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•

Fixed-dimensional parallel linear programming via relative e-approximations

[...]

Michael T. Goodrich

28 Jan 1996

TL;DR: It is shown that linear programming in IRd can be solved deterministically in O(logn(loglogn)d-l) time using linear work in the PRAM model of computation, for any fixed constant d.

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Abstract: We show that linear programming in IRd can be solved deterministically in O((loglogn)d) time using linear work in the PRAM model of computation, for any fixed constant d. Our method is developed for the CRCW variant of the PRAM parallel computation model, and can be easily implemented to run in O(logn(loglogn)d-l) time using linear work on an EREW PRAM. A key component in these algorithms is a new, efficient parallel method for constructing c-nets and c-approximations (which have wide applicability in computational geometry). In addition, we introduce a new deterministic set approximation for range spaces with finite VC-exponent, which we call the b-relative c-approtimation, and we show how such approximations can be efficiently constructed in parallel.

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19 citations

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

...Like previous arguments, which are based upon mutual independence (e.g., see [4, 34 ]), our k-wise independent analysis is based upon a doublesampling technique....
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...The study of random sampling in the design of efficient computational geometry methods really began in earnest with some outstanding early work of Clarkson [19], Haussler and Welzl [ 34 ], and Clarkson and Shor [al]....
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Book Chapter•DOI•

Teaching and Compressing for Low VC-Dimension

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Shay Moran¹, Shay Moran², Amir Shpilka³, Avi Wigderson, Amir Yehudayoff² - Show less +1 more•Institutions (3)

Max Planck Society¹, Technion – Israel Institute of Technology², Tel Aviv University³

01 Jan 2017-Electronic Colloquium on Computational Complexity

TL;DR: In this paper, the quality of sample compression schemes and of teaching sets for classes of low VC-dimension was studied, and the best known upper bounds for both parameters were log(m), while the best lower bounds are linear in d.

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Abstract: In this work we study the quantitative relation between VC-dimension and two other basic parameters related to learning and teaching. Namely, the quality of sample compression schemes and of teaching sets for classes of low VC-dimension. Let C be a binary concept class of size m and VC-dimension d. Prior to this work, the best known upper bounds for both parameters were log(m), while the best lower bounds are linear in d. We present significantly better upper bounds on both as follows. Set k = O(d2 d loglog | C | ).

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19 citations

Journal Article•DOI•

Exact learning from an honest teacher that answers membership queries

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Nader H. Bshouty¹•Institutions (1)

Technion – Israel Institute of Technology¹

15 Jul 2018-Theoretical Computer Science

TL;DR: In this paper, the authors present some of the results known from the literature, different techniques used, some new problems, and open problems for the problem of finding a minimum number of queries, optimal time complexity, and optimal resources.

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18 citations

Posted Content•

Low Rank Matrix Approximation in Linear Time.

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Sariel Har-Peled

31 Oct 2014-arXiv: Computational Geometry

TL;DR: This algorithm is the first linear-time algorithm to achieve multiplicative approximation and succeeds with constant probability.

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Abstract: $ ewcommand{\MatA}{\mathcal{M}}$ $ ewcommand{\eps}{\varepsilon}$ $ ewcommand{\NSize}{\mathsf{N}{}}$ $ ewcommand{\MatB}{\mathcal{B}}$ $ ewcommand{\Fnorm}[1]{\left\| {#1} \right\|_F}$ $ ewcommand{\PrcOpt}[2]{\mu_{\mathrm{opt}}\pth{#1, #2}}$ $ ewcommand{\pth}[1]{\left(#1\right)}$ Given a matrix $\MatA$ with $n$ rows and $d$ columns, and fixed $k$ and $\eps$, we present an algorithm that in linear time (i.e., $O(\NSize )$) computes a $k$-rank matrix $\MatB$ with approximation error $\Fnorm{\MatA - \MatB}^2 \leq (1+\eps) \PrcOpt{\MatA}{k}$, where $\NSize = n d$ is the input size, and $\PrcOpt{\MatA}{k}$ is the minimum error of a $k$-rank approximation to $\MatA$. This algorithm succeeds with constant probability, and to our knowledge it is the first linear-time algorithm to achieve multiplicative approximation.

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18 citations

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

...As such, we can use the ε-net theorem of Haussler and Welzl [HW87], which implies that |U ′ ∩ E| ≥ (23/24) |U ′|, with probability at least 1− δ/4....
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Proceedings Article•DOI•

Randomized geometric algorithms and pseudo-random generators

[...]

Ketan Mulmuley¹•Institutions (1)

University of Chicago¹

24 Oct 1992

TL;DR: The author shows that the expected running times of most of the randomized incremental algorithms in computational geometry do not change (up to a constant factor), when the sequence of additions is not truly random but is instead generated using only O(log n) random bits.

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Abstract: The so called randomized incremental algorithms in computational geometry can be thought of as a generalization of Quicksort to higher dimensional geometric problems. They all construct the geometric complex in the given problem, such as a Voronoi diagram or a convex polytope, by adding the objects in the input set, one at a time, in a random order. The author shows that the expected running times of most of the randomized incremental algorithms in computational geometry do not change (up to a constant factor), when the sequence of additions is not truly random but is instead generated using only O(log n) random bits. The pseudo-random generator used is a generalization of the well known linear congruential generator. >

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18 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 ]....
[...]
...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.

...read moreread less

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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