\Re{R} \eps{\varepsilon} \Net{S} \tldO{{O}} \body{C} We revisit the problem of building weak%DIFDELCMD < \eps%%% -nets for convex ranges over a point set in \Re^d. Unfortunately, the known constructions of weak%DIFDELCMD < \eps%%% -nets yields sets that are of size \Omega(%DIFDELCMD < \eps%%% ^{-d e^{c d^2} ) }%DIFDELCMD < \Bigr.%%% , where c is some constant. We offer two alternative schemes that yield significantly smaller sets, and two related results, as follows: (A)Let%DIFDELCMD < \Net %%% be a sample of size ] \tldO(d ^2%DIFDELCMD < \eps%%% ^{-1 , where \tldO hides polylogarithmic terms. Given a convex body%DIFDELCMD < \body%%% , via a separation oracle, the algorithm performs a small sequence of (oracle) stabbing queries (computed from%DIFDELCMD < \Net%%% ) -- if none of the query points hits%DIFDELCMD < \body%%% , then%DIFDELCMD < \body %%% contains less than an%DIFDELCMD < \eps%%% -fraction of the input points. The number of stabbing queries performed is O( d^2\log%DIFDELCMD < \eps%%% ^{-1 \tldO(d ^9%DIFDELCMD < \eps%%% ^{-1 is the first weak%DIFDELCMD < \eps%%% -net related construction where all constants/bounds are polynomial in the dimension. (B) If one is allowed to expand the convex range before checking if it intersects the sample, then a sample of size%DIFDELCMD < \Bigl.\tldO%%% (%DIFDELCMD < \eps%%% ^{-(d+1)/2), from the original point set, is sufficient to form a net. (C) We show a construction of weak }%DIFDELCMD < \eps%%% -nets which have the following additional property: For a heavy body, there is a net point that stabs the body, and it is also a good centerpoint for the points contained inside the body. (D) We present a variant of a known algorithm for approximating a centerpoint, improving the running time from%DIFDELCMD < \tldO%%% (d^9) to%DIFDELCMD < \tldO%%% (d^7). Our analysis of this algorithmis arguably cleaner than the previous version .
\Re{R} \eps{\varepsilon} \Net{S} \tldO{{O}} \body{C} We revisit %DIFDELCMD < \eps%%% %DIFDELCMD < \eps%%% %DIFDELCMD < \eps%%% e^{c d^2} ) }%DIFDELCMD < \Bigr.%%% %DIFDELCMD < \Net %%% an algorithm of Clarkson etal CEMST96], that computes (roughly) a 1/(4d^2)-centerpoint in \tldO(d %DIFDELCMD < \eps%%% ^9) time, for a point set in \Re^d , where \tldO hides polylogarithmic terms. %DIFDELCMD < \body%%% %DIFDELCMD < \Net%%% %DIFDELCMD < \body%%% %DIFDELCMD < \body %%% %DIFDELCMD < \eps%%% %DIFDELCMD < \eps%%% We present an improved algorithm that computes (roughly) a 1/d^2-centerpoint with running time \tldO(d %DIFDELCMD < \eps%%% ^7). While the improvements are (arguably) mild, it is the first %DIFDELCMD < \eps%%% %DIFDELCMD < \Bigl.\tldO%%% %DIFDELCMD < \eps%%% ), from the original point set, is sufficient to form a net. (C) We show a construction of weak }%DIFDELCMD < \eps%%% %DIFDELCMD < \tldO%%% %DIFDELCMD < \tldO%%% progress on this well known problem in over twenty years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm .
D
the problem of building weak
D
-nets for convex ranges over a point set in \Re^d. Unfortunately, the known constructions of weak
D
-nets yields sets that are of size \Omega(
D
^{-d
D
, where c is some constant. We offer two alternative schemes that yield significantly smaller sets, and two related results, as follows: (A)Let
R
be a sample of size
an algorithm of Clarkson etal
A
CEMST96
A
, that computes (roughly) a 1/(4d^2)-centerpoint in
D
^2
R
^{-1
^9) time, for a point set in \Re^d
D
Given a convex body
D
, via a separation oracle, the algorithm performs a small sequence of (oracle) stabbing queries (computed from
D
) -- if none of the query points hits
D
, then
D
contains less than an
D
-fraction of the input points. The number of stabbing queries performed is O( d^2\log
R
^{-1
We present an improved algorithm that computes (roughly) a 1/d^2-centerpoint with running time
D
^9
R
^{-1
^7). While the improvements are (arguably) mild, it
D
weak
D
-net related construction where all constants/bounds are polynomial in the dimension. (B) If one is allowed to expand the convex range before checking if it intersects the sample, then a sample of size
D
(
D
^{-(d+1)/2
D
-nets which have the following additional property: For a heavy body, there is a net point that stabs the body, and it is also a good centerpoint for the points contained inside the body. (D) We present a variant of a known algorithm for approximating a centerpoint, improving the running time from
D
(d^9) to
R
(d^7). Our analysis of this algorithmis arguably cleaner than the previous version
progress on this well known problem in over twenty years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm