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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Book ChapterDOI
An Abstract Approach to Polychromatic Coloring: Shallow Hitting Sets in ABA-free Hypergraphs and Pseudohalfplanes
Balázs Keszegh,Dömötör Pálvölgyi +1 more
TL;DR: An abstract version of a framework by Smorodinsky and Yuditsky, used for polychromatic coloring halfplanes, is introduced, and applied to so-called ABA-free hypergraphs, which are a generalization of interval graphs.
Proceedings ArticleDOI
Output-sensitive construction of the union of triangles
Esther Ezra,Micha Sharir +1 more
TL;DR: This work presents an efficient algorithm, which combines a variety of techniques related to range-searching in two dimensions, and shows that it does indeed run in subquadratic time (for a reasonable range of ζ).
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Pac Learning, Noise, and Geometry
TL;DR: This paper describes the probably approximately correct model of concept learning, paying special attention to the case where instances are points in Euclidean n-space.
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On Regular Vertices of the Union of Planar Convex Objects
TL;DR: It is shown that the maximum number of regular vertices on the boundary of the union U of the collection of n compact convex sets in the plane is O*(n4/3, which improves earlier bounds due to Aronov et al.
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Using ε-Nets for Linear Separation of Two Sets in a Euclidean Space Rd
M. A. Ivanchuk,I. V. Malyk +1 more
TL;DR: In this paper, the concept of -separability was introduced and necessary and sufficient conditions of separation were proved for disjoint?-nets, and it was proved that the problem of separability of two disjunctive sets can be reduced to the trivial problem of separation of their disjoins.
References
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