Low Diameter Decompositions (LDDs) are invaluable tools in the design of combinatorial graph algorithms. While historically they have been applied mainly to undirected graphs, in the recent breakthrough for the negative-length Single Source Shortest Path problem, Bernstein, Nanongkai, and Wulff-Nilsen [FOCS '22] extended the use of LDDs to directed graphs for the first time. Specifically, their LDD deletes each edge with probability at most $O(\frac{1}{D} \cdot \log^2 n)$, while ensuring that each strongly connected component in the remaining graph has a (weak) diameter of at most $D$. In this work, we make further advancements in the study of directed LDDs. We reveal a natural and intuitive (in hindsight) connection to Expander Decompositions, and leveraging this connection along with additional techniques, we establish the existence of an LDD with an edge-cutting probability of $O(\frac{1}{D} \cdot \log n \log\log n)$. This improves the previous bound by nearly a logarithmic factor and closely approaches the lower bound of $\Omega(\frac{1}{D} \cdot \log n)$. With significantly more technical effort, we also develop two efficient algorithms for computing our LDDs: a deterministic algorithm that runs in time $\tilde O(m \cdot poly(D))$ and a randomized algorithm that runs in near-linear time $\tilde O(m)$. We believe that our work provides a solid conceptual and technical foundation for future research relying on directed LDDs, which will undoubtedly follow soon.

翻译：低直径分解（LDD）是组合图算法设计中不可或缺的工具。尽管历史上它们主要应用于无向图，但在负权单源最短路径问题最近的突破性工作中，Bernstein、Nanongkai 和 Wulff-Nilsen [FOCS '22] 首次将 LDD 的应用扩展到了有向图。具体而言，他们的 LDD 以最多 $O(\frac{1}{D} \cdot \log^2 n)$ 的概率删除每条边，同时确保剩余图中每个强连通分量具有至多 $D$ 的（弱）直径。在本工作中，我们在有向 LDD 的研究中取得了进一步进展。我们揭示了一个（事后看来）自然且直观的与扩展图分解的联系，并利用这一联系以及额外的技术，我们证明了一种边切割概率为 $O(\frac{1}{D} \cdot \log n \log\log n)$ 的 LDD 的存在性。这比先前的界改进了近一个对数因子，并接近了 $\Omega(\frac{1}{D} \cdot \log n)$ 的下界。通过付出显著更多的技术努力，我们还开发了两种计算我们 LDD 的高效算法：一种在 $\tilde O(m \cdot poly(D))$ 时间内运行的确定性算法，以及一种在近线性时间 $\tilde O(m)$ 内运行的随机算法。我们相信，我们的工作为未来依赖有向 LDD 的研究（无疑将很快跟进）奠定了坚实的理论与技术基础。