Abstract
We consider a problem of hyper-minimisation of an automaton [2,3]: given a DFA M we want to compute a smallest automaton N such that the language L(M) ΔL(N) is finite, where Δ denotes the symmetric difference. We improve the previously known \(\mathcal O (|\Sigma|n^2)\) solution by giving an expected \(\mathcal O (|\delta|\log n)\) time algorithm for this problem, where |δ| is the size of the (potentially partial) transition function. We also give a slightly slower deterministic \(\mathcal O(|\delta|\log^2 n)\) version of the algorithm.
Then we introduce a similar problem of k-minimisation: for an automaton M and number k we want to find a smallest automaton N such that L(M) ΔL(N) ⊆ Σ< k, i.e. the languages they recognize differ only on words of length less than k. We characterise such minimal automata and give algorithm with a similar complexity for this problem.
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References
Andersson, A., Thorup, M.: Dynamic ordered sets with exponential search trees. J. ACM 54(3), 13 (2007)
Badr, A.: Hyper-minimization in \(\mathcal{O}(n^2\)). In: Ibarra, O.H., Ravikumar, B. (eds.) CIAA 2008. LNCS, vol. 5148, pp. 223–231. Springer, Heidelberg (2008)
Badr, A., Geffert, V., Shipman, I.: Hyper-minimizing minimized deterministic finite state automata. RAIRO - Theoretical Informatics and Applications 43(1), 69–94 (2009)
Câmpeanu, C., Santean, N., Yu, S.: Minimal cover-automata for finite languages. Theor. Comput. Sci. 267(1-2), 3–16 (2001)
Castiglione, G., Restivo, A., Sciortino, M.: Hopcroft’s algorithm and cyclic automata, pp. 172–183 (2008)
Dietzfelbinger, M., Karlin, A.R., Mehlhorn, K., auf der Heide, F.M., Rohnert, H., Tarjan, R.E.: Dynamic perfect hashing: Upper and lower bounds. SIAM J. Comput. 23(4), 738–761 (1994)
Gries, D.: Describing an algorithm by Hopcroft. Acta Informatica 2, 97–109 (1973)
Holzer, M., Maletti, A.: An n log n algorithm for hyper-minimizing states in a (minimized) deterministic automaton. In: CIAA, pp. 4–13 (2009)
Hopcroft, J.: An n logn algorithm for minimizing states in a finite automaton. Technical Report, CS-190 (1970)
Körner, H.: On minimizing cover automata for finite languages in \(\mathcal{O}(n \log n)\) time. In: Champarnaud, J.-M., Maurel, D. (eds.) CIAA 2002. LNCS, vol. 2608, pp. 117–127. Springer, Heidelberg (2003)
Pagh, R., Rodler, F.F.: Cuckoo hashing. J. Algorithms 51(2), 122–144 (2004)
Valmari, A., Lehtinen, P.: Efficient minimization of DFAs with partial transition. In: Albers, S., Weil, P. (eds.) STACS. Dagstuhl Seminar Proceedings, vol. 08001, pp. 645–656. Internationales Begegnungs- und Forschungszentrum fuer Informatik (IBFI), Schloss Dagstuhl (2008)
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Gawrychowski, P., Jeż, A. (2009). Hyper-minimisation Made Efficient. In: Královič, R., Niwiński, D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03816-7_31
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DOI: https://doi.org/10.1007/978-3-642-03816-7_31
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