A State-Efficient Mapping Scheme for Designing Two-Dimensional Firing Squad Synchronization Algorithms

Abstract

A simple and state-efficient mapping scheme is proposed for embedding any one-dimensional (1-D) firing squad synchronization algorithm onto two-dimensional (2-D) arrays, and some new 2-D synchronization algorithms based on the mapping scheme are presented. The proposed mapping scheme can be readily applied to the design of 2-D synchronization algorithms with fault tolerance, algorithms operating on multi-dimensional cellular arrays, and for the generalized case where the general is located at an arbitrary position on the 2-D array. First a six-state algorithm is developed that can synchronize any m×n rectangular array in 2(m+n)−4 steps. Then, a fourteen-state generalized algorithm is also developed that can synchronize any m×n rectangular array with the general at an arbitrary initial position (r,s) in m+n+max(r+s,m+n−r−s+2)−4 steps. The latter algorithm is interesting in that it includes a new optimum-time synchronization algorithm as a special case where the general is located at one corner. We progressively reduce the number of internal states of each cellular automaton on rectangular arrays, achieving six states and fourteen states for a rectangular array in non-optimum and optimum-time complexities, respectively. These are the smallest number of states reported to date for synchronizing rectangular arrays.