覆盖在匿名网络中的作用：生成树构造与拓扑识别
Revisiting the Role of Coverings in Anonymous Networks: Spanning Tree Construction and Topology Recognition
摘要：
本文从图覆盖理论的角度重新审视了匿名网络中两个经典的分布式问题：生成树的构造和拓扑识别。对于这两个问题，我们刻画了通信图的有向对称覆盖的充分必要条件。这些刻画回答了Yamashita和Kameda [YK96]提出的一个长期存在的问题，并对覆盖与由同一作者开发的视图和商图的概念之间的联系提供了新的见解。用覆盖刻画条件是很重要的，因为它将这个领域与图论和代数拓扑学中大量的经典文献联系起来。特别是，它提供了强大的工具，如Reidemeister定理和Mazurkiewicz算法。结合在一起，这些工具使我们能够提出优雅的证明，否则复杂的结果，其建设性的性质，使他们有效地在算法中使用。本文还使我们有机会以教学的方式介绍覆盖理论领域，重点关注上述两种工具，其潜在影响超出了本工作中考虑的具体问题。
英文摘要：
This paper revisits two classical distributed problems in anonymous networks, namely spanning tree construction and topology recognition, from the point of view of graph covering theory. For both problems, we characterize necessary and sufficient conditions on the communication graph in terms of directed symmetric coverings. These characterizations answer along-standing open question posed by Yamashita and Kameda [YK96], and shed new light on the connection between coverings and the concepts of views and quotient graphs developed by the same authors. Characterizing conditions in terms of coverings is significant because it connects the field with a vast body of classical literature in graph theory and algebraic topology. In particular, it gives access to powerful tools such as Reidemeister's theorem and Mazurkiewicz's algorithm. Combined together, these tools allow us to present elegant proofs of otherwise intricate results, and their constructive nature makes them effectively usable in the algorithms. This paper also gives us the opportunity to present the field of covering theory in a pedagogical way, with a focus on the two aforementioned tools, whose potential impact goes beyond the specific problems considered in this work.
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