Systematic extraction and implementation of divide-and-conquer parallelism

Abstract

Homomorphisms are functions that match the divide-and-conquer paradigm and thus can be computed in parallel. Two problems are studied for homomorphisms on lists: (1) parallelism extraction: finding a homomorphic representation of a given function; (2) parallelism implementation: deriving an efficient parallel program that computes the function. A systematic approach to parallelism extraction proceeds by generalization of two sequential representations based on traditional cons lists and dual snoc lists. For some non-homomorphic functions, e.g., the maximum segment sum problem, our method provides an embedding into a homomorphism. The implementation is addressed by introducing a subclass of distributable homomorphisms and deriving for them a parallel program schema, which is time optimal on the hypercube architecture. The derivation is based on equational reasoning in the Bird-Meertens formalism, which guarantees the correctness of the parallel target program. The approach is illustrated with function scan (parallel prefix), for which the combination of our two systematic methods yields the “folklore” hypercube algorithm, usually presented ad hoc in the literature.