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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.

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.