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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Proceedings ArticleDOI
Selection and Orientation of Directional Sensors for Coverage Maximization
Giordano Fusco,Himanshu Gupta +1 more
TL;DR: A simple greedy algorithm is designed that delivers a solution that k-covers at least half of the target points using at most M log(k|C|) sensors, where |C| is the maximum number of target points covered by a sensor and M is the minimum number of sensor required to k-cover all the given points.
Journal ArticleDOI
Lower bounds on the complexity of polytope range searching
TL;DR: It is proved that the worst case query time is Q(n/l/Hi) in the Euclidean plane, and more generally, Q((n/ log n)/m'l/d) in d-space, for d > 3, where n is the number of points and m is the amount of storage available.
Journal ArticleDOI
Geometric Separators for Finite-Element Meshes
TL;DR: A class of graphs that would occur naturally in finite-element and finite-difference problems is proposed and it is proved a bound on separators for this class of graph is O(n(d-1)d), which is the best possible bound.
Proceedings ArticleDOI
Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling
TL;DR: This work provides a geometry-inspired algorithm whose approximation guarantee depends solely on an instance-specific combinatorial property known as shallow cell complexity (SCC), and settles an open question by showing that weighted instances of the capacitated covering problem with underlying network structure have O(1)-approximations.
Proceedings ArticleDOI
How to net a lot with little: small ε-nets for disks and halfspaces
TL;DR: In this paper, it was shown that disks and pseudo-disks in the plane as well as halfspaces in R3 allow e-nets of size only O(1/e), which is best possible up to a multiplicative constant.
References
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Book ChapterDOI
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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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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