We consider an extension to the geometric amoebot model that allows amoebots to form so-called circuits. Given a connected amoebot structure, a circuit is a subgraph formed by the amoebots that permits the instant transmission of signals. We show that such an extension allows for significantly faster solutions to a variety of problems related to programmable matter. More specifically, we provide algorithms for leader election, consensus, compass alignment, chirality agreement, and shape recognition. Leader election can be solved in Θ(logn) rounds, with high probability (w.h.p.), consensus in O(1) rounds, and both, compass alignment and chirality agreement, can be solved in O(logn) rounds, w.h.p. For shape recognition, the amoebots have to decide whether the amoebot structure forms a particular shape. We show that the amoebots can detect a shape composed of triangles within O(1) rounds. Finally, we show how the amoebots can detect a parallelogram with linear and polynomial side ratio within Θ(logn) rounds, w.h.p. 

Coordinating Amoebots via Reconfigurable Circuits

We consider geometric collision-detection problems for modular reconfigurable robots. Assuming the nodes (modules) are connected squares on a grid, we investigate the complexity of deciding whether collisions may occur, or can be avoided, if a set of expansion and contraction operations is executed. We study both discrete- and continuous-time models, and allow operations to be coupled into a single parallel group. Our algorithms to decide if a collision may occur run in $O(n^2\log^2 n)$ time, $O(n^2)$ time, or $O(n\log^2 n)$ time, depending on the presence and type of coupled operations, in a continuous-time model for a modular robot with $n$ nodes. To decide if collisions can be avoided, we show that a very restricted version is already NP-complete in the discrete-time model, while the same problem is polynomial in the continuous-time model. A less restricted version is NP-hard in the continuous-time model.

pdf/collision-detection-for-modular-robots-it-is-easy-to-cause-22svmdik.pdf

Collision Detection for Modular Robots - it is easy to cause collisions and hard to avoid them

Several different distributed computational universes have been considered and studied within the interdisciplinary field called Programmable Matter. In this field, the matter is envisioned as a very large number of micro and nano-sized computational entities with limited capabilities programmed to collectively perform a task without the need for any central or external intervention. Within distributed computing, several theoretical active and hybrid models for programmable matter have been proposed. Within these models, a central concern has been the formation of geometric shapes; among them, the line is especially important. An important requirement, common to most research, is the connectivity of the operating elements at all times. In the extensive literature on the problem of shape formation in programmable matter systems, it is almost generally assumed that the system elements never fail. Hence the problem of reconfiguring the shape following the failure of some elements has been neglected. In this thesis we studied the problem of handling failures when the shape is the line. We considered first of all the Connected Line Recovery problem requiring the non-faulty elements to restore the line shape following the failures of some of the elements. We examined the instance of this problem in the programmable matter systems defined by the Metamorphic Robots and Amoebot models. We then studied the more complex Dynamic Line Maintenance problem when the faults are fully dynamic (i.e., can occur at any time). We examined the instance of this problem in the systems defined by the Amoebot and the Hybrid Programmable Matter models. For both problems and the systems considered, we provided a near complete feasibility characterization of problems, identifying the conditions necessary for their solvability, and constructively proving the sufficiency of those conditions. In particular, we presented solution protocols that operate correctly, maintain connectivity of the non-faulty entities, without constraints on the number of entities that will become faulty, nor on the location, nor (in the dynamic case) on the time of the occurrence of each fault. Our impossibility results hold even under the weak fully-synchronous scheduler, while the possibility results hold under the more difficult semi-synchronous one 

/pdf/connected-line-recovery-and-maintenance-by-programmable-21t6cxnl.pdf

Connected Line Recovery and Maintenance by Programmable Matter

Motivated by the prospect of nano-robots that assist human physiological functions at the nanoscale, we investigate the coating problem in the three-dimensional model for hybrid programmable matter. In this model, a single agent with strictly limited viewing range and the computational capability of a deterministic finite automaton can act on passive tiles by picking up a tile, moving, and placing it at some spot. The goal of the coating problem is to fill each node of some surface graph of size $n$ with a tile. We first solve the problem on a restricted class of graphs with a single tile type, and then use constantly many tile types to encode this graph in certain surface graphs capturing the surface of 3D objects. Our algorithm requires $\mathcal{O}(n^2)$ steps, which is worst-case optimal compared to an agent with global knowledge and no memory restrictions.

Universal Coating in the 3D Hybrid Model

AbstractWe study a model of programmable matter systems consisting of n devices lying on the cells of a 2-dimensional square grid, which are able to perform the minimal mechanical operation of rotating around each other. The goal is to transform an initial shape of devices A into a target shape B. We are interested in characterising the class of shapes which can be transformed into each other in such a scenario, under the additional constraint of maintaining global connectivity at all times. This was one of the main problems left open by [Michail et al., JCSS’19]. Note that the considered question is about structural feasibility of transformations, which we exclusively deal with via centralised constructive proofs. Distributed solutions are left for future work and form an interesting research direction. Past work made some progress for the special class of colour-consistent nice shapes. We here consider the class of orthogonal convex shapes, where for any two nodes u, v in a horizontal or vertical line on the grid, there is no empty cell between u and v. We develop a generic centralised transformation and prove that, for any pair A, B of colour-consistent orthogonal convex shapes, it can transform A into B. In light of the existence of blocked shapes in the considered class, we use a minimal 3-node seed (additional nodes placed at the start) to trigger the transformation. The running time of our transformation is an optimal $O(n^2)$ sequential moves, where $n=|A|=|B|$. We leave as an open problem the existence of a universal connectivity-preserving transformation with a small seed. Our belief is that the techniques developed in this paper might prove useful to answer this.KeywordsProgrammable matterTransformationReconfigurable roboticsShape formationCentralised algorithms 

/pdf/centralised-connectivity-preserving-transformations-by-3u8fh17g.pdf

Centralised Connectivity-Preserving Transformations by Rotation: 3 Musketeers for All Orthogonal Convex Shapes

We consider an extension to the geometric amoebot model that allows amoebots to form so-called \emph{circuits}. Given a connected amoebot structure, a circuit is a subgraph formed by the amoebots that permits the instant transmission of signals. We show that such an extension allows for significantly faster solutions to a variety of problems related to programmable matter. More specifically, we provide algorithms for leader election, consensus, compass alignment, chirality agreement and shape recognition. Leader election can be solved in $\Theta(\log n)$ rounds, w.h.p., consensus in $O(1)$ rounds and both, compass alignment and chirality agreement, can be solved in $O(\log n)$ rounds, w.h.p. For shape recognition, the amoebots have to decide whether the amoebot structure forms a particular shape. We show how the amoebots can detect a parallelogram with linear and polynomial side ratio within $\Theta(\log{n})$ rounds, w.h.p. Finally, we show that the amoebots can detect a shape composed of triangles within $O(1)$ rounds, w.h.p.

Accelerating Amoebots via Reconfigurable Circuits.

https://export.arxiv.org/pdf/2503.18663v3.pdf

Efficient Distributed Algorithms for Shape Reduction via Reconfigurable Circuits

In this paper, we study the computation of shortest paths within the \emph{geometric amoebot model}, a commonly used model for programmable matter. Shortest paths are essential for various tasks and therefore have been heavily investigated in many different contexts. For example, in the programmable matter context, which is the focus of this paper, Kostitsyna et al. have utilized shortest path trees to transform one amoebot structure into another [DISC, 2023]. We consider the \emph{reconfigurable circuit extension} of the model where this amoebot structure is able to interconnect amoebots by so-called circuits. These circuits permit the instantaneous transmission of simple signals between connected amoebots. We propose two distributed algorithms for the \emph{shortest path forest problem} where, given a set of $k$ sources and a set of $\ell$ destinations, the amoebot structure has to compute a forest that connects each destination to its closest source on a shortest path. For hole-free structures, the first algorithm constructs a shortest path tree for a single source within $O(\log \ell)$ rounds, and the second algorithm a shortest path forest for an arbitrary number of sources within $O(\log n \log^2 k)$ rounds. The former algorithm also provides an $O(1)$ rounds solution for the \emph{single pair shortest path problem} (SPSP) and an $O(\log n)$ rounds solution for the \emph{single source shortest path problem} (SSSP) since these problems are special cases of the considered problem.

Polylogarithmic Time Algorithms for Shortest Path Forests in Programmable Matter

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