Automatic generation of injective modular mappings

Abstract

Many optimizations (of programs with loops) used in parallelizing compilers and systolic array design are based on linear transformations of loop iteration spaces. Additional important optimizations and designs are possible by using recently proposed modular mappings, which are described by linear transformations modulo a constant vector. Previous work on modular mappings focused an conditions that guarantee injectivity of a modular mapping for algorithms with rectangular index sets. This paper generalizes previous work by providing new injectivity conditions that cover the cases when the program index set has arbitrary shape and size, and the target processor array and the mapping moduli are of arbitrary size. A systematic technique to efficiently generate modular mappings is also proposed. The complexity of the proposed generation technique is O(n/sup 2/n!) for a nested loop of depth n with a rectangular index set and a target processor array with as many processors as required. A bounded search scheme is also provided for general cases. Each trial is formulated as an integer linear programming problem with at most 3n variables.