Strongly connected component

In the mathematical theory of directed graphs, a graph is said to be strongly connected or diconnected if every vertex is reachable from every other vertex. The strongly connected components or diconnected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(V+E)). In the mathematical theory of directed graphs, a graph is said to be strongly connected or diconnected if every vertex is reachable from every other vertex. The strongly connected components or diconnected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(V+E)). A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair to the second, and another path exists from the second vertex to the first.In a directed graph G that may not itself be strongly connected, a pair of vertices u and v are said to be strongly connected to each other if there is a path in each direction between them. The binary relation of being strongly connected is an equivalence relation, and the induced subgraphs of its equivalence classes are called strongly connected components.Equivalently, a strongly connected component of a directed graph G is a subgraph that is strongly connected, and is maximal with this property: no additional edges or vertices from G can be included in the subgraph without breaking its property of being strongly connected. The collection of strongly connected components forms a partition of the set of vertices of G. If each strongly connected component is contracted to a single vertex, the resulting graph is a directed acyclic graph, the condensation of G. A directed graph is acyclic if and only if it has no strongly connected subgraphs with more than one vertex, because a directed cycle is strongly connected and every nontrivial strongly connected component contains at least one directed cycle. Several algorithms based on depth first search compute strongly connected components in linear time. Although Kosaraju's algorithm is conceptually simple, Tarjan's and the path-based algorithm require only one depth-first search rather than two. Previous linear-time algorithms are based on depth-first search which is generally considered hard to parallelize. Fleischer et al. in 2000 proposed a divide-and-conquer approach based on reachability queries, and such algorithms are usually called reachability-based SCC algorithms. The idea of this approach is to pick a random pivot vertex and apply forward and backward reachability queries from this vertex. The two queries partition the vertex set into 4 subsets: vertices reached by both, either one, or none of the searches. One can show that a strongly connected component has to be contained in one of the subsets. The vertex subset reached by both searches forms a strongly connected components, and the algorithm then recurses on the other 3 subsets. The expected sequential running time of this algorithm is shown to be O(n log n), a factor of O(log n) more than the classic algorithms. The parallelism comes from: (1) the reachability queries can be parallelized more easily (e.g. by a BFS, and it can be fast if the diameter of the graph is small); and (2) the independence between the subtasks in the divide-and-conquer process.This algorithm performs well on real-world graphs, but does not have theoretical guarantee on the parallelism (consider if a graph has no edges, the algorithm requires O(n) levels of recursions). Blelloch et al. in 2016 shows that if the reachability queries are applied in a random order, the cost bound of O(n log n) still holds. Furthermore, the queries then can be batched in a prefix-doubling manner (i.e. 1, 2, 4, 8 queries) and run simultaneously in one round. The overall span of this algorithm is log2 n reachability queries, which is probably the optimal parallelism that can be achieved using the reachability-based approach.