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

Given a stream with frequencies fd, for d ∈ [n], we characterize the space necessary for approximating the frequency negative moments Fp = ∑ |fd|, where p < 0 and the sum is taken over all items d ∈ [n] with nonzero frequency, in terms of n, , andm = ∑ |fd|. 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± )-approximation to ∑ g(|fd|), where again the sum is over items with nonzero frequency. 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± )-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). 1998 ACM Subject Classification F.1.2 Models of Computation, F.2.1 Numerical Algorithms and Problems