Revisiting Frequency Moment Estimation in Random Order Streams

We revisit one of the classic problems in the data stream literature, namely, that of estimating the frequency moments Fp for 0 < p < 2 of an underlying n-dimensional vector presented as a sequence of additive updates in a stream. It is well-known that using p-stable distributions one can approximate any of these moments up to a multiplicative (1+ε)-factor using O(ε-2 log n) bits of space, and this space bound is optimal up to a constant factor in the turnstile streaming model. We show that surprisingly, if one instead considers the popular random-order model of insertion-only streams, in which the updates to the underlying vector arrive in a random order, then one can beat this space bound and achieve tildeO(ε-2 + log n) bits of space, where the tildeO hides poly(log(1/ε) + log log n) factors. If ε-2 ≈ log n, this represents a roughly quadratic improvement in the space achievable in turnstile streams. Our algorithm is in fact deterministic, and we show our space bound is optimal up to poly(log(1/ε) + log log n) factors for deterministic algorithms in the random order model. We also obtain a similar improvement in space for p = 2 whenever F2 gtrsim log n· F1.