Push–relabel maximum flow algorithm

The Goldberg - Tarjan algorithm, also push- relabel algorithm, called an algorithm from graph theory for computing a maximum flow in a network. It was developed by Andrew Goldberg and Robert Endre Tarjan and published in 1988.

The algorithm

Hereinafter referred to the network the directed graph, the capacity function ( the capacitance indicates an edge ), the node, from which the flow starts, and the destination node of the river. denotes the node set of the graph, the edge set and the set of edges leaving the node.

The algorithm computes a maximum st- flow by modifying a river until he is a st- flow. When this occurs, the resulting st- flow is maximum. The algorithm further uses a distance marker, ie with a function, and for all edges. One edge of the Residualgraphen called allowed if they met.

The algorithm works as follows:

A modification of the river in step 3 is also called " push", a modification of the distance marker " relabel ". Therefore, the name derives push relabel algorithm.

Is a maximum at the end of S- t- flow. For each time the node is the only source and the algorithm continues until the node is the only sink. As always remains a distance marker and thus satisfies the properties described above, it is ensured that the Residualgraphen the node is never reachable from the source. This ensures that the computed by the algorithm st- flow is actually a maximum st- flow.

Term

So generally as indicated above, the algorithm has a running time of.

Whatever one chooses in step 3 of the algorithm, an active node for the active among all nodes in the distance mark the maximum value ( ie with a ) can be a term of prove. However, in the implementation, this requires that for each value of up to and a list of active nodes is performed with (that is, for each value that can take the theoretical), in addition to the respective current maximum of the amount of the active nodes have to be tracked. This is necessary so that in each iteration of the loop, an active node can be selected with maximum duration without loss.

With more sophisticated implementations can also be a term of

And

Reach. Herein, the maximum value of capacitance function.

Swell

  • Dieter Jungnickel: Graphs, Networks and Algorithms, Springer (1998) ISBN 978-3-540-72779-8
  • Bernhard Korte, Jens Vygen: Combinatorial Optimization: Theory and Algorithms. Translated from English by Rabe von Randow. Springer -Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76918-7
  • Alexander Schrijver: Combinatorial Optimization: Polyhedra and Efficiency, Springer- Verlag, 2003, ISBN 3-540-44389-4
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