Polylogarithmic Time Algorithms for Shortest Path Forests in Programmable Matter
Andreas Padalkin, Christian Scheideler
TL;DR
This work studies shortest-path computation in the geometric amoebot model with a reconfigurable-circuit extension, introducing two deterministic polylogarithmic-time distributed algorithms for the (k,ℓ)-SPF problem. The single-source SPF achieves $O(\log \ell)$ rounds, while the multi-source SPF runs in $O(\log n \log^2 k)$ rounds, leveraging a suite of tree primitives (Euler Tour, Root-and-Prune, Election, Centroid Decomposition) and portal-graph techniques (PASC, implicit portals) to enable efficient divide-and-conquer and merging across regions. SPSP and SSSP emerge as direct corollaries in $O(1)$ and $O(\log n)$ rounds, respectively, highlighting the polylogarithmic-time potential in this constrained, highly distributed setting. The results demonstrate the power of long-range circuit signaling to overcome diameter-based lower bounds and open avenues for rapid reconfiguration tasks such as energy distribution and shape transformation in programmable matter.
Abstract
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.