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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].

Cite As Get BibTex

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.

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