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The Grandmama de Bruijn Sequence for Binary Strings

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9644)

Abstract

A de Bruijn sequence is a binary string of length \(2^n\) which, when viewed cyclically, contains every binary string of length n exactly once as a substring. Knuth refers to the lexicographically least de Bruijn sequence for each n as the “granddaddy” and Fredricksen et al. showed that it can be constructed by concatenating the aperiodic prefixes of the binary necklaces of length n in lexicographic order. In this paper we prove that the granddaddy has a lexicographic partner. The “grandmama” sequence is constructed by instead concatenating the aperiodic prefixes in co-lexicographic order. We explain how our sequence differs from the previous sequence and why it had not previously been discovered.

Keywords

  • de Bruijn sequence
  • Lexicographic order
  • Necklace
  • Lyndon word
  • FKM construction
  • Ford sequence

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Notes

  1. 1.

    Note: In the grandmama construction the necklaces are still the lexicographically least representatives for their rotational equivalence class, as clarified in Sect. 4.1.

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Dragon, P.B., Hernandez, O.I., Williams, A. (2016). The Grandmama de Bruijn Sequence for Binary Strings. In: Kranakis, E., Navarro, G., Chávez, E. (eds) LATIN 2016: Theoretical Informatics. LATIN 2016. Lecture Notes in Computer Science(), vol 9644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49529-2_26

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  • DOI: https://doi.org/10.1007/978-3-662-49529-2_26

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