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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Improved bounds on weak ź-nets for convex sets
Bernard Chazelle,Herbert Edelsbrunner,Michelangelo Grigni,Leonidas J. Guibas,Micha Sharir,Emo Welzl +5 more
TL;DR: In the case whereS consists of the vertices of a regular polygon, an argument from hyperbolic geometry is used to exhibit an optimal net of sizeO(1/ε), which improves a previous bound of Capoyleas.
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Convex Hulls under Uncertainty
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Polling: a new randomized sampling technique for computational geometry
John H. Reif,Sandeep Sen +1 more
TL;DR: A new randomized sampling technique, called Polling, is introduced which has applications to deriving efficient parallel algorithms for fundamental problems like the convex hull in three dimensions, Voronoi diagram of point sites on a plane and Euclidean minimal spanning tree.
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The discrete yet ubiquitous theorems of Carath\'eodory, Helly, Sperner, Tucker, and Tverberg
TL;DR: In this paper, the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath-eodory, Helly, and Tverberg are discussed.
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Testing of Clustering
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References
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