Faster Sparse Interpolation of Straight-Line Programs

  • Andrew Arnold
  • Mark Giesbrecht
  • Daniel S. Roche
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8136)


We give a new probabilistic algorithm for interpolating a “sparse” polynomial f given by a straight-line program. Our algorithm constructs an approximation f * of f, such that f − f * probably has at most half the number of terms of f, then recurses on the difference f − f *. Our approach builds on previous work by Garg and Schost (2009), and Giesbrecht and Roche (2011), and is asymptotically more efficient in terms of the total cost of the probes required than previous methods, in many cases.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ben-Or and Tiwari(1988)]
    Ben-Or, M., Tiwari, P.: A deterministic algorithm for sparse multivariate polynomial interpolation. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 301–309. ACM (1988)Google Scholar
  2. [Bläser et al.(2009)Bläser, Hardt, Lipton, and Vishnoi]
    Bläser, M., Hardt, M., Lipton, R.J., Vishnoi, N.K.: Deterministically testing sparse polynomial identities of unbounded degree. Information Processing Letters 109(3), 187–192 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [Bruno et al.(2002)Bruno, Heintz, Matera, and Wachenchauzer]
    Bruno, N., Heintz, J., Matera, G., Wachenchauzer, R.: Functional programming concepts and straight-line programs in computer algebra. Mathematics and Computers in Simulation 60(6), 423–473 (2002), doi:10.1016/S0378-4754(02)00035-6MathSciNetzbMATHCrossRefGoogle Scholar
  4. [Bürgisser et al.(1997)Bürgisser, Clausen, and Shokrollahi]
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Grundlehren der mathematischen Wissenschaften, vol. 315. Springer (1997)Google Scholar
  5. [Cantor and Kaltofen(1991)]
    Cantor, D.G., Kaltofen, E.: On fast multiplication of polynomials over arbitrary algebras. Acta Informatica 28, 693–701 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [de Prony(1795)]
    de Prony, R.: Essai expérimental et analytique sur les lois de la dilabilité et sur celles de la force expansive de la vapeur de l’eau et de la vapeur de l’alkool, à différentes températures. J. de l’École Polytechnique 1, 24–76 (1795)Google Scholar
  7. [Garg and Schost(2009)]
    Garg, S., Schost, É.: Interpolation of polynomials given by straight-line programs. Theor. Comput. Sci. 410(27-29), 2659–2662 (2009),, doi:10.1016/j.tcs.2009.03.030, ISSN 0304-3975MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Gathen and Gerhard(2003)]
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, New York (2003) ISBN 0521826462zbMATHGoogle Scholar
  9. [Giesbrecht and Roche(2011)]
    Giesbrecht, M., Roche, D.S.: Diversification improves interpolation. In: ISSAC 2011, pp. 123–130 (2011),, doi:10.1145/1993886.1993909
  10. [Giesbrecht et al.(2009)Giesbrecht, Labahn, and Lee]
    Giesbrecht, M., Labahn, G., Lee, W.-S.: Symbolic– numeric sparse interpolation of multivariate polynomials. Journal of Symbolic Computation 44(8), 943–959 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. Kaltofen(1989).
    Kaltofen, E.: Factorization of polynomials given by straight-line programs. In: Randomness and Computation, pp. 375–412. JAI Press (1989)Google Scholar
  12. Kaltofen et al.(1990)Kaltofen, Lakshman, and Wiley.
    Kaltofen, E., Lakshman, Y.N., Wiley, J.M.: Modular rational sparse multivariate polynomial interpolation. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC 1990, pp. 135–139. ACM, New York (1990), doi:10.1145/96877.96912CrossRefGoogle Scholar
  13. [Pritchard(1982)]
    Pritchard, P.: Explaining the wheel sieve. Acta Informatica 17(4), 477–485 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Rosser and Schoenfeld(1962)]
    Barkley Rosser, J., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Illinois J. Math. 6, 64–94 (2082) ISSN 0019-2082Google Scholar
  15. [Sturtivant and Zhang(1990)]
    Sturtivant, C., Zhang, Z.-L.: Efficiently inverting bijections given by straight line programs. In: Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pp. 327–334. IEEE (October 1990), doi:10.1109/FSCS.1990.89551Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Andrew Arnold
    • 1
  • Mark Giesbrecht
    • 1
  • Daniel S. Roche
    • 2
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooCanada
  2. 2.United States Naval AcademyUSA

Personalised recommendations