We consider the theoretical model of Crystalline robots, which have been introduced and prototyped by the robotics community. These robots consist of independently manipulable unit-square atoms that can extend/contract arms on each side and attach/detach from neighbors. These operations suffice to reconfigure between any two given (connected) shapes. The worst-case number of sequential moves required to transform one connected configuration to another is known to be Theta(n). 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(nlogn). The result is the first (theoretically) almost-instantaneous universally reconfigurable robot built from simple units.
@article{arxiv.0908.2440,
title = {Reconfiguration of 3D Crystalline Robots Using O(log n) Parallel Moves},
author = {Greg Aloupis and Sebastien Collette and Erik D. Demaine and Stefan Langerman and Vera Sacristan and Stefanie Wuhrer},
journal= {arXiv preprint arXiv:0908.2440},
year = {2009}
}