what is strongly connected components? | strongly connected components |
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Strongly connected components (SCCs) are a fundamental concept in graph theory and are useful in many applications. In a directed graph, an SCC is a subset of vertices where there is a path from each vertex to every other vertex in the subset.
Depth-First Search (DFS) is a common algorithm used to find SCCs in a directed graph. The idea is to perform a DFS traversal of the graph and mark the vertices in the order they are visited. After the first DFS traversal, the graph is transposed, meaning that all the edges are reversed. A second DFS traversal is performed on the transposed graph, but the vertices are visited in reverse order of their finishing times from the first traversal.
During the second DFS traversal, the vertices are explored in a way that allows us to identify SCCs. If a vertex is encountered that has already been marked as visited in the first traversal, then it must be part of an SCC. The SCC is then identified by exploring all the vertices that can be reached from the current vertex via a path consisting of only marked vertices.
Once all the SCCs have been identified, they can be represented as separate subgraphs of the original graph. Each SCC will have at least one cycle, which is a path that starts and ends at the same vertex.
In summary, DFS can be used to find SCCs in a directed graph by performing two DFS traversals - one on the original graph and one on the transposed graph. During the second traversal, SCCs can be identified by exploring the vertices in reverse order of their finishing times from the first traversal.
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