/**
* Splits recursively the points of the graph while the value of the best cut found is less of a
* specified limit (the alpha star factor).
*
* @param W the weight matrix of the graph
* @param alpha_star the alpha star factor
* @return an array of sets of points (partitions)
*/
protected int[][] partition(DoubleMatrix2D W, double alpha_star) {
numPartitions++;
// System.out.println("!");
// If the graph contains only one point
if (W.columns() == 1) {
int[][] p = new int[1][1];
p[0][0] = 0;
return p;
// Otherwise
} else {
// Computes the best cut
int[][] cut = bestCut(W);
// Computes the value of the found cut
double cutVal = Ncut(W, cut[0], cut[1], null);
// System.out.println("cutVal = "+cutVal +"\tnumPartitions = "+numPartitions);
// If the value is less than alpha star
if (cutVal < alpha_star && numPartitions < 2) {
// Recursively partitions the first one found ...
DoubleMatrix2D W0 = W.viewSelection(cut[0], cut[0]);
int[][] p0 = partition(W0, alpha_star);
// ... and the second one
DoubleMatrix2D W1 = W.viewSelection(cut[1], cut[1]);
int[][] p1 = partition(W1, alpha_star);
// Merges the partitions found in the previous recursive steps
int[][] p = new int[p0.length + p1.length][];
for (int i = 0; i < p0.length; i++) {
p[i] = new int[p0[i].length];
for (int j = 0; j < p0[i].length; j++) p[i][j] = cut[0][p0[i][j]];
}
for (int i = 0; i < p1.length; i++) {
p[i + p0.length] = new int[p1[i].length];
for (int j = 0; j < p1[i].length; j++) p[i + p0.length][j] = cut[1][p1[i][j]];
}
return p;
} else {
// Otherwise returns the partitions found in current step
// w/o recursive invocation
int[][] p = new int[1][W.columns()];
for (int i = 0; i < p[0].length; i++) p[0][i] = i;
return p;
}
}
}
/**
* Returns the best cut of a graph w.r.t. the degree of dissimilarity between points of different
* partitions and the degree of similarity between points of the same partition.
*
* @param W the weight matrix of the graph
* @return an array of two elements, each of these contains the points of a partition
*/
protected static int[][] bestCut(DoubleMatrix2D W) {
int n = W.columns();
// Builds the diagonal matrices D and D^(-1/2) (represented as their diagonals)
DoubleMatrix1D d = DoubleFactory1D.dense.make(n);
DoubleMatrix1D d_minus_1_2 = DoubleFactory1D.dense.make(n);
for (int i = 0; i < n; i++) {
double d_i = W.viewRow(i).zSum();
d.set(i, d_i);
d_minus_1_2.set(i, 1 / Math.sqrt(d_i));
}
DoubleMatrix2D D = DoubleFactory2D.sparse.diagonal(d);
// System.out.println("DoubleMatrix2D :\n"+D.toString());
DoubleMatrix2D X = D.copy();
// System.out.println("DoubleMatrix2D copy :\n"+X.toString());
// X = D^(-1/2) * (D - W) * D^(-1/2)
X.assign(W, Functions.minus);
// System.out.println("DoubleMatrix2D X: (D-W) :\n"+X.toString());
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
X.set(i, j, X.get(i, j) * d_minus_1_2.get(i) * d_minus_1_2.get(j));
// Computes the eigenvalues and the eigenvectors of X
EigenvalueDecomposition e = new EigenvalueDecomposition(X);
DoubleMatrix1D lambda = e.getRealEigenvalues();
// Selects the eigenvector z_2 associated with the second smallest eigenvalue
// Creates a map that contains the pairs <index, eigenvalue>
AbstractIntDoubleMap map = new OpenIntDoubleHashMap(n);
for (int i = 0; i < n; i++) map.put(i, Math.abs(lambda.get(i)));
IntArrayList list = new IntArrayList();
// Sorts the map on the value
map.keysSortedByValue(list);
// Gets the index of the second smallest element
int i_2 = list.get(1);
// y_2 = D^(-1/2) * z_2
DoubleMatrix1D y_2 = e.getV().viewColumn(i_2).copy();
y_2.assign(d_minus_1_2, Functions.mult);
// Creates a map that contains the pairs <i, y_2[i]>
map.clear();
for (int i = 0; i < n; i++) map.put(i, y_2.get(i));
// Sorts the map on the value
map.keysSortedByValue(list);
// Search the element in the map previuosly ordered that minimizes the cut
// of the partition
double best_cut = Double.POSITIVE_INFINITY;
int[][] partition = new int[2][];
// The array v contains all the elements of the graph ordered by their
// projection on vector y_2
int[] v = list.elements();
// For each admissible splitting point i
for (int i = 1; i < n; i++) {
// The array a contains all the elements that have a projection on vector
// y_2 less or equal to the one of i-th element
// The array b contains the remaining elements
int[] a = new int[i];
int[] b = new int[n - i];
System.arraycopy(v, 0, a, 0, i);
System.arraycopy(v, i, b, 0, n - i);
double cut = Ncut(W, a, b, v);
if (cut < best_cut) {
best_cut = cut;
partition[0] = a;
partition[1] = b;
}
}
// System.out.println("Partition:");
// UtilsJS.printMatrix(partition);
return partition;
}
