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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Proceedings ArticleDOI
Near-linear approximation algorithms for geometric hitting sets
TL;DR: This work presents near-linear time approximation algorithms for the hitting set problem, under the following geometric settings: R is a set of planar regions with small union complexity, and X is either the entire d-dimensional space or a finite set of points in d-space.
Geometric graph theory and wireless sensor networks
János Pach,Deniz Sariöz +1 more
TL;DR: This work proves that this problem of finding a minimum obstacle representation of a given straight-line graph drawing is NP-complete, and provides a O(log OPT)-factor approximation algorithm by showing that the corresponding hypergraph family has bounded Vapnik-Chervonenkis dimension.
Proceedings ArticleDOI
Multi-resolution Hashing for Fast Pairwise Summations
Moses Charikar,Paris Siminelakis +1 more
TL;DR: This work provides a general framework for designing data structures through hashing that reaches far beyond what previous techniques allowed, and leads to data structures with sub-linear query time that significantly improve upon random sampling and can be used for Kernel Density, Partition Function Estimation and sampling.
Book ChapterDOI
Dynamic partition trees
Haijo Schipper,Mark H. Overmars +1 more
TL;DR: This paper studies dynamic variants of conjugation trees and related structures introduced for performing various types of queries on sets of points and line segments, like half-planar range searching, shooting, intersection queries, etc, and obtains an output-sensitive method for hidden surface removal in a set of n triangles.
Some results in computational and combinatorial geometry
TL;DR: An O(n log n) time algorithm to find points z1 6= z2, at least one of which must meet any convex set of “size” greater than 4n/7; z1 and z2 comprise a hitting set of size two for such convex ranges, the first algorithm for computing the hitting sets of fixed size.
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.