Acta Informatica

, Volume 24, Issue 6, pp 595–632 | Cite as

The derivation of systolic implementations of programs

  • Chua-Huang Huang
  • Christian Lengauer


We present a mathematically rigorous and, at the same time, convenient method for systolic design and derive systolic designs for three matrix computation problems. Each design is synthesized from a simple program and a proposed layout of processors. The synthesis derives a systolic parallel execution, channel connections for the proposed processor layout, and an arrangement of data streams such that the systolic execution can begin. Our choices of designs are governed by formal theorems. The synthesis method is implementable and is particularly effective if implemented with graphics capability. Our implementation on the Symbolics 3600 displays the resulting designs and simulated executions graphically on the screen. The method's centerpiece, a transformation of sequential program computations into systolic parallel ones, has been mechanically proved correct.


Computational Mathematic Computer System System Organization Data Stream Synthesis Method 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boyer, R.S., Moore, J.S.: A Computational Logic. ACM Monograph Series. New York: Academic Press 1979Google Scholar
  2. 2.
    Cappello, P.R., Steiglitz, K.: Unifying VLSI Array Design with Linear Transformations of Space-time. In: Advances in Computing Research, F.P. Preparata, (ed.) VLSI Theory, Vol. 2, pp. 23–65. Greenwich, CT: JAI Press Inc. 1984Google Scholar
  3. 3.
    Chandy, M.: Concurrent Programming for the Masses. Proc. 4th Ann. ACM Symp. on Principles of Distributed Computing, pp. 1–12 (1985)Google Scholar
  4. 4.
    Chandy, K.M., Misra, J.: Systolic Algorithms as Programs. Distrib. Comput. 1, 3, 177–183 (1986)Google Scholar
  5. 5.
    Chen, M.C.: Synthesizing Systolic Designs. YALEU/DCS/RR-374, Department of Computer Science, Yale University, March 1985Google Scholar
  6. 6.
    Chen, M.C.: A Parallel Language and Its Compilation to Multiprocessor Machines or VLSI. Proc. 13th Ann. ACM Symp. on Principles of Programming Languages, pp. 131–139 (1986)Google Scholar
  7. 7.
    Delosme, J.-M., Ipsen, I.C.F.: Design Methodology for Systolic Arrays. Proc. SPIE Symposium, Vol. 696, Advanced Algorithms and Architectures for Signal Processing, pp. 245–259 (1986)Google Scholar
  8. 8.
    Fortes, J.A.B., Moldovan, D.I.: Parallelism Detection and Transformation Techniques for VLSI Algorithms. J. Parallel Distrib. Comput. 2, 277–301 (1985)Google Scholar
  9. 9.
    Huang, C.-H., Lengauer, C.: An Incremental, Mechanical Development of Systolic Solutions to the Algebraic Path Problem. TR-86-28, Department of Computer Sciences, The University of Texas at Austin, December 1986Google Scholar
  10. 10.
    Huang, C.-H., Lengauer, C.: An Implemented Method for Incremental Systolic Design. In: Parallel Architectures and Languages Europe (PARLE). J.W. de Bakker, A.J. Nijman, P.C. Treleaven, (eds.) Parallel Architectures, Vol. 1, pp. 160–177, Lecture Notes in Computer Science 258. Berlin, Heidelberg, New York, Tokyo: Springer 1987Google Scholar
  11. 11.
    Knuth, D.E.: The Art of Computer Programming, Vol. 3: Sorting and Searching Sect. 5.3.4. Reading, MA: Addison-Wesley 1973Google Scholar
  12. 12.
    Kung, H.T., Leiserson, C.E.: Algorithms for VLSI Processor Arrays. In: Introduction to VLSI Systems, Sect. 8.3., C. Mead, L. Conway (eds.). Reading, MA: Addison-Wesley 1980Google Scholar
  13. 13.
    Lam, M.S., Mostow, J.: A Transformational Model of VLSI Systolic Design. Computer 18, 42–52 (1985)Google Scholar
  14. 14.
    Leiserson, C.E.: Systolic and Semisystolic Design (Extended Abstract). Proc. IEEE Int. Conf. on Computer Design/VLSI in Computers, pp. 627–632 (ICCD '83), 1983Google Scholar
  15. 15.
    Leiserson, C.E., Saxe, J.B.: Optimizing Synchronous Systems. J. VLSI Comput. Syst. 1, 41–67 (1983)Google Scholar
  16. 16.
    Lengauer, C.: A Methodology for Programming with Concurrency: The Formalism. Sci. Comput. Programming 2, 19–52 (1982)Google Scholar
  17. 17.
    Lengauer, C., Huang, C.-H.: A Mechanically Certified Theorem about Optimal Concurrency of Sorting Networks. Proc. 13th Ann. ACM Symp. on Principles of Programming Languages, pp. 307–317 (1986)Google Scholar
  18. 18.
    Li, G.-H., Wah, B.W.: The Design of Optimal Systolic Arrays. IEEE Trans. Comput. 34, 66–77 (1985)Google Scholar
  19. 19.
    Miranker, W.L., Winkler, A.: Spacetime Representations of Computational Structures. Computing 32, 93–114 (1984)Google Scholar
  20. 20.
    Moldovan, D.I.: On the Design of Algorithms for VLSI Systolic Arrays. Proc. IEEE 71, 113–120 (1983)Google Scholar
  21. 21.
    Moldovan, D.I., Fortes, J.A.B.: Partitioning and Mapping Algorithms into Fixed Size Systolic Arrays. IEEE Trans. Comput. 35, 1–12 (1986)Google Scholar
  22. 22.
    Quinton, P.: The Systematic Design of Systolic Arrays. TR84-11, Microelectronics Center of North Carolina, May 1984Google Scholar
  23. 23.
    Rao, S.K.: Regular Iterative Algorithms and their Implementations on Processor Arrays. Ph.D. Thesis, Department of Electrical Engineering, Stanford University, October 1985Google Scholar
  24. 24.
    Rote, G.: A Systolic Array Algorithm for the Algebraic Path Problem (Shortest Paths; Matrix Inversion). Computing 34, 191–219 (1985)Google Scholar
  25. 25.
    van de Snepscheut, J.L.A.: A Derivation of a Distributed Implementation of Warshall's Algorithm. Sci. Comput. Programming 7, 55–60 (1986)Google Scholar
  26. 26.
    Weiser, U., Davis, A.: A Wavefront Notation Tool for VLSI Array Design. In: VLSI Systems and Computations, H.T. Kung, B. Sproull, G. Steele (eds.) Rockville, MA: Computer Science, pp. 226–234. Press 1981Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Chua-Huang Huang
    • 1
  • Christian Lengauer
    • 1
  1. 1.Department of Computer SciencesThe University of Texas at AustinAustinUSA

Personalised recommendations