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 (i.e. the system is homogeneous). Moreover, we assume pairwise interactions between the processes that are scheduled by an adversary. The only constraint on the adversary scheduler is that it must be fair. 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 the 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. We provide proofs of correctness for all of our protocols and analyze the expected time to convergence of most of them under a uniform random scheduler that selects the next pair of interacting processes uniformly at random from all such pairs. 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.
Nature has an intrinsic ability to form complex structures and networks via a process known as . By self-assembly, small components (like e.g. molecules) automatically assemble into large, and usually complex structures (like e.g. a crystal). There is an abundance of such examples in the physical world. Lipid molecules form a cell's membrane, ribosomal proteins and RNA coalesce into functional ribosomes, and bacteriophage virus proteins self-assemble a capsid that allows the virus to invade bacteria @cite_19 . Mixtures of RNA fragments that self-assemble into self-replicating ribozymes spontaneously form cooperative catalytic cycles and networks. Such cooperative networks grow faster than selfish autocatalytic cycles indicating an intrinsic ability of RNA populations to evolve greater complexity through cooperation @cite_12 . Through billions of years of prebiotic molecular selection and evolution, nature has produced a basic set of molecules. By combining these simple elements, natural processes are capable of fashioning an enormously diverse range of fabrication units, which can further self-organize into refined structures, materials and molecular machines that not only have high precision, flexibility and error-correction capacity, but are also self-sustaining and evolving. In fact, nature shows a strong preference for bottom-up design.
@cite_19 @cite_12
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Self-assembly is the process by which small components automatically assemble themselves into large, complex structures. Examples in nature abound: lipids self-assemble a cell's membrane, and bacteriophage virus proteins self-assemble a capsid that allows the virus to invade other bacteria. Even a phenomenon as simple as crystal formation is a process of self-assembly. How could such a process be described as "algorithmic?" The key word in the first sentence is automatically. Algorithms automate a series of simple computational tasks. Algorithmic self-assembly systems automate a series of simple growth tasks, in which the object being grown is simultaneously the machine controlling its own growth. Two complementary strategies can be used in the fabrication of molecular biomaterials. In the 'top-down' approach, biomaterials are generated by stripping down a complex entity into its component parts (for example, paring a virus particle down to its capsid to form a viral cage). This contrasts with the 'bottom-up' approach, in which materials are assembled molecule by molecule (and in some cases even atom by atom) to produce novel supramolecular architectures. The latter approach is likely to become an integral part of nanomaterials manufacture and requires a deep understanding of individual molecular building blocks and their structures, assembly properties and dynamic behaviors. Two key elements in molecular fabrication are chemical complementarity and structural compatibility, both of which confer the weak and noncovalent interactions that bind building blocks together during self-assembly. Using natural processes as a guide, substantial advances have been achieved at the interface of nanomaterials and biology, including the fabrication of nanofiber materials for three-dimensional cell culture and tissue engineering, the assembly of peptide or protein nanotubes and helical ribbons, the creation of living microlenses, the synthesis of met al nanowires on DNA templates, the fabrication of peptide, protein and lipid scaffolds, the assembly of electronic materials by bacterial phage selection, and the use of radiofrequency to regulate molecular behaviors. We describe a computational model for studying the complexity of self-assembled structures with active molecular components. Our model captures notions of growth and movement ubiquitous in biological systems. The model is inspired by biology's fantastic ability to assemble biomolecules that form systems with complicated structure and dynamics, from molecular motors that walk on rigid tracks and proteins that dynamically alter the structure of the cell during mitosis, to embryonic development where large scale complicated organisms efficiently grow from a single cell. Using this active self-assembly model, we show how to efficiently self-assemble shapes and patterns from simple monomers. For example we show how to grow a line of monomers in time and number of monomer states that is merely logarithmic in its length. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time polylogarithmic in the size of the shape plus roughly the time required to run a Turing machine deciding whether or not a given pixel is in the shape. We do this while keeping the number of monomer types logarithmic in shape size, plus monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the fundamental efficiency advantage of active self-assembly over passive self-assembly and motivates experimental effort to construct self-assembly systems with active molecular components. We describe a computational model for studying the complexity of self-assembled structures with active molecular components. Our model captures notions of growth and movement ubiquitous in biological systems. The model is inspired by biology's fantastic ability to assemble biomolecules that form systems with complicated structure and dynamics, from molecular motors that walk on rigid tracks and proteins that dynamically alter the structure of the cell during mitosis, to embryonic development where large-scale complicated organisms efficiently grow from a single cell. Using this active self-assembly model, we show how to efficiently self-assemble shapes and patterns from simple monomers. For example, we show how to grow a line of monomers in time and number of monomer states that is merely logarithmic in the length of the line. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time that is polylogarithmic in the size of the shape, plus roughly the time required to run a Turing machine deciding whether or not a given pixel is in the shape. We do this while keeping the number of monomer types logarithmic in shape size, plus those monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the efficiency advantages of active self-assembly over passive self-assembly and motivates experimental effort to construct general-purpose active molecular self-assembly systems.