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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Journal Article•DOI•

Approximate Halfspace Range Counting

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Boris Aronov, Micha Sharir

01 May 2010-SIAM Journal on Computing

TL;DR: A simple scheme extending the shallow partitioning data structures of Matousek, which supports efficient approximate halfspace range-counting queries in $\mathbb{R}^d$ with relative error $\varepsilon$.

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Abstract: We present a simple scheme extending the shallow partitioning data structures of Matousek, which supports efficient approximate halfspace range-counting queries in $\mathbb{R}^d$ with relative error $\varepsilon$. Specifically, the problem is, given a set $P$ of $n$ points in $\mathbb{R}^d$, to preprocess them into a data structure that returns, for a query halfspace $h$, a number $t$ so that $(1-\varepsilon)|h\cap P|\leq t\leq(1+\varepsilon)|h\cap P|$. One of our data structures requires linear storage and $O(n^{1+\delta})$ preprocessing time, for any $\delta>0$, and answers a query in time $O(\varepsilon^{-\gamma}n^{1-1/\lfloor d/2\rfloor}2^{b\log^\ast n})$ for any $\gamma>2/\lfloor d/2\rfloor$; the choice of $\gamma$ and $\delta$ affects $b$ and the implied constants. Several variants and extensions are also discussed. As presented, the construction of the structure is mostly deterministic, except for one critical randomized step, and so are the query, storage, and preprocessing costs. The quality of approximation, for every query, is guaranteed with high probability. The construction can also be fully derandomized, at the expense of increasing preprocessing time.

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22 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 nite (i.e., independent of n) VC-dimension which is indeed the case in many geometric applications; see [10, 19 , 25, 27] for denitions and more details....
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...q , for a suciently large absolute constant "-approximation with probability at least 1 q. See [10, 11, 19 , 26] for more details....
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Proceedings Article•DOI•

Construction of epsilon nets

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Jiří Matoušek¹•Institutions (1)

Charles University in Prague¹

05 Jun 1989

22 citations

k-Sets and k-Facets

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Uli Wagner

01 Jan 2007

TL;DR: In this article, the authors survey problems, results, and methods concerning k-facets and k-sets of finite point sets in real ane d-space and the dual notion of levels in arrangements of hyperplanes.

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Abstract: We survey problems, results, and methods concerning k-facets and k-sets of finite point sets in real ane d-space and the dual notion of levels in arrangements of hyperplanes.

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22 citations

Journal Article•DOI•

Good splitters for counting points in triangles

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Jiří Matoušek¹, Jiří Matoušek², Emo Welzl²•Institutions (2)

Charles University in Prague¹, Free University of Berlin²

01 Jun 1992-Journal of Algorithms

TL;DR: A solution to the problem of storing a set A of n points in the plane in such a way that for any query triangle t the number of points of A inside t can be computed efficiently is presented.

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22 citations

Journal Article•DOI•

Constructing arrangements optimally in parallel

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Michael T. Goodrich¹•Institutions (1)

Johns Hopkins University¹

01 Dec 1993-Discrete and Computational Geometry

TL;DR: This second result solves a well-known open problem in parallel computational geometry, and involves the use of a new algorithmic technique, the construction of an ɛ-pseudocutting.

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Abstract: We give two optimal parallel algorithms for constructing the arrangement ofn lines in the plane. The first nethod is quite simple and runs inO(log2n) time usingO(n2) work, and the second method, which is more sophisticated, runs inO(logn) time usingO(n2) work. This second result solves a well-known open problem in parallel computational geometry, and involves the use of a new algorithmic technique, the construction of an ?-pseudocutting. Our results immediately imply that the arrangement ofn hyperplanes in ?d inO(logn) time usingO(nd) work, for fixedd, can be optimally constructed. Our algorithms are for the CREW PRAM.

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22 citations

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

...This is closely related to the random sampling techniques of Clarkson [18] and Haussler and Welzl [ 32 ], which state that if the arrangement of r randomly chosen hyperplanes from X is triangulated, a decomposition of the space into O(r a) constant-size polytope is produced such that the expected number of hyperplanes intersecting any polytope is O((n/r) log r). These approaches have turned out to be very powerful, having applications to a ......
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...Recent developments on the theory of e-cuttings have proven useful for solving a number of problems in computational geometry (e.g., see [1], [2], [14], [ 32 ], and [39])....
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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....
[...]

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

[...]

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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