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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Matching Polyhedral Terrains Using Overlays of Envelopes
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Family complexity and VC-Dimension
Christian Mauduit,András Sárközy +1 more
TL;DR: This paper presents a survey of papers on the connection of family complexity and VC-dimension in Ahlswede, Khachatrian, Mauduit and Sarkozy 2006 and several further related papers published on this subject since that.
Proceedings ArticleDOI
Massively parallel algorithms for computing TIN DEMs and contour trees for large terrains
TL;DR: An efficient algorithm in the MPC model for computing the contour tree of the resulting DEM by computing the Delaunay triangulation of the xy-projections of points in S, which is also stored across multiple machines.
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Learning Convex Sets Under Uniform Distribution
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New -Net Constructions
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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.