Abstract
In cellular automata with multiple speeds for each cell i there is a positive integer p_i such that this cell updates its state still periodically but only at times which are a multiple of p_i. Additionally there is a finite upper bound on all p_i. Manzoni and Umeo have described an algorithm for these (one-dimensional) cellular automata which solves the Firing Squad Synchronization Problem. This algorithm needs linear time (in the number of cells to be synchronized) but for many problem instances it is slower than the optimum time by some positive constant factor. In the present paper we derive lower bounds on possible synchronization times and describe an algorithm which is never slower and in some cases faster than the one by Manzoni and Umeo and which is close to a lower bound (up to a constant summand) in more cases.
Highlights
The Firing Squad Synchronization Problem (FSSP) has a relatively long history in the field of cellular automata
Readers interested in more recent developments concerning several specialized problems and questions are referred to the survey [4]
In a “really” asynchronous setting it is impossible to achieve synchronization
Summary
The Firing Squad Synchronization Problem (FSSP) has a relatively long history in the field of cellular automata. As a middle ground the FSSP has been considered in what Manzoni and Umeo [2] have called CA with multiple speeds, abbreviated in the following as MS- CA. In these CA different cells may update their states at different times. There is a finite upper bound on all pi , so that it can be assumed that each cell has stored pi as part of its state This means that there are always times (namely the multiples of the least common multiple of all periods) when all cells update their states simultaneously.
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