Strong Connectivity in Directed Graphs under Failures, with Applications

Abstract

Let G be a directed graph (digraph) with m edges and n vertices, and let G \ e (resp., G \ v) be the digraph obtained after deleting edge e (resp., vertex v) from G. We show how to compute in O(m + n) worst-case time:• The total number of strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G.• The size of the largest and of the smallest strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G. Let G be strongly connected. We say that edge e (resp., vertex v) separates two vertices x and y, if x and y are no longer strongly connected in G \ e (resp., G \ v). We also show how to build in O(m+n) time O(n)-space data structures that can answer in optimal time the following basic connectivity queries on digraphs:• Report in O(n) worst-case time all the strongly connected components of G \ e (resp., G \ v), for a query edge e (resp., vertex v).• Test whether an edge or a vertex separates two query vertices in O(1) worst-case time.• Report all edges (resp., vertices) that separate two query vertices in optimal worst-case time, i.e., in time O(k), where k is the number of separating edges (resp., separating vertices). (For k = 0, the time is O(1)). All our bounds are tight and are obtained with a common algorithmic framework, based on a novel compact representation of the decompositions induced by 1-edge and 1-vertex cuts in digraphs, which might be of independent interest. With the help of our data structures we can design efficient algorithms for several other connectivity problems on digraphs and we can also obtain in linear time a strongly connected spanning subgraph of G with O(n) edges that maintains the 1-connectivity cuts of G and the decompositions induced by those cuts.

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

“…To that end, we conducted an extensive empirical analysis, using realworld and synthetic graphs, in order to asses the different aspects of each approach. Our main findings suggest that for the computation of the 2-vertex-and the 2-edge-connected components the new loop-nesting-based algorithms from [11] perform substantially faster compared to previous approaches, and moreover use significantly less memory, especially in sparse graphs. This makes them suitable for large-scale real-world graphs, which are known to be inherently sparse.…”

“…For the computation of the 2-vertexand the 2-edge-connected components, our experiments indicate that the new loop-nesting-based algorithms from [11] are not only substantially faster in practice than previous state-of-the-art implementations, but they are also much more efficient in terms of memory usage, especially for sparse graphs. This makes them particularly suitable to large-scale realworld graphs, which are known to be inherently sparse.…”

“…Loop nesting forests. Very recently, [11] presented new linear-time algorithms to compute the 2-edge-and 2-vertex-connected components, based on loop nesting forests. We refer to these algorithms as 2E-LNF and 2V-LNF, respectively.…”

“…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.…”

“…We do this by comparing the new implementations against the fastest implementations in [7] and carry out a thorough empirical analysis that highlights the merits and weaknesses of each technique. 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.…”