### Abstract

This paper reports our work on parallelizing an algorithm computing Gröbner bases on a distributed memory parallel machine. When computing Gröbner bases, the efficiency of computation is dominated by the total number of S-polynomials. To decrease the total number of S-polynomials it is necessary to apply a selection strategy that selects the minimum polynomial as a new element of an intermediate base. On a distributed memory parallel machine, as opposed to a shared memory parallel machine, we have to take into account non-trivial communication costs between processors. To reduce such communication costs, it is better to employ coarse grained parallelism rather than fine grained parallelism. We adopt a manager-worker model. S-polynomials are reduced in worker processes in parallel, and the minimum polynomial is selected in the manager process. To implement the selection strategy in this parallel model, synchronization between worker processes is required for every selection of a new element of the intermediate base. However, in spite of synchronization, introducing the selection strategy produces not only a better absolute computation speed but also better speedup with multi-processors. We achieved about 8 times speedup with 64 processors for large problems, T-6 and Ex-17.

Original language | English |
---|---|

Pages (from-to) | 207-222 |

Number of pages | 16 |

Journal | Journal of Symbolic Computation |

Volume | 18 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1994 Sep |

Externally published | Yes |

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### ASJC Scopus subject areas

- Computational Mathematics
- Algebra and Number Theory

### Cite this

*Journal of Symbolic Computation*,

*18*(3), 207-222. https://doi.org/10.1006/jsco.1994.1045

**Parallel Computation of Gröbner Bases on Distributed Memory Machines.** / Sawada, Hiroyuki; Terasaki, Satoshi; Aiba, Akira.

Research output: Contribution to journal › Article

*Journal of Symbolic Computation*, vol. 18, no. 3, pp. 207-222. https://doi.org/10.1006/jsco.1994.1045

}

TY - JOUR

T1 - Parallel Computation of Gröbner Bases on Distributed Memory Machines

AU - Sawada, Hiroyuki

AU - Terasaki, Satoshi

AU - Aiba, Akira

PY - 1994/9

Y1 - 1994/9

N2 - This paper reports our work on parallelizing an algorithm computing Gröbner bases on a distributed memory parallel machine. When computing Gröbner bases, the efficiency of computation is dominated by the total number of S-polynomials. To decrease the total number of S-polynomials it is necessary to apply a selection strategy that selects the minimum polynomial as a new element of an intermediate base. On a distributed memory parallel machine, as opposed to a shared memory parallel machine, we have to take into account non-trivial communication costs between processors. To reduce such communication costs, it is better to employ coarse grained parallelism rather than fine grained parallelism. We adopt a manager-worker model. S-polynomials are reduced in worker processes in parallel, and the minimum polynomial is selected in the manager process. To implement the selection strategy in this parallel model, synchronization between worker processes is required for every selection of a new element of the intermediate base. However, in spite of synchronization, introducing the selection strategy produces not only a better absolute computation speed but also better speedup with multi-processors. We achieved about 8 times speedup with 64 processors for large problems, T-6 and Ex-17.

AB - This paper reports our work on parallelizing an algorithm computing Gröbner bases on a distributed memory parallel machine. When computing Gröbner bases, the efficiency of computation is dominated by the total number of S-polynomials. To decrease the total number of S-polynomials it is necessary to apply a selection strategy that selects the minimum polynomial as a new element of an intermediate base. On a distributed memory parallel machine, as opposed to a shared memory parallel machine, we have to take into account non-trivial communication costs between processors. To reduce such communication costs, it is better to employ coarse grained parallelism rather than fine grained parallelism. We adopt a manager-worker model. S-polynomials are reduced in worker processes in parallel, and the minimum polynomial is selected in the manager process. To implement the selection strategy in this parallel model, synchronization between worker processes is required for every selection of a new element of the intermediate base. However, in spite of synchronization, introducing the selection strategy produces not only a better absolute computation speed but also better speedup with multi-processors. We achieved about 8 times speedup with 64 processors for large problems, T-6 and Ex-17.

UR - http://www.scopus.com/inward/record.url?scp=0001170914&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001170914&partnerID=8YFLogxK

U2 - 10.1006/jsco.1994.1045

DO - 10.1006/jsco.1994.1045

M3 - Article

VL - 18

SP - 207

EP - 222

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

SN - 0747-7171

IS - 3

ER -