ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
TLDR
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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Journal ArticleDOI
Quasi-optimal range searching in spaces of finite VC-dimension
Bernard Chazelle,Emo Welzl +1 more
TL;DR: It is proved that any set ofn points inEd admits a spanning tree which cannot be cut by any hyperplane (or hypersphere) through more than roughlyn1−1/d edges, and this result yields quasi-optimal solutions to simplex range searching in the arithmetic model of computation.
Journal ArticleDOI
A singly exponential stratification scheme for real semi-algebraic varieties and its applications
TL;DR: This paper describes an effective procedure for stratifying a real semi-algebraic set into cells of constant description size that compares favorably with the doubly exponential size of Collins' decomposition.
Journal Article
Forest Fire Modeling and Early Detection using Wireless Sensor Networks
Mohamed Hefeeda,Majid Bagheri +1 more
TL;DR: This work designs a constant-factor centralized algorithm, and a fully distributed version which does not require sensors know their locations and proposes approximation algorithms for the node k-coverage problem which is shown to be NP-hard.
Book ChapterDOI
On Rectangular Partitionings in Two Dimensions: Algorithms, Complexity, and Applications
TL;DR: In this article, the problem of partitioning a two-dimensional array into rectangular tiles of arbitrary size in a way that minimizes the number of tiles required to satisfy a given constraint is studied.
Proceedings Article
Predicting {0,1}-Functions on Randomly Drawn Points (Extended Abstract)
TL;DR: In this article, the authors consider the problem of predicting (0, l)valued functions on R" and smaller domains, based on their values on randomly drawn points, and construct prediction strategies that are optimal to within a constant factor for any reasonable class F of target functions.
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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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.