  Recently, five quasi-polynomial-time algorithms solving parity games were
proposed. We elaborate on one of the algorithms, by Lehtinen (2018).
  Czerwi\'nski et al. (2019) observe that four of the algorithms can be
expressed as constructions of separating automata (of quasi-polynomial size),
that is, automata that accept all plays decisively won by one of the players,
and rejecting all plays decisively won by the other player. The separating
automata corresponding to three of the algorithms are deterministic, and it is
clear that deterministic separating automata can be used to solve parity games.
The separating automaton corresponding to the algorithm of Lehtinen is
nondeterministic, though. While this particular automaton can be used to solve
parity games, this is not true for every nondeterministic separating automaton.
As a first (more conceptual) contribution, we specify when a nondeterministic
separating automaton can be used to solve parity games.
  We also repeat the correctness proof of the Lehtinen's algorithm, using
separating automata. In this part, we prove that her construction actually
leads to a faster algorithm than originally claimed in her paper: its
complexity is $n^{O(\log n)}$ rather than $n^{O(\log d \cdot \log n)}$ (where
$n$ is the number of nodes, and $d$ the number of priorities of a considered
parity game), which is similar to complexities of the other
quasi-polynomial-time algorithms.
