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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Proceedings ArticleDOI
Methods for achieving fast query times in point location data structures
TL;DR: A re-examination of the issues that arise in three different classes of point-location query sequences, focusing on sequences that are reasonably uniform spatially and temporally, as well as an experimental analysis over a range of subdivision and query domains.
Journal ArticleDOI
Quasi-Polynomial Time Approximation Scheme for Weighted Geometric Set Cover on Pseudodisks and Halfspaces
TL;DR: Even for the unweighted case, a polynomial time approximation scheme (PTAS) for a fundamental class of objects called pseudodisks (which includes halfspaces, disks, unit-height rectangles, translates of convex sets, etc.) is curren...
Proceedings ArticleDOI
An efficient algorithm for terrain simplification
TL;DR: A randomized algorithm is presented that computes an {epsilon}-approximation of size O(c{sup 2} log{Sup 2} c) in O(n {sup 2+{delta}} + c{sup 3} log {Sup 3}c log n/c) expected time, where c is the size of the {Epsilon]- approximation with the minimum number of vertices and {delta} is any arbitrarily small positive
Proceedings ArticleDOI
Discrepancy and in -approximations for bounded VC-dimension
TL;DR: It is shown that if for any m-point subset Y contained in X the number of distinct subsets induced by R on Y is bounded by O(m/sup d/) for a fixed integer d, then there is a coloring with discrepancy at most O(n/sup 1/2-1/2d/ n), implying improved upper bounds on the size of in -approximations for (X, R.
Journal ArticleDOI
Randomized Algorithms for Binary Search and Load Balancing on Fixed Connection Networks with Geometric Applications
John H. Reif,Sandeep Sen +1 more
TL;DR: These algorithms involve tackling some of the very basic problems, like binary search and load balancing, that are taken for granted in PRAM models and are the first nontrivial geometric algorithms that attain this performance on fixed connection networks.
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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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.