  An attractor decomposition meta-algorithm for solving parity games is given
that generalises the classic McNaughton-Zielonka algorithm and its recent
quasi-polynomial variants due to Parys (2019), and to Lehtinen, Schewe, and
Wojtczak (2019). The central concepts studied and exploited are attractor
decompositions of dominia in parity games and the ordered trees that describe
the inductive structure of attractor decompositions. The universal algorithm
yields McNaughton-Zielonka, Parys, and Lehtinen-Schewe-Wojtczak algorithms as
special cases when suitable universal trees are given to it as inputs. The main
technical results provide a unified proof of correctness and structural
insights into those algorithms. Suitably adapting the universal algorithm for
parity games to fixpoint games gives a quasi-polynomial time algorithm to
compute nested fixpoints over finite complete lattices. The universal
algorithms for parity games and nested fixpoints can be implemented
symbolically. It is shown how this can be done with $O(\lg d)$ symbolic space
complexity, improving the $O(d \lg n)$ symbolic space complexity achieved by
Chatterjee, Dvo\v{r}\'{a}k, Henzinger, and Svozil (2018) for parity games,
where $n$ is the number of vertices and $d$ is the number of distinct
priorities in a parity game.
