Repeated depth-first search: Difference between revisions
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== Correctness == | == Correctness == | ||
Obviously, all nodes are finished eventually. For two nodes <math>v,w\in V</math> that are finished in the same DFS, the specific characteristic follows from the fact that DFS fulfills it. So assume <math>v</math> is seen in an earlier DFS than <math>w</math>. Since <math>v</math> is also earlier finished than <math>w</math>, the specific characteristic is fulfilled for <math>v</math> and <math>w</math> unless there is a path from <math>v</math> to <math>w</math>. However, if such a path existed, <math>w</math> had been seen in the same DFS as <math>v</math> | Obviously, all nodes are finished eventually. For two nodes <math>v,w\in V</math> that are finished in the same DFS, the specific characteristic follows from the fact that DFS fulfills it. So assume <math>v</math> is seen in an earlier DFS than <math>w</math>. Since <math>v</math> is also earlier finished than <math>w</math>, the specific characteristic is fulfilled for <math>v</math> and <math>w</math> unless there is a path from <math>v</math> to <math>w</math>. However, if such a path existed, <math>w</math> had been seen in the same DFS as <math>v</math>. | ||
== Complexity == | == Complexity == | ||
Revision as of 12:24, 10 October 2014
General information
Algorithmic problem: Exhaustive graph traversal
Type of algorithm: loop
Additional output: cf. DFS.
Specific characteristic: Let [math]\displaystyle{ v,w\in V }[/math] such that [math]\displaystyle{ v }[/math] is seen before [math]\displaystyle{ w }[/math]. If there is a path from [math]\displaystyle{ v }[/math] to [math]\displaystyle{ w }[/math], [math]\displaystyle{ w }[/math] is finished prior to [math]\displaystyle{ v }[/math] (cf. DFS).
Abstract View
While there are nodes not yet seen:
- Select a start node [math]\displaystyle{ s }[/math] from the unseen nodes.
- Apply a DFS starting at [math]\displaystyle{ s }[/math], second type of output order.
Correctness
Obviously, all nodes are finished eventually. For two nodes [math]\displaystyle{ v,w\in V }[/math] that are finished in the same DFS, the specific characteristic follows from the fact that DFS fulfills it. So assume [math]\displaystyle{ v }[/math] is seen in an earlier DFS than [math]\displaystyle{ w }[/math]. Since [math]\displaystyle{ v }[/math] is also earlier finished than [math]\displaystyle{ w }[/math], the specific characteristic is fulfilled for [math]\displaystyle{ v }[/math] and [math]\displaystyle{ w }[/math] unless there is a path from [math]\displaystyle{ v }[/math] to [math]\displaystyle{ w }[/math]. However, if such a path existed, [math]\displaystyle{ w }[/math] had been seen in the same DFS as [math]\displaystyle{ v }[/math].
Complexity
Statement: The asymptotic complexity is in [math]\displaystyle{ \Theta(|V|+|A|) }[/math].
Proof: Follows immediately from the linear asymptotic complexity of DFS and the fact that each node and each arc is touched in only one DFS.