Proceedings of ISSAC ' 96 ( International Symposium on Symbolic and Algebraic Computation )

We investigate sources of parallelism in the Grr obner Basis algorithm for their practical use on the desk-top. Our execution environment is a standard multi-processor workstation, and our parallel programming environment is PARSAC-2 on top of a multi-threaded operating system. We investigate the performance of two main variants of our master parallel algorithm on a standard set of examples. The rst version exploits only work parallelism in a strategy compliant way. The second version investigates search parallelism in addition, where large super-linear speedups can be obtained. These speedups are due to improved S-polynomial selection behavior and therefore carry over to single processor machines. Since we obtain our parallel variants by a controlled variation of only a few parameters in the master algorithm, we obtain new insights into the way in which different sources of parallelism interact in Grr obner Basis completion. 1 Introduction 1.1 Overall Goals The purpose of our work is to systematically study sources of parallelism in the Grr obner Basis computation which can be prootably exploited in practice on desk-top parallel machines. We try hard to produce parallel software that out-performs established sequential software and that researchers in the eld could and would run on their workstations on a daily basis. While we have not yet fully succeeded in this respect, we have found a form for the Grr obner Basis algorithm which is largely compatible with traditional implementations, including modern selection heuristics and deletion criteria, yet is parallelizable and produces signii-cant speedups on many examples, even when run on a traditional uniprocessor. This work is part of our overall eeort to create PARSAC-2 17], a parallel Computer Algebra library targeted for execution on networks of multi-processor workstations. A comprehensive introduction and motivation for this approach has been given in 19].