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. Like the data stream model, the most important parameter for this model is the amount of internal memory used by such an algorithm. The other key parameters are the number of streams the algorithm uses and the number of passes it makes on these streams. We consider how the addition of multiple streams impacts the ability of algorithms to approximate the frequency moments of the input 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 Ω(N ^1−4/k−δ) for any δ > 0. For comparison, it is known that with a single read-only one-pass data stream there is a randomized constant-factor approximation for F_k using Õ(N^1−2/k) space, and that by sorting, which can be done deterministically in O(log N) passes using 3 read/write streams, F_k can be computed exactly. Therefore, although the ability to manipulate multiple read/write streams can add substantial power to the data stream model, with a sublogarithmic number of passes this does not significantly improve the ability to approximate higher frequency moments efficiently. We obtain our results by showing a new connection between the read/write streams model and the multiparty number-in-hand communication model.