Abstract
The problem of coding labeled trees has been widely studied in the literature and several bijective codes that realize associations between labeled trees and sequences of labels have been presented. k-trees are one of the most natural and interesting generalizations of trees and there is considerable interest in developing efficient tools to manipulate this class of graphs, since many NP-Complete problems have been shown to be polynomially solvable on k-trees and partial k-trees. In 1970 Rényi and Rényi generalized the Prüfer code, the first bijective code for trees, to a subset of labeled k-trees. Subsequently, non redundant codes that realize bijection between k-trees (or Rényi k-trees) and a well defined set of strings were produced. In this paper we introduce a new bijective code for labeled k-trees which, to the best of our knowledge, produces the first coding and decoding algorithms running in linear time with respect to the size of the k-tree.
Similar content being viewed by others
References
Beineke, L.W., Pippert, R.E.: On the number of k-dimensional trees. J. Comb. Theory 6, 200–205 (1969)
Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybern. 11, 1–21 (1993)
Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comp. Sci. 209, 1–45 (1998)
Caminiti, S., Finocchi, I., Petreschi, R.: A unified approach to coding labeled trees. In: Proceedings of the 6th Latin American Symposium on Theoretical Informatics (LATIN’04). LNCS, vol. 2976, pp. 339–348. Springer, Berlin (2004)
Caminiti, S., Finocchi, I., Petreschi, R.: On coding labeled trees. Theor. Comp. Sci. 382(2), 97–108 (2007)
Caminiti, S., Fusco, E.G., Petreschi, R.: A bijective code for k-trees with linear time encoding and decoding. In: Proceedings of the International Symposium on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies (ESCAPE’07). LNCS, vol. 4614, pp. 408–420. Springer, Berlin (2007)
Caminiti, S., Petreschi, R.: String coding of trees with locality and heritability. In: Proceedings of the 11th International Conference on Computing and Combinatorics (COCOON’05). LNCS, vol. 3595, pp. 251–262. Springer, Berlin (2005)
Cayley, A.: A theorem on trees. Q. J. Math. 23, 376–378 (1889)
Chen, W.Y.C.: A general bijective algorithm for trees. Proc. Nat. Acad. Sci., USA 87, 9635–9639 (1990)
Chen, W.Y.C.: A Coding algorithm for Rényi trees. J. Comb. Theory A 63, 11–25 (1993)
Crochemore, M., Rytter, W.: Jewels of Stringology. World Scientific, Singapore (2002)
Deo, N., Kumar, N., Kumar, V.: Parallel generation of random trees and connected graphs. Congr. Numer. 130, 7–18 (1998)
Deo, N., Micikevičius, P.: A new encoding for labeled trees employing a stack and a queue. Bull. Inst. Comb. Appl. 34, 77–85 (2002)
Edelson, W., Gargano, M.L.: Feasible encodings for GA solutions of constrained minimal spanning tree problems. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’00), p. 754. Las Vegas, Nevada, USA (2000)
Eğecioğlu, Ö., Remmel, J.B.: Bijections for Cayley trees, spanning trees, and their q-analogues. J. Comb. Theory A 42(1), 15–30 (1986)
Eğecioğlu, Ö., Shen, L.P.: A bijective proof for the number of labeled q-trees. Ars Comb. B 25, 3–30 (1988)
Foata, D.: Enumerating k-trees. Discrete Math. 1(2), 181–186 (1971)
Greene, C., Iba, G.A.: Cayley’s formula for multidimensional trees. Discrete Math. 13, 1–11 (1975)
Harary, F., Palmer, E.M.: On acyclic simplicial complexes. Mathematika 15, 115–122 (1968)
Kelmans, A., Pak, I., Postnikov, A.: Tree and forest volumes of graphs. Technical Report, DIMACS 2000-03 (2000)
Markenzon, L., Costa Pereira, P.R., Vernet, O.: The reduced Prüfer code for rooted labelled k-trees. In: Proceedings of 7th International Colloquium on Graph Theory. Electronic Notes in Discrete Mathematics, vol. 22, pp. 135–139 (2005)
Moon, J.W.: The number of labeled k-trees. J. Comb. Theory 6, 196–199 (1969)
Moon, J.W.: Counting Labeled Trees. William Clowes and Sons, London (1970)
Neville, E.H.: The codifying of tree-structure. In: Proceedings of Cambridge Philosophical Society, vol. 49, pp. 381–385 (1953)
Picciotto, S.: How to encode a tree. Ph.D. Thesis, University of California, San Diego (1999)
Prüfer, H.: Neuer Beweis eines Satzes über Permutationen. Arch. Math. Phys. 27, 142–144 (1918)
Rényi, A., Rényi, C.: The Prüfer code for k-trees. In: Erdös, P. et al. (eds.) Combinatorial Theory and its Applications, pp. 945–971. North-Holland, Amsterdam (1970)
Rose, D.J.: On simple characterizations of k-trees. Discrete Math. 7, 317–322 (1974)
Vardi, I.: Computational Recreations in Mathematica. Benjamin-Cummings, Redwood City (1991). Chap. Computing Binomial Coefficients
Author information
Authors and Affiliations
Corresponding author
Additional information
Work partially supported by MIUR: Italian Ministry for University and Scientific Research. A preliminary version of this paper appeared in the Proceedings of the International Symposium on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies (ESCAPE’07) 6.
Rights and permissions
About this article
Cite this article
Caminiti, S., Fusco, E.G. & Petreschi, R. Bijective Linear Time Coding and Decoding for k-Trees. Theory Comput Syst 46, 284–300 (2010). https://doi.org/10.1007/s00224-008-9131-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-008-9131-0