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Open AccessJournal ArticleDOI

ź-nets and simplex range queries

TLDR
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 Â-net with any desired probability are given.
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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Citations
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Journal ArticleDOI

On the number of points in general position in the plane

TL;DR: In this article, it was shown that there is a planar point set of n points such that no four are collinear but no matter how we choose a subset of size $n^{5/6+o(1)} $ it contains a collincear triple.
Journal ArticleDOI

Small weak epsilon-nets

TL;DR: In this article, the smallest epsilon that can be guaranteed for any P when |Q|=i, for small values of i, is determined. But the bounds are not applicable to the case where the set P of points in the plane is a weak @e-net.
Journal ArticleDOI

Dense forests and Danzer sets

TL;DR: In this paper, it was shown that natural candidates, such as discrete sets that arise from substitutions and from cut-and-project constructions, are not Danzer sets.
Journal ArticleDOI

Approximation algorithm for minimum power partial multi-coverage in wireless sensor networks

TL;DR: In this article, the authors considered the problem of minimizing the total power of each sensor in a wireless sensor network such that the coverage of targets meets the partial multi-cover requirement.
Posted Content

Black-box Certification and Learning under Adversarial Perturbations

TL;DR: This work formally study the problem of classification under adversarial perturbations, both from the learner's perspective, and from the viewpoint of a third-party who aims at certifying the robustness of a given black-box classifier.
References
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Book ChapterDOI

On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities

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.
Book

Algorithms in Combinatorial Geometry

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.
Journal ArticleDOI

On the density of families of sets

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.
Journal ArticleDOI

Central Limit Theorems for Empirical Measures

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.
Journal ArticleDOI

The power of geometric duality

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.