A Quadratic Upper Bound on the Size of a Synchronizing Word in One-Cluster Automata

  • Marie-Pierre Béal
  • Dominique Perrin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5583)


Černý’s conjecture asserts the existence of a synchronizing word of length at most (n − 1)2 for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p,q, one has p ·ar = q ·as for some integers r,s (for a state p and a word w, we denote by p ·w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n2). This applies in particular to Huffman codes.


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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Marie-Pierre Béal
    • 1
  • Dominique Perrin
    • 1
  1. 1.Université Paris-Est, LIGM CNRSMarne-la-Vallée Cedex 2France

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