Strong Connectivity in Directed Graphs under Failures, with Application

In this paper, we investigate some basic connectivity problems in directed graphs (digraphs). Let $G$ be a digraph with $m$ edges and $n$ vertices, and let $G\setminus e$ be the digraph obtained after deleting edge $e$ from $G$. As a first result, we show how to compute in $O(m+n)$ worst-case time: $(i)$ The total number of strongly connected components in $G\setminus e$, for all edges $e$ in $G$. $(ii)$ The size of the largest and of the smallest strongly connected components in $G\setminus e$, for all edges $e$ in $G$. Let $G$ be strongly connected. We say that edge $e$ separates two vertices $x$ and $y$, if $x$ and $y$ are no longer strongly connected in $G\setminus e$. As a second set of results, we 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: $(i)$ Report in $O(n)$ worst-case time all the strongly connected components of $G\setminus e$, for a query edge $e$. $(ii)$ Test whether an edge separates two query vertices in $O(1)$ worst-case time. $(iii)$ Report all edges that separate two query vertices in optimal worst-case time, i.e., in time $O(k)$, where $k$ is the number of separating edges. (For $k=0$, the time is $O(1)$). All of the above results extend to vertex failures. All our bounds are tight and are obtained with a common algorithmic framework, based on a novel compact representation of the decompositions induced by the $1$-connectivity (i.e., $1$-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.Comment: An extended abstract of this work appeared in the SODA 201

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

We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple $O(m n)$-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 $O(m n)$ 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 $O(n^2 \log^2 n)$ for edges and $O(n^3)$ for vertices

Dominating sets and ego-centered decompositions in social networks

Our aim here is to address the problem of decomposing a whole network into a minimal number of ego-centered subnetworks. For this purpose, the network egos are picked out as the members of a minimum dominating set of the network. However, to find such an efficient dominating ego-centered construction, we need to be able to detect all the minimum dominating sets and to compare all the corresponding dominating ego-centered decompositions of the network. To find all the minimum dominating sets of the network, we are developing a computational heuristic, which is based on the partition of the set of nodes of a graph into three subsets, the always dominant vertices, the possible dominant vertices and the never dominant vertices, when the domination number of the network is known. To compare the ensuing dominating ego-centered decompositions of the network, we are introducing a number of structural measures that count the number of nodes and links inside and across the ego-centered subnetworks. Furthermore, we are applying the techniques of graph domination and ego=centered decomposition for six empirical social networks.Comment: 17 pages, 7 figure