Sliding Squares in Parallel

TL;DR

This work extends the sliding squares model to a parallel, connectivity-preserving setting, analyzing both unlabeled and labeled variants. It establishes tight complexity results: in the unlabeled case, solving in a single parallel step is NP-complete, while an in-place algorithm achieves worst-case optimal $O(P)$ parallel steps; in the labeled case, makespan $1$ is poly-time but makespan $2$ is NP-complete, with related approximation hardness. The authors develop scale-based techniques to canonical intermediate forms (notably $3$-scaled configurations) and present a robust three-phase algorithm that achieves a worst-case optimal reconfiguration in $O(P_1+P_2)$ steps, extendable to the labeled setting with mild relaxations. The approach leverages skeleton/backbone structures and sweep-line meta-modules to coordinate parallel motions while maintaining connectivity, enabling both theoretical optimality and practical scheduling insights for modular robotics. Overall, the work advances understanding of parallel, connectivity-preserving reconfiguration and provides algorithmic frameworks transferable to universal-in-place and scaffold-based strategies.

Abstract

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.

Figures (7)

• Figure 1: Our model allows for two types of moves to occur in parallel, collision-free transformations. In this paper, we show the symmetric difference of a transformation using turquoise and yellow.
• Figure 2: We illustrate the relation of the integer grid and configurations.
• Figure 3: Illegal transformations: Moving non-free modules causes disconnections.
• Figure 4: Collisions: (a) Modules cannot swap places or (b) enter the same cell simultaneously, (c) moves cannot meet at endpoints orthogonally, and (d) speed mismatches must be avoided.
• Figure 5: An overview of our hardness reduction and the two types of gadget used.

Theorems & Definitions (14)

• Proposition 1
• Theorem 2
• Corollary 5
• Corollary 6
• Theorem 7
• Corollary 10
• Theorem 11
• Lemma 12
• Lemma 13
• Corollary 14