A Lower Bound for Estimating High Moments of a Data Stream

We show an improved lower bound for the Fp estimation problem in a data stream setting for p > 2. A data stream is a sequence of items from the domain [n] with possible repetitions. The frequency vector x is an n-dimensional non-negative integer vector x such that x(i) is the number of occurrences of i in the sequence. Given an accuracy parameter Ω(n) < ǫ < 1, the problem of estimating the pth moment of frequency is to estimate ‖x‖p = ∑ i∈[n]|x(i)| correctly to within a relative accuracy of 1 ± ǫ with high constant probability in an online fashion and using as little space as possible. The current lower bound for space for this problem is Ω ( nǫ+nǫ/ log(n)+ (ǫ+ log(n)) ) . The first term in the lower bound expression was proved in [2, 3], the second in [6] and the third in [5]. In this note, we show an Ω(pnǫ/ log(n)) bits space bound, for Ω(pn) ≤ ǫ ≤ 1/10.