arXiv subjects

Gabriele Fici

Publications and source records attributed to Gabriele Fici.

59 records · Page 4Linked to original sources

arXiv2012

A Characterization of Bispecial Sturmian Words

A finite Sturmian word w over the alphabet {a,b} is left special (resp. right special) if aw and bw (resp. wa and wb) are both Sturmian words. A bispecial Sturmian word is a Sturmian word that is both left and right special. We show as a main result that bispecial Sturmian words are exactly the maximal internal factors of Christoffel words, that are words coding the digital approximations of segments in the Euclidean plane. This result is an extension of the known relation between central words and primitive Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the set of Sturmian words.

cs.FL↗

arXiv2012

Automata and Differentiable Words

We exhibit the construction of a deterministic automaton that, given k > 0, recognizes the (regular) language of k-differentiable words. Our approach follows a scheme of Crochemore et al. based on minimal forbidden words. We extend this construction to the case of C\infinity-words, i.e., words differentiable arbitrary many times. We thus obtain an infinite automaton for representing the set of C\infinity-words. We derive a classification of C\infinity-words induced by the structure of the automaton. Then, we introduce a new framework for dealing with \infinity-words, based on a three letter alphabet. This allows us to define a compacted version of the automaton, that we use to prove that every C\infinity-word admits a repetition in C\infinity whose length is polynomially bounded.

cs.DM↗

arXiv2011

A Classification of Trapezoidal Words

Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n>=0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A trapezoidal word is closed if its longest repeated prefix has exactly two occurrences in the word, the second one being a suffix of the word. Otherwise it is open. We show that open trapezoidal words are all primitive and that closed trapezoidal words are all Sturmian. We then show that trapezoidal palindromes are closed (and therefore Sturmian). This allows us to characterize the special factors of Sturmian palindromes. We end with several open problems.

cs.FL↗

arXiv2011

Algorithms for Jumbled Pattern Matching in Strings

The Parikh vector p(s) of a string s is defined as the vector of multiplicities of the characters. Parikh vector q occurs in s if s has a substring t with p(t)=q. We present two novel algorithms for searching for a query q in a text s. One solves the decision problem over a binary text in constant time, using a linear size index of the text. The second algorithm, for a general finite alphabet, finds all occurrences of a given Parikh vector q and has sub-linear expected time complexity; we present two variants, which both use a linear size index of the text.

cs.DS↗

arXiv2010

On the Minimal Uncompletable Word Problem

Let S be a finite set of words over an alphabet Sigma. The set S is said to be complete if every word w over the alphabet Sigma is a factor of some element of S*, i.e. w belongs to Fact(S*). Otherwise if S is not complete, we are interested in finding bounds on the minimal length of words in Sigma* which are not elements of Fact(S*) in terms of the maximal length of words in S.

cs.FL↗