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• W2979234330 abstract "One of the oldest problems in the data stream model is to approximate the p th moment (Vert mathbf {X}Vert _p^p = sum _{i=1}^n mathbf {X}_i^p) of an underlying non-negative vector (mathbf {X}in mathbb {R}^n) , which is presented as a sequence of (mathrm{poly}(n)) updates to its coordinates. Of particular interest is when (p in (0,2]) . Although a tight space bound of (Theta (epsilon ^{-2} log n)) bits is known for this problem when both positive and negative updates are allowed, surprisingly, there is still a gap in the space complexity of this problem when all updates are positive. Specifically, the upper bound is (O(epsilon ^{-2} log n)) bits, while the lower bound is only (Omega (epsilon ^{-2} + log n)) bits. Recently, an upper bound of (tilde{O}(epsilon ^{-2} + log n)) bits was obtained under the assumption that the updates arrive in a random order . We show that for (p in (0, 1]) , the random order assumption is not needed. Namely, we give an upper bound for worst-case streams of (tilde{O}(epsilon ^{-2} + log n)) bits for estimating (Vert mathbf {X}Vert _p^p) . Our techniques also give new upper bounds for estimating the empirical entropy in a stream. However, we show that for (p in (1,2]) , in the natural coordinator and blackboard distributed communication topologies, there is an (tilde{O}(epsilon ^{-2})) bit max-communication upper bound based on a randomized rounding scheme. Our protocols also give rise to protocols for heavy hitters and approximate matrix product. We generalize our results to arbitrary communication topologies G , obtaining an (tilde{O}(epsilon ^{2} log d)) max-communication upper bound, where d is the diameter of G . Interestingly, our upper bound rules out natural communication complexity-based approaches for proving an (Omega (epsilon ^{-2} log n)) bit lower bound for (p in (1,2]) for streaming algorithms. In particular, any such lower bound must come from a topology with large diameter." @default.
• W2979234330 created "2019-10-10" @default.
• W2979234330 creator A5024805876 @default.
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• W2979234330 date "2023-06-24" @default.
• W2979234330 modified "2023-09-24" @default.
• W2979234330 title "Towards Optimal Moment Estimation in Streaming and Distributed Models" @default.
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• W2979234330 doi "https://doi.org/10.1145/3596494" @default.
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