A design of symmetrical six-state 3n-step firing squad synchronization algorithms and their implementations

Abstract

In 1994, Yunès [19] began to explore 3n-step firing squad synchronization algorithms and developed two seven-state synchronization algorithms for one-dimensional cellular arrays. His algorithms were so interesting in that he progressively decreased the number of internal states of each cellular automaton. In this paper, we propose a new symmetrical six-state 3n-step firing squad synchronization algorithm. Our result improves the seven-state 3n-step synchronization algorithms developed by Yunès [19]. The number six is the smallest one known at present in the class of 3n-step synchronization algorithms. A non-trivial and new symmetrical six-state 3n-step generalized firing squad synchronization algorithm is also given. In addition, we study a state-change complexity in 3n-step firing squad synchronization algorithms. We show that our algorithms have O(n2) state-change complexity, on the other hand, the thread-like 3n-step algorithms developed so far have O(n log n) state-change complexity. © Springer-Verlag Berlin Heidelberg 2006.