Parallel Buchberger Algorithms on Virtual Shared Memory KSR1

We develop parallel versions of Buchbergers Gröbner Basis algorithm for a virtual shared memory KSR1 computer. A coarse grain version does S-polynomial reduction concurrently and respects the same critical pair selection strategy as the sequential algorithm. A fine grain version parallelizes polynomial reduction in a pipeline and can be combined with the parallel S-polynomial reduction. The algorithms are designed for a virtual shared memory architecture and a dynamic memory management with concurrent garbage collection implemented in the MAS computer algebra system. We discuss the achieved speedup figures for up to 24 processors on some standard examples.