Streaming space complexity of nearly all functions of one variable on frequency vectors

A central problem in the theory of algorithms for data streams is to determine which functions on a stream can be approximated in sublinear, and especially sub-polynomial or polylogarithmic, space. Given a function g, we study the space complexity of approximating Σn=1 g(|fi|), where f ∈ ℤn is the frequency vector of a turnstile stream. This is a generalization of the well-known frequency moments problem, and previous results apply only when g is monotonic or has a special functional form. Our contribution is to give a condition such that, except for a narrow class of functions g, there is a space-efficient approximation algorithm for the sum if and only if g satisfies the condition. The functions g that we are able to characterize include all convex, concave, monotonic, polynomial, and trigonometric functions, among many others, and is the first such characterization for non-monotonic functions. Thus, for nearly all functions of one variable, we answer the open question from the celebrated paper of Alon, Matias and Szegedy (1996).