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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.7
URN: urn:nbn:de:0030-drops-94118
URL: http://drops.dagstuhl.de/opus/volltexte/2018/9411/
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Braverman, Vladimir ; Grigorescu, Elena ; Lang, Harry ; Woodruff, David P. ; Zhou, Samson

Nearly Optimal Distinct Elements and Heavy Hitters on Sliding Windows

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LIPIcs-APPROX-RANDOM-2018-7.pdf (0.6 MB)


Abstract

We study the distinct elements and l_p-heavy hitters problems in the sliding window model, where only the most recent n elements in the data stream form the underlying set. We first introduce the composable histogram, a simple twist on the exponential (Datar et al., SODA 2002) and smooth histograms (Braverman and Ostrovsky, FOCS 2007) that may be of independent interest. We then show that the composable histogram{} along with a careful combination of existing techniques to track either the identity or frequency of a few specific items suffices to obtain algorithms for both distinct elements and l_p-heavy hitters that are nearly optimal in both n and epsilon. Applying our new composable histogram framework, we provide an algorithm that outputs a (1+epsilon)-approximation to the number of distinct elements in the sliding window model and uses O{1/(epsilon^2) log n log (1/epsilon)log log n+ (1/epsilon) log^2 n} bits of space. For l_p-heavy hitters, we provide an algorithm using space O{(1/epsilon^p) log^2 n (log^2 log n+log 1/epsilon)} for 0<p <=2, improving upon the best-known algorithm for l_2-heavy hitters (Braverman et al., COCOON 2014), which has space complexity O{1/epsilon^4 log^3 n}. We also show complementing nearly optimal lower bounds of Omega ((1/epsilon) log^2 n+(1/epsilon^2) log n) for distinct elements and Omega ((1/epsilon^p) log^2 n) for l_p-heavy hitters, both tight up to O{log log n} and O{log 1/epsilon} factors.

BibTeX - Entry

@InProceedings{braverman_et_al:LIPIcs:2018:9411,
  author =	{Vladimir Braverman and Elena Grigorescu and Harry Lang and David P. Woodruff and Samson Zhou},
  title =	{{Nearly Optimal Distinct Elements and Heavy Hitters on Sliding Windows}},
  booktitle =	{Approximation, Randomization, and Combinatorial  Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{7:1--7:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Eric Blais and Klaus Jansen and Jos{\'e} D. P. Rolim and David Steurer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9411},
  URN =		{urn:nbn:de:0030-drops-94118},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.7},
  annote =	{Keywords: Streaming algorithms, sliding windows, heavy hitters, distinct elements}
}

Keywords: Streaming algorithms, sliding windows, heavy hitters, distinct elements
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)
Issue Date: 2018
Date of publication: 02.08.2018


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