Universal Sketches for the Frequency Negative Moments and Other Decreasing Streaming Sums

Abstract

Given a stream with frequency vector f in n dimensions, we characterize the space necessary for approximating the frequency negative moments Fp, where p<0, in terms of n, the accuracy, and the L_1 length of the vector f. To accomplish this, we actually prove a much more general result. Given any nonnegative and nonincreasing function g, we characterize the space necessary for any streaming algorithm that outputs a (1 +/- eps)-approximation to the sum of the coordinates of the vector f transformed by g. The storage required is expressed in the form of the solution to a relatively simple nonlinear optimization problem, and the algorithm is universal for (1 +/- eps)-approximations to any such sum where the applied function is nonnegative, nonincreasing, and has the same or smaller space complexity as g. This partially answers an open question of Nelson (IITK Workshop Kanpur, 2009).