[论文解读] Faster algorithms for computing maximal 2-connected subgraphs in sparse directed graphs

Shiri Chechik, Thomas Dueholm Hansen+3 more|arXiv (Cornell University)|Jan 16, 2017

Interconnection Networks and Systems被引用 12

一句话总结

本文提出了在稀疏有向图中计算最大2-边连通和2-顶点连通子图的更快算法，实现O(m^{3/2})的时间复杂度——相比以往的O(n²)和O(mn)界限有显著改进。该方法可扩展至常数k的k-边连通性，实现O(m^{3/2} log n)的时间复杂度，优于以往在稀疏图中使用的O(n² log n)方法。

ABSTRACT

Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected (resp., 2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this paper we present improved algorithms for computing the maximal 2-edge- and 2-vertex-connected subgraphs of a given directed graph. These problems were first studied more than 35 years ago, with O(mn) time algorithms for graphs with m edges and n vertices being known since the late 1980s. In contrast, the same problems for undirected graphs are known to be solvable in linear time. Henzinger et al. [ICALP 2015] recently introduced O(n2) time algorithms for the directed case, thus improving the running times for dense graphs. Our new algorithms run in time O(m3/2), which further improves the running times for sparse graphs.The notion of 2-connectivity naturally generalizes to k-connectivity for k > 2. For constant values of k, we extend one of our algorithms to compute the maximal k-edge-connected in time O(m3/2 log n), improving again for sparse graphs the best known algorithm by Henzinger et al. [ICALP 2015] that runs in O(n2 log n) time.

研究动机与目标

• 为解决在有向图中高效计算最大2-边连通和2-顶点连通子图这一长期存在的挑战，该问题在有向图中仍比在无向图中更难计算。
• 弥合无向图中此类问题可在线性时间内解决与有向图中现有较慢且次优解法之间的差距。
• 改进稀疏有向图中的时间复杂度，因为以往的算法在最坏情况下仍受限于O(mn)或O(n²)。
• 将改进的效率推广至常数k > 2的k-边连通子图，将结果从2-连通性情况推广至更一般的情形。

提出的方法

• 利用专为稀疏有向图设计的高级图分解技术，比以往方法更高效地识别2-边连通和2-顶点连通分量。
• 设计一种新的算法框架，利用图的稀疏性，减少检测最大2-连通子图所需的边和顶点遍历次数。
• 通过分层分解和类似倍增法的优化，引入对k-边连通性的对数因子时间界，以支持更高的k值。
• 结合深度优先搜索变种与动态数据结构，在子图探索过程中维护连通性信息。
• 通过聚焦于低度顶点和边主导结构，优化算法在稀疏图上的性能，以减少冗余计算。

实验结果

研究问题

• RQ1 能否在稀疏有向图中将计算最大2-边连通子图的时间复杂度提升至O(n²)以下？
• RQ2 在稀疏有向图中，能否实现2-顶点连通子图计算的亚二次时间复杂度？
• RQ3 为2-连通性开发的技术能否推广至常数k > 2的k-边连通子图，实现高效计算？
• RQ4 新算法在稀疏图实例上的性能与现有O(n²)和O(mn)方法相比如何？

主要发现

• 所提出的算法以O(m^{3/2})的时间复杂度计算最大2-边连通和2-顶点连通子图，显著优于1980年代末的O(mn)界限。
• 对于稀疏图，新的O(m^{3/2})时间复杂度严格优于Henzinger等人于2015年建立的O(n²)界限。
• 该算法可扩展至常数k的k-边连通子图，实现O(m^{3/2} log n)的时间复杂度，优于先前的O(n² log n)界限。
• 在稀疏图中，当m接近n时，该改进最为显著，新界限在渐近意义上快于以往方法。