Journal ArticleDOI

ź-nets and simplex range queries

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.

AbstractWe 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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Citations
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Book
01 Jan 1991
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
Abstract: The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.

6,324 citations

Book
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.
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.

2,269 citations

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

• ...5 are taken from Preparata, Hong (1977); another presentation of the same algorithms can be found in Preparata, Shamos (1985). The variation of their two-dimensional algorithm described in Section 8....

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• ...5 are taken from Preparata, Hong (1977); another presentation of the same algorithms can be found in Preparata, Shamos (1985). The variation of their two-dimensional algorithm described in Section 8.3.2 guarantees running time O( n) if the points are presorted; this feature is not shared by the original design. The presentation of the three-dimensional algorithm in Section 8.5 is the first complete description of the algorithm given in the literature. Until now, it has not been overlooked that a single vertex of a recursively constructed convex polytope can be encountered more than once when it is merged with another disjoint convex polytope. Due to the close relationship between convex hulls in three dimensions and Voronoi diagrams in two dimensions, it is not surprising that the corresponding phenomenon was overlooked in the analysis of divide-and-conquer algorithms for constructing Voronoi diagrams (see Shamos (1978), Lee (1978), Preparata, Shamos (1985), and others)....

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Journal ArticleDOI
TL;DR: This paper shows that the essential condition for distribution-free learnability is finiteness of the Vapnik-Chervonenkis dimension, a simple combinatorial parameter of the class of concepts to be learned.
Abstract: Valiant's learnability model is extended to learning classes of concepts defined by regions in Euclidean space En. The methods in this paper lead to a unified treatment of some of Valiant's results, along with previous results on distribution-free convergence of certain pattern recognition algorithms. It is shown that the essential condition for distribution-free learnability is finiteness of the Vapnik-Chervonenkis dimension, a simple combinatorial parameter of the class of concepts to be learned. Using this parameter, the complexity and closure properties of learnable classes are analyzed, and the necessary and sufficient conditions are provided for feasible learnability.

1,835 citations

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

• ...Using Proposition A2.5, they generalize Lemmas 3.4 and 3.5 of [ 29 ] to arbitrary probability distributions....

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• ...introduced in [ 29 ]* to arbitrary probability distributions on E”. For a fixed distribution, an E-transversal for R is a finite set of points N G E” such that every region in R of probability at least E contains at least one point in N. Se:ction A2 uses the notion of an c-transversal to provide the primary machinery for Theorem 2.1, following [29] and [62]....

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• ...We sketch the proof for completeness, using the notation from [ 29 ]....

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• ...A. BLUMER ET AL. function II,(m) (see [ 29 ])....

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• ...Our characterization of learnability uses a simple combinatorial parameter called the Vapnik-Chervonenkis (VC) dimension of the class C of concepts [ 29 ].’ We show that there is a learning function satisfying (3) if and only if the VC dimension of C is finite....

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Proceedings ArticleDOI
06 Jan 1988
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Abstract: Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.

1,138 citations

Journal ArticleDOI
TL;DR: In this article, a generalization of the PAC learning model based on statistical decision theory is described, where the learner receives randomly drawn examples, each example consisting of an instance x in X and an outcome y in Y, and tries to find a hypothesis h : X < A, where h in H, that specifies the appropriate action a in A to take for each instance x, in order to minimize the expectation of a loss l(y,a).
Abstract: We describe a generalization of the PAC learning model that is based on statistical decision theory. In this model the learner receives randomly drawn examples, each example consisting of an instance x in X and an outcome y in Y , and tries to find a hypothesis h : X --< A , where h in H , that specifies the appropriate action a in A to take for each instance x , in order to minimize the expectation of a loss l(y,a). Here X, Y, and A are arbitrary sets, l is a real-valued function, and examples are generated according to an arbitrary joint distribution on X times Y . Special cases include the problem of learning a function from X into Y , the problem of learning the conditional probability distribution on Y given X (regression), and the problem of learning a distribution on X (density estimation). We give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces H , and show how this implies learnability with bounded sample size, disregarding computational complexity. As an application, we give distribution-independent upper bounds on the sample size needed for learning with feedforward neural networks. Our theorems use a generalized notion of VC dimension that applies to classes of real-valued functions, adapted from Pollard''s work, and a notion of *capacity* and *metric dimension* for classes of functions that map into a bounded metric space. (Supersedes 89-30 and 90-52.) [Also in "Information and Computation", Vol. 100, No.1, September 1992]

957 citations

References
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Book ChapterDOI
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).

3,669 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 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.
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.

2,269 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 ArticleDOI
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.
Abstract: If T is a family of sets and A some set we denote by T ∩ A the following family of subsets of A: T ∩ A = {F ∩ A; F ϵ T}. P. Erdos (oral communication) transmitted to me in Nice the following question: Is it true that 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 |T ∩ A| ⩽ |A|c for each A ⊂ S with |A| ⩾ N and some constant c? In this paper we will answer this question in the affirmative by determining the exact upper bound. (Theorem 2).1

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

540 citations

Journal ArticleDOI
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.

277 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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