A balanced separator of a graph $G$ is a set of vertices whose removal disconnects the graph into connected components that are a constant factor smaller than $G$. Lipton and Tarjan [FOCS'77] famously proved that every planar graph admits a balanced separator of size $O(\sqrt{n})$, as well as a balanced separator of size $O(D)$ that is a simple path (where $D$ is $G$'s diameter). In the centralized setting, both separators can be found in linear time. In the distributed setting, $D$ is a universal lower bound for the round complexity of solving many optimization problems, so, separators of size $O(D)$ are preferable. It was not until [DISC'17] that a distributed algorithm was devised by Ghaffari and Parter to compute such an $O(D)$-size separator in $\tilde O(D)$ rounds, by adapting the Lipton-Tarjan algorithm to the distributed model. Since then, this algorithm was used in several distributed algorithms for planar graphs, e.g., [GP, DISC'17], [LP, STOC'19], [AEDPW, PODC'25]. However, the algorithm is randomized, deeming the algorithms that use it to be randomized as well. Obtaining a deterministic algorithm remained an interesting open question until [PODC'25], when a (complex) deterministic separator algorithm was given by Jauregui, Montealegre and Rapaport. We present a much simpler deterministic separator algorithm with the same (near-optimal) $\tilde O(D)$-round complexity. While previous works devised either complicated or randomized ways of transferring weights from vertices to faces of $G$, we show that a straightforward way also works: Each vertex simply transfers its weight to one arbitrary face it lies on. That's it! We note that a deterministic separator algorithm directly derandomizes the state-of-the-art distributed algorithms for classical problems on planar graphs such as single-source shortest-paths, maximum-flow, directed global min-cut, and reachability.

翻译：图$G$的平衡分隔子是指一个顶点集合，移除这些顶点后，图被分割成若干个连通分量，这些分量的大小相对于$G$而言是一个常数因子。Lipton和Tarjan [FOCS'77] 著名地证明了每个平面图都存在一个大小为$O(\sqrt{n})$的平衡分隔子，以及一个大小为$O(D)$且为简单路径的平衡分隔子（其中$D$是$G$的直径）。在集中式设置中，这两种分隔子都可以在线性时间内找到。在分布式设置中，$D$是解决许多优化问题的轮复杂度的通用下界，因此，大小为$O(D)$的分隔子更受青睐。直到[DISC'17]，Ghaffari和Parter才设计出一种分布式算法，通过将Lipton-Tarjan算法适配到分布式模型，在$\tilde O(D)$轮内计算出这样一个$O(D)$大小的分隔子。此后，该算法被用于多个平面图的分布式算法中，例如[GP, DISC'17]、[LP, STOC'19]、[AEDPW, PODC'25]。然而，该算法是随机化的，导致使用它的算法也变成随机化的。获得一个确定性算法一直是一个有趣的开放性问题，直到[PODC'25]，Jauregui、Montealegre和Rapaport给出了一个（复杂的）确定性分隔子算法。我们提出了一种更简单的确定性分隔子算法，具有相同的（接近最优的）$\tilde O(D)$轮复杂度。先前的工作设计了复杂或随机化的方式将权重从顶点转移到$G$的面，而我们证明一种直接的方式同样有效：每个顶点只需将其权重转移到它所在的任意一个面上。仅此而已！我们注意到，确定性分隔子算法直接对平面图上经典问题（如单源最短路径、最大流、有向全局最小割和可达性）的最先进分布式算法进行了去随机化。