# Separations and Equivalences between Turnstile Streaming and Linear Sketching

## TL;DR

This paper explores the relationship between turnstile streaming algorithms and linear sketching, establishing new separations and equivalences under various conditions, and providing explicit algorithms for deterministic cases.

## Contribution

It demonstrates new separations when stream length or maximum value are restricted and removes previous limitations by providing explicit algorithms for deterministic streaming.

## Key findings

- Linear sketching can be exponentially harder than turnstile streaming under certain restrictions.
- Turnstile algorithms are equivalent to linear sketches for arbitrarily long streams with large values.
- An explicit small-space algorithm is constructed for deterministic streaming to compute equivalent modules.

## Abstract

A longstanding observation, which was partially proven in \cite{LNW14,AHLW16}, is that any turnstile streaming algorithm can be implemented as a linear sketch (the reverse is trivially true). We study the relationship between turnstile streaming and linear sketching algorithms in more detail, giving both new separations and new equivalences between the two models. It was shown in \cite{LNW14} that, if a turnstile algorithm works for arbitrarily long streams with arbitrarily large coordinates at intermediate stages of the stream, then the turnstile algorithm is equivalent to a linear sketch. We show separations of the opposite form: if either the stream length or the maximum value of the stream are substantially restricted, there exist problems where linear sketching is exponentially harder than turnstile streaming. A further limitation of the \cite{LNW14} equivalence is that the turnstile sketching algorithm is neither explicit nor uniform, but requires an exponentially long advice string. We show how to remove this limitation for deterministic streaming algorithms: we give an explicit small-space algorithm that takes the streaming algorithm and computes an equivalent module.