Abstract
In this paper we study the complexity of rational functions and multirational functions. These include 1) functions containing the absolute value, max and min functions, 2) data structure functions such as sort, insert and merge 3) integer functions such as the gcd (greater common divisor), modulo, bitwise ‘and’ and 4) polynomial functions such as the gcd and modulo of two polynomials.
We prove tight lower bounds for the above functions over a finite input domain in a RAM that uses arithmetic operations and that has unlimited power for answering YES/NO questions.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
A. V. Aho, J. E. Hopcroft, J. D. Ullman. The design and Analysis of Computer Algorithms, Addison Wesley, 1974.
M. Ben-Or, Lower bound for algebraic computation trees, In Proc. 15th ACM Symp. on Theory of Computing, pp. 80–86, May 1983.
N. H. Bshouty, Lower bounds for the complexity of functions in random access machines. In preparation.
N. H. Bshouty, Euclidean GCD algorithm is not optimal. I. Upper bound. In preparation.
N. H. Bshouty, Euclidean GCD algorithm is not optimal. I. Lower bound. In preparation.
N. H. Bshouty, Y. Mansour, B. Schieber, P. Tiwari, The complexity of approximating the square root. In preparation.
W. Baur, V. Strassen, The complexity of partial derivatives, Theoretical Computer Science 22, 317–330, (1983).
E. Dittert, M. O'Donnell, Lower bounds for sorting with realistic instruction sets, IEEE Transactions on Computers 34(4), 311–317, (1985).
N. Friedman, Some results on the effect of arithmetics on comparison problem, In Proc. IEEE Symp. on Foundation of Computer Science, pp. 139–143, (1972).
M. C. Golumbic, Combinatorial merging, IEEE Trans. Comp. 25, 11, 1164–1167, (1976).
O. H. Ibarra, S. Moran, Some independence results in complexity theory, Intern. J. Computer Math. 17, 113–122, (1985).
M. Kaminski, N. H. Bshouty, Multiplicative complexity of polynomials multiplication over finite fields, J. of ACM, 36, 1, pp. 150–170, (1989).
Y. Mansour, B. Schieber, P. Tiwari, Lower bounds for integer greatest common divisor computation. In Proc. 29th IEEE Symp. on Foundation of Computer Science, pp. 54–63, October (1982).
W. Paul, J. Simon, Decision trees and random access machines, In Monographie de L'Enseigment Mathematique, No. 30, pp 331–340, (1981).
C. P. Schnorr, An extension of strassen's degree bound, SIAM J. Comute,10, 371–382, (1981).
J. T. Schwartz, Fast probabilistic algorithms for verification of polynomial identities, J. of ACM, 27, 701–717, (1980).
V. Strassen, Die Berechnungskomplexitadotdott von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten, Numer. Math. 20, 238–251, (1973).
V. Strassen, The computational complexity of continued fractions. SIAM J. of Computing, 12(1), 1–27, (1983).
A. Yao, Lower bounds for Algebraic Computation Trees with Integer Inputs, Proc. 30th IEEE Symp. on Foundations of Computer Science, 1989.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bshouty, N.H. (1991). Lower bounds for algebraic computation trees of functions with finite domains. In: Dehne, F., Fiala, F., Koczkodaj, W.W. (eds) Advances in Computing and Information — ICCI '91. ICCI 1991. Lecture Notes in Computer Science, vol 497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54029-6_154
Download citation
DOI: https://doi.org/10.1007/3-540-54029-6_154
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54029-8
Online ISBN: 978-3-540-47359-6
eBook Packages: Springer Book Archive