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× 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 εapproximation. 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( ε log 1 ε ) exists for any P with respect to orthogonal ranges, and the best lower bound is Ω( 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( ε log 1 ε log log 1 ε logn) 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 logn) 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.