On the Value of Multiple Read/Write Streams for Approximating Frequency Moments

We consider the read/write streams model, an extension of the standard data stream model in which an algorithm can create and manipulate multiple read/write streams in addition to its input data stream. We show that any randomized read/write stream algorithm with a fixed number of streams and a sublogarithmic number of passes that produces a constant factor approximation of the k-th frequency moment F_k of an input sequence of length of at most N from {1, ..., N} requires space Omega(N^1-4/k-delta) for any delta > 0. For comparison, it is known that with a single read-only data stream there is a randomized constant- factor approximation for F_k using O(N^1-2/k) space and that there is a deterministic algorithm computing F_k exactly using 3 read/write streams, O(log N) passes, and O(log N) space. Therefore, although the ability to manipulate multiple read/write streams can add substantial power to the data stream model, with a sub-logarithmic number of passes this does not significantly improve the ability to approximate higher frequency moments efficiently. Our lower bounds also apply to (1 + epsi)-approximations of F_k for epsi ges 1/N.