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Local interactions of uncoordinated individuals produce the collective behaviors of many biological systems, inspiring much of the current research in programmable matter. A striking example is the spontaneous assembly of fire ants into "bridges" comprising their own bodies to traverse obstacles and reach sources of food. Experiments and simulations suggest that, remarkably, these ants always form one bridge - instead of multiple, competing bridges - despite a lack of central coordination. We argue that the reliable formation of a single bridge does not require sophistication on behalf of the individuals by provably reproducing this behavior in a self-organizing particle system. We show that the formation of a single bridge by the particles is a statistical inevitability of their preferences to move in a particular direction, such as toward a food source, and their preference for more neighbors. Two parameters, η and β, reflect the strengths of these preferences and determine the Gibbs stationary measure of the corresponding particle system’s Markov chain dynamics. We show that a single bridge almost certainly forms when η and β are sufficiently large. Our proof introduces an auxiliary Markov chain, called an "occupancy chain," that captures only the significant, global changes to the system. Through the occupancy chain, we abstract away information about the motion of individual particles, but we gain a more direct means of analyzing their collective behavior. Such abstractions provide a promising new direction for understanding many other systems of programmable matter. 

Throughput is a main performance objective in communication networks. This paper considers a fundamental maximum throughput routing problem-the all-or-nothing multicommodity flow (ANF) problem - in arbitrary directed graphs and in the practically relevant but challenging setting where demands can be (much) larger than the edge capacities. Hence, in addition to assigning requests to valid flows for each routed commodity, an admission control mechanism is required which prevents overloading the network when routing commodities. 

We present and rigorously analyze the behavior of a distributed, stochastic algorithm for separation and integration in self-organizing particle systems, an abstraction of programmable matter. Such systems are composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational power. We consider heterogeneous particle systems of two different colors and prove that these systems can collectively separate into different color classes or integrate, indifferent to color. We accomplish both behaviors with the same fully distributed, local, stochastic algorithm. Achieving separation or integration depends only on a single global parameter determining whether particles prefer to be next to other particles of the same color or not; this parameter is meant to represent external, environmental influences on the particle system. The algorithm is a generalization of a previous distributed, stochastic algorithm for compression (PODC '16), which can be viewed as a special case of separation where all particles have the same color. It is significantly more challenging to prove that the desired behavior is achieved in the heterogeneous setting, however, even in the bichromatic case we focus on. This requires combining several new techniques, including the cluster expansion from statistical physics, a new variant of the bridging argument of Miracle, Pascoe and Randall (RANDOM '11), the high-temperature expansion of the Ising model, and careful probabilistic arguments. 

This is a chapter in Book "Distributed Computing by Mobile Entities: Current Research in Moving and Computing", Springer Nature. The vision for programmable matter is to realize a physical substance that is scalable, versatile, instantly reconfigurable, safe to handle, and robust to failures. Programmable matter could be deployed in a variety of domain spaces to address a wide gamut of problems, including applications in construction, environmental science, synthetic biology, and space exploration. However, there are considerable engineering and computational challenges that must be overcome before such a system could be implemented. Towards developing efficient algorithms for novel programmable matter behaviors, the amoebot model for self-organizing particle systems and its variant, hybrid programmable matter, provide formal computational frameworks that facilitate rigorous algorithmic research. In this chapter, we discuss distributed algorithms under these models for shape formation, shape recognition, object coating, compression, shortcut bridging, and separation in addition to some underlying algorithmic primitives.