Inverse Lyndon words and Inverse Lyndon factorizations of words

Motivated by applications to string processing, we introduce variants of the Lyndon factorization called inverse Lyndon factorizations. Their factors, named inverse Lyndon words, are in a class that strictly contains anti-Lyndon words, that is Lyndon words with respect to the inverse lexicographic order. We prove that any nonempty word w admits a canonical inverse Lyndon factorization, named ICFL(w), that maintains the main properties of the Lyndon factorization of w: it can be computed in linear time, it is uniquely determined, it preserves a compatibility property for sorting suffixes. In particular, the compatibility property of ICFL(w) is a consequence of another result: any factor in ICFL(w) is a concatenation of consecutive factors of the Lyndon factorization of w with respect to the inverse lexicographic order. As for the applications, experimental results on biological datasets shown that ICFL(w) combined with the Lyndon factorization is intermediate between the Lyndon factorization and the LZ factorization with respect to the size of the factors. Moreover ICFL(w) allows us to handle too long or too short factors in the Lyndon factorization.