A synchronization problem on 1-bit communication cellular automata

• Authors: Hiroshi Umeo;Koshi Michisaka;Naoki Kamikawa
• Venue: ICCS'03 Proceedings of the 1st international conference on Computational science: PartI
• Year: 2003

We study a classical firing squad synchronization problem for a large sale 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 cell 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 × n square and m × 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 × n rectangular arrays with the general located at an arbitrary position (r, s) of the array, where 1 ≤ r ≤ m and 1 ≤ s ≤ 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 algoritms on CA1-bit although more than 40 years have passed since the development of the problem.