smart_ini_x.c

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/*************************************************************************
 * Copyright (c) 2011 AT&T Intellectual Property 
 * All rights reserved. This program and the accompanying materials
 * are made available under the terms of the Eclipse Public License v1.0
 * which accompanies this distribution, and is available at
 * https://www.eclipse.org/legal/epl-v10.html
 *
 * Contributors: Details at https://graphviz.org
 *************************************************************************/
 
#include <float.h>
#include <math.h>
#include <neatogen/digcola.h>
#include <util/alloc.h>
#ifdef DIGCOLA
#include <neatogen/kkutils.h>
#include <neatogen/matrix_ops.h>
#include <neatogen/conjgrad.h>
#include <stdbool.h>
 
static void
standardize(double* orthog, int nvtxs) 
{
        double len, avg = 0;
    int i;
        for (i=0; i<nvtxs; i++)
                avg+=orthog[i];
        avg/=nvtxs;
        
        /* centralize: */
        for (i=0; i<nvtxs; i++)
                orthog[i]-=avg;
        
        /* normalize: */
        len = norm(orthog, nvtxs-1);
 
        // if we have a degenerate length, do not attempt to scale by it
        if (fabs(len) < DBL_EPSILON) {
                return;
        }
 
        vectors_scalar_mult(nvtxs, orthog, 1.0 / len, orthog);
}
 
static void
mat_mult_vec_orthog(float** mat, int dim1, int dim2, double* vec, 
    double* result, double* orthog)
{
        /* computes mat*vec, where mat is a dim1*dim2 matrix */
        int i,j;
        double sum;
        
        for (i=0; i<dim1; i++) {
                sum=0;
                for (j=0; j<dim2; j++) {
                        sum += mat[i][j]*vec[j];
                }
                result[i]=sum;
        }
        assert(orthog != NULL);
        double alpha = -vectors_inner_product(dim1, result, orthog);
        scadd(result, dim1 - 1, alpha, orthog); 
}
 
static void
power_iteration_orthog(float** square_mat, int n, int neigs, 
     double** eigs, double* evals, double* orthog, double p_iteration_threshold)
{
  // Power-Iteration with
  //   (I - orthog × orthogᵀ) × square_mat × (I - orthog × orthogᵀ)
 
        int i,j;
        double *tmp_vec = gv_calloc(n, sizeof(double));
        double *last_vec = gv_calloc(n, sizeof(double));
        double *curr_vector;
        double len;
        double angle;
        double alpha;
        int largest_index;
        double largest_eval;
 
        double tol=1-p_iteration_threshold;
 
        if (neigs>=n) {
                neigs=n;
        }
 
        for (i=0; i<neigs; i++) {
                curr_vector = eigs[i];
                /* guess the i-th eigen vector */
choose:
                for (j=0; j<n; j++) {
                        curr_vector[j] = rand()%100;
                }
 
                assert(orthog != NULL);
                alpha = -vectors_inner_product(n, orthog, curr_vector);
                scadd(curr_vector, n - 1, alpha, orthog);       
                        // orthogonalize against higher eigenvectors
                for (j=0; j<i; j++) {
                        alpha = -vectors_inner_product(n, eigs[j], curr_vector);
                        scadd(curr_vector, n-1, alpha, eigs[j]);
            }
                len = norm(curr_vector, n-1);
                if (len<1e-10) {
                        /* We have chosen a vector colinear with prvious ones */
                        goto choose;
                }
                vectors_scalar_mult(n, curr_vector, 1.0 / len, curr_vector);    
                do {
                        copy_vector(n, curr_vector, last_vec);
                        
                        mat_mult_vec_orthog(square_mat,n,n,curr_vector,tmp_vec,orthog);
                        copy_vector(n, tmp_vec, curr_vector);
                                                
                        /* orthogonalize against higher eigenvectors */
                        for (j=0; j<i; j++) {
                                alpha = -vectors_inner_product(n, eigs[j], curr_vector);
                                scadd(curr_vector, n-1, alpha, eigs[j]);
                        }
                        len = norm(curr_vector, n-1);
                        if (len<1e-10) {
                            /* We have reached the null space (e.vec. associated 
                 * with e.val. 0)
                 */
                                goto exit;
                        }
 
                        vectors_scalar_mult(n, curr_vector, 1.0 / len, curr_vector);
                        angle = vectors_inner_product(n, curr_vector, last_vec);
                } while (fabs(angle)<tol);
        /* the Rayleigh quotient (up to errors due to orthogonalization):
         * u*(A*u)/||A*u||)*||A*u||, where u=last_vec, and ||u||=1
         */
                evals[i]=angle*len;
        }
exit:
        for (; i<neigs; i++) {
                /* compute the smallest eigenvector, which are 
                 * probably associated with eigenvalue 0 and for
                 * which power-iteration is dangerous
         */
                curr_vector = eigs[i];
                /* guess the i-th eigen vector */
                for (j=0; j<n; j++)
                        curr_vector[j] = rand()%100;
                /* orthogonalize against higher eigenvectors */
                for (j=0; j<i; j++) {
                        alpha = -vectors_inner_product(n, eigs[j], curr_vector);
                        scadd(curr_vector, n-1, alpha, eigs[j]);
            }
                len = norm(curr_vector, n-1);
                vectors_scalar_mult(n, curr_vector, 1.0 / len, curr_vector);
                evals[i]=0;
                
        }
 
        /* sort vectors by their evals, for overcoming possible mis-convergence: */
        for (i=0; i<neigs-1; i++) {
                largest_index=i;
                largest_eval=evals[largest_index];
                for (j=i+1; j<neigs; j++) {
                        if (largest_eval<evals[j]) {
                                largest_index=j;
                                largest_eval=evals[largest_index];
                        }
                }
                if (largest_index!=i) { // exchange eigenvectors:
                        copy_vector(n, eigs[i], tmp_vec);
                        copy_vector(n, eigs[largest_index], eigs[i]);
                        copy_vector(n, tmp_vec, eigs[largest_index]);
 
                        evals[largest_index]=evals[i];
                        evals[i]=largest_eval;
                }
        }
        
        free (tmp_vec); free (last_vec);
 
}
 
static float* 
compute_avgs(DistType** Dij, int n, float* all_avg) 
{
        float* row_avg = gv_calloc(n, sizeof(float));
        int i,j;
        double sum=0, sum_row;
 
        for (i=0; i<n; i++) {
                sum_row=0;
                for (j=0; j<n; j++) {
                        sum+=(double)Dij[i][j]*(double)Dij[i][j];
                        sum_row+=(double)Dij[i][j]*(double)Dij[i][j];
                }
                row_avg[i]=(float)sum_row/n;
        }
        *all_avg=(float)sum/(n*n);
    return row_avg;
}
 
static float**
compute_Bij(DistType** Dij, int n)
{
        int i,j;
        float *storage = gv_calloc(n * n, sizeof(float));
        float **Bij = gv_calloc(n, sizeof(float *));
        float* row_avg; 
    float all_avg;
 
        for (i=0; i<n; i++) 
                Bij[i] = storage+i*n;
 
        row_avg = compute_avgs(Dij, n, &all_avg);       
        for (i=0; i<n; i++) {
                for (j=0; j<=i; j++) {
                        Bij[i][j]=-(float)Dij[i][j]*Dij[i][j]+row_avg[i]+row_avg[j]-all_avg;
                        Bij[j][i]=Bij[i][j];
                }
        }
    free (row_avg);
    return Bij;
}
 
static void
CMDS_orthog(int n, int dim, double** eigs, double tol,
            double* orthog, DistType** Dij)
{
        int i,j;
        float** Bij = compute_Bij(Dij, n);
        double *evals = gv_calloc(dim, sizeof(double));
        
        assert(orthog != NULL);
        double *orthog_aux = gv_calloc(n, sizeof(double));
        for (i=0; i<n; i++) {
                orthog_aux[i]=orthog[i];
        }
        standardize(orthog_aux,n);
    power_iteration_orthog(Bij, n, dim, eigs, evals, orthog_aux, tol);
        
        for (i=0; i<dim; i++) {
                for (j=0; j<n; j++) {
                        eigs[i][j]*=sqrt(fabs(evals[i])); 
                }
        }
        free (Bij[0]); free (Bij);
        free (evals); free (orthog_aux);
}
 
#define SCALE_FACTOR 256
 
int IMDS_given_dim(vtx_data* graph, int n, double* given_coords, 
       double* new_coords, double conj_tol)
{
        int iterations2;
        int i,j, rv = 0;
        DistType** Dij;
        double* x = given_coords;       
        double uniLength;
        double* y = new_coords;
        float **lap = gv_calloc(n, sizeof(float *));
        float degree;
        double pos_i;
        double *balance = gv_calloc(n, sizeof(double));
        double b;
        bool converged;
 
        Dij = compute_apsp(graph, n);
        
        /* scaling up the distances to enable an 'sqrt' operation later 
     * (in case distances are integers)
     */
        for (i=0; i<n; i++)
                for (j=0; j<n; j++)
                        Dij[i][j]*=SCALE_FACTOR;
        
        assert(x!=NULL);
        {
                double sum1, sum2;
        
                for (sum1=sum2=0,i=1; i<n; i++) {
                        for (j=0; j<i; j++) {           
                                sum1+=1.0/(Dij[i][j])*fabs(x[i]-x[j]);
                                sum2+=1.0/(Dij[i][j]*Dij[i][j])*fabs(x[i]-x[j])*fabs(x[i]-x[j]);
                        }
                }
                uniLength = isinf(sum2) ? 0 : sum1 / sum2;
                for (i=0; i<n; i++)
                        x[i]*=uniLength;
        }
 
        /* smart ini: */
        CMDS_orthog(n, 1, &y, conj_tol, x, Dij);
        
        /* Compute Laplacian: */
        float *f_storage = gv_calloc(n * n, sizeof(float));
        
        for (i=0; i<n; i++) {
                lap[i]=f_storage+i*n;
                degree=0;
                for (j=0; j<n; j++) {
                        if (j==i)
                                continue;
                        degree-=lap[i][j]=-1.0f/((float)Dij[i][j]*(float)Dij[i][j]); // w_{ij}
                        
                }
                lap[i][i]=degree;
        }
        
 
        /* compute residual distances */
        /* if (x!=NULL)  */
    {
                double diff;
                for (i=1; i<n; i++) {
                        pos_i=x[i];             
                        for (j=0; j<i; j++) {
                                diff=(double)Dij[i][j]*(double)Dij[i][j]-(pos_i-x[j])*(pos_i-x[j]);
                                Dij[i][j]=Dij[j][i]=diff>0 ? (DistType)sqrt(diff) : 0;
                        }
                }
        }
        
        /* Compute the balance vector: */
        for (i=0; i<n; i++) {
                pos_i=y[i];
                balance[i]=0;
                for (j=0; j<n; j++) {
                        if (j==i)
                                continue;
                        if (pos_i>=y[j]) {
                                balance[i]+=Dij[i][j]*(-lap[i][j]); // w_{ij}*delta_{ij}
                        }
                        else {
                                balance[i]-=Dij[i][j]*(-lap[i][j]); // w_{ij}*delta_{ij}
                        }
                }
        }
 
        for (converged=false,iterations2=0; iterations2<200 && !converged; iterations2++) {
                if (conjugate_gradient_f(lap, y, balance, n, conj_tol, n, true) < 0) {
                    rv = 1;
                    goto cleanup;
                }
                converged = true;
                for (i=0; i<n; i++) {
                        pos_i=y[i];
                        b=0;
                        for (j=0; j<n; j++) {
                                if (j==i)
                                        continue;
                                if (pos_i>=y[j]) {
                                        b+=Dij[i][j]*(-lap[i][j]);
                                        
                                }
                                else {
                                        b-=Dij[i][j]*(-lap[i][j]);
                                        
                                }
                        }
                        if ((b != balance[i]) && (fabs(1-b/balance[i])>1e-5)) {
                                converged = false;
                                balance[i]=b;
                        }
                }
        }
        
        for (i = 0; !(fabs(uniLength) < DBL_EPSILON) && i < n; i++) {
                x[i] /= uniLength;
                y[i] /= uniLength;
        }
        
cleanup:
 
        free (Dij[0]); free (Dij);      
        free (lap[0]); free (lap);      
        free (balance); 
        return rv;
}
 
#endif /* DIGCOLA */