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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Book ChapterDOI
Hitting Set Problem for Axis-Parallel Squares Intersecting a Straight Line Is Polynomially Solvable for Any Fixed Range of Square Sizes
TL;DR: An improved algorithm for unit squares intersecting a straight line is proposed, whose complexity bound is more than 100 orders of magnitude less and the geometric HSP for axis-parallel squares intersected by a line is polynomially solvable for any fixed range of squares to hit.
Journal ArticleDOI
Random Sampling with Removal
TL;DR: For many relevant values of r, δ, and k, it is proved matching upper and lower bounds for the expected number of constraints violated by a random sample, after the removal of k constraints.
Journal ArticleDOI
Danzer's problem, effective constructions of dense forests and digital sequences
TL;DR: In this paper , the authors provide deterministic constructions of dense and optical forests which yield the best known results in any dimension d ⩾ 2 $d \geqslant 2$ in terms of visibility and density bounds, respectively.
Book ChapterDOI
Fast Range Searching with Delaunay Triangulations
TL;DR: It is shown that when the query polygons are well-shaped the expected query time is O(n1/3 + Q + n ⁗ area(Q)), which improves the best known average query time for range searching achieved by the bucketing method.
Posted Content
Random hyperplane search trees in high dimensions
Luc Devroye,James King +1 more
TL;DR: For any fixed dimension d, a random hyperplane search tree with height at most (1 + O(1/sqrt(d)) log 2 n and average element depth at most 2 n with high probability as n \rightarrow \infty was shown in this article.
References
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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.
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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.