Estimating hybrid frequency moments of data streams

  • Authors:

  • Sumit Ganguly;Mohit Bansal;Shruti Dube

  • Affiliations:

  • Department of Computer Science and Engineering, Indian Institute of Technology, Kanpur, Kanpur, India 208016;Department of Computer Science and Engineering, Indian Institute of Technology, Kanpur, Kanpur, India 208016 and Department of EECS, University of California, Berkeley, USA;Department of Computer Science and Engineering, Indian Institute of Technology, Kanpur, Kanpur, India 208016 and McKinsey & Company, New Delhi, India

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2012

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Abstract

We consider the problem of estimating hybrid frequency moments of two dimensional data streams. In this model, data is viewed to be organized in a matrix form (A i,j )1驴i,j,驴n . The entries A i,j are updated coordinate-wise, in arbitrary order and possibly multiple times. The updates include both increments and decrements to the current value of A i,j . The hybrid frequency moment F p,q (A) is defined as $\sum_{j=1}^{n}(\sum_{i=1}^{n}{A_{i,j}}^{p})^{q}$ and is a generalization of the frequency moment of one-dimensional data streams.We present the first $\tilde{O}(1)$ space algorithm for the problem of estimating F p,q for p驴[0,2] and q驴[0,1] to within an approximation factor of 1卤驴. The $\tilde{O}$ notation hides poly-logarithmic factors in the size of the stream m, the matrix size n and polynomial factors of 驴 驴1. We also present the first $\tilde{O}(n^{1-1/q})$ space algorithm for estimating F p,q for p驴[0,2] and q驴(1,2].