Revisiting Norm Estimation in Data Streams
  The problem of estimating the pth moment F_p (p nonnegative and real) in data streams is as follows. There is a vector x which starts at 0, and many updates of the form x_i <-- x_i + v come sequentially in a stream. The algorithm also receives an error parameter 0 < eps < 1. The goal is then to output an approximation with relative error at most eps to F_p = ||x||_p^p.   Previously, it was known that polylogarithmic space (in the vector length n) was achievable if and only if p <= 2. We make several new contributions in this regime, including:   (*) An optimal space algorithm for 0 < p < 2, which, unlike previous algorithms which had optimal dependence on 1/eps but sub-optimal dependence on n, does not rely on a generic pseudorandom generator.   (*) A near-optimal space algorithm for p = 0 with optimal update and query time.   (*) A near-optimal space algorithm for the "distinct elements" problem (p = 0 and all updates have v = 1) with optimal update and query time.   (*) Improved L_2 --> L_2 dimensionality reduction in a stream.   (*) New 1-pass lower bounds to show optimality and near-optimality of our algorithms, as well as of some previous algorithms (the "AMS sketch" for p = 2, and the L_1-difference algorithm of Feigenbaum et al.).   As corollaries of our work, we also obtain a few separations in the complexity of moment estimation problems: F_0 in 1 pass vs. 2 passes, p = 0 vs. p > 0, and F_0 with strictly positive updates vs. arbitrary updates. 
Daniel M. Kane, Jelani Nelson, David P. Woodruff