Simple 8-state minimal time solution to the firing squad synchronization problem

The firing squad synchronization problem with sub-generals

Synchronizing large scale networks is an important and fundamental computing primitive in parallel and distributed systems. The synchronization in cellular automata has been known as the firing squad synchronization problem (FSSP). The FSSP has been studied extensively for more than forty years, and a rich variety of synchronization algorithms have been proposed so far. In the present paper, we construct a survey on recent developments in FSSP algorithms for not only one-dimensional but two-dimensional, even multi-dimensional cellular automata, focusing on smaller solutions to the FSSP.

The firing squad synchronization problem (FSSP) has been studied extensively for more than fifty years, and a rich variety of synchronization algorithms has been proposed. In the present paper, we give a lower bound of FSSP algorithms for two-dimensional (2D) torus and present a minimum-time FSSP algorithm that can synchronize any 2D torus of size m × n in max (m, n) + floor min(m, n)/2 floor steps. It is shown that the lower bound and the algorithm given for the 2D torus is a generalization of the known results on one-dimensional (1D) rings (A. Berthiaume et al. [2004]) and it can be generalized to multi-dimensional rings.

The Firing Squad Synchronization Problem is one of the classical problems for cellular automata. In this paper we consider the case of more than one general. A synchronous and an asynchronous version of the problem are considered. In the latter case the generals may start their activities at different times. In the synchronous case there are optimumtime solutions. Very simple and elegant techniques for constructing one of them are the main contribution of this paper on the algorithmic side. For the asynchronous case an exact formula for the optimum synchronization time of each instance is derived. We prove that no CA can solve all instances in optimum time, but we describe a CA whose running time is very close to it; it only needs additional log n steps.

An Iterated Local Search to find many solutions of the 6-states Firing Squad Synchronization Problem

The firing squad synchronization problem 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 new and simpler optimum-time synchronization algorithm that can synchronize any rectangle array of size m ×n with a general at one corner in m + n + max (m, n) − 3 steps. An implementation for the algorithm in terms of local transition rules is also given.

Traditional cellular automata (CA) assume the presence of a single global clock regulating the update of all their cells. When this assumption is dropped, cells can update with different speeds, thus increasing the difficulty of solving synchronization problems. Here we solve the traditional and the generalized Firing Squad Synchronization Problem in dimension two and higher on multiple updating cycle CA.

A class of time-optimum FSSP algorithms for multi-dimensional cellular arrays

The generalized firing squad synchronization problem (GFSSP) is the well-known firing squad synchronization problem extended to arbitrarily connected networks of finite automata. When the transmission delays associated with the links of a network are allowed to be arbitrary nonnegative integers, the problem is called GFSSP-NUD (GFSSP with nonuniform delays). A solution of GFSSP-NUD is given for the first time. The solution is independent of the structure of the network and the actual delays of the links. The firing time of the solution is bounded by O( Delta /sup 3/+ tau /sub max/), where tau /sub max/ is the maximum transmission delay of any single link and Delta is the maximum transmission delay between the general and any other node of a given network. Extensions of GFSSP and GFSSP-NUD to networks with more than one general are presented. >

The firing squad synchronization problem on CA with multiple updating cycles

We consider several problems relating to strongly-connected directed networks of identical finite-state processors that work synchronously in discrete time steps. The conceptually simplest of these problems is the Wake Up and Report Problem; this is the problem of having a unique root processor send a signal to all other processors in the network and then enter a special done state only when all other processors have received the signal. The most difficult of the problems we consider is the classic Firing Squad Synchronization Problem; this is the much-studied problem of achieving macro-synchrorization in a network given micro-synchronization. We show via a complex algorithmic application of the snake data stucture first introduced in Even, Litman, and Winkler[6] that these two problems in particular are asymptotically time-equivalent up to a constant factor. This result leads immediately to the inclusion of several other related problems into this new asymptotic time-class.

We present an improved solution for the Firing Squad Synchronization Problem (FSSP) for digraph-structured P systems. We improve our previous FSSP algorithm by allowing the general to delegate a more central cell in the P system to send the final command to synchronize. With e being the eccentricity of the general and r denoting the radius of the underlying digraph, our new algorithm guarantees to synchronize all cells of the system, between e + 2r + 3 steps (for all trees structures and many digraphs) and up to 3e + 7 steps, in the worst case for any digraph. Empirical results show our new algorithm for tree-structured P systems yields at least 20% reduction in the number of steps needed to synchronize over the previous best-known algorithm.Keywordscellular automatafiring squad synchronizationP systems

Firing Squad Synchronization Problem on 2-D Arrays

Vehicle-to-vehicle (V2V) communication enables intention sharing among neighboring vehicles and thereby vehicles' motion can be coordinated to incorporate collision (or conflict) avoidance. In this paper, we propose a general framework to distribute the computational burden for coordinating multiple vehicles' stop-or-go motion. We formulate the multi-vehicle motion coordination problem as a total stopping time minimization problem under the constraint of mutual collision avoidance. The minimal stopping time solution, if such exists, is found using the A∗ search algorithm in the coordination diagram. The solution is executed efficiently by placing temporary virtual obstacles on the desired path. The communication latency is analyzed for practical applications. The simulations show the correctness and efficacy of our algorithm. An on-road experiment involving two autonomous vehicles, each equipped with V2V communication devices, was performed to demonstrate how deadlocks are successfully avoided.

Analysis of modern scientific literature reveals a tendency towards the widespread implementation of artificial intelligence, often without sufficient consideration of indirect efficiency factors such as economic costs, implementation complexity, maintenance, and information security. These studies focus more on the accuracy and performance metrics of artificial intelligence systems, while ignoring indirect but critically important efficiency factors. The aim of this article was to investigate the suitability of applying Artificial Intelligence technologies compared to simple algorithmic solutions within the context of developing software applications for personal asset management. The research methodology was based on a comprehensive comparative analysis of a developed simple algorithm for predicting the time of the next product order and the statistical Auto Regressive Integrated Moving Average (ARIMA) model, as a representative of more complex, albeit not deep, intelligent methods for time series forecasting. Based on the implementation and experiment using data that simulated a real-world scenario, the performance of both approaches was evaluated using key metrics, including accuracy, required computational resources, and implementation complexity. It was found that for tasks with limited data volumes and relatively simple behavioral patterns, which are characteristic of small personal asset management projects, the simple algorithm demonstrated comparable accuracy to the artificial intelligence ARIMA model. It was revealed that the simple algorithm operated with lower computational costs, measured in nanoseconds, and was characterised by lower implementation and subsequent maintenance complexity. The analysis showed that the use of ARIMA, despite its statistical power, was less justified under such conditions, requiring greater computational expenditures and deeper knowledge for its configuration. It was demonstrated that the execution time of ARIMA on small samples was higher (in microseconds), and its reliability was significantly dependent on the volume and quality of the input data. Thus, the necessity of a reasoned choice of technologies, based on the real needs and resource constraints of the project, was emphasised