Abstract
The firing squad synchronization problem on cellular automata has been studied extensively for more than fifty years, and a rich variety of synchronization algorithms have been proposed for not only one-dimensional arrays but two-dimensional arrays. In the present paper, we propose a seven-state optimum-time synchronization algorithm that can synchronize any square arrays of size n Ćn with a general at one corner in 2nāāā2 steps, which is a smallest realization of time-optimum square synchronizer known at present. The implementation is based on a new, simple zebra-like mapping scheme which embeds synchronization operations on one-dimensional arrays onto square arrays.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Balzer, R.: An 8-state minimal time solution to the firing squad synchronization problem. Information and ControlĀ 10, 22ā42 (1967)
Beyer, W.T.: Recognition of topological invariants by iterative arrays. Ph.D. Thesis, p. 144. MIT (1969)
Goto, E.: A minimal time solution of the firing squad problem. Dittoed course notes for Applied Mathematics, vol.Ā 298, pp. 52ā59. Harvard University, Cambridge (1962)
Mazoyer, J.: A six-state minimal time solution to the firing squad synchronization problem. Theoretical Computer ScienceĀ 50, 183ā238 (1987)
Moore, E.F.: The firing squad synchronization problem. In: Moore, E.F. (ed.) Sequential Machines, Selected Papers, pp. 213ā214. Addison-Wesley, Reading (1964)
Schmid, H.: Synchronisationsprobleme fĆ¼r zellulƤre Automaten mit mehreren GenerƤlen. Diplomarbeit, UniversitƤt Karsruhe (2003)
Shinahr, I.: Two- and three-dimensional firing squad synchronization problems. Information and ControlĀ 24, 163ā180 (1974)
Szwerinski, H.: Time-optimum solution of the firing-squad-synchronization-problem for n-dimensional rectangles with the general at an arbitrary position. Theoretical Computer ScienceĀ 19, 305ā320 (1982)
Umeo, H.: Firing squad synchronization algorithms for two-dimensional cellular automata. Journal of Cellular AutomataĀ 4, 1ā20 (2008)
Umeo, H.: Firing squad synchronization problem in cellular automata. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and System Science, vol.Ā 4, pp. 3537ā3574. Springer, Heidelberg (2009)
Umeo, H., Hisaoka, M., Akiguchi, S.: Twelve-state optimum-time synchronization algorithm for two-dimensional rectangular cellular arrays. In: Calude, C.S., Dinneen, M.J., PÄun, G., JesĆŗs PĆ©rez-JĆmenez, M., Rozenberg, G. (eds.) UC 2005. LNCS, vol.Ā 3699, pp. 214ā223. Springer, Heidelberg (2005)
Umeo, H., Maeda, M., Fujiwara, N.: An efficient mapping scheme for embedding any one-dimensional firing squad synchronization algorithm onto two-dimensional arrays. In: Bandini, S., Chopard, B., Tomassini, M. (eds.) ACRI 2002. LNCS, vol.Ā 2493, pp. 69ā81. Springer, Heidelberg (2002)
Umeo, H., Maeda, M., Hisaoka, M., Teraoka, M.: A state-efficient mapping scheme for designing two-dimensional firing squad synchronization algorithms. Fundamenta InformaticaeĀ 74(4), 603ā623 (2006)
Umeo, H., Uchino, H.: A new time-optimum synchronization algorithm for rectangle arrays. Fundamenta InformaticaeĀ 87(2), 155ā164 (2008)
Umeo, H., Yamawaki, T., Shimizu, N., Uchino, H.: Modeling and simulation of global synchronization processes for large-scale-of two-dimensional cellular arrays. In: Proc. of Intern. Conf. on Modeling and Simulation, AMS 2007, pp. 139ā144 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Umeo, H., Kubo, K. (2010). A Seven-State Time-Optimum Square Synchronizer. In: Bandini, S., Manzoni, S., Umeo, H., Vizzari, G. (eds) Cellular Automata. ACRI 2010. Lecture Notes in Computer Science, vol 6350. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15979-4_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-15979-4_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15978-7
Online ISBN: 978-3-642-15979-4
eBook Packages: Computer ScienceComputer Science (R0)