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Exploration of Dynamic Cactuses with Sub-logarithmic Overhead

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Abstract

We study the problem of exploration by a mobile entity (agent) of a class of dynamic networks, namely constantly connected dynamic graphs. This problem has already been studied in the case where the agent knows the dynamics of the graph and the underlying graph is a ring of n vertices (Ilcinkas and Wade 2018). In this paper, we consider the same problem and we suppose that the underlying graph is a cactus graph (a connected graph in which any two simple cycles have at most one vertex in common). We propose an algorithm that allows the agent to explore these dynamic graphs in at most \(O(n\frac {\log n}{\log \log n})\) time units. We show that the lower bound of the algorithm is \({\varOmega }(n\frac {\log n}{(\log \log n)^{2}})\) time units (for infinitely many n).

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Notes

  1. In [5], the authors define a ring of size two as a two-node path if the graph is simple, or as two nodes linked by two bidirectional edges otherwise.

  2. The actual definition of the “clockwise” direction does not really matter as long as it is any fixed direction.

  3. To simplify the notation, we define fM(Ci) as 0 when i > ℓ.

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Acknowledgements

The authors would like to thank Arnaud Casteigts for insightful and valuable discussions regarding the topic of this paper.

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Authors and Affiliations

  1. LaBRI, CNRS, University of Bordeaux, Bordeaux, France

    David Ilcinkas

  2. École Polytechnique de Thiès, Thies, Senegal

    Ahmed M. Wade

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  1. David Ilcinkas
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  2. Ahmed M. Wade
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Correspondence to Ahmed M. Wade.

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A (weaker) preliminary version of this paper appeared in the Proceedings of the 21st International Colloquium on Structural Information and Communication Complexity (SIROCCO 2014) [18]. D. Ilcinkas is partially supported by the ANR projects DESCARTES (ANR-16-CE40-0023) and EState (ANR-16-CE25-0009), and the “Investments for the future” Programme IdEx Bordeaux – CPU (ANR-10-IDEX-03-02). A. M. Wade is partially supported by the African Center of Excellence in Mathematics, Computer Science and ICT (CEA-MITIC).

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Ilcinkas, D., Wade, A.M. Exploration of Dynamic Cactuses with Sub-logarithmic Overhead. Theory Comput Syst 65, 257–273 (2021). https://doi.org/10.1007/s00224-020-10001-0

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