The idea behind universal coating is to have a thin layer of a specific substance covering an object of any shape so that one can measure a certain condition (like temperature or cracks) at any spot on the surface of the object without requiring direct access to that spot. We study the universal coating problem in the context of self-organizing programmable matter consisting of simple computational elements, called particles, that can establish and release bonds and can actively move in a self-organized way. Based on that matter, we present a worst-case work-optimal universal coating algorithm that uniformly coats any object of arbitrary shape and size that allows a uniform coating. Our particles are anonymous, do not have any global information, have constant-size memory, and utilize only local interactions.
In a recent paper @cite_18 , Michail and Spirakis propose a model for network construction that is inspired by population protocols @cite_9 . The population protocol model relates to self-organizing particles systems, but is also intrinsically different: agents (which would correspond to our particles) freely move in space and can establish connections to any other agent in the system at any point in time, following the respective probabilistic distribution. In the paper the authors focus on network construction for specific topologies (e.g., spanning line, spanning star, etc.). However, in principle, it would be possible to adapt their approach also for studying coating problems under the population protocol model.
@cite_9 @cite_18
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We extend here the Population Protocol (PP) model of (2004, 2006) [2,4] in order to model more powerful networks of resource-limited agents that are possibly mobile. The main feature of our extended model, called the Mediated Population Protocol (MPP) model, is to allow the edges of the interaction graph to have states that belong to a constant-size set. We then allow the protocol rules for pairwise interactions to modify the corresponding edge state. The descriptions of our protocols preserve both the uniformity and anonymity properties of PPs, that is, they do not depend on the size of the population and do not use unique identifiers. We focus on the computational power of the MPP model on complete interaction graphs and initially identical edges. We provide the following exact characterization of the class MPS of stably computable predicates: a predicate is in MPS iff it is symmetric and is in NSPACE(n^2). We describe and analyze a 3-state one-way population protocol for approximate majority in the model in which pairs of agents are drawn uniformly at random to interact. Given an initial configuration of x's, y's and blanks that contains at least one non-blank, the goal is for the agents to reach consensus on one of the values x or y. Additionally, the value chosen should be the majority non-blank initial value, provided it exceeds the minority by a sufficient margin. We prove that with high probability n agents reach consensus in O(n log n) interactions and the value chosen is the majority provided that its initial margin is at least ω(√n log n). This protocol has the additional property of tolerating Byzantine behavior in o(√n) of the agents, making it the first known population protocol that tolerates Byzantine agents. Turning to the register machine construction from [2], we apply the 3-state approximate majority protocol and other techniques to speed up the per-step parallel time overhead of the simulation from O(log4 n) to O(log2 n). To increase the robustness of the phase clock at the heart of the register machine, we describe a consensus version of the phase clock and present encouraging simulation results; its analysis remains an open problem. Population protocols are used as a theoretical model for a collection (or population) of tiny mobile agents that interact with one another to carry out a computation. The agents are identically programmed finite state machines. Input values are initially distributed to the agents, and pairs of agents can exchange state information with other agents when they are close together. The movement pattern of the agents is unpredictable, but subject to some fairness constraints, and computations must eventually converge to the correct output value in any schedule that results from that movement. This framework can be used to model mobile ad hoc networks of tiny devices or collections of molecules undergoing chemical reactions. This chapter surveys results that describe what can be computed in various versions of the population protocol model. In this work, we study protocols so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol. Moreover, we assume pairwise interactions between the processes that are scheduled by a fair adversary. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of their connection and updates all of them. Initially all connections are inactive and the goal is for the processes, after interacting and activating deactivating connections for a while, to end up with a desired stable network. We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. The expected time to convergence of our protocols is analyzed under a uniform random scheduler. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks. We additionally show how to partition the population into k supernodes, each being a line of log k nodes, for the largest such @math . This amount of local memory is sufficient for the supernodes to obtain unique names and exploit their names and their memory to realize nontrivial constructions.