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Algorithmica

, Volume 81, Issue 5, pp 2072–2091 | Cite as

Succinct Summing over Sliding Windows

  • Ran Ben Basat
  • Gil EinzigerEmail author
  • Roy Friedman
  • Yaron Kassner
Article
  • 32 Downloads

Abstract

This paper considers the problem of estimating the sum the last \(W\) elements of a stream of integers in \(\left\{ 0,1,\ldots , R \right\} \). Specifically, we study the memory requirements for computing a \( R W\varepsilon \)-additive approximation for the window’s sum. We derive a lower bound of \(W\log \left\lfloor {\frac{1}{2W\varepsilon } + 1}\right\rfloor \) bits when \(\varepsilon \le 1/2W\) and show a matching succinct algorithm that uses \((1+o(1)) \left( {W\log \left\lfloor {\frac{1}{2W\varepsilon } + 1}\right\rfloor }\right) \) bits. Next, we prove a \((1-o(1)) \varepsilon ^{-1} /2\) bits lower bound when \(\varepsilon =\omega \left( {W^{-1}}\right) \wedge \varepsilon =o(\log ^{-1}W)\) and provide a succinct algorithm that requires \((1+o(1)) \varepsilon ^{-1} /2\) bits. We show that when \(\varepsilon =\varOmega \left( {\log ^{-1}W}\right) \) any solution to the problem must consume at least \((1-o(1))\cdot \left( {{ \varepsilon ^{-1} /2}+\log W}\right) \) bits, while our algorithm needs \((1+o(1))\cdot \left( {{ \varepsilon ^{-1} /2}+2\log W}\right) \) bits. Finally, we show that our lower bounds generalize to randomized algorithms as well, while our algorithms are deterministic and can process elements and answer queries in O(1) worst-case time.

Keywords

Basic summing Counting Sliding window Approximate counting Additive approximation 

Notes

Acknowledgements

We thank Dror Rawitz for helpful comments. This work was partially funded by MOST Grant #3-10886 and the Technion-HPI research school.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceTechnionHaifaIsrael
  2. 2.Nokia Bell LabbsKfar SavaIsrael

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