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Given a stream with frequencies $f_d$, for $d\in[n]$, we characterize thespace necessary for approximating the frequency negative moments $F_p=\sum|f_d|^p$, where $p<0$ and the sum is taken over all items $d\in[n]$ withnonzero frequency, in terms of $n$, $\epsilon$, and $m=\sum |f_d|$. Toaccomplish this, we actually prove a much more general result. Given anynonnegative and nonincreasing function $g$, we characterize the space necessaryfor any streaming algorithm that outputs a $(1\pm\epsilon)$-approximation to$\sum g(|f_d|)$, where again the sum is over items with nonzero frequency. Thestorage required is expressed in the form of the solution to a relativelysimple nonlinear optimization problem, and the algorithm is universal for$(1\pm\epsilon)$-approximations to any such sum where the applied function isnonnegative, nonincreasing, and has the same or smaller space complexity as$g$. This partially answers an open question of Nelson (IITK Workshop Kanpur,2009).

Universal sketches for the frequency negative moments and other decreasing streaming sums