Serge Dulucq

Analysis of tree edit distance algorithms

In this article, we study the behaviour of dynamic programming methods for the tree edit distance problem, such as [4] and [2]. We show that those two algorithms may be described in a more general framework of cover strategies. This analysis allows us to define a new tree edit distance algorithm, that is optimal for cover strategies.

RNA Secondary structure comparison: exact analysis of the Zhang--Shasha tree edit algorithm

We are interested in RNA secondary structure comparison, using an approach which consists to represent these structures by labeled ordered trees. Following the problem considered, this tree representation can be rough (considering only the structural patterns), or refined until an exact coding of the structure is obtained. After some preliminary definitions and the description of the Zhang?Shasha (SIAM J. Comput. 18 (6) (1989) 1245) tree edit algorithm, which is on the one hand the reference when dealing with ordered labeled trees comparison, and on the other hand the starting point of our work, this article will present an exact analysis of its complexity. The purpose of this work is also to lead us to a better comprehension of the parameters of this algorithm, in order to be able to modify it more easily without changing its time complexity to take into account biological constraints that occur when comparing RNA secondary structures.

Permutations with forbidden subsequences and nonseparable planar maps

The goal of the present work is to connect combinatorially a family of maps to a family of permutations with forbidden subsequences. We obtain a generating tree of nonseparable planar rooted maps and show that this tree is the generating tree of a family of permutations. The distribution of these permutations is then obtained. Finally, the different steps leading to the combinatorial proof of Wests conjecture are listed.

Shuffle of parenthesis systems and Baxter permutations

Abstract A formula for the number alternating Baxter permutations is given. The proof of this formula is given by constructing bijection between permutations, trees, and words. This gives also a combinatorial proof of a formula appearing in the enumerative theory of planar maps.

A combinatorial proof of J. West's conjecture

Abstract We give what we believe to be the first combinatorial proof of J. Wests conjecture that is to say we show that two-stack sortable permutations are in correspondence with rooted nonseparable planar maps. The generating tree of a set of permutations with forbidden subsequences and its characterization by a rewriting system are used to obtain this correspondence. Other results on the number of two-stack sortable permutations with given classical parameters are presented.

Alignments of RNA Structures

We describe a theoretical unifying framework to express the comparison of RNA structures, which we call alignment hierarchy. This framework relies on the definition of common supersequences for arc-annotated sequences and encompasses the main existing models for RNA structure comparison based on trees and arc-annotated sequences with a variety of edit operations. It also gives rise to edit models that have not been studied yet. We provide a thorough analysis of the alignment hierarchy, including a new polynomial-time algorithm and an NP-completeness proof. The polynomial-time algorithm involves biologically relevant edit operations such as pairing or unpairing nucleotides. It has been implemented in a software, called gardenia, which is available at the Web server http://bioinfo.lifl.fr/RNA/gardenia.

Stack words, standard tableaux and Baxter permutations

Abstract The origin of this work is based on the enumeration of stack sortable permutations [11, 17, 18]. The problem, particularly in case of two stacks, exhibits classical objects in combinatorics such as permutations with forbidden subsequences, nonseparable planar maps [4, 5], and also standard Young tableaux if we are interested in the movements of stacks. So, we show that the number of 3 × n rectangular standard Young tableaux which avoid two consecutive integers on second row is c2n (where cn = (2n)!/(n + 1)!n!) and there is a one-to-one correspondence between the same tableaux which avoid two consecutive integers on the same row and Baxter permutations which are enumerated by . We also give formulas enumerating these objects according to various parameters.

Decomposition algorithms for the tree edit distance problem

Abstract We study the behavior of dynamic programming methods for the tree edit distance problem, such as [P. Klein, Computing the edit-distance between unrooted ordered trees, in: Proceedings of 6th European Symposium on Algorithms, 1998, p. 91–102; K. Zhang, D. Shasha, SIAM J. Comput. 18 (6) (1989) 1245–1262]. We show that those two algorithms may be described as decomposition strategies. We introduce the general framework of cover strategies, and we provide an exact characterization of the complexity of cover strategies. This analysis allows us to define a new tree edit distance algorithm, that is optimal for cover strategies.

Combinatorial properties of smooth infinite words

We describe some combinatorial properties of an intriguing class of infinite words, called smooth, connected with the Kolakoski one, K, defined as the fixed point of the run-length encoding Δ. It is based on a bijection on the free monoid over Σ = {1, 2}, that shows some surprising mixing properties. All words contain the same finite number of square factors, and consequently they are cube-free. This suggests that they have the same complexity as confirmed by extensive computations. We further investigate the occurrences of palindromic subwords. Finally, we show that there exist smooth words obtained as fixed points of substitutions (realized by transducers) as in the case of K.

Local similarity between quotiented ordered trees

In this paper we propose a dynamic programming algorithm to evaluate local similarity between ordered quotiented trees using a constrained edit scoring scheme. A quotiented tree is a tree defined with an additional equivalent relation on vertices and such that the quotient graph is also a tree. The core of the method relies on two adaptations of an algorithm proposed by Zhang et al. [K. Zhang, D. Shasha, Simple fast algorithms for the editing distance between trees and related problems (1989) 1245-1262] for comparing ordered rooted trees. After some preliminary definitions and the description of this tree edit algorithm, we propose extensions to globally and locally compare two quotiented trees. This last method allows to find the region in each tree with the highest similarity. Algorithms are currently being used in genomic analysis to evaluate variability between RNA secondary structures.