The Strahler number of a rooted tree is the largest height of a perfect binary tree that is its minor. The Strahler number of a parity game is proposed to be defined as the smallest Strahler number of the tree of any of its attractor decompositions. It is proved that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices~$n$ and linear in $({d}/{2k})^k$, where $d$ is the number of priorities and $k$ is the Strahler number. This complexity is quasi-polynomial because the Strahler number is at most logarithmic in the number of vertices. The proof is based on a new construction of small Strahler-universal trees. It is shown that the Strahler number of a parity game is a robust parameter: it coincides with its alternative version based on trees of progress measures and with the register number defined by Lehtinen~(2018). It follows that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices and linear in $({d}/{2k})^k$, where $k$ is the register number. This significantly improves the running times and space achieved for parity games of bounded register number by Lehtinen (2018) and by Parys (2020). The running time of the algorithm based on small Strahler-universal trees yields a novel trade-off $k \cdot \lg(d/k) = O(\log n)$ between the two natural parameters that measure the structural complexity of a parity game, which allows solving parity games in polynomial time. This includes as special cases the asymptotic settings of those parameters covered by the results of Calude, Jain Khoussainov, Li, and Stephan (2017), of Jurdzi\'nski and Lazi\'c (2017), and of Lehtinen (2018), and it significantly extends the range of such settings, for example to $d = 2^{O\left(\sqrt{\lg n}\right)}$ and $k = O\!\left(\sqrt{\lg n}\right)$.

中文翻译：

---

平价游戏的 Strahler 数

有根树的斯特拉勒数是其次要的完美二叉树的最大高度。建议将奇偶博弈的 Strahler 数定义为其任何吸引子分解的树的最小 Strahler 数。证明了奇偶博弈可以在拟线性空间和时间上求解，其中顶点数为多项式~$n$，线性为$({d}/{2k})^k$，其中$d$是优先级数，$k$ 是 Strahler 数。这种复杂性是拟多项式的，因为 Strahler 数至多是顶点数的对数。该证明基于小型 Strahler 通用树的新构造。结果表明，平价游戏的 Strahler 数是一个鲁棒参数：它与基于进度树的替代版本和 Lehtinen~(2018) 定义的注册号一致。因此，奇偶游戏可以在准线性空间和时间上求解，顶点数为多项式，$({d}/{2k})^k$ 为线性，其中 $k$ 是寄存器编号。这显着改善了 Lehtinen (2018) 和 Parys (2020) 为有界寄存器数的奇偶游戏实现的运行时间和空间。基于小型 Strahler 通用树的算法的运行时间在测量结构复杂性的两个自然参数之间产生了一种新颖的权衡 $k \cdot \lg(d/k) = O(\log n)$奇偶游戏，它允许在多项式时间内解决奇偶游戏。这包括作为特殊情况的 Calude 结果涵盖的那些参数的渐近设置，