Self-stabilizing Numerical Iterative Computation

  • Ezra N. Hoch
  • Danny Bickson
  • Danny Dolev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5340)


Many challenging tasks in sensor networks, including sensor calibration, ranking of nodes, monitoring, event region detection, collaborative filtering, collaborative signal processing, etc., can be formulated as a problem of solving a linear system of equations. Several recent works propose different distributed algorithms for solving these problems, usually by using linear iterative numerical methods.

In this work, we extend the settings of the above approaches, by adding another dimension to the problem. Specifically, we are interested in self-stabilizing algorithms, that continuously run and converge to a solution from any initial state. This aspect of the problem is highly important due to the dynamic nature of the network and the frequent changes in the measured environment.

In this paper, we link together algorithms from two different domains. On the one hand, we use the rich linear algebra literature of linear iterative methods for solving systems of linear equations, which are naturally distributed with rapid convergence properties. On the other hand, we are interested in self-stabilizing algorithms, where the input to the computation is constantly changing, and we would like the algorithms to converge from any initial state. We propose a simple novel method called SS-Iterative as a self-stabilizing variant of the linear iterative methods. We prove that under mild conditions the self-stabilizing algorithm converges to a desired result. We further extend these results to handle the asynchronous case.

As a case study, we discuss the sensor calibration problem and provide simulation results to support the applicability of our approach.


Sensor Network Wireless Sensor Network Input Sequence Atomic Step Circle Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bell, R.M., Koren, Y.: Scalable collaborative filtering with jointly derived neighborhood interpolation weights. In: ICDM 2007 (2007)Google Scholar
  2. 2.
    Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Calculation. Numerical Methods. Prentice Hall, Englewood Cliffs (1989)zbMATHGoogle Scholar
  3. 3.
    Bickson, D., Malkhi, D.: A unifying framework for rating users and data items in peer-to-peer and social networks. Peer-to-Peer Networking and Applications (PPNA) Journal (2008)Google Scholar
  4. 4.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    Davidovitch, L., Dolev, S., Rajsbaum, S.: Stability of multivalued continuous consensus. SIAM Journal on Computing 37(4), 1057–1076 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dolev, D., Hoch, E.N.: Ocd: Obsessive consensus disorder (or repetitive consensus). In: Proc. of the 27st Int. Symposium on Principles of Distributed Computing (PODC 2008), Tornoto, Canada (August 2008)Google Scholar
  7. 7.
    Dolev, S.: Self-Stabilization. MIT Press, Cambridge (2000)zbMATHGoogle Scholar
  8. 8.
    Dolev, S., Rajsbaum, S.: Stability of long-lived consensus. J. Comput. Syst. Sci. 67(1), 26–45 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fang, J., Li, H.: Distributed event-region detection in wireless sensor networks. In: EURASIP J. Adv. Signal Process, New York, NY, United States, vol. 2008, pp. 1–10. Hindawi Publishing Corp. (January 2008)Google Scholar
  10. 10.
    Langendoen, K., Reijers, N.: Distributed localization in wireless sensor networks: a quantitative comparison. In: Comput. Networks, New York, NY, USA, November 2003, vol. 43, pp. 499–518. Elsevier North-Holland, Inc., Amsterdam (2003)Google Scholar
  11. 11.
    Marzullo, K.: Tolerating failures of continuous-valued sensors. ACM Trans. Comput. Syst. 8(4), 284–304 (1990)CrossRefGoogle Scholar
  12. 12.
    Olfati-Saber, R.: Distributed Kalman Filtering for Sensor Networks. In: Proc. of the 46th IEEE Conference on Decision and Control (December 2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ezra N. Hoch
    • 1
  • Danny Bickson
    • 2
  • Danny Dolev
    • 1
  1. 1.School of Computer Science and EngineeringThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.IBM Haifa Research LabHaifaIsrael

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