Optimal Canonization of All Substrings of a String

Any word can be decomposed uniquely into lexicographically nonincreasing factors each one of which is a Lyndon word. This paper addresses the relationship between the Lyndon decomposition of a word x and a canonical rotation of x, i.e., a rotation w ofx that is lexicographically smallest among all rotations ofx. The main combinatorial result is a characterization of the Lyndon factor of x with which w must stan. As an application, faster on-line algorithms for finding the canonical rotation(s) of x are developed by nontrivial extension of known Lyndon factorization strategies. Unlike their predecessors, the new algorithms lend themselves to incremental variants that compute, in linear time, the canonical rotations of all prefixes of x. The fastest such variant represents the main algorithmic contribution of the paper. It performs within the same 3lxl character-comparisons bound as that of the fastest previous on-line algorithms for the canonization of a single string. This leads to the canonization of all substrings of a string in optimal quadratic time, within less than 31x12 character comparisons and using linear auxiliary space.