Terminating Distributed Construction of Shapes and Patterns in a Fair Solution of Automata

In this work, we consider a solution of automata similar to Population Protocols and Network Constructors. The au-tomata, also called nodes, move passively in a well-mixed solution and can cooperate by interacting in pairs. Dur-ing every such interaction, the nodes, apart from updating their states, may also choose to connect to each other in order to start forming some required structure. The model introduced here is a more applied version of Network Con-structors, imposing geometrical constraints on the permissi-ble connections. Each node can connect to other nodes only via a very limited number of local ports, which implies that at any given time it has only a bounded number of neigh-bors. Connections are always made at unit distance and are perpendicular to connections of neighboring ports. Thoug

Terminating Distributed Construction of Shapes and Patterns in a Fair Solution of Automata

In this work, we consider a solution of automata (or nodes) that move passively in a well-mixed solution without being capable of controlling their movement. Nodes can cooperate by interacting in pairs and every such interaction may result in an update of their local states. Additionally, the nodes may also choose to connect to each other in order to start forming some required structure. Such nodes can be thought of as small programmable pieces of matter, like tiny nanorobots or programmable molecules. The model that we introduce here is a more applied version of network constructors, imposing physical (or geometric) constraints on the connections that the nodes are allowed to form. Each node can connect to other nodes only via a very limited number of local ports. Connections are always made at unit distance and are perpendicular to connections of neighboring ports, which makes the model capable of forming 2D or 3D shapes. We provide direct constructors for some basic shape construction problems, like spanning line, spanning square, and self-replication. We then develop new techniques for determining the computational and constructive capabilities of our model. One of the main novelties of our approach is that of exploiting the assumptions that the system is well-mixed and has a unique leader, in order to give terminating protocols that are correct with high probability. This allows us to develop terminating subroutines that can be sequentially composed to form larger modular protocols. One of our main results is a terminating protocol counting the size n of the system with high probability. We then use this protocol as a subroutine in order to develop our universal constructors, establishing that it is possible for the nodes to become self-organized with high probability into arbitrarily complex shapes while still detecting termination of the construction

On the Distributed Construction of Stable Networks in Polylogarithmic Parallel Time

We study the class of networks, which can be created in polylogarithmic parallel time by network constructors: groups of anonymous agents that interact randomly under a uniform random scheduler with the ability to form connections between each other. Starting from an empty network, the goal is to construct a stable network that belongs to a given family. We prove that the class of trees where each node has any k≥2 children can be constructed in O(logn) parallel time with high probability. We show that constructing networks that are k-regular is Ω(n) time, but a minimal relaxation to (l,k)-regular networks, where l=k−1, can be constructed in polylogarithmic parallel time for any fixed k, where k&gt;2. We further demonstrate that when the finite-state assumption is relaxed and k is allowed to grow with n, then k=loglogn acts as a threshold above which network construction is, again, polynomial time. We use this to provide a partial characterisation of the class of polylogarithmic time network constructors.</jats:p

Connectivity preserving network transformers

The Population Protocol model is a distributed model that concerns systems of very weak computational entities that cannot control the way they interact. The model of Network Constructors is a variant of Population Protocols capable of (algorithmically) constructing abstract networks. Both models are characterized by a fundamental inability to terminate. In this work, we investigate the minimal strengthenings of the latter that could overcome this inability. Our main conclusion is that initial connectivity of the communication topology combined with the ability of the protocol to transform the communication topology plus a few other local and realistic assumptions are sufficient to guarantee not only termination but also the maximum computational power that one can hope for in this family of models. The technique is to transform any initial connected topology to a less symmetric and detectable topology without ever breaking its connectivity during the transformation. The target topology of all of our transformers is the spanning line and we call Terminating Line Transformation the corresponding problem. We first study the case in which there is a pre-elected unique leader and give a time-optimal protocol for Terminating Line Transformation. We then prove that dropping the leader without additional assumptions leads to a strong impossibility result. In an attempt to overcome this, we equip the nodes with the ability to tell, during their pairwise interactions, whether they have at least one neighbor in common. Interestingly, it turns out that this local and realistic mechanism is sufficient to make the problem solvable. In particular, we give a very efficient protocol that solves Terminating Line Transformation when all nodes are initially identical. The latter implies that the model computes with termination any symmetric predicate computable by a Turing Machine of space $\Theta(n^2)$

Distributed Systems and Mobile Computing

The book is about Distributed Systems and Mobile Computing. This is a branch of Computer Science devoted to the study of systems whose components are in different physical locations and have limited communication capabilities. Such components may be static, often organized in a network, or may be able to move in a discrete or continuous environment. The theoretical study of such systems has applications ranging from swarms of mobile robots (e.g., drones) to sensor networks, autonomous intelligent vehicles, the Internet of Things, and crawlers on the Web. The book includes five articles. Two of them are about networks: the first one studies the formation of networks by agents that interact randomly and have the ability to form connections; the second one is a study of clustering models and algorithms. The three remaining articles are concerned with autonomous mobile robots operating in continuous space. One article studies the classical gathering problem, where all robots have to reach a common location, and proposes a fast algorithm for robots that are endowed with a compass but have limited visibility. The last two articles deal with the evacuations problem, where two robots have to locate an exit point and evacuate a region in the shortest possible time