Andreas Padalkin

Henning Hillebrandt, Andreas Padalkin, Christian Scheideler et al.

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.

Manish Kumar, Othon Michail, Andreas Padalkin et al.

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.