Reconfiguration of Cube-Style Modular Robots Using O(log n) Parallel Moves

We consider a model of recofigurable robot, introduced and prototyped by the robotics community. The robot consists of independently manipulable unit-square atoms that can extend/contract arms on each side and attach/detach from neighbors. The optimal worst-case number of sequential moves required to transform one connected configuration to another was shown to be Theta(n) at ISAAC 2007. However, in principle, atoms can all move simultaneously. We develop a parallel algorithm for reconfiguration that runs in only O(log n) parallel steps, although the total number of operations increases slightly to Theta(n log n). The result is the first (theoretically) almost-instantaneous universally reconfigurable robot built from simple units.