In this paper, we prove (cf. Theorem 23) that for any m2 one can synchronize in time Dlogm(D) all lines of total communication delay >m9 (shorter lines being synchronized in time 4D). A result which extends to bounded degree connected graphs using Rosensthiel's technique [P. Rosenstiehl, Existence d’automates capables de s’accorder bien qu’arbitrairement connectés et nombreux, Internat. Comput. Center Bull. 5 (1966) 245–261, P. Rosenstiehl, J.R. Fiksel, A. Holliger, Intelligent graphs: networks of finite automata capable of solving graph problems, in: R.C. Read (Ed.), Graph Theory and Computing, Academic Press, New York, 1972, pp. 219–265]. As shown by Vivien [Cellular Automata: A Geometrical Approach, to appear], this result is already optimal for lines of two cells with arbitrary communication delay.
The method relies heavily on Jiang technique of circuit with revolving information.