We consider algorithmic problems motivated by modular robotic reconfiguration, for which we are given $n$ square-shaped modules (or robots) 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 from Computational Geometry has aimed at minimizing the total number of moves, resulting in schedules that can perform reconfigurations in $\mathcal{O}(n^2)$ moves, or $\mathcal{O}(nP)$ for an arrangement of bounding box perimeter size $P$, but are fully sequential. Here we provide first results in the sliding square model that exploit parallel robot motion, resulting in an optimal speedup to achieve reconfiguration in worst-case optimal makespan of $\mathcal{O}(P)$. We also provide tight bounds on the complexity of the problem by showing that even deciding the possibility of reconfiguration within makespan $1$ is NP-complete in the unlabeled case; for the labeled case, deciding reconfiguration within makespan $2$ is NP-complete, while makespan $1$ can be decided in polynomial time.

翻译：我们研究受模块化机器人重构启发的算法问题，其中给定$n$个方形模块（或机器人）处于（带标记或无标记的）初始构型，需要寻找通过滑动操作将其转换为目标构型的调度方案，且在整个过程中始终保持构型的连通性。近期计算几何领域的研究致力于最小化总操作步数，实现了$\mathcal{O}(n^2)$步或$\mathcal{O}(nP)$步（其中$P$为包围盒周长）的重构调度，但这些方案均为完全串行执行。本文首次在滑动方块模型中利用机器人并行运动特性，实现了最优加速，在$\mathcal{O}(P)$的最坏情况最优完工时间内完成重构。我们还通过证明以下结论给出了该问题复杂度的紧确界：在无标记情况下，即使仅判定能否在完工时间$1$内完成重构也是NP完全的；对于带标记情况，判定完工时间$2$内的重构可能性是NP完全的，而完工时间$1$的情况可在多项式时间内判定。