# Lower bounds for randomized read/write stream algorithms

## Abstract

Motivated by the capabilities of modern storage architectures, we consider the following generalization of the data stream model where the algorithm has sequential access to multiple streams. Unlike the data stream model, where the stream is read only, in this new model (introduced in [8, 9]) the algorithms can also write onto streams. There is no limit on the size of the streams but the number of passes made on the streams is restricted. On the other hand, the amount of internal memory used by the algorithm is scarce, similar to data stream model. We resolve the main open problem in [7] of proving lower bounds in this model for algorithms that are allowed to have 2-sided error. Previously, such lower bounds were shown only for deterministic and 1-sided error randomized algorithms [9, 7]. We consider the classical set disjointness problem that has proved to be invaluable for deriving lower bounds for many other problems involving data streams and other randomized models of computation. For this problem, we show a near-linear lower bound on the size of the internal memory used by a randomized algorithm with 2-sided error that is allowed to have o(log N/ log logN) passes over the streams. This bound is almost optimal since there is a simple algorithm that can solve this problem using logarithmic memory if the number of passes over the streams is allowed to be O(logN). Applications include near-linear lower bounds on the internal memory for well-known problems in the literature: (1) approximately counting the number of distinct elements in the input (F0); (2) approximating the frequency of the mode of an input sequence (F*8 ); (3) computing the join of two relations; and (4) deciding if some node of an XML document matches an XQuery (or XPath) query. Our techniques involve a novel direct-sum type of argument that yields lower bounds for many other problems. Our results asymptotically improve previously known bounds for any problem even in deterministic and 1-sided error models of computation. Copyright 2007 ACM.