We study the power of Arthur-Merlin probabilistic proof systems in the data stream model. We show a canonical @math 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 @math streaming algorithm for the problem. Given a data stream of length @math over alphabet of size @math , the algorithm uses @math space and a proof of size @math , for every @math such that @math (where @math hides a @math factor). We also prove a lower bound, showing that every @math streaming algorithm for the problem that uses @math bits of space and a proof of size @math , satisfies @math . As a part of the proof of the lower bound for the problem, we show a new lower bound of @math on the @math communication complexity of the problem, and prove its tightness.
The frequency moments of a sequence containing mi elements of type i, for 1 i n, are the numbers Fk = P n=1 m k . We consider the space complexity of randomized algorithms that approximate the numbers Fk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbers F0;F1 and F2 can be approximated in logarithmic space, whereas the approximation of Fk for k 6 requires n (1) space. Applications to data bases are mentioned as well.