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
Dynamic partition trees
TL;DR: An output-sensitive method for hidden surface removal in a set ofn triangles that runs in timeO(nlogn+n·kγ) whereγ=log2((1+√5)/2) ≈ 0.695 andk is the size of the visibility map obtained.
Journal ArticleDOI
VC-dimensions of random function classes
Bernard Ycart,Joel Ratsaby +1 more
TL;DR: The typical behavior of classes generated randomly by two methods, repeated biased coin flips on the n-dimensional hypercube or uniform sampling over the space of all possible classes of cardinality k on [n] is much more similar to the necessary condition; the Cardinality k need only be larger than a threshold of 2d for its VC-dimension to be at least d with high probability.
Book ChapterDOI
Concentration of Measure for Randomized Algorithms: Techniques and Analysis
Devdatt Dubhashi,Sandeep Sen +1 more
On weak $ε$-nets and the Radon number
Shay Moran,Amir Yehudayoff +1 more
TL;DR: In this paper, the Radon number characterizes the existence of weak nets in separable convexity spaces, and it is shown that weak nets can be constructed using the chromatic number of the Kneser graph.
Posted Content
Beyond the Richter-Thomassen Conjecture
TL;DR: The Crossing Lemma for closed Jordan curves in the plane has been proved in this article, showing that the number of intersection points between any pairwise intersecting simple closed curves is at least (1-o(1))n^2.
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.
Book
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.
Journal ArticleDOI
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.
Journal ArticleDOI
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.