Hyperbolicity is a graph parameter which indicates how much the shortest-path distance metric of a graph deviates from a tree metric. It is used in various fields such as networking, security, and bioinformatics for the classification of complex networks, the design of routing schemes, and the analysis of graph algorithms. Despite recent progress, computing the hyperbolicity of a graph remains challenging. Indeed, the best known algorithm has time complexity $O(n^{3.69})$, which is prohibitive for large graphs, and the most efficient algorithms in practice have space complexity $O(n^2)$. Thus, time as well as space are bottlenecks for computing hyperbolicity. In this paper, we design a tool for enumerating all far-apart pairs of a graph by decreasing distances. A node pair $(u, v)$ of a graph is far-apart if both $v$ is a leaf of all shortest-path trees rooted at $u$ and $u$ is a leaf of all shortest-path trees rooted at $v$. This notion was previously used to drastically reduce the computation time for hyperbolicity in practice. However, it required the computation of the distance matrix to sort all pairs of nodes by decreasing distance, which requires an infeasible amount of memory already for medium-sized graphs. We present a new data structure that avoids this memory bottleneck in practice and for the first time enables computing the hyperbolicity of several large graphs that were far out-of-reach using previous algorithms. For some instances, we reduce the memory consumption by at least two orders of magnitude. Furthermore, we show that for many graphs, only a very small fraction of far-apart pairs have to be considered for the hyperbolicity computation, explaining this drastic reduction of memory. As iterating over far-apart pairs in decreasing order without storing them explicitly is a very general tool, we believe that our approach might also be relevant to other problems.

翻译：超斜度是一个图形参数, 它表明一个图形最短距离的测量值与树度值差多少。 它用于网络、 安全和生物信息学等不同领域, 用于复杂网络的分类、 路由计划的设计以及图形算法的分析。 尽管最近取得了进展, 计算图的双向性仍然具有挑战性。 事实上, 最知名的算法具有时间复杂性 $O (n ⁇ 3. 69}) 。 这对大图来说是令人望而却步的, 而实践中最有效的算法则具有空间复杂性 $O (n ⁇ 2) 。 因此, 时间和空间是计算超偏斜度的瓶颈。 在本文中, 我们设计一个工具来计算远方的图。 一个不偏差的对面的算法( $ $, v) 仍然具有挑战性。 最远的算法是, 美元是所有最短方向的树叶子, 其根基为$1美元, 美元。 最短的算法是所有最短直径的树的叶子, $5美元。 。 。 这个概念以前用来大幅降低的算算算算算法, 里数的直数的算算算算算法, 。