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Separations for Estimating Large Frequency Moments on Data Streams

Authors David P. Woodruff, Samson Zhou

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Author Details

David P. Woodruff
  • Carnegie Mellon University, Pittsburgh, PA, USA
Samson Zhou
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

The authors would like to thank support from NSF grant No. CCF-181584 and a Simons Investigator Award.

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David P. Woodruff and Samson Zhou. Separations for Estimating Large Frequency Moments on Data Streams. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 112:1-112:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.112

Abstract

We study the classical problem of moment estimation of an underlying vector whose n coordinates are implicitly defined through a series of updates in a data stream. We show that if the updates to the vector arrive in the random-order insertion-only model, then there exist space efficient algorithms with improved dependencies on the approximation parameter ε. In particular, for any real p > 2, we first obtain an algorithm for F_p moment estimation using 𝒪̃(1/(ε^{4/p})⋅ n^{1-2/p}) bits of memory. Our techniques also give algorithms for F_p moment estimation with p > 2 on arbitrary order insertion-only and turnstile streams, using 𝒪̃(1/(ε^{4/p})⋅ n^{1-2/p}) bits of space and two passes, which is the first optimal multi-pass F_p estimation algorithm up to log n factors. Finally, we give an improved lower bound of Ω(1/(ε²)⋅ n^{1-2/p}) for one-pass insertion-only streams. Our results separate the complexity of this problem both between random and non-random orders, as well as one-pass and multi-pass streams.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • streaming algorithms
  • frequency moments
  • random order
  • lower bounds

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