FSSP Algorithms for Square and Rectangular Arrays

Conference paper


The synchronization in cellular automata has been known as firing squad synchronization problem (FSSP) since its development. The firing squad synchronization problem on cellular automata has been studied extensively for more than fifty years, and a rich variety of synchronization algorithms has been proposed not only for one-dimensional arrays but also for two-dimensional arrays. In the present paper, we focus our attention to the two-dimensional array synchronizers that can synchronize any square/rectangle arrays and construct a survey on recent developments in their designs and implementations of optimum-time and non-optimum-time synchronization algorithms for two-dimensional arrays.



A part of this work is supported by Grant-in-Aid for Scientific Research (C) 21500023.


  1. 1.
    Balzer, R.: An 8-state minimal time solution to the firing squad synchronization problem. Inf. Control 10, 22–42 (1967)CrossRefGoogle Scholar
  2. 2.
    Beyer, W.T.: Recognition of topological invariants by iterative arrays. Ph.D. Thesis, MIT, pp. 144 (1969)Google Scholar
  3. 3.
    Gerken, H.D.: Über Synchronisations problem bei Zellularautomaten, Diplomarbeiten, Institut für Theoretische Informatik, Technische Universität Braunschweig, pp. 50, (1987)Google Scholar
  4. 4.
    Goto, E.: A minimal time solution of the firing squad problem. Dittoed Course Notes for Applied Mathematics 298, Harvard University, pp. 52–59 (1962)Google Scholar
  5. 5.
    Gruska, J., Torre, S.L., Parente, M.: The firing squad synchronization problem on squares, toruses and rings. Int. J. Found. Comput. Sci. 18(3), 637–654 (2007)MATHCrossRefGoogle Scholar
  6. 6.
    Ishii, S., Yanase, H., Maeda, M., Umeo, H.: State-efficient implementations of time-optimum synchronization algorithms for square arrays. Technical Report of IEICE, Circuit and Systems, pp. 13–18 (2006)Google Scholar
  7. 7.
    Mazoyer, J.: A six-state minimal time solution to the firing squad synchronization problem. Theor. Comput. Sci. 50, 183–238 (1987)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Schmid, H.: Synchronisationsprobleme für zelluläre Automaten mit mehreren Generälen. Diplomarbeit, Universität Karsruhe, (2003)Google Scholar
  10. 10.
    Shinahr, I.: Two- and three-dimensional firing squad synchronization problems. Inf. Control 24, 163–180 (1974)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Szwerinski, H.: Time-optimum solution of the firing-squad-synchronization-problem for n-dimensional rectangles with the general at an arbitrary position. Theor. Comput. Sci. 19, 305–320 (1982)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Umeo, H.: Firing squad synchronization algorithms for two-dimensional cellular automata. J. Cell. Autom. 4, 1–20, (2008)MathSciNetGoogle Scholar
  13. 13.
    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, Berlin, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Umeo, H., Hisaoka, M., Akiguchi, S.: Twelve-state optimum-time synchronization algorithm for two-dimensional rectangular cellular arrays. In: Proceedings of 4th International Conference on Unconventional Computing: UC 2005, Sevilla. LNCS 3699, pp. 214–223 (2005)MathSciNetGoogle Scholar
  15. 15.
    Umeo, H., Hisaoka, M., Teraoka, M., Maeda, M.: Several new generalized linear- and optimum-time synchronization algorithms for two-dimensional rectangular arrays. In: Margenstern, M. (Ed.) Proceedings of 4th International Conference on Machines, Computations and Universality: MCU 2004, Saint Petersburg. LNCS 3354, pp. 223–232 (2005)MathSciNetGoogle Scholar
  16. 16.
    Umeo, H., Ishida, K., Tachibana, K., Kamikawa, N.: A transition rule set for the first 2-D optimum-time synchronization algorithm. In: Proceedings of the 4th International Workshop on Natural Computing, PICT 2, Himeji, pp. 333–341. Springer (2009)Google Scholar
  17. 17.
    Umeo, H., Kamikawa, N., Nishioka, K., Akiguchi, S.: Generalized firing squad synchronization protocols for one-dimensional cellular automata – a survey. Acta Phys. Pol. B, Proc. Suppl. 3, 267–289 (2010)Google Scholar
  18. 18.
    Umeo, H., Kubo, K.: A seven-state time-optimum square synchronizer. In: Proceedings of the 9th International Conference on Cellular Automata for Research and Industry, Ascoli Piceno. LNCS 6350, pp. 219–230. Springer (2010)Google Scholar
  19. 19.
    Umeo, H., Kubo, K.: Recent developments in constructing square synchronizers. In: Proceedings of the 10th International Conference on Cellular Automata for Research and Industry, Santorini. LNCS 7495, pp. 171–183. Springer (2012)Google Scholar
  20. 20.
    Umeo, H., Maeda, M., Fujiwara, N.: An efficient mapping scheme for embedding any one-dimensional firing squad synchronization algorithm onto two-dimensional arrays. In: Proceedings of the 5th International Conference on Cellular Automata for Research and Industry, Geneva. LNCS 2493, pp. 69–81. Springer (2002)Google Scholar
  21. 21.
    Umeo, H., Maeda, M., Hisaoka, M., Teraoka, M.: A state-efficient mapping scheme for designing two-dimensional firing squad synchronization algorithms. Fundam. Inform. 74, 603–623 (2006)MathSciNetMATHGoogle Scholar
  22. 22.
    Umeo, H., Nishide, K., Yamawaki, T.: A new optimum-time firing squad synchronization algorithm for two-dimensional rectangle arrays – one-sided recursive halving based. In: Löwe, B. et al. (Eds.) Proceedings of the International Conference on Models of Computation in Context, Computability in Europe 2011, CiE 2011, Sofia. LNCS 6735, pp. 290–299 (2011)Google Scholar
  23. 23.
    Umeo, H., Nomura, A.: Zebra-like mapping for state-efficient implementation of two-dimensional synchronization algorithms (2014, manuscript in preparation)Google Scholar
  24. 24.
    Umeo, H., Uchino, H.: A new time-optimum synchronization algorithm for rectangle arrays. Fundam. Inform. 87(2), 155–164 (2008)MathSciNetMATHGoogle Scholar
  25. 25.
    Umeo, H., Uchino, H., Nomura, A.: How to synchronize square arrays in optimum-time. In: Proceedings of the 2011 International Conference on High Performance Computing and Simulation (HPCS 2011), Istanbul, pp. 801–807. IEEE (2011)Google Scholar
  26. 26.
    Umeo, H., Yanagihara, T.: Smallest implementations of optimum-time firing squad synchronization algorithms for one-bit-communication cellular automata. In: Proceedings of the 2011 International Conference on Parallel Computing and Technology, PaCT 2011, Kazan. LNCS 6873, pp. 210–223 (2011)Google Scholar
  27. 27.
    Umeo, H., Yamawaki, T., Shimizu, N., Uchino, H.: Modeling and simulation of global synchronization processes for large-scale-of two-dimensional cellular arrays. In: Proceedings of International Conference on Modeling and Simulation, AMS 2007, Phuket, pp. 139–144 (2007)Google Scholar
  28. 28.
    Waksman, A.. An optimum solution to the firing squad synchronization problem. Inf. Control 9, 66–78 (1966)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of Osaka Electro-CommunicationNeyagawa-shiJapan

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