void compute_new_weights(vtx_data * graph, int n)
{
/* Reweight graph so that high degree nodes will be separated */
int i, j;
int nedges = 0;
float *weights;
int *vtx_vec = N_GNEW(n, int);
int deg_i, deg_j, neighbor;
for (i = 0; i < n; i++) {
nedges += graph[i].nedges;
}
weights = N_GNEW(nedges, float);
for (i = 0; i < n; i++) {
vtx_vec[i] = 0;
}
for (i = 0; i < n; i++) {
graph[i].ewgts = weights;
fill_neighbors_vec_unweighted(graph, i, vtx_vec);
deg_i = graph[i].nedges - 1;
for (j = 1; j <= deg_i; j++) {
neighbor = graph[i].edges[j];
deg_j = graph[neighbor].nedges - 1;
weights[j] =
(float) (deg_i + deg_j -
2 * common_neighbors(graph, i, neighbor,
vtx_vec));
}
empty_neighbors_vec(graph, i, vtx_vec);
weights += graph[i].nedges;
}
free(vtx_vec);
}
static int
maxmatch(v_data * graph, /* array of vtx data for graph */
ex_vtx_data * geom_graph, /* array of vtx data for graph */
int nvtxs, /* number of vertices in graph */
int *mflag, /* flag indicating vtx selected or not */
int dist2_limit
)
/*
Compute a matching of the nodes set.
The matching is not based only on the edge list of 'graph',
which might be too small,
but on the wider edge list of 'geom_graph' (which includes 'graph''s edges)
We match nodes that are close both in the graph-theoretical sense and
in the geometry sense (in the layout)
*/
{
int *order; /* random ordering of vertices */
int *iptr, *jptr; /* loops through integer arrays */
int vtx; /* vertex to process next */
int neighbor; /* neighbor of a vertex */
int nmerged = 0; /* number of edges in matching */
int i, j; /* loop counters */
float max_norm_edge_weight;
double inv_size;
double *matchability = N_NEW(nvtxs, double);
double min_edge_len;
double closest_val = -1, val;
int closest_neighbor;
float *vtx_vec = N_NEW(nvtxs, float);
float *weighted_vtx_vec = N_NEW(nvtxs, float);
float sum_weights;
// gather statistics, to enable normalizing the values
double avg_edge_len = 0, avg_deg_2 = 0;
int nedges = 0;
for (i = 0; i < nvtxs; i++) {
avg_deg_2 += graph[i].nedges;
for (j = 1; j < graph[i].nedges; j++) {
avg_edge_len += ddist(geom_graph, i, graph[i].edges[j]);
nedges++;
}
}
avg_edge_len /= nedges;
avg_deg_2 /= nvtxs;
avg_deg_2 *= avg_deg_2;
// the normalized edge weight of edge <v,u> is defined as:
// weight(<v,u>)/sqrt(size(v)*size(u))
// Now we compute the maximal normalized weight
if (graph[0].ewgts != NULL) {
max_norm_edge_weight = -1;
for (i = 0; i < nvtxs; i++) {
inv_size = sqrt(1.0 / geom_graph[i].size);
for (j = 1; j < graph[i].nedges; j++) {
if (graph[i].ewgts[j] * inv_size /
sqrt((float) geom_graph[graph[i].edges[j]].size) >
max_norm_edge_weight) {
max_norm_edge_weight =
(float) (graph[i].ewgts[j] * inv_size /
sqrt((double)
geom_graph[graph[i].edges[j]].size));
}
}
}
} else {
max_norm_edge_weight = 1;
}
/* Now determine the order of the vertices. */
iptr = order = N_NEW(nvtxs, int);
jptr = mflag;
for (i = 0; i < nvtxs; i++) {
*(iptr++) = i;
*(jptr++) = -1;
}
// Option 1: random permutation
#if 0
int temp;
for (i=0; i<nvtxs-1; i++) {
// use long_rand() (not rand()), as n may be greater than RAND_MAX
j=i+long_rand()%(nvtxs-i);
temp=order[i];
order[i]=order[j];
order[j]=temp;
}
#endif
// Option 2: sort the nodes begining with the ones highly approriate for matching
#ifdef DEBUG
srand(0);
#endif
for (i = 0; i < nvtxs; i++) {
vtx = order[i];
matchability[vtx] = graph[vtx].nedges; // we less want to match high degree nodes
matchability[vtx] += geom_graph[vtx].size; // we less want to match large sized nodes
min_edge_len = 1e99;
for (j = 1; j < graph[vtx].nedges; j++) {
min_edge_len =
MIN(min_edge_len,
ddist(geom_graph, vtx,
graph[vtx].edges[j]) / avg_edge_len);
}
matchability[vtx] += min_edge_len; // we less want to match distant nodes
matchability[vtx] += ((double) rand()) / RAND_MAX; // add some randomness
}
quicksort_place(matchability, order, 0, nvtxs - 1);
free(matchability);
// Start determining the matched pairs
for (i = 0; i < nvtxs; i++) {
vtx_vec[i] = 0;
}
for (i = 0; i < nvtxs; i++) {
weighted_vtx_vec[i] = 0;
}
// relative weights of the different criteria
for (i = 0; i < nvtxs; i++) {
vtx = order[i];
if (mflag[vtx] >= 0) { /* already matched. */
continue;
}
inv_size = sqrt(1.0 / geom_graph[vtx].size);
sum_weights = fill_neighbors_vec(graph, vtx, weighted_vtx_vec);
fill_neighbors_vec_unweighted(graph, vtx, vtx_vec);
closest_neighbor = -1;
/*
We match node i with the "closest" neighbor, based on 4 criteria:
(1) (Weighted) fraction of common neighbors (measured on orig. graph)
(2) AvgDeg*AvgDeg/(deg(vtx)*deg(neighbor)) (degrees measured on orig. graph)
(3) AvgEdgeLen/dist(vtx,neighbor)
(4) Weight of normalized direct connection between nodes (measured on orig. graph)
*/
for (j = 1; j < geom_graph[vtx].nedges; j++) {
neighbor = geom_graph[vtx].edges[j];
if (mflag[neighbor] >= 0) { /* already matched. */
continue;
}
// (1):
val =
A * unweighted_common_fraction(graph, vtx, neighbor,
vtx_vec);
if (val == 0 && (dist2_limit || !dist3(graph, vtx, neighbor))) {
// graph theoretical distance is larger than 3 (or 2 if '!dist3(graph, vtx, neighbor)' is commented)
// nodes cannot be matched
continue;
}
// (2)
val +=
B * avg_deg_2 / (graph[vtx].nedges *
graph[neighbor].nedges);
// (3)
val += C * avg_edge_len / ddist(geom_graph, vtx, neighbor);
// (4)
val +=
(weighted_vtx_vec[neighbor] * inv_size /
sqrt((float) geom_graph[neighbor].size)) /
max_norm_edge_weight;
if (val > closest_val || closest_neighbor == -1) {
closest_neighbor = neighbor;
closest_val = val;
}
}
if (closest_neighbor != -1) {
mflag[vtx] = closest_neighbor;
mflag[closest_neighbor] = vtx;
nmerged++;
}
empty_neighbors_vec(graph, vtx, vtx_vec);
empty_neighbors_vec(graph, vtx, weighted_vtx_vec);
}
free(order);
free(vtx_vec);
free(weighted_vtx_vec);
return (nmerged);
}