXiv:1712.02913
, 2017), and are adapted here (in simplified form) for the linear decision tree model. Several applications of our algorithm are presented, such as the k-SUM problem and the Knapsack and SubsetSum problems. However, these applications have been superseded by the more recent result of Kane et al. (in: Proceedings of the 50th ACM symposium on theory of computing, 2018), obtained after the original submission (and acceptance) of the conference version of our paper (Ezra and Sharir in: Proceedings of the 33rd international symposium on computational geometry, 2017). This result only applies to ‘low-complexity’ hyperplanes (for which the $$\ell _1$$
-norm of their coefficient vector is a small integer), which arise in the aforementioned applications. Still, our algorithm has currently the best performance for arbitrary hyperplanes.
15 citations
01 Aug 2011
TL;DR: These summaries improve over the accuracy of existing structure-oblivious sampling schemes on range queries while retaining the benefits of sample-based summaries: flexible summaries, with high accuracy on both range queries and arbitrary subset queries.
Abstract: In processing large quantities of data, a fundamental problem is to obtain a summary which supports approximate query answering Random sampling yields flexible summaries which naturally support subset-sum queries with unbiased estimators and well-understood confidence bounds Classic sample-based summaries, however, are designed for arbitrary subset queries and are oblivious to the structure in the set of keys The particular structure, such as hierarchy, order, or product space (multi-dimensional), makes range queries much more relevant for most analysis of the dataDedicated summarization algorithms for range-sum queries have also been extensively studied They can outperform existing sampling schemes in terms of accuracy on range queries per summary size Their accuracy, however, rapidly degrades when, as is often the case, the query spans multiple ranges They are also less flexible---being targeted for range sum queries alone---and are often quite costly to build and useIn this paper we propose and evaluate variance optimal sampling schemes that are structure-aware These summaries improve over the accuracy of existing structure-oblivious sampling schemes on range queries while retaining the benefits of sample-based summaries: flexible summaries, with high accuracy on both range queries and arbitrary subset queries
14 citations
TL;DR: In this paper, the authors studied the problem of maximizing the overlap of two convex polytopes under translation in R^d for some constant d>=3, and presented an algorithm that finds an overlap at least the optimum minus @e and reports the translation realizing it.
Abstract: We study the problem of maximizing the overlap of two convex polytopes under translation in R^d for some constant d>=3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any @e>0, finds an overlap at least the optimum minus @e and reports the translation realizing it. The running time is O(n^@?^d^/^2^@?^+^1log^dn) with probability at least 1-n^-^O^(^1^), which can be improved to O(nlog^3^.^5n) in R^3. The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes. The perturbation causes an additive error @e, which can be made arbitrarily small by decreasing the perturbation magnitude. Our algorithm in fact computes the maximum overlap of the perturbed polytopes. The running time bounds, the probability bound, and the big-O constants in these bounds are independent of @e.
14 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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