ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
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.read more
Citations
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Book ChapterDOI
Unsplittable Coverings in the Plane
János Pach,Dömötör Pálvölgyi +1 more
TL;DR: In this article, it was shown that for any positive integer m, every m-fold covering of a region with unit disks splits into two coverings, provided that every point is covered by at most c 2 m/2 sets.
Book ChapterDOI
ε-Net Approach to Sensor k-Coverage
Giordano Fusco,Himanshu Gupta +1 more
TL;DR: In this article, the authors present an algorithm based on an extension of the classical e-net technique, which gives a O(logM )-approximation, where M is the number of sensors in an optimal solution.
Proceedings ArticleDOI
Scalable spatial scan statistics through sampling
TL;DR: These algorithms randomly sample data at two scales, one to define regions and the other to approximate the counts in these regions, allowing spatial scan statistics to run on a million or more data points without making assumptions on the spatial distribution of the data.
Journal ArticleDOI
On counting pairs of intersecting segments and off-line triangle range searching
TL;DR: New efficientdeterministic algorithms for counting pairs of intersecting segments, and for answering off-line triangle range queries are obtained, based on properties of the sparse nets introduced by Chazelle.
Journal ArticleDOI
On computing the diameter of a point set in high dimensional Euclidean space
TL;DR: An algorithm that in time O(dnlogn + n2) finds with high probability an arbitrarily close approximation of the diameter is described, which is a substantial improvement over the complexity bounds of previously known exact algorithms.
References
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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.
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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.
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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.
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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.