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Concept
Tarjan's strongly connected components algorithm
Summary
Tarjan's strongly connected components algorithm is an algorithm in graph theory for finding the strongly connected components (SCCs) of a directed graph. It runs in linear time, matching the time bound for alternative methods including Kosaraju's algorithm and the path-based strong component algorithm. The algorithm is named for its inventor, Robert Tarjan.
The algorithm takes a directed graph as input, and produces a partition of the graph's vertices into the graph's strongly connected components. Each vertex of the graph appears in exactly one of the strongly connected components. Any vertex that is not on a directed cycle forms a strongly connected component all by itself: for example, a vertex whose in-degree or out-degree is 0, or any vertex of an acyclic graph.
The basic idea of the algorithm is this: a depth-first search (DFS) begins from an arbitrary start node (and subsequent depth-first searches are conducted on any nodes that have not yet been found). As usual with depth-first search, the search visits every node of the graph exactly once, declining to revisit any node that has already been visited. Thus, the collection of search trees is a spanning forest of the graph. The strongly connected components will be recovered as certain subtrees of this forest. The roots of these subtrees are called the "roots" of the strongly connected components. Any node of a strongly connected component might serve as a root, if it happens to be the first node of a component that is discovered by search.
Nodes are placed on a stack in the order in which they are visited. When the depth-first search recursively visits a node v and its descendants, those nodes are not all necessarily popped from the stack when this recursive call returns. The crucial invariant property is that a node remains on the stack after it has been visited if and only if there exists a path in the input graph from it to some node earlier on the stack. In other words, it means that in the DFS a node would be only removed from the stack after all its connected paths have been traversed.
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