Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time

Abstract

We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs with m edges and n vertices only rather simple O(mn)-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time O(n 2 ). For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an O(m 2 / log n)-time algorithm for 2-edge strongly connected components, and thus improve over the O(mn) running time also when m = O(n). Our approach extends to k-edge and k-vertex strongly connected components for any constant k with a running time of O(n 2 log n) for k-edge-connectivity and O(n 3 ) for k-vertex-connectivity.improve upon the previous O(mn)-time algorithms for both 2eSCCs [NW93, GIL + 15a] and 2vSCCs [Jab14]. For 2eSCCs the previous upper bound stood for 20 years. Our approach immediately generalizes to computing the k-edge strongly connected components (keSCCs) and the k-vertex strongly connected components (kvSCCs). We give algorithms that, for any integral constant k > 2, compute (1) the keSCCs in time O(n 2 log n) (improving upon the previous upper bound of O(mn) [NW93]) and (2) the kvSCCs in time O(n 3 ) (improving upon the previous upper bound of O(mn 2 ) [Mak88]).Related Work. The 2-edge and 2-vertex connected components of an undirected graph can be determined in linear time [Tar72,HT73]. In directed graphs several related problems can be solved in linear time: Testing whether a graph is 2-edge or 2-vertex strongly connected [Tar76, GT85, Geo10], finding all strong bridges and strong articulation points [ILS12], and determining the 2-edge and 2-vertex strongly connected blocks [GIL + 15a, GIL + 15b]. An edge is a strong bridge and a vertex is a strong articulation point, respectively, if its removal from the graph increases the number of strongly connected components (SCCs) of the graph. Note the difference between 'blocks' and 'components' in directed graphs: In a 2-edge strongly connected block every pair of distinct vertices is 2-edge strongly connected; however, as opposed to a 2eSCC, the paths to connect the vertices in a block might use vertices that are not in the same block. Each 2eSCC is completely contained in one 2-edge strongly connected block, i.e., the 2eSCCs refine the 2-edge strongly connected blocks. In Appendix C we provide a construction that shows that knowing the 2-edge strongly connected blocks of a graph does not help in finding its 2eSCCs. The relation between blocks and components for vertex connectivity is analogous.Georgiadis et al. [GIL + 15a] and Jaberi [Jab14] described simple algorithms to compute the 2eSCCs and 2vSCCs in O(mn)-time, respectively, and posed as an open problem whether this can be improved to linear time as well. An O(mn) running time for computing the 2eSCCs was already achieved by Nagamochi and Watanabe...

Computing 2-Connected Components and Maximal 2-Connected Subgraphs in Directed Graphs: An Experimental Study

“…We thus propose a variant of this computation that alleviates the problem, which may be of independent interest, since loop nesting information is useful in a variety of applications (see, e.g., [21,25]). For the computation of the maximal 2-vertex-and 2-edgeconnected subgraphs, our experimental results reveal that, although asymptotically faster, in real-world graphs the new more sophisticated algorithms [5,12] are not competitive with the previous algorithm based on dominator tree decomposition [7]. On the other hand, we show that our carefully engineered version of the algorithm by Chechik et al [5] performs closely to the previous algorithms from [7] in real-world graphs (within a factor of 3.7, on average), and moreover in pathological worst-case graphs it performs more than two orders of magnitude faster than previous methods.…”

“…In the case of digraphs, however, only O(mn) algorithms were known (see e.g., [14,15,18,20]). It was shown only recently how to compute the 2-vertex-and 2-edge-connected components in linear time [9,10], and the best current bound for computing the maximal 2-vertex-and the 2-edge-connected subgraphs is O(min{m 3/2 , n 2 }) [5,12]. Throughout, we refer to the problems of computing the 2-vertex-and 2-edge-connected components, respectively as 2VCC and 2ECC, and to the problems of computing the maximal 2-vertex-and 2-edge-connected subgraphs, respectively as Max2VCS and Max2ECS.…”

“…Our results. In this paper we revisit the problem of computing the 2-vertex-and the 2-edge-connected components and the maximal 2-vertex-and the 2-edge-connected subgraphs of a directed graph G in practice, by taking into account the bulk of recent theoretical advances in this area (see, e.g., [5,11,12]). In particular, we explore the design space for algorithms that perform well in practice, by implementing and engineering the newly proposed algorithms.…”

“…Specifically, we present an efficient implementation of a new linear-time algorithm for computing the 2-vertex and the 2-edge-connected components of G, based on loop nesting information [11], and compare it against the algorithms based on a twolevel decomposition of G using auxiliary graphs [9,10] implemented in [7]. Then, we consider the computation of the maximal 2-vertex-and 2-edgeconnected subgraphs of G, and investigate how the recent O(n 2 )-time algorithms by Henzinger et al [12], and the recent O(m 3/2 )-time algorithms by Chechik et al [5] perform in practice. Although the asymptotic running time of those algorithms is better than O(mn), a straightforward implementation of those algorithms makes them non-competitive to the simpler O(mn) algorithms [7].…”

“…Although the asymptotic running time of those algorithms is better than O(mn), a straightforward implementation of those algorithms makes them non-competitive to the simpler O(mn) algorithms [7]. We engineer the algorithms from [5,12], providing implementations that run faster by 2-3 orders of magnitude, compared to their straightforward implementations.…”