Let PT-DFA mean a deterministic finite automaton whose transition relation is a partial function. We present an algorithm for minimizing a PT-DFA in $O(m lg n)$ time and $O(m+n+alpha)$ memory, where $n$ is the number of states, $m$ is the number of defined transitions, and $alpha$ is the size of the alphabet. Time consumption does not depend on $alpha$, because the $alpha$ term arises from an array that is accessed at random and never initialized. It is not needed, if transitions are in a suitable order in the input. The algorithm uses two instances of an array-based data structure for maintaining a refinable partition. Its operations are all amortized constant time. One instance represents the classical blocks and the other a partition of transitions. Our measurements demonstrate the speed advantage of our algorithm on PT-DFAs over an $O(alpha n lg n)$ time, $O(alpha n)$ memory algorithm.
@InProceedings{valmari_et_al:LIPIcs.STACS.2008.1328, author = {Valmari, Antti and Lehtinen, Petri}, title = {{Efficient Minimization of DFAs with Partial Transition}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {645--656}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1328}, URN = {urn:nbn:de:0030-drops-13286}, doi = {10.4230/LIPIcs.STACS.2008.1328}, annote = {Keywords: Deterministic finite automaton, sparse adjacency matrix, partition refinement} }
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