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Parallel arithmetic computations: A survey

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Mathematical Foundations of Computer Science 1986 (MFCS 1986)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 233))

Abstract

A survey of parallel algorithms for algebraic problems is presented.

Part of this work was done while the author was visiting Universität Zürich, and supported by Schweizerischer Nationalfonds, grant 2175-0.83, and by NSERC, grant 3-650-126-40. An extended version will appear as Technical Report, Department of Computer Science, University of Toronto, 1986.

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References

  • M. Ben-Or, Lower bounds for algebraic computation trees. Proc. 15th Ann. ACM Symp. Theory of Computing, Boston MA, 1983, 80–86.

    Google Scholar 

  • M. Ben-Or, E. Feig, D. Kozen, and P. Tiwari, A fast parallel algorithm for determining all roots of a polynomial with real roots. To appear in Proc. 18th Ann. ACM Symp. Theory of Computing, Berkeley CA, 1986.

    Google Scholar 

  • M. Ben-Or, D. Kozen, and J. Reif, The complexity of elementary algebra and geometry. Proc. 16th Ann. ACM Symp. Theory of Computing, Washington DC, 1984, 457–464.

    Google Scholar 

  • D. Bini, Parallel solution of certain Toeplitz linear systems. SIAM J. Computing 13 (1984), 268–276.

    Google Scholar 

  • D. Bini and V. Pan, Fast parallel algorithms for polynomial division over arbitrary field of constants. Nota interna, Dipartimento di Informatica, Università di Pisa, 1984.

    Google Scholar 

  • D. Bini and V. Pan, Polynomial division and its computational complexity. Tech. Rep. TR 86-2, Computer Science Department, SUNY Albany NY, 1986.

    Google Scholar 

  • S.J. Berkowitz, On computing the determinant in small parallel time using a small number of processors. Information Processing Letters 18 (1984), 147–150.

    Google Scholar 

  • A. Borodin, On relating time and space to size and depth. SIAM J. Comput. 6 (1977), 733–744.

    Google Scholar 

  • A. Borodin, J. von zur Gathen, and J. Hopcroft, Fast parallel matrix and GCD computations. Information and Control 52 (1982), 241–256.

    Google Scholar 

  • R.P. Brent, The parallel evaluation of general arithmetic expressions. J. ACM 21 (1974), 201–206.

    Google Scholar 

  • A.L. Chistov, Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic. Proc. Int. Conf. Foundations of Computation Theory, Springer Lecture Notes in Computer Science 199, 1985, 63–69.

    Google Scholar 

  • A.L. Chistov and D.Yu. Grigoryev, Fast decomposition of polynomials into irreducible ones and the solution of systems of algebraic equations. Soviet Math. Dokl 29 (1984), 380–383.

    Google Scholar 

  • S.A. Cook, A taxonomy of problems with fast parallel algorithms. Information and Control 64 (1985), 2–22.

    Google Scholar 

  • L. Csanky, Fast parallel matrix inversion algorithms. SIAM J. Comput. 5 (1976), 618–623.

    Google Scholar 

  • W. Eberly, Very fast parallel matrix and polynomial arithmetic. Proc. 25th Ann. IEEE Symp. Foundations of Computer Science, Singer Island FL, 1984, 21–30. Tech. Rep. #178/85, Department of Computer Science, University of Toronto.

    Google Scholar 

  • W. Eberly, Very fast parallel polynomial arithmetic. Preprint, University of Toronto, Canada, 1986.

    Google Scholar 

  • F. Fich and M. Tompa, The parallel complexity of exponentiating polynomials over finite fields, Proc. 17th Ann. ACM Sympos. Theory of Computing, Providence RI, 1985, 38–47.

    Google Scholar 

  • M.R. Garey and D.S. Johnson, Computers and intractability: A guide to the theory of NP-completeness. W.H. Freeman, San Francisco, 1979.

    Google Scholar 

  • J. von zur Gathen [1984a], Parallel algorithms for algebraic problems. SIAM J. Comput. 13 (1984), 802–824.

    Google Scholar 

  • J. von zur Gathen [1984b], Computing powers in parallel. Proc. 25th Ann. IEEE Symp. Foundations of Computer Science, Singer Island FL, 1984, 31–36.

    Google Scholar 

  • J. von zur Gathen [1985a], Irreducibility of multivariate polynomials. J. Computer System Sciences 31 (1985), 225–264.

    Google Scholar 

  • J. von zur Gathen [1985b], Factoring polynomials and primitive elements for special primes. Preprint, University of Toronto, April 1985.

    Google Scholar 

  • J. von zur Gathen, Representations and parallel computations for rational functions. SIAM J. Comput 15 (1986), 432–452.

    Google Scholar 

  • J. von zur Gathen and G. Seroussi, Boolean circuits versus arithmetic circuits. To appear in Proc. 6th Int. Conf. Computer Science, Santiago, Chile, 1986.

    Google Scholar 

  • R. Glover, Simultaneous Padé approximation. M. Sc. Thesis, University of Toronto, September 1984.

    Google Scholar 

  • L. Hyafil, On the parallel evaluation of multivariate polynomials. SIAM J. Computing 8 (1979), 120–123.

    Google Scholar 

  • O.H. Ibarra, S. Moran and L.E. Rosier, A note on the parallel complexity of computing the rank of order n matrices. Information Processing Letters 11 (1980), 162.

    Google Scholar 

  • T.L. Jordan, A guide to parallel computation and some Cray-1 experiences. In: Parallel computations, ed. by G. Rodrigues, Academic Press, New York, 1982, 1–50.

    Google Scholar 

  • E. Kaltofen, Fast parallel absolute irreducibility testing. J. Symb. Computation 1 (1985), 57–67.

    Google Scholar 

  • E. Kaltofen, Uniform closure properties of p-computable functions. To appear in Proc. 18th Ann. ACM Symp. Theory of Computing, Berkeley CA, 1986.

    Google Scholar 

  • E. Kaltofen, M. Krishnamoorthy, and B.D. Saunders [1986a], Fast parallel computation of Hermite and Smith forms of polynomial matrices. To appear in SIAM J. Algebraic and Discrete Methods, 1986.

    Google Scholar 

  • E. Kaltofen, M. Krishnamoorthy, and B.D. Saunders [1986b], Fast parallel algorithms for similarity of matrices. To appear in Proc. ACM Symp. Symbolic and Algebraic Computation, Waterloo, Canada, 1986.

    Google Scholar 

  • H.T. Kung, New algorithms and lower bounds for the parallel evaluation of certain rational expressions and recurrences. J. ACM 23 (1976), 252–261.

    Google Scholar 

  • A.K. Lenstra, H.W. Lenstra, and L. Lovász, Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), 515–534.

    Google Scholar 

  • E.W. Mayr and A.R. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals. Adv. Math. 46 (1982), 305–329.

    Google Scholar 

  • G.L. Miller, E. Kaltofen, and V. Ramachandran, Efficient parallel evaluation of straight-line code. To appear in Proc. Aegean Workshop on Computing, VLSI Algorithms and Architectures, Attica, Greece, 1986.

    Google Scholar 

  • D.E. Muller and F.P. Preparata, Restructuring of arithmetic expressions for parallel evaluation. J. ACM 23 (1976), 534–543.

    Google Scholar 

  • K. Mulmuley, Computing the rank of a matrix over an arbitrary field is in NC (2). To appear in Proc. 18th Ann. ACM Symp. Theory of Computing, Berkeley CA, 1986.

    Google Scholar 

  • J. Reif, Logarithmic depth circuits for algebraic functions. Proc. 24th Ann. IEEE Symp. Foundations of Computer Science, Tucson AZ, 1983, 138–145.

    Google Scholar 

  • V. Strassen, Vermeidung von Divisionen. J. reine u. angew. Math. 264 (1973), 182–202.

    Google Scholar 

  • V. Strassen, The Computational Complexity of Continued Fractions. SIAM J. Comput. 12 (1983), 1–27.

    Google Scholar 

  • V. Strassen, Algebraische Berechnungskomplexität. In: Perspectives in Mathematics, Birkhäuser Verlag, Basel, 1984.

    Google Scholar 

  • L.G. Valiant, Completeness classes in algebra. Proc. 11th Ann. ACM Symp. Theory of Computing, Atlanta GA, 1979, 249–261.

    Google Scholar 

  • L.G. Valiant, Computing multivariate polynomials in parallel. Information Processing Letters 11 (1980), 44–45, and 12, 54.

    Google Scholar 

  • L. Valiant, S. Skyum, S. Berkowitz, and C. Rackoff, Fast parallel computation of polynomials using few processors. SIAM J. Comput. 12 (1983), 641–644.

    Google Scholar 

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Jozef Gruska Branislav Rovan Juraj Wiedermann

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© 1986 Springer-Verlag Berlin Heidelberg

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von zur Gathen, J. (1986). Parallel arithmetic computations: A survey. In: Gruska, J., Rovan, B., Wiedermann, J. (eds) Mathematical Foundations of Computer Science 1986. MFCS 1986. Lecture Notes in Computer Science, vol 233. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0016236

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  • DOI: https://doi.org/10.1007/BFb0016236

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  • Print ISBN: 978-3-540-16783-9

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