Before revision:  Given a   B\"uchi  automaton with n states and a finite degree of ambiguity, we show that the number of states in the complementary B\"uchi automaton constructed by the classical Rank-based and Slice-based complementation constructions can be improved, respectively, to 2^{ \mathcal{O (n)} from 2^{ \mathcal{O (n\log n)} and to \mathcal{O (4^n) from \mathcal{O ((3n)^n) , based on reduced run DAGs. To the best of our knowledge, the improved complexity is exponentially better than best known result of \mathcal{O 21 for complementing B\"uchi automata with a finite degree of ambiguity .
After revision:  Given a B\"uchi automaton with n states and a finite degree of ambiguity, we show that the number of states in the complementary B\"uchi automaton constructed by the classical Rank-based and Slice-based complementation constructions can be improved, respectively, to 2^{ \mathcal{O (n)} from 2^{ \mathcal{O (n\log n)} and to \mathcal{O (4^n) from \mathcal{O ((3n)^n) 21 for complementing B\"uchi automata with a finite degree of ambiguity .