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
Triangles in Space or Building (and Analyzing) Castles in the Air
Boris Aronov,Micha Sharir +1 more
TL;DR: In this article, it was shown that the combinatorial complexity of all nonconvex cells in an arrangement of n (possibly intersecting) triangles in 3-space is O(n7/3+d), for any d>0, and that this bound is almost tight in the worst case.
Proceedings ArticleDOI
Efficient randomized algorithms for the repeated median line estimator
TL;DR: This paper presents the best known theoretical algorithm and a practical subquadratic algorithm for computing a 50% breakdown point line estimators, the Siegel or repeated median line estimator, and presents an O(n\log n) randomized expected-time algorithm, where n is the number of given points.
Journal ArticleDOI
Compression Schemes, Stable Definable Families, and o-Minimal Structures
TL;DR: The combinatorial complexity of any definable family in a structure with a o-minimal theory is bounded by the number of parameters in the defining formula, and extended compression schemes for uniformly definable families corresponding to stable formulas are shown to exist.
Journal ArticleDOI
Derandomization in Computational Geometry
TL;DR: Techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time are surveyed.
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.