The class of robot convergence tasks has been shown to capture fundamental aspects of fault-tolerant computability. A set of asynchronous robots that may fail by crashing, start from unknown places in some given space, and have to move towards positions close to each other. In this article, we study the case where the space is uni-dimensional, modeled as a graph G. In graph convergence, robots have to end up on one or two vertices of the same edge. We consider also a variant of robot convergence on graphs, edge covering, where additionally, it is required that not all robots end up on the same vertex. Remarkably, these two similar problems have very different computability properties, related to orthogonal fundamental issues of distributed computations: agreement and symmetry breaking. We characterize the graphs on which each of these problems is solvable, and give optimal time algorithms for the solvable cases. Although the results can be derived from known general topology theorems, the presentation serves as a self-contained introduction to the algebraic topology approach to distributed computing, and yields concrete algorithms and impossibility results.

中文翻译：

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免等待机器人图的收敛和覆盖

机器人收敛任务类别已被证明可以捕获容错可计算性的基本方面。一组可能因碰撞而失败的异步机器人，从某个给定空间中的未知位置开始，并且必须向彼此靠近的位置移动。在本文中，我们研究空间是一维的情况，建模为图 G。在图收敛中，机器人必须在同一边的一两个顶点上结束。我们还考虑了机器人在图上收敛的变体，边缘覆盖，此外，要求并非所有机器人都在同一顶点上结束。值得注意的是，这两个相似的问题具有非常不同的可计算性，与分布式计算的正交基本问题相关：一致性和对称性破坏。我们描述了这些问题中的每一个都可以解决的图，并为可解决的情况提供了最佳时间算法。虽然结果可以从已知的一般拓扑定理导出，但该演示文稿作为对分布式计算的代数拓扑方法的独立介绍，并产生具体的算法和不可能的结果。