Smaller solutions for the firing squad

In this paper we improve the bounds on the complexity of solutions to the ̄ring squad problem, also known as the ̄ring synchronization problem. In the ̄ring synchronization problem we consider a one-dimensional array of n identical ̄nite automata. Initially all automata are in the same state except for one automaton designated as the initiator for the synchronization. Our results hold for the original problem, where the initiator may be located at either endpoint, and for the variant where any one of the automata may be the initiator, called the generalized problem. In both cases, the goal is to de ̄ne the set of states and transition rules for the automata so that all machines enter a special ̄re state simultaneously and for the ̄rst time during the ̄nal round of the computation. In our work we improve the construction for the best known minimal-time solution to the generalized problem by reducing the number of states needed and give non-minimaltime solutions to the original and generalized problem that use fewer states than the corresponding minimal-time solutions.