A synchronization problem on 1-bit communication cellular automata

Abstract

We study a classical firing squad synchronization problem for a large scale of one- and two-dimensional cellular automata having 1-bit inter-cell communications (CA1-bit). First, it is shown that there exists a one-dimensional CA1-bit that can synchronize n cells with the general on the kth bell in n + max(k, n - k + 1) steps, where the performance is two steps larger than the optimum one that was developed for O(1)-bit communication model. Next, we give a two-dimensional CA1-bit which can synchronize any n x n square and m x n rectangular arrays in 2n - 1 and m + n + max(m, n) steps, respectively. Lastly, we propose a generalized synchronization algorithm that operates in m + n + max(r + s, m + n - r - s) + O(1) steps on two-dimensional m x n rectangular arrays with the general located at an arbitrary position (r, s) of the array, where 1 ≤ r ≤ m and 1 ≤ 3 ≤ n. The time complexities for the first three algorithms developed are one to four steps larger than optimum ones proposed for O(1)-bit communication models. We show that there still exist several new interesting synchronization algorithms on CA1-bit although more than 40 years have passed since the development of the problem. © Springer-Verlag Berlin Heidelberg 2003.