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A loopless algorithm for generating (k, m)-ary trees in Gray code order

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Abstract

A family of (k, m)-ary trees was firstly introduced by Du and Liu when they studied hook length polynomial for plane trees. Recently, Amani and Nowzari-Dalini presented a generation algorithm to produce (k, m)-ary trees of order n encoding by Z-sequences in reverse lexicographic order. In this paper, we propose a loopless algorithm to generate all such Z-sequences in Gray code order. Hence, the worst-case time complexity of generating one Z-sequence is \({\mathcal {O}}(1)\), and the space requirement of our algorithm is \(2n+{\mathcal {O}}(1)\). Moreover, based on this ordering, we also provide ranking and unranking algorithms. The ranking algorithm can be run in \({\mathcal {O}}(\max \{kmn,n^2\})\) time and \({\mathcal {O}}(kmn)\) space, whereas the unranking algorithm requires \({\mathcal {O}}(kmn^2)\) time and space.

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Acknowledgements

The authors express their sincere thanks to the anonymous associate editor and referees for their valuable suggestions that greatly improved the original manuscript.

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Authors and Affiliations

  1. Institute of Information and Decision Sciences, National Taipei University of Business, Taipei, Taiwan

    Yu-Hsuan Chang & Jou-Ming Chang

  2. Department of Industrial Management, Lunghwa University of Science and Technology, Taoyuan, Taiwan

    Ro-Yu Wu

  3. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

    Cheng-Kuan Lin

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  1. Yu-Hsuan Chang
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  2. Ro-Yu Wu
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  4. Jou-Ming Chang
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Correspondence to Jou-Ming Chang.

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This research was partially supported by MOST grants 108-2221-M-262-001 (R.-Y. Wu) and 107-2221-E-141-001-MY3 (J.-M. Chang) from the Ministry of Science and Technology, Taiwan.

Appendices

Appendix A: The pseudo codes of the loopless algorithm

figure a

The algorithm generates the Z-sequences of all (k, m)-ary trees in Gray code order. The first sequence is generated in \({\mathcal {O}}(n)\) time (see Lines 2–6 in the main program). And, each subsequent one is generated in constant time by calling the procedure NextZ(), where the complexity does not include the output of a sequence.

Appendix B: The pseudo codes of the ranking function

figure b

Appendix C: The pseudo codes of the unranking function

figure c

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Chang, YH., Wu, RY., Lin, CK. et al. A loopless algorithm for generating (k, m)-ary trees in Gray code order. Optim Lett 15, 1133–1154 (2021). https://doi.org/10.1007/s11590-020-01613-z

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  • DOI: https://doi.org/10.1007/s11590-020-01613-z

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