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Edit distance between unlabeled ordered trees

Published online by Cambridge University Press:  08 November 2006

Anne Micheli  and
Dominique Rossin
Anne Micheli
Affiliation:
CNRS, LIAFA, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France; amicheli@liafa.jussieu.fr; rossin@liafa.jussieu.fr
Dominique Rossin
Affiliation:
CNRS, LIAFA, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France; amicheli@liafa.jussieu.fr; rossin@liafa.jussieu.fr
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Abstract

There exists a bijection between one-stack sortable permutations (permutations which avoid the pattern (231)) and rooted plane trees. We define an edit distance between permutations which is consistent with the standard edit distance between trees. This one-to-one correspondence yields a polynomial algorithm for the subpermutation problem for (231) pattern-avoiding permutations. Moreover, we obtain the generating function of the edit distance between ordered unlabeled trees and some special ones. For the general case we show that the mean edit distance between a rooted plane tree and all other rooted plane trees is at least n/ln(n). Some results can be extended to labeled trees considering colored Dyck paths or, equivalently, colored one-stack sortable permutations.

Keywords

Edit distancetrees.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 40 , Issue 4 , October 2006 , pp. 593 - 609
Copyright
© EDP Sciences, 2006

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References

Bose, P., Buss, J.F. and Lubiw, A., Pattern matching for permutations. Inf. Proc. Lett. 65 (1998) 277–283. CrossRef
Bousquet-Mélou, M., Sorted and/or sortable permutations. Disc. Math. 225 (2000) 25–50. CrossRef
N.G. De Bruijn, D.E. Knuth and S.O. Rice, Graph theory and Computing. Academic Press (1972) 15–22.
Deutsch, E., Hildebrand, A.J. and Wilf, H.S., Longest increasing subsequences in pattern-restricted permutations. Elect. J. Combin. 9 (2003) R12.
M. Garofalakis and A. Kumar, Correlating XML data streams using tree-edit distance embeddings, in Proc. PODS'03 (2003).
P.N. Klein, Computing the edit-distance between unrooted ordered trees, in ESA '98 (1998) 91–102.
D.E. Knuth, The Art of Computer Programming: Fundamental Algorithms. Addison-Wesley (1973) 533.
P.A. MacMahon, Combinatorial Analysis 1–2. Cambridge University Press (reprinted by Chelsea in 1960) 1915–1916.
Narayana, T.V., Sur les treillis formés par les partitions d'un entier et leurs applications à la théorie des probabilités. C. R. Acad. Sci. Paris 240 (1955) 1188–1189.
Narayana, T.V., A partial order and its application to probability theory. Sankhyā 21 (1959) 91–98.
A. Reifegerste, On the diagram of 132-avoiding permutations. Technical Report 0208006, Math. CO (2002).
Roblet, E. and Viennot, X.G., Théorie combinatoire des t-fractions et approximants de Padé en deux points. Discrete Math. 153 (1996) 271–288. CrossRef
J. West, Permutations and restricted subsequences and Stack-sortable permutations. Ph.D. thesis, M.I.T., 1990.
Zhang, K. and Shasha, D., Simple fast algorithms for the editing distance between trees and related problems. SIAM J. Comput. 18 (1989) 1245–1262. CrossRef