Algorithmic Redistribution Methods for Block-Cyclic Decompositions

This research aims at creating and providing a framework to describe algorithmic redistribution methods for various block cyclic decompositions. To do so properties of this data distribution scheme are formally exhibited. The examination of a number of basic dense linear algebra operations illustrates the application of those properties. This study analyzes the extent to which the general two-dimensional block cyclic data distribution allows for the expression of e cient as well as exible matrix operations. This study also quanti es theoretically and practically how much of the e ciency of optimal block cyclic data layouts can be maintained. The general block cyclic decomposition scheme is shown to allow for the expression of exible basic matrix operations with little impact on the performance and e ciency delivered by optimal and restricted kernels available today. Second, block cyclic data layouts, such as the purely scattered distribution, which seem less promising as far as performance is concerned, are shown to be able to achieve optimal performance and e ciency for a given set of matrix operations. Consequently, this research not only demonstrates that the restrictions imposed by the optimal block cyclic data layouts can be alleviated, but also that e ciency and exibility are not antagonistic features of the block cyclic mappings. These results are particularly relevant to the design of dense linear algebra software libraries as well as to data parallel compiler technology. vi