Efficient Construction of Long Orientable Sequences

Authors Daniel Gabrić , Joe Sawada



PDF
Thumbnail PDF

File

LIPIcs.CPM.2024.15.pdf
  • Filesize: 0.81 MB
  • 12 pages

Document Identifiers

Author Details

Daniel Gabrić
  • University of Guelph, Canada
Joe Sawada
  • University of Guelph, Canada

Cite AsGet BibTex

Daniel Gabrić and Joe Sawada. Efficient Construction of Long Orientable Sequences. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 15:1-15:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CPM.2024.15

Abstract

An orientable sequence of order n is a cyclic binary sequence such that each length-n substring appears at most once in either direction. Maximal length orientable sequences are known only for n ≤ 7, and a trivial upper bound on their length is 2^{n-1} - 2^{⌊(n-1)/2⌋}. This paper presents the first efficient algorithm to construct orientable sequences with asymptotically optimal length; more specifically, our algorithm constructs orientable sequences via cycle-joining and a successor-rule approach requiring O(n) time per bit and O(n) space. This answers a longstanding open question from Dai, Martin, Robshaw, Wild [Cryptography and Coding III (1993)]. Our sequences are applied to find new longest-known orientable sequences for n ≤ 20.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • orientable sequence
  • de Bruijn sequence
  • concatenation tree
  • cycle-joining
  • universal cycle

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. D. Adamson, V. V. Gusev, I. Potapov, and A. Deligkas. Ranking bracelets in polynomial time. In Paweł Gawrychowski and Tatiana Starikovskaya, editors, 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021), volume 191 of Leibniz International Proceedings in Informatics (LIPIcs), pages 4:1-4:17, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CPM.2021.4.
  2. A. Alhakim, C. J. Mitchell, J. Szmidt, and P. R. Wild. Orientable sequences over non-binary alphabets. manuscript, 2023. Google Scholar
  3. K. S. Booth. Lexicographically least circular substrings. Inform. Process. Lett., 10(4/5):240-242, 1980. URL: https://doi.org/10.1016/0020-0190(80)90149-0.
  4. S. Brlek, S. Hamel, M. Nivat, and C. Reutenauer. On the palindromic complexity of infinite words. Internat. J. Found. Comp. Sci., 15(02):293-306, 2004. URL: https://doi.org/10.1142/S012905410400242X.
  5. J. Burns and C. J. Mitchell. Position sensing coding schemes. In Cryptography and Coding III (M.J.Ganley, ed.), pages 31-66. Oxford University Press, 1993. Google Scholar
  6. J. Currie and P. Lafrance. Avoidability index for binary patterns with reversal. Electronic J. Combinatorics, 23((1) P1.36):1-14, 2016. URL: https://doi.org/10.37236/5483.
  7. Z. D. Dai, K. M. Martin, M. J. B. Robshaw, and P. R. Wild. Orientable sequences. In Cryptography and Coding III (M.J.Ganley, ed.), pages 97-115. Oxford University Press, 1993. Google Scholar
  8. T. Etzion. An algorithm for generating shift-register cycles. Theoret. Comput. Sci., 44(2):209-224, 1986. URL: https://doi.org/10.1016/0304-3975(86)90118-0.
  9. T. Etzion. Self-dual sequences. J. Combin. Theory Ser. A, 44(2):288-298, 1987. URL: https://doi.org/10.1016/0097-3165(87)90035-5.
  10. T. Etzion and A. Lempel. Algorithms for the generation of full-length shift-register sequences. IEEE Trans. Inform. Theory, 30(3):480-484, 1984. URL: https://doi.org/10.1109/TIT.1984.1056919.
  11. L. Fleischer and J. O. Shallit. Words that avoid reversed factors, revisited. Arxiv preprint arXiv:1911.11704 [cs.FL], available at http://arxiv.org/abs/1911.11704, 2019.
  12. H. Fredricksen. A survey of full length nonlinear shift register cycle algorithms. SIAM Review, 24(2):195-221, 1982. URL: https://doi.org/10.1137/1024041.
  13. D. Gabrić, J. Sawada, A. Williams, and D. Wong. A framework for constructing de Bruijn sequences via simple successor rules. Discrete Math., 241(11):2977-2987, 2018. URL: https://doi.org/10.1016/j.disc.2018.07.010.
  14. D. Gabrić, J. Sawada, A. Williams, and D. Wong. A successor rule framework for constructing k-ary de Bruijn sequences and universal cycles. IEEE Trans. Inform. Theory, 66(1):679-687, 2020. URL: https://doi.org/10.1109/TIT.2019.2928292.
  15. C. Hierholzer. Ueber die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Math. Annalen, 6:30-32, 1873. URL: https://doi.org/10.1007/BF01442866.
  16. Y. Huang. A new algorithm for the generation of binary de Bruijn sequences. J. Algorithms, 11(1):44-51, 1990. URL: https://doi.org/10.1016/0196-6774(90)90028-D.
  17. C. J. A. Jansen, W. G. Franx, and D. E. Boekee. An efficient algorithm for the generation of DeBruijn cycles. IEEE Trans. Inform. Theory, 37(5):1475-1478, 1991. URL: https://doi.org/10.1109/18.133272.
  18. A. Lempel. On a homomorphism of the de Bruijn graph and its applications to the design of feedback shift registers. IEEE Trans. Comput., C-19(12):1204-1209, 1970. URL: https://doi.org/10.1109/T-C.1970.222859.
  19. R. Mercaş. On the aperiodic avoidability of binary patterns with variables and reversals. Theoret. Comput. Sci., 682:180-189, 2017. URL: https://doi.org/10.1016/j.tcs.2016.12.022.
  20. C. J. Mitchell and P. R. Wild. Constructing orientable sequences. IEEE Trans. Inform. Theory, 68(7):4782-4789, 2022. URL: https://doi.org/10.1109/TIT.2022.3158645.
  21. N. Rampersad and J. O. Shallit. Words that avoid reversed subwords. J. Combin. Math. Combin. Comput., 54:157-164, 2005. Google Scholar
  22. J. Sawada, J. Sears, A. Trautrim, and A. Williams. Concatenation trees: A framework for efficient universal cycle and de Bruijn sequence constructions. Arxiv preprint arXiv:2308.12405 [math.CO], available at https://arxiv.org/abs/2308.12405, 2023.
  23. J. Sawada and A. Williams. Constructing the first (and coolest) fixed-content universal cycle. Algorithmica, 85(6):1754-1785, 2023. URL: https://doi.org/10.1007/s00453-022-01047-2.
  24. J. Sawada and D. Wong. Efficient universal cycle constructions for weak orders. Discrete Math., 343(10):112022, 2020. URL: https://doi.org/10.1016/j.disc.2020.112022.
  25. N. J. A. Sloane et al. OEIS Foundation Inc. (2024), The On-Line Encyclopedia of Integer Sequences, URL: https://oeis.org.