Revisiting the Role of Coverings in Anonymous Networks: Spanning Tree Construction and Topology Recognition

TL;DR

This paper explores how graph covering theory can characterize and solve fundamental anonymous network problems like spanning tree construction and topology recognition, providing new insights and tools for distributed computing.

Contribution

It provides necessary and sufficient conditions using directed symmetric coverings for these problems, answering longstanding open questions and connecting to classical graph theory and algebraic topology.

Findings

Characterization of problems via graph coverings

Application of Reidemeister's theorem and Mazurkiewicz's algorithm

Constructive proofs enabling practical algorithms

Abstract

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.