One of the most basic techniques in algorithm design consists of breaking a problem into subproblems and then proceeding recursively. In the case of graph algorithms, one way to implement this approach is through separator sets. Given a graph $G=(V,E)$, a subset of nodes $S \subseteq V$ is called a separator set of $G$ if the size of each connected component of $G-S$ is at most $2/3 \cdot |V|$. The most useful separator sets are those that satisfy certain restrictions of cardinality or structure. For over 40 years, various efficient algorithms have been developed for computing separators of different kinds, particularly in planar graphs. Separator sets, combined with a divide and conquer approach, have been fundamental in the design of efficient algorithms in various settings. In this work, we present the first deterministic algorithm in the distributed CONGEST model that recursively computes a cycle separator in planar graphs in $\tilde{\mathcal{O}}(D)$ rounds. This result, as in the centralized setting, has significant implications for distributed planar algorithms. In fact, from this result, we can construct a deterministic algorithm that computes a DFS tree in $\tilde{\mathcal{O}}(D)$ rounds. This matches both the best-known randomized algorithm of Ghaffari and Parter (DISC'17) and, up to polylogarithmic factors, the trivial lower bound of $Ω(D)$ rounds. Besides DFS, our deterministic cycle separator algorithm can be used to derandomize several planar-graph algorithms whose only randomized ingredient is the computation of a cycle separator, such as maximum flow (Abd-Elhaleem, Dory, Parter and Weimann, PODC'25), single-source shortest path (Li and Parter, STOC'19), and reachability (Parter, DISC'20).

翻译：算法设计中最基本的技术之一是将问题分解为子问题，然后递归求解。在图算法中，实现此方法的一种途径是通过分隔集。给定图 $G=(V,E)$，若节点子集 $S \subseteq V$ 满足 $G-S$ 的每个连通分量大小至多为 $2/3 \cdot |V|$，则称 $S$ 为 $G$ 的分隔集。最实用的分隔集是那些在基数或结构上满足特定约束的集合。四十余年来，学界已针对不同类型的分隔器（尤其是平面图中的分隔器）开发了多种高效算法。分隔集与分治策略相结合，已成为各类场景下高效算法设计的基石。本工作首次在分布式CONGEST模型中提出确定性算法，可在 $\tilde{\mathcal{O}}(D)$ 轮内递归计算平面图中的环分隔器。与集中式场景类似，该结果对分布式平面图算法具有重要影响。基于此结果，我们可构建在 $\tilde{\mathcal{O}}(D)$ 轮内计算DFS树的确定性算法，这既匹配了Ghaffari与Parter（DISC'17）已知的最佳随机算法，又在多对数因子内达到了平凡下界 $Ω(D)$ 轮。除DFS外，我们的确定性环分隔器算法可用于对多个平面图算法进行去随机化——这些算法唯一的随机性成分正是环分隔器的计算，例如最大流（Abd-Elhaleem, Dory, Parter and Weimann, PODC'25）、单源最短路径（Li and Parter, STOC'19）以及可达性问题（Parter, DISC'20）。