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Generating a Cyclic 2-Gray Code for Lucas Words in Constant Amortized Time

Authors Bowie Liu , Dennis Wong , Chan-Tong Lam , Sio-Kei Im

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Lucas word
  • Fibonacci word
  • Fibonacci sequence
  • q-decreasing sequence
  • Gray code
  • CAT algorithm

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Abstract

We present a two-stage algorithm to generate a cyclic 2-Gray code for Lucas words. The first step involves a simple recursive algorithm that generates a cyclic 2-Gray code for Fibonacci words starting with a 0, which are strings that avoid p consecutive ones starting with a 0. Then, by considering the first and last blocks of 1s and concatenating lists of Fibonacci words starting with a 0 of different length n, we construct the first cyclic 2-Gray code for Lucas words. By using a dynamic programming approach, our algorithm generates each Lucas word and Fibonacci word in constant amortized time per string, using O(n³) space. The algorithm can also be modified to produce the first efficient algorithm for generating q-decreasing sequences in constant amortized time per string, also using O(n³) space. Our work extends a previous result on generating a cyclic 2-Gray code for q-decreasing sequences [Conference proceeding: International Conference and Workshop on Algorithms and Computation (WALCOM), LNCS 14549:91-102, 2024].

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Bowie Liu, Dennis Wong, Chan-Tong Lam, and Sio-Kei Im. Generating a Cyclic 2-Gray Code for Lucas Words in Constant Amortized Time. In 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 331, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CPM.2025.22

Author Details

Bowie Liu
  • Faculty of Applied Sciences, Macao Polytechnic University, Macao
Dennis Wong
  • Faculty of Applied Sciences, Macao Polytechnic University, Macao
Chan-Tong Lam
  • Faculty of Applied Sciences, Macao Polytechnic University, Macao
Sio-Kei Im
  • Macao Polytechnic University, Macao

Funding

This research is supported by the Macao Science and Technology Development Fund FDCT/0033/2023/RIA1.

Acknowledgements

The second author would also like to thank his parents for their invaluable assistance in helping him to move house while he was busy drafting this paper.

References

  1. J. Azarija, S. Klavžar, J. Lee, and Y. Rho. Connectivity of Fibonacci cubes, Lucas cubes and generalized cubes. Discret. Math. Theor. Comput. Sci. , Vol. 17 no. 1, 2015. URL: https://doi.org/10.46298/DMTCS.2115.
  2. J.-L. Baril, S. Kirgizov, and V. Vajnovszki. Asymptotic bit frequency in Fibonacci words. Pure Math. Appl. , 30(1):23-30, 2022. Google Scholar
  3. J.-L. Baril, S. Kirgizov, and V. Vajnovszki. Gray codes for Fibonacci q-decreasing words. Theor. Comput. Sci., 927:120-132, 2022. URL: https://doi.org/10.1016/J.TCS.2022.06.003.
  4. J.-L. Baril and C. Moreira dos Santos. Gray code for compositions of n with parts 1 and p. Appl. Discret. Math., 3(1):67-84, 2009. Google Scholar
  5. J.-L. Baril and V. Vajnovszki. Gray codes for order p Lucas strings. In International Conference on Combinatorics on Words, Turku, Finland, 2003. Google Scholar
  6. J.-L. Baril and V. Vajnovszki. Minimal change list for Lucas strings and some graph theoretic consequences. Theor. Comput. Sci., 346(2):189-199, 2005. URL: https://doi.org/10.1016/J.TCS.2005.08.020.
  7. T. Chinburg, C. Savage, and H. Wilf. Combinatorial families that are exponentially far from being listable in Gray code sequence. Trans. Am. Math. Soc., 351:379-402, 1999. Google Scholar
  8. P. Codara and O. M. D’Antona. Generalized Fibonacci and Lucas cubes arising from powers of paths and cycles. Discrete Math., 339(1):270-282, 2016. URL: https://doi.org/10.1016/J.DISC.2015.08.012.
  9. O. Eǧecioǧlu and V. Iršič. Fibonacci-run graphs I: Basic properties. Discret. Appl. Math., 295:70-84, 2021. URL: https://doi.org/10.1016/J.DAM.2021.02.025.
  10. O. Eǧecioǧlu and V. Iršič. Fibonacci-run graphs II: Degree sequences. Discret. Appl. Math., 300:56-71, 2021. URL: https://doi.org/10.1016/J.DAM.2021.05.018.
  11. O. Eǧecioǧlu, S. Klavžar, and M. Mollard. Fibonacci Cubes With Applications And Variations. World Scientific Publishing Company, 2023. Google Scholar
  12. M. Feinberg. Fibonacci–Tribonacci. Fibonacci Q., 1(3):71-74, 1963. Google Scholar
  13. I. Flores. Direct calculation of k-generalized Fibonacci numbers. Fibonacci Q., 5(3):259-266, 1967. Google Scholar
  14. I. Goulden and D. Jackson. Combinatorial Enumeration. A Wiley-Interscience Publication. John Wiley & Sons Inc., New York, 1983. Google Scholar
  15. N. Hassler, V. Vajnovszki, and D. Wong. Efficient generation of some greedy Gray codes. Korean Mathematical Society Annual Meeting, 2024. Google Scholar
  16. N. Hassler, V. Vajnovszki, and D. Wong. Greedy Gray codes for some restricted classes of binary words. Electronic Proceedings in Theoretical Computer Science, EPTCS, 403:108-112, June 2024. Publisher Copyright: © N. Hassler, V. Vajnovszki, D. Wong; 13th Conference on Random Generation of Combinatorial Structures. Polyominoes and Tilings, GASCom 2024; Conference date: 24-06-2024 Through 28-06-2024. URL: https://doi.org/10.4204/EPTCS.403.23.
  17. W.-J. Hsu. Fibonacci cubes - a new interconnection topology. IEEE Trans. Parallel Distrib. Syst., 4(1):3-12, 1993. Google Scholar
  18. A. Ilić, S. Klavžar, and Y. Rho. Generalized Lucas cubes. Appl. Anal. Discrete Math., 6(1):82-94, 2012. Google Scholar
  19. S. Kirgizov. Q-bonacci words and numbers. Fibonacci Q., 60(5):187-195, 2022. Google Scholar
  20. S. Kirgizov and J. Ramírez. Polyominoes and graphs built from Fibonacci words. Fibonacci Q., 60(5):196-211, 2022. Google Scholar
  21. S. Klavžar and M. Mollard. Asymptotic properties of Fibonacci cubes and Lucas cubes. Ann. Comb., 18(3):447-457, 2014. Google Scholar
  22. D. Knuth. The Art of Computer Programming, Volume 4A, Combinatorial Algorithms. Addison-Wesley Professional, 2011. Google Scholar
  23. J. Liu, W.-J. Hsu, and M. J. Chung. Generalized Fibonacci cubes are mostly Hamiltonian. J. Graph Theory, 18(8):817-829, 1994. URL: https://doi.org/10.1002/JGT.3190180806.
  24. E. Miles. Generalized Fibonacci numbers and associated matrices. Amer. Math. Monthly, 67(8):745-752, 1960. Google Scholar
  25. E. Munarini, C. P. Cippo, and N. Z. Salvi. On the Lucas cubes. Fibonacci Q., 39(1):12-21, 2001. Google Scholar
  26. T. Mütze. Combinatorial Gray codes — an updated survey. Electron. J. Comb., 30:3, 2023. Google Scholar
  27. T. Noe and J. Post. Primes in Fibonacci n-step and Lucas n-step sequences. J. of Integer Sequences, 8:Article 05.4.4, 2005. Google Scholar
  28. OEIS Foundation Inc. The on-line encyclopedia of integer sequences, published electronically at http://oeis.org. Sequences A000032, A001644, A073817, A074048, A074584, A074584, A105754, and A105755, 2023.
  29. F. Ruskey. Combinatorial Generation. Book under preparation, 2003. Google Scholar
  30. C. Savage. A survey of combinatorial Gray codes. SIAM Review, (4):605-629, 1997. URL: https://doi.org/10.1137/S0036144595295272.
  31. J. Sawada, A. Williams, and D. Wong. Inside the binary reflected Gray code: Flip-swap languages in 2-Gray code order. In Thierry Lecroq and Svetlana Puzynina, editors, Combinatorics on Words, pages 172-184, Cham, 2021. URL: https://doi.org/10.1007/978-3-030-85088-3_15.
  32. J. Sawada, A. Williams, and D. Wong. Flip-swap languages in binary reflected Gray code order. Theor. Comput. Sci., 933:138-148, 2022. URL: https://doi.org/10.1016/J.TCS.2022.08.024.
  33. D. Stanton and D. White. Constructive Combinatorics. Springer Science & Business Media, 2012. Google Scholar
  34. V. Vajnovszki. A loopless generation of bitstrings without p consecutive ones. In C. S. Calude, M. J. Dinneen, and S. Sburlan, editors, Combinatorics, Computability and Logic, pages 227-240, London, 2001. Springer London. URL: https://doi.org/10.1007/978-1-4471-0717-0_19.
  35. V. Vajnovszki. Gray code order for Lyndon words. Discret. Math. Theor. Comput. Sci., 9(2), 2007. URL: https://doi.org/10.46298/DMTCS.393.
  36. J. Wei and Y. Yang. Fibonacci and Lucas p-cubes. Discret. Appl. Math., 322:365-383, 2022. URL: https://doi.org/10.1016/J.DAM.2022.09.004.
  37. D. Wong, B. Liu, C.-T. Lam, and M. Im. Generating cyclic 2-Gray codes for Fibonacci q-decreasing words. In Ryuhei Uehara, Katsuhisa Yamanaka, and Hsu-Chun Yen, editors, WALCOM: Algorithms and Computation, pages 91-102, Singapore, 2024. Springer Nature Singapore. URL: https://doi.org/10.1007/978-981-97-0566-5_8.
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