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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Journal ArticleDOI
EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs.
Marthe Bonamy,Édouard Bonnet,Nicolas Bousquet,Pierre Charbit,Panos Giannopoulos,Eun Jung Kim,Paweł Rzążewski,Florian Sikora,Stéphan Thomassé +8 more
TL;DR: In this article, it was shown that the disjoint union of two odd cycles is not the complement of a disk graph nor of a unit (3-dimensional) ball graph.
Proceedings ArticleDOI
Visualization of Big Spatial Data using Coresets for Kernel Density Estimates
TL;DR: A method for subsampling of spatial data suitable for creating kernel density estimates from very large data is described and it is demonstrated that it results in less error than random sampling.
Product Range Spaces, Sensitive Sampling, and Derandomization
TL;DR: The concept of a sensitive E-approximation is introduced, and by extending the method to the intersection of a set of balls with the same radius, an O(n log3 n) deterministic algorithm for computing the diameter of an n-point set in 3-dimensional space is obtained.
Proceedings ArticleDOI
Small-size relative (p,ε)-approximations for well-behaved range spaces
TL;DR: Improved upper bounds for the size of relative (p,ε)-approximation for range spaces with the following property are presented: For any (finite) range space projected onto a ground set of size n and for any parameter 1 ≤ k ≤ n, the number of ranges of size at most k is only nearly-linear in n and polynomial in k.
Journal ArticleDOI
An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams
TL;DR: This study study covers many concrete order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and proposes a randomized incremental construction algorithm that runs in O(k(n-k)log2n+nlog3n) steps.
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.
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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.
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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.