Connected Line Recovery and Maintenance by Programmable Matter

Abstract

Several different distributed computational universes have been considered and studied within the interdisciplinary field called Programmable Matter. In this field, the matter is envisioned as a very large number of micro and nano-sized computational entities with limited capabilities programmed to collectively perform a task without the need for any central or external intervention. Within distributed computing, several theoretical active and hybrid models for programmable matter have been proposed. Within these models, a central concern has been the formation of geometric shapes; among them, the line is especially important. An important requirement, common to most research, is the connectivity of the operating elements at all times. In the extensive literature on the problem of shape formation in programmable matter systems, it is almost generally assumed that the system elements never fail. Hence the problem of reconfiguring the shape following the failure of some elements has been neglected. In this thesis we studied the problem of handling failures when the shape is the line. We considered first of all the Connected Line Recovery problem requiring the non-faulty elements to restore the line shape following the failures of some of the elements. We examined the instance of this problem in the programmable matter systems defined by the Metamorphic Robots and Amoebot models. We then studied the more complex Dynamic Line Maintenance problem when the faults are fully dynamic (i.e., can occur at any time). We examined the instance of this problem in the systems defined by the Amoebot and the Hybrid Programmable Matter models. For both problems and the systems considered, we provided a near complete feasibility characterization of problems, identifying the conditions necessary for their solvability, and constructively proving the sufficiency of those conditions. In particular, we presented solution protocols that operate correctly, maintain connectivity of the non-faulty entities, without constraints on the number of entities that will become faulty, nor on the location, nor (in the dynamic case) on the time of the occurrence of each fault. Our impossibility results hold even under the weak fully-synchronous scheduler, while the possibility results hold under the more difficult semi-synchronous one