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 {\em 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 {\em 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。我们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，以及一些新的局限性。1. 在静态情况下，我们给出了时间复杂度为$n^{3+o(1)}/D^2$和$\tilde{O}(n\cdot D^2)$的算法，用于计算直径为$D$的平面图$G$的直径。当$D<n^{1/3}$或$D>n^{2/3}$时，这些算法比当前最优的$\tilde{O}(n^{5/3})$算法更快。2. 在容错设置中，我们给出时间复杂度为$n^{7/3+o(1)}$的算法，用于计算$G$中每条边$e$对应的$G\setminus \{e\}$的直径（替换直径问题）。这相比对每条边运行静态算法的朴素$\tilde{O}(n^{8/3})$算法有所提升。3. 在增量设置中（我们希望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。这相比每次更新后运行静态算法的朴素$\tilde{O}(n^{8/3})$算法有所改进。4. 我们给出一个下界（基于SETH假设），排除了在{\em 加权}平面图中以摊还$O(n^{1-\varepsilon})$更新时间复杂度维护直径的可能性。该下界对增量或减量更新均成立。我们的上界通过对Voronoi图的新颖应用和操作获得，包括：在图的边被删除时维护Voronoi图，允许Voronoi图的站点位于BFS树层级（而非$r$-划分的边界上），以及从增量直径到增量{\em 距离预言机}的新归约——该归约可能对平面图之外的问题也有价值。我们的下界是首个基于SETH假设的动态平面图问题下界。