Streaming with Catalytic Memory

Abstract

We introduce a streaming model that uses both catalytic and regular memory. In this model, we show how to exactly compute the frequency moments using a logarithmic number of bits of regular memory and a polynomial number of bits of catalytic memory. More generally, we show how to compute arbitrary polynomials of the item frequencies exactly within the same space bounds. As an application, we obtain catalytic streaming algorithms that exactly compute the number of distinct elements in a stream, count the number of triangles (or any other small subgraph) in a graph whose edges arrive in a stream, and identify heavy hitters.
Our algorithms for frequency moments perform a constant number of passes over the stream, and for polynomial evaluation, we require one more pass than the degree of the polynomial. In particular, for the second moment, we perform three passes over the stream. By relating our catalytic streaming model to the catalytic communication model introduced in [Pyne et al., 2025], we show that catalytic memory is not useful for any one pass streaming algorithms. For lower bounds on multi pass streaming algorithms, the impossibility results of [Pyne et al., 2025] are not strong enough. However, using a different technique, we show that computing the second frequency moment cannot be achieved by a two pass catalytic streaming algorithm that satisfies certain natural assumptions.