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
Planar point sets determine many pairwise crossing segments
TL;DR: It is shown that any set of n points in general position in the plane determines n1−o(1) pairwise crossing segments and the proof is fully constructive, and extends to dense geometric graphs.
Proceedings ArticleDOI
Intersection queries for curved objects (extended abstract)
TL;DR: Most of the data structures presented here use linear or near to linear space and have query time near to 0(/E+K) or 0(n2/3+K), where K is the size of the output.
Journal ArticleDOI
Output-Sensitive Construction of the Union of Triangles
Esther Ezra,Micha Sharir +1 more
TL;DR: In this article, the problem of finding a set cover in a set system of finite VC-dimension is solved in randomized expected time, which is subquadratic for ∆ = o(n/ ∆ log 2 n + ∆ n/∆ log n) for any ∆ > 0.
Proceedings Article
Proceedings of the 12th Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW 2013)
TL;DR: A special issue of Discrete AppliedMathematics devoted to the 12th Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW 2013), held at the University of Twente, Enschede, The Netherlands, between May 21 and May 23, 2013 as mentioned in this paper.
Proceedings ArticleDOI
Faster output-sensitive parallel convex hulls for d≤3: optimal sublogarithmic algorithms for small outputs
Neelima Gupta,Sandeep Sen +1 more
TL;DR: A fast randomized algorithm for planar hulls that runs in expected time O(log H . log log n) and does optimal O(rz log H) work where n and ET are the input and output sizes respectively and a very simple O (log nlog H) time optimal deterministic algorithm which is an improvement for small outputs is described.
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.
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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.
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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.
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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.