path-based strong component algorithm
{{Short description|Graph 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.{{harvtxt|Sedgewick|2004}}. Versions of this algorithm have been proposed by {{harvtxt|Purdom|1970}}, {{harvtxt|Munro|1971}}, {{harvtxt|Dijkstra|1976}}, {{harvtxt|Cheriyan|Mehlhorn|1996}}, and {{harvtxt|Gabow|2000}}; of these, Dijkstra's version was the first to achieve linear time.[http://www.cs.colorado.edu/~hal/Papers/DFS/pbDFShistory.html History of Path-based DFS for Strong Components], Harold N. Gabow, accessed 2012-04-24.
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:
- Set the preorder number of v to C, and increment C.
- Push v onto S and also onto P.
- For each edge from v to a neighboring vertex w:
- * If the preorder number of w has not yet been assigned (the edge is a tree edge), recursively search w;
- *Otherwise, if w has not yet been assigned to a strongly connected component (the edge is a forward/back/cross edge):
- **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.
- 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
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References
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| last1 = Cheriyan | first1 = J.
| last2 = Mehlhorn | first2 = K. | author2-link = Kurt Mehlhorn
| doi = 10.1007/BF01940880
| journal = Algorithmica
| pages = 521–549
| title = Algorithms for dense graphs and networks on the random access computer
| volume = 15
| year = 1996| issue = 6
| s2cid = 8930091
}}.
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| location = NJ
| at = Ch. 25
| publisher = Prentice Hall
| title = A Discipline of Programming
| year = 1976}}.
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| doi = 10.1016/S0020-0190(00)00051-X
| issue = 3–4
| journal = Information Processing Letters
| mr = 1761551
| pages = 107–114
| title = Path-based depth-first search for strong and biconnected components
| volume = 74
| year = 2000
| url = https://www.cs.princeton.edu/courses/archive/spr04/cos423/handouts/path%20based...pdf}}.
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| pages = 56–58
| title = Efficient determination of the transitive closure of a directed graph
| volume = 1
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| title = A transitive closure algorithm
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| year = 1970
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| url = http://digital.library.wisc.edu/1793/57514
}}.
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| contribution = 19.8 Strong Components in Digraphs
| edition = 3rd
| location = Cambridge MA
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| publisher = Addison-Wesley
| title = Algorithms in Java, Part 5 – Graph Algorithms
| year = 2004}}.