Class GoldbergMaximumDensitySubgraphAlgorithmBase<V,E>

Type Parameters:
V - Type of vertices
E - Type of edges
All Implemented Interfaces:
Direct Known Subclasses:
GoldbergMaximumDensitySubgraphAlgorithm, GoldbergMaximumDensitySubgraphAlgorithmNodeWeightPerEdgeWeight, GoldbergMaximumDensitySubgraphAlgorithmNodeWeights

public abstract class GoldbergMaximumDensitySubgraphAlgorithmBase<V,E> extends Object implements MaximumDensitySubgraphAlgorithm<V,E>
This abstract base class computes a maximum density subgraph based on the algorithm described by Andrew Vladislav Goldberg in Finding Maximum Density Subgraphs, 1984, University of Berkley. Each subclass decides which concrete definition of density is used by implementing getEdgeWeightFromSourceToVertex(Object) and getEdgeWeightFromVertexToSink(Object) as proposed in the paper. After the computation the density is computed using MaximumDensitySubgraphAlgorithm.getDensity().
The basic concept is to construct a network that can be used to compute the maximum density subgraph using a binary search approach.

In the simplest case of an unweighted graph $G=(V,E)$ the density of $G$ can be defined to be \[\frac{\left|{E}\right|}{\left|{V}\right|}\], where a directed graph can be considered as undirected. Therefore it is in this case equal to half the average vertex degree. This variant is implemented in GoldbergMaximumDensitySubgraphAlgorithm; because the following math translates directly to other variants the full math is only fully explained once.

The idea of the algorithm is to construct a network based on the input graph $G=(V,E)$ and some guess $g$ for the density. This network $N=(V_N, E_N)$ is constructed as follows: \[V_N=V\cup {s,t}\] \[E_N=\{(i,j)| \{i,j\} \in E\} \cup \{(s,i)| i\in V\} \cup \{(i,t)| i \in V\}\]
Additionally one defines the following weights for the network: \[c_{ij}=1 \forall \{i,j\}\in E\] \[c_{si}=m \forall i \in V\] \[c_{it}=m+2g-d_i \forall i \in V\] where $m=\left|{E}\right|$ and $d_i$ is the degree of vertex $i$.
As seen later these weights depend on the definition of the density. Therefore these weights and the following applies to the definition of density from above. Definitions suitable for other cases in can be found in the corresponding subclasses.

Using this network one can show some important properties, that are essential for the algorithm to work. The capacity of a s-t of N is given by: \[C(S,T) = m\left|{V}\right| + 2\left|{V_1}\right|\left(g - D_1\right)\] where $V_1 \dot{\cup} V_2=V$ and $V_1 = S\setminus \{s\}, V_2= T\setminus \{t\}$ and $D_1$ shall be the density of the induced subgraph of $V_1$ regarding $G$.

Especially important is the capacity of minimum s-t Cut. Using the above equation, one can derive that given a minimum s-t Cut of $N$ and the maximum density of $G$ to be $D$, then $g\geq D$ if $V_1=\emptyset$,otherwise $g\leq D$. Moreover the induced subgraph of $V_1$ regarding G is guaranteed to have density greater $g$, otherwise it can be used to proof that there can't exist any subgraph of $G$ greater $g$. Based on this property one can use a binary search approach to shrink the possible interval which contains the solution.

Because the density is per definition guaranteed to be rational, the distance of 2 possible solutions for the maximum density can't be smaller than $\frac{1}{n(n-1)}$. This means shrinking the binary search interval to this size, the correct solution is found. The runtime can in this case be given by $O(M(n,n+m)\log{n}$, where $M(n,m)$ is the runtime of the internally used MinimumSTCutAlgorithm. Especially for large networks it is advised to use a MinimumSTCutAlgorithm whose runtime doesn't depend on the number of edges, because the network $N$ has $O(n+m)$ edges. Preferably one should use PushRelabelMFImpl, leading to a runtime of $O(n^{3}\log{n})$.

Similar to the above explanation the same argument can be applied for other definitions of density by adapting the definitions and the network accordingly. Some generalizations can be found in the paper. As these more general variants including edge weights are only guaranteed to terminate for integer edge weights, instead of using the natural termination property, the algorithm needs to be called with $\varepsilon$ in the constructor. The computation then ensures, that the returned maximum density only differs at most $\varepsilon$ from the correct solution. This is why subclasses of this class might have a little different runtime analysis regarding the $\log{n}$ part.

Andre Immig
  • Field Details

    • guess

      protected double guess
    • graph

      protected final Graph<V,E> graph
  • Constructor Details

    • GoldbergMaximumDensitySubgraphAlgorithmBase

      public GoldbergMaximumDensitySubgraphAlgorithmBase(Graph<V,E> graph, V s, V t, boolean checkWeights, double epsilon, Function<Graph<V,DefaultWeightedEdge>,MinimumSTCutAlgorithm<V,DefaultWeightedEdge>> algFactory)
      graph - input for computation
      s - additional source vertex
      t - additional target vertex
      checkWeights - if true implementation will enforce all internal weights to be positive
      epsilon - to use for internal computation
      algFactory - function to construct the subalgorithm
  • Method Details

    • calculateDensest

      public Graph<V,E> calculateDensest()
      Algorithm to compute max density subgraph Performs binary search on the initial interval lower-upper until interval is smaller than epsilon In case no solution is found because epsilon is too big, the computation continues until a (first) solution is found, thereby avoiding to return an empty graph.
      Specified by:
      calculateDensest in interface MaximumDensitySubgraphAlgorithm<V,E>
      max density subgraph of the graph
    • getDensity

      public double getDensity()
      Computes density of a maximum density subgraph.
      Specified by:
      getDensity in interface MaximumDensitySubgraphAlgorithm<V,E>
      the actual density of the maximum density subgraph
    • getEdgeWeightFromSourceToVertex

      protected abstract double getEdgeWeightFromSourceToVertex(V vertex)
      Getter for network weights of edges su for u in V
      vertex - of V
      weight of the edge (s,v)
    • getEdgeWeightFromVertexToSink

      protected abstract double getEdgeWeightFromVertexToSink(V vertex)
      Getter for network weights of edges ut for u in V
      vertex - of V
      weight of the edge (v,t)
    • computeDensityNumerator

      protected abstract double computeDensityNumerator(Graph<V,E> g)
      g - the graph to compute the numerator density from
      numerator part of the density
    • computeDensityDenominator

      protected abstract double computeDensityDenominator(Graph<V,E> g)
      g - the graph to compute the denominator density from
      numerator part of the density