A Tight Lower Bound for High Frequency Moment Estimation with Small Error

We show an Ω((n 1 − 2/p logM)/e 2) bits of space lower bound for (1 + e)-approximating the p-th frequency moment \(F_p = \|x\|_p^p = \sum_{i=1}^n |x_i|^p\) of a vector x ∈ { − M, − M + 1, …, M} n with constant probability in the turnstile model for data streams, for any p > 2 and e ≥ 1/n 1/p (we require e ≥ 1/n 1/p since there is a trivial O(n logM) upper bound). This lower bound matches the space complexity of an upper bound of Ganguly for any e 2 and e ≥ 1/n 1/p . This is again optimal for e < 1/log O(1) n.