/**
   * 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;
  }