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
More filters
Journal ArticleDOI
Point Location in Arrangements of Hyperplanes
TL;DR: A solution to the point location problem in arrangements of hyperplanes in Ed with running time O(d5 log n) and space O(nd+?) for arbitrary ? > 0, where n is the number ofhyperplanes.
Journal ArticleDOI
Learning conjunctive concepts in structural domains
TL;DR: It is shown that heuristic methods for learning from larger scenes are likely to give an accurate hypothesis if they produce a simple hypothesis consistent with a large enough random sample and that this class of concepts is polynomiaIIy learnable from random examples in the sense of Valiant.
Journal ArticleDOI
Prediction-preserving reducibility
Leonard Pitt,Manfred K. Warmuth +1 more
TL;DR: A model of polynomial-time concept prediction is investigated which is a relaxation of the distribution-independent model of concept learning due to Valiant and prediction-preserving reductions are defined and are shown to be effective tools for comparing the relative difficulty of solving various prediction problems.
Journal ArticleDOI
Geometric range searching
TL;DR: A survey of theoretical results and the main techniques in geometric range searching is presented, which can be used as subroutines in solutions to many seemingly unrelated problems.
Automatic Mesh Partitioning
TL;DR: In this paper, the authors describe an efficient approach to partitioning unstructured meshes that occur naturally in the finite element and finite difference methods, making use of the underlying geometric structure of a given mesh and finding a provably good partition in random O(n) time.
References
More filters
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.