Before revision:  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:  To the best of our knowledge, the improved complexity is exponentially better than best known result of \mathcal{O  for complementing B\"uchi automata with a finite degree of ambiguity .