Languages with mismatches

Abstract

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 | + | o c c ( x ) | ) for any word x , where occ is the list of all occurrences of x in S up to errors.