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Tracking the l_2 Norm with Constant Update Time

Authors Chi-Ning Chou, Zhixian Lei, Preetum Nakkiran

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ACM Subject Classification
  • Theory of computation → Sketching and sampling
Keywords
  • Streaming algorithms
  • Sketching algorithms
  • Tracking
  • CountSketch

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Abstract

The l_2 tracking problem is the task of obtaining a streaming algorithm that, given access to a stream of items a_1,a_2,a_3,... from a universe [n], outputs at each time t an estimate to the l_2 norm of the frequency vector f^{(t)}in R^n (where f^{(t)}_i is the number of occurrences of item i in the stream up to time t). The previous work [Braverman-Chestnut-Ivkin-Nelson-Wang-Woodruff, PODS 2017] gave a streaming algorithm with (the optimal) space using O(epsilon^{-2}log(1/delta)) words and O(epsilon^{-2}log(1/delta)) update time to obtain an epsilon-accurate estimate with probability at least 1-delta. We give the first algorithm that achieves update time of O(log 1/delta) which is independent of the accuracy parameter epsilon, together with the nearly optimal space using O(epsilon^{-2}log(1/delta)) words. Our algorithm is obtained using the Count Sketch of [Charilkar-Chen-Farach-Colton, ICALP 2002].

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Chi-Ning Chou, Zhixian Lei, and Preetum Nakkiran. Tracking the l_2 Norm with Constant Update Time. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.2

Author Details

Chi-Ning Chou
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, USA
Zhixian Lei
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, USA
Preetum Nakkiran
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, USA

Funding

  • Chou, Chi-Ning: Supported by NSF awards CCF 1565264 and CNS 1618026.
  • Lei, Zhixian: Supported by NSF awards CCF 1565264 and CNS 1618026.
  • Nakkiran, Preetum: Work supported in part by a Simons Investigator Award, NSF Awards CCF 1565641 and CCF 1715187, and the NSF Graduate Research Fellowship Grant No. DGE1144152.

Acknowledgements

The authors wish to thank Jelani Nelson for invaluable advice throughout the course of this research. We also thank Mitali Bafna and Jarosław Błasiok for useful discussion and thank Boaz Barak for many helpful comments on an earlier draft of this article. We are also grateful to reviewers' comments.

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