An O( nlogn) implementation of the standard method for minimizing n-state finite automata

Abstract

More than 20 years ago, Hopcroft (1971) has given an algorithm for minimizing an n-state finite automaton in O( kn log n) time where k is the size of the alphabet. This contrasts to the usual O( kn 2) algorithms presented in text books. Since Hopcroft's algorithm changes the standard method, a nontrivial correctness proof for its method is needed. In lectures given at university, mostly the standard method and its straightforward O( Kn 2) implementation is presented. We show that a slight modification of the O( kn 2) implementation combined with the use of a simple data structure composed of chained lists and four arrays of pointers (essentially the same as Hopcroft's data structure) leads to an O( kn log n) implementation of the standard method. Thus, it is possible to present in lectures, with a little additional amount of time, an O( kn log n) time algorithm for minimizing n-state finite automata.