On extremal cases of Hopcroft’s algorithm

Abstract

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.

Average Case Analysis of Moore’s State Minimization Algorithm

“…Using appropriate data structures, its worst-case complexity is O(kn log n). It is the best known minimization algorithm and therefore it has been studied intensively: in [3,4] different proofs of its correctness are given, in [5][6][7] the tightness of the complexity upper bound for various families of automata is proved, in [8,9] a precise description of the data structures that are needed to reach the O(kn log n) complexity is given. In [10,11] two O(m log n) solutions for incomplete automata are given, where m denotes the number of defined transitions, using improvements in Hopcroft's strategy together with advanced data structures.…”

“…(a) u = abbaa is the smallest word of length 5, for the lexicographic order, such that 3·u = 3 and 7·u = 8. The set F 5(3,7,3,8) is not empty, since it contains at least {4, 8}. The bold transitions are the ones followed when reading u from p and from q.…”

Epichristoffel Words and Minimization of Moore Automata

“…For more details we refer to [2,8,7,9]. As described in Section 7, such a variant becomes fundamental for a simpler study of the tightness of the Hopcroft's minimization on Moore automata over more than two letters output alphabet.…”

“…[1,6,8,7]) the study of combinatorial properties of Christoffel classes (or circular sturmian words) allowed to face open problems regarding Hopcroft's minimization algorithm. Indeed, as well known, Hopcroft's algorithm presents some degree of freedom that make more difficult the study of the tightness of its time complexity.…”

Hopcroft's algorithm and tree-like automata

“…But, the solution does not seem to extend to larger alphabet. Motivated by this fact, in [6,7] we analyzed the tightness of the algorithm when the alphabet contains more than one letter together with the uniqueness of the refinement process of the set of states. We defined an infinite family of binary automata associated to particular finite labeled binary trees, called standard tree, for which the refinement process during Hopcroft's algorithm is unique even if there could be different executions.…”