Abstract
This paper takes up a remark in the well-known paper of Alon, Matias, and Szegedy (J. Comput. Syst. Sci. 58(1):137–147, 1999) about the computation of the frequency moments of data streams and shows in detail how any F k with k≥1 can be approximately computed using space O(km 1−1/k(k+log m+log log n)) based on approximate counting. An important building block for this, which may be interesting in its own right, is a new approximate variant of reservoir sampling using space O(log log n) for constant error parameters.
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Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. J. Comput. Syst. Sci. 58(1), 137–147 (1999)
Babcock, B., Babu, S., Datar, M., Motwani, R., Widom, J.: Models and issues in data stream systems. In: Proceedings of 21st PODS, pp. 1–16 (2002)
Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci. 68(4), 702–732 (2004)
Bhuvanagiri, L., Ganguly, S., Kesh, D., Saha, C.: Simpler algorithm for estimating frequency moments of data streams. In: Proceedings of 17th SODA, pp. 708–713 (2006)
Chakrabarti, M., Khot, S., Sun, X.: Near-optimal lower bounds on the multi-party communication complexity of set disjointness. In: Proceedings of 18th Conference on Computational Complexity, pp. 107–117 (2003)
Coppersmith, D., Kumar, R.: An improved data stream algorithm for frequency moments. In: Proceedings of 15th SODA, pp. 151–156 (2004)
Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1991)
Flajolet, P.: Approximate counting: a detailed analysis. In: BIT, pp. 113–134 (1985)
Ganguly, S.: Estimating frequency moments of update streams using random linear combinations. In: Proceedings of 8th RANDOM, pp. 369–380 (2004)
Ganguly, S.: A hybrid technique for estimating frequency moments over data streams, 2004. Manuscript, available at http://www.cse.iitk.ac.in/users/sganguly/HybridFk.pdf
Hofri, M., Kechris, N.: Probabilistic counting of a large number of events—revisited, 1995. Manuscript, available at http://www.cs.wpi.edu/~hofri/
Indyk, P., Woodruff, D.: Optimal approximations of the frequency moments. In: Proceedings of 37th STOC, pp. 202–208 (2005)
Morris, R.: Counting large numbers of events in small registers. Commun. ACM 21(10), 840–842 (1978)
Muthukrishnan, S.: Data streams: algorithms and applications. In: Proceedings of 14th SODA, pp. 413–413 (2003). Online version: http://athos.rutgers.edu/~muthu/stream-1-1.ps
Muthukrishnan, S.: Data streams: Algorithms and Applications. Now Publishers, Hanover (2005)
Vitter, J.S.: Random sampling with a reservoir. ACM Trans. Math. Softw. 11(1), 37–57 (1985)
Wegener, I.: The Complexity of Boolean Functions. Wiley–Teubner, Stuttgart (1987)
Woodruff, D.: Optimal space lower bounds for all frequency moments. In: Proceedings of 15th SODA, pp. 167–175 (2004)
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Gronemeier, A., Sauerhoff, M. Applying Approximate Counting for Computing the Frequency Moments of Long Data Streams. Theory Comput Syst 44, 332–348 (2009). https://doi.org/10.1007/s00224-007-9048-z
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DOI: https://doi.org/10.1007/s00224-007-9048-z