Arthur-Merlin Streaming Complexity

TL;DR

This paper explores Arthur-Merlin streaming algorithms, providing a general framework, an application to the Distinct Elements problem with a space-proof tradeoff, and establishing lower bounds for related problems.

Contribution

It introduces a canonical AM streaming algorithm with a space-proof tradeoff and proves new lower bounds for the Distinct Elements and Gap Hamming Distance problems.

Findings

An AM streaming algorithm for Distinct Elements with space-proof tradeoff.

Lower bounds showing s·w ≥ n for Distinct Elements in MA streaming algorithms.

A new Ω(√n) lower bound for MA communication complexity of Gap Hamming Distance.

Abstract

We study the power of Arthur-Merlin probabilistic proof systems in the data stream model. We show a canonical $\mathcal{AM}$ streaming algorithm for a wide class of data stream problems. The algorithm offers a tradeoff between the length of the proof and the space complexity that is needed to verify it. As an application, we give an $\mathcal{AM}$ streaming algorithm for the \emph{Distinct Elements} problem. Given a data stream of length $m$ over alphabet of size $n$, the algorithm uses $\tilde O(s)$ space and a proof of size $\tilde O(w)$, for every $s,w$ such that $s \cdot w \ge n$ (where $\tilde O$ hides a $\polylog(m,n)$ factor). We also prove a lower bound, showing that every $\mathcal{MA}$ streaming algorithm for the \emph{Distinct Elements} problem that uses $s$ bits of space and a proof of size $w$, satisfies $s \cdot w = \Omega(n)$. As a part of the proof of the lower bound for the \emph{Distinct Elements} problem, we show a new lower bound of $\Omega(\sqrt n)$ on the $\mathcal{MA}$ communication complexity of the \emph{Gap Hamming Distance} problem, and prove its tightness.