Skip to main content
SpringerLink
Log in

Smallest Implementations of Optimum-Time Firing Squad Synchronization Algorithms for One-Bit-Communication Cellular Automata

  • Conference paper
Parallel Computing Technologies (PaCT 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6873))

Included in the following conference series:

Abstract

Synchronization of large-scale networks is an important and fundamental computing primitive in parallel and distributed systems. The firing squad synchronization problem (FSSP) on cellular automata (CA) has been studied extensively for more than fifty years, and a rich variety of synchronization algorithms has been proposed for not only one-dimensional but two-dimensional arrays. In the present paper, we study the FSSP on 1-bit-communication cellular automata, CA1 − bit. The CA1 − bitis a weakest subclass of CAs in which the amount of inter-cell communication bits transferred among neighboring cells at one step is restricted to 1-bit. We propose two state-efficient implementations of optimum-time FSSP algorithms for the CA1 − bitand show that the communication restriction has no influence on the design of optimum-time FSSP algorithms. The implementations proposed are the smallest ones, known at present.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Balzer, R.: An 8-state minimal time solution to the firing squad synchronization problem. Information and Control 10, 22–42 (1967)

    Article  MATH  Google Scholar 

  2. Beyer, W.T.: Recognition of topological invariants by iterative arrays. Ph.D. Thesis, MIT, p. 144 (1969)

    Google Scholar 

  3. Fischer, P.C.: Generation of primes by a one-dimensional real-time iterative array. J. of ACM 12(3), 388–394 (1965)

    Article  MATH  Google Scholar 

  4. Gerken, H.D.: Über Synchronisations - Probleme bei Zellularautomaten. Diplom arbeit,Institut für Theoretische Informatik, Technische Universität Braunschweig, p. 50 (1987)

    Google Scholar 

  5. Goto, E.: A minimal time solution of the firing squad problem. Dittoed course notesfor Applied Mathematics, vol. 298, pp. 52–59. Harvard University, Cambridge (1962)

    Google Scholar 

  6. Gruska, J., Torre, S.L., Parente, M.: The firing squad synchronization problem on squares, toruses and rings. Intern. J. of Foundations of Computer Science 18(3), 637–654 (2007)

    Article  MATH  Google Scholar 

  7. Mazoyer, J.: A six-state minimal time solution to the firing squad synchronizationproblem. Theoretical Computer Science 50, 183–238 (1987)

    Article  MATH  Google Scholar 

  8. Mazoyer, J.: On optimal solutions to the firing squad synchronization problem. Theoretical Computer Science 168, 367–404 (1996)

    Article  MATH  Google Scholar 

  9. Minsky, M.L.: Computation: Finite and infinite machines, pp. 28–29. Prentice-Hall, Englewood Cliffs (1967)

    MATH  Google Scholar 

  10. Moore, E.F.: The firing squad synchronization problem. In: Moore, E.F. (ed.) Sequential Machines, Selected Papers, pp. 213–214. Addison-Wesley, Reading (1964)

    Google Scholar 

  11. Nishimura, J., Sogabe, T., Umeo, H.: A design of o ptimum-time firing squad synchronizationalgorithm on 1-bit cellular automaton. In: Proc. of the 8th International Symposium on Artificial Life and Robotics, pp. 381–386 (2003)

    Google Scholar 

  12. Shinahr, I.: Two- and three-dimensional firing squad synchronization problems. Information and Control 24, 163–180 (1974)

    Article  MATH  Google Scholar 

  13. La Torre, S., Napoli, M., Parente, M.: Firing squad synchronization problem on bidimensional cellular automata with communication constraints. In: Margenstern, M., Rogozhin, Y. (eds.) MCU 2001. LNCS, vol. 2055, pp. 264–275. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  14. Umeo, H.: Firing squad synchronization problem in cellular automata. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and System Science, vol. 4, pp. 3537–3574. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  15. Umeo, H.: Problem solving on one-bit-communication cellular automata. In: Hoekstra, A.G., Kroc, J., Sloot, P.M.A. (eds.) Simulating Complex Systems by Cellular Automata, ch. 6, pp. 117–144. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  16. Umeo, H., Hisaoka, M., Sogabe, T.: A survey on optimum-time firing squad synchronization algorithms for one-dimensional cellular automata. Intern. J. of Unconventional Computing 1, 403–426 (2005)

    MATH  Google Scholar 

  17. Umeo, H., Maeda, M., Fujiwara, N.: An efficient mapping scheme for embeddingany one-dimensional firing squad synchronization algorithm onto two-dimensional arrays. In: Bandini, S., Chopard, B., Tomassini, M. (eds.) ACRI 2002. LNCS, vol. 2493, pp. 69–81. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  18. Umeo, H., Maeda, M., Hongyo, K.: A design of symmetrical six-state 3n-step firing squad synchronization algorithms and their implementations. In: El Yacoubi, S., Chopard, B., Bandini, S. (eds.) ACRI 2006. LNCS, vol. 4173, pp. 157–168. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  19. Umeo, H., Kubo, K.: A seven-state time-optimum square synchronizer. In: Bandini, S., Manzoni, S., Umeo, H., Vizzari, G. (eds.) ACRI 2010. LNCS, vol. 6350, pp. 219–230. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  20. Umeo, H., Maeda, M., Hisaoka, M., Teraoka, M.: A state -efficient mapping schemefor designing two-dimensional firing squad synchronization algorithms. Fundamenta Informaticae 74(4), 603–623 (2006)

    MATH  Google Scholar 

  21. Umeo, H., Michisaka, K., Kamikawa, N., Kanazawa, M.: State-efficient one-bit communication solutions for some classical cellular automata problems. Fundamenta Informaticae 78(3), 449–465 (2007)

    MATH  Google Scholar 

  22. Umeo, H., Yanagihara, T., Kanazawa, M.: State-efficient firing squad synchronization protocols for communication-restricted cellular automata. In: El Yacoubi, S., Chopard, B., Bandini, S. (eds.) ACRI 2006. LNCS, vol. 4173, pp. 169–181. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  23. Waksman, A.: An optimum solution to the firing squad synchronization problem. Information and Control 9, 66–78 (1966)

    Article  MATH  Google Scholar 

  24. Yunès, J.B.: Seven-state solution to the firing squad synchronization problem. Theoretical Computer Science 127, 313–332 (1994)

    Article  MATH  Google Scholar 

  25. Yunès, J.B.: An intrinsically non minimal-time Minsky-like 6 states solution to the firing squad synchronization problem. Theoretical Informatics and Applications 42(1), 55–68 (2008)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Univ. of Osaka Electro-Communication, Neyagawa-shi, Hastu-cho, 18-8, Osaka, 572-8530, Japan

    Hiroshi Umeo & Takashi Yanagihara

Authors
  1. Hiroshi Umeo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takashi Yanagihara
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Computational Mathematics and Mathematical Geophysics, Supercomputer Software Department, Russian Academy of Sciences Pr. Lavrentieva, ICM&MG RAS, 630090, Novosibirsk, Russia

    Victor Malyshkin

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Umeo, H., Yanagihara, T. (2011). Smallest Implementations of Optimum-Time Firing Squad Synchronization Algorithms for One-Bit-Communication Cellular Automata. In: Malyshkin, V. (eds) Parallel Computing Technologies. PaCT 2011. Lecture Notes in Computer Science, vol 6873. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23178-0_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-23178-0_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23177-3

  • Online ISBN: 978-3-642-23178-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation