An nlogn Algorithm for Hyper-minimizing States in a (Minimized) Deterministic Automaton
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.
KeywordsRegular Language Finite Automaton Input Symbol State Automaton Lossless Compression
Unable to display preview. Download preview PDF.
- 12.Paun, A., Paun, M., Rodríguez-Patón, A.: On the Hopcroft minimization technique for DFA and DCFA. Theoret. Comput. Sci. (to appear, 2009), http://dx.doi.org/10.1016/j.tcs.2009.02.034
- 16.Kosaraju, S.R.: Strong-connectivity algorithm (unpublished manuscript 1978)Google Scholar