Sublinear-Time Reconfiguration of Programmable Matter with Joint Movements

TL;DR

The findings demonstrate that the joint movement model supports sublinear reconfiguration without auxiliary assumptions, and affirmatively answer the open problem by Padalkin et al. whether a within-the-model sublinear-time universal reconfiguration algorithm is possible, by proving that any structure can be reconfigured into a canonical line-segment structure in O(\sqrt{n}\log n) rounds.

Abstract

We study centralized reconfiguration problems for geometric amoebot structures. A set of $n$ amoebots occupy nodes on the triangular grid and can reconfigure via expansion and contraction operations. We focus on the joint movement extension, where amoebots may expand and contract in parallel, enabling coordinated motion of larger substructures. Prior work introduced this extension and analyzed reconfiguration under additional assumptions such as metamodules. In contrast, we investigate the intrinsic dynamics of reconfiguration without such assumptions by restricting attention to centralized algorithms, leaving distributed solutions for future work. We study the reconfiguration problem between two classes of amoebot structures $A$ and $B$: For every structure $S\in A$, the goal is to compute a schedule that reconfigures $S$ into some structure $S'\in B$. Our focus is on sublinear-time algorithms. We affirmatively answer the open problem by Padalkin et al. (Auton. Robots, 2025) whether a within-the-model sublinear-time universal reconfiguration algorithm is possible, by proving that any structure can be reconfigured into a canonical line-segment structure in $O(\sqrt{n}\log n)$ rounds. Additionally, we give a constant-time algorithm for reconfiguring any spiral structure into a line segment. These results are enabled by new constant-time primitives that facilitate efficient parallel movement. Our findings demonstrate that the joint movement model supports sublinear reconfiguration without auxiliary assumptions. A central open question is whether universal reconfiguration within this model can be achieved in polylogarithmic or even constant time.

Figures (33)

• Figure 1: Amoebot model and axes. The black nodes and edges indicate the nodes and edges occupied by amoebots. The red edges indicate bonds between the amoebots. The gray edges indicate the triangular grid $G_{\Delta}$. We omit the grid in all other figures.
• Figure 2: Expansion.
• Figure 3: Contraction.
• Figure 4: Handover.
• Figure 6: Results. Classes with a thicker frame indicate canonical classes. Red arrows point to superclasses and with that indicate trivial solutions. Note that we do not depict all superclasses. Black arrows indicate results by Padalkin et al.DBLP:journals/arobots/PadalkinKS25. Note that their superconstant algorithms simulate algorithms from DBLP:conf/isaac/AloupisCDLAW08DBLP:journals/arobots/HurtadoMRA15. They define rhombical and hexagonal metamodules. Rhombical ( Hexagonal) contains all amoebot structure that consist of rhombical (hexagonal) metamodules. RhombicalLine ( HexagonalLine) contains all amoebot structure forming a line of rhombical (hexagonal) metamodules. Ring contains canonical amoebot structures forming a ring (see DBLP:conf/isaac/AloupisCDLAW08 for details). $m$ denotes the number of metamodules. Blue edges indicate our results. Note that our algorithm that reconfigures bounded structures into histograms simulates an algorithm from DBLP:journals/comgeo/AloupisCDDFLORAW09.

Theorems & Definitions (18)

• Lemma 1
• Lemma 2
• Remark 3
• Lemma 4
• Lemma 5
• Lemma 6
• Remark 7
• Theorem 8
• Lemma 9
• Theorem 10