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)


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 \(\mathcal {S}, \mathcal {T} \subset V\).


Directed acyclic graph ST-reachable DAG Self-stabilization 



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


  1. 1.
    Chaudhuri, P., Thompson, H.: A self-stabilizing algorithm for the st-order problem. Int. J. Parallel Emergent Distrib. Syst. 23(3), 219–234 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Karaata, M.H., Chaudhuri, P.: A dynamic self-stabilizing algorithm for constructing a transport net. Computing 68(2), 143–161 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Kim, Y., Aono, H., Katayama, Y., Masuzawa, T.: A self-stabilizing algorithm for constructing a maximal (2,2)-directed acyclic mixed graph. In: the 6th International Symposium on Computing and Networking (CANDAR) (2018)Google Scholar
  4. 4.
    Kim, Y., Ohno, H., Katayama, Y., Masuzawa, T.: A self-stabilizing algorithm for constructing a maximal (1, 1)-directed acyclic mixed graph. Int. J. Netw. Comput. 8(1), 53–72 (2018)CrossRefGoogle Scholar
  5. 5.
    Kim, Y., Shibata, M., Sudo, Y., Nakamura, J., Katayama, Y., Masuzawa, T.: A self-stabilizing algorithm for constructing an \(\cal{ST}\)-reachable directed acyclic graph when \(|\cal{S}| \le 2\) and \(|\cal{T}| \le 2\). In: Proceedings of the 39th IEEE International Conference on Distributed Computing Systems (ICDCS), pp. 2228–2237 (2019)Google Scholar

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

Personalised recommendations