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Computing Parameters of Sequence-Based Dynamic Graphs

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Abstract

We present a general framework for computing parameters of dynamic networks which are modelled as a sequence \({\mathcal {G}}=(G_{1},G_{2},\ldots ,G_{\delta })\) of static graphs such that \(G_{i}=(V,E_{i})\) represents the network topology at time i and changes between consecutive static graphs are arbitrary. The framework operates at a high level, manipulating the graphs in the sequence as atomic elements with two types of operations: a composition operation and a test operation. The framework allows us to compute different parameters of dynamic graphs using a common high-level strategy by using composition and test operations that are specific to the parameter. The resulting algorithms are optimal in the sense that they use only \(O(\delta )\) composition and test operations, where \(\delta \) is the length of the sequence. We illustrate our framework with three minimization problems, bounded realization of the footprint, temporal diameter, and round trip temporal diameter, and with T-interval connectivity which is a maximization problem. We prove that the problems are in NC by presenting polylogarithmic-time parallel versions of the algorithms. Finally, we show that the algorithms can operate online with amortized complexity \({\Theta }(1)\) composition and test operations for each graph in the sequence.

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Acknowledgements

We thank the anonymous referees for their careful reading and valuable comments which helped to improve the presentation of the paper.

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Authors and Affiliations

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

    Arnaud Casteigts & Ralf Klasing

  2. LAAS-CNRS, INSA-Toulouse, Toulouse, France

    Yessin M. Neggaz

  3. School of Computing Science, Simon Fraser University, Burnaby, BC, Canada

    Joseph G. Peters

Authors
  1. Arnaud Casteigts
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  2. Ralf Klasing
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  3. Yessin M. Neggaz
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  4. Joseph G. Peters
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Corresponding author

Correspondence to Ralf Klasing.

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Preliminary results concerning this work have been presented at CIAC 2015 [13] and at SIROCCO 2017 [14]. Part of this work was done while Joseph Peters was visiting the LaBRI as aguest professor of the University of Bordeaux (IdEx Bordeaux -ANR-10-IDEX-03-02). This work was also partially funded by ANR projects DISPLEXITY (ANR-11-BS02-014), ESTATE (ANR-16-CE25-0009-03), and NSERC of Canada.

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Casteigts, A., Klasing, R., Neggaz, Y.M. et al. Computing Parameters of Sequence-Based Dynamic Graphs. Theory Comput Syst 63, 394–417 (2019). https://doi.org/10.1007/s00224-018-9876-z

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