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
More filters
Posted Content
Information Width
TL;DR: To answer these questions Kolmogorov’s definition is extended and a new concept termed information width which is similar to n-widths in approximation theory is introduced.
Book ChapterDOI
Paul Erdős and Probabilistic Reasoning
TL;DR: This short paper describes some of the beautiful applications of the Probabilistic Method, focusing on the long-term impact of the work, questions and results of Erdős.
Posted Content
Convex Set Disjointness, Distributed Learning of Halfspaces, and LP Feasibility
TL;DR: These protocols are based on a Container Lemma for Halfspaces and on two variants of Caratheodory's Theorem, which may be of independent interest, and derive upper and lower bounds of $\tilde O(d^2\log n)$ and~$Omega(d\ log n)$.
Proceedings Article
The VC-dimension of graphs with respect to k-connected subgraphs.
TL;DR: In this article, the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its k-connected subgraphs was studied and tight upper and lower bounds were given.
Journal ArticleDOI
Small strong epsilon nets
TL;DR: It is proved that a strong centerpoint exists for axis-parallel boxes in R^d and given exact bounds and this is extended to small strong @e-nets in the plane.
References
More filters
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.