Strongly connected components: Difference between revisions

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== Definition ==
== Basic definitions ==
[[File:Scc.png|300px|thumb|right|Graph with marked strongly connected components]]
 
Let <math>G=(V,A)</math> be a [[Basic graph definitions|directed graph]]. Consider the following equivalence relation on the nodes: <math>v\in V</math> and <math>w\in V</math> are equivalent if, and only if, there is a path from <math>v</math> to <math>w</math> and a path from <math>w</math> to <math>v</math> in <math>G</math>. The equivalence classes are called the '''strongly connected components (SCC)''' of <math>G</math>.
# [[Basic graph definitions]]


== Input ==
== Input ==
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== Output ==
== Output ==


A set of sets of nodes. Each set of nodes contains exactly the nodes of one SCC. The correspndence between SCC and sets of nodes is one-to-one.
A set of sets of nodes. Each set of nodes contains exactly the nodes of one [[Basic graph definitions#Connectedness|SCC]]. The correspondence between the [[Basic graph definitions#Connectedness|SCC]] and these sets of nodes is one-to-one.
 
== Pseudocode ==
 
====STRONGLY-CONNECTED-COMPONENTS(''D'')====
 
<code>
STRONGLY-CONNECTED-COMPONENTS(''D'')
1 call '''DFS'''(''D'') to compute finishing times ''f''[v] for each vertex ''v'' &isin; ''V''
2 compute ''D''<sup>''T''</sup> (w.r.t. step 3)
3 call '''DFS'''(''D''<sup>''T''</sup>), but in the main loop of '''DFS''', consider the vertices in order of decreasing ''f''[v] as computed in step 1
4 output the vertices of each tree in the '''DFS''' forest of step 3 as a separate strongly connected component
</code>


== Known algorithms ==
== Known algorithms ==


# [[Kosaraju]]
# [[Kosaraju]]
== Further information ==
''STRONGLY-CONNECTED-COMPONENTS'' runs in linear time.

Latest revision as of 19:07, 9 November 2014

Basic definitions

  1. Basic graph definitions

Input

A directed graph [math]\displaystyle{ G=(V,A) }[/math].

Output

A set of sets of nodes. Each set of nodes contains exactly the nodes of one SCC. The correspondence between the SCC and these sets of nodes is one-to-one.

Known algorithms

  1. Kosaraju
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