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The Cost of Cyclic Permutations and Remainder Sums in the Euclidean Algorithm

Authors Valentin Blomer , Kai-Uwe Bux

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LIPIcs.AofA.2026.14.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • generic algorithms
  • average case analysis
  • Euclidean algorithm
  • exponential sums
  • cyclic permutations

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Abstract

We discuss a modification to the Gries-Mills block swapping scheme for in-place rotation with average costs of 1.85 moves per element and worst case performance still at 3 moves per element. The analysis of the average case relies on the asymptotic behavior of the sum of remainders in the Euclidean algorithm.

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Valentin Blomer and Kai-Uwe Bux. The Cost of Cyclic Permutations and Remainder Sums in the Euclidean Algorithm. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.14

Author Details

Valentin Blomer
  • University of Bonn, Germany
Kai-Uwe Bux
  • Bielefeld University, Germany

Funding

This collaboration was funded by the DFG via CRC TRR 358 - Project-ID 491392403

Acknowledgements

We thank Markus Nebel, Jens Stoye and Pascal Schweizer for helpful conversations.

Supplementary Materials

References

  1. Valentin Blomer and Kai-Uwe Bux. k-u-bux/rotate-paper. Software, (visited on 2026-06-30). URL: https://github.com/k-u-bux/rotate-paper
    Software Heritage Logo archived version
    full metadata available at: https://doi.org/10.4230/artifacts.26939
  2. Igor Van den Hoven. scandum/rotate. https://github.com/scandum/rotate, 2021. Google Scholar
  3. David Gries and Harlan Mills. Swapping sections, January 1981. URL: http://hdl.handle.net/1813/6292.
  4. Mervat Hashem, Kenli Li, and Ahmad Salah. Pi-sqrt: novel parallel implementations of in-place sequence rotation on multicore systems. Clust. Comput., 26(1):539-557, 2023. URL: https://doi.org/10.1007/s10586-022-03591-6.
  5. H. Heilbronn. On the average length of a class of finite continued fractions. In Paul Turán, editor, Number Theory and Analysis (Papers in Honor of Edmund Landau), pages 87-96. Plenum, New York, 1969. Google Scholar
  6. Ching-Kuang Shene. An analysis of two in-place array rotation algorithms. Comput. J., 40(9):541-546, 1997. URL: https://doi.org/10.1093/comjnl/40.9.541.
  7. Don Zagier. Values of zeta functions and their applications. In First European Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages 497-512. Birkhäuser, Basel, 1994. Google Scholar
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