Complexity of testing morphic primitivity

The word u = abaaba satisfies f(u) = u where f maps b to aba and cancels a. Such words, which are fixed points of a nontrivial morphism, are called morphically imprimitive. On the other hand, the word u′ = abba can be easily verified to be morphically primitive, which means that the only morphism satisfying f(u′) = u′ defined on {a, b}∗ is the identity. Fixed points of word morphisms and morphically (im)primitive words are studied in [2, 3, 6, 5]. In [4], the first polynomial algorithm is presented (called MorphicFactorization) that decides whether a given word w is morphically primitive. Moreover, given the input word w, it finds a corresponding morphism satisfying f(w) = w with minimal number of letters mapped to a nonempty word (that is, not cancelled). The complexity of MorphicFactorization is estimated as O(m+ log n) · n in [4]. Here we provide a more detailed analysis of the algorithm and improve the estimate to O(|E| · n), where E is the set of those letters x for which f(x) is nonempty.