Surrounding Theorem: Developing Parallel Programs for Matrix-Convolutions

Abstract

Computations on two-dimensional arrays such as matrices and images are one of the most fundamental and ubiquitous things in computational science and its vast application areas, but development of efficient parallel programs on two-dimensional arrays is known to be hard. To solve this problem, we have proposed a skeletal framework on two-dimensional arrays based on the theory of constructive algorithmics. It supports users, even with little knowledge about parallel machines, to develop systematically both correct and efficient parallel programs on two-dimensional arrays. In this paper, we apply our framework to the matrix-convolutions often used in image filters and difference methods. We show the efficacy of the framework by giving a general parallel program for the matrix-convolutions described with the skeletons, and a theorem that optimizes the general program into an application-specific one.

Map, reduce and mapreduce, the skeleton way

“…List homomorphism have nice compositional properties, can be efficiently implemented in parallel and it can be shown that the useful functions on lists can be expressed as homomorphism [13]. These properties also hold in case of homomorphisms defined on different data structures, such as trees or arrays [14]. Map and reduce can be defined themselves as homomorphisms.…”