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
A 4-approximation algorithm for guarding 1.5-dimensional terrains
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Proceedings ArticleDOI
The complexity of many faces in arrangements of lines of segments
TL;DR: A (randomized) algorithm is given that produces these faces and, with high probability, takes time that is within a log 2-supscrpt factor of the combinatorial bound.
Journal ArticleDOI
Tight lower bounds for the size of epsilon-nets
János Pach,Gábor Tardos +1 more
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Improved Bounds for the Union of Locally Fat Objects in the Plane
TL;DR: It is shown that, for any $\gamma > 0$, the combinatorial complexity of the union of $n$ locally $\Gamma$-fat objects of constant complexity in the plane is $\frac{n}{\gamma^4} 2^{O(\log^*n)}$.
Journal ArticleDOI
Coloring Geometric Range Spaces
TL;DR: The goal of this paper is to bound these two functions for several types of region families, such as halfplanes, halfspaces, disks, and pseudo-disks, related to previous results of Pach, Tardos, and Tóth on decompositions of coverings.
References
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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.
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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.