Abstract
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.
Highlights
We study a synchronization problem that gives a finite-state protocol for synchronizing large scale cellular automata
The synchronization in cellular automata has been known as the firing squad synchronization problem (FSSP, for short) which was originally proposed by J
The first optimum-time synchronization algorithm developed by Beyer [1969] and Shinahr [1974] for rectangle arrays operates as follows: We assume that an initial general is located on C11 on a rectangular array of size m×n
Summary
We study a synchronization problem that gives a finite-state protocol for synchronizing large scale cellular automata. The synchronization in cellular automata has been known as the firing squad synchronization problem (FSSP, for short) which was originally proposed by J. We propose a simple and stateefficient mapping scheme: zebra-like mapping for implementing 2D synchronization algorithms. The zebra-like mapping we propose embeds two types of configurations alternately onto a 2D array like a zebra-like 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. The mapping gives us a smallest, known at present, implementation of 2D FSSP algorithms. H. Umeo et al, A State-Efficient Zebra-Like Implementation of Synchronization Algorithms.
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