Formal Derivation of Divide-and-Conquer Programs: A Case Study in the Multidimensional FFT's

This paper reports a case study in the development of parallel programs in the Bird-Meertens formalism (BMF), starting from divide-and-conquer algorithm speciications. The contribution of the paper is twofold: (1) we classify divide-and-conquer algorithms and formally derive a parameterized family of parallel implementations for an important subclass of divide-and-conquer, called DH (distributable homomor-phisms); (2) we systematically adjust the mathematical speciication of the Fast Fourier Transform (FFT) to the DH format and thereby obtain a generic SPMD program, well suited for implementation under MPI. The target program includes the eecient FFT solutions used in practice the binary-exchange and the 2D-and 3D-transpose implementations as its special cases. 1. Motivation Formal methods ooer a way to cope with the problems of programming parallel machines. They can be viewed as an alternative to the current practice of low-level programming, when the correctness and performance of programs strongly depend on the program-mer's experience and require a cumbersome process of debugging and testing, followed by the performance monitoring and tuning to a particular architecture. We follow the general approach of transformational program derivation from an intuitively clear and correct speciication via exclusive use of semantically sound rules in a suitable formalism. Our formal base is the Bird-Meertens formalism (BMF), where higher-order functions on lists are used for speciication, and equational reasoning is used for program derivation. More speciically, we take the skeleton-based approach 4] which captures the ubiquitous patterns of paral-lelism in a mathematically rigorous, higherrorder way. The transformational, skeleton-based approach splits the task of parallel programming between two groups of experts: The system experts, rst, identify a set of parallel skeletons which adequately express non-trivial applications and, second, develop provably correct, portable parallel skeleton implementations with a predictable performance. This work requires a deep knowledge of both the design formalism and the target parallel software. The application experts should be shielded from both the parallel machine details and the underlying formalism (in our case BMF). Their knowledge of the application domain is used to do non-trivial changes and optimizations in the process of problem speciication and the adjustment of the speci-cation to the available set of skeletons. In the paper, we present a case study, which demonstrates this rrle distribution in the case of divide-and-conquer algorithms. On the system side, we classify algorithms, which are based on the divide-and-conquer strategy, and identify a subclass of divide-and-conquer functions, called DH (Distributable Homomorphisms). For this class, a …