An Optimal Algorithm for Large Frequency Moments Using O(n^(1-2/k)) Bits

In this paper, we provide the first optimal algorithm for the remaining open question from the seminal paper of Alon, Matias, and Szegedy: approximating large frequency moments. We give an upper bound on the space required to find a k-th frequency moment of O(n^(1-2/k)) bits that matches, up to a constant factor, the lower bound of Woodruff et. al for constant epsilon and constant k. Our algorithm makes a single pass over the stream and works for any constant k > 3. It is based upon two major technical accomplishments: first, we provide an optimal algorithm for finding the heavy elements in a stream; and second, we provide a technique using Martingale Sketches which gives an optimal reduction of the large frequency moment problem to the all heavy elements problem. We also provide a polylogarithmic improvement for frequency moments, frequency based functions, spatial data streams, and measuring independence of data sets