## On Extremal Cases of the Hopcroft's Algorithm

**Authors:**Castiglione, G.; Restivo, A.; Sciortino, M.**Publication year:**2010**Type:**Articolo in rivista (Articolo in rivista)**Key words:**Deterministic finite state automata; Hopcroftâ€™s minimization algorithm; Standard trees; Word trees**OA Link:**http://hdl.handle.net/10447/59192

### Abstract

In this paper we consider the problem of minimization of deterministic finite automata (DFA) with reference to Hopcroftâ€™s algorithm. Hopcroftâ€™s 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 Hopcroftâ€™s 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 Hopcroftâ€™s algorithm when the alphabet contains more than one letter. In particular we define an infinite family of binary automata representing the worst case of Hopcroftâ€™s algorithm, for each execution. They are automata associated with particular trees and we deepen the connection between the refinement process of Hopcroftâ€™s algorithm and the combinatorial properties of such trees.