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A de Bruijn Sequence Construction by Concatenating Cycles of the Complemented Cycling Register

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Combinatorics on Words (WORDS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10432))

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Abstract

We present a new de Bruijn sequence construction based on co-necklaces and the complemented cycling register (CCR). A co-necklace is the lexicographically smallest string in an equivalence class of strings induced by the CCR. We prove that a concatenation of the cycles of the CCR forms a de Bruijn sequence when the cycles are ordered in colexicographic order with respect to their co-necklace representatives. We also give an algorithm that produces the de Bruijn sequence in O(1)-time per bit. Finally, we prove that our construction has a discrepancy bounded above by 2n.

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Notes

  1. 1.

    Because a longest run of the form \(0^*\) or \(1^*\) must be at the start of a co-necklace, the algorithm can be further optimized by keeping track of the longest current run of the form \(1^*\). However, it will not affect the asymptotic analysis.

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Authors and Affiliations

  1. University of Guelph, Guelph, Canada

    Daniel Gabric & Joe Sawada

Authors
  1. Daniel Gabric
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  2. Joe Sawada
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Corresponding author

Correspondence to Joe Sawada .

Editor information

Editors and Affiliations

  1. Université du Québec à Montréal, Montreal, Québec, Canada

    Srečko Brlek

  2. Université du Québec à Montréal, Montreal, Québec, Canada

    Francesco Dolce

  3. Université du Québec à Montréal, Montreal, Québec, Canada

    Christophe Reutenauer

  4. Université du Québec à Montréal, Montreal, Québec, Canada

    Élise Vandomme

Appendix - C Code

Appendix - C Code

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Gabric, D., Sawada, J. (2017). A de Bruijn Sequence Construction by Concatenating Cycles of the Complemented Cycling Register. In: Brlek, S., Dolce, F., Reutenauer, C., Vandomme, É. (eds) Combinatorics on Words. WORDS 2017. Lecture Notes in Computer Science(), vol 10432. Springer, Cham. https://doi.org/10.1007/978-3-319-66396-8_6

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  • DOI: https://doi.org/10.1007/978-3-319-66396-8_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66395-1

  • Online ISBN: 978-3-319-66396-8

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