Improved bounds on weak &egr;-nets for convex sets
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Lectures on Discrete Geometry
New Constructions of Weak ε-Nets
Discrete & Computational Geometry
Weak ε-nets and interval chains
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Weak ε-nets and interval chains
Journal of the ACM (JACM)
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Given a set P of n points in Rd and ε 0, we consider the problemof constructing weak ε-nets for P.We show the following: pick a random sample Q of size O(1/ε log (1/ε)) from P. Then, with constant probability, a weak ε-net of P can be constructed from only the points of Q. This shows that weak ε-nets in Rd can be computed from a subset of P of size O(1/ε log(1/ε)) with only the constant of proportionality depending on the dimension, unlike all previous work where the size of the subset had the dimension in the exponent of 1/ε. However, our final weak ε-nets still have a large size (with the dimension appearing in the exponent of 1/ε).