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•

Tight lower bounds for the size of epsilon-nets

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János Pach¹, Gábor Tardos²•Institutions (2)

École Polytechnique Fédérale de Lausanne¹, Simon Fraser University²

13 Jun 2011

TL;DR: In this article, it was shown that any range space of bounded VC-dimension admits an e-net of size O(1/e log 1/e) where g is an extremely slowly growing function related to the inverse Ackermann function.

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Abstract: According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an e-net of size O(1/e log 1/e). Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VC-dimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest e-nets is superlinear in 1/e, were found by Alon (2010). In his examples, every e-net is of size Ω(1/eg(1/e)), where g is an extremely slowly growing function, related to the inverse Ackermann function.We show that there exist geometrically defined range spaces, already of VC-dimension 2, in which the size of the smallest e-nets is Ω(1/e log 1/e). We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest e-nets is Ω(1/e log log 1/e). By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.

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69 citations

Book•

Implicitly Representing Arrangements of Lines or Segments

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Herbert Edelsbrunner¹, Leonidas J. Guibas², John Hershberger, Raimund Seidel³, Micha Sharir⁴ - Show less +1 more•Institutions (4)

University of Illinois at Urbana–Champaign¹, Stanford University², University of California, Berkeley³, New York University⁴

26 Aug 2011

TL;DR: This paper studies the problem of calculating and storing arrangements using subquadratic space and preprocessing, so that, given any query point, the face containing p can be calculated efficiently, and reports faces in an arrangement of line segments in time.

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Abstract: An arrangement of n lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists of O(n2) regions, called faces. In this paper we study the problem of calculating and storing arrangements implicitly, using subquadratic space and preprocessing, so that, given any query point p, we can calculate efficiently the face containing p. First, we consider the case of lines and show that with L(n) space1 and L(n3/2) preprocessing time, we can answer face queries in L(√n) + O(K) time, where K is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: 1) given a set of n points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, 2) given a simple polygonal path G, form a data structure from which we can find the convex hull of any subpath of G quickly, and 3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a trade-off between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in L(n1/3) time, given L(n4/3) space and L(n5/3) preprocessing time.Lastly, we note that our techniques allow us to compute m faces in an arrangement of n lines in time L(m2/3n2/3 + n), which is nearly optimal.

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68 citations

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

...We choose a random sample L' of r of the lines in L and, independently, a random sample G' of r 2 of the points in G. We now triangulate the arrangement of the lines in L' with the additional points of G' thrown in--see Fig. 9. By using O(r 2) triangles we can obtain a triangulation such that no triangle contains one of the points in G' in its interior, or has its interior intersected by one of the lines in L'. Now the ......
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...Proof. In order to minimize the total number of line-edge intersections as we build our tree, we employ random sampling techniques due to Clarkson [C], which are closely related to the e-nets used by Haussler and Welzl [We], [ HW ]....
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...The e-net theory of Haussler and Welzl [ HW ] (or the probabilistic lemma of Clarkson [C]) then guarantees that with high probability each of the O(r 2) triangles thus created is intersected by at most O(n log r/r) of the arrangement lines....
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Journal Article•DOI•

Lines in space: Combinatorics and algorithms

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Bernard Chazelle¹, Herbert Edelsbrunner², Leonidas J. Guibas³, Micha Sharir⁴, Jorge Stolfi - Show less +1 more•Institutions (4)

Princeton University¹, University of Illinois at Urbana–Champaign², Stanford University³, New York University⁴

01 May 1996-Algorithmica

TL;DR: A tight Θ( n2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to then given lines and an algorithm that tests the “towering property” inO(n2+ɛ) time.

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Abstract: Questions about lines in space arise frequently as subproblems in three-dimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements ofn lines in three-dimensional space. Our main results include:1.A tight ź(n2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to then given lines.2.A similar bound of ź(n3) for the complexity of the set of all lines passing above then given lines.3.A preprocessing procedure usingO(n2+ź) time and storage, for anyź>0, that builds a structure supportingO(logn)-time queries for testing if a line lies above all the given lines.4.An algorithm that tests the "towering property" inO(n2+ź) time, for anyź>0; don given red lines lie all aboven given blue lines? The tools used to obtain these and other results include Plucker coordinates for lines in space andź-nets for various geometric range spaces.

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67 citations

Book Chapter•DOI•

A Singly-Expenential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications

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Bernard Chazelle¹, Herbert Edelsbrunner², Leonidas J. Guibas³, Micha Sharir⁴•Institutions (4)

Princeton University¹, University of Illinois at Urbana–Champaign², Stanford University³, New York University⁴

11 Jul 1989

TL;DR: An effective procedure for stratifying a real semi-algebraic set into cells of constant description size that compares favorably with the doubly exponential size of Collins’ decomposition and is able to apply in interesting ways to problems of point location and geometric optimization.

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Abstract: Chazelle, B., H. Edelsbrunner, L.J. Guibas and M. Sharir, A singly exponential stratification scheme for real semi-algebraic varieties and its applications, Theoretical Computer Science 84 (1991) 77-105. This paper describes an effective procedure for stratifying a real semi-algebraic set into cells of constant description size. The attractive feature of our method is that the number of cells produced is singly exponential in the number of input variables. This compares favorably with the doubly exponential size of Collins’ decomposition. Unlike Collins’ construction, however, our scheme does not produce a cell complex but only a smooth stratification. Nevertheless, we are able to apply our results in interesting ways to problems of point location and geometric optimization.

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66 citations

Proceedings Article•DOI•

Finding small balanced separators

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Uriel Feige¹, Mohammad Mahdian¹•Institutions (1)

Microsoft¹

21 May 2006

TL;DR: This paper gives a randomized algorithm that finds an α-separator of size k in the given graph, unless the graph contains an (α+ε)-separators of size strictly less than k, in which case the algorithm finds one such separator.

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Abstract: Let G be an n-vertex graph that has a vertex separator of size k that partitions the graph into connected components of size smaller than α n, for some fixed 2/3 ≤ α

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66 citations

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

...Lemma 2.6 (for the case of †-nets) is from [4], and improves over bounds given in [ 13 ] (as well as in [1], for example) by a factor of O(logd)....
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...We are not aware of a reference with an explicit proof for the case of †-sample, but such a proof follows by straightforward modifications to the proofs in [ 13 , 4, 16]....
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...The concept of †-nets (Definition 2.4) was introduced by Haussler and Welzl [ 13 ]....
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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....
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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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