Buchi Determinization Made Tighter.

By separating the principal acceptance mechanism from the concrete acceptance condition of a given B\"{u}chi automaton with $n$ states,Schewe presented the construction of an equivalent deterministic Rabin transition automaton with $o((1.65n)^n)$ states via \emph{history trees}, which can be simply translated to a standard Rabin automaton with $o((2.26n)^n)$ states. Apart from the inherent simplicity, Schewe's construction improved Safra's construction (which requires $12^nn^{2n}$ states). However, the price that is paid is the use of $2^{n-1}$ Rabin pairs (instead of $n$ in Safra's construction). Further, by introducing the \emph{later introduction record} as a record tailored for ordered trees, deterministic automata with Parity acceptance condition is constructed which exactly resembles Piterman's determinization with Parity acceptance condition where the state complexity is $O((n!)^2)$ and the index complexity is $2n$.In this paper, we improve Schewe's construction to $2^{\lceil (n-1)/2\rceil}$ Rabin pairs with the same state complexity. Meanwhile, we give a new determinization construction of Parity automata with the state complexity being $o(n^2(0.69n\sqrt{n})^n)$ and index complexity being $n$.