AI 中文总结
该研究针对割查询模型下有向图可达性研究不足的问题，提出一种基于拓扑排序的确定性DAG单源可达性算法，查询复杂度达$O(n \text{log}n)$，还可适配计算DAG单源最短路径。
AI 中文摘要
在割查询模型中，我们通过一个预言机访问（有向）图，可查询给定顶点子集的（有向）割的大小。该模型中最基础的任务之一是判断两个固定顶点$s$和$t$之间是否存在路径。虽然无向图的相关研究已有诸多成果，但割查询模型下有向图的研究仍远未充分。即便对于$s$-$t$可达性这一基础任务，目前已知的最优随机算法是利用Grebinski和Kucherov提出的技术重构整个图，需$O(n^2 / \text{log}n)$次查询[Grebinski and Kucherov, 2000]。我们将研究范围限定在有向无环图（DAG），得到了一个确定性单源可达性算法，仅需$O(n \text{log}n)$次查询。该成果基于拓扑排序算法，还可适配用于计算DAG中的单源最短路径。
英文摘要
In the cut-query model, we have access to a (directed) graph via an oracle and we can query the size of the (directed) cut of a given subset of the vertices. One of the most elementary tasks in this model is to decide if there is a path two fixed vertices $s$ and $t$. While many results are known for undirected graphs, much less in understood for directed graphs in the cut query model. Even for the basic task of $s$-$t$ reachability, the best known randomized algorithm, is to reconstruct the entire graph with a technique by Grebinski and Kucherov using $O(n^2 / \log n)$ queries [Grebinski and Kucherov, 2000]. We restrict our attention to directed acyclic graphs (DAGs) and obtain a deterministic single-source reachability algorithm using $O(n \sqrt{n \log n})$ queries. The result is based on a topological sort algorithm, and can also be adapted to compute single-source shortest paths in DAGs.
