State-efficient firing squad synchronization protocols for communication-restricted cellular automata

Abstract

In this paper, we study a trade-off between internal states and communication bits in firing squad synchronization protocols for k-bit communication-restricted cellular automata (CAk-bit) and propose several time-optimum state-efficient bit-transfer-based synchronization protocols. It is shown that there exists a 1-state CA5-bit that can synchronize any n cells in 2n -2 optimum-step. The result is interesting, since we know that there exists no 4-state synchronization algorithm on conventional O(1)-bit communication cellular automata. A bit-transfer complexity is also introduced to measure the efficiency of synchronization protocols. We show that Ω(n log n) bit-transfer is a lower-bound for synchronizing n cells in (2n - 2) steps. In addition, each optimum-time/non-optimum-time synchronization protocols, presented in this paper, has an O(n2) bit-transfer complexity, respectively. © Springer-Verlag Berlin Heidelberg 2006.