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Generalizing the Layering Method of Indyk and Woodruff: Recursive Sketches for Frequency-Based Vectors on Streams

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2013, RANDOM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8096))

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Abstract

In their ground-breaking paper, Indyk and Woodruff (STOC 05) showed how to compute the k-th frequency moment F k (for k > 2) in space O(poly − log(n,m) · n\(^{1-{2} \over{k}})\), giving the first optimal result up to poly-logarithmic factors in n and m (here m is the length of the stream and n is the size of the domain.) The method of Indyk and Woodruff reduces the problem of F k to the problem of computing heavy hitters in the streaming manner. Their reduction only requires polylogarithmic overhead in term of the space complexity and is based on the fundamental idea of “layering.” Since 2005 the method of Indyk and Woodruff has been used in numerous applications and has become a standard tool for streaming computations.

We propose a new recursive sketch that generalizes and improves the reduction of Indyk and Woodruff. Our method works for any non-negative frequency-based function in several models, including the insertion-only model, the turnstile model and the sliding window model. For frequency-based functions with sublinear polynomial space complexity our reduction only requires log(c)(n) overhead, where log(c)(n) is the iterative log function. Thus, we improve the reduction of Indyk and Woodruff by polylogarithmic factor. We illustrate the generality of our method by several applications: frequency moments, frequency based functions, spatial data streams and measuring independence of data sets.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Johns Hopkins University, USA

    Vladimir Braverman

  2. Department of Computer Science and Department of Mathematics, University of California Los Angeles, USA

    Rafail Ostrovsky

Authors
  1. Vladimir Braverman
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  2. Rafail Ostrovsky
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Editor information

Editors and Affiliations

  1. University of California, Berkeley, CA, USA

    Prasad Raghavendra

  2. Dept. of computer Science and Engineering, Pennsylvania State University, University Park, PA, USA

    Sofya Raskhodnikova

  3. Institute of Computer Science, University of Kiel, 24118, Kiel, Germany

    Klaus Jansen

  4. University of Geneva, Centre Universitaire d’Informatique, 1227, Carouge, Switzerland

    José D. P. Rolim

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Braverman, V., Ostrovsky, R. (2013). Generalizing the Layering Method of Indyk and Woodruff: Recursive Sketches for Frequency-Based Vectors on Streams. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2013 2013. Lecture Notes in Computer Science, vol 8096. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40328-6_5

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  • DOI: https://doi.org/10.1007/978-3-642-40328-6_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40327-9

  • Online ISBN: 978-3-642-40328-6

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