An nlogn algorithm for hyper-minimizing a (minimized) deterministic automaton

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Abstract

We improve a recent result [A. Badr, Hyper-minimization in O(n2), Internat. J. Found. Comput. Sci. 20 (4) (2009) 735–746] for hyper-minimized finite automata. Namely, we present an O(nlogn) algorithm that computes for a given deterministic finite 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) 69–94] and by Badr. Moreover, we show that minimization linearly reduces to hyper-minimization, which shows that the time-bound O(nlogn) is optimal for hyper-minimization. Independently, similar results were obtained in [P. Gawrychowski, A. Jeż, Hyper-minimisation made efficient, in: Proc. 34th Int. Symp. Mathematical Foundations of Computer Science, in: LNCS, vol. 5734, Springer, 2009, pp. 356–368].

Keywords

Deterministic finite automaton
Minimization
Lossy compression

Cited by (0)

This is an extended and revised version of [M. Holzer, A. Maletti, An nlogn algorithm for hyper-minimizing states in a (minimized) deterministic automaton, in: Proc. 14th Int. Conf. Implementation and Application of Automata, in: LNCS, vol. 5642, Springer, 2009, pp. 4–13].