An nlogn Algorithm for Hyper-minimizing States in a (Minimized) Deterministic Automaton

  • Markus Holzer
  • Andreas Maletti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5642)


We improve a recent result [A. Badr: Hyper-Minimization in O(n 2). In Proc. CIAA, LNCS 5148, 2008] for hyper-minimized finite automata. Namely, we present an O(nlogn) algorithm that computes for a given finite deterministic automaton (dfa) an almost equivalent dfa that is as small as possible—such an automaton is called hyper-minimal. Here two finite automata are almost equivalent if and only if the symmetric difference of their languages is finite. In other words, two almost-equivalent automata disagree on acceptance on finitely many inputs. In this way, we solve an open problem stated in [A. Badr, V. Geffert, I. Shipman: Hyper-minimizing minimized deterministic finite state automata. RAIRO Theor. Inf. Appl. 43(1), 2009] and by Badr. Moreover, we show that minimization linearly reduces to hyper-minimization, which shows that the time-bound O(n logn) is optimal for hyper-minimization.


Regular Language Finite Automaton Input Symbol State Automaton Lossless Compression 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Markus Holzer
    • 1
  • Andreas Maletti
    • 2
  1. 1.Institut für InformatikUniversität GiessenGiessenGermany
  2. 2.Departament de Filologies RomàniquesUniversitat Rovira i VirgiliTarragonaSpain

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