Repeated depth-first search: Difference between revisions
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'''Type of algorithm:''' loop | '''Type of algorithm:''' loop | ||
'''Additional output:''' cf. [[Depth-first search|DFS]]. | '''Additional output:''' the same as for [[Depth-first search|DFS]] with one modification: Instead of an arborescence, the algorithm returns a [[Basic graph definitions#Forests, trees, branchings, arborescences|branching]] that is a [[Basic graph definitions#Subgraphs|spanning subgraph]] of <math>G</math> (cf. [[Depth-first search|DFS]]). | ||
'''Specific characteristic:''' | '''Specific characteristic:''' | ||
Let <math>v,w\in V</math> such that <math>v</math> is seen before <math>w</math>. If there is a path from <math>v</math> to <math>w</math>, <math>w</math> is finished prior to <math>v</math> | analogous to the parenthetical output order of [[Depth-first search|DFS]]: Let <math>v,w\in V</math> such that <math>v</math> is seen before <math>w</math>. If there is a path from <math>v</math> to <math>w</math>, <math>w</math> is finished prior to <math>v</math>. | ||
== Abstract View == | == Abstract View == | ||
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While there are nodes not yet seen: | While there are nodes not yet seen: | ||
# Select a start node <math>s</math> from the unseen nodes. | # Select a start node <math>s</math> from the unseen nodes. | ||
# Apply a [[Depth-first search|DFS]] starting at <math>s</math>, | # Apply a [[Depth-first search|DFS]] starting at <math>s</math>, parenthetical output order. | ||
== 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 == | ||
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'''Statement:''' The asymptotic complexity is in <math>\Theta(|V|+|A|)</math>. | '''Statement:''' The asymptotic complexity is in <math>\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]]. | '''Proof:''' Follows immediately from the linear asymptotic complexity of [[Depth-first search|DFS]] and the fact that each node and each arc is touched in only one [[DFS]]. | ||
Latest revision as of 07:28, 3 November 2014
General information
Algorithmic problem: Exhaustive graph traversal
Type of algorithm: loop
Additional output: the same as for DFS with one modification: Instead of an arborescence, the algorithm returns a branching that is a spanning subgraph of [math]\displaystyle{ G }[/math] (cf. DFS).
Specific characteristic: analogous to the parenthetical output order of DFS: 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].
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], parenthetical 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.