Tight Bounds for Adversarially Robust Streams and Sliding Windows via Difference Estimators
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We introduce difference estimators for data stream computation, which provide approximations to F(v)-F(u) for frequency vectors v≽ u and a given function F. We show how to use such estimators to carefully trade error for memory in an iterative manner. The function F is generally non-linear, and we give the first difference estimators for the frequency moments F_p for p∈[0,2], as well as for integers p>2. Using these, we resolve a number of central open questions in adversarial robust streaming and sliding window models. For adversarially robust streams, we obtain a (1+ϵ)-approximation to F_p using 𝒪̃(log n/ϵ^2) bits of space for p∈[0,2] and using 𝒪̃(1/ϵ^2n^1-2/p) bits of space for integers p>2. We also obtain an adversarially robust algorithm for the L_2-heavy hitters problem using 𝒪̃(log n/ϵ^2) bits of space. Our bounds are optimal up to poly(loglog n+log(1/ϵ)) factors, and improve the 1/ϵ^3 dependence of Ben-Eliezer et al. (PODS 2020, best paper award) and the 1/ϵ^2.5 dependence of Hassidim et al. (NeurIPS 2020, oral presentation). For sliding windows, we obtain a (1+ϵ)-approximation to F_p for p∈(0,2], resolving a longstanding question of Braverman and Ostrovsky (FOCS 2007). For example, for p = 2 we improve the dependence on ϵ from 1/ϵ^4 to an optimal 1/ϵ^2. For both models, our dependence on ϵ shows, up to log1/ϵ factors, that there is no overhead over the standard insertion-only data stream model for any of these problems.
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