Abstract
The firing squad synchronization problem has been studied extensively for more than 40 years [1-18]. The present authors are involved in research on firing squad synchronization algorithms on two-dimensional (2-D) rectangular cellular arrays. Several synchronization algorithms on 2-D arrays have been proposed, including Beyer [2], Grasselli [3], Kobayashi [4], Shinahr [10], Szwerinski [12] and Umeo et al. [13, 15]. To date, the smallest number of cell states for which an optimum-time synchronization algorithm has been developed is 14 for rectangular array, achieved by Umeo et al. [15]. In the present paper, we propose a new optimum-time synchronization algorithm that can synchronize any 2-D m × n rectangular arrays in m + n + max(m, n) –3 steps. We progressively reduce the number of internal states of each cellular automaton on rectangular arrays, achieving twelve states. This is the smallest number of states reported to date for synchronizing rectangular arrays in optimum-step.
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References
Balzer, R.: An 8-state minimal time solution to the firing squad synchronization problem. Information and Control 10, 22–42 (1967)
Beyer, W.T.: Recognition of topological invariants by iterative arrays. Ph.D. Thesis, MIT, p. 144 (1969)
Grasselli, A.: Synchronization of cellular arrays: The firing squad problem in two dimensions. Information and Control 28, 113–124 (1975)
Kobayashi, K.: The firing squad synchronization problem for two-dimensional arrays. Information and Control 34, 177–197 (1977)
Mazoyer, J.: A six-state minimal time solution to the firing squad synchronization problem. Theoretical Computer Science 50, 183–238 (1987)
Minsky, M.: Computation: Finite and infinite machines, pp. 28–29. Prentice Hall, Englewood Cliffs (1967)
Moore, E.F.: The firing squad synchronization problem. In: Moore, E.F. (ed.) Sequential Machines, Selected Papers, pp. 213–214. Addison-Wesley, Reading (1964)
Moore, F.R., Langdon, G.G.: A generalized firing squad problem. Information and Control 12, 212–220 (1968)
Nguyen, H.B., Hamacher, V.C.: Pattern synchronization in two-dimensional cellular space. Information and Control 26, 12–23 (1974)
Shinahr, I.: Two- and three-dimensional firing squad synchronization problems. Information and Control 24, 163–180 (1974)
Settle, A., Simon, J.: Smaller solutions for the firing squad. Theoretical Computer Science 276, 83–109 (2002)
Szwerinski, H.: Time-optimum solution of the firing-squad-synchronization-problem for n-dimensional rectangles with the general at an arbitrary position. Theoretical Computer Science 19, 305–320 (1982)
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. 2493, pp. 69–81. Springer, Heidelberg (2002)
Umeo, H., Hisaoka, M., Michisaka, K., Nishioka, K., Maeda, M.: Some new generalized synchronization algorithms and their implementations for large scale cellular automata. In: Calude, C.S., Dinneen, M.J., Peper, F. (eds.) UMC 2002. LNCS, vol. 2509, pp. 276–286. Springer, Heidelberg (2002)
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.) MCU 2004. LNCS, vol. 3354, pp. 223–232. Springer, Heidelberg (2005)
Umeo, H., Hisaoka, M., Sogabe, T.: A survey on optimum-time firing squad synchronization algorithms for one-dimensional cellular automata. International Journal of Unconventional Computing 1(3), 1–24 (2005)
Varshavsky, V.I., Marakhovsky, V.B., Peschansky, V.A.: Synchronization of interacting automata. Mathematical Systems Theory 4(3), 212–230 (1970)
Waksman, A.: An optimum solution to the firing squad synchronization problem. Information and Control 9, 66–78 (1966)
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Umeo, H., Hisaoka, M., Akiguchi, S. (2005). A Twelve-State Optimum-Time Synchronization Algorithm for Two-Dimensional Rectangular Cellular Arrays. In: Calude, C.S., Dinneen, M.J., Păun, G., Pérez-Jímenez, M.J., Rozenberg, G. (eds) Unconventional Computation. UC 2005. Lecture Notes in Computer Science, vol 3699. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11560319_20
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DOI: https://doi.org/10.1007/11560319_20
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