ź-nets and simplex range queries

doi:10.1007/BF02187876

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ź-nets and simplex range queries

Journal Article•DOI•

ź-nets and simplex range queries

David Haussler¹, Emo Welzl²•Institutions (2)

University of California, Santa Cruz¹, University of Graz²

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 Ã‚-net with any desired probability are given.

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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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Book Chapter•DOI•

Deterministic Parallel Computational Geometry.

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Mikhail J. Atallah¹, Danny Z. Chen²•Institutions (2)

Purdue University¹, University of Notre Dame²

01 Jan 2000

TL;DR: This work describes general methods for designing deterministic parallel algorithms in computational geometry and focuses on techniques for shared-memory parallel machines.

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Abstract: We describe general methods for designing deterministic parallel algorithms in computational geometry. We focus on techniques for shared-memory parallel machines, which we describe and illustrate with examples. We also discuss some open problems in this area.

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16 citations

Book Chapter•DOI•

Quantifying the amount of verboseness (extended abstract)

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Richard Beigel¹, Martin Kummer², Frank Stephan²•Institutions (2)

Yale University¹, Karlsruhe Institute of Technology²

20 Jul 1992

TL;DR: A complete characterization is obtained relating the question to finite combinatorics of the classification of sets of natural numbers A according to the number of queries which are needed to compute the n-fold characteristic function of A.

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Abstract: We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the n-fold characteristic function of A. A complete characterization is obtained relating the question to finite combinatorics. In order to obtain an explicit description we encounter several interesting combinatorial problems.

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16 citations

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

...It has important applications in probability theory [42], computational learning theory [10], and computational geometry [22]....
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Proceedings Article•DOI•

Lenses in arrangements of pseudo-circles and their applications

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Eran Nevo¹, János Pach², Rom Pinchasi³, Micha Sharir², Shakhar Smorodinsky⁴ - Show less +1 more•Institutions (4)

Hebrew University of Jerusalem¹, New York University², Massachusetts Institute of Technology³, Tel Aviv University⁴

05 Jun 2002

TL;DR: It is shown that any collection of n pseudo-circles can be cut into $\bx$ arcs so that any two intersect at most once, provided that the given pseudo- circles are x-monotone and admit an algebraic representation by three real parameters.

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Abstract: (MATH) A collection of simple closed Jordan curves in the plane is called a family of pseudo-circles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct pseudo-circles is said to be an empty lens if it does not intersect any other member of the family. We establish a linear upper bound on the number of empty lenses in an arrangement of n pseudo-circles with the property that any two curves intersect precisely twice. Enhancing this bound in several ways, and combining it with the technique of Tamaki and Tokuyama [16], we show that any collection of n pseudo-circles can be cut into $\bx$ arcs so that any two intersect at most once, provided that the given pseudo-circles are x-monotone and admit an algebraic representation by three real parameters; here $\alpha(n)$ is the inverse Ackermann function, and s is a constant that depends on the algebraic degree of the representation of the pseudo-circles (s=2 for circles and parabolas). For arbitrary collections of pseudo-circles, any two of which intersect twice, the number of necessary cuts reduces to O(n 4/3). As applications, we obtain improved bounds for the number of point-curve incidences, the complexity of a single level, and the complexity of many faces in arrangements of circles, pairwise intersecting pseudo-circles, parabolas, and families of homothetic copies of a fixed convex curve. We also obtain a variant of the Gallai-Sylvester theorem for arrangements of pairwise intersecting pseudo-circles, and a new lower bound for the number of distinct distances among n points in the plane under any simply-defined norm or convex distance function.

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16 citations

Book Chapter•DOI•

Applying valiant's learning framework to Al concept-learning problems

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David Haussler¹•Institutions (1)

University of California, Santa Cruz¹

01 Jun 1990-Machine Learning

TL;DR: An overview of some recent theoretical results in the learning framework introduced by Valiant and further developed and a comparison to the work of Mitchell on version spaces are presented.

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Abstract: We present an overview of some recent theoretical results in the learning framework introduced by Valiant in [Valiant, 1984] and further developed in [Valiant 1985; Blumer, et al., 1987; 1989; Pitt and Valiant, 1988; Haussler, 1988; Angluin and Laird, 1988; Angluin, 1988; Rivest, 1987; Haussler, 1989; Kearns, et al., 1987a; 1987b]. Our focus is on applications to AI problems of learning from examples as given in [Haussler, 1988; 1989] and [Kearns, et al., 1987a; 1987b], along with a comparison to the work of Mitchell on version spaces [Mitchell, 1982]. We discuss learning problems for both attribute-based and structural domains. This is a revised and expanded version of [Haussler, 1987].

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16 citations

Journal Article•DOI•

On the transversal number and VC-dimension of families of positive homothets of a convex body

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Márton Naszódi¹, Steven Taschuk¹•Institutions (1)

University of Alberta¹

01 Jan 2010-Discrete Mathematics

TL;DR: It is shown that the maximum dual VC-dimension of a family of positive homothets of a given convex body K in R^n is n+1 and it is proved that vcdim(F)@?3 if n=2 and is infinite for some F if n>=3.

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15 citations

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

...Our main motivation in studying the VC-dimension is its involvement in upper bounds on transversal numbers (see the Epsilon Net Theorem of Haussler and Welzl [7] and Corollary 10.2.7 of [11]) and related phenomena (see [12], for example)....
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...Our main motivation in studying the VC-dimension is its involvement in upper bounds on transversal numbers (see the Epsilon Net Theorem of Haussler and Welzl [7] and Corollary 10....
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References

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Book Chapter•DOI•

On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities

[...]

Vladimir Vapnik, A. Ya. Chervonenkis

01 Jan 1971-Theory of Probability and Its Applications

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.

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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).

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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....
[...]
...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])....
[...]

Book•

Algorithms in Combinatorial Geometry

[...]

Herbert Edelsbrunner¹•Institutions (1)

University of Illinois at Urbana–Champaign¹

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.

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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.

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

Journal Article•DOI•

On the density of families of sets

[...]

Norbert Sauer¹•Institutions (1)

University of Calgary¹

01 Jul 1972-Journal of Combinatorial Theory, Series A

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.

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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'. []...
[...]

Journal Article•DOI•

Central Limit Theorems for Empirical Measures

[...]

Richard M. Dudley

01 Dec 1978-Annals of Probability

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.

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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 $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.

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555 citations

Journal Article•DOI•

The power of geometric duality

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Bernard Chazelle¹, Leonidas J. Guibas², Der-Tsai Lee³•Institutions (3)

Brown University¹, PARC², Northwestern University³

01 Jun 1985-Bit Numerical Mathematics

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.

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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.

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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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