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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Bounding the vertex cover number of a hypergraph
TL;DR: It is shown that τ(H), the minimumk such that some set ofk vertices meets all the edges, is bounded above by a function of ν (H) and λ (H), and indeed that if λ(H) is bounded by a constant then τ( H) is at most a polynomial function of μ(H).
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A randomized algorithm for fixed-dimensional linear programming
Martin Dyer,Alan Frieze +1 more
TL;DR: A (Las Vegas) randomized algorithm for linear programming in a fixed dimension for which the expected computation time is O(d^{(3 + \varepsilon _d )d} n)$$, where limd→∞εd = 0.5 improves the corresponding worst-case complexity.
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Approximation algorithms for projective clustering
TL;DR: In this paper, the authors considered the projective clustering problem and gave a randomized algorithm that computes O(k log k) strips of width at most w* that cover a set S of n points in road and an integer k > 0.
Proceedings ArticleDOI
Testing of clustering
TL;DR: The benefit of the algorithms is that they construct an implicit representation of such clusterings in time independent of |X|, and the implicit representation can be used to answer queries concerning the cluster any given point belongs to.
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New existence proofs ε-nets
Evangelia Pyrga,Saurabh Ray +1 more
TL;DR: In this article, a technique for proving the existence of small μ-nets for hypergraphs satisfying certain simple conditions is described. But the technique is not suitable for proving o(1/μ log 1/μ) upper bounds which the standard VC-dimension theory does not allow.
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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Algorithms in Combinatorial Geometry
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
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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.