The Wake Up Dominating Set Problem

  • Amir Bannoura
  • Christian Ortolf
  • Christian Schindelhauer
  • Leonhard Reindl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8243)

Abstract

Recently developed wake-up receivers pose a viable alternative for duty-cycling in wireless sensor networks. Here, a special radio signal can wake up close-by nodes. We model the wake-up range by the unit-disk graph. Such wake-up radio signals are very energy expensive and limited in range. Therefore, the number of signals must be minimized. So, we revisit the Connected Dominating Set (CDS) problem for unit-disk graphs and consider an online variant, where starting from an initial node all nodes need to be woken up, while the online algorithm knows only the nodes woken up so far and has no information about the number and location of the sleeping nodes.

We show that in general this problem cannot be solved effectively, since a worst-case setting exists where the competitive ratio, i.e. the number of wake-up signals divided by the size of the minimum CDS, is \(\varTheta (n)\) for \(n\) nodes. For dense random uniform placements, this problem can be solved within a constant factor competitive ratio with high probability, i.e. \(1-n^{-c}\).

For a restricted adversary with a reduced wake-up range of \(1-\epsilon \) we present a deterministic wake-up algorithm with a competitive ratio of \(O(\epsilon ^{-\frac{1}{2}})\) for the general problem in two dimensions.

In the case of random placement without any explicit position information we present an \(O(\log n)\)-competitive epidemic algorithm with high probability to wake up all nodes. Simulations show that a simplified version of this oblivious online algorithm already produces reasonable results, that allows its application in the real world.

Keywords

Wake-up receivers Online algorithm Connected dominating set Unit-disk graph Epidemic algorithms 

Notes

Acknowledgements

This work has been supported by the Ministry for Education and Research (Bundesministerium für Bildung und Forschung, BMBF) within the Research project AURIS (13N11746).

The authors would like to thank SmartExergy WMS GmbH for providing the wake-up receiver’s board.

References

  1. 1.
    Bar-Yehuda, R., Goldreich, O., Itai, A.: On the time-complexity of broadcast in radio networks: an exponential gap between determinism randomization. In: Proceedings of the Sixth Annual ACM Symposium on Principles of Distributed Computing, PODC ’87, pp. 98–108. ACM, New York (1987)Google Scholar
  2. 2.
    Boukerche, A.: Algorithms and Protocols for Wireless Sensor Networks, vol. 62. Wiley-IEEE, New York (2008)Google Scholar
  3. 3.
    Cheng, X., Huang, X., Li, D., Wu, W., Du, D.-Z.: A polynomial-time approximation scheme for the minimum-connected dominating set in ad hoc wireless networks. Networks 42(4), 202–208 (2003)Google Scholar
  4. 4.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1991)MathSciNetGoogle Scholar
  5. 5.
    Das, B., Bharghavan, V.: Routing in ad-hoc networks using minimum connected dominating sets. In: IEEE International Conference on Communications, ICC ’97 Montreal, Towards the Knowledge Millennium, vol.1, pp. 376–380 (1997)Google Scholar
  6. 6.
    Ding, L., Gao, X., Wu, W., Lee, W., Zhu, X., Du, D.-Z.: Distributed construction of connected dominating sets with minimum routing cost in wireless networks. In: Proceedings of the 2010 IEEE 30th International Conference on Distributed Computing Systems, ICDCS ’10, pp. 448–457. IEEE Computer Society, Washington, DC, USA (2010)Google Scholar
  7. 7.
    Du, H., Ye, Q., Zhong, J., Wang, Y., Lee, W., Park, H.: PTAS for minimum connected dominating set with routing cost constraint in wireless sensor networks. In: Wu, W., Daescu, O. (eds.) COCOA 2010, Part I. LNCS, vol. 6508, pp. 252–259. Springer, Heidelberg (2010)Google Scholar
  8. 8.
    Stephan, E.: Online dominating set and variations on restricted graph classes. Technical report, Institute of Theoretical Computer Science, ETH Zürich (2002)Google Scholar
  9. 9.
    Gamm, G.U., Sippel, M., Kostic, M., Reindl, L.M.: Low power wake-up receiver for wireless sensor nodes. In: 2010 Sixth International Conference on Intelligent Sensors, Sensor Networks and Information Processing (ISSNIP), pp. 121–126 (2010)Google Scholar
  10. 10.
    Guha, S., Khuller, S.: Approximation algorithms for connected dominating sets. Algorithmica 20(4), 374–387 (1998)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Gupta, P., Kumar, P.R.: Critical power for asymptotic connectivity. In: Proceedings of the 37th IEEE Conference on Decision and Control, vol. 1, pp. 1106–1110 (1998)Google Scholar
  12. 12.
    Karp, R., Schindelhauer, C., Shenker, S., Vöcking, B.: Randomized rumor spreading. In: IEEE Symposium on Foundations of Computer, Science, pp. 565–574 (2000)Google Scholar
  13. 13.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Rührup, S., Schindelhauer, C.: Competitive time and traffic analysis of position-based routing using a cell structure. In: Proceedings of the 19th IEEE International Parallel and Distributed Processing Symposium, pp. 8 (2005)Google Scholar
  15. 15.
    Wan, P.-J., Alzoubi, K.M., Frieder, O.: Distributed construction of connected dominating set in wireless ad hoc networks. In: Proceedings of the IEEE Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM, vol. 3, pp. 1597–1604 (2002)Google Scholar
  16. 16.
    Wu, W., Du, H., Jia, X., Li, Y., Huang, S.C.-H.: Minimum connected dominating sets and maximal independent sets in unit disk graphs. Theor. Comput. Sci. 352(1), 1–7 (2006)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Amir Bannoura
    • 1
  • Christian Ortolf
    • 1
  • Christian Schindelhauer
    • 1
  • Leonhard Reindl
    • 1
  1. 1.Faculty of EngineeringUniversity of FreiburgFreiburgGermany

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