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.Comment: 19 pages, 5 figure

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 x n grid and an error parameter ε, 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 ∩ R with additive error εn. A well-known solution for this problem is the ε-approximation. Informally speaking, an ε-approximation of P is a subset A ⊆ P that allows us to estimate the number of points in P ∩ R by counting the number of points in A ∩ R. It is known that an ε-approximation of size O(1/ε log2.5 1/ε) exists for any P with respect to orthogonal ranges, and the best lower bound is Ω(1/ε log 1/ε). The ε-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(1/ε log 1/ε log log 1/ε log n) bits that answers queries with error εn. We then prove a lower bound that any data structure that answers queries with error O(log n) must use Ω(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 1/ε) factor. Copyright © SIAM