Logarithmic-Time Geodesically Convex Decomposition in Programmable Matter

TL;DR

This paper presents an efficient logarithmic-time algorithm for decomposing complex amoebot structures into geodesically convex regions, improving previous methods and applicable to arbitrary structures.

Contribution

It introduces a novel decomposition method into convex regions for any amoebot structure and achieves $O( ext{log } n)$ computation rounds, extending prior restricted algorithms.

Findings

Decomposition into $O(| ext{holes}|)$ convex regions for arbitrary structures.

Achieves $O( ext{log } n)$ rounds for decomposition.

Improves global maxima and spanning tree algorithms to $O( ext{log } n)$ rounds.

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.