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 p [superscript th] 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(n [superscript 1 − 2/p] logn) words of space [AKO11,BO10], whereas the lower bound is of Ω(n [superscript 1 − 2/p]) [BJKS04]. In this paper, we show a tight lower bound of Ω(n [superscript 1 − 2/p] logn) 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.