Fast moment estimation in data streams in optimal space

We give a space-optimal streaming algorithm with update time O(log 2(1/ε)loglog(1/ε)) for approximating the pth frequency moment, 0 < p < 2, of a length-n vector updated in a data stream up to a factor of 1 ± ε. This provides a nearly exponential improvement over the previous space optimal algorithm of [Kane-Nelson-Woodruff, SODA 2010], which had update time Ω(1/ε2). When combined with the work of [Harvey-Nelson-Onak, FOCS 2008], we also obtain the first algorithm for entropy estimation in turnstile streams which simultaneously achieves near-optimal space and fast update time. © 2011 ACM.