Repeated depth-first search

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:

1. Select a start node [math]\displaystyle{ s }[/math] from the unseen nodes.
2. 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.