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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Proceedings Article•DOI•

On approximate halfspace range counting and relative epsilon-approximations

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Boris Aronov¹, Sariel Har-Peled², Micha Sharir³•Institutions (3)

New York University¹, Urbana University², Tel Aviv University³

06 Jun 2007

TL;DR: A construction of smaller-size relative ε-approximations for range spaces that involve points and halfspaces in two and higher dimensions is given, based on a new structure--spanning trees with small relative crossing number, which it is believed to be of independent interest.

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Abstract: The paper consists of two major parts. In the first part, we re-examine relative e-approximations, previously studied in [12, 13, 18, 25], and their relation to certain geometric problems, most notably to approximate range counting. We give a simple constructive proof of their existence in general range spaces with finite VC dimension, and of a sharp bound on their size, close to the best known one. We then give a construction of smaller-size relative e-approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure--spanning trees with small relative crossing number, which we believe to be of independent interest. In the second part, we consider the approximate halfspace range-counting problem in Rd with relative error e, and show that relative e-approximations, combined with the shallow partitioning data structures of Matousek, yields efficient solutions to this problem. For example, one of our data structures requires linear storage and O(n1+δ) preprocessing time, for any δ>0, and answers a query in time O(e-γn1-1/⌊ d/2 ⌋ 2b log* n), for any γ > 2/⌊ d/2⌋ the choice of γ and δ affects b and the implied constants. Several variants and extensions are also discussed.

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27 citations

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

...We make the standard assumption that the range space (X,R) (or, in fact, (U,RU)) has finite (i.e., independent of n) VC dimension � which is indeed the case in many geometric applications; see [8, 14 , 21, 24] for definitions and more details....
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Book Chapter•DOI•

Minimizing interference of a wireless ad-hoc network in a plane

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Magnús M. Halldórsson¹, Takeshi Tokuyama²•Institutions (2)

University of Iceland¹, Tohoku University²

15 Jul 2006

TL;DR: A method based on quad-tree decomposition and bucketing is given that has another provable interference bound in terms of the ratio of the minimum distance to the radius of a uniform-radius ad-hoc network.

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Abstract: We consider the problem of topology control of a wireless ad-hoc network on a given set of points in the plane, where we aim to minimize the maximum interference by assigning a suitable transmission radius to each point. By using computational geometric ideas and e-net theory, we attain an $O(\sqrt{\Delta})$ bound for the maximum interference where Δ is the interference of a uniform-radius ad-hoc network. This generalizes a result given in [8] for the special case of highway model (i.e., one-dimensional problem) to the two-dimensional case. We also give a method based on quad-tree decomposition and bucketing that has another provable interference bound in terms of the ratio of the minimum distance to the radius of a uniform-radius ad-hoc network.

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27 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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Journal Article•DOI•

The Vapnik-Chervonenkis dimension of a random graph

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Martin Anthony¹, Graham Brightwell¹, Colin Cooper²•Institutions (2)

London School of Economics and Political Science¹, University of North London²

06 Mar 1995-Discrete Mathematics

TL;DR: The main result gives the exact threshold function for a random graph G ( n, p) to have VC dimension d, which is defined to be the largest cardinality of a shattered set of vertices.

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27 citations

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

...set of D. In this case, each subset of D can be realised as a dichotomy of D by F. The Vapnik-Chervonenkis dimension [11, 7 ] (or VC dimension) ofF, denoted VCdim(F), is the maximald such that some subset ofX of cardinalityd is shattered byF....
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...Haussler and Welzl [ 7 ] noted that if a graph has VC dimension at least 5 then it contains a subgraph homeomorphic to the complete graph on ve vertices....
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...Following Haussler and Welzl [ 7 ], we dene the VC dimension of a graph as follows....
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...The Vapnik-Chervonenkis dimension has proved useful in a number of areas of mathematics and computer science; in probability theory [11, 10, 8], in computational geometry [ 7 ] and in the theory of machine learning [4, 2], for example....
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...The VC dimension of a graph was dened by Haussler and Welzl [ 7 ] and is an interesting special case of the more general and well-established notion of the Vapnik-Chervonenkis dimension of a set system, rst introduced in [11]....
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Journal Article•DOI•

Efficient randomized algorithms for robust estimation of circular arcs and aligned ellipses

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David M. Mount¹, Nathan S. Netanyahu², Nathan S. Netanyahu¹•Institutions (2)

University of Maryland, College Park¹, Bar-Ilan University²

01 Jun 2001-Computational Geometry: Theory and Applications

TL;DR: In this paper, the authors introduce nonlinear Theil-Sen and repeated median (RM) variants for estimating the center and radius of a circular arc, and for estimating center and horizontal and vertical radii of an axis-aligned ellipse.

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Abstract: Fitting two-dimensional conic sections (e.g., circular and elliptical arcs) to a finite collection of points in the plane is an important problem in statistical estimation and has significant industrial applications. Recently there has been a great deal of interest in robust estimators, because of their lack of sensitivity to outlying data points. The basic measure of the robustness of an estimator is its breakdown point, that is, the fraction (up to 50%) of outlying data points that can corrupt the estimator. In this paper we introduce nonlinear Theil–Sen and repeated median (RM) variants for estimating the center and radius of a circular arc, and for estimating the center and horizontal and vertical radii of an axis-aligned ellipse. The circular arc estimators have breakdown points of ≈ 21% and 50%, respectively, and the ellipse estimators have breakdown points of ≈ 16% and 50%, respectively. We present randomized algorithms for these estimators, whose expected running times are O (n 2 log n) for the circular case and O (n 3 log n) for the elliptical case. All algorithms use O (n) space in the worst case.

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27 citations

Journal Article•DOI•

The VC-dimension of set systems defined by graphs

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Evangelos Kranakis¹, Danny Krizanc¹, Berthold Ruf², Jorge Urrutia³, Gerhard J. Woeginger² - Show less +1 more•Institutions (3)

Carleton University¹, Graz University of Technology², University of Ottawa³

22 Aug 1997-Discrete Applied Mathematics

TL;DR: It is shown that the VC-dimension for set systems induced by stars is computable in polynomial time, and the extremal graphs G with the minimum number of edges such that VC P ( G ) ⩾ k .

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27 citations

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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....
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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•

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....
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..., [11] for a general treatment of arrangements....
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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'. []...
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Journal Article•DOI•

Central Limit Theorems for Empirical Measures

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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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