计算（双曲）Helly图所有偏心率的快速确定性算法
Fast deterministic algorithms for computing all eccentricities in (hyperbolic) Helly graphs
摘要：
一个图是Helly图，如果每一族两两相交的球都有一个非空的公共交。Helly图类是超凸度量空间类的离散模拟。每个图都等距嵌入到一个Helly图中，使得后者成为度量图论中的一类重要图。我们研究直径，半径和所有偏心率计算内的Helly图。在合理的复杂性假设下，一般图的直径和半径都不能在真正的次二次时间内计算。与这些否定的结果相反，最近表明，$n$ -顶点$m$ -边Helly图$G$的半径和直径可以在$\tilde{\mathcal O}(m\sqrt{n})$时间内以高概率计算（即，$n+m$中的次二次）。在本文中，我们提出了一个确定性的${\mathcal O}(m\sqrt{n})$时间算法，不仅计算半径和直径，但也在Helly图的所有顶点偏心率改进的结果。此外，我们给出了一个参数化的线性时间算法在Helly图上，与参数的Gromov双曲$δ$。更具体地说，我们证明了$m$ -边$δ$ -双曲Helly图$G$的半径和中心顶点可以在$\mathcal O(δm)$时间内计算，并且$G$中所有顶点的偏心率都可以在$\mathcal O(δ^2 m)$时间内计算.为了证明这个更一般的结果，我们大量使用我们的新的结构性质获得的Helly图。
英文摘要：
A graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. The class of Helly graphs is the discrete analogue of the class of hyperconvex metric spaces. It is also known that every graph isometrically embeds into a Helly graph, making the latter an important class of graphs in Metric Graph Theory. We study diameter, radius and all eccentricity computations within the Helly graphs. Under plausible complexity assumptions, neither the diameter nor the radius can be computed in truly subquadratic time on general graphs. In contrast to these negative results, it was recently shown that the radius and the diameter of an $n$-vertex $m$-edge Helly graph $G$ can be computed with high probability in $\tilde{\mathcal O}(m\sqrt{n})$ time (i.e., subquadratic in $n+m$). In this paper, we improve that result by presenting a deterministic ${\mathcal O}(m\sqrt{n})$ time algorithm which computes not only the radius and the diameter but also all vertex eccentricities in a Helly graph. Furthermore, we give a parameterized linear-time algorithm for this problem on Helly graphs, with the parameter being the Gromov hyperbolicity $δ$. More specifically, we show that the radius and a central vertex of an $m$-edge $δ$-hyperbolic Helly graph $G$ can be computed in $\mathcal O(δm)$ time and that all vertex eccentricities in $G$ can be computed in $\mathcal O(δ^2 m)$ time. To show this more general result, we heavily use our new structural properties obtained for Helly graphs.
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