Synonyms
Lp distances; Lp norms
Definition
Consider a stream (i.e., an ordered list) = a1 , a2 … , an of elements ai ∈ [m]def = {1, 2, … , m}. For i ∈ [m], its frequency fi is the number of times it occurs in . The kth frequency moment Fk of , for real k > 0, is defined to be \( {F}_k({\cal S})={\displaystyle {\sum}_{i\in \left[m\right]}{f}_i^k} \). Interpreting 00 as 0, one can also define F0 this way, so that it equals the number of distinct elements in . Observe that F1 = n is the length of S. In the database community, F2 is known as the repeat rate or Gini’s index of homogeneity. It is also natural to define F∞ = max1 ≤ i ≤ mfi.
It is usually assumed that nis very large and that algorithms which compute the frequency moments do not have enough storage to keep the entire stream in memory. It is also common to assume that they are only given a constant (usually one) number of passes over the data. It is further assumed that the stream is presented in an arbitrary, possibly...
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Recommended Reading
Alon N, Matias Y, Szegedy M. The space complexity of approximating the frequency moments. J Comput Syst Sci. 1999;58(1):137–47.
Bar-Yossef Z, Jayram T, Kumar R, Sivakumar D. An information statistics approach to data stream and communication complexity. In: Proceedings of the 43rd Annual Symposium on Foundations of Computer Science; 2002. p. 209–18.
Indyk P, Woodruff D. Optimal approximations of the frequency moments of data streams. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing; 2005. p. 202–8.
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Woodruff, D. (2018). Frequency Moments. In: Liu, L., Özsu, M.T. (eds) Encyclopedia of Database Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8265-9_167
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