Generic Functional Parallel Algorithms: Scan and FFT

Parallel programming, whether imperative or functional, has long focused on arrays as the central data type. Meanwhile, typed functional programming has explored a variety of data types, including lists and various forms of trees. Generic functional programming decomposes these data types into a small set of fundamental building blocks: sum, product, composition, and their associated identities. Definitions over these few fundamental type constructions then automatically assemble into algorithms for an infinite set of data types--some familiar and some new. This paper presents generic functional formulations for two important and well-known classes of parallel algorithms: parallel scan (generalized prefix sum) and Fast Fourier Transform (FFT). Notably, arrays play no role in these formulations. Consequent benefits include a simpler and more compositional style, much use of common algebraic patterns--such as Functor, Applicative, Foldable, and Traversable--and freedom from possibility of run-time indexing errors. The functional generic style also clearly reveals deep commonality among what otherwise appears to be quite different algorithms. Instantiating the generic formulations to "top-down" and "bottom-up" trees as well as "bushes", two well-known algorithms for each of parallel scan and FFT naturally emerge, as well as two possibly new algorithms.