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
Capacitated Discrete Unit Disk Cover
TL;DR: The geometric version of the capacitated covering problem, where the set of customers and set of service centers are two point sets in the Euclidean plane, is considered, and it is proved that the \((\alpha, P, Q)-covering problem is NP-complete for \(\alpha \ge 3\) and it admits a PTAS.
Posted Content
On the Decision Tree Complexity of $k$-SUM.
Jean Cardinal,Aurélien Ooms +1 more
TL;DR: It is proved that there exists $o(n)$-linear decision trees of depth $O(n^c)$ for some constant $c$.
Book ChapterDOI
Sketched History: VC Combinatorics, 1826 up to 1975
TL;DR: The earliest published instance of combinatorial quantities later occurring in the work of Vapnik and Chervonenkis was in a paper of Jakob Steiner in 1826 as mentioned in this paper.
Journal ArticleDOI
On the VC-Dimension of Binary Codes
TL;DR: This work investigates the asymptotic rates of length-n binary codes with VC-dimension at most dn and minimum distance at least δn with Gilbert-Varshamov type arguments over constant-weight and Markov-type sets.
Journal ArticleDOI
Approximating a Convex Body by a Polytope Using the Epsilon-Net Theorem
TL;DR: It is proved that roughly d (1-\vartheta ) dln1(1-ϑ)d points chosen uniformly and independently from a centered convex body K in Rd yield a polytope P for whichϑK⊆P ⊆K holds with large probability.
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.