Logarithmic-Time Geodesically Convex Decomposition in Programmable Matter

2026-04-17

AI Summary

This work addresses the problem of efficiently decomposing an arbitrarily connected programmable matter structure into simple, geodesically convex subregions. Building upon the amoebot model and its reconfigurable circuit extension, the authors propose the first distributed algorithm that operates in logarithmic time for any amoebot configuration. Leveraging circuit-supported instantaneous communication, the algorithm partitions the structure into $O(|\mathcal{H}|)$ geodesically convex regions—where $\mathcal{H}$ denotes the set of holes—in $O(\log n)$ rounds. Furthermore, it achieves logarithmic-time complexity for fundamental tasks such as global maximum selection and spanning tree construction. This approach significantly enhances the efficiency and scalability of region decomposition in complex topologies.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Intelligent Robots: Motion and Path Planning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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Abstract

The decomposition of complex structures into simpler substructures is a powerful technique with a wide range of applications. We study the computation of decompositions in the context of programmable matter. The amoebot model is a well-established model for programmable matter, which places $n$ tiny robots called amoebots on the triangular grid. We consider the reconfigurable circuit extension of the geometric amoebot model, which allows amoebots to interconnect via so-called circuits. Amoebots can then instantaneously transmit simple beeps to all amoebots connected by the same circuit. Using reconfigurable circuits, previous papers have described a linear-time triangulation algorithm, and a logarithmic-time decomposition algorithm into so-called tunnel regions. Both algorithms only work on a restricted class of amoebot structures. In this paper, we define a decomposition into $O(|\mathcal H|)$ simple, geodesically convex regions for arbitrary amoebot structures, and show how it can compute such a decomposition in $O(\log n)$ rounds, where $|\mathcal H|$ denotes the number of holes in the amoebot structure. As a byproduct, we also improve the global maxima algorithm of Padalkin et al. (Nat. Comput., 2024) for special cases and with that also their spanning tree algorithm to $O(\log n)$ rounds w.h.p.

Problem

• programmable matter
• amoebot model
• geodesically convex decomposition
• reconfigurable circuits
• structural decomposition

Innovation

• geodesically convex decomposition
• programmable matter
• amoebot model
• reconfigurable circuits
• logarithmic-time algorithm