static void Multilevel_coarsen(SparseMatrix A, SparseMatrix *cA, real *node_wgt, real **cnode_wgt,
SparseMatrix *P, SparseMatrix *R, Multilevel_control ctrl, int *coarsen_scheme_used){
int *matching = NULL, nmatch, nc, nzc, n, i;
int *irn = NULL, *jcn = NULL, *ia = NULL, *ja = NULL;
real *val = NULL;
SparseMatrix B = NULL;
int *vset = NULL, nvset, ncov, j;
int *cluster, *clusterp, ncluster;
assert(A->m == A->n);
*cA = NULL;
*P = NULL;
*R = NULL;
n = A->m;
*coarsen_scheme_used = ctrl->coarsen_scheme;
switch (ctrl->coarsen_scheme){
case COARSEN_HYBRID:
#ifdef DEBUG_PRINT
if (Verbose)
fprintf(stderr, "hybrid scheme, try COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE_LEAVES_FIRST first\n");
#endif
*coarsen_scheme_used = ctrl->coarsen_scheme = COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE_LEAVES_FIRST;
Multilevel_coarsen(A, cA, node_wgt, cnode_wgt, P, R, ctrl, coarsen_scheme_used);
if (!(*cA)) {
#ifdef DEBUG_PRINT
if (Verbose)
fprintf(stderr, "switching to COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE_SUPERNODES_FIRST\n");
#endif
*coarsen_scheme_used = ctrl->coarsen_scheme = COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE_SUPERNODES_FIRST;
Multilevel_coarsen(A, cA, node_wgt, cnode_wgt, P, R, ctrl, coarsen_scheme_used);
}
if (!(*cA)) {
#ifdef DEBUG_PRINT
if (Verbose)
fprintf(stderr, "switching to COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_CLUSTER_PERNODE_LEAVES_FIRST\n");
#endif
*coarsen_scheme_used = ctrl->coarsen_scheme = COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_CLUSTER_PERNODE_LEAVES_FIRST;
Multilevel_coarsen(A, cA, node_wgt, cnode_wgt, P, R, ctrl, coarsen_scheme_used);
}
if (!(*cA)) {
#ifdef DEBUG_PRINT
if (Verbose)
fprintf(stderr, "switching to COARSEN_INDEPENDENT_VERTEX_SET\n");
#endif
*coarsen_scheme_used = ctrl->coarsen_scheme = COARSEN_INDEPENDENT_VERTEX_SET;
Multilevel_coarsen(A, cA, node_wgt, cnode_wgt, P, R, ctrl, coarsen_scheme_used);
}
if (!(*cA)) {
#ifdef DEBUG_PRINT
if (Verbose)
fprintf(stderr, "switching to COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE\n");
#endif
*coarsen_scheme_used = ctrl->coarsen_scheme = COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE;
Multilevel_coarsen(A, cA, node_wgt, cnode_wgt, P, R, ctrl, coarsen_scheme_used);
}
ctrl->coarsen_scheme = COARSEN_HYBRID;
break;
case COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE_SUPERNODES_FIRST:
case COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_CLUSTER_PERNODE_LEAVES_FIRST:
case COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE_LEAVES_FIRST:
if (ctrl->coarsen_scheme == COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE_LEAVES_FIRST) {
maximal_independent_edge_set_heavest_edge_pernode_leaves_first(A, ctrl->randomize, &cluster, &clusterp, &ncluster);
} else if (ctrl->coarsen_scheme == COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE_SUPERNODES_FIRST) {
maximal_independent_edge_set_heavest_edge_pernode_supernodes_first(A, ctrl->randomize, &cluster, &clusterp, &ncluster);
} else {
maximal_independent_edge_set_heavest_cluster_pernode_leaves_first(A, 4, ctrl->randomize, &cluster, &clusterp, &ncluster);
}
assert(ncluster <= n);
nc = ncluster;
if (nc > ctrl->min_coarsen_factor*n || nc < ctrl->minsize) {
#ifdef DEBUG_PRINT
if (Verbose)
fprintf(stderr, "nc = %d, nf = %d, minsz = %d, coarsen_factor = %f coarsening stops\n",nc, n, ctrl->minsize, ctrl->min_coarsen_factor);
#endif
goto RETURN;
}
irn = N_GNEW(n,int);
jcn = N_GNEW(n,int);
val = N_GNEW(n,real);
nzc = 0;
for (i = 0; i < ncluster; i++){
for (j = clusterp[i]; j < clusterp[i+1]; j++){
assert(clusterp[i+1] > clusterp[i]);
irn[nzc] = cluster[j];
jcn[nzc] = i;
val[nzc++] = 1.;
}
}
assert(nzc == n);
*P = SparseMatrix_from_coordinate_arrays(nzc, n, nc, irn, jcn, (void *) val, MATRIX_TYPE_REAL);
*R = SparseMatrix_transpose(*P);
B = SparseMatrix_multiply(*R, A);
if (!B) goto RETURN;
*cA = SparseMatrix_multiply(B, *P);
if (!*cA) goto RETURN;
SparseMatrix_multiply_vector(*R, node_wgt, cnode_wgt, FALSE);
*R = SparseMatrix_divide_row_by_degree(*R);
SparseMatrix_set_symmetric(*cA);
SparseMatrix_set_pattern_symmetric(*cA);
*cA = SparseMatrix_remove_diagonal(*cA);
break;
case COARSEN_INDEPENDENT_EDGE_SET:
maximal_independent_edge_set(A, ctrl->randomize, &matching, &nmatch);
case COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE:
if (ctrl->coarsen_scheme == COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE)
maximal_independent_edge_set_heavest_edge_pernode(A, ctrl->randomize, &matching, &nmatch);
case COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE_DEGREE_SCALED:
if (ctrl->coarsen_scheme == COARSEN_INDEPENDENT_EDGE_SET_HEAVEST_EDGE_PERNODE_DEGREE_SCALED)
maximal_independent_edge_set_heavest_edge_pernode_scaled(A, ctrl->randomize, &matching, &nmatch);
nc = nmatch;
if (nc > ctrl->min_coarsen_factor*n || nc < ctrl->minsize) {
#ifdef DEBUG_PRINT
if (Verbose)
fprintf(stderr, "nc = %d, nf = %d, minsz = %d, coarsen_factor = %f coarsening stops\n",nc, n, ctrl->minsize, ctrl->min_coarsen_factor);
#endif
goto RETURN;
}
irn = N_GNEW(n,int);
jcn = N_GNEW(n,int);
val = N_GNEW(n,real);
nzc = 0; nc = 0;
for (i = 0; i < n; i++){
if (matching[i] >= 0){
if (matching[i] == i){
irn[nzc] = i;
jcn[nzc] = nc;
val[nzc++] = 1.;
} else {
irn[nzc] = i;
jcn[nzc] = nc;
val[nzc++] = 1;
irn[nzc] = matching[i];
jcn[nzc] = nc;
val[nzc++] = 1;
matching[matching[i]] = -1;
}
nc++;
matching[i] = -1;
}
}
assert(nc == nmatch);
assert(nzc == n);
*P = SparseMatrix_from_coordinate_arrays(nzc, n, nc, irn, jcn, (void *) val, MATRIX_TYPE_REAL);
*R = SparseMatrix_transpose(*P);
B = SparseMatrix_multiply(*R, A);
if (!B) goto RETURN;
*cA = SparseMatrix_multiply(B, *P);
if (!*cA) goto RETURN;
SparseMatrix_multiply_vector(*R, node_wgt, cnode_wgt, FALSE);
*R = SparseMatrix_divide_row_by_degree(*R);
SparseMatrix_set_symmetric(*cA);
SparseMatrix_set_pattern_symmetric(*cA);
*cA = SparseMatrix_remove_diagonal(*cA);
break;
case COARSEN_INDEPENDENT_VERTEX_SET:
case COARSEN_INDEPENDENT_VERTEX_SET_RS:
if (ctrl->coarsen_scheme == COARSEN_INDEPENDENT_VERTEX_SET){
maximal_independent_vertex_set(A, ctrl->randomize, &vset, &nvset, &nzc);
} else {
maximal_independent_vertex_set_RS(A, ctrl->randomize, &vset, &nvset, &nzc);
}
ia = A->ia;
ja = A->ja;
nc = nvset;
if (nc > ctrl->min_coarsen_factor*n || nc < ctrl->minsize) {
#ifdef DEBUG_PRINT
if (Verbose)
fprintf(stderr, "nc = %d, nf = %d, minsz = %d, coarsen_factor = %f coarsening stops\n",nc, n, ctrl->minsize, ctrl->min_coarsen_factor);
#endif
goto RETURN;
}
irn = N_GNEW(nzc,int);
jcn = N_GNEW(nzc,int);
val = N_GNEW(nzc,real);
nzc = 0;
for (i = 0; i < n; i++){
if (vset[i] == MAX_IND_VTX_SET_F){
ncov = 0;
for (j = ia[i]; j < ia[i+1]; j++){
if (vset[ja[j]] >= MAX_IND_VTX_SET_C){
ncov++;
}
}
assert(ncov > 0);
for (j = ia[i]; j < ia[i+1]; j++){
if (vset[ja[j]] >= MAX_IND_VTX_SET_C){
irn[nzc] = i;
jcn[nzc] = vset[ja[j]];
val[nzc++] = 1./(double) ncov;
}
}
} else {
assert(vset[i] >= MAX_IND_VTX_SET_C);
irn[nzc] = i;
jcn[nzc] = vset[i];
val[nzc++] = 1.;
}
}
*P = SparseMatrix_from_coordinate_arrays(nzc, n, nc, irn, jcn, (void *) val, MATRIX_TYPE_REAL);
*R = SparseMatrix_transpose(*P);
B = SparseMatrix_multiply(*R, A);
if (!B) goto RETURN;
*cA = SparseMatrix_multiply(B, *P);
if (!*cA) goto RETURN;
SparseMatrix_multiply_vector(*R, node_wgt, cnode_wgt, FALSE);
SparseMatrix_set_symmetric(*cA);
SparseMatrix_set_pattern_symmetric(*cA);
*cA = SparseMatrix_remove_diagonal(*cA);
break;
default:
goto RETURN;
}
RETURN:
if (matching) FREE(matching);
if (vset) FREE(vset);
if (irn) FREE(irn);
if (jcn) FREE(jcn);
if (val) FREE(val);
if (B) SparseMatrix_delete(B);
}
Multilevel_MQ_Clustering Multilevel_MQ_Clustering_establish(Multilevel_MQ_Clustering grid, int maxcluster) {
int *matching = grid->matching;
SparseMatrix A = grid->A;
int n = grid->n, level = grid->level, nc = 0, nclusters = n;
real mq = 0, mq_in = 0, mq_out = 0, mq_new, mq_in_new, mq_out_new, mq_max = 0, mq_in_max = 0, mq_out_max = 0;
int *ia = A->ia, *ja = A->ja;
real *a, amax = 0;
real *deg_intra = grid->deg_intra, *wgt = grid->wgt;
real *deg_intra_new, *wgt_new = NULL;
int i, j, k, jj, jc, jmax;
real *deg_inter, gain = 0, *dout = grid->dout, *dout_new, deg_in_i, deg_in_j, wgt_i, wgt_j, a_ij, dout_i, dout_j, dout_max = 0, wgt_jmax = 0;
int *mask;
real maxgain = 0;
real total_gain = 0;
SingleLinkedList *neighbors = NULL, lst;
neighbors = MALLOC(sizeof(SingleLinkedList)*n);
for (i = 0; i < n; i++) neighbors[i] = NULL;
mq = grid->mq;
mq_in = grid->mq_in;
mq_out = grid->mq_out;
deg_intra_new = MALLOC(sizeof(real)*n);
wgt_new = MALLOC(sizeof(real)*n);
deg_inter = MALLOC(sizeof(real)*n);
mask = MALLOC(sizeof(int)*n);
dout_new = MALLOC(sizeof(real)*n);
for (i = 0; i < n; i++) mask[i] = -1;
assert(n == A->n);
for (i = 0; i < n; i++) matching[i] = UNMATCHED;
/* gain in merging node A into cluster B is
mq_in_new = mq_in - |E(A,A)|/(V(A))^2 - |E(B,B)|/(V(B))^2 + (|E(A,A)|+|E(B,B)|+|E(A,B)|)/(|V(A)|+|V(B)|)^2
. = mq_in - deg_intra(A)/|A|^2 - deg_intra(B)/|B|^2 + (deg_intra(A)+deg_intra(B)+a(A,B))/(|A|+|B|)^2
mq_out_new = mq_out - |E(A,B)|/(|V(A)|*V(B)|)-\sum_{C and A connected, C!=B} |E(A,C)|/(|V(A)|*|V(C)|)-\sum_{C and B connected,C!=B} |E(B,C)|/(|V(B)|*|V(C)|)
. + \sum_{C connected to A or B, C!=A, C!=B} (|E(A,C)|+|E(B,C)|)/(|V(C)|*(|V(A)|+|V(B)|)
. = mq_out + a(A,B)/(|A|*|B|)-\sum_{C and A connected} a(A,C)/(|A|*|C|)-\sum_{C and B connected} a(B,C)/(|B|*|C|)
. + \sum_{C connected to A or B, C!=A, C!=B} (a(A,C)+a(B,C))/(|C|*(|A|+|B|))
Denote:
dout(i) = \sum_{j -- i} a(i,j)/|j|
then
mq_out_new = mq_out - |E(A,B)|/(|V(A)|*V(B)|)-\sum_{C and A connected, C!=B} |E(A,C)|/(|V(A)|*|V(C)|)-\sum_{C and B connected,C!=B} |E(B,C)|/(|V(B)|*|V(C)|)
. + \sum_{C connected to A or B, C!=A, C!=B} (|E(A,C)|+|E(B,C)|)/(|V(C)|*(|V(A)|+|V(B)|)
. = mq_out + a(A,B)/(|A|*|B|)-dout(A)/|A| - dout(B)/|B|
. + (dout(A)+dout(B))/(|A|+|B|) - (a(A,B)/|A|+a(A,B)/|B|)/(|A|+|B|)
. = mq_out -dout(A)/|A| - dout(B)/|B| + (dout(A)+dout(B))/(|A|+|B|)
after merging A and B into cluster AB,
dout(AB) = dout(A) + dout(B);
dout(C) := dout(C) - a(A,C)/|A| - a(B,C)/|B| + a(A,C)/(|A|+|B|) + a(B, C)/(|A|+|B|)
mq_new = mq_in_new/(k-1) - mq_out_new/((k-1)*(k-2))
gain = mq_new - mq
*/
a = (real*) A->a;
for (i = 0; i < n; i++) {
if (matching[i] != UNMATCHED) continue;
/* accumulate connections between i and clusters */
for (j = ia[i]; j < ia[i+1]; j++) {
jj = ja[j];
if (jj == i) continue;
if ((jc=matching[jj]) != UNMATCHED) {
if (mask[jc] != i) {
mask[jc] = i;
deg_inter[jc] = a[j];
} else {
deg_inter[jc] += a[j];
}
}
}
deg_in_i = deg_intra[i];
wgt_i = wgt[i];
dout_i = dout[i];
maxgain = 0;
jmax = -1;
for (j = ia[i]; j < ia[i+1]; j++) {
jj = ja[j];
if (jj == i) continue;
jc = matching[jj];
if (jc == UNMATCHED) {
a_ij = a[j];
wgt_j = wgt[jj];
deg_in_j = deg_intra[jj];
dout_j = dout[jj];
} else if (deg_inter[jc] < 0) {
continue;
} else {
a_ij = deg_inter[jc];
wgt_j = wgt_new[jc];
deg_inter[jc] = -1; /* so that we do not redo the calulation when we hit another neighbor in cluster jc */
deg_in_j = deg_intra_new[jc];
dout_j = dout_new[jc];
}
mq_in_new = mq_in - deg_in_i/pow(wgt_i, 2) - deg_in_j/pow(wgt_j,2)
+ (deg_in_i + deg_in_j + a_ij)/pow(wgt_i + wgt_j,2);
mq_out_new = mq_out - dout_i/wgt_i - dout_j/wgt_j + (dout_i + dout_j)/(wgt_i + wgt_j);
if (nclusters > 2) {
mq_new = 2*(mq_in_new/(nclusters - 1) - mq_out_new/((nclusters - 1)*(nclusters - 2)));
} else {
mq_new = 2*mq_in_new/(nclusters - 1);
}
#ifdef DEBUG
{ int ncluster;
double mq2, mq_in2, mq_out2, *dout2;
int *matching2, nc2 = nc;
matching2 = MALLOC(sizeof(int)*A->m);
matching2 = MEMCPY(matching2, matching, sizeof(real)*A->m);
if (jc != UNMATCHED) {
matching2[i] = jc;
} else {
matching2[i] = nc2;
matching2[jj] = nc2;
nc2++;
}
for (k = 0; k < n; k++) if (matching2[k] == UNMATCHED) matching2[k] =nc2++;
mq2 = get_mq(A, matching2, &ncluster, &mq_in2, &mq_out2, &dout2);
fprintf(stderr," {dout_i, dout_j}={%f,%f}, {predicted, calculated}: mq = {%f, %f}, mq_in ={%f,%f}, mq_out = {%f,%f}\n",dout_i, dout_j, mq_new, mq2, mq_in_new, mq_in2, mq_out_new, mq_out2);
mq_new = mq2;
}
#endif
gain = mq_new - mq;
if (Verbose) fprintf(stderr,"gain in merging node %d with node %d = %f-%f = %f\n", i, jj, mq, mq_new, gain);
if (j == ia[i] || gain > maxgain) {
maxgain = gain;
jmax = jj;
amax = a_ij;
dout_max = dout_j;
wgt_jmax = wgt_j;
mq_max = mq_new;
mq_in_max = mq_in_new;
mq_out_max = mq_out_new;
}
}
/* now merge i and jmax */
if (maxgain > 0 || (nc >= 1 && nc > maxcluster)) {
total_gain += maxgain;
jc = matching[jmax];
if (jc == UNMATCHED) {
fprintf(stderr, "maxgain=%f, merge %d, %d\n",maxgain, i, jmax);
neighbors[nc] = SingleLinkedList_new_int(jmax);
neighbors[nc] = SingleLinkedList_prepend_int(neighbors[nc], i);
dout_new[nc] = dout_i + dout_max;
matching[i] = matching[jmax] = nc;
wgt_new[nc] = wgt[i] + wgt[jmax];
deg_intra_new[nc] = deg_intra[i] + deg_intra[jmax] + amax;
nc++;
} else {
fprintf(stderr,"maxgain=%f, merge with existing cluster %d, %d\n",maxgain, i, jc);
neighbors[jc] = SingleLinkedList_prepend_int(neighbors[jc], i);
dout_new[jc] = dout_i + dout_max;
wgt_new[jc] += wgt[i];
matching[i] = jc;
deg_intra_new[jc] += deg_intra[i] + amax;
}
mq = mq_max;
mq_in = mq_in_max;
mq_out = mq_out_max;
nclusters--;
} else {
fprintf(stderr,"gain: %f -- no gain, skip merging node %d\n", maxgain, i);
assert(maxgain <= 0);
neighbors[nc] = SingleLinkedList_new_int(i);
matching[i] = nc;
deg_intra_new[nc] = deg_intra[i];
wgt_new[nc] = wgt[i];
nc++;
}
/* update scaled outdegree of neighbors of i and its merged node/cluster jmax */
jc = matching[i];
lst = neighbors[jc];
do {
mask[*((int*) SingleLinkedList_get_data(lst))] = n+i;
lst = SingleLinkedList_get_next(lst);
} while (lst);
lst = neighbors[jc];
do {
k = *((int*) SingleLinkedList_get_data(lst));
for (j = ia[k]; j < ia[k+1]; j++) {
jj = ja[j];
if (mask[jj] == n+i) continue;/* link to within cluster */
if ((jc = matching[jj]) == UNMATCHED) {
if (k == i) {
dout[jj] += -a[j]/wgt_i + a[j]/(wgt_i + wgt_jmax);
} else {
dout[jj] += -a[j]/wgt_jmax + a[j]/(wgt_i + wgt_jmax);
}
} else {
if (k == i) {
dout_new[jc] += -a[j]/wgt_i + a[j]/(wgt_i + wgt_jmax);
} else {
dout_new[jc] += -a[j]/wgt_jmax + a[j]/(wgt_i + wgt_jmax);
}
}
}
lst = SingleLinkedList_get_next(lst);
} while (lst);
}
fprintf(stderr,"verbose=%d\n",Verbose);
if (Verbose) fprintf(stderr,"mq = %f new mq = %f level = %d, n = %d, nc = %d, gain = %g, mq_in = %f, mq_out = %f\n", mq, mq + total_gain,
level, n, nc, total_gain, mq_in, mq_out);
#ifdef DEBUG
{ int ncluster;
mq = get_mq(A, matching, &ncluster, &mq_in, &mq_out, &dout);
fprintf(stderr," mq = %f\n",mq);
}
#endif
if (nc >= 1 && (total_gain > 0 || nc < n)) {
/* now set up restriction and prolongation operator */
SparseMatrix P, R, R0, B, cA;
real one = 1.;
Multilevel_MQ_Clustering cgrid;
R0 = SparseMatrix_new(nc, n, 1, MATRIX_TYPE_REAL, FORMAT_COORD);
for (i = 0; i < n; i++) {
jj = matching[i];
SparseMatrix_coordinate_form_add_entries(R0, 1, &jj, &i, &one);
}
R = SparseMatrix_from_coordinate_format(R0);
SparseMatrix_delete(R0);
P = SparseMatrix_transpose(R);
B = SparseMatrix_multiply(R, A);
if (!B) goto RETURN;
cA = SparseMatrix_multiply(B, P);
if (!cA) goto RETURN;
SparseMatrix_delete(B);
grid->P = P;
grid->R = R;
level++;
cgrid = Multilevel_MQ_Clustering_init(cA, level);
deg_intra_new = REALLOC(deg_intra_new, nc*sizeof(real));
wgt_new = REALLOC(wgt_new, nc*sizeof(real));
cgrid->deg_intra = deg_intra_new;
cgrid->mq = grid->mq + total_gain;
cgrid->wgt = wgt_new;
dout_new = REALLOC(dout_new, nc*sizeof(real));
cgrid->dout = dout_new;
cgrid = Multilevel_MQ_Clustering_establish(cgrid, maxcluster);
grid->next = cgrid;
cgrid->prev = grid;
} else {
/* no more improvement, stop and final clustering found */
for (i = 0; i < n; i++) matching[i] = i;
FREE(deg_intra_new);
FREE(wgt_new);
FREE(dout_new);
}
RETURN:
for (i = 0; i < n; i++) SingleLinkedList_delete(neighbors[i], free);
FREE(neighbors);
FREE(deg_inter);
FREE(mask);
return grid;
}
