Leader election is a fundamental problem in distributed computing, particularly within programmable matter systems, where coordination among simple computational entities is crucial for solving complex tasks. In these systems, particles (i.e., constant-memory computational entities) operate in a regular triangular grid as described in the geometric Amoebot model. While leader election has been extensively studied in non self-stabilising settings, self-stabilising solutions remain more limited. In this work, we study the problem of self-stabilising leader election in connected (but not necessarily simply connected) configurations. We present the first self-stabilising algorithm for connected programmable matter systems that guarantees the election of a unique leader under an unfair scheduler, for oblivious particles (i.e., particles with no persistent memory) that share a common sense of direction. Our approach leverages particle movement, a capability not previously exploited in the self-stabilising context. We show that movement in conjunction with particles sharing a sense of orientation and operating in a grid can overcome classical impossibility results for constant-memory systems established by Dolev, Gouda and Schneider (1999).

翻译：领导者选举是分布式计算中的一个基本问题，在可编程物质系统中尤为重要，其中简单计算实体间的协调对于解决复杂任务至关重要。在这些系统中，粒子（即恒定内存计算实体）在几何阿米巴模型所描述的正三角形网格中运行。尽管领导者选举在非自稳定设置中已得到广泛研究，但自稳定解决方案仍然较为有限。在本工作中，我们研究了在连通（但不一定是单连通）配置下的自稳定领导者选举问题。我们提出了首个适用于连通可编程物质系统的自稳定算法，该算法在非公平调度器下保证为具有共同方向感、无记忆粒子（即无持久内存粒子）选举出唯一领导者。我们的方法利用了粒子移动这一能力，该能力在自稳定研究背景下尚未被充分利用。我们证明，在网格中运行的粒子通过结合移动与共享方向感，能够克服Dolev、Gouda和Schneider（1999）为恒定内存系统建立的经典不可能性结果。