Optimality of Linear Sketching under Modular Updates

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.