Path-based strong component algorithm

In graph theory, the strongly connected components of a directed graph may be found using an algorithm that uses depth-first search in combination with two stacks, one to keep track of the vertices in the current component and the second to keep track of the current search path.[1] Versions of this algorithm have been proposed by Purdom (1970), Munro (1971), Dijkstra (1976), Cheriyan & Mehlhorn (1996), and Gabow (2000); of these, Dijkstra's version was the first to achieve linear time.[2]

Description

The algorithm performs a depth-first search of the given graph G, maintaining as it does two stacks S and P (in addition to the normal call stack for a recursive function). Stack S contains all the vertices that have not yet been assigned to a strongly connected component, in the order in which the depth-first search reaches the vertices. Stack P contains vertices that have not yet been determined to belong to different strongly connected components from each other. It also uses a counter C of the number of vertices reached so far, which it uses to compute the preorder numbers of the vertices.

When the depth-first search reaches a vertex v, the algorithm performs the following steps:

  1. Set the preorder number of v to C, and increment C.
  2. Push v onto S and also onto P.
  3. For each edge from v to a neighboring vertex w:
    • If the preorder number of w has not yet been assigned, recursively search w;
    • Otherwise, if w has not yet been assigned to a strongly connected component:
      • Repeatedly pop vertices from P until the top element of P has a preorder number less than or equal to the preorder number of w.
  4. If v is the top element of P:
    • Pop vertices from S until v has been popped, and assign the popped vertices to a new component.
    • Pop v from P.

The overall algorithm consists of a loop through the vertices of the graph, calling this recursive search on each vertex that does not yet have a preorder number assigned to it.

Related algorithms

Like this algorithm, Tarjan's strongly connected components algorithm also uses depth first search together with a stack to keep track of vertices that have not yet been assigned to a component, and moves these vertices into a new component when it finishes expanding the final vertex of its component. However, in place of the stack P, Tarjan's algorithm uses a vertex-indexed array of preorder numbers, assigned in the order that vertices are first visited in the depth-first search. The preorder array is used to keep track of when to form a new component.

Notes

References

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