Sliding Squares in Parallel

We consider algorithmic problems motivated by modular robotic reconfiguration in the sliding square model, in which we are given $n$ square-shaped modules in a (labeled or unlabeled) start configuration and need to find a schedule of sliding moves to transform it into a desired goal configuration, maintaining connectivity of the configuration at all times. Recent work has aimed at minimizing the total number of moves, resulting in fully sequential schedules that perform reconfiguration in $\mathcal{O}(nP)$ moves for arrangements of bounding box perimeter size $P$, or a number of moves linear in the sum of module coordinates in the start and target arrangements. We extend the model to leverage the possibility of parallel motion, thereby reducing worst-case makespans by a factor linear in $n$. Our work presents tight results both in terms of complexity and algorithms: We show that deciding the existence of a single parallel reconfiguration step that solves an instance is NP-complete for unlabeled modules, but can be solved efficiently in the labeled setting. Nevertheless, deciding whether a labeled instance can be solved in two parallel steps is NP-complete. Finally, we describe an algorithm to perform in-place reconfiguration in worst-case optimal $\mathcal{O}(P)$ parallel steps for the unlabeled setting. This algorithm has a straight-forward extension to the labeled setting with slight relaxations to either the reconfiguration time or space constraint.