Advertisement

The mapping of linear recurrence equations on regular arrays

  • Patrice Quinton
  • Vincent van Dongen
Article

Abstract

The parallelization of many algorithms can be obtained using space-time transformations which are applied on nested do-loops or on recurrence equations. In this paper, we analyze systems of linear recurrence equations, a generalization of uniform recurrence equations. The first part of the paper describes a method for finding automatically whether such a system can be scheduled by an affine timing function, independent of the size parameter of the algorithm. In the second part, we describe a powerful method that makes it possible to transform linear recurrences into uniform recurrence equations. Both parts rely on results on integral convex polyhedra. Our results are illustrated on the Gauss elimination algorithm and on the Gauss-Jordan diagonalization algorithm.

Keywords

Recurrence Equation Dependence Graph Systolic Array Pointed Cone Convex Polyhedron 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. M. Karp, R. E. Miller, and S. Winograd. The organization of computations for uniform recurrence equations.JACM, 14(3): 563–590, July 1967.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    L. Lamport. The parallel execution of Do-Loops.Comm. ACM, Vol. 17, N. 2, pp. 83–93, Feb. 1974.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    P.R. Cappello and K. Steiglitz. Digital signal processing applications of systolic algorithms. InVLSI Systems and Computation, H.T. Kung, B. Sproull, and G. Steel editors, Computer Science Press (1981), 245–254.Google Scholar
  4. 4.
    M.C. Chen, Synthesizing systolic designs.1985 International Symposium on VLSI Technology, Systems and Applications, Taipei, Taiwan, R.O.C., 8–10 May 1985.Google Scholar
  5. 5.
    J.M. Delosme and I.C.F. Ipsen. Efficient systolic arrays for the solution of Toeplitz systems: An illustration of a methodology for the construction of systolic architectures for VLSI. In W. Moore, A. McCabe, and R. Urquhart, editors,International Workshop on Systolic Arrays, pages 37–46, Adam Hilger, University of Oxford, UK, July 2–4 1986.Google Scholar
  6. 6.
    S.Y. Kung. VLSI array processors.IEEE ASSP Magazine, 2(3):4–22, July 1985.CrossRefGoogle Scholar
  7. 7.
    G.H. Li and B.W. Wah. The design of optimal systolic arrays,IEEE Trans. Computers 34, 1 (1985), 66–77.zbMATHGoogle Scholar
  8. 8.
    W.L. Miranker and A. Winkler. Space-time representations of systolic computational structures.Computing, 32:93–114, 1984.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D.I. Moldovan. On the analysis and synthesis of VLSI algorithms,IEEE Trans. On Computer C-31 (11), November 1982, pages 1121–1126.CrossRefGoogle Scholar
  10. 10.
    P. Quinton.The Systematic Design of Systolic Arrays. Technical Report 193, Publication Interne IRISA, April 1983.Google Scholar
  11. 11.
    S.K. Rao,Regular iterative algorithms and their implementations on processor arrays, PhD Thesis, Information Systems Lab., Stanford Univerity, October 1985.Google Scholar
  12. 12.
    P. Quinton, Automatic synthesis of systolic arrays from uniform recurrent equationsProc. IEEE 11th Int. Sym. on Computer Architecture, Ann Arbor, MI, 1984, 208–214.Google Scholar
  13. 13.
    H.T. Kung. Let’s design algorithms for VLSI systems.Proc. of the Caltech conference on VLSI, January 1979.Google Scholar
  14. 14.
    D.I. Moldovan and J.A.B. Fortes. Partitioning and mapping algorithms into fixed size systolic arrays.IEEE Transaction on Computers, C-35(1):1–12, January 1986.CrossRefGoogle Scholar
  15. 15.
    V. Van Dongen and P. Quinton. The mapping of linear recurrence equations on regular arrays, Manuscript M 264, Philips Research Lab. Brussels, October 1988.Google Scholar
  16. 16.
    Y. Yaacoby and R. Cappello. Scheduling a system of affine recurrence equations onto a systolic array.International Conference on Systolic Arrays, San Diego, pages 373–381, May 1988.Google Scholar
  17. 17.
    P. Gachet and B. Joinnault. Conception d’alorithmes et d’architectures systoliques. Thèse de l’Université de Rennes I, Sept 1987.Google Scholar
  18. 18.
    Y. Saouter and P. Quinton. Computability of recurrence equations, IRISA Research Report, to appear, 1989.Google Scholar
  19. 19.
    S.V. Rajopadhye and R.M. Fujimoto. Systolic array synthesis by static analysis of program dependencies. In J.W. de Bakker, A.J. Nijman, and PC. Treleaven, editors,Parallel Architectures and Languages Europe, pages 295–310, Springer-Verlag, June 1987.Google Scholar
  20. 20.
    J.A.B. Fortes and D.I. Moldovan. Data broadcasting in linearly scheduled array processors.Proc. 11th Annual Symp. on Computer Architecutre, pages 224–231, 1984.Google Scholar
  21. 21.
    Y. Wong and J.M. Delosme. Broadcast removal in systolic algorithms.International Conference on Systolic Arrays, San Diego, pages 403–412, May 1988.Google Scholar
  22. 22.
    R.H. Kuhn. Optimization and interconnection complexity for: Parallel processors, single-stage networks, and decision trees. Technical Report UIUCDCS-R 80-1009, Univeristy of Illinois at Urbana-Champaign, February 1980.Google Scholar
  23. 23.
    P. Gachet, B. Joinnault, and P. Quinton. Synthesizing systolic arrays using DIASTOL. In W. Moore, A. McCabe, and R. Urquhart, editors,International Workshop on Systolic Arrays, pages 25–36, Adam Hilger, University of Oxford, UK, July 2–4 1986.Google Scholar
  24. 24.
    V. Van Dongen and P. Quinton. Uniformization of linear recurrence equations: a step toward the automatic synthesis of systolic arrays.International Conference on Systolic Arrays, San Diego, pages 473–482, May 1988.Google Scholar
  25. 25.
    B.W. Wah, M. Aboelaze, and W. Shang. Systematic designs of buffers in macropipelines of systolic arrays.Journal of Parallel and Distributed Computing 5, pp. 1–25, 1988.CrossRefGoogle Scholar
  26. 26.
    Y. Robert and D. Trystram. Systolic solution of the algebraic path problem. In W. Moore, A. McCabe, and R. Urquhart, editors,International Workshop of Systolic Arrays, pages 171–180, Adam Hilger, University of Oxford, UK, July 2–4, 1986.Google Scholar
  27. 27.
    P.S. Lewis and S.Y. Kung. Dependence graph based design of systolic arrays for the algebraic path problem.20th Asilomar Conf., Nov. 10–12, 1986.Google Scholar
  28. 28.
    C. Mongenet and G.R. Perrin, Synthesis of systolic arrays for inductive problems. In J.W. de Bakker, A.J. Nijmann, and P.C. Treleaven, editors,Parallel Architectures and Languages Europe, pages 260–277, Springer-Verlag, June 1987.Google Scholar
  29. 29.
    P. Clauss. Contribution à la synthèse de rèseaux systoliques.Rapport de DEA, Université de Besançon 1987.Google Scholar
  30. 30.
    C. Guerra and R. Melhem, Synthesizing non-uniform systolic designs.Proc. of the 1986 International Conference on Parallel Processing, pages 765–772, August 1986.Google Scholar
  31. 31.
    M.C. Chen. Synthesizing VLSI architectures: Dynamic programming solver.Proc. of the 1986 International Conference on Parallel Processing, pages 776–784, August 1986.Google Scholar
  32. 32.
    P. Gachet, P. Quinton, C. Mauras and Y. Saouter. Alpha du Centaur: A prototype environment for the design of parallel regular algorithms. IRISA Research Report number 439, November 1988.Google Scholar
  33. 33.
    V. Van Dongen. Presage, a tool for the design of low-cost systolic arrays.ISCAS 88, pages 2765–2768, 1988.Google Scholar
  34. 34.
    A. Schrijver.Theory of linear and integer programming. Wiley-Interscience series in Discrete Mathematics, John Wiley and Sons, 1986.Google Scholar
  35. 35.
    J. Dieudonné.Algèbre linéaire et géométrie élémentaire, Hermann, Paris, 1964.zbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • Patrice Quinton
    • 1
  • Vincent van Dongen
    • 2
  1. 1.IRISARennes-CedexFrance
  2. 2.PRLBBrusselsBelgium

Personalised recommendations