Proceedings ArticleDOI
Improved bounds on weak e-nets for convex sets
Bernard Chazelle,Herbert Edelsbrunner,Michelangelo Grigni,Leonidas J. Guibas,Micha Sharir,Emo Welzl +5 more
- pp 495-504
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TLDR
In the case whereS consists of the vertices of a regular polygon, this work uses an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/e), which improves a previous bound of Capoyleas.Abstract:
LetS be a set ofn points in ℝ d . A setW is aweak e-net for (convex ranges of)S if, for anyT⊆S containing en points, the convex hull ofT intersectsW. We show the existence of weak e-nets of size\(O((1/\varepsilon ^d )\log ^{\beta _d } (1/\varepsilon ))\), whereβ2=0,β3=1, andβ d ≈0.149·2d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/e) log1.6(1/e)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/e), which improves a previous bound of Capoyleas.read more
Citations
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Journal ArticleDOI
Almost optimal set covers in finite VC-dimension
TL;DR: A deterministic polynomial-time method for finding a set cover in a set system (X, ℛ) of dual VC-dimensiond such that the size of the authors' cover is at most a factor ofO(d log(dc)) from the optimal size,c.
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.
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Streaming Algorithms for Extent Problems in High Dimensions
TL;DR: The notion of blurred ball cover is introduced and used for answering approximate farthest-point queries and maintaining approximate minimum enclosing ball and diameter of S and lower bounds on the worst-case approximation ratio are obtained.
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Computational geometry: a retrospective
TL;DR: This work will survey some of its principal accomplishments, and in light of recent developments, it will discuss the profound transformations the field has begun to undergo.
Journal ArticleDOI
A Non-linear Lower Bound for Planar Epsilon-nets
TL;DR: It is shown that the minimum possible size of an ε-net for point objects and line (or rectangle)-ranges in the plane is (slightly) bigger than linear in $\frac{1}{\epsilon}$.
References
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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
ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
TL;DR: 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.
Proceedings ArticleDOI
Epsilon-nets and simplex range queries
David Haussler,Emo Welzl +1 more
TL;DR: A new technique for half-space and simplex range query using random sampling to build a partition-tree structure and introduces the concept of anε-net for an abstract set of ranges to describe the desired result of this random sampling.
Journal ArticleDOI