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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Intersection Queries in Sets of Disks
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Recent Developments in the Theory of Arrangements of Surfaces
Micha Sharir,Micha Sharir +1 more
TL;DR: Applications of the new results of the study of arrangements of surfaces in higher dimensions to a variety of problems in computational geometry and its applications, including motion planning, Voronoi diagrams, union of geometric objects, visibility, and geometric optimization are presented.
Range Searching: Emptiness, Reporting, and Approximate Counting
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A size-sensitive discrepancy bound for set systems of bounded primal shatter dimension
TL;DR: In this paper, Lovett and Meka showed that there exists a coloring χ with discrepancy bound O(|S|1/2-d1/(2d)n(d1-1)/(2d)), where O*(·) hides a polylogarithmic factor in n.
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CHAPTER 7 – Computational Geometry
TL;DR: The fastest known algorithm for triangulating a simple polygon requires 0 (n log n) time, while no nonlinear lower bound to this problem is known as discussed by the authors. But the problem of testing whether n points in the plane are in general position has an upper bound of 0(n2) and a lower bound of Ω(n log N).
References
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On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
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Algorithms in Combinatorial Geometry
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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.
Journal ArticleDOI
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.