The Voronoi diagrams technique was introduced by Cabello to compute the diameter of planar graphs in subquadratic time. We present novel applications of this technique in static, fault-tolerant, and partially-dynamic undirected unweighted planar graphs, as well as some new limitations. 1. In the static case, we give $n^{3+o(1)}/D^2$ and $\tilde{O}(n\cdot D^2)$ time algorithms for computing the diameter of a planar graph $G$ with diameter $D$. These are faster than the state of the art $\tilde{O}(n^{5/3})$ when $D<n^{1/3}$ or $D>n^{2/3}$. 2. In the fault-tolerant setting, we give an $n^{7/3+o(1)}$ time algorithm for computing the diameter of $G\setminus \{e\}$ for every edge $e$ in $G$ the replacement diameter problem. Compared to the naive $\tilde{O}(n^{8/3})$ time algorithm that runs the static algorithm for every edge. 3. In the incremental setting, where we wish to maintain the diameter while while adding edges, we present an algorithm with total running time $n^{7/3+o(1)}$. Compared to the naive $\tilde{O}(n^{8/3})$ time algorithm that runs the static algorithm after every update. 4. We give a lower bound (conditioned on the SETH) ruling out an amortized $O(n^{1-\varepsilon})$ update time for maintaining the diameter in *weighted* planar graph. The lower bound holds even for incremental or decremental updates. Our upper bounds are obtained by novel uses and manipulations of Voronoi diagrams. These include maintaining the Voronoi diagram when edges of the graph are deleted, allowing the sites of the Voronoi diagram to lie on a BFS tree level (rather than on boundaries of $r$-division), and a new reduction from incremental diameter to incremental distance oracles that could be of interest beyond planar graphs. Our lower bound is the first lower bound for a dynamic planar graph problem that is conditioned on the SETH.

翻译：Voronoi图技术由Cabello引入，用于在次二次时间内计算平面图的直径。本文展示了该技术在静态、容错和部分动态无向无权平面图中的新应用，并提出了若干新的局限性。1. 在静态情形下，我们给出了计算直径为$D$的平面图$G$直径的$n^{3+o(1)}/D^2$和$\tilde{O}(n\cdot D^2)$时间算法。当$D<n^{1/3}$或$D>n^{2/3}$时，这些算法比现有最优的$\tilde{O}(n^{5/3})$算法更快。2. 在容错设置中，我们给出了$n^{7/3+o(1)}$时间算法，用于计算$G$中每条边$e$的替换直径问题（即$G\setminus \{e\}$的直径）。相比对每条边运行静态算法的朴素$\tilde{O}(n^{8/3})$时间算法有所改进。3. 在增量设置中（需在添加边时维护直径），我们提出总运行时间为$n^{7/3+o(1)}$的算法。相比每次更新后运行静态算法的朴素$\tilde{O}(n^{8/3})$时间算法有所改进。4. 我们给出了一个下界（基于SETH假设），排除了在*带权*平面图中维护直径的均摊$O(n^{1-\varepsilon})$更新时间的可能性。该下界对增量或减量更新均成立。我们的上界通过Voronoi图的新颖运用与操作实现，包括在删除图边时维护Voronoi图、允许Voronoi图站点位于BFS树层级（而非$r$-划分边界），以及从增量直径到增量距离预言机的新归约——这一归约可能对非平面图也有参考价值。本文的下界是首个基于SETH假设的动态平面图问题下界。