A Simple Optimum-Time FSSP Algorithm for Multi-Dimensional Cellular Automata

TL;DR

This paper introduces a simple recursive-halving synchronization algorithm for multi-dimensional cellular automata, achieving optimal timing for rectangle arrays and extendable to higher dimensions with a general at any position.

Contribution

It presents a new recursive-halving based algorithm that optimally synchronizes multi-dimensional arrays, extending classical FSSP solutions to higher dimensions and arbitrary positions.

Findings

Synchronizes rectangle arrays in m+n+max(m, n)-3 steps

Extensible to three-dimensional and multi-dimensional arrays

General can be placed at any position in the array

Abstract

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.