Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T22:17:12.376Z Has data issue: false hasContentIssue false
  • English
  • Français

Hopcroft's algorithmand tree-like automata

Published online by Cambridge University Press:  15 March 2011

G. Castiglione ,
A. Restivo  and
M. Sciortino
G. Castiglione
Affiliation:
Università di Palermo, Dipartimento di Matematica e Applicazioni, via Archirafi, 34, 90123 Palermo, Italy; giusi@math.unipa.it
A. Restivo
Affiliation:
Università di Palermo, Dipartimento di Matematica e Applicazioni, via Archirafi, 34, 90123 Palermo, Italy; giusi@math.unipa.it
M. Sciortino
Affiliation:
Università di Palermo, Dipartimento di Matematica e Applicazioni, via Archirafi, 34, 90123 Palermo, Italy; giusi@math.unipa.it
Get access

Abstract

Minimizing a deterministic finite automata (DFA) is a very important problem in theory of automata and formal languages.Hopcroft's algorithm represents the fastest known solution to the such a problem. In this paper we analyze the behavior of this algorithm on a family binary automata, called tree-like automata, associated to binary labeled trees constructed by words. We prove that all the executions of the algorithm on tree-like automata associated to trees, constructed by standard words, have running time with the same asymptotic growth rate. In particular, we provide a lower and upper bound for the running time of the algorithm expressed in terms of combinatorial properties of the trees. We consider also tree-like automata associated to trees constructed by de Brujin words,and we prove that a queue implementation of the waiting set gives a Θ(n log n) execution while a stack implementation produces a linear execution. Such a result confirms the conjecture given in [A. Paun, M. Paun and A. Rodríguez-Patón. Theoret. Comput. Sci.410 (2009) 2424–2430.] formulated for a family of unary automata and, in addition, gives a positive answer also for the binary case.

Keywords

Automata minimizationHopcroft's algorithmword trees
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 45 , Issue 1: ICTCS 09 , January 2011 , pp. 59 - 75
Copyright
© EDP Sciences, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berstel, J. and Carton, O., On the complexity of Hopcroft's state minimization algorithm, in CIAA. Lecture Notes in Computer Science 3317 (2004) 35–44. CrossRef
Berstel, J., Boasson, L. and Carton, O., Continuant polynomials and worst-case behavior of Hopcrofts minimization algorithm. Theoret. Comput. Sci. 410 (2009) 2811–2822. CrossRef
Berstel, J., Boasson, L., Carton, O. and Fagnot, I., Sturmian trees. Theor. Comput. Syst. 46 (2010) 443–478. CrossRef
Borel, J.P. and Reutenauer, C., Christoffel, On classes. RAIRO-Theor. Inf. Appl. 450 (2006) 15–28. CrossRef
Castiglione, G., Restivo, A. and Sciortino, M., Hopcroft's algorithm and cyclic automata, in LATA. Lecture Notes in Computer Science 5196 (2008) 172–183. CrossRef
Castiglione, G., Restivo, A. and Sciortino, M., On extremal cases of hopcroft's algorithm, in CIAA. Lecture Notes in Computer Science 5642 (2009) 14–23. CrossRef
Castiglione, G., Restivo, A. and Sciortino, M., On extremal cases of hopcroft's algorithm. Theoret. Comput. Sci. 411 (2010) 3414–3422 . CrossRef
J.E. Hopcroft, An $n$ log $n$ algorithm for mimimizing the states in a finite automaton, in Theory of machines and computations (Proc. Internat. Sympos. Technion, Haifa, 1971). Academic Press, New York (1971), 189–196.
Knuutila, T., Re-describing an algorithm by Hopcroft. Theoret. Comput. Sci. 250 (2001) 333–363. CrossRef
M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications 90. Cambridge University Press (2002).
E.F. Moore, Gedaken experiments on sequential, in Automata Studies. Annals of Mathematical Studies 34 (1956) 129–153.
Paige, R., Tarjan, R.E. and Bonic, R., A linear time solution to the single function coarsest partition problem. Theoret. Comput. Sci. 40 (1985) 67–84 . CrossRef
Paun, A., Paun, M. and Rodríguez-Patón, A., Hopcroft's minimization technique: Queues or stacks? in CIAA. Lecture Notes in Computer Science 5148 (2008) 78–91. CrossRef
Paun, A., Paun, M. and Rodríguez-Patón, A., On the hopcroft's minimization technique for dfa and dfca. Theoret. Comput. Sci. 410 (2009) 2424–2430. CrossRef
B. Watson, A taxonomy of finite automata minimization algorithms. Technical Report 93/44, Eindhoven University of Technology, Faculty of Mathematics and Computing Science (1994).