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.Comment: 21 pages, 10 figure