Tight lower bound for linear sketches of moments

Abstract

The problem of estimating frequency moments of a data stream has attracted a lot of attention since the onset of streaming algorithms [AMS99]. While the space complexity for approximately computing the pth moment, for p ∈ (0,2] has been settled [KNW10], for p > 2 the exact complexity remains open. For p > 2 the current best algorithm uses O(n1-2/p log n) words of space [AKO11,BO10], whereas the lower bound is of Ω(n 1-2/p) [BJKS04]. In this paper, we show a tight lower bound of Ω(n1-2/p log n) words for the class of algorithms based on linear sketches, which store only a sketch Ax of input vector x and some (possibly randomized) matrix A. We note that all known algorithms for this problem are linear sketches. © 2013 Springer-Verlag.