Optimality of Linear Sketching under Modular Updates

TL;DR

This paper establishes a strong connection between streaming algorithms and linear sketching for binary updates, showing that efficient algorithms in one domain imply efficiency in the other, with improvements over previous exponential bounds.

Contribution

It demonstrates that efficient streaming algorithms with many updates lead to efficient linear sketching algorithms, extending previous results to modular updates and approximate computations.

Findings

Efficient streaming algorithms imply efficient linear sketching algorithms.

Improved bounds over previous work requiring triple-exponential updates.

Extended results to modular arithmetic and approximation scenarios.

Abstract

We study the relation between streaming algorithms and linear sketching algorithms, in the context of binary updates. We show that for inputs in $n$ dimensions, the existence of efficient streaming algorithms which can process $\Omega(n^2)$ updates implies efficient linear sketching algorithms with comparable cost. This improves upon the previous work of Li, Nguyen and Woodruff [LNW14] and Ai, Hu, Li and Woodruff [AHLW16] which required a triple-exponential number of updates to achieve a similar result for updates over integers. We extend our results to updates modulo $p$ for integers $p \ge 2$, and to approximation instead of exact computation.