A new reconstruction and the first implementation of Goto's FSSP algorithm

The firing squad synchronization problem (FSSP) on cellular automata has been studied extensively for more than fifty years, and a rich variety of synchronization algorithms has been proposed. Goto's FSSP algorithm (Goto 1962) has been known as the first minimum-time FSSP algorithm, however the paper itself had been a completely unknown one in the research community of cellular automata for a long time due to its hard accessibility. In the present paper, we reconstruct the Goto's FSSP algorithm and present the first small-state implementation. The implementation is realized on a cellular automaton having 165-state and 4378 state-transition rules and the realization is far smaller than Goto (1962) imagined, where he thought that it would require many thousands of thousands states. It is shown that the reconstructed algorithm uses a quite different synchronization mechanism in comparison with the designs employed in Waksman (1966), Balzer (1967), Gerken (1987) and Mazoyer (1987). We show that the algorithm has Θ(nlog n) minimum-state-change complexity for synchronizing n cells. The algorithm is optimum not only in time but also in state-change complexities. We show that the reconstructed algorithm can be generalized as to the initial general's position and its implementation on a cellular automaton with 434 internal states and 13,328 state-transition rules is also given. The general purpose of this investigation is to achieve more insights into the structure of the classical minimum-time FSSP solutions and such insights would be helpful in the design of new FSSP algorithms.