Parallel implementations of Brunotte’s algorithm

In this paper the author make a comprehensive comparison of different parallelizations of a sequential number theoretic algorithm having large memory requirements. Brunotte’s algorithm is one of the currently known best methods for the decision of the canonical number system (or more generally shift radix system) property. Still, it can be very space-consuming in some cases. Pushing the algorithm to its limits may hopefully shed light on mathematical patterns that would otherwise not be discernible. The algorithm contains many nn-dimensional vector operations and set operations like insert, find, clear, etc. The parallel algorithms encounter two difference kinds of concurrency problems. First, they need computationally intensive arithmetic vector operations, second, the set implementations require a huge amount of memory and general purpose processors. The algorithms described in this article are basically designed for two platforms. The first platform is a generic symmetric multiprocessing (SMP) architecture without any vector processor extension, the second is the Cell Broadband Engine. The SMP platforms have several general purpose processors in contrast with the Cell Broadband Engine where the processors have Synergistic vector processors.

Research highlights
► Parallelizations of Brunnotte algorithm.
► An SMP base parallel version is described.
► A Cell BEA architecture base version is described.
► Proofs for deadlock freedom and correctness.
► Comparison of running times.