All pairs shortest paths: Difference between revisions

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'''Remark:'''
'''Remark:'''
Obviously, if no <math>(v,w)</math>-path meets any negative cycle, a shortest path exists, and at least one shortest path is [[Basic graph definitions#Paths|simple]]  (because such a path may only contain cycles of zero total length, which may be removed). This path has at most <math>|V|-1</math> arcs.
Obviously, if no <math>(v,w)</math>-path meets any negative cycle, a shortest path exists, and at least one shortest path is [[Basic graph definitions#Paths|simple]]  (because such a path may only contain cycles of zero total length, which may be removed). This path has at most <math>|V|-1</math> arcs. On the other hand, if there are negative cycles, there is, evidently, at least one simple negative cycle. A simple cycle hast at most <math>|V|</math> arcs. Therefore, the distance from <math>v</math> in the output is negative. Of course, the case <math>v=w</math> is included.
 
On the other hand, if there are negative cycles, there is, evidently, at least one simple negative cycle. A simple cycle hast at most <math>|V|</math> arcs. Therefore, the distance from <math>v</math> in the output is negative. Of course, the case <math>v=w</math> is included.


== Complexity ==
== Complexity ==

Revision as of 10:54, 22 October 2014

Input

  1. A directed graph [math]\displaystyle{ G = (V,A) }[/math]
  2. An arc length [math]\displaystyle{ l(a) \in \mathbb{R} }[/math] for each arc [math]\displaystyle{ a \in A }[/math]

Output

For each pair [math]\displaystyle{ (v,w) \in A }[/math] with [math]\displaystyle{ v,w \in V }[/math], the length [math]\displaystyle{ \Delta(v,w) }[/math] of a shortest [math]\displaystyle{ (v,w) }[/math]-path in [math]\displaystyle{ G }[/math] with respect to [math]\displaystyle{ \ell }[/math] among all paths that have at most [math]\displaystyle{ |V| }[/math] arcs.

Remark: Obviously, if no [math]\displaystyle{ (v,w) }[/math]-path meets any negative cycle, a shortest path exists, and at least one shortest path is simple (because such a path may only contain cycles of zero total length, which may be removed). This path has at most [math]\displaystyle{ |V|-1 }[/math] arcs. On the other hand, if there are negative cycles, there is, evidently, at least one simple negative cycle. A simple cycle hast at most [math]\displaystyle{ |V| }[/math] arcs. Therefore, the distance from [math]\displaystyle{ v }[/math] in the output is negative. Of course, the case [math]\displaystyle{ v=w }[/math] is included.

Complexity

Polynomial

Known algorithms

  1. Floyd-Warshall
  2. Bellman-Ford
  3. Shortest paths by repeated squaring (variant of Bellman-Ford)