On Estimating Frequency Moments of Data Streams

Space-economical estimation of the pth frequency moments, defined asFp = Pn i=1|fi| , for p > 0, are of interest in estimating all-pairs distances in a large data matrix [14], machine learning, and in data stream computation. Random sketches formed by the inner product of the frequency vector f1, . . . , fn with a suitably chosen random vector were pioneered by Alon, Matias and Szegedy [1], and have since played a central role in estimating Fp and for data stream computations in general. The concept of p-stable sketches formed by the inner product of the frequency vector with a random vector whose components are drawn from a p-stable distribution, was proposed by Indyk [11] for estimating Fp, for 0 < p < 2, and has been further studied in Li [13]. In this paper, we consider the problem of estimating Fp, for 0 < p < 2. A disadvantage of the stable sketches technique and its variants is that they require O( 1 2 ) inner-products of the frequency vector with dense vectors of stable (or nearly stable [14, 13]) random variables to be maintained. This means that each stream update can be quite time-consuming. We present algorithms for estimating Fp, for 0 < p < 2, that does not require the use of stable sketches or its approximations. Our technique is elementary in nature, in that, it uses simple randomization in conjunction with well-known summary structures for data streams, such as the COUNT-MIN sketch [7] and the COUNTSKETCH structure [5]. Our algorithms require space Õ( 1 2+p ) 3 to estimate Fp to within 1± factors and requires expected time O(logF1 log 1δ ) to process each update. Thus, our technique trades an O( 1 p ) factor in space for much more efficient processing of stream updates. We also present a stand-alone iterative estimator for F1.