On the size of minimal automata for approximate string matching

A natural way to solve the problem of string matching with k mismatches, is to construct a nite automaton recognizing the language L P;k of all strings being at a distance k of the searched pattern P, and to use it for a linear search in the text. The problem of this approach is a high space complexity. In this paper, we show that, even if we consider the minimal DFA recognizing L P;k , the memory space required remains large, and the number of states C of the minimal automaton increases quickly with the size m of P. For a pattern composed of the repetition of one character, the exact number of states is C = ? m+1 k+1. For a pattern composed of characters that are all diierent, an accurate lower bound on C is P k i=1 i b i , where, for all i, b i is the (i + 1) th Catalan number and i is a positive integer depending on m and k. For a random pattern P, a lower bound is obtained by considering the longest preex of P consisting, either in the repetition of a single character, or in characters that are all diierent.