V- Vertex type
E- Edge type
public class DulmageMendelsohnDecomposition<V,E> extends Object
This class computes a Dulmage-Mendelsohn Decomposition of a bipartite graph. A Dulmage–Mendelsohn decomposition is a partition of the vertices of a bipartite graph into subsets, with the property that two adjacent vertices belong to the same subset if and only if they are paired with each other in a perfect matching of the graph. This particular implementation is capable of computing both a coarse and a fine Dulmage-Mendelsohn Decomposition.
The Dulmage-Mendelsohn Decomposition is based on a maximum-matching of the graph $G$. This implementation uses the Hopcroft-Karp maximum matching algorithm by default.
A coarse Dulmage-Mendelsohn Decomposition is a partitioning into three subsets. Where $D$ is the set of vertices in G that are not matched in the maximum matching of $G$, these subsets are:
A fine Dulmage-Mendelsohn Decomposition further partitions the remaining vertices into strongly-connected sets. This implementation uses Kosaraju's algorithm for the strong-connectivity analysis.
The Dulmage-Mendelsohn Decomposition was introduced in:
Dulmage, A.L., Mendelsohn, N.S. Coverings of bipartitegraphs, Canadian J. Math., 10, 517-534, 1958.
The implementation of this class is based on:
Bunus P., Fritzson P., Methods for Structural Analysis and Debugging of Modelica Models, 2nd International Modelica Conference 2002
The runtime complexity of this class is $O(V + E)$.
|Modifier and Type||Class and Description|
The output of a decomposition operation
|Constructor and Description|
Construct the algorithm for a given bipartite graph $G=(V_1,V_2,E)$ and it's partitions $V_1$ and $V_2$, where $V_1\cap V_2=\emptyset$.
|Modifier and Type||Method and Description|
Perform the decomposition, using a pre-calculated bipartite matching
Perform the decomposition, using the Hopcroft-Karp maximum-cardinality matching algorithm to perform the matching.
public DulmageMendelsohnDecomposition(Graph<V,E> graph, Set<V> partition1, Set<V> partition2)
graph- bipartite graph
partition1- the first partition, $V_1$, of vertices in the bipartite graph
partition2- the second partition, $V_2$, of vertices in the bipartite graph
public DulmageMendelsohnDecomposition.Decomposition<V,E> getDecomposition(boolean fine)
fine- true if the fine decomposition is required, false if the coarse decomposition is required
public DulmageMendelsohnDecomposition.Decomposition<V,E> decompose(MatchingAlgorithm.Matching<V,E> matching, boolean fine)
matching- the matching from a
fine- true if the fine decomposition is required
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