# Lyndon Word and Lyndon Factorization

We came across a very interesting paper, but before we get to it, let us define few terms with help from wikipedia.

Lyndon Word

In mathematics, in the areas of combinatorics and computer science, a Lyndon word is a string that is strictly smaller in lexicographic order than all of its rotations.

**Connection to de Bruijn sequences**

If one concatenates together, in lexicographic order, all the Lyndon words that have length dividing a given number n, the result is a de Bruijn sequence, a circular sequence of symbols such that each possible length-n sequence appears exactly once as one of its contiguous subsequences. For example, the concatenation of the binary Lyndon words whose length divides four is

0 0001 0011 01 0111 1

This construction, together with the efficient generation of Lyndon words, provides an efficient method for constructing a particular de Bruijn sequence in linear time and logarithmic space.

A Lyndon word over a totally ordered alphabet A is a word which is lexicographically less than all its rotations. The ChenFoxLyndon theorem states that every string may be formed in a unique way by concatenating a non- increasing sequence of Lyndon words. Hence taking Xl to be the singleton set {l} for each Lyndon word l, with the index set L of Lyndon words ordered lexicographically, we obtain a factorisation of A*. Such a factorisation can be found in linear time.

Now, we can discuss the paper. Giovanna Rosone, one of the authors, is a contributor to BCR algorithm that we covered earlier, when Heng Li brought them to our attention in a comment. Please enjoy the beautiful math of the new paper.

Sorting su?xes of a text via its Lyndon Factorization

The process of sorting the su?xes of a text plays a fundamental role in Text Algorithms. They are used for instance in the constructions of the Burrows Wheeler transform and the su?x array, widely used in several ?elds of Computer Science. For this reason, several recent researches have been devoted to ?nding new strategies to obtain e?ective methods for such a sorting. In this paper we introduce a new methodology in which an important role is played by the Lyndon factorization, so that the local su?xes inside factors detected by this factorization keep their mutual order when extended to the su?xes of the whole word. This property suggests a versatile technique that easily can be adapted to di?erent implementative scenarios.

“The goal of this paper is to introduce a new strategy for the sorting of the su?xes of a word that opens new scenarios of the computation of the SA and the bwt.”

Do not forget to sit with the following relevant papers (and we presume you have all relevant BCR papers already):

This factorization was introduced in  and a linear time algorithm is due to Duval . The intuition that the knowledge of Lyndon factorization of a text can be used for the computation of the su?x array of the text itself has been introduced in . Conversely, a way to ?nd the Lyndon factorization from the su?x array can be found in .

 S. Bonomo, S. Mantaci, A. Restivo, G. Rosone, and M. Sciortino: Su?xes, Conjugates and Lyndon words, in DLT, vol. 7907 of LNCS, Springer, 2013, pp. 131142.

 K. T. Chen, R. H. Fox, and R. C. Lyndon: Free di?erential calculus. IV. The quotient groups of the lower central series. Ann. of Math. (2), 68 1958, pp. 8195.

 J.-P. Duval: Factorizing words over an ordered alphabet. Journal of Algorithms, 4(4) 1983, pp. 363 381.

 . C. Hohlweg and C. Reutenauer: Lyndon words, permutations and trees. Theoretical Computer Science, 307(1) 2003, pp. 173 178.

Tomorrow, we will start again with more interesting math and 10002nd commandment from NIH !

Written by M. //