Component-level parallelization of triangular decompositions

  • Authors:

  • Marc Moreno Maza;Yuzhen Xie

  • Affiliations:

  • ORCCA, University of Western Ontario (UWO) London, Ontario, Canada;ORCCA, University of Western Ontario (UWO) London, Ontario, Canada

  • Venue:
  • Proceedings of the 2007 international workshop on Parallel symbolic computation
  • Year:
  • 2007

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Abstract

We discuss the parallelization of algorithms for solving poly-nomial systems symbolically by way of triangular decompositions. We introduce a component-level parallelism for which the number of processors in use depends on the geometry of the solution set of the input system. Our long term goal is to achieve an efficient multi-level parallelism: coarse grained (component) level for tasks computing geometric objects in the solution sets, and medium/fine grained level for polynomial arithmetic such as GCD/resultant computation within each task. Component-level parallelization of triangular decompositions belongs to the class of dynamic irregular parallel applications, which leads us to address the following question: How to exploit geometrical information at an early stage of the solving process that would be favorable to parallelization? We report on the effectiveness of the approaches that we have applied, including "modular methods", "solving by decreasing order of dimension", "task pool with dimension and rank guided scheduling". We have extended the Aldor programming language to support multiprocessed parallelism on SMPs and realized a preliminary implementation. Our experimentation shows promising speedups for some well-known problems and proves that our component-level parallelization is practically efficient. We expect that this speedup would add a multiplicative factor to the speedup of medium/fine grained level parallelization as parallel GCD and resultant computations.