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In this paper we investigate similarity measures based on minimal absent words, introduced by Chairungsee and Crochemore in [1]. They make use of a length-weighted index on a sample set corresponding to the symmetric difference M(x)ΔM(y) of the minimal absent words M(x) and M(y) of two sequences x and y, respectively. We first propose a variant of this measure by choosing as a sample set a proper subset (x, y) of M(x)ΔM(y), which appears to be more appropriate for distinguishing x and y. From the algebraic point of view, we prove that (x, y) is the base of the ideal generated by M(x)ΔM(y). We then remark that such measures are able to recognize whether the sequences x and y share a common structure, but they are not able to detect the difference on the number of occurrences of such a structure in the two sequences. In order to take into account such a multiplicity, we introduce the notion of multifactor, and define a new measure that uses both absent words and multifactors. Surprisingly, we prove that this similarity measure coincides with a distance on sequences introduced by Ehrenfeucht and Haussler in [2], in the context of block-moves strategies. In this way, our result creates a non trivial bridge between similarity measures based on absent words and those based on the block-moves approach.

The main goal of this paper is to highlight the relationship between the suffix array of a text and its Lyndon factorization. It is proved in [15] that one can obtain the Lyndon factorization of a text from its suffix array. Conversely, here we show a new method for constructing the suffix array of a text that takes advantage of its Lyndon factorization. The surprising consequence of our results is that, in order to construct the suffix array, the local suffixes inside each Lyndon factor can be separately processed, allowing different implementative scenarios, such as online, external and internal memory, or parallel implementations. Based on our results, the algorithm that we propose sorts the suffixes by starting from the leftmost Lyndon factors, even if the whole text or the complete Lyndon factorization are not yet available.

We study balance properties of circular words over alphabets of size greater than two. We give some new characterizations of balanced words connected to the Kawasaki-Ising model and to the notion of derivative of a word. Moreover we consider two different generalizations of the notion of balance, and we find some relations between them. Some of our results can be generalized to non periodic infinite words as well.

Sequence comparison has become a very essential tool in modern molecular biology. In fact, in biomolecular sequences high similarity usually implies significant functional or structural similarity. Traditional approaches use techniques that are based on sequence alignment able to measure character level differences. However, the recent developments of whole genome sequencing technology give rise to need of similarity measures able to capture the rearrangements involving large segments contained in the sequences. This paper is devoted to illustrate different methods recently introduced for the alignment-free comparison of biological sequences. Goal of the paper is both to highlight the peculiarities of each of such approaches by focusing on its advantages and disadvantages and to find the common features of all these different methods.

The objective of this paper is to study, by new formal methods, the notion of tree code introduced by Nivat in (Tree Automata and Languages, Elsevier, Amsterdam, 1992, pp. 1–19). In particular, we consider the notion of stability for sets of trees closed under concatenation. This allows us to give a characterization of tree codes which is very close to the algebraic characterization of word codes in terms of free monoids. We further define the stable hull of a set of trees and derive a defect theorem for trees, which generalizes the analogous result for words. As a consequence, we obtain some properties of tree codes having two elements. Moreover, we propose a new algorithm to test whether a finite set of trees is a tree code. The running time of the algorithm is polynomial in the size of the input. We also introduce the notion of tree equation as complementary view to tree codes. The main problem emerging in this approach is to decide the satisfiability of tree equations: it is a special case of second-order unification, and it remains still open.

In this paper we are interested in the study of the combinatorial aspects connecting three important constructions in the field of string algorithms: the suffix array, the Burrows-Wheeler transform (BWT) and the extended Burrows-Wheeler transform (EBWT). Such constructions involve the notions of suffixes and conjugates of words and are based on two different order relations, denoted by $\plex$ and $\pom$, that, even if strictly connected, are quite different from the computational point of view. In this study an important role is played by Lyndon words. In particular, we improve the upper bound on the number of symbol comparisons needed to establish the $\pom$ order between two primitive words by using a preliminary knowledge of the $\plex$ order of the corresponding Lyndon conjugates. Moreover, we propose an algorithm that efficiently sorts, according to the $\pom$ order, the list of conjugates of a multiset of Lyndon words. Finally, we show that the Lyndon factorization of a word helps the construction of its suffix array, allowing a reduction of the number of symbol comparisons needed to lexicographically sort the suffixes of the word.

In this paper we generalize an encoding method due to Girod (cf. [6]) using prefix codes, that allows a bidirectional decoding of the encoded messages. In particular we generalize it to any finite alphabet A, to any operation defined on A, to any code with finite deciphering delay and to any key x ∈ A+ , on a length depending on the deciphering delay. We moreover define, as in [4], a deterministic transducer for such generalized method. We prove that, fixed a code X ∈ A* with finite deciphering delay and a key x ∈ A *, the transducers associated to different operations are isomorphic as unlabelled graphs. We also prove that, for a fixed code X with finite deciphering delay, transducers associated to different keys have an isomorphic non trivial strongly connected component.

In this paper we introduce a new alignment-free method for comparing sequences which is combinatorial by nature and does not use any compressor nor any information-theoretic notion. Such a method is based on an extension of the Burrows-Wheeler Transform, a transformation widely used in the context of Data Compression. The new extended transformation takes as input a multiset of sequences and produces as output a string obtained by a suitable rearrangement of the characters of all the input sequences. By using such a transformation we give a general method for comparing sequences that takes into account how much the characters coming from the different input sequences are mixed in the output string. Such a method is tested on a real data set for the whole mitochondrial genome phylogeny problem. However, the goal of this paper is to introduce a new and general methodology for automatic categorization of sequences.

Summary form only given. We introduce an extension of the Burrows-Wheeler transform to a multiset of primitive words. Primitiveness is not actually a restrictive hypothesis, since in practice almost all the processed texts are primitive (or become primitive by adding an end-of-string symbol). We prove that such a transformation as the BWT is reversible. We show how to use the transformation as a preprocessing for the simultaneous compression of different texts.

We generalize different notions of a rank of a set of words to sets of trees. We prove that almost all of those ranks can be used to formulate a defect theorem. However, as we show, the prefix rank forms an exception.

We consider the problem of solving equations over k-ary trees. Here an equation is a pair of labeled α-ary trees, where α is a function associating an arity to each label. A solution to an equation is a morphism from α-ary trees to k-ary trees that maps the left and right hand side of the equation to the same k-ary tree.

We introduce the notion of equation on trees, generalizing the corresponding notion for words, and we develop the first steps of a theory of tree equations. The main result of the paper states that, if a pair of trees is the solution of a tree equation with two indeterminates, then the two trees are both powers of the same tree. As an application, we show that a tree can be expressed in a unique way as a power of a primitive tree. This extends a basic result of combinatorics on words to trees. Some open problems are finally proposed.