ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
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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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Improved Results on Geometric Hitting Set Problems
Nabil H. Mustafa,Saurabh Ray +1 more
TL;DR: This work gives the first PTAS for this problem when the geometric objects are half-spaces in ℝ3 and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2- admissible).
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Improved Approximation Algorithms for Geometric Set Cover
TL;DR: It is shown that polynomial-time approximation algorithms with provable performance exist, under a certain general condition: that for a random subset $R\subset S$ and nondecreasing function f(·), there is a decomposition of the complement ${Bbb U}\backslash\bigcup (R)$ into an expected at most f(|R|) regions, each region of a particular simple form.
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Equivalence of models for polynomial learnability
David Haussler,Michael Kearns +1 more
TL;DR: Comparisons and equivalences are given between Valiant's model and the prediction learning models of Haussler, Littlestone, and Warmuth and show that several simplifying assumptions on polynomial learning algorithms can be made without loss of generality.
Proceedings ArticleDOI
Approximations and optimal geometric divide-and-conquer
TL;DR: A new deterministic algorithm which for a given collection H of n hyperplanes in Ed and a parameter r ?
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On range searching with semialgebraic sets
Pankaj K. Agarwal,Jirí Matousek +1 more
TL;DR: A solution with nearly linear space and preprocessing time and withO(n1−1/b+δ) query time is given, whered≤b≤2d−3 and δ>0 is an arbitrarily small constant.
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.