Reductions in streaming algorithms, with an application to counting triangles in graphs

We introduce reductions in the streaming model as a tool in the design of streaming algorithms. We develop the concept of list-efficient streaming algorithms that are essential to the design of efficient streaming algorithms through reductions. Our results include a suite of list-efficient streaming algorithms for basic statistical primitives. Using the reduction paradigm along with these tools, we design streaming algorithms for approximately counting the number of triangles in a graph presented as a stream. A specific highlight of our work is the first algorithm for the number of distinct elements in a data stream that achieves arbitrary approximation factors. (Independently, Trevisan [Tre01] has solved this problem via a different approach; our algorithm has the advantage of being list-efficient.)