Generalized FSSP on Hexagonal Tiling: Towards Arbitrary Regular Spaces

  • Luidnel Maignan
  • Jean-Baptiste YunèsEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8996)


Here we present a solution to the generalized firing squad synchronization problem that works on some class of shapes in the hexagonal tiling of the plane. The solution is obtained from a previous solution which works on grids with either a von Neumann or a Moore neighborhood. Analyzing the construction of this previous solution, we were able to exhibit a parameter that leads us to abstract the solution. First, and for an arbitrary considered neighborhood, we focus our attention on a class of shapes built from this neighborhood, and determine the corresponding parameter value for them. Second, we apply our previous solution with the determined parameter value for the hexagonal neighborhood and show that, indeed, all the considered shapes on the hexagonal tiling synchronizes.


Cellular Automaton Synchronization Time Border Cell Space Shape Distance Field 
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.


  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.
    Grasselli, A.: Synchronization of cellular arrays: the firing squad problem in two dimensions. Inf. Control 28, 113–124 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kobayashi, K.: The firing squad synchronization problem for two-dimensional arrays. Inf. Control 34, 177–197 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Maignan, L., Yunès, J.-B.: A spatio-temporal algorithmic point of view on firing squad synchronisation problem. In: Sirakoulis, G.C., Bandini, S. (eds.) ACRI 2012. LNCS, vol. 7495, pp. 101–110. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  5. 5.
    Maignan, L., Yunès, J.B.: Moore and von Neumann neighborhood n-dimensional generalized firing squad solutions using fields. In: AFCA 2013 Workshop, CANDAR 2013 Conference, Matsuyama, Japan, 4–6 December 2013Google Scholar
  6. 6.
    Mazoyer, J.: A six-state minimal time solution to the firing squad synchronization problem. Theoret. Comput. Sci. 50, 183–238 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Moore, E.E.: Sequential Machines, Selected Papers, pp. 213–214. Addison-Wesley, Reading (1964) zbMATHGoogle Scholar
  8. 8.
    Moore, E.E., Langdon, G.: A generalized firing squad problem. Inf. Control 12, 212–220 (1968)CrossRefzbMATHGoogle Scholar
  9. 9.
    Noguchi, K.: Simple 8-state minimal time solution to the firing squad synchronization problem. Theoret. Comput. Sci. 314(3), 303–334 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Róka, Z.: The firing squad synchronization problem on Cayley graphs. In: Hájek, Petr, Wiedermann, Jiří (eds.) MFCS 1995. LNCS, vol. 969, pp. 402–411. Springer, Heidelberg (1995) CrossRefGoogle Scholar
  11. 11.
    Romani, F.: Cellular automata synchronization. Inf. Sci. 10, 299–318 (1976)CrossRefzbMATHGoogle Scholar
  12. 12.
    Schmidt, H., Worsch, T.: The firing squad synchronization problem with many generals for one-dimensional ca. In: Levy, J.J., Mayr, E.W., Mitchell, J.C. (eds.) TCS 2004. IFIP, vol. 155, pp. 111–124. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Settle, A., Simon, J.: Smaller solutions for the firing squad. Theoret. Comput. Sci. 276(1), 83–109 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shinahr, I.: Two- and three-dimensional firing-squad synchronization problems. Inf. Control 24, 163–180 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Szwerinski, H.: Time-optimum solution of the firing-squad-synchronization-problem for \(n\)-dimensional rectangles with the general at an arbitrary position. Theoret. Comput. Sci. 19, 305–320 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Umeo, H.: Recent developments in firing squad synchronization algorithms for two-dimensional cellular automata and their state-efficient implementations. In: AFL, pp. 368–387 (2011)Google Scholar
  17. 17.
    Yamakawi, T., Amesara, T., Umeo, H.: A note on three-dimensional firing squad synchronization algorithm. In: ITC-CSCC, pp. 773–776 (2008)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.LACLUniversité Paris-Est-CréteilCréteilFrance
  2. 2.LIAFAUniversité Paris-DiderotParisFrance

Personalised recommendations