# Improved Complexity Analysis of Quasi-Polynomial Algorithms Solving Parity Games

We improve the complexity of solving parity games (with priorities in vertices) for d=ω(log n) by a factor of θ(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 θ(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 θ(d).

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