C. G. Ponder, Evaluation of "Performance Enhancements" in Algebraic Manipulation Systems, University of California at Berkeley Computer Science Report UCB/CSD 88/438, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....computation is summarized. Buchberger [7] first proposed the parallel L Machine to compute GrSbner basis in parallel. His idea was based on manager worker approach. This scheme was not implemented but the critical pair set manager can become a bottleneck in this type of computation [27] Ponder [15] proposed two corrected parallel algorithms which were inherently less parallel because of the serialization in keeping the basis set reduced. Senechaud [20, 21, 22] developed parallel algorithms for a ring of processors and a hypercube of processors. For large enough data set (sequential ....

C. Ponder. Evaluation of "Performance Enhancements" in Algebraic Manipulation Systems. In Della Dora and Fitch, editors, Computer Algebra and Parallelism, pages 51 73. Academic Press, 1989.

....by any other polynomial in G. Buchberger [49] gives an algorithm which computes a reduced Grobner basis. The computation has two phases: the first produces a Grobner basis G and the second produces the reduced basis from G. There have been various attempts at parallelizing Buchberger s algorithm [194, 61, 225], but our aim is slightly different. We want to produce a reasonably efficient parallel program using existing software components. Since our focus is not on algorithm design, we adopt the same approaches as others, in parallelizing only the first phase which produces the Grobner basis. 4.7.1 ....

Carl Glen Ponder. Evaluation of "Performance Enhancements" in Algebraic Manipulation Systems. PhD thesis, University of California at Berkeley, 1988. Also available as Report No. UCB/CSD 88/438, August, 1988, Computer Science Division (EECS), University of California, Berkeley, CA 94720.

....computation is summarized. Buchberger [7] first proposed the parallel L Machine to compute Grobner basis in parallel. His idea was based on manager worker approach. This scheme was not implemented but the critical pair set manager can become a bottleneck in this type of computation [27] Ponder [15] proposed two corrected parallel algorithms which were inherently less parallel because of the serialization in keeping the basis set reduced. Senechaud [20, 21, 22] developed parallel algorithms for a ring of processors and a hypercube of processors. For large enough data set (sequential ....

C. Ponder. Evaluation of "Performance Enhancements" in Algebraic Manipulation Systems. In Della Dora and Fitch, editors, Computer Algebra and Parallelism, pages 51--73. Academic Press, 1989.

....Many approaches have already been performed to parallelize Groebner Basis computation . The parallelization of the (non factoring) Groebner Basis computation has attracted much attention, going back to the mid 1980 s [Wat86, Buc87] However, it has posed considerable difficulties in practice [Pon89] Firstly, as Groebner Basis completion both creates new polynomials and deletes old polynomials, there is much demand for synchronization. Vidal [Vid90] Melenk and Neun [MN89] were among the first who used shared memory parallelism and achieved speed ups in practice; their work was combined by ....

Carl G. Ponder. Evaluation of "performance enhancements" in algebraic manipulation systems. In Della Dora and Fitch [DDF89], pages 51--73. (Proc. CAP'88, Grenoble, France, June 1988).

....method (proposed by [ Watt, 1986 ] divides the process into alternating phases of S polynomial computation and reduction of the basis, using parallelism in each stage. But Watt s implementation of the idea was incorrect. Two corrected versions of Watt s version of the algorithm are presented in [ Ponder, 1988 ] but does not show any speed up. In fact Buchberger himself sketched a parallel version of his algorithm in an early paper [ Buchberger, 1987 ] using an central manager. An implementation of this method using an Shared memory machine was presented in [ Vidal, 1989 ] showing that it is possible to ....

C. G. Ponder. Evaluation of "Performance Enhancements" in Algebraic Manipulation Systems. Technical Report UCB/CSD 88/438, Computer Science Division (EECS), University of California, August 1988.

....stage. Unfortunately Watt s implementation of the idea was incorrect. Two corrected versions of Watt s Supported by: Austrian Science Foundation (FWF) Proj. Parallel Symbolic Computation, No S5302 PHY, Proj. POSSO, No. 9181 TEC (ESPRIT BRA No. 6846) version of the algorithm are presented in [ Ponder, 1988 ] but they do not show any speed up. In fact Buchberger himself sketched a parallel version of his algorithm in an early paper [ Buchberger, 1987 ] using a central manager. An implementation of this method using a shared memory machine was presented in [ Vidal, 1989 ] showing that it is possible ....

C. G. Ponder. Evaluation of "Performance Enhancements" in Algebraic Manipulation Systems. Technical Report UCB/CSD 88/438, Computer Science Division (EECS), University of California, August 1988.

....as it is done in Watt s basis reduction phase. A basis polynomial that is used to reduce other ones, can not be reduced by other processes simultaneously. Thus the reductions of the basis has some implicit serialization. Two corrected versions of Watt s version of the algorithm are presented in [Ponder, 1988]: 4.4.2 Parallel Reduction of the S polynomials One corrected method uses parallelism only during the first phase of Watt s algorithm. The reduction of the basis is done in serial afterwards. CHAPTER 4. PARALLEL GR OBNER BASES ALGORITHM 50 Problems with this method are: ffl Reducing the basis ....

....the basis in serial will add a big serial phase to the process, limiting the effect of parallelization. ffl The method forces an (unnecessary) synchronization after each parallel phase. ffl Each new S polynomial might reduce the basis such that other S polynomial operations are redundant. ffl [Ponder, 1988] does not apply the criteria for detecting unnecessary Spolynomials. This will increase the achievable parallelism but compared with a good sequential form involving the criteria the performance will be quite bad. ffl With the application of the criteria, the average number of S polynomials which ....

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C. G. Ponder. Evaluation of "Performance Enhancements" in Algebraic Manipulation Systems. Technical Report UCB/CSD 88/438, Computer Science Division (EECS), University of California, August 1988.

....the consistency requirement on the basis is rather lax: a processor can do significant amounts of useful work while having an incomplete or even inconsistent copy of the basis. Allowing Inconsistent Copies An example of a consistency problem that may occur is the following race condition [16]. Suppose processors P 1 and P 2 both have copies of polynomials g; h, which happen to be equal. InterReduce fires on P 1 and P 2 . Say g is reduced by h to 0 on P 1 . Processor P 2 does not modify its copy of g, instead it reduces h by g to 0. Subsequent invalidation messages lead both ....

C. G. Ponder. Evaluation of "performance enhancements" in algebraic manipulation systems. Technical Report UCB/CSD 88/438, University of California, Berkeley, 1988. Chapter 7, Parallel Algorithms for Grobner Basis Reduction.

....frequently written on Lisp type languages for which restructuring compiler technology is much less developed. While research for the discovery of better SAC algorithms is continuing, improvements in speed and usability are expected as good compilers for the underlying languages become available [153], 154] examples are the parallelizing compiler for sequential Scheme [96] multiprocessing extensions for Lisp [88] 199] the effort of [75] for constructing a compilation driven parallel REDUCE system for loosely coupled, distributed architectures. There have also been efforts to provide ....

C. G. Ponder. Evaluation of "Performance Enhancements" in algebraic manipulation systems. PhD thesis, University of California, Berkeley, August 1988. Also Tech. Rep. UCB 88/438.

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C. G. Ponder, Evaluation of "Performance Enhancements" in Algebraic Manipulation Systems, University of California at Berkeley Computer Science Report UCB/CSD 88/438, 1988.

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C. G. Ponder. Evaluation of "performance enhancements" in algebraic manipulation systems. In Della Dora and Fitch 23 , pages 51--73.

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