// This is an improved version of Algorithm 12 of the PhD thesis by Sadia Sharmin, which runs in O(n) instead of O(n^2)
// The RelativeNeighborhood selection method picks the next vertex based on the ratio of neighbors that are already contained in N(Left)
// While in the PhD thesis the algorithm loops over all vertices w in N(v) * right to determine if they are internal/external, we can make this faster
// by realizing that |((N(v) * right) \ (N(left) * right)| is already equal to the size of the external set. This is also equal to the symmetric difference.
private static int RelativeNeighborhood(Graph graph, BitSet left, BitSet right)
{
// Minimum ratio found so far
double minRatio = double.MaxValue;
// Vertex that we will choose
int chosen = -1;
// Set of vertices in right that are adjacent to vertices in left
BitSet nl = graph.Neighborhood(left) * right;
Dictionary<int, double> ratios = new Dictionary<int, double>();
foreach (int v in nl)
{
// Neighborhood of v
BitSet nv = graph.OpenNeighborhood(v) * right;
// Number of vertices outside of N(left) * right, but inside N(v) * right
double external_v = (nv - nl).Count;
// Number of vertices inside of both N(left) * right and N(v) * right
double internal_v = nv.Count - external_v;
if (internal_v == 0) internal_v = 1; // cant devide by zero!
// Possibly update the minimum ratio found so far
double ratio = external_v / internal_v;
ratios[v+1] = ratio;
if (ratio < minRatio)
{
chosen = v;
minRatio = ratio;
}
}
return chosen;
}
/*************************/
// Candidate strategy
/*************************/
public static BitSet Candidates(Graph graph, BitSet left, BitSet right, CandidateStrategy candidateStrategy)
{
BitSet nl = graph.Neighborhood(left) * right;
return candidateStrategy == CandidateStrategy.All ?
right.Copy() : (nl + graph.Neighborhood(nl)) * right;
}
/*************************/
// Selection algorithms for the next vertex in the sequence
/*************************/
// Implementation of Algorithm 11 of the PhD thesis by Sadia Sharmin
// The LeastUncommonNeighbor selection method finds the vertex in right that has the fewest neighbors in right that are
// not adjacent to a vertex in left. In other words, we add as few as possibly new vertices to the neighborhoods.
private static int LeastUncommonNeighbor(Graph graph, BitSet left, BitSet right)
{
// Minimum symmetric difference found so far
int minSymmDiff = int.MaxValue;
// Vertex that we will choose
int chosen = -1;
// Set of vertices in right that are adjacent to vertices in left
BitSet nl = graph.Neighborhood(left) * right;
foreach (int v in right)
{
// Neighborhood of v in right
BitSet nv = graph.OpenNeighborhood(v) * right;
// Count the difference between N(v) and N(Left)
int symmDiff = (nv - nl).Count;
// Possibly update the best choice so far
if (symmDiff < minSymmDiff)
{
minSymmDiff = symmDiff;
chosen = v;
}
}
return chosen;
}
