Abstract
Coset enumeration is a most important procedure for investigating finitely presented groups. We present a practical parallel procedure for coset enumeration on shared memory processors. The shared memory architecture is particularly interesting because such parallel computation is both faster and cheaper. The lower cost comes when the program requires large amounts of memory, and additional CPU’s allow us to lower the time that the expensive memory is being used.
Rather than report on a suite of test cases, we take a single, typical case, and analyze the performance factors in-depth. The parallelization is achieved through a master-slave architecture. This results in an interesting phenomenon, whereby the CPU time is divided into a sequential and a parallel portion, and the parallel part demonstrates a speedup that is linear in the number of processors. We describe an early version for which only 40% of the program was parallelized, and we describe how this was modified to achieve 90% parallelization while using 15 slave processors and a master. In the latter case, a sequential time of 158 seconds was reduced to 29 seconds using 15 slaves.
Supported in part by NSF Grant CCR-9509783.
Supported in part by the Australian Research Council.
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References
S. Akl, G. Labontè, M. Leeder and K. Qiu, “On doing Todd-Coxeter coset enumeration in parallel”, Discrete Applied Math. 34: 27–35 (1991).
R.P. Brent, “The Parallel Evaluation of General Arithmetic Expressions”, J. of ACM 21: 201–208 (1974).
John J. Cannon, Lucien A. Dimino, George Havas and Jane M. Watson, “Implementation and analysis of the Todd-Coxeter algorithm”, Math. Comput. 27: 463–490 (1973).
G. Cooperman, “STAR/MPI: Binding a Parallel Library to Interactive Symbolic Algebra Systems”, Proc. International Symposium on Symbolic and Algebraic Computation (ISSAC ‘85), ACM Press, 1995, pp. 126–132.
G. Cooperman, “TOP-C: A Task-Oriented Parallel C Interface”, 5 th International Symposium on High Performance Distributed Computing (HPDC-5), IEEE Press, 1996, pp. 141–150.
G. Cooperman, L. Finkelstein, B. York and M. Tselman, “Constructing Permutation Representations for Large Matrix Groups”, Proc. International Symposium on Symbolic and Algebraic Computation (ISSAC ‘84), ACM Press, 1994, pp. 134–138.
A. Diaz, E. Kaltofen, K. Schmitz and T. Valente, “A System for Distributed Symbolic Computation”, ISSAC’91 (Proceedings of the 1991 International Symposium on Syvnbolic and Algebraic Computation), ACM Press, 1991, pp. 323–332.
H.W. Gollan, A new existence proof for Ly, the sporadic simple group of R. Lyons, Preprint 30, Institut für Experimentelle Mathematik, Universität GH Essen, 1995.
George Havas, “Coset enumeration strategies”, ISSAC91 (Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation), ACM Press, 1991, pp. 191–199.
George Havas and M.F. Newman, “Applications of computers to questions like those of Burnside”, Burnside Groups, Lecture Notes in Math. 806, Springer, 1980, pp. 211–230.
N. Kajler, “CAS/PI: a Portable and Extensible Interface for Computer Algebra Systems”, Proc. of Internat. Symp. on Symbolic and Algebraic Computation (ISSAC-92), ACM Press, 1992, pp. 376–386.
John Leech, “Coset enumeration”, Computational Group Theory,Academic Press, 1984, pp. 3–18.
J. Neubüser, “An elementary introduction to coset-table methods in computational group theory”, Groups — St Andrews 1981, London Math. Soc. Lecture Note Ser. 71, Cambridge University Press, 1984, pp. 1–45.
M. Schönert et al., GAP — Groups, Algorithms and Programming. Lehrstuhl D für Mathematik, RWTH, Aachen, 1996.
C.C. Sims, “The Existence and Uniqueness of Lyons Group”, Finite Groups ’72, North-Holland, 1973, pp. 138–141.
C.C. Sims, Computation with finitely presented groups, Cambridge University Press, 1994.
J.A. Todd and H.S.M. Coxeter, “A practical method for enumerating cosets of a finite abstract group”, Proc. Edinburgh Math. Soc. 5: 26–34 (1936).
Michael Vaughan-Lee, “Engel-4 groups of exponent 5”, Proc. London Math. Soc. (to appear).
D.A. Wood and M.D. Hill, “Cost-effective parallel computing”, IEEE Computer 28: 69–72 (1995).
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© 1997 Springer-Verlag London
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Cooperman, G., Havas, G. (1997). Practical parallel coset enumeration. In: Cooperman, G., Michler, G., Vinck, H. (eds) Workshop on High Performance Computing and Gigabit Local Area Networks. Lecture Notes in Control and Information Sciences, vol 226. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3540761691_3
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DOI: https://doi.org/10.1007/3540761691_3
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