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

We improve a recent result [Badr: Hyper-minimization inO(n). Int. J. Found. Comput. Sci. 20, 2009] for hyper-minimized nite automata. Namely, we present an O(n log n) algorithm that computes for a given deterministic nite automaton (dfa) an almostequivalent dfa that is as small as possible such an automaton is called hyper-minimal. Here two nite automata are almost-equivalent if and only if the symmetric di erence of their languages is nite. In other words, two almost-equivalent automata disagree on acceptance on nitely many inputs. In this way, we solve an open problem stated in [Badr, Geffert, Shipman: Hyper-minimizing minimized deterministic nite state automata. RAIRO Theor. Inf. Appl. 43, 2009] and by Badr. Moreover, we show that minimization linearly reduces to hyper-minimization, which shows that the time-bound O(n log n) is optimal for hyper-minimization. Independently, similar results were obtained in [Gawrychowski, Je»: Hyper-minimisation made e cient. Proc. MFCS, LNCS 5734, 2009].