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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State of the Union (of geometric objects)
TL;DR: Pach's number: [221] Reference DCG-CHAPTER-2008-040 Record created on 2008-11-18, modified on 2017-05-12 as mentioned in this paper ].
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Multi-Pass Geometric Algorithms
Timothy M. Chan,Eric Y. Chen +1 more
TL;DR: This paper proposes the study of exact geometric algorithms that require limited storage and make only a small number of passes over the input.
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Generalizing the PAC model: sample size bounds from metric dimension-based uniform convergence results
TL;DR: The probably approximately correct (PAC) model of learning from examples is generalized, and a distribution-independent uniform convergence result for certain classes of functions computed by feedforward neural nets is obtained.
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Quasi-Monte-Carlo methods and the dispersion of point sequences
Günter Rote,Robert F. Tichy +1 more
TL;DR: The dispersion of a point set with respect to various classes of range spaces, like balls, squares, triangles, axis-parallel and arbitrary rectangles, spherical caps and slices, is the area of the largest empty range, and it is a measure for the distribution of the points as mentioned in this paper.
Proceedings ArticleDOI
Geometric partitioning made easier, even in parallel
TL;DR: A simple approach for constructing geometric partitions in a way that is easy to apply to new problems, which leads to asymptotically faster and more-efficient EREW PRAM parallel algorithms for a number of computational geometry problems, including the development of the first optimal-work NC algorithm for the well-known 3-dimensional convex hull problem.
References
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On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
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