Julian Werthmann

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.

John Augustine, Henning Hillebrandt, Manish Kumar et al.

We consider a recently proposed \emph{supervised distributed computing} paradigm \cite{augustine2025supervised} that extends and refines the standard master-worker paradigm for parallel computations. In this paradigm, there is a supervisor, a source, a target, and a collection of workers. The distributed computation is given as an acyclic task graph that is known to the supervisor. The source initially stores the input and the target is supposed to store the output of the computation. The individual tasks of the computation are supposed to be executed by the workers under the guidance of the supervisor. The source, target and supervisor are assumed to be reliable, while a $β$-fraction of the workers might be adversarial, for some $β\in [0,1)$. This covers, for example, the case where a supervisor has to work with untrusted volunteers. In the standard master-worker approach, the master checks whether the workers correctly execute the assigned tasks, creating a severe bottleneck, whereas in the supervised approach, the supervisor outsources this checking to the workers. Prior to this work, only supervised solutions were known for the case that $β$ is a sufficiently small constant. We show that robust and efficient supervised solutions are possible for \emph{any} constant $β<1$ while the expected work for the honest workers is close to a \emph{single} execution per task, given that there is a lightweight verification mechanism that allows honest workers to check the correctness of task outputs, which is significantly better than all robust master-worker as well as peer-to-peer approaches known so far.