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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Proceedings ArticleDOI
Two Proofs for Shallow Packings
TL;DR: Two proofs are presented, the first is an extension of Haussler's approach, and the second extends the proof of Chazelle, originally presented as a simplification for Hausser's proof.
Proceedings ArticleDOI
Randomized algorithms for binary search and load balancing with geometric applications
John H. Reif,Sandeep Sen +1 more
TL;DR: In this paper, randomized parallel algorithms for trapezoidal decomposition, visibility, triangulation, and Z-D convex hull are presented. But these algorithms are based on some previous work of the authors on PRAM algorithms.
Journal ArticleDOI
Independence number and the complexity of families of sets
Daniel Q. Naiman,Henry P. Wynn +1 more
TL;DR: The theory of Independence number (IN) is developed and examples of exact and asymptotic evaluations are given, which give improvements over Sauer's Lemma for several examples.
Proceedings ArticleDOI
Counting and representing intersections among triangles in three dimensions
Esther Ezra,Micha Sharir +1 more
TL;DR: An algorithm that efficiently counts all intersecting triples among a collection T of triangles in ℝ3 in nearly-quadratic time is presented, and it is proved that this counting problem belongs to the 3SUM-hard family, and thus the algorithm is likely to be nearly optimal.
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.