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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Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles
Pankaj K. Agarwal,Rolf Klein,Christian Knauer,Stefan Langerman,Pat Morin,Micha Sharir,Michael Soss +6 more
Dissertation
PAC-learning geometrical figures
TL;DR: This thesis shows that it is hard to find a single geometrical figure which is close to the positive examples but not to the negative examples, and finds a general technique for prediction which is able to predict the status of further unlabelled examples in polynomial time.
Applications of Graph and Hypergraph Theory in Geometry
TL;DR: In this article, a survey of geometric results whose proof uses graph or hypergraph theory is presented, including forbidden configurations, geometric constructions, saturated hypergraphs in geometry, independent sets in graphs, the regularity lemma, and VC-dimension.
Proceedings ArticleDOI
Efficient parallel computation of arrangements of hyperplanes in d dimensions
Torben Hagerup,H. Jung,Emo Welzl +2 more
TL;DR: The jirst optimal parallel algorithm computing arrangements of hyperplanes in Ed is proposed and computes the arrangement of n hyperplanes within expected logarithmic time on a CRCW-PRAM with O(nd/ log n) processors.
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.