License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2016.11
URN: urn:nbn:de:0030-drops-60241
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6024/
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Ben Basat, Ran ; Einziger, Gil ; Friedman, Roy ; Kassner, Yaron

Efficient Summing over Sliding Windows

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LIPIcs-SWAT-2016-11.pdf (0.6 MB)

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.

BibTeX - Entry

@InProceedings{benbasat_et_al:LIPIcs:2016:6024,
  author =	{Ran Ben Basat and Gil Einziger and Roy Friedman and Yaron Kassner},
  title =	{{Efficient Summing over Sliding Windows}},
  booktitle =	{15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)},
  pages =	{11:1--11:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-011-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{53},
  editor =	{Rasmus Pagh},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6024},
  URN =		{urn:nbn:de:0030-drops-60241},
  doi =		{10.4230/LIPIcs.SWAT.2016.11},
  annote =	{Keywords: Streaming, Statistics, Lower Bounds}
}

Keywords: Streaming, Statistics, Lower Bounds
Collection: 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)
Issue Date: 2016
Date of publication: 22.06.2016

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