The Space Complexity of 2-Dimensional Approximate Range Counting

We study the problem of $2$-dimensional orthogonal range counting with additive error. Given a set $P$ of $n$ points drawn from an $n\times n$ 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 $P\cap R$ with additive error $\eps n$. A well-known solution for this problem is the {\em $\eps$-approximation}, which is a subset $A\subseteq P$ that can estimate the number of points in $P\cap R$ with the number of points in $A\cap R$. 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 $2^{{\Omega(n\log} n)}$ point sets with large {\em union combinatorial discrepancy}, and thus are hard to distinguish unless we use $\Omega(n\log n)$ bits.