## Class GusfieldGomoryHuCutTree<V,​E>

• Type Parameters:
V - the graph vertex type
E - the graph edge type
All Implemented Interfaces:
FlowAlgorithm<V,​E>, MaximumFlowAlgorithm<V,​E>, MinimumSTCutAlgorithm<V,​E>

public class GusfieldGomoryHuCutTree<V,​E>
extends Object
implements MaximumFlowAlgorithm<V,​E>, MinimumSTCutAlgorithm<V,​E>
This class computes a Gomory-Hu tree (GHT) using the algorithm proposed by Dan Gusfield. For a definition of GHTs, refer to: Gomory, R., Hu, T. Multi-terminal network flows. Journal of the Socieity for Industrial and Applied mathematics, 9(4), p551-570, 1961. GHTs can be used to efficiently query the maximum flows and minimum cuts for all pairs of vertices. The algorithm is described in: Gusfield, D, Very simple methods for all pairs network flow analysis. SIAM Journal on Computing, 19(1), p142-155, 1990
In an undirected graph, there exist $\frac{n(n-1)}{2}$ different vertex pairs. This class computes the maximum flow/minimum cut between each of these pairs efficiently by performing exactly $(n-1)$ minimum $s-t$ cut computations. If your application needs fewer than $n-1$ flow/cut computations, consider computing the maximum flows/minimum cuts manually through MaximumFlowAlgorithm/MinimumSTCutAlgorithm.

The runtime complexity of this class is $O((V-1)Q)$, where $Q$ is the runtime complexity of the algorithm used to compute $s-t$ cuts in the graph. By default, this class uses the PushRelabelMFImpl implementation to calculate minimum s-t cuts. This class has a runtime complexity of $O(V^3)$, resulting in a $O(V^4)$ runtime complexity for the overall algorithm.

Note: this class performs calculations in a lazy manner. The GHT is not calculated until the first invocation of getMaximumFlowValue(Object, Object) or getGomoryHuTree(). Moreover, this class only calculates the value of the maximum flow between a source-destination pair; it does not calculate the corresponding flow per edge. If you need to know the exact flow through an edge, use one of the alternative MaximumFlowAlgorithm implementations.

In contrast to an Equivalent Flow Tree (GusfieldEquivalentFlowTree), Gomory-Hu trees also provide all minimum cuts for all pairs of vertices!

This class does not support changes to the underlying graph. The behavior of this class is undefined when the graph is modified after instantiating this class.

Author:
Joris Kinable
• ### Constructor Detail

• #### GusfieldGomoryHuCutTree

public GusfieldGomoryHuCutTree​(Graph<V,​E> network)
Constructs a new GusfieldEquivalentFlowTree instance.
Parameters:
network - input graph
• #### GusfieldGomoryHuCutTree

public GusfieldGomoryHuCutTree​(Graph<V,​E> network,
double epsilon)
Constructs a new GusfieldEquivalentFlowTree instance.
Parameters:
network - input graph
epsilon - precision
• #### GusfieldGomoryHuCutTree

public GusfieldGomoryHuCutTree​(Graph<V,​E> network,
MinimumSTCutAlgorithm<V,​E> minimumSTCutAlgorithm)
Constructs a new GusfieldEquivalentFlowTree instance.
Parameters:
network - input graph
minimumSTCutAlgorithm - algorithm used to compute the minimum s-t cuts
• ### Method Detail

• #### getGomoryHuTree

public SimpleWeightedGraph<V,​DefaultWeightedEdge> getGomoryHuTree()
Returns the Gomory-Hu Tree as an actual tree (graph). Note that this tree is not necessarily unique. The edge weights represent the flow values/cut weights. This method runs in $O(n)$ time.
Returns:
Gomory-Hu Tree
• #### getMaximumFlowValue

public double getMaximumFlowValue​(V source,
V sink)
Returns the Maximum flow between source and sink. The algorithm is only executed once; successive invocations of this method will return in $O(1)$ time.
Specified by:
getMaximumFlowValue in interface MaximumFlowAlgorithm<V,​E>
Parameters:
source - source vertex
sink - sink vertex
Returns:
the Maximum flow between source and sink.
• #### getFlowDirection

public V getFlowDirection​(E e)
Unsupported operation
Specified by:
getFlowDirection in interface FlowAlgorithm<V,​E>
Parameters:
e - edge
Returns:
nothing
• #### calculateMinCut

public double calculateMinCut​(V source,
V sink)
Description copied from interface: MinimumSTCutAlgorithm
Computes a minimum capacity $s-t$ cut.
Specified by:
calculateMinCut in interface MinimumSTCutAlgorithm<V,​E>
Parameters:
source - s
sink - t
Returns:
capacity of the cut
• #### calculateMinCut

public double calculateMinCut()
Calculates the minimum cut in the graph, that is, the minimum cut over all $s-t$ pairs. The same result can be obtained with the StoerWagnerMinimumCut implementation. After invoking this method, the source/sink partitions corresponding to the minimum cut can be queried through the getSourcePartition() and getSinkPartition() methods. After computing the Gomory-Hu Cut tree, this method runs in $O(N)$ time.
Returns:
weight of the minimum cut in the graph
• #### getCutEdges

public Set<E> getCutEdges()
Description copied from interface: MinimumSTCutAlgorithm
Returns the set of edges which run from $S$ to $T$, in the $s-t$ cut obtained after the last invocation of MinimumSTCutAlgorithm.calculateMinCut(Object, Object) In case of a directed graph, only the edges with their tail in $S$ and their head in $T$ are returned. In cased of a undirected graph, all edges with one endpoint in $S$ and one endpoint in $T$ are returned.
Specified by:
getCutEdges in interface MinimumSTCutAlgorithm<V,​E>
Returns:
set of edges which run from $S$ to $T$