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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03 Jan 1988
TL;DR: Several efficient algorithms involving the analysis of the connectivity and other useful properties of arrangements of line segments are presented, including an O(n4/3+δ+k)-time randomized algorithm for finding all k intersections of n line segments in the plane.
Abstract: We present a variety of applications of certain techniques, based on partition trees, that were originally developed for range searching problems. Our results are obtained by enhancing and extending these techniques, and include: (i) An O(n4/3+δ+k)-time (for any δ>0), O(n)-space randomized algorithm for finding all k intersections of n line segments in the plane (we can count the number of these intersections in O(n4/3+δ) time and linear space). (ii) Preprocessing a collection of n (possibly intersecting) segments in the plane so that, given any query ray, we can find quickly the first segment it hits. Other applications concern “implicit” point location, hidden surface removal in three dimensions, polygon placement queries, and problems involving overlapping planar maps. We also present several efficient algorithms involving the analysis of the connectivity and other useful properties of arrangements of line segments.
33 citations
Cites background or methods from "ź-nets and simplex range queries"
...The second best known value for ~/ is 2/3+8, for any 8>0 [ HW ], and the third prize goes to ~/ ~ 0.695 [EW]....
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...In this paper we present efficient algorithms for solving a variety of problems in computational geometry, using several enhancements and extensions of techniques that have been developed originally for half-plane or polygon range searching (see ~], [EW], [ HW ])....
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...Let v be a node of T. If mv>-n2v we do not continue the construction of T below v (so v is a leaf of T), and instead apply the procedure described below to obtain all intersections within Qv. Otherwise we partition Qv into a collection of subregions, using a technique akin to the t-net approach of Haussler and Welzl [ HW ] or the random sampling technique of Clarkson [C1]....
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01 Feb 1989
TL;DR: The tools used to obtain these results include Plücker coordinates for lines in space, random sampling in geometric problems, and a new variant of segment trees.
Abstract: We study combinatorial and algorithmic problems involving arrangements of n lines in 3-dimensional space, and then present applications of our results to a variety of problems on polyhedral terrains. Our main results include: A tight T(n2) bound on the complexity of the space of all lines passing above all the n given lines (their “upper envelope”) and satisfying a certain orientation consistency constraint.A preprocessing procedure using near-quadratic time and storage that builds a structure supporting O(log n) time queries for testing if a line lies above all the given lines.An O(n4/3+e) randomized expected time algorithm, for any fixed e > 0, that tests the “towering property”: do n given red lines lie all above n given blue lines?A preprocessing procedure for a polyhedral terrain S with n edges, that uses near-quadratic time and storage and builds a structure supporting O(log2n) time rayshooting queries for computing the first intersection of an arbitrary query ray with S.Finding the smallest vertical distance between two disjoint polyhedral terrains with a total of n edges, in time O(n4/3+e), for any e > 0.Computing the upper envelope (pointwise maximum) of two polyhedral terrains with a total of n edges, in time O(n1.5+e + klog2n), for any e > 0, where k is the size of the output envelope.The tools used to obtain these results include Plucker coordinates for lines in space, random sampling in geometric problems, and a new variant of segment trees.
33 citations
05 Jun 1989
TL;DR: This paper presents a deterministic algorithm which is faster than Matousk's recent algorithm [Ma] for large values of r, and applies it to several problems involving lines or segments in the plane, and obtain deterministic algorithms which are faster than any previously known algorithms.
Abstract: In this paper we consider the following problem: Given a set l of n lines in the plane, partition the plane into O(r2) triangles so that no triangle intersects more than O(n/r) lines of l. We present a deterministic algorithm for this problem with O(nr log n logor) running time, where o is a constant
32 citations
01 Sep 1991
TL;DR: This problem can be solved substantially more efficiently than the more general simplex range searching problem and a data structure for halfspace range reporting in dimensions d>or=4 is given.
Abstract: The author considers the halfspace range reporting problem: Given a finite set P of points in E/sup d/, preprocess it so that given a query halfspace gamma , the points of p intersection gamma can be reported efficiently. It is shown that, with almost linear storage, this problem can be solved substantially more efficiently than the more general simplex range searching problem. A data structure for halfspace range reporting in dimensions d>or=4 is given. It uses O(n log log n) space and O (n log n) deterministic preprocessing time. The query time is also given. Results for the halfspace emptiness problem, where one only wants to know whether P intersection gamma is empty, are also presented. >
32 citations
Book•
12 Nov 2020TL;DR: This comprehensive introduction to data summarization, aimed at practitioners and students, showcases the algorithms, their behavior, and the mathematical underpinnings of their operation that have been incorporated in systems from companies such as Google, Apple, Microsoft, Netflix and Twitter.
Abstract: The massive volume of data generated in modern applications can overwhelm our ability to conveniently transmit, store, and index it. For many scenarios, building a compact summary of a dataset that is vastly smaller enables flexibility and efficiency in a range of queries over the data, in exchange for some approximation. This comprehensive introduction to data summarization, aimed at practitioners and students, showcases the algorithms, their behavior, and the mathematical underpinnings of their operation. The coverage starts with simple sums and approximate counts, building to more advanced probabilistic structures such as the Bloom Filter, distinct value summaries, sketches, and quantile summaries. Summaries are described for specific types of data, such as geometric data, graphs, and vectors and matrices. The authors offer detailed descriptions of and pseudocode for key algorithms that have been incorporated in systems from companies such as Google, Apple, Microsoft, Netflix and Twitter.
32 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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