Beating CountSketch for heavy hitters in insertion streams

Given a stream p1,..., pm of items from a universe U, which, without loss of generality we identify with the set of integers {1,2,...,n}, we consider the problem of returning all l₂-heavy hitters, i.e., those items j for which f_j ≥ ϵ√F2, where f_j is the number of occurrences of item j in the stream, and F2 = Σ_i∈[n] f²_i. Such a guarantee is considerably stronger than the l₁-guarantee, which finds those j for which f_j ≥ ϵm. In 2002, Charikar, Chen, and Farach-Colton suggested the CountSketch data structure, which finds all such j using Θ(log² n) bits of space (for constant ϵ > 0). The only known lower bound is Ω(log n) bits of space, which comes from the need to specify the identities of the items found. In this paper we show one can achieve O(log n log log n) bits of space for this problem. Our techniques, based on Gaussian processes, lead to a number of other new results for data streams, including: (1) The first algorithm for estimating F² simultaneously at all points in a stream using only O(log n log log n) bits of space, improving a natural union bound. (2) A way to estimate the l_∞ norm of a stream up to additive error ϵ√F² with O (log n log log n) bits of space, resolving Open Question 3 from the IITK 2006 list for insertion only streams.