A State-Efficient Zebra-Like Implementation of Synchronization Algorithms for 2D Rectangular Cellular Arrays

• Published in: BIOMATH Vol. 1; no. 1
• Main Authors: Umeo, Hiroshi, Nomura, A.
• Format: Journal Article
• Language: English; Japanese
• Published: Institute of Mathematics and Informatics Bulgarian Academy of Sciences 25.09.2012; Bulgarian Academy of Sciences, Institute of Mathematics and Informatics
• Subjects: Biology (General); cellular automaton; firing squad synchronization problem; FSSP; Mathematics; QA1-939; QH301-705.5
• ISSN: 1314-684X; 1314-7218; 1314-7218
• DOI: 10.11145/j.biomath.2012.09.023

The firing squad synchronization problem on cellular automata has been studied extensively for more than forty years, and a rich variety of synchronization algorithms has been proposed for not only one-dimensional (1D) but two-dimensional (2D) arrays. In the present paper, we propose a simple and state-efficient mapping scheme: zebra-like mapping for implementing 2D synchronization algorithms for rectangular arrays. The zebra-like mapping we propose embeds two types of configurations alternately onto a 2D array like a zebra pattern, one configuration is a synchronization configuration of 1D arrays and the other is a stationary configuration which keeps its state unchanged until the final synchronization. It is shown that the mapping gives us a smallest, known at present, implementation of 2D FSSP algorithms for rectangular arrays. The implementation itself has a nice property that the correctness of the constructed transition rule set is clear and transparent. It is shown that there exists a nine-state 2D cellular automaton that can synchronize any (m x n) rectangle in (m+n+max(m,n)-3) steps.