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Constructing the first (and coolest) fixed-content universal cycle

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Abstract

We explicitly construct the first universal cycles for strings with fixed-content—also known as strings with the same Parikh vector, or multiset permutations—using their shorthand encoding, which omits the final symbol as it is redundant. For example, 112312131132 is a universal cycle for content \(S = \{1,1,2,3\}\). Its first three windows—112, 123, and 231—are the shorthand representatives of 1123, 1231, and 2311, respectively. Our first construction \(\mathcal {V}(S)\) applies the classic cycle-joining approach on the first-inversion tree of necklace cycles with content S. For example, when \(S = \{1,1,2,3\}\) the root is the necklace cycle 1123 and its children are \(1\underline{21}3\) and \(11\underline{32}\) by swapping their first (i.e., leftmost) inversions. From this construction, we derive a successor rule to generate successive symbols of \(\mathcal {V}(S)\) in O(n)-time, where \(n=|S|\) is the cardinality of S. Our second construction \(\mathcal {U}(S)\) concatenates fixed-content necklaces together in a cool-lex order using the necklace-prefix algorithm. For example, \(\mathcal {U}(S) = 1123 \cdot 1213 \cdot 1132\) for \(S = \{1,1,2,3\}\). Central to this construction is the first shift Gray code for fixed-content necklaces, and a new efficient algorithm for generating these strings. From this construction, we can generate successive symbols of \(\mathcal {U}(S)\) in O(1)-amortized time while using O(n)-space. We complete our investigation with a pleasant surprise: \(\mathcal {V}(S) = \mathcal {U}(S)\). Our new results simultaneously generalize universal cycle constructions of shorthand permutations by Ruskey et al. (Algorithmica 64, 2012) and shorthand fixed-weight binary strings by Ruskey et al. (SIAM J Disc Math 26(2):605–617, 2012). They also provide a prefix-shift Gray code for multiset permutations in which the first symbol moves into the last or second-last position, which tightens the previous prefix-shift Gray code by Williams (Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2009). Finally, we draw parallels between our constructions and the well-known granddaddy de Bruijn sequence for binary strings.

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Notes

  1. Universal cycles can also be defined for objects other than strings (e.g., graphs [3]).

  2. This generalizes the decreasing spanning tree presented in [24], but with the relative values of the symbols inverted.

  3. In this presentation, we use the lexicographically smallest representative for necklaces rather than the lexicographically largest.

  4. Note that the aperiodic prefix of a necklace of length n is a Lyndon word whose length divides n.

  5. In nature, a scut is a short tail. Here, it is a suffix of tail(S) with a small symbol prepended.

  6. In contrast, the swaps used in the Steinhaus-Johnson-Trotter order [48] cause up to three edges to be changed.

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Acknowledgements

We’d like to thank the anonymous referees for helpful comments and corrections.

Funding

Joe Sawada: Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN 400673-2012

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  1. University of Guelph, Guelph, ON, Canada

    Joe Sawada

  2. Williams College, Williamstown, MA, USA

    Aaron Williams

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  1. Joe Sawada
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Correspondence to Aaron Williams.

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The research of Joe Sawada is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN-2018-04211.

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Appendices

Appendix A: Successor-rule approach to construct \(\mathcal {V}(S)\)

figure g

Appendix B: Concatenation approach to construct the reverse of \(\mathcal {U}(S)\)

figure h

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Sawada, J., Williams, A. Constructing the first (and coolest) fixed-content universal cycle. Algorithmica 85, 1754–1785 (2023). https://doi.org/10.1007/s00453-022-01047-2

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