The firing squad synchronization problem with sub-generals

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

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

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 firing squad synchronization problem on CA with multiple updating cycles

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.

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. >

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.

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.

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

Safe and simultaneous arrival of constant speed, constant altitude unmanned air vehicles (UAVs) on target is solved by design of paths of equal lengths. The starting point for our solution is the well-known Dubins path, which is composed of circular arc and line segments, thus requiring only one simple maneuver—constant rate turn. An explicit bound can be imposed on the rate during the design and the resulting paths are the minimum time solution of the problem. However, transition between arc and line segments entails discontinuous changes in lateral acceleration (latax), making this approach impractical for real fixed wing UAVs. Therefore, we replace the Dubins solution with a novel one, based on quintic Pythagorean hodograph (PH) curves, whose latax demand is continuous. The PH paths are designed to have lengths close to the lengths of the Dubins paths to stay close to the minimum time solution. To derive the PH paths, the Dubins solution is first interpreted in terms of differential geometry of curves using the path length and curvature as the key parameters. The curvature of a Dubins path is a piecewise constant and discontinuous function of its path length, which is a differential geometric expression of the discontinuous latax demand involved in transitions between the arc and the line segment. By contrast, the curvature of the PH path is a fifth order polynomial of its path length. This is not only continuous but also has enough design parameters (polynomial coefficients) to meet the latax (curvature) constraints (bounds) and make the PH solution close to the minimum time one. The solution involves the design of paths meeting the curvature constraint and is followed by producing multiple paths of equal length by increasing the lengths of the shorter paths to match the longest one. The safety constraint, intercollision avoidance is achieved by satisfying two conditions: minimum separation distance and nonintersection at equal distance. The offset curves of the PH path are used to design a safety region along each path.