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The firing squad synchronization problem (FSSP) on cellular automata has been studied extensively for more than forty years, and a rich variety of synchronization algorithms have been proposed for not only one-dimensional arrays but two-dimensional arrays. In the present paper, we propose a simple recursive-halving based optimum-time synchronization algorithm that can synchronize any rectangle arrays of size m*n with a general at one corner in m+n+max(m, n)-3 steps. The algorithm is a natural expansion of the well-known FSSP algorithm proposed by Balzer [1967], Gerken [1987], and Waksman [1966] and it can be easily expanded to three-dimensional arrays, even to multi-dimensional arrays with a general at any position of the array.

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.