Suffix Automata and Standard Sturmian Words

Abstract

Blumer et al. showed (cf. [3,2]) that the suffix automaton of a word w must have at least |w| + 1 states and at most 2|w| − 1 states. In this paper we characterize the language L of all binary words w whose minimal suffix automaton $\mathcal{S}(w)$ has exactly |w| + 1 states; they are precisely all prefixes of standard Sturmian words. In particular, we give an explicit construction of suffix automaton of words that are palindromic prefixes of standard words. Moreover, we establish a necessary and sufficient condition on $\mathcal{S}(w)$ which ensures that if w ∈ L and a ∈ {0,1} then wa ∈ L. By using such a condition, we show how to construct the automaton $\mathcal{S}(wa)$ from $\mathcal{S}(w)$. More generally, we provide a simple construction that by starting from an automaton recognizing all suffixes of a word w over a finite alphabet A, allows to obtain an automaton that recognizes the suffixes of wa, a ∈ A.