The dual of a planar graph $G$ is a planar graph $G^*$ that has a vertex for each face of $G$ and an edge for each pair of adjacent faces of $G$. The profound relationship between a planar graph and its dual has been the algorithmic basis for solving numerous (centralized) classical problems on planar graphs. In the distributed setting however, the only use of planar duality is for finding a recursive decomposition of $G$ [DISC 2017, STOC 2019]. We extend the distributed algorithmic toolkit to work on the dual graph $G^*$. These tools can then facilitate various algorithms on $G$ by solving a suitable dual problem on $G^*$. Given a directed planar graph $G$ with positive and negative edge-lengths and hop-diameter $D$, our key result is an $\tilde{O}(D^2)$-round algorithm for Single Source Shortest Paths on $G^*$, which then implies an $\tilde{O}(D^2)$-round algorithm for Maximum $st$-Flow on $G$. Prior to our work, no $\tilde{O}(\text{poly}(D))$-round algorithm was known for Maximum $st$-Flow. We further obtain a $D\cdot n^{o(1)}$-rounds $(1-\epsilon)$-approximation algorithm for Maximum $st$-Flow on $G$ when $G$ is undirected and $st$-planar. Finally, we give a near optimal $\tilde O(D)$-round algorithm for computing the weighted girth of $G$. The main challenges in our work are that $G^*$ is not the communication graph (e.g., a vertex of $G$ is mapped to multiple vertices of $G^*$), and that the diameter of $G^*$ can be much larger than $D$ (i.e., possibly by a linear factor). We overcome these challenges by carefully defining and maintaining subgraphs of the dual graph $G^*$ while applying the recursive decomposition on the primal graph $G$. The main technical difficulty, is that along the recursive decomposition, a face of $G$ gets shattered into (disconnected) components yet we still need to treat it as a dual node.

翻译：平面图$G$的对偶图$G^*$是一个平面图，其中$G$的每个面对应$G^*$的一个顶点，$G$中每对相邻面对应$G^*$的一条边。平面图与其对偶图之间的深刻关系，已成为解决平面图上众多（集中式）经典问题的算法基础。然而在分布式计算环境中，平面对偶性的唯一应用是寻找$G$的递归分解[DISC 2017, STOC 2019]。我们将分布式算法工具扩展至对偶图$G^*$上。这些工具随后可通过在$G^*$上求解合适的对偶问题，为$G$上的各类算法提供支持。给定具有正负边长度及跳直径$D$的有向平面图$G$，我们的核心成果是在$G^*$上实现$\tilde{O}(D^2)$轮的单源最短路径算法，进而推导出$G$上最大$st$流问题的$\tilde{O}(D^2)$轮算法。在本工作之前，最大$st$流问题尚未存在$\tilde{O}(\text{poly}(D))$轮算法。当$G$为无向$st$平面图时，我们进一步获得了$D\cdot n^{o(1)}$轮的$(1-\epsilon)$近似最大$st$流算法。最后，我们提出了计算$G$加权围长的近乎最优的$\tilde O(D)$轮算法。本研究的主要挑战在于：$G^*$并非通信图（例如$G$的一个顶点可能映射到$G^*$的多个顶点），且$G^*$的直径可能远大于$D$（可能存在线性倍数关系）。我们通过在原始图$G$上应用递归分解的同时，精确定义并维护对偶图$G^*$的子图来克服这些挑战。主要技术难点在于：在递归分解过程中，$G$的一个面会被分割为（不连通的）分量，但我们仍需将其视为对偶节点进行处理。