On Generalizing Divide and Conquer Parallel Programming Pattern

Abstract

(1) Background: Structuring is important in parallel programming in order to master its complexity, and this structuring could be achieved through programming patterns and skeletons. Divide-and-conquer computation is essentially defined by a recurrence relation that links the solution of a problem to the solutions of subproblems of the same type, but of smaller sizes. This pattern allows the specification of different types of computations, and so it is important to provide a general specification that comprises all its cases. We intend to prove that the divide-and-conquer pattern could be generalized such that to comprise many of the other parallel programming patterns, and in order to prove this, we provide a general formulation of it. (2) Methods: Starting from the proposed generalized specification of the divide-and-conquer pattern, the computation of the pattern is analyzed based on its stages: decomposition, base-case and composition. Examples are provided, and different execution models are analyzed. (3) Results: a general functional specification is provided for a divide-and-conquer pattern and based on it, and we prove that this general formulation could be specialized through parameters’ instantiating into other classical parallel programming patterns. Based on the specific stages of the divide-and-conquer, three classes of computations are emphasized. In this context, an equivalent efficient bottom-up computation is formally proved. Associated models of executions are emphasized and analyzed based on the three classes of divide-and-conquer computations. (4) Conclusion: A more general definition of the divide-and-conquer pattern is provided, and this includes an arity list for different decomposition degrees, a level of recursion, and also an alternative solution for the cases that are not trivial but allow other approaches (sequential or parallel) that could lead to better performance. Together with the associated analysis of patterns equivalence and optimized execution models, this provides a general formulation that is useful both at the semantic level and implementation level.