Class GoldbergMaximumDensitySubgraphAlgorithmBase<V,E>
 Type Parameters:
V
 Type of verticesE
 Type of edges
 All Implemented Interfaces:
MaximumDensitySubgraphAlgorithm<V,
E>
 Direct Known Subclasses:
GoldbergMaximumDensitySubgraphAlgorithm
,GoldbergMaximumDensitySubgraphAlgorithmNodeWeightPerEdgeWeight
,GoldbergMaximumDensitySubgraphAlgorithmNodeWeights
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.
Additionally one defines the following weights for the network: \[c_{ij}=1 \forall \{i,j\}\in E\] \[c_{si}=m \forall i \in V\] \[c_{it}=m+2gd_i \forall i \in V\] 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 st 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 st Cut. Using the above equation, one can derive that given a minimum st 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 $\frac{1}{n(n1)}$. 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.
 Author:
 Andre Immig

Field Summary

Constructor Summary
ConstructorDescriptionGoldbergMaximumDensitySubgraphAlgorithmBase
(Graph<V, E> graph, V s, V t, boolean checkWeights, double epsilon, Function<Graph<V, DefaultWeightedEdge>, MinimumSTCutAlgorithm<V, DefaultWeightedEdge>> algFactory) Constructor 
Method Summary
Modifier and TypeMethodDescriptionAlgorithm to compute max density subgraph Performs binary search on the initial interval lowerupper 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.protected abstract double
protected abstract double
double
Computes density of a maximum density subgraph.protected abstract double
getEdgeWeightFromSourceToVertex
(V vertex) Getter for network weights of edges su for u in Vprotected abstract double
getEdgeWeightFromVertexToSink
(V vertex) Getter for network weights of edges ut for u in V

Field Details

guess
protected double guess 
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) Constructor Parameters:
graph
 input for computations
 additional source vertext
 additional target vertexcheckWeights
 if true implementation will enforce all internal weights to be positiveepsilon
 to use for internal computationalgFactory
 function to construct the subalgorithm


Method Details

calculateDensest
Algorithm to compute max density subgraph Performs binary search on the initial interval lowerupper 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 interfaceMaximumDensitySubgraphAlgorithm<V,
E>  Returns:
 max density subgraph of the graph

getDensity
public double getDensity()Computes density of a maximum density subgraph. Specified by:
getDensity
in interfaceMaximumDensitySubgraphAlgorithm<V,
E>  Returns:
 the actual density of the maximum density subgraph

getEdgeWeightFromSourceToVertex
Getter for network weights of edges su for u in V Parameters:
vertex
 of V Returns:
 weight of the edge (s,v)

getEdgeWeightFromVertexToSink
Getter for network weights of edges ut for u in V Parameters:
vertex
 of V Returns:
 weight of the edge (v,t)

computeDensityNumerator
 Parameters:
g
 the graph to compute the numerator density from Returns:
 numerator part of the density

computeDensityDenominator
 Parameters:
g
 the graph to compute the denominator density from Returns:
 numerator part of the density
