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.
Categories: cs.CC cs.DS
