  In this work, we exploit the power of \emph{finite ambiguity} for the
complementation problem of B\"uchi automata by using reduced run directed
acyclic graphs (DAGs) over infinite words, in which each vertex has at most one
predecessor; these reduced run DAGs have only a finite number of infinite runs,
thus obtaining the finite ambiguity in B\"uchi complementation. We show how to
use this type of reduced run DAGs as a unified tool to optimize both rank-based
and slice-based complementation constructions for B\"uchi automata with a
finite degree of ambiguity. As a result, given a B\"uchi automaton with $n$
states and a finite degree of ambiguity, the number of states in the
complementary B\"uchi automaton constructed by the classical rank-based and
slice-based complementation constructions can be improved from $2^{\mathsf{O}(n
\log n)}$ and $\mathsf{O}((3n)^{n})$ to $\mathsf{O}(6^{n}) \subseteq
2^{\mathsf{O}(n)}$ and $\mathsf{O}(4^{n})$, respectively. We further show how
to construct such reduced run DAGs for limit deterministic B\"uchi automata and
obtain a specialized complementation algorithm, thus demonstrating the
generality of the power of finite ambiguity.
