Abstract
Time-varying graphs model in a natural way infrastructure-less highly dynamic systems, such as wireless ad-hoc mobile networks, robotic swarms, vehicular networks, etc. In these systems, a path from a node to another might still exist over time, rendering computing possible, even though at no time the path exists in its entirety. Some of these systems allow waiting (i.e., provide the nodes with store-carry-forward-like mechanisms such as local buffering) while others do not.
In this paper, we focus on the structure of the time-varying graphs modelling these highly dynamical environments. We examine the complexity of these graphs, with respect to waiting, in terms of their expressivity; that is in terms of the language generated by the feasible journeys (i.e., the “paths over time”).
We prove that the set of languages \({\cal L}_{nowait}\) when no waiting is allowed contains all computable languages. On the other end, using algebraic properties of quasi-orders, we prove that \({\cal L}_{wait}\) is just the family of regular languages, even if the presence of edges is controlled by some arbitrary function of the time. In other words, we prove that, when waiting is allowed, the power of the accepting automaton drops drastically from being as powerful as a Turing machine, to becoming that of a Finite-State machine. This large gap provides a measure of the impact of waiting.
We also study bounded waiting; that is when waiting is allowed at a node for at most d time units. We prove that \({\cal L}_{wait[d]} = {\cal L}_{nowait}\); that is, the complexity of the accepting automaton decreases only if waiting is unbounded.
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References
Alur, R., Dill, D.L.: A theory of timed automata. Theoretical Comp. Sci. 126(2), 183–235 (1994)
Avin, C., Koucký, M., Lotker, Z.: How to explore a fast-changing world (Cover time of a simple random walk on evolving graphs). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 121–132. Springer, Heidelberg (2008)
Baumann, H., Crescenzi, P., Fraigniaud, P.: Parsimonious flooding in dynamic graphs. In: Proc. 28th Symp. Princ. Distr. Comput., pp. 260–269 (2009)
Bui-Xuan, B., Ferreira, A., Jarry, A.: Computing shortest, fastest, and foremost journeys in dynamic networks. Intl. J. Found. Comp. Science 14(2), 267–285 (2003)
Casteigts, A., Chaumette, S., Ferreira, A.: Characterizing topological assumptions of distributed algorithms in dynamic networks. In: Kutten, S., Žerovnik, J. (eds.) SIROCCO 2009. LNCS, vol. 5869, pp. 126–140. Springer, Heidelberg (2010)
Casteigts, A., Flocchini, P., Mans, B., Santoro, N.: Measuring temporal lags in delay-tolerant networks. IEEE Transactions on Computers (2013)
Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs and dynamic networks. International Journal of Parallel, Emergent and Distributed Systems 27(5), 387–408 (2012)
Clementi, A., Monti, A., Pasquale, F., Silvestri, R.: Information spreading in stationary markovian evolving graphs. IEEE Transactions on Parallel and Distributed Systems 22(9), 1425–1432 (2011)
Ehrenfeucht, A., Haussler, D., Rozenberg, G.: On regularity of context-free languages. Theoretical Computer Science 27(3), 311–332 (1983)
Ferreira, A.: Building a reference combinatorial model for MANETs. IEEE Network 18(5), 24–29 (2004)
Flocchini, P., Mans, B., Santoro, N.: On the exploration of time-varying networks. Theoretical Computer Science 469, 53–68 (2013)
Harju, T., Ilie, L.: On quasi orders of words and the confluence property. Theoretical Computer Science 200(1-2), 205–224 (1998)
Higman, G.: Ordering by divisibility in abstract algebras. Proceedings of the London Mathematical Society s3-2, 326–336 (1952)
Ilcinkas, D., Wade, A.M.: On the power of waiting when exploring public transportation systems. In: Fernàndez Anta, A., Lipari, G., Roy, M. (eds.) OPODIS 2011. LNCS, vol. 7109, pp. 451–464. Springer, Heidelberg (2011)
Kossinets, G., Kleinberg, J., Watts, D.: The structure of information pathways in a social communication network. In: Proc. 14th Intl. Conf. Knowledge Discovery Data Mining, pp. 435–443 (2008)
Kuhn, F., Lynch, N., Oshman, R.: Distributed computation in dynamic networks. In: Proc. 42nd Symp. Theory Comp, pp. 513–522 (2010)
Liu, C., Wu, J.: Scalable routing in cyclic mobile networks. IEEE Trans. Parallel Distrib. Syst. 20(9), 1325–1338 (2009)
C. St. Nash-Williams, J. A.: On well-quasi-ordering finite trees. Mathematical Proceedings of the Cambridge Philosophical Society 59(04), 833–835 (1963)
Zhang, Z.: Routing in intermittently connected mobile ad hoc networks and delay tolerant networks: Overview and challenges. IEEE Communications Surveys & Tutorials 8(1), 24–37 (2006)
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Casteigts, A., Flocchini, P., Godard, E., Santoro, N., Yamashita, M. (2013). Expressivity of Time-Varying Graphs. In: Gąsieniec, L., Wolter, F. (eds) Fundamentals of Computation Theory. FCT 2013. Lecture Notes in Computer Science, vol 8070. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40164-0_12
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DOI: https://doi.org/10.1007/978-3-642-40164-0_12
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