Search results

...</math>-path in <math>G</math> with respect to <math>\ell</math> among all paths that have at most <math>|V|</math> arcs. # The last arc (pointing to <math>w</math>) of one of these paths.

== Negative cost factors == The case that some arcs have negative cost factors, may be transformed as follows into the above version, in whic

File:V12 negative Kostenfaktoren Beispiel.jpg|negative Kostenfaktoren Beispiel File:V12 negative Kostenfaktoren konzeptionell.jpg|negative Kostenfaktoren konzeptionell

# The '''length''' of an [[Basic graph definitions#Paths|ordinary path]] (incl. [[Basic graph definitions#Cycles|ordinary cycles]]) ...ons#Cycles|generalized cycles]]) is either defined identically to ordinary paths or, alternatively, the lengths of the backward arcs are not added but subtr

...''' There is no more [[Basics of shortest paths#Path lengths and distances|negative cycle]] in the residual network. # If all arcs <math>(s,v)</math> are [[Basic flow definitions#Flow-augmenting paths and saturated arcs|saturated]], return the restriction of this flow to the

...ts of the invariant of the [[Successive shortest paths|successive shortest paths]] algorithm. ...ep|induction step]] of the [[Successive shortest paths|successive shortest paths algorithm]]. The essential modification is that <math>c</math> is replaced

# There is no negative cycle in the [[Basic flow definitions#Residual network|residual network]]  ...[Dijkstra|Dijkstra's]] are not an option. However, a [[Successive shortest paths with reduced costs|variant]] of the algorithm works on a nonnegative variat

* [[Shortest Paths Problems]] ** [[All pairs shortest paths]]

...enting paths and saturated arcs|flow-augmenting paths]] <math>(s,t)</math>-paths, always choose one with smallest number of arcs. Consequently, a [[Breadth- .../math> according to [[Ford-Fulkerson]] does not create new flow-augmenting paths of length <math>k</math> or less than <math>k</math>.

[[Category:All Pairs Shortest Paths]] [[Category:Shortest Paths Problems]]

...d by <math>f(a)\geq\ell(a)</math>. In particular, <math>f(a)</math> may be negative if <math>\ell(a)</math> is so. == Flow-augmenting paths and saturated arcs ==