Bimonotone enumeration

Author:
Michael Eisermann

Journal:
Math. Comp. **78** (2009), 591-613

MSC (2000):
Primary 68P10; Secondary 11Y50, 68W10, 11Y16, 11D45

DOI:
https://doi.org/10.1090/S0025-5718-08-02162-5

Published electronically:
June 16, 2008

MathSciNet review:
2448723

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Abstract: Solutions of a diophantine equation , with in some finite range, can be efficiently enumerated by sorting the values of and in ascending order and searching for collisions. This article considers functions that are bimonotone in the sense that whenever and . A two-variable polynomial with non-negative coefficients is a typical example. The problem is to efficiently enumerate all pairs such that the values appear in increasing order. We present an algorithm that is memory-efficient and highly parallelizable. In order to enumerate the first values of , the algorithm only builds up a priority queue of length at most . In terms of bit-complexity this ensures that the algorithm takes time and requires memory , which considerably improves on the memory bound provided by a naïve approach, and extends the semimonotone enumeration algorithm previously considered by R.L.Ekl and D.J.Bernstein.

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Additional Information

**Michael Eisermann**

Affiliation:
Institut Fourier, Université Grenoble I, France

Email:
Michael.Eisermann@ujf-grenoble.fr

DOI:
https://doi.org/10.1090/S0025-5718-08-02162-5

Keywords:
Sorting and searching,
diophantine equation,
bimonotone function,
sorted enumeration,
semimonotone enumeration,
bimonotone enumeration

Received by editor(s):
July 27, 2005

Received by editor(s) in revised form:
June 22, 2007

Published electronically:
June 16, 2008

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.