On sparse graphs, Roditty and Williams [2013] proved that no \(\varvec{O(n^{2-\varepsilon })}\)-time algorithm achieves an approximation factor smaller than \(\varvec{\frac{3}{2}}\) for the diameter problem unless SETH fails. In this article, we solve an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier? We propose the first combinatorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for all eccentricities in median graphs with bounded dimension \(\varvec{d}\), i.e. the dimension of the largest induced hypercube. This prerequisite on \(\varvec{d}\) is not necessary anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is \(\varvec{O(n^{1.6456}\log ^{O(1)} n)}\). We provide also some satellite outcomes related to this general result. In particular, restricted to simplex graphs, this algorithm enumerates all eccentricities with a quasilinear running time. Moreover, an algorithm is proposed to compute exactly all reach centralities in time \(\varvec{O(2^{3d}n\log ^{O(1)}n)}\).

在稀疏图上，Roditty 和 Williams [2013] 证明没有\(\varvec{O(n^{2-\varepsilon })}\)时间算法能够实现小于\(\varvec{\frac{3) 的近似因子}{2}}\)解决直径问题，除非 SETH 失败。在本文中，我们解决了文献中提出的一个悬而未决的问题：我们可以使用中值图的结构特性来打破这个全局二次障碍吗？我们提出了第一个组合算法，可以在真正的次二次时间内精确计算中值图的所有偏心率。中值图构成了度量图论中研究最多的图族，因为它们的结构代表了许多其他离散和几何概念，例如 CAT(0) 立方体复合体。我们的结果概括了最近的一个结果，指出对于具有有界维度\(\varvec{d}\) （即最大诱导超立方体的维度）的中值图中的所有偏心率，存在一种线性时间算法。不再需要\(\varvec{d}\)的先决条件来确定次二次时间内的所有偏心率。我们算法的执行时间是\(\varvec{O(n^{1.6456}\log ^{O(1)} n)}\)。我们还提供了与此总体结果相关的一些卫星结果。特别是，仅限于单纯形图，该算法以拟线性运行时间枚举所有偏心率。此外，提出了一种算法来精确计算时间上的所有到达中心性\(\varvec{O(2^{3d}n\log ^{O(1)}n)}\)。