基于边连通度的层次图分解的近优算法
A Near-optimal Algorithm for Edge Connectivity-based Hierarchical Graph Decomposition
摘要：
在图分析的许多应用的推动下，针对用户给定的$k$，计算图$G$的$k$-边连通分支（$k$-ECC）的问题最近得到了广泛的研究.本文研究了基于边连通性的图分解的层次结构的构造问题，该层次结构可紧表示图的所有可能的$k$值的$k$-ECC.这是基于每个$k$-ECC完全包含在$(k-1)$-ECC中的事实。与现有的方法进行计算，无论是在一个自下而上或自上而下的方式，我们提出了一个基于二进制搜索的框架，调用一个$k$-ECC计算算法作为一个黑盒子。令$T_{kecc}(G)$为针对特定$k$值计算$G$的所有$k$-ECC的时间复杂度。我们证明了该框架的时间复杂度为${\cal O}\big( (\log δ(G))\times T_{kecc}(G)\big)$，其中$δ(G)$是$G$的退化度，等于$G$所有子图的最小顶点度中的最大值.由于$δ(G)$对于真实世界的图来说通常很小，因此该时间复杂度最大为对数因子。
英文摘要：
Driven by many applications in graph analytics, the problem of computing $k$-edge connected components ($k$-ECCs) of a graph $G$ for a user-given $k$ has been extensively studied recently. In this paper, we investigate the problem of constructing the hierarchy of edge connectivity-based graph decomposition, which compactly represents the $k$-ECCs of a graph for all possible $k$ values. This is based on the fact that each $k$-ECC is entirely contained in a $(k-1)$-ECC. In contrast to the existing approaches that conduct the computation either in a bottom-up or a top-down manner, we propose a binary search-based framework which invokes a $k$-ECC computation algorithm as a black box. Let $T_{kecc}(G)$ be the time complexity of computing all $k$-ECCs of $G$ for a specific $k$ value. We prove that the time complexity of our framework is ${\cal O}\big( (\log δ(G))\times T_{kecc}(G)\big)$, where $δ(G)$ is the degeneracy of $G$ and equals the maximum value among the minimum vertex degrees of all subgraphs of $G$. As $δ(G)$ is typically small for real-world graphs, this time complexity is optimal up to a logarithmic factor.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
