Revisiting Frequency Moment Estimation in Random Order Streams
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We revisit one of the classic problems in the data stream literature, namely, that of estimating the frequency moments F_p 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 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 Õ(ϵ^-2 + n) bits of space, where the Õ hides poly((1/ϵ) + n) factors. If ϵ^-2≈ 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((1/ϵ) + n) factors for deterministic algorithms in the random order model. We also obtain a similar improvement in space for p = 2 whenever F_2 ≳ n· F_1.
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