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
On the Zarankiewicz Problem for Intersection Hypergraphs
Nabil H. Mustafa,János Pach +1 more
TL;DR: The flexibility of this technique is demonstrated by extending the proof of the planar version of the theorem to intersection graphs of x-monotone curves, which works in any dimension and uses size-sensitive cuttings, a variant of random sampling.
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Minimum Enclosing Ball Revisited: Stability and Sub-linear Time Algorithms.
TL;DR: This paper revisits the Minimum Enclosing Ball problem and its robust version, MEB with outliers, in Euclidean space and introduces the notion of stability for MEB (with outliers), which is natural and easy to understand.
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Visualization of Big Spatial Data Using Coresets for Kernel Density Estimates
Abstract: The size of large, geo-located datasets has reached scales where visualization of all data points is inefficient. Random sampling is a method to reduce the size of a dataset, yet it can introduce unwanted errors. We describe a method for subsampling of spatial data suitable for creating kernel density estimates from very large data and demonstrate that it results in less error than random sampling. We also introduce a method to ensure that thresholding of low values based on sampled data does not omit any regions above the desired threshold when working with sampled data. We demonstrate the effectiveness of our approach using both, artificial and real-world large geospatial datasets.
Regions, distances and graphs
Sébastien Collette,Jean Cardinal +1 more
TL;DR: It is shown that every monotone property has at least one corresponding tight region; it is also shown that various operations, such as path and point queries using geometric graphs as data structures, have complexities which can be expressed as local properties.
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A Lower Bound for Families of Natarajan Dimension d
Paul Fischer,Jiří Matoušek +1 more
TL;DR: A lower bound of cdkdnd is improved for the maximum size of F of Natarajan dimension at most d by a factor somewhat smaller than k (e.g., by k for d=1).
References
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On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
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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.