a b s t r a c tIn this paper we consider the problem of minimization of deterministic finite automata (DFA) with reference to Hopcroft's algorithm. Hopcroft'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 Hopcroft'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 Hopcroft'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 Hopcroft's algorithm, for each execution. They are automata associated with particular trees and we deepen the connection between the refinement process of Hopcroft's algorithm and the combinatorial properties of such trees.
In this paper, we consider the class of L-convex polyominoes, i.e. the convex polyominoes in which any two cells can be connected by a path of cells in the polyomino that switches direction between the vertical and the horizontal at most once.Using the ECO method, we prove that the number f n of L-convex polyominoes with perimeter 2(n + 2) satisfies the rational recurrence relation f n = 4f n−1 − 2f n−2 , with f 0 = 1, f 1 = 2, f 2 = 7. Moreover, we give a combinatorial interpretation of this statement. In the last section, we present some open problems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.