ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
Reads0
Chats0
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
Proceedings ArticleDOI
Dynamic point location in arrangements of hyperplanes
Ketan Mulmuley,Sandeep Sen +1 more
TL;DR: Algorithms for maintaining data structures supporting fast (polylogarithmic) point-location and ray-shooting queries in arrangements of hyperplanes using random bits are presented, which has a versatile quality and is likely to have further applications to other dynamic algorithms.
Dissertation
Algorithms for Geometric Covering and Piercing Problems
TL;DR: This thesis involves the study of a range of geometric covering and piercing problems, where the unifying thread is approximation using disks, and outlines approximation algorithms for the general DUDC problem which make use of the algorithms for LSDUDC and WSDUDC.
Proceedings ArticleDOI
Output sensitive construction of levels and Voronoi diagrams in Rd of order 1 to k
TL;DR: With this method, the recoverable coal is in physical contact with a series of two or more positive electrodes while it is isolated from a single negative electrode by an electrolytic solution.
Journal ArticleDOI
Approximate testing and its relationship to learning
TL;DR: A model for approximate testing of concepts is developed, which relates to the PAC (probably almost correct) model of learning as well as other learning models, and a new measure that is similar to the Vapnik-Chervonenkis dimension of a concept class is defined, and it is shown how it yields untestability results for certain concept classes.
Journal ArticleDOI
Speeding Up Sparsification using Inner Product Search Data Structures
TL;DR: The heart of the work is the design of a variety of different inner product search data structures that have efficient initialization, query and update time, compatible to dimensionality reduction and robust against adaptive adversary.
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.