Abstract
Du and Liu (2007) introduced (k, m)-ary trees as a generalization of k-ary trees. In a (k, m)-ary tree, every node on even level has degree k (i.e., has k children), and every node on odd level has degree m (which is called a crucial node) or is a leaf. In particular, a (k, m)-ary tree of order n has exactly n crucial nodes. Recently, Amani and Nowzari-Dalini (2019) presented a generation algorithm to produce all (k, m)-ary trees of order n in B-order using Zaks’ encoding, and show that the generated ordering of this encoding results in a reverse-lexicographical ordering. They also proposed the corresponding ranking and unranking algorithms for (k, m)-ary trees according to such a generated ordering. These algorithms take \({\mathcal {O}}(kmn^2)\) time and space for building a precomputed table in which (k, m)-Catalan numbers (i.e., a kind of generalized Catalan numbers) are stored in advance. In this paper, we revisit the ranking and unranking problems. With the help of an encoding scheme called “right-distance” introduced by Wu et al. (2011), we propose new ranking and unranking algorithms for (k, m)-ary trees of order n in B-order using Zaks’ encoding. We show that both algorithms can be improved in \({\mathcal {O}}(kmn)\) time and \({\mathcal {O}}(n)\) space without building the precomputed table.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Amani, M.: Gap terminology and related combinatorial properties for AVL trees and Fibonacci-isomorphic trees. AKCE Int. J. Graphs Comb. 15, 14–21 (2018)
Amani, M., Nowzari-Dalini, A.: Efficient generation, ranking, and unranking of \((k,m)\)-ary trees in B-order. Bull. Iranian Math. Soc. (2019). https://doi.org/10.1007/s41980-018-0190-y
Amani, M., Nowzari-Dalini, A.: Ranking and unranking algorithm for neuronal trees in B-order. J. Phys. Sci. 20, 19–34 (2015)
Amani, M., Nowzari-Dalini, A.: Generation, ranking and unranking of ordered trees with degree bounds. In: Proceedings of DCM 2015. Electronic Proceedings in Theoretical Computer Science, vol. 204, pp. 31–45 (2015)
Amani, M., Nowzari-Dalini, A., Ahrabian, H.: Generation of neuronal trees by a new three letters encoding. Comput. Inform. J. 33, 1428–1450 (2014)
Du, R.R.X., Liu, F.: \((k, m)\)-Catalan numbers and hook length polynomials for plane trees. Euro. J. Combin. 28, 1312–1321 (2007)
Li, L.: Ranking and unranking AVL trees. SIAM J. Comput. 15, 1025–1035 (1986)
Pai, K.-J., Chang, J.-M., Wu, R.-Y., Chang, S.-C.: Amortized efficiency of generation, ranking and unranking left-child sequences in lexicographic order. Discrete Appl. Math. (2018). https://doi.org/10.1016/j.dam.2018.09.035
Pallo, J.: Generating trees with \(n\) nodes and \(m\) leaves. Int. J. Comput. Math. 21, 133–144 (1987)
Seyedi-Tabari, E., Ahrabian, H., Nowzari-Dalini, A.: A new algorithm for generation of different types of RNA. Int. J. Comput. Math. 87, 1197–1207 (2010)
Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)
Wu, R.-Y., Chang, J.-M., Chan, H.-C., Pai, K.-J.: A loopless algorithm for generating multiple binary tree sequences simultaneously. Theor. Comput. Sci. 556, 25–33 (2014)
Wu, R.-Y., Chang, J.-M., Chang, C.-H.: Ranking and unranking of non-regular trees with a prescribed branching sequence. Math. Comput. Model. 53, 1331–1335 (2011)
Wu, R.-Y., Chang, J.-M., Chen, A.-H., Liu, C.-L.: Ranking and unranking \(t\)-ary trees in a Gray-code order. Comput. J. 56, 1388–1395 (2013)
Wu, R.-Y., Chang, J.-M., Wang, Y.-L.: A linear time algorithm for binary tree sequences transformation using left-arm and right-arm rotations. Theor. Comput. Sci. 355, 303–314 (2006)
Wu, R.-Y., Chang, J.-M., Wang, Y.-L.: Loopless generation of non-regular trees with a prescribed branching sequence. Comput. J. 53, 661–666 (2010)
Wu, R.-Y., Chang, J.-M., Wang, Y.-L.: Ranking and unranking of \(t\)-ary trees using RD-sequences. IEICE Trans. Inform. Syst. E94–D, 226–232 (2011)
Zaks, S.: Lexicographic generation of ordered trees. Theor. Comput. Sci. 10, 63–82 (1980)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Chang, YH., Wu, RY., Chang, RS., Chang, JM. (2019). Improved Algorithms for Ranking and Unranking (k, m)-Ary Trees. In: Du, DZ., Li, L., Sun, X., Zhang, J. (eds) Algorithmic Aspects in Information and Management. AAIM 2019. Lecture Notes in Computer Science(), vol 11640. Springer, Cham. https://doi.org/10.1007/978-3-030-27195-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-27195-4_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-27194-7
Online ISBN: 978-3-030-27195-4
eBook Packages: Computer ScienceComputer Science (R0)