# The Space Complexity of 2-Dimensional Approximate Range Counting

## William Eric Devanny

(a SODA 2013 paper by by Zhewei Wei and Ke Yi)

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 $R$, the data structure 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}. Informally speaking, an $\eps$-approximation of $P$ is a subset $A\subseteq P$ that allows us to estimate the number of points in $P\cap R$ by counting 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 a subset of points in $P$. In this paper, we explore what can be achieved without any restriction on the data structure. We first describe a data structure that uses $O(\frac{1}{\eps} \log \frac{1} {\eps} \log\log \frac{1}{\eps} \log n)$ bits that answers queries with error $\eps n$. We then prove a lower bound that any data structure that answers queries with error $O(\log n)$ must use $\Omega(n\log n)$ bits. This lower bound has two consequences: 1) answering queries with error $O(\log n)$ is as hard as answering the queries exactly; and 2) our upper bound cannot be improved in general by more than an $O(\log \log \frac{1}{\eps})$ factor.