平面图的一个简单分布式确定性分割器
A Simple Distributed Deterministic Planar Separator
AI总结：
本文提出一个简单确定性算法，用于平面图的分布式平衡分割，具有与之前算法相同的轮复杂度。
摘要：
平面图$G$的平衡分割器是指一个顶点集合，其移除会将图分割成多个连通分量，每个分量的大小都是原图的常数因子更小。Lipton和Tarjan在FOCS'77中证明，每个平面图都存在一个大小为$O(\sqrt{n})$的平衡分割器，以及一个大小为$O(D)$的平衡分割器，其中$D$是图的直径。在集中式环境下，这两种分割器都可以在线性时间内找到。在分布式环境下，$D$是许多优化问题解决轮数的下界，因此大小为$O(D)$的分割器更优。直到DISC'17，Ghaffari和Parter才设计出一个分布式算法，通过将Lipton-Tarjan算法适应到分布式模型中，计算出大小为$O(D)$的分割器，耗时$\tilde O(D)$轮。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，因此使用它的算法也是随机化的。直到PODC'25，Jauregui、Montealegre和Rapaport给出了一个复杂的确定性分割器算法。我们提出一个更简单的确定性分割器算法，具有相同的（近最优）$\tilde O(D)$轮复杂度。虽然先前的工作设计了复杂或随机化的方法将权重从顶点转移到图的面，我们表明一个简单的方法也有效：每个顶点将权重转移到它所处的任意一个面。这就是全部！我们注意到，一个确定性的分割器算法可以直接去随机化平面图上经典问题的分布式算法，如单源最短路径、最大流、有向全局最小割和可达性。
英文摘要：
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.
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