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Brief Announcement: Self-stabilizing Construction of a Minimal Weakly \(\mathcal {ST}\)-Reachable Directed Acyclic Graph

  • Junya NakamuraEmail author
  • Masahiro Shibata
  • Yuichi Sudo
  • Yonghwan Kim
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11914)

Abstract

In this paper, we propose a self-stabilizing algorithm to construct a minimal weakly \(\mathcal {ST}\)-reachable directed acyclic graph (DAG). Given an arbitrary simple, connected, and undirected graph \(G=(V, E)\) and two sets of vertices, senders \(\mathcal {S} (\subset V)\) and targets \(\mathcal {T} (\subset V)\), a directed subgraph \(\overrightarrow{G}\) of G is a weakly \(\mathcal {ST}\)-reachable DAG on G if \(\overrightarrow{G}\) is a DAG and every sender can reach at least one target, and every target is reachable from at least one sender in \(\overrightarrow{G}\). We say that a weakly \(\mathcal {ST}\)-reachable DAG \(\overrightarrow{G}\) on G is minimal if any proper subgraph of \(\overrightarrow{G}\) is no longer a weakly \(\mathcal {ST}\)-reachable DAG. The weakly \(\mathcal {ST}\)-reachable DAG on G, which we consider here, is a relaxed version of the original (or strongly) \(\mathcal {ST}\)-reachable DAG on G where all targets are reachable from all senders. A strongly \(\mathcal {ST}\)-reachable DAG G does not always exist; even if we focus on the case \(|\mathcal {S}|=|\mathcal {T}|=2\), some G has no strongly \(\mathcal {ST}\)-reachable DAG. On the other hand, the proposed algorithm always construct a weakly \(\mathcal {ST}\)-reachable DAG for any given graph \(G=(V, E)\) and any \(\mathcal {S}, \mathcal {T} \subset V\).

Keywords

Directed acyclic graph ST-reachable DAG Self-stabilization 

Notes

Acknowledgments

This work was supported by JSPS KAKENHI Grant Numbers 18K18000, 18K18029, and 18K18031.

References

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Junya Nakamura
    • 1
    Email author
  • Masahiro Shibata
    • 2
  • Yuichi Sudo
    • 3
  • Yonghwan Kim
    • 4
  1. 1.Toyohashi University of TechnologyToyohashiJapan
  2. 2.Kyushu Institute of TechnologyIizukaJapan
  3. 3.Osaka UniversitySuitaJapan
  4. 4.Nagoya Institute of TechnologyNagoyaJapan

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