Acta Informatica

, Volume 3, Issue 2, pp 123–133 | Cite as

A unified view of the complexity of evaluation and interpolation

  • Ellis Horowitz


Four problems are considered: 1) from an n-precision integer compute its residues modulo n single precision primes; 2) from an n-degree polynomial compute its values at n points; 3) from n residues compute the unique n-precision integer congruent to the residues; 4) from n points compute the unique interpolating polynomial through those points. If M (n) is the time for n-precision integer multiplication, then the time for problems 1 and 2 is shown to be M (n) log n and for problems 3 and 4 to be M (n) (log n) 2. Moreover it is shown that each of the four algorithms are really all instances of the same general algorithm. Finally it is shown how preconditioning or a change of domain will reduce the time for problems 3 and 4 to M (n) (log n).


Information System Operating System Data Structure Communication Network Information Theory 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Borodin, A., Munro, I.: Evaluation polynomials at many points. Information Processing Letters 1, 66–68 (1971)Google Scholar
  2. 2.
    Brown, W. S.: On Euclid's algorithm and the computation of polynomial gratest common divisors. J. ACM 18, 478–504 (1971)Google Scholar
  3. 3.
    Fiduccia, C.: Polynomial evaluation via the division algorithm: The fast fourier transform revisited. Proceeding of the 4th Annual ACM Symposium on Theory of Computing, Denver (Col.) May 1972, pp. 88–93Google Scholar
  4. 4.
    Heindel, L., Horowitz, E.: On decreasing the computing time for modular arithmetic. Proceedings of the 12th Symposium on Switching and Automata Theory, Oct. 1971, pp. 126–128Google Scholar
  5. 5.
    Horowitz, E.: A fast method for interpolation using preconditioning. Information Processing Letters 1, 157–163 (1972)Google Scholar
  6. 6.
    Horowitz, E.: The efficient calculation of powers of polynomials. J. Computer and System Sciences 7, 469–481 (1973)Google Scholar
  7. 7.
    Knuth, D. E.: The art of computer programming, Vol. 2. 2nd edition. Reading (Mass.): Addison-Wesley 1971Google Scholar
  8. 8.
    Kung, H. T.: Fast evaluation and interpolation. Computer Science Technical Report, Carnegie-Mellon University, Jan. 1973Google Scholar
  9. 9.
    Moenck, R., Borodin, A.: Fast modular transform via division. Proceedings of the 13th Symposium on Switching and Automata Theory, IEEE Computer Society, Oct. 1972, pp. 90–96Google Scholar
  10. 10.
    Schönhage, A.: Fast computation of continued fraction expansions. English translation of: “Schnelle Berechnung von Kettenbruchentwicklungen”. Acta Informatica 1, 139–144 (1971)Google Scholar
  11. 11.
    Sieveking, M.: An algorithm for division of power series. Computing 10, 153–156 (1972)Google Scholar
  12. 12.
    Aho, A., Hopcroft, J., Ullman, J.: Analysis of algorithms. Book in preparationGoogle Scholar

Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • Ellis Horowitz
    • 1
  1. 1.Computer Science Program University of Southern California Powell HallLos AngelesUSA

Personalised recommendations