A generalized minimum-time minimum-state-change FSSP algorithm

Abstract

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. Here we consider the FSSP from a view point of state-change-complexity that models the energy consumption of SRAM-type storage with which cellular automata might be built. In the present paper, we construct an n − 2 + max(k, n − k + 1) minimum-time, Θ(n log n) minimum-statechange generalized FSSP (GFSSP, for short) algorithm for synchronizing any one-dimensional (1D) cellular automaton of length n, where the synchronization operations are started from any position k (1 ≤ k ≤ n) in the array. The realized minimum-time GFSSP algorithm can be implemented on a cellular automaton with 215 internal states and 4077 state-transition rules and has a minimum-state-change complexity. The algorithm is optimum not only in time but also in state-change complexity. The implemented minimum-time GFSSP algorithm is the first one having the minimum-state-change complexity. In addition, we also present a six-state 145-rule non-minimum-time, minimum-state-change GFSSP algorithm. The implemented GFSSP algorithm is a smallest one, known at present, in number of states of the finite state automaton.