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
Sample Compression, Learnability, and the Vapnik-Chervonenkis Dimension
Sally Floyd,Manfred K. Warmuth +1 more
TL;DR: It is demonstrated that the existence of a sample compression scheme of fixed-size for aclass C is sufficient to ensure that the class C is pac-learnable, and the relationship between sample compression schemes and the VC dimension is explored.
Journal ArticleDOI
Reporting points in halfspaces
TL;DR: The halfspace itrange itreporting problem, given a finite set P of points in R d, can be solved substantially more efficiently that the more general simplex range searching problem.
Journal ArticleDOI
Separators for sphere-packings and nearest neighbor graphs
TL;DR: This result implies that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, which gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions.
Journal ArticleDOI
A randomized algorithm for closest-point queries
TL;DR: This result approaches the $\Omega (n^{\lceil {{d / 2}} \rceil } )$ worst-case time required for any algorithm that constructs the Voronoi...
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Las Vegas algorithms for linear and integer programming when the dimension is small
TL;DR: An algorithm for solving linear programming problems with n constraints and d variables and the number of bits required to specify the rational numbers defining an input constraint or the objective function vector is given.
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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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.