Generation of injective and reversible modular mappings

A modular mapping consists of a linear transformation followed by modulo operations. It is characterized by a transformation matrix and a vector of moduli, called the modulus vector. Modular mappings are useful to derive parallel versions of algorithms with commutative operations and algorithms intended for execution on processor arrays with toroidal networks. In order to preserve algorithm correctness, modular mappings must be injective. Results of previous work characterize injective modular mappings of rectangular index sets. This paper provides a technique to generate modular mappings that satisfy these injective conditions and extends the results to general index sets. For an n-dimensional rectangular index set, the technique has O(n²n!) complexity. To facilitate generation of efficient code, modular mappings must also be reversible (i.e., have easily described inverses). An O(n²) method is provided to generate reversible modular mappings. This method reduces the search space by fixing entries of the modulus vector while attempting to minimize the number of entries to exclude few solutions. For general index sets defined by linear inequalities, injectivity can be checked by formulating and solving a set of linear inequalities. A modified Fourier-Motzkin elimination is proposed to solve these inequalities. To generate an injective modular mapping of an index set defined by linear inequalities, this paper proposes a technique that attempts to minimize the values of the entries of the modulus vector. Several examples are provided to illustrate the application of the above mentioned methods, including the case of BLAS routines.