M assively parallel distributed-memory multicomputers can achieve the high performance levels required to solve the Grand Challenge computational science problems (a class of computational applications, identified by the 1992 US Presidential Initiative in High-Performance Computing and Communications, that would require a significant increase in computing power). Multicomputers such as the Intel Paragon, the IBM SP-l/SP-2 (Scalable PowerParallel 1 and 2) and the Thinking Machines CM-5 (Connection Machine 5) offer significant cost and scalability advantages over shared-memory multiprocessors. However, to harness these machines' computational power, users must write efficient software. This process is laborious because of the absence of global address space. The programmer must manually distribute computations and data across processors and explicitly manage communication. The Paradigm (Parallelizing Compiler for Distributed-Memory, General-Purpose Multicomputers) project at the University of Illinois addresses this problem by developing automatic methods for efficient parallelization of sequential programs.

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The PARADIGM compiler project provides an automated means to parallelize programs, written in a serial programming model, for efficient execution on distributed-memory muiticomputers. A previous implementation of the compiler based on the PTD representation allowed symbolic array sizes, affine loop bounds and array subscripts, and variable number of processors, provided that arrays were singleor multi-dimensionally block distributed. The techniques .presented here extend the compiler to also accept multidimensional cyclic and block-cyclic distributions within a uniform symbolic framework. These extensions demand more sophisticated symbolic manipulation capabilities. A novel aspect of our approach is to meet this demand by interfacing PARADIGM with a powerful off-the-shelf symbolic package. Mathematical™. This paper describes some of the Mathematical™ routines that performs various transformations, shows how they are invoked and used by the compiler . to overcome the new challenges, and presents experimental results for code involving cyclic and block-cyclic arrays as evidence of the feasibility of the approach.

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