线性时间的安全交替DFS与SCCs
Linear-Time Safe-Alternating DFS and SCCs
AI总结：
研究交替图的强连通性，提出安全交替可达性概念及线性时间分解算法，将Update Game求解复杂度降至Θ(m+n)，并将Explicit McNaughton-Müller Games复杂度从立方级降至平方级。
摘要：
交替图是一种有向图，其顶点集被划分为存在和全称两类。这构成了大量无限时长双人博弈的基本竞技场，其中玩家□和○交替进行基于回合的、沿其控制弧滑动棋子的动作。我们将交替强连通性研究作为有向图强连通性的一种推广，旨在提供线性时间的分解和合理的结构图特征。为此引入了一种改进的交替可达性概念：玩家□试图在不离开顶点预设子集的情况下到达顶点，而玩家○则起阻碍作用。这被称为安全交替可达性。结果表明，每个竞技场都唯一分解为安全交替强连通分量，其中玩家□可以在给定分量内无限次访问每个顶点，而永远不必离开该分量本身。我们的主要结果是一个用于计算这种交替图分解的线性时间算法。其底层的图结构和算法都推广了有向图分解为强连通分量的经典方法。该算法建立在交替的深度优先搜索的线性时间推广之上，从 Tarjan 1972 的机制中汲取了灵感。我们的理论在更快求解著名的无限时长棋子博弈方面有直接应用。Dinneen 和 Khoussainov 在 1999 年表明，判定给定的 Update Game 的代价为 O(mn) 时间，其中 n 是顶点数，m 是弧数。我们在 Θ(m+n) 线性时间内解决了该任务。Explicit McNaughton-Müller Games 的复杂度也从立方级降低到了平方级。
英文摘要：
An alternating graph is a directed graph whose vertex set is partitioned into two classes, existential and universal. This forms the basic arena for a plethora of infinite duration two-player games where Player~$\square$ and~$\ocircle$ alternate in a turn-based sliding of a pebble along the arcs they control. We study alternating strongly-connectedness as a generalization of strongly-connectedness in directed graphs, aiming at providing a linear time decomposition and a sound structural graph characterization. For this a refined notion of alternating reachability is introduced: Player~$\square$ attempts to reach vertices without leaving a prescribed subset of the vertices, while Player~$\ocircle$ works against. This is named \emph{safe alternating reachability}. It is shown that every arena uniquely decomposes into safe alternating strongly-connected components where Player~$\square$ can visit each vertex within a given component infinitely often, without having to ever leave out the component itself. Our main result is a linear time algorithm for computing this alternating graph decomposition. Both the underlying graph structures and the algorithm generalize the classical decomposition of a directed graph into strongly-connected components. The algorithm builds on a linear time generalization of the depth-first search on alternation, taking inspiration from Tarjan 1972 machinery. Our theory has direct applications in solving well-known infinite duration pebble games faster. Dinneen and Khoussainov showed in 1999 that deciding a given Update Game costs $O(mn)$ time, where $n$ is the number of vertices and $m$ is that of arcs. We solve the task in $Θ(m+n)$ linear~time. The complexity of Explicit McNaughton-Müller Games also improves from cubic to quadratic.
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