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Efficient Summing over Sliding Windows

Authors Ran Ben Basat, Gil Einziger, Roy Friedman, Yaron Kassner

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Ran Ben Basat
Gil Einziger
Roy Friedman
Yaron Kassner

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Ran Ben Basat, Gil Einziger, Roy Friedman, and Yaron Kassner. Efficient Summing over Sliding Windows. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.SWAT.2016.11

Abstract

This paper considers the problem of maintaining statistic aggregates over the last W elements of a data stream. First, the problem of counting the number of 1's in the last W bits of a binary stream is considered. A lower bound of Omega(1/epsilon + log(W)) memory bits for Wepsilon-additive approximations is derived. This is followed by an algorithm whose memory consumption is O(1/epsilon + log(W)) bits, indicating that the algorithm is optimal and that the bound is tight. Next, the more general problem of maintaining a sum of the last W integers, each in the range of {0, 1, ..., R}, is addressed. The paper shows that approximating the sum within an additive error of RW epsilon can also be done using Theta(1/epsilon + log(W)) bits for epsilon = Omega(1/W). For epsilon = o(1/W), we present a succinct algorithm which uses B(1 + o(1)) bits, where B = Theta(W*log(1/(W*epsilon))) is the derived lower bound. We show that all lower bounds generalize to randomized algorithms as well. All algorithms process new elements and answer queries in O(1) worst-case time.
Keywords
  • Streaming
  • Statistics
  • Lower Bounds

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