منابع مشابه
When are epsilon-nets small?
Given a range space (X,R), where X is a set equipped with probability measure P and the family of measurable subsets R ⊂ 2X , and ǫ > 0, an ǫ-net is a subset of X in the support of P , which intersects each R ∈ R with P (R) ≥ ǫ. In many interesting situations the size of ǫ-nets depends only on ǫ together with different complexity measures. The aim of this paper is to give a systematic treatment...
متن کاملSmall strong epsilon nets
Let P be a set of n points in R. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than dn d+1 points of P . We call a point x a strong centerpoint for a family of objects C if x ∈ P is contained in every object C ∈ C that contains more than a constant fraction of points of P . A strong centerpoint does not exist even for halfspaces in R. We ...
متن کاملSmall weak epsilon nets
Given a set P of points in the plane, a set of points Q is a weak ε-net with respect to a family of sets S (e.g., rectangles, disks, or convex sets) if every set of S containing ε|P | points contains a point of Q. In this paper, we determine bounds on εi , the Preprint submitted to Elsevier Science 10 February 2008 smallest epsilon that can be guaranteed for any P when |Q| = i, for small values...
متن کاملSmall Weak Epsilon-Nets in Three Dimensions
We study the problem of finding small weak ε-nets in three dimensions and provide new upper and lower bounds on the value of ε for which a weak ε-net of a given small constant size exists. The range spaces under consideration are the set of all convex sets and the set of all halfspaces in R3.
متن کاملEpsilon-approximations and epsilon-nets
The use of random samples to approximate properties of geometric configurations has been an influential idea for both combinatorial and algorithmic purposes. This chapter considers two related notions—ǫ-approximations and ǫ-nets—that capture the most important quantitative properties that one would expect from a random sample with respect to an underlying geometric configuration. An example pro...
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ژورنال
عنوان ژورنال: Journal of Computer and System Sciences
سال: 2020
ISSN: 0022-0000
DOI: 10.1016/j.jcss.2019.12.006