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.
After revision:  We then show how to use this type of reduced run DAGs as 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.