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
Epsilon-Nets for Halfspaces Revisited.
TL;DR: It is shown that, for any $\varepsilon >0$, there exists an $\vARpsilon-net of $P$ for halfspace ranges, of size $O(1/\varpsilon)$, and five proofs of this result are given.
Journal ArticleDOI
Optimal, output-sensitive algorithms for constructing planar hulls in parallel
Sandeep Sen,Neelima Gupta +1 more
TL;DR: This paper describes a very simple O(lognlogH) time optimal deterministic algorithm for the planar hulls which is an improvement over the previously known Ω( log 2 n) time algorithm for small outputs and presents a fast randomized algorithm that runs in expected time O(logH·loglogn) and does optimal O(n logH) work.
Posted Content
Small Strong Epsilon Nets
TL;DR: In this article, it was shown that strong centerpoints do not exist even for axis-parallel boxes in the plane, and that strong centers do not even exist for halfspaces in the Euclidean plane.
Dissertation
Small and stable descriptors of distributions for geometric statistical problems
TL;DR: This thesis explores how to sparsely represent distributions of points for geometric statistical problems, and shows how to create distributions of e-kernels and e-samples for these uncertain data sets.
Proceedings ArticleDOI
A Nearly Quadratic Bound for the Decision Tree Complexity of k-SUM
Esther Ezra,Micha Sharir +1 more
TL;DR: The approach relies on a new point-location mechanism, exploiting "Epsilon-cuttings" that are based on vertical decompositions in hyperplane arrangements in high dimensions, and results in a sharper bound on the complexity of the vertical decomposition of such an arrangement (in terms of its dependence on the dimension).
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.