Improved Complexity Analysis of Quasi-Polynomial Algorithms Solving Parity Games

We improve the complexity of solving parity games (with priorities in vertices) for $d={\omega}(\log n)$ by a factor of ${\theta}(d^2)$: the best complexity known to date was $O(mdn^{1.45+\log_2(d/\log_2(n))})$, while we obtain $O(mn^{1.45+\log_2(d/\log_2(n))}/d)$, where $n$ is the number of vertices, $m$ is the number of edges, and $d$ is the number of priorities. We base our work on existing algorithms using universal trees, and we improve their complexity. We present two independent improvements. First, an improvement by a factor of ${\theta}(d)$ comes from a more careful analysis of the width of universal trees. Second, we perform (or rather recall) a finer analysis of requirements for a universal tree: while for solving games with priorities on edges one needs an $n$-universal tree, in the case of games with priorities in vertices it is enough to use an $n/2$-universal tree. This way, we allow to solve games of size $2n$ in the time needed previously to solve games of size $n$; such a change divides the quasi-polynomial complexity again by a factor of ${\theta}(d)$.