Universal Streaming of Subset Norms

Most known algorithms in the streaming model of computation aim to approximate a single function such as an $\ell_p$-norm. In 2009, Nelson [\url{https://sublinear.info}, Open Problem 30] asked if it possible to design \emph{universal algorithms}, that simultaneously approximate multiple functions of the stream. In this paper we answer the question of Nelson for the class of \emph{subset $\ell_0$-norms} in the insertion-only frequency-vector model. Given a family of subsets $\mathcal{S}\subset 2^{[n]}$, we provide a single streaming algorithm that can $(1\pm \epsilon)$-approximate the subset-norm for every $S\in\mathcal{S}$. Here, the subset-$\ell_p$-norm of $v\in \mathbb{R}^n$ with respect to set $S\subseteq [n]$ is the $\ell_p$-norm of vector $v_{|S}$ (which denotes restricting $v$ to $S$, by zeroing all other coordinates). Our main result is a near-tight characterization of the space complexity of every family $\mathcal{S}\subset 2^{[n]}$ of subset-$\ell_0$-norms in insertion-only streams, expressed in terms of the "heavy-hitter dimension" of $\mathcal{S}$, a new combinatorial quantity that is related to the VC-dimension of $\mathcal{S}$. In contrast, we show that the more general turnstile and sliding-window models require a much larger space usage. All these results easily extend to $\ell_1$. In addition, we design algorithms for two other subset-$\ell_p$-norm variants. These can be compared to the Priority Sampling algorithm of Duffield, Lund and Thorup [JACM 2007], which achieves additive approximation $\epsilon\|{v}\|$ for all possible subsets ($\mathcal{S}=2^{[n]}$) in the entry-wise update model. One of our algorithms extends this algorithm to handle turnstile updates, and another one achieves multiplicative approximation given a family $\mathcal{S}$.