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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.531
URN: urn:nbn:de:0030-drops-47217
URL: http://drops.dagstuhl.de/opus/volltexte/2014/4721/
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Braverman, Vladimir ; Katzman, Jonathan ; Seidell, Charles ; Vorsanger, Gregory

An Optimal Algorithm for Large Frequency Moments Using O(n^(1-2/k)) Bits

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Abstract

In this paper, we provide the first optimal algorithm for the remaining open question from the seminal paper of Alon, Matias, and Szegedy: approximating large frequency moments. We give an upper bound on the space required to find a k-th frequency moment of O(n^(1-2/k)) bits that matches, up to a constant factor, the lower bound of Woodruff et. al for constant epsilon and constant k. Our algorithm makes a single pass over the stream and works for any constant k > 3. It is based upon two major technical accomplishments: first, we provide an optimal algorithm for finding the heavy elements in a stream; and second, we provide a technique using Martingale Sketches which gives an optimal reduction of the large frequency moment problem to the all heavy elements problem. We also provide a polylogarithmic improvement for frequency moments, frequency based functions, spatial data streams, and measuring independence of data sets.

BibTeX - Entry

@InProceedings{braverman_et_al:LIPIcs:2014:4721,
  author =	{Vladimir Braverman and Jonathan Katzman and Charles Seidell and Gregory Vorsanger},
  title =	{{An Optimal Algorithm for Large Frequency Moments Using O(n^(1-2/k)) Bits}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{531--544},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Klaus Jansen and Jos{\'e} D. P. Rolim and Nikhil R. Devanur and Cristopher Moore},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2014/4721},
  URN =		{urn:nbn:de:0030-drops-47217},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.531},
  annote =	{Keywords: Streaming Algorithms, Randomized Algorithms, Frequency Moments, Heavy Hitters}
}

Keywords: Streaming Algorithms, Randomized Algorithms, Frequency Moments, Heavy Hitters
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)
Issue Date: 2014
Date of publication: 02.09.2014


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