ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
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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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Journal ArticleDOI
Transversal numbers for hypergraphs arising in geometry
TL;DR: A ( p,q) theorem for abstract set systems F is proved and a topological (p,d+1) theorem is obtained for convex lattice sets in Z^d and examples illustrating that some of the assumptions cannot be weakened are provided.
Proceedings ArticleDOI
Partition trees for triangle counting and other range searching problems
TL;DR: Bounds for space and query time are optimal up to polylog — factors and the preprocessing time for the data structures is polynomial according to recent results by Chazelle.
Journal ArticleDOI
Relative ( p , ε )-Approximations in Geometry
Sariel Har-Peled,Micha Sharir +1 more
TL;DR: The notion of relative (p,ε)-approximations is re-examine, and upper bounds on their size are established, in general range spaces of finite VC-dimension, using the sampling theory developed in Li et al. (Manuscript, 2006), and new structures are applied to obtain efficient solutions for approximate range counting in three dimensions.
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Cutting hyperplane arrangements
TL;DR: A deterministic algorithm for finding a (1/r)-cutting withO(rd) simplices with asymptotically optimal running time is given, which has numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly.
Proceedings ArticleDOI
Optimal partition trees
TL;DR: A new method is given that achieves simultaneously O(n log n) preprocessing time, O (n) space, and O( n1-1/d) query time with high probability and leads to more efficient multilevel partition trees, which are important in many data structural applications.
References
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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
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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.
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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.