On Extremal Cases of the Hopcroft's Algorithm

In this paper we consider the problem of minimization of deterministic finite automata (DFA) with reference to Hopcroft\u2019s algorithm. Hopcroft\u2019s algorithm has several degrees of freedom, so there can exist different executions that can lead to different sequences of refinements of the set of the states up to the final partition. We find an infinite family of binary automata for which such a process is unique, whatever strategy is chosen. Some recent papers (cf. Berstel and Carton (2004) [3], Castiglione et al. (2008) [6] and Berstel et al. (2009) [1]) have been devoted to find families of automata for which Hopcroft\u2019s algorithm has its worst execution time. They are unary automata associated with circular words. However, automata minimization can be achieved also in linear time when the alphabet has only one letter (cf. Paige et al. (1985) [14]), but such a method does not seem to extend to larger alphabet. So, in this paper we face the tightness of Hopcroft\u2019s algorithm when the alphabet contains more than one letter. In particular we define an infinite family of binary automata representing the worst case of Hopcroft\u2019s algorithm, for each execution. They are automata associated with particular trees and we deepen the connection between the refinement process of Hopcroft\u2019s algorithm and the combinatorial properties of such trees

Standard Sturmian words and automata minimization algorithms

The study of some close connections between the combinatorial properties of words and the performance of the automata minimization process constitutes the main focus of this paper. These relationships have been, in fact, the basis of the study of the tightness and the extremal cases of Hopcroft's algorithm, that is, up to now, the most efficient minimization method for deterministic finite state automata. Recently, increasing attention has been paid to another minimization method that, unlike the approach proposed by Hopcroft, is not based on refinement of the set of states of the automaton, but on automata operations such as determinization and reverse, and is also applicable to non-deterministic finite automata. However, even for deterministic automata, it was proved that the method incurs, almost surely, in an explosion of the number of the states. Very recently, some polynomial variants of Brzozowski's method have been introduced. In this paper, by using some combinatorial properties of words, we analyze the performance of one of such algorithms when applied to a particular infinite family of automata, called standard word automata, constructed by using standard sturmian words. In particular, Θ( nlog n) is the worst case time complexity when the algorithm is applied to the infinite family of word automata associated to Fibonacci words representing also the worst case of Hopcroft's minimization algorithm