ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
Reads0
Chats0
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
Random inscribed polytopes
TL;DR: In this article, a divide-and-conquer martingale technique was used to obtain the upper bound on the variance of Vold(Kn) for convex bodies with C2 boundary and everywhere positive Gaus-Kronecker curvature.
Journal ArticleDOI
Subquadratic algorithms for some 3Sum-hard geometric problems in the algebraic decision-tree model
TL;DR: Agarwal et al. as mentioned in this paper presented subquadratic algorithms in the algebraic decision-tree model for several 3Sum -hard geometric problems, all of which can be reduced to the following question: given two sets A , B , each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the planes, we want to count, for each triangle Δ ∈ C , the number of intersection points between the segments of A and those of B that lie in Δ.
Posted Content
A short proof of the first selection lemma and weak $\frac{1}{r}$-nets for moving points.
TL;DR: It is established that one can find a kinetic analog $N$ of a weak $\frac{1}{r}$-net of cardinality $O(r^{\frac{d(d+1)}{2}}\log^{d}r)$ whose points are moving with coordinates that are rational functions with bounded description complexity.
Proceedings Article
On Coresets for Fair Regression and Individually Fair Clustering
TL;DR: This paper defines coresets for Fair Regression with Statistical Parity (SP) constraints and for Individually Fair Clustering and shows that to obtain such coresets, it is sufficient to sample points based on the probabilities dependent on combination of sensitivity score and a carefully chosen term according to the fairness constraints.
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.