Jean-Pierre Duval

Relationship between the period of a finite word and the length of its unbordered segments

A nonempty word is unbordered if and only if it has no proper period. We study in this paper the relationship between the smallest period of a word and the maximum length of its unbordered segments. We established: If the length of a word is greater or equal to four times the maximum length of its unbordered segments, then this maximum length is the smallest period of the word. This gives the answer announced in [4] to the question asked in [6].

Periodes et repetitions des mots du monoide libre

Abstract We study the relationship between the global periodicity of a word and its (local) repetitions, that is to say its successive equal factors. Our main result generalizes the theorem of Cesari Vincent according to which the period of a word is the maximum of the minimal repetitions. Our generalization is twofold, it does not take account of any sides to define repetitions (the above result does) and calculates the maximum only on a special set of points. It allows a sharpened version of the solution to a problem settled by Schutzenberger in which the theorem of Cesari Vincent originates.

Linear-time computation of local periods

We present a linear-time algorithm for computing all local periods of a given word. This subsumes (but is substantially more powerful than) the computation of the (global) period of the word and on the other hand, the computation of a critical factorization, implied by the Critical Factorization Theorem.

Efficient validation and construction of border arrays and validation of string matching automata

We present an on-line linear time and space algorithm to check if an integer array f is the border array of at least one string w built on a bounded or unbounded size alphabet Σ . First of all, we show a bijection between the border array of a string w and the skeleton of the DFA recognizing Σ*ω, called a string matching automaton (SMA). Different strings can have the same border array but the originality of the presented method is that the correspondence between a border array and a skeleton of SMA is independent from the underlying strings. This enables to design algorithms for validating and generating border arrays that outperform existing ones. The validating algorithm lowers the delay (maximal number of comparisons on one element of the array) from O(|w|) to 1 + min{|Σ|,1 + log 2 |ω|} compared to existing algorithms. We then give results on the numbers of distinct border arrays depending on the alphabet size. We also present an algorithm that checks if a given directed unlabeled graph G is the skeleton of a SMA on an alphabet of size s in linear time. Along the process the algorithm can build one string w for which G is the SMA skeleton.

Unbordered factors and Lyndon words

A primitive word w is a Lyndon word if w is minimal among all its conjugates with respect to some lexicographic order. A word w is bordered if there is a nonempty word u such that w=uvu for some word v. A right extension of a word w of length n is a word wu where all factors longer than n are bordered. A right extension wu of w is called trivial if there exists a positive integer k such that w^k=uv for some word v. We prove that Lyndon words have only trivial right extensions. Moreover, we give a conjecture which characterizes a property of every word w which has a nontrivial right extension of length 2|w|-2.

Local periods and propagation of periods in a word

Abstract A word of length n over an alphabet A is a sequence a 1 … a n of letters of A . It is convenient to consider a “long enough” word over A as an infinite right word, that is an infinite sequence a 1 … a i … of elements of A . An integer λ is a period of the word in the interval [ j … k ] if we have a i = a i + λ for those indices i and i + λ in the considered interval. The period of a word designates its smallest period over its whole length. A point p of a word is the cut ( a 1 … a p , a p +1 …). A non-negative integer λ is a local period at point p iff λ is a period in the interval [ p − λ + 1 … p + λ ]. According to the critical points theorem [1,2], the period of a “long enough (or not)” word is the maximum of the minimal local periods taken in each point of this word. M.P. Schutzenberger, who was at the origin of the research work on the relations between local periods and periods of a word obtained by concatenation of periodical words, and our ability to characterize its period from the observation of the local period at the concatenation points only. This is the formulation of the unpublished answer we offered him that we suggest here.

Parsing with a finite dictionary

We address the following issue: given a word w ∈ A* and a set of n nonempty words X, how does one determine efficiently whether w ∈ X* or not? We discuss several methods including an O(r × |w| + |X|) algorithm for this problem where r ≤ n is the length of a longest suffix chain of X and |X| is the sum of the lengths of words in X. We also consider the more general problem of providing all the decompositions of w in words of X.

Mots de Lyndon et périodicité

— We show in this note how,byordering the freemonoïd X*, gêner atedby an alphabet X, we can establish a caractérisation of periodicity of the worâs on this alphabet.

Linear computation of unbordered conjugate on unordered alphabet

We present an algorithm that, given a word w of length n on an unordered alphabet, computes one of its unbordered conjugates. If such a conjugate does not exist, the algorithm computes one of its conjugates that is a power of an unbordered word. The time complexity of the algorithm is O(n): the number of comparisons between letters of w is bounded by 4n.

Linear-Time Computation of Local Periods

We present a linear-time algorithm for computing all local periods of a given word. This subsumes (but is substantially more powerful than) the computation of the (global) period of the word and on the other hand, the computation of a critical factorization, implied by the Critical Factorization Theorem.