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Exploration of the T-Interval-Connected Dynamic Graphs: the Case of the Ring

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Abstract

In this paper, we study the T-interval-connected dynamic graphs from the point of view of the time necessary and sufficient for their exploration by a mobile entity (agent). A dynamic graph (more precisely, an evolving graph) is T-interval-connected (T ≥ 1) if, for every window of T consecutive time steps, there exists a connected spanning subgraph that is stable (always present) during this period. This property of connection stability over time was introduced by Kuhn, Lynch and Oshman (Kuhn et al. 14) (STOC 2010). We focus on the case when the underlying graph is a ring of size n, and we show that the worst-case time complexity for the exploration problem is 2n − T − Θ(1) time units if the agent knows the dynamics of the graph, and \(n+ \frac {n}{\max \{1, T-1\} } (\delta -1) \pm {\Theta }(\delta )\) time units otherwise, where δ is the maximum time between two successive appearances of an edge.

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Notes

  1. Note that several specializations of this problem exist, depending on whether the agent has to eventually detect termination (exploration with stop), return to its starting position (exploration with return), or even visit each vertex infinitely often (perpetual exploration). The rest of the paper just considers the general version of the problem.

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Authors and Affiliations

  1. LaBRI, CNRS, Univ., Bordeaux, France

    David Ilcinkas

  2. École Polytechnique de Thiès, Thiès, Senegal

    Ahmed M. Wade

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

Additional information

A preliminary version of this paper appeared in the Proceedings of the 20th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2013) [13]. Partially supported by the ANR projects DISPLEXITY (ANR-11-BS02-014) and MACARON (ANR-13-JS02-002).

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Ilcinkas, D., Wade, A.M. Exploration of the T-Interval-Connected Dynamic Graphs: the Case of the Ring. Theory Comput Syst 62, 1144–1160 (2018). https://doi.org/10.1007/s00224-017-9796-3

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