We study the problem of 2-dimensional orthogonal range counting with
additive error. Given a set P of n points drawn from an n×n grid
and an error parameter \eps, the goal is to build a data structure, such that
for any orthogonal range R, it can return the number of points in P∩R
with additive error \eps n. A well-known solution for this problem is the
{\em \eps-approximation}, which is a subset A⊆P that can estimate
the number of points in P∩R with the number of points in A∩R. It is
known that an \eps-approximation of size O(\frac{1}{\eps} \log^{2.5}
\frac{1}{\eps}) exists for any P with respect to orthogonal ranges, and the
best lower bound is \Omega(\frac{1}{\eps} \log \frac{1}{\eps}). The
\eps-approximation is a rather restricted data structure, as we are not
allowed to store any information other than the coordinates of the points in
P. In this paper, we explore what can be achieved without any restriction on
the data structure. We first describe a simple data structure that uses
O(\frac{1}{\eps}(\log^2\frac{1} {\eps} + \log n) ) bits and answers queries
with error \eps n. We then prove a lower bound that any data structure that
answers queries with error \eps n must use
\Omega(\frac{1}{\eps}(\log^2\frac{1} {\eps} + \log n) ) bits. Our lower bound
is information-theoretic: We show that there is a collection of
2Ω(nlogn) point sets with large {\em union combinatorial
discrepancy}, and thus are hard to distinguish unless we use Ω(nlogn)
bits.Comment: 19 pages, 5 figure