A Systematic Framework for the Dependence Cycle Removal in Practical Loops

Abstract

Traditionally research in dependence testing considered the most general linear subscripted DO loops permitted by standard Fortran. However, the existing program transformations are discussed in isolation without developing a unified and formal theory for the most general loops. Consequently all the available parallelism in the serial programs is not exploited by the existing techniques; for example, the existing techniques can neither completely remove the output dependence cycles nor recognize all types of reduction loops. In this paper, based on the empirical studies of Goff et al. ( Proc. ACM SIGPLAN ′91 Conference on Programming Language Design and Implementation, Toronto, Ontario, Canada, June 26-28, 1991, pp. 15-29), first we consider a DO loop model which encompasses 97% of the practical Fortran DO loops. By analyzing their dependence characteristics, we develop the formal theory for the application of various parallelizing transformations, specifically the dependence cycle breaking transformations. By this formalization, we contribute the following useful results: (1) define the scope of the existing cycle breaking techniques and illustrate the shortcomings in them; (2) present the dependence characteristics of reduction loops, which will help in designing general reduction loop recognition algorithms in contrast to the existing pattern matching techniques; and (3) propose a new transformation called node replication and prove that by applying this transformation all the output dependence cycles can be completely removed. Finally, we discuss how the most general linear subscripted DO loops can be converted into the above-mentioned loop model, showing that the results presented in this paper are applicable to all the linear subscripted DO loops permitted by standard Fortran.