Abstract
We propose several new generalized synchronization algorithms for 2-D cellular arrays. Firstly, a generalized linear-time synchronization algorithm and its 14-state implementation are given. It is shown that there exists a 14-state 2-D CA that can synchronize any m × n rectangular array in m + n + max(r + s , m + n – r – s + 2) – 4 steps with the general at an arbitrary initial position (r, s),where 1 ≤ r ≤ m, 1 ≤ s ≤ n. The generalized linear-time synchronization algorithm is interesting in that it includes an optimum-step synchronization algorithm as a special case where the general is located at one corner. In addition, we propose a noveloptimum-time generalized synchronization scheme that can synchronize any m × n array in m + n + max (m, n) − min (r, m − r + 1) − min (s, n − s + 1) − 1 optimum steps.
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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, p. 144, MIT, Cambridge (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 generalized synchronization algorithms and their implementations for a 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.: A simple design of time-efficient firing squad synchronization algorithms with fault-tolerance. IEICE Trans. on Information and Systems E-87-D(3), 733–739 (2004)
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., Teraoka, M., Maeda, M. (2005). Several New Generalized Linear- and Optimum-Time Synchronization Algorithms for Two-Dimensional Rectangular Arrays. In: Margenstern, M. (eds) Machines, Computations, and Universality. MCU 2004. Lecture Notes in Computer Science, vol 3354. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31834-7_18
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DOI: https://doi.org/10.1007/978-3-540-31834-7_18
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