The firing squad synchronization problem on Cayley graphs
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The Firing Squad Synchronization Problem (FSSP for short) has been intensively studied in the one-dimensional space. The problem consists in the synchronization of a segment of automata. We generalize this problem on Cayley graphs. We give minimal time solutions for (a) synchronizing all cells in all minimal paths between any pair of cells of a Cayley graph; (b) synchronizing all cells in all minimal paths starting at a given cell G (the “general”) and leading to all cells at a given distance from G in a Cayley graph. In solutions for (b), in some cases, all cells of a ball in a Cayley graph will be synchronized, in other cases this is not possible because of the existence of “culs-de-sac”.
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- 1.R. Balzer. A 8 state minimal time solution to the Firing Squad Synchronization Problem. Information and Control, 10:22–42, 1967.Google Scholar
- 2.C. Culik. Variation of the Firing Squad Synchronization Problem. Information Processing Letter, pages 152–157, 1989.Google Scholar
- 3.A. Machí and F. Mignosi. Garden of Eden configurations for cellular automata on Cayley graphs of groups. SIAM Journal on Discrete Mathematics, 6(1):44–56, 1993.Google Scholar
- 4.J. Mazoyer. A six states minimal time solution to the Firing Squad Synchronization Problem. Theoretical Computer Science, 50:183–238, 1987.Google Scholar
- 5.F.R. Moore and G.G. Langdon. A generalized firing squad problem. Information and Control, (12):212–220, 1968.Google Scholar
- 6.D.E. Müller and P.E. Schupp. Pushdown automata, ends, second-order logic and reachability problems. In 13th Annual ACM Symp. on the Theory of Computing, Milwaukee, 1981.Google Scholar
- 7.A. Holliger P. Rosenstiehl, J.R. Fiksel. Graph Theory and Computing, chapter Intelligent graphs: Networks of finite automata capable of solving graph problems. Lectures in Computer Science. R. C. Read, academic press edition, 1972.Google Scholar
- 8.Zs. Róka. Cellular automata on Cayley graphs. PhD thesis, Ecole Normale Supérieure de Lyon, France, July 1994.Google Scholar
- 9.Zs. Róka. The Firing Squad Synchronization Problem on Cayley graphs, preprint, 1995.Google Scholar
- 10.H. Szwerinsky. Time optimal solution for the Firing Squad Synchronization Problem for n-dimensional rectangles with the general at any position. Theoretical Computer Science, 19:305–320, 1982.Google Scholar