Coordinating Amoebots via Reconfigurable Circuits

We consider an extension to the geometric amoebot model that allows amoebots to form so-called circuits. Given a connected amoebot structure, a circuit is a subgraph formed by the amoebots that permits the instant transmission of signals. We show that such an extension allows for significantly faster solutions to a variety of problems related to programmable matter. More specifically, we provide algorithms for leader election, consensus, compass alignment, chirality agreement, and shape recognition. Leader election can be solved in Θ(logn) rounds, with high probability (w.h.p.), consensus in O(1) rounds, and both, compass alignment and chirality agreement, can be solved in O(logn) rounds, w.h.p. For shape recognition, the amoebots have to decide whether the amoebot structure forms a particular shape. We show that the amoebots can detect a shape composed of triangles within O(1) rounds. Finally, we show how the amoebots can detect a parallelogram with linear and polynomial side ratio within Θ(logn) rounds, w.h.p.