We show how to preprocess a weighted undirected $n$-vertex planar graph in $\tilde O(n^{4/3})$ time, such that the distance between any pair of vertices can then be reported in $\tilde O(1)$ time. This improves the previous $\tilde O(n^{3/2})$ preprocessing time [JACM'23]. Our main technical contribution is a near optimal construction of \emph{additively weighted Voronoi diagrams} in undirected planar graphs. Namely, given a planar graph $G$ and a face $f$, we show that one can preprocess $G$ in $\tilde O(n)$ time such that given any weight assignment to the vertices of $f$ one can construct the additively weighted Voronoi diagram of $f$ in near optimal $\tilde O(|f|)$ time. This improves the $\tilde O(\sqrt{n |f|})$ construction time of [JACM'23].

翻译：本文提出一种预处理带权无向$n$顶点平面图的方法，预处理时间为$\tilde O(n^{4/3})$，此后任意顶点对之间的距离可在$\tilde O(1)$时间内查询。该结果改进了先前$\tilde O(n^{3/2})$的预处理时间[JACM'23]。我们的核心技术贡献在于无向平面图中\emph{加性加权Voronoi图}的近似最优构造。具体而言，给定平面图$G$及其某个面$f$，我们证明可在$\tilde O(n)$时间内预处理$G$，使得对于$f$顶点的任意权重分配，均可在近似最优的$\tilde O(|f|)$时间内构造出$f$的加性加权Voronoi图。这改进了[JACM'23]中$\tilde O(\sqrt{n |f|})$的构造时间。