A twelve-state optimum-time synchronization algorithm for two-dimensional rectangular cellular arrays

Abstract

The firing squad synchronization problem has been studied extensively for more than 40 years [1-18]. The present authors are involved in research on firing squad synchronization algorithms on two-dimensional (2-D) rectangular cellular arrays. Several synchronization algorithms on 2-D arrays have been proposed, including Beyer [2], Grasselli [3], Kobayashi [4], Shinahr [10], Szwerinski [12] and Umeo et al. [13, 15]. To date, the smallest number of cell states for which an optimum-time synchronization algorithm has been developed is 14 for rectangular array, achieved by Umeo et al. [15]. In the present paper, we propose a new optimum-time synchronization algorithm that can synchronize any 2-D m × n rectangular arrays in m+n+max(m,n) -3 steps. We progressively reduce the number of internal states of each cellular automaton on rectangular arrays, achieving twelve states. This is the smallest number of states reported to date for synchronizing rectangular arrays in optimum-step. © Springer-Verlag Berlin Heidelberg 2005.