TY - GEN

T1 - Weak -nets have basis of size o(1/ log (1/)) in any dimension

AU - Mustafa, Nabil

AU - Ray, Saurabh

PY - 2007

Y1 - 2007

N2 - Given a set P of n points in Rd and > 0, we consider the problemof constructing weak -nets for P.We show the following: pick a random sample Q of size O(1/ log (1/)) from P. Then, with constant probability, a weak -net of P can be constructed from only the points of Q. This shows that weak -nets in Rd can be computed from a subset of P of size O(1/ log(1/)) 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/. However, our final weak -nets still have a large size (with the dimension appearing in the exponent of 1/).

AB - Given a set P of n points in Rd and > 0, we consider the problemof constructing weak -nets for P.We show the following: pick a random sample Q of size O(1/ log (1/)) from P. Then, with constant probability, a weak -net of P can be constructed from only the points of Q. This shows that weak -nets in Rd can be computed from a subset of P of size O(1/ log(1/)) 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/. However, our final weak -nets still have a large size (with the dimension appearing in the exponent of 1/).

KW - Combinatorial geometry

KW - Discrete geometry

KW - Hitting convex sets

KW - Weak epsilon nets

UR - http://www.scopus.com/inward/record.url?scp=35348925348&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35348925348&partnerID=8YFLogxK

U2 - 10.1145/1247069.1247113

DO - 10.1145/1247069.1247113

M3 - Conference contribution

AN - SCOPUS:35348925348

SN - 1595937056

SN - 9781595937056

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 239

EP - 244

BT - Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07

T2 - 23rd Annual Symposium on Computational Geometry, SCG'07

Y2 - 6 June 2007 through 8 June 2007

ER -