Chiara Epifanio

On a conjecture on bidimensional words

We prove that, given a double sequence w over the alphabet A (i.e. a mapping from Z2 to A), if there exists a pair (n0, m0) ∈ Z2 such that pw(n0, m0) < 1/100n0m0, then w has a periodicity vector, where pw is the complexity function in rectangles of w.

A Trie-Based Approach for Compacting Automata

We describe a new technique for reducing the number of nodes and symbols in automata based on tries. The technique stems from some results on anti-dictionaries for data compression and does not need to retain the input string, differently from other methods based on compact automata. The net effect is that of obtaining a lighter automaton than the directed acyclic word graph (DAWG) of Blumer et al., as it uses less nodes, still with arcs labeled by single characters.

Languages with mismatches

In this paper we study some combinatorial properties of a class of languages that represent sets of words occurring in a text S up to some errors. More precisely, we consider sets of words that occur in a text S with k mismatches in any window of size r. The study of this class of languages mainly focuses both on a parameter, called repetition index, and on the set of the minimal forbidden words of the language of factors of S with errors. The repetition index of a string S is defined as the smallest integer such that all strings of this length occur at most in a unique position of the text S up to errors. We prove that there is a strong relation between the repetition index of S and the maximal length of the minimal forbidden words of the language of factors of S with errors. Moreover, the repetition index plays an important role in the construction of an indexing data structure. More precisely, given a text S over a fixed alphabet, we build a data structure for approximate string matching having average size O(|S|log k+1|S|) and answering queries in time O(|x|+|occ(x)|) for any word x, where occ is the list of all occurrences of x in S up to errors.

Languages with mismatches and an application to approximate indexing

In this paper we describe a factorial language, denoted by L(S,k,r), that contains all words that occur in a string S up to k mismatches every r symbols. Then we give some combinatorial properties of a parameter, called repetition index and denoted by R(S,k,r), defined as the smallest integer h≥ 1 such that all strings of this length occur at most in a unique position of the text S up to k mismatches every r symbols. We prove that R(S,k,r) is a non-increasing function of r and a non-decreasing function of k and that the equation r=R(S,k,r) admits a unique solution. The repetition index plays an important role in the construction of an indexing data structure based on a trie that represents the set of all factors of L(S,k,r) having length equal to R(S,k,r). For each word x∈ L(S,k,r) this data structure allows us to find the list occ(x) of all occurrences of the word x in a text S up to k mismatches every r symbols in time proportional to |x|+|occ(x)|.

Sturmian graphs and integer representations over numeration systems

In this paper we consider a numeration system, originally due to Ostrowski, based on the continued fraction expansion of a real number @a. We prove that this system has deep connections with the Sturmian graph associated with @a. We provide several properties of the representations of the natural integers in this system. In particular, we prove that the set of lazy representations of the natural integers in this numeration system is regular if and only if the continued fraction expansion of @a is eventually periodic. The main result of the paper is that for any number i the unique path weighted i in the Sturmian graph associated with @a represents the lazy representation of i in the Ostrowski numeration system associated with @a.

Sturmian graphs and a conjecture of moser

In this paper we define Sturmian graphs and we prove that all of them have a “counting” property. We show deep connections between this counting property and two conjectures, by Moser and by Zaremba, on the continued fraction expansion of real numbers. These graphs turn out to be the underlying graphs of CDAWGs of central Sturmian words. We show also that, analogously to the case of Sturmian words, these graphs converge to infinite ones.

Linear-size suffix tries

Suffix trees are highly regarded data structures for text indexing and string algorithms MCreight 76, Weiner 73. For any given string w of length n = | w | , a suffix tree for w takes O ( n ) nodes and links. It is often presented as a compacted version of a suffix trie for w, where the latter is the trie (or digital search tree) built on the suffixes of w. Here the compaction process replaces each maximal chain of unary nodes with a single arc. For this, the suffix tree requires that the labels of its arcs are substrings encoded as pointers to w (or equivalent information). On the contrary, the arcs of the suffix trie are labeled by single symbols but there can be ź ( n 2 ) nodes and links for suffix tries in the worst case because of their unary nodes. It is an interesting question if the suffix trie can be stored using O ( n ) nodes. We present the linear-size suffix trie, which guarantees O ( n ) nodes. We use a new technique for reducing the number of unary nodes to O ( n ) , that stems from some results on antidictionaries. For instance, by using the linear-size suffix trie, we are able to check whether a pattern p of length m = | p | occurs in w in O ( m log ź | Σ | ) time and we can find the longest common substring of two strings w 1 and w 2 in O ( ( | w 1 | + | w 2 | ) log ź | Σ | ) time for an alphabet Σ.

On the suffix automaton with mismatches

In this paper we focus on the construction of the minimal deterministic finite automaton Sk that recognizes the set of suffixes of a word w up to k errors. We present an algorithm that makes use of Sk in order to accept in an efficient way the language of all suffixes of w up to k errors in every window of size r, where r is the value of the repetition index of w. Moreover, we give some experimental results on some wellknown words, like prefixes of Fibonacci and Thue-Morse words, and we make a conjecture on the size of the suffix automaton with mismatches.

A multidimensional critical factorization theorem

The Critical Factorization Theorem is one of the principal results in combinatorics on words. It relates local periodicities of a word to its global periodicity. In this paper we give a multidimensional extension of it. More precisely, we give a new proof of the Critical Factorization Theorem, but in a weak form, where the weakness is due to the fact that we loose the tightness of the local repetition order. In exchange, we gain the possibility of extending our proof to the multidimensional case. Indeed, this new proof makes use of the Theorem of Fine and Wilf, that has several classical generalizations to the multidimensional case.

On lazy representations and sturmian graphs

In this paper we establish a strong relationship between the set of lazy representations and the set of paths in a Sturmian graph associated with a real number α. We prove that for any non-negative integer i the unique path weighted i in the Sturmian graph associated with α represents the lazy representation of i in the Ostrowski numeration system associated with α. Moreover, we provide several properties of the representations of the natural integers in this numeration system.