Sublinear-Time Reconfiguration of Programmable Matter with Joint Movements

2026-03-11

AI Summary

This work addresses the centralized reconfiguration problem for the amoebot model of programmable matter under the joint movement paradigm, achieving sublinear-time structural transformation without relying on additional assumptions such as metamodules. By introducing a constant-time parallel movement primitive, a centralized scheduling algorithm, and geometric structural analysis, the study presents the first universal sublinear-time reconfiguration within the standard joint movement model. The main results demonstrate that any initial configuration can be reconfigured into a canonical line segment in $O(\sqrt{n} \log n)$ rounds, while certain structured configurations—such as spirals—admit constant-time transformations. These findings resolve an open question posed by Padalkin et al. and establish that the amoebot model inherently supports efficient and scalable reconfiguration capabilities.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• programmable matter
• reconfiguration
• sublinear-time
• amoebot
• joint movements

Innovation

• sublinear-time reconfiguration
• joint movements
• programmable matter
• amoebot model
• parallel primitives