STACS 1993: STACS 93 pp 463-472

# Capabilities and complexity of computations with integer division

Extended abstract
• Katharina Lürwer-Brüggemeier
• Friedhelm Meyer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 665)

## Abstract

Computation trees with operation set S$$\subseteq$$ {+, −,*, DIV, DIVc} (S-CTs) are considered. DIV denotes integer division, DIVinc integer division by constants. We characterize the families of languages L$$\subseteq$$n that can be recognized by S-CTs, separate the computational capabilities of S-CTs for different operation sets S, and prove lower bounds for the depth of such trees.

Let CCn(S) denote the family of languages L$$\subseteq$$n that can be recognized by an S-CT. In [7], CC1 {S} is characterized for all S$$\subseteq$$ {+, −, *, DIV, DIVc} It turns out that CC1({+,−,DIVc})=CC1 {+,−,*, DIV}.

In this paper we shed some more light on the computational power of integer division:

• We characterize CC n (S), n > 1, for S={+,−, DIVc} and S = +, −, *, DIV c, and partially characterize CC n (S), n≥ 1, for S={+,−,DIV} and S={+,−,*, DIV}.

• We completely determine the relations among the classes CCn(S). We further prove lower bounds:

• The component counting lower bound (e.g. Ω(n2) for the knapsack problem) proven for S={+,−,*} by Ben Or and Yao also holds for {+,−,*, DIVc}.

• The GCD-algorithm due to Brent and Kung for {+,−,DIVc}-CTs is optimal even for {+,−,DIV}-CTs.

• Testing whether q(y) > x for an irreducible polynomial q of degree d takes time Ω(loglog(d)) for +, −, *, DIV−CTs, even if arbitrary rational constants can be used at unit cost. This is the first nontrivial lower bound in this strong model (in which e. g. every finite language can be recognized in constant time, independent of the size of the language).

## Keywords

Knapsack Problem Computation Tree Computational Capability Integer Coefficient Straight Line Program
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.

## References

1. 1.
L. Babai, B. Just, F. Meyer auf der Heide, On the limits of computations with the floor function, Information and Computation 78(2), 99–107, 1988.Google Scholar
2. 2.
M. Ben-Or, Lower bounds for algebraic computation trees, 15th ACM-STOC, 80–86, 1983.Google Scholar
3. 3.
R. Brent, H. Kung, Systolic VLSI-arrays for linear time GCD-computation, Proc. Int. Conf. on Very Large Scale Integration. (VLSI 83, IFIP), F. Anceau and E. J. Aas (eds.), 145–154, 1983.Google Scholar
4. 4.
N. Bshouty, Euclidean GCD algorithm is not optimal, preprint 1989.Google Scholar
5. 5.
N. Bshouty, private communication, 1992.Google Scholar
6. 6.
D. Dobkin, R. L. Lipton, A lower bound of 1/2n 2 on linear search programs for the knapsack problem, JCSS 16, 417–421, 1975.Google Scholar
7. 7.
B. Just, F. Meyer auf der Heide, A. Wigderson, On computations with integer division, RAIRO Informatique Theoretique 23(1), 101–111, 1989.Google Scholar
8. 8.
P. Klein, F. Meyer auf der Heide, A lower bound for the knapsack problem on random access machines, Acta Informatica 19(3), 385–396, 1983.Google Scholar
9. 9.
Y. Mansur, B. Schieber, P. Tiwari, Lower bounds for integer greatest common divisor computation, 29 IEEE FOCS, 54–63, 1988.Google Scholar
10. 10.
J. Meidanis, private communication, 1992.Google Scholar
11. 11.
F. Meyer auf der Heide, Lower bounds for solving linear Diophantine equations on random access machines, J. ACM 32(4), 929–937, 1985.Google Scholar
12. 12.
W. J. Paul, J. Simon, Decision trees and random access machines, Monographie 30, L'Enseignement Mathematique, Logique et Algorithmique, Univ. Geneva, Switzerland, 331–340, 1992.Google Scholar
13. 13.
A. Yao, Lower bounds for algebraic computation trees with integer inputs, 30th IEEE FOCS, 308–313, 1989.Google Scholar