On minimizing cover automata for finite languages in O(n log n) time

Abstract

A deterministic finite automaton (DFA) A is called a cover automaton (DFCA) for a finite language L over some alphabet Σ if L = L(A) ∩ Σ≤l, with l being the length of some longest word in L. Thus a word w ∈ Σ* is in L if and only if |w| ≤ l and w ∈ L(A). The DFCA A is minimal if no DFCA for L has fewer states. In this paper, we present an algorithm which converts an n-state DFA for some finite language L into a corresponding minimal DFCA, using only O(n log n) time and O(n) space. The best previously known algorithm [2] requires O(n2) time and space. Furthermore, the new algorithm can also be used to minimize any DFCA, where the best previous method [1] takes O(n4) time and space. © Springer-Verlag Berlin Heidelberg 2003.