Acta Informatica

, Volume 55, Issue 3, pp 213–225 | Cite as

Big data interpolation using functional representation

  • Hadassa Daltrophe
  • Shlomi Dolev
  • Zvi Lotker
Original Article


Given a large set of measurement data, in order to identify a simple function that captures the essence of the data, we suggest representing the data by an abstract function, in particular by polynomials. We interpolate the datapoints to define a polynomial that would represent the data succinctly. The interpolation is challenging, since in practice the data can be noisy and even Byzantine where the Byzantine data represents an adversarial value that is not limited to being close to the correct measured data. We present two solutions, one that extends the Welch-Berlekamp technique (Error correction for algebraic block codes, 1986) to eliminate the outliers appearance in the case of multidimensional data, and copes with discrete noise and Byzantine data; and the other solution is based on Arora and Khot (J Comput Syst Sci 67(2):325–340, 2003) method which handles noisy data, and we have generalized it in the case of multidimensional noisy and Byzantine data.


Big data Data aggregation Sampling Data interpolation Representation 



The research was partially supported by the Rita Altura Trust Chair in Computer Sciences; grant of the Ministry of Science, Technology and Space, Israel, and the National Science Council (NSC) of Taiwan; the Ministry of Foreign Affairs, Italy; the Ministry of Science, Technology and Space, Infrastructure Research in the Field of Advanced Computing and Cyber Security; and the Israel National Cyber Bureau.


  1. 1.
    Ar, S., Lipton, R.J., Rubinfeld, R., Sudan, M.: Reconstructing algebraic functions from mixed data. In FOCS. IEEE Computer Society, pp. 503–512 (1992)Google Scholar
  2. 2.
    Arora, S., Khot, S.: Fitting algebraic curves to noisy data. J. Comput. Syst. Sci. 67(2), 325–340 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertino, E., Bernstein, P., Agrawal, D., Davidson, S., Dayal, U., Franklin, M., Gehrke, J., Haas, L., Halevy, A., Han, J., et al.: Challenges and opportunities with big data. (2011)Google Scholar
  4. 4.
    Daltrophe, H., Dolev, S., Lotker, Z.: Big data interpolation: an efficient sampling alternative for sensor data aggregation. Algo. Sensors 2012(2013), 66–77 (2013)Google Scholar
  5. 5.
    Davis, P.J.: Interpolation and approximation. Dover Publications, New York (1975)zbMATHGoogle Scholar
  6. 6.
    Ditzian, Z.: Multivariate Bernstein and Markov inequalities. J. Approx. Theory 70(3), 273–283 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fasolo, E., Rossi, M., Widmer, J., Zorzi, M.: In-network aggregation techniques for wireless sensor networks: a survey. IEEE Wirel. Commun. 14(2), 70–87 (2007)CrossRefGoogle Scholar
  8. 8.
    Jesus, P., Baquero, C., Almeida, P.S.: A survey of distributed data aggregation algorithms. arXiv preprint arXiv:1110.0725 (2011)
  9. 9.
    Kahn, Joseph M., Katz, Randy H., Pister, Kristofer SJ.: Next century challenges: mobile networking for “Smart Dust”. In Proceedings of the 5th annual ACM/IEEE international conference on Mobile computing and networking. ACM, pp. 271–278 (1999)Google Scholar
  10. 10.
    Madden, S.: From databases to big data. IEEE Internet Comput. 16, 3 (2012)CrossRefGoogle Scholar
  11. 11.
    Nürnberger, G.: Approximation by spline functions, vol. 1. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar
  12. 12.
    Pinkus, A.: Weierstrass and approximation theory. J. Approx. Theory 107(1), 1–66 (2000). doi: 10.1006/jath.2000.3508 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rajagopalan, R., Varshney, P.K.: 2006. A survey, Data aggregation techniques in sensor networks (2006)Google Scholar
  14. 14.
    Rivlin, T.J.: An introduction to the approximation of functions. Dover Publications, New York (2003)zbMATHGoogle Scholar
  15. 15.
    Saniee, R.: A simple expression for multivariate lagrange interpolation. (2008)Google Scholar
  16. 16.
    Sudan, M.: Decoding of Reed Solomon codes beyond the error-correction bound. J. Complex. 13(1), 180–193 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ullman, Jeffrey D., Aho, Alfred V., Hopcroft, John E.: The design and analysis of computer algorithms. Addison-Wesley, Reading (1974)zbMATHGoogle Scholar
  18. 18.
    Welch, L.R., Berlekamp, E.R.: Error correction for algebraic block codes. US Patent 4,633,470, 30 Dec 1986Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Ben-Gurion University of the NegevBeer-shevaIsrael

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