Efficient Distributed Algorithms for Shape Reduction via Reconfigurable Circuits

Abstract

Autonomous reconfiguration of agent-based systems is a key challenge in the study of programmable matter, distributed robotics, and molecular self-assembly. While substantial prior work has focused on size-preserving transformations, much less is known about size-changing transformations. Such transformations find application in natural processes, active self-assembly, and dynamical systems, where structures may evolve through the addition or removal of components controlled by local rules. In this paper, we study efficient distributed algorithms for transforming 2D geometric configurations of simple agents, called shapes, using only local size-changing operations. A novelty of our approach is the use of reconfigurable circuits as the underlying communication model, a recently proposed model enabling instant node-to-node communication via primitive signals. Unlike previous work, we integrate collision avoidance as a core responsibility of the distributed algorithm. We consider two graph update models: connectivity and adjacency. Let $n$ denote the number of agents and $k$ the number of turning points in the initial shape. In the connectivity model, we show that any tree-shaped configuration can be reduced to a single agent using only shrinking operations in $O(k \log n)$ rounds w.h.p., and to its incompressible form in $O(\log n)$ rounds w.h.p. given prior knowledge of the incompressible nodes, or in $O(k \log n)$ rounds otherwise. When both shrinking and growth operations are available, we give an algorithm that transforms any tree to a topologically equivalent one in $O(k \log n + \log^2 n)$ rounds w.h.p. On the negative side, we show that one cannot hope for $o(\log^2 n)$-round transformations for all shapes of $\Theta(\log n)$ turning points. In the adjacency model, we show that any connected shape can reduce itself to a single node using only shrinking in $O(\log n)$ rounds w.h.p.