Leader Election is an important primitive for programmable matter, since it is often an intermediate step for the solution of more complex problems. Although the leader election problem itself is well studied even in the specific context of programmable matter systems, research on fault tolerant approaches is more limited. We consider the problem in the previously studied Amoebot model on a triangular grid, when the configuration is connected but contains nodes the particles cannot move to (e.g., obstacles). We assume that particles agree on a common direction (i.e., the horizontal axis) but do not have chirality (i.e., they do not agree on the other two directions of the triangular grid). We begin by showing that an election algorithm with explicit termination is not possible in this case, but we provide an implicitly terminating algorithm that elects a unique leader without requiring any movement. These results are in contrast to those in the more common model with chirality but no agreement on directions, where explicit termination is always possible but the number of elected leaders depends on the symmetry of the initial configuration. Solving the problem under the assumption of one common direction allows for a unique leader to be elected in a stationary and deterministic way, which until now was only possible for simply connected configurations under a sequential scheduler.

翻译：领导者选举是可编程物质领域的重要基础原语，常作为解决更复杂问题的中间步骤。尽管领导者选举问题本身已在可编程物质系统的特定场景中得到充分研究，但关于容错方法的探索仍较为有限。本文在先前研究的三角形网格Amoebot模型中探讨该问题，假定配置是连通的但包含粒子无法移动的节点（如障碍物）。我们假设粒子间就共同方向（即水平轴）达成一致，但不具备手性（即无法就三角形网格的另外两个方向达成一致）。首先证明在此情形下无法实现具有显式终止条件的选举算法，但提出了一种无需任何移动即可选举出唯一领导者的隐式终止算法。该结果与更常见的具有手性但无方向一致性的模型形成对比——在后者中显式终止始终可行，但被选举出的领导者数量取决于初始配置的对称性。在共同方向假设下解决该问题，使得能够以静止且确定性的方式选举出唯一领导者，而此前仅能在顺序调度器下针对简单连通配置实现该目标。