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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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Katharina Lürwer-Brüggemeier
    • 1
  • Friedhelm Meyer
    • 1
  1. 1.Department for Mathematics and Computer Science and Heinz Nixdorf InstituteUniversity of PaderbornPaderbornGermany

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