On Optimizing a Class of Multi-Dimensional Loops with Reductions for Parallel Execution

This paper addresses the compile-time optimization of a form of nested-loop computation that is motivated by a computational physics application. The computations involve multi-dimensional surface and volume integrals where the integrand is a product of a number of array terms. Besides the issue of optimal distribution of the arrays among the processors, there is also scope for reordering of the operations using the commutativity and associativity properties of addition and multiplication, and the application of the distributive law to significantly reduce the number of operations executed. A formalization of the operation minimization problem and proof of its NPcompleteness is provided. A pruning search strategy for determination of an optimal form is developed. An analysis of the communication requirements and a polynomial-time algorithm for determination of optimal distribution of the arrays are also provided.