We consider programmable matter that consists of computationally limited devices (which we call particles) that are able to self-organize in order to achieve some collective goal without the need for central control or external intervention. We use the geometric amoebot model to describe such self-organizing particle systems, which defines how particles can actively move and establish or release bonds with one another. Under this model, we investigate the feasibility of solving fundamental problems relevant to programmable matter. In this paper, we present an efficient local-control algorithm which solves the leader election problem in O(n) asynchronous rounds with high probability, where n is the number of particles in the system. Our algorithm relies only on local information (e.g., particles do not have unique identifiers, any knowledge of n, or any sort of global coordinate system), and requires only constant memory per particle.
The model @cite_10 is a model for self-organizing programmable matter that aims to provide a framework for rigorous algorithmic research for nano-scale systems. @cite_8 , the authors describe a leader election algorithm for an abstract (synchronous) version of the amoebot model that decides the problem in expected linear time. Recently, a universal shape formation algorithm @cite_0 , a universal coating algorithm @cite_21 and a Markov chain algorithm for the compression problem @cite_19 were introduced, showing that there is potential to investigate a wide variety of problems under this model.
@cite_8 @cite_21 @cite_0 @cite_19 @cite_10
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Molecular self-assembly presents a bottom-up' approach to the fabrication of objects specified with nanometre precision. DNA molecular structures and intermolecular interactions are particularly amenable to the design and synthesis of complex molecular objects. We report the design and observation of two-dimensional crystalline forms of DNA that self-assemble from synthetic DNA double-crossover molecules. Intermolecular interactions between the structural units are programmed by the design of sticky ends' that associate according to Watson-Crick complementarity, enabling us to create specific periodic patterns on the nanometre scale. The patterned crystals have been visualized by atomic force microscopy. Imagine coating buildings and bridges with smart particles (also coined smart paint) that monitor structural integrity and sense and report on traffic and wind loads, leading to technology that could do such inspection jobs faster and cheaper and increase safety at the same time. In this paper, we study the problem of uniformly coating objects of arbitrary shape in the context of self-organizing programmable matter, i.e., programmable matter which consists of simple computational elements called particles that can establish and release bonds and can actively move in a self-organized way. Particles are anonymous, have constant-size memory, and utilize only local interactions in order to coat an object. We continue the study of our universal coating algorithm by focusing on its runtime analysis, showing that our algorithm terminates within a linear number of rounds with high probability. We also present a matching linear lower bound that holds with high probability. We use this lower bound to show a linear lower bound on the competitive gap between fully local coating algorithms and coating algorithms that rely on global information, which implies that our algorithm is also optimal in a competitive sense. Simulation results show that the competitive ratio of our algorithm may be better than linear in practice. We envision programmable matter consisting of systems of computationally limited devices (which we call particles) that are able to self-organize in order to achieve a desired collective goal without the need for central control or external intervention. Central problems for these particle systems are shape formation and coating problems. In this paper, we present a universal shape formation algorithm which takes an arbitrary shape composed of a constant number of equilateral triangles of unit size and lets the particles build that shape at a scale depending on the number of particles in the system. Our algorithm runs in O(√n) asynchronous execution rounds, where @math is the number of particles in the system, provided we start from a well-initialized configuration of the particles. This is optimal in a sense that for any shape deviating from the initial configuration, any movement strategy would require Ω(√n) rounds in the worst case (over all asynchronous activations of the particles). Our algorithm relies only on local information (e.g., particles do not have ids, nor do they know n, or have any sort of global coordinate system), and requires only a constant-size memory per particle. We consider programmable matter as a collection of simple computational elements (or particles) with limited (constant-size) memory that self-organize to solve system-wide problems of movement, configuration, and coordination. Here, we focus on the compression problem, in which the particle system gathers as tightly together as possible, as in a sphere or its equivalent in the presence of some underlying geometry. More specifically, we seek fully distributed, local, and asynchronous algorithms that lead the system to converge to a configuration with small perimeter. We present a Markov chain based algorithm that solves the compression problem under the geometric amoebot model, for particle systems that begin in a connected configuration with no holes. The algorithm takes as input a bias parameter λ, where λ > 1 corresponds to particles favoring inducing more lattice triangles within the particle system. We show that for all λ > 5, there is a constant α > 1 such that at stationarity with all but exponentially small probability the particles are α-compressed, meaning the perimeter of the system configuration is at most α ⋅ pmin, where pmin is the minimum possible perimeter of the particle system. We additionally prove that the same algorithm can be used for expansion for small values of λ in particular, for all 0 The term programmable matter refers to matter which has the ability to change its physical properties (shape, density, moduli, conductivity, optical properties, etc.) in a programmable fashion, based upon user input or autonomous sensing. This has many applications like smart materials, autonomous monitoring and repair, and minimal invasive surgery, so there is a high relevance of this topic to industry and society in general. While programmable matter has just been science fiction more than two decades ago, a large amount of research activities can now be seen in this field in the recent years. Often programmable matter is envisioned, as a very large number of small locally interacting computational . We propose the Amoebot model, a new model which builds upon this vision of programmable matter. Inspired by the behavior of amoeba, the Amoebot model offers a versatile framework to model self-organizing particles and facilitates rigorous algorithmic research in the area of programmable matter.