Arthur-Merlin streaming complexity

Abstract

We study the power of Arthur-Merlin probabilistic proof systems in the data stream model. We show a canonical 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 AM streaming algorithm for the Distinct Elements problem. Given a data stream of length m over alphabet of size n, the algorithm uses Õ(s) space and a proof of size Õ(w), for every s,w such that s·w ≥ n (where Õ hides a polylog(m,n) factor). We also prove a lower bound, showing that every MA streaming algorithm for the Distinct Elements problem that uses s bits of space and a proof of size w, satisfies s·w = Ω(n). As a part of the proof of the lower bound for the Distinct Elements problem, we show a new lower bound of Ω(√n) on the MA communication complexity of the Gap Hamming Distance problem, and prove its tightness. © 2013 Springer-Verlag.