Automatic generation of modular mappings

Modular mappings have been recently proposed for optimization of algorithms that cannot be efficiently mapped by affine mappings. This paper addresses the problem of generating modular mappings that satisfy conditions for validity and optimality. In general, this is a difficult problem due to the presence of non-linear constraints. Hence, a method of O(n²) complexity is provided to assign values to some entries of a transformation matrix so that non-linear constraints are transformed into linear ones, where n is the dimension of a computation domain. The proposed heuristic attempts to reduce the number of value-assigned entries and exclude as few solutions as possible. This paper also considers the issue of deriving the inverse transformation of a given modular mapping. It identifies a class of modular functions whose inverses result directly from computing the inverse of the (coefficient) matrix used to specify a modular mapping. An efficient method of O(n²) complexity is provided to formulate the problem of generating such modular mappings as an integer linear programming problem