Polylogarithmic Time Algorithms for Shortest Path Forests in Programmable Matter

TL;DR

This paper introduces polylogarithmic time distributed algorithms for computing shortest path forests in the geometric amoebot model of programmable matter, enabling efficient path computations for multiple sources and destinations.

Contribution

It presents two novel distributed algorithms for shortest path forests in the amoebot model, achieving polylogarithmic time complexity for various configurations.

Findings

Shortest path tree for a single source in O(log ℓ) rounds

Shortest path forest for multiple sources in O(log n log^2 k) rounds

O(1) rounds for single pair shortest path problem

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.