All pairs shortest paths: Difference between revisions
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== Input == | == Input == | ||
# A directed Graph <math>G = (V,A)</math> | # A [[directed Graph]] <math>G = (V,A)</math> | ||
# An arc length <math>l(a) \in \mathbb{R}</math> for each arc <math>a \in A</math> | # An arc length <math>l(a) \in \mathbb{R}</math> for each arc <math>a \in A</math> | ||
Revision as of 14:02, 4 October 2014
Input
- A directed Graph [math]\displaystyle{ G = (V,A) }[/math]
- An arc length [math]\displaystyle{ l(a) \in \mathbb{R} }[/math] for each arc [math]\displaystyle{ a \in A }[/math]
Ouptut
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{ l }[/math].
Complexity
Polynomial
Known algorithms
- Floyd-Warshall
- Bellman-Ford
- Shortest oaths by repeated squaring (varian of Bellman-Ford)