Minimisation of acyclic deterministic automata in linear time
Theoretical Computer Science - Selected papers of the Combinatorial Pattern Matching School
Re-describing an algorithm by Hopcroft
Theoretical Computer Science
Repetitions in Sturmian strings
Theoretical Computer Science
Automata, Languages, and Machines
Automata, Languages, and Machines
Concrete Math
Theory of Codes
Characterisations of balanced words via orderings
Theoretical Computer Science
On an involution of Christoffel words and Sturmian morphisms
European Journal of Combinatorics
Incremental construction of minimal acyclic finite state automata and transducers
FSMNLP '09 Proceedings of the International Workshop on Finite State Methods in Natural Language Processing
On the complexity of hopcroft’s state minimization algorithm
CIAA'04 Proceedings of the 9th international conference on Implementation and Application of Automata
Continuant polynomials and worst-case behavior of Hopcroft's minimization algorithm
Theoretical Computer Science
On Extremal Cases of Hopcroft's Algorithm
CIAA '09 Proceedings of the 14th International Conference on Implementation and Application of Automata
On extremal cases of Hopcroft's algorithm
Theoretical Computer Science
A challenging family of automata for classical minimization algorithms
CIAA'10 Proceedings of the 15th international conference on Implementation and application of automata
Nondeterministic Moore automata and Brzozowski's algorithm
CIAA'11 Proceedings of the 16th international conference on Implementation and application of automata
A graph theoretic approach to automata minimality
Theoretical Computer Science
Characteristic Sturmian words are extremal for the Critical Factorization Theorem
Theoretical Computer Science
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In order to analyze some extremal cases of Hopcroft's algorithm, we investigate the relationships between the combinatorial properties of a circular sturmian word (x) and the run of the algorithm on the cyclic automaton A"x associated to (x). The combinatorial properties of words taken into account make use of sturmian morphisms and give rise to the notion of reduction tree of a circular sturmian word. We prove that the shape of this tree uniquely characterizes the word itself. The properties of the run of Hopcroft's algorithm are expressed in terms of the derivation tree of the automaton, which is a tree that represents the refinement process that, in the execution of Hopcroft's algorithm, leads to the coarsest congruence of the automaton. We prove that the shape of the reduction tree of a circular sturmian word (x) coincides with that of the derivation tree T(A"x) of the automaton A"x. From this we derive a recursive formula to compute the running time of Hopcroft's algorithm on the automaton A"x, expressed in terms of parameters of the reduction tree of (x). As a special application, we obtain the time complexity @Q(nlogn) of the algorithm in the case of automata associated to Fibonacci words.