Parity games have witnessed several new quasi-polynomial algorithms since the breakthrough result of Calude et al. (2017). The central combinatorial object underlying these approaches is a universal tree, as identified by Czerwi\'nski et al. (2019). By providing a quasi-polynomial lower bound on the size of universal trees, they have highlighted a barrier that must be overcome by all existing approaches to attain polynomial runtime. This is due to the existence of worst case instances which force these algorithms to explore a large portion of the tree. As an attempt to overcome this barrier, we propose a strategy iteration framework which can be applied on any universal tree. It is at least as fast as its value iteration counterparts, while allowing one to take bigger leaps in the universal tree. Value iteration - asymptotically the fastest known algorithm for parity games - is a repeated application of operators associated with arcs in the game graph to obtain the least fixed point. Our main technical contribution is an efficient method for computing the least fixed point of operators associated with arcs in a strategy subgraph. This is achieved via a careful adaptation of shortest path algorithms to the setting of ordered trees. By plugging in the universal tree of Jurdzi\'nski and Lazi\'c (2017), or the Strahler universal tree of Daviaud et al. (2020), we obtain instantiations of the general framework that take time $O(mn^2\log n\log d)$ and $O(mn^2\log^3 n \log d)$ respectively per iteration.

翻译：自Calude等人(2017年)的突破成果以来,Paity游戏已经见证了几个新的准球效算法。这些方法背后的核心组合对象是一个通用树(Czerwi\'nski等人(2019年)),Czerwi\'nski等人(2019年)指出,它是一个通用树(Czerwi\'nski等人),它是一个通用树(Czerwi\'nski等人(2019年)),它提供了一种准球质下对全树大小的较低约束。它们突出了一种障碍,所有现有方法都必须克服这种障碍,才能达到多球运行时间。这是因为存在一些最坏的情况,迫使这些算法不得不探索大树的一大部分。为了克服这一屏障,我们提出了一个战略树圈中最小固定的操作者点,我们通过仔细调整了普通树值的直线上的直线上的直线上的直线上的直径。 (20) 和直径直径的直径直径的直径直径的直径直径的直径直径直径直径。(20)