Universal Streaming of Subset Norms
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Most known algorithms in the streaming model of computation aim to approximate a single function such as an ℓ_p-norm. In 2009, Nelson [<https://sublinear.info>, Open Problem 30] asked if it possible to design universal algorithms, that simultaneously approximate multiple functions of the stream. In this paper we answer the question of Nelson for the class of subset ℓ_0-norms in the insertion-only frequency-vector model. Given a family of subsets S⊂ 2^[n], we provide a single streaming algorithm that can (1±ϵ)-approximate the subset-norm for every S∈S. Here, the subset-ℓ_p-norm of v∈R^n with respect to set S⊆ [n] is the ℓ_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 S⊂ 2^[n] of subset-ℓ_0-norms in insertion-only streams, expressed in terms of the "heavy-hitter dimension" of S, a new combinatorial quantity that is related to the VC-dimension of 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 ℓ_1. In addition, we design algorithms for two other subset-ℓ_p-norm variants. These can be compared to the Priority Sampling algorithm of Duffield, Lund and Thorup [JACM 2007], which achieves additive approximation ϵv for all possible subsets (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 S.
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