Tight lower bounds for the distinct elements problem

We prove strong lower bounds for the space complexity of (epsilon, delta)-approximating the number of distinct elements F-0 in a data stream. Let m be the size of the universe from which the stream elements are drawn. We show that any one-pass streaming algorithm for (epsilon, delta)-approximating F-0 must use Omega (1/epsilon(2)) space when epsilon = Omega (m(-1/9+k)). for any k > 0, improving upon the known lower bound of Omega (1/epsilon)for this range of epsilon. This lower bound is tight up to a factor of log log m for small epsilon and log 1/epsilon for large epsilon. Our lower bound is derived from a reduction from the one-way communication complexity of approximating a booleanfunction in Euclidean space. The reduction makes use of a low-distortion embedding from an l(2) to an l(1) norm.