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Annals of Operations Research

, Volume 264, Issue 1–2, pp 267–286 | Cite as

And/or-convexity: a graph convexity based on processes and deadlock models

  • Carlos V. G. C. Lima
  • Fábio Protti
  • Dieter Rautenbach
  • Uéverton S. Souza
  • Jayme L. Szwarcfiter
Original Paper

Abstract

Deadlock prevention techniques are essential in the design of robust distributed systems. However, despite the large number of different algorithmic approaches to detect and solve deadlock situations, yet there remains quite a wide field to be explored in the study of deadlock-related combinatorial properties. In this work we consider a simplified AND-OR model, where the processes and their communication are given as a graph G. Each vertex of G is labelled AND or OR, in such a way that an AND-vertex (resp., OR-vertex) depends on the computation of all (resp., at least one) of its neighbors. We define a graph convexity based on this model, such that a set \(S \subseteq V(G)\) is convex if and only if every AND-vertex (resp., OR-vertex) \(v \in V(G){\setminus }S\) has at least one (resp., all) of its neighbors in \(V(G) {\setminus } S\). We relate some classical convexity parameters to blocking sets that cause deadlock. In particular, we show that those parameters in a graph represent the sizes of minimum or maximum blocking sets, and also the computation time until system stability is reached. Finally, a study on the complexity of combinatorial problems related to such graph convexity is provided.

Keywords

Graph convexity Deadlock AND-OR model And/or graphs 

Notes

Acknowledgements

Partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)—Brazilian research agencies.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Carlos V. G. C. Lima
    • 1
  • Fábio Protti
    • 2
  • Dieter Rautenbach
    • 3
  • Uéverton S. Souza
    • 2
  • Jayme L. Szwarcfiter
    • 1
  1. 1.PESC, COPPEFederal University of Rio de JaneiroRio de JaneiroBrazil
  2. 2.Institute of ComputingFluminense Federal UniversityNiteróiBrazil
  3. 3.Institute of Optimization and Operations ResearchUlm UniversityUlmGermany

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