Hopcroft's algorithm and tree-like automata

G. Castiglione; A. Restivo; M. Sciortino

RAIRO - Theoretical Informatics and Applications (2011)

  • Volume: 45, Issue: 1, page 59-75
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Castiglione, G., Restivo, A., and Sciortino, M.. "Hopcroft's algorithm and tree-like automata." RAIRO - Theoretical Informatics and Applications 45.1 (2011): 59-75. <http://eudml.org/doc/222033>.

@article{Castiglione2011,
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. },
author = {Castiglione, G., Restivo, A., Sciortino, M.},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Automata minimization; Hopcroft's algorithm; word trees; automata minimization},
language = {eng},
month = {3},
number = {1},
pages = {59-75},
publisher = {EDP Sciences},
title = {Hopcroft's algorithm and tree-like automata},
url = {http://eudml.org/doc/222033},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Castiglione, G.
AU - Restivo, A.
AU - Sciortino, M.
TI - Hopcroft's algorithm and tree-like automata
JO - RAIRO - Theoretical Informatics and Applications
DA - 2011/3//
PB - EDP Sciences
VL - 45
IS - 1
SP - 59
EP - 75
AB - 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.
LA - eng
KW - Automata minimization; Hopcroft's algorithm; word trees; automata minimization
UR - http://eudml.org/doc/222033
ER -

References

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  7. G. Castiglione, A. Restivo and M. Sciortino, On extremal cases of hopcroft's algorithm. Theoret. Comput. Sci.411 (2010) 3414–3422 .  Zbl1214.68193
  8. 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.  
  9. T. Knuutila, Re-describing an algorithm by Hopcroft. Theoret. Comput. Sci.250 (2001) 333–363.  Zbl0952.68077
  10. M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications90. Cambridge University Press (2002).  Zbl1001.68093
  11. E.F. Moore, Gedaken experiments on sequential, in Automata Studies. Annals of Mathematical Studies 34 (1956) 129–153.  
  12. R. Paige, R.E. Tarjan and R. Bonic, A linear time solution to the single function coarsest partition problem. Theoret. Comput. Sci.40 (1985) 67–84 .  Zbl0574.68060
  13. A. Paun, M. Paun and A. Rodríguez-Patón, Hopcroft's minimization technique: Queues or stacks? in CIAA. Lecture Notes in Computer Science5148 (2008) 78–91.  
  14. A. Paun, M. Paun and A. Rodríguez-Patón, On the hopcroft's minimization technique for dfa and dfca. Theoret. Comput. Sci.410 (2009) 2424–2430.  Zbl1168.68028
  15. B. Watson, A taxonomy of finite automata minimization algorithms. Technical Report 93/44, Eindhoven University of Technology, Faculty of Mathematics and Computing Science (1994).  

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