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With the objective of minimizing the total execution time of a parallel program on a distributed memory parallel computer, this paper discusses the selection of an optimal supernode shape of a supernode transformation (also known as tiling). We identify three parameters of a supernode transformation: supernode size, relative side lengths, and cutting hyperplane directions. For supernode transformations on algorithms with perfectly nested loops and uniform dependencies, we prove the optimality of a constant linear schedule vector and give a necessary and sufficient condition for optimal relative side lengths. We also prove that the total running time is minimized by a cutting hyperplane direction matrix from a particular subset of all valid directions and we discuss the cases where this subset is unique. The results are derived in continuous space and should be considered approximate. Our model does not include cache effects and assumes an unbounded number of available processors, the communication cost approximated by a constant, uniform dependences, and loop bounds known at compile time. A comprehensive example is discussed with an application of the results to the Jacobi algorithm.

With the objective of minimizing the total execution time of a parallel program on a distributed memory parallel computer, this paper discusses how to nd an optimal supernode size and optimal supernode relative side lengths of a supernode transformation (also known as tiling). We identify three parameters of supernode transformation: supernode size, relative side lengths, and cutting hyperplane directions. For algorithms with perfectly nested loops and uniform dependencies, for su ciently large supernodes and number of processors, and for the case where multiple supernodes are mapped to a single processor, we give an order n polynomial whose real positive roots include the optimal supernode size. For two special cases: (1) two dimensional algorithm problems and (2) n-dimensional algorithm problems where the communication cost is dominated by the startup penalty and therefore, can be approximated by a constant, we give a closed form expression for the optimal supernode size, which is independent of the supernode relative side lengths and cutting hyperplanes. For the case where the algorithm iteration index space and the supernodes are hyperrectangular, we give closed form expressions for the optimal supernode relative side lengths. Our experiment shows a good match of the closed form expressions with experimental data.

Most existing methods of mapping algorithms into processor arrays are restricted to the case where n-dimensional algorithms, or algorithms with 71 nested loops, are mapped into ( n -1)-dimensional arrays. However, in practice, it is interesting to map n-dimensional algorithms into ( k -1)-dimensional arrays where I; < 1 1 . For example, many algorithms at bit level are at least four-dimensional (matrix multiplication, convolution, LU decomposition, etc.) and most existing bit level processor arrays are two-dimensional. A computational conflict occurs if two or more computations of an algorithm are mapped into the same processor and the same execution time. In this paper, based on the Hermite normal form of the mapping matrix, necessary and sufficient conditions are derived to identify mappings without computational conflicts. These conditions are used to find time mappings of n-dimensional algorithms into ( k -1)-dimensional arrays, k < t i , without computational conflicts. For some applications, the mapping is time-optimal.Zndex Terms-Bit level algorithm, conflict-free, nested loops, optimal time mapping, processor array.

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