Distributed computation of coverage in sensor networks by homological methods

Open Access
Original Paper

Abstract

Recent work on algebraic-topological methods for verifying coverage in planar sensor networks relied exclusively on centralized computation: a limiting constraint for large networks. This paper presents a distributed algorithm for homology computation over a sensor network, for purposes of verifying coverage. The techniques involve reduction and coreduction of simplicial complexes, and are of independent interest. Verification of the ensuing algorithms is proved, and simulations detail the improved network efficiency and performance.

Keywords

Homology Sensor network Coverage Distributed computation 

Mathematics Subject Classification

55-04 55N99 52B99 

Notes

Acknowledgments

The first, third and fourth author are partially supported by Polish MNiSW, Grant N201 037 31/3151 and N N201 419639. The first author is partially supported by Polish MNiSW Grant N N206 625439. The second author is supported by the ONR and by DARPA SToMP # HR0011-07-1-0002.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

References

  1. 1.
    Arai Z., Hayashi K., Hiraoka Y.: Mayer-Vietoris sequences and coverage problems in sensor networks. Jpn. J. Ind. Appl. Math. 28, 237–250 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Awerbuch B., Gallager R.: A new distributed algorithm to find breadth first search trees. IEEE Trans. Inf. Theory. 33(3), 315–322 (1987)MATHCrossRefGoogle Scholar
  3. 3.
    Barrière, L., Fraigniaud, P., Narayanan, L.: Robust position-based routing in wireless ad hoc networks with unstable transmission ranges. In: Proc. Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications. (2001)Google Scholar
  4. 4.
    Carlsson, G., de Silva, V.: Zigzag persistence. In: Proc. Found. of Computational Mathematics. (2009)Google Scholar
  5. 5.
    Carlsson, G., de Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: Proc. Symp. on Comput. Geometry. (2009)Google Scholar
  6. 6.
    Chambers E., de Silva V., Erickson J., Ghrist R.: Rips complexes of planar point sets. Discret. Comput. Geom. 44(1), 75–90 (2010)MATHCrossRefGoogle Scholar
  7. 7.
    Cortes, J., Martinez, S., Karatas, T., Bullo, F.: Coverage control for mobile sensing networks. In: Proc. IEEE Int. Conf. Robot. Autom., vol. 2, pp. 1327–1332. Washington, DC (2002)Google Scholar
  8. 8.
    Damian, M., Pandit, S., Pemmaraju, S.: Local approximation schemes for topology control. In: Proc. ACM Symp. on Prin. of Dist. Comput. (PODC), pp. 208–217 (2006)Google Scholar
  9. 9.
    de Silva V., Ghrist R.: Coordinate-free coverage in sensor networks with controlled boundaries via homology. Int. J. Robot. Res. 25, 1205–1222 (2006)MATHCrossRefGoogle Scholar
  10. 10.
    de Silva V., Ghrist R.: Homological sensor networks. Notices Am. Math. Soc. 54(1), 10–17 (2007)MATHGoogle Scholar
  11. 11.
    Eckmann B.: Harmonische funktionen und randwertaufgaben einem komplex. Comment. Math. Helvetici. 17, 240–245 (1945)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Estrin D., Culler D., Pister K., Sukhatme G.: Connecting the physical world with pervasive networks. IEEE Pervasive Comput. 1(1), 59–69 (2002)CrossRefGoogle Scholar
  13. 13.
    Fekete, S., Kröller, A., Pfisterer, D., Fischer, S.: Deterministic boundary recongnition and topology extraction for large sensor networks. In: Algorithmic Aspects of Large and Complex Networks. (2006)Google Scholar
  14. 14.
    Gelfand S., Manin Y.: Methods of Homological Algebra, 2nd edn. Springer, Berlin (2003)MATHGoogle Scholar
  15. 15.
    Ghrist, R., Hiraoka, Y.: Applications of sheaf cohomology and exact sequences to network coding. Proc. NOLTA: Nonlinear Theory Appl. 266–269 (2011)Google Scholar
  16. 16.
    Kempe, D., Dobra, A., Gehrke, J.: Computing aggregate information using gossip. In: Proc. Foundations of Computer Science, Cambridge, MA (2003)Google Scholar
  17. 17.
    Koskinen, H.: On the coverage of a random sensor network in a bounded domain. In: Proceedings of 16th ITC Specialist Seminar, pp. 11–18. (2004)Google Scholar
  18. 18.
    Kuhn F., Wattenhofer R., Zollinger A.: Ad-hoc networks beyond unit disk graphs. Wirel. Netw. 14(5), 715–729 (2008)CrossRefGoogle Scholar
  19. 19.
    Li X.-Y., Wan P.-J., Frieder O.: Coverage in wireless ad-hoc sensor networks. IEEE Trans. Comput. 52(6), 753–763 (2003)CrossRefGoogle Scholar
  20. 20.
    Meguerdichian, S., Koushanfar, F., Potkonjak, M., Srivastava, M.: Coverage problems in wireless ad-hoc sensor networks. In: IEEE INFOCOM, pp. 1380–1387 (2001)Google Scholar
  21. 21.
    Mrozek M., Batko B.: Coreduction homology algorithm. Discret. Comput. Geom. 41, 96–118 (2009)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Mrozek M., Wanner Th.: Coreduction homology algorithm for inclusions and persistent homology. Comput. Math. Appl. 60(10), 2812–2833 (2010). doi: 10.1016/j.camwa.2010.09.036 MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Muhammad, A., Egerstedt, M.: Control using higher order Laplacians in network topologies, In Proc. of the 17th International Symposium on Mathematical Theory of Networks and Systems, pp. 1024–1038. (2006)Google Scholar
  24. 24.
    Muhammad, A., Jadbabaie, A.: Decentralized computation of homology groups in networks by gossip. In: Proc. of American Control Conference, pp. 3438–3443 (2007)Google Scholar
  25. 25.
    Robinson, M.: Inverse problems in geometric graphs using internal measurements. arXiv:1008.2933v1 (2010)Google Scholar
  26. 26.
    Robinson, M.: Asynchronous logic circuits and sheaf obstructions. arXiv:1008.2729v1 (2010)Google Scholar
  27. 27.
    Tahbaz Salehi A., Jadbabaie A.: Distributed coverage verification in sensor networks without location information. IEEE Trans. Autom. Control 55(8), 1837–1849 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tahbaz Salehi, A., Jadbabaie, A.: Distributed coverage verification in sensor networks without location information. In: IEEE Conference on Decision and Control (2008)Google Scholar
  29. 29.
    Xue F., Kumar P.R.: The number of neighbors needed for connectivity of wireless networks. Wirel. Netw. 10(2), 169–181 (2004)CrossRefGoogle Scholar
  30. 30.
    Zhang, H., Hou, J.: Maintaining coverage and Connectivity in large sensor networks. In: International Workshop on Theoretical and Algorithmic Aspects of Sensor, Ad hoc Wireless and Peer-to-Peer Networks, Florida (2004)Google Scholar
  31. 31.
    The RedHom homology algorithms library: http://redhom.ii.uj.edu.pl
  32. 32.
    Computational Homology Project: http://chomp.rutgers.edu
  33. 33.
    Sensor Network Simulator: http://redhom.ii.uj.edu.pl/sensors/

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Institute of Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Departments of Mathematics and Electrical & Systems EngineeringUniversity of PennsylvaniaPhiladelphiaUSA
  3. 3.Division of Computational MathematicsWSB-NLUNowy Sa̧czPoland

Personalised recommendations