Determinization of $\omega$-automata unified

We describe a uniform construction for converting $\omega$-automata with arbitrary acceptance conditions (based on the notion of infinity sets i.e. the set of states visited infinitely often in a run of the automaton) to equivalent deterministic parity automata (DPW). Given a non-deterministic automaton with $n$ states, our construction gives a DPW with at most $2^{O(n^2 \log n)}$ states and $O(n^2)$ parity indices. The corresponding bounds when the original automaton is deterministic are O(n!) and O(n), respectively. Our algorithm gives better asymptotic bounds on the number of states and parity indices vis-a-vis the best known technique when determinizing Rabin or Streett automata with $\Omega{(2^n)}$ acceptance pairs, where $n > 1$. We demonstrate this by describing a family of Streett (and Rabin) automata with $2^{n}$ non-redundant acceptance pairs, for which the best known determinization technique gives a DPW with at least $\Omega{(2^{(n^3)})}$ states, while our construction constructs a DRW/DPW with $2^{O(n^2\log n)}$ states. An easy corollary of our construction is that an $\omega$-language with Rabin index $k$ cannot be recognized by any $\omega$-automaton (deterministic or non-deterministic) with fewer than $O(\sqrt{k})$ states.