igraph_vector_t * ggen_analyze_longest_antichain(igraph_t *g)
{
/* The following steps are implemented :
* - Convert our DAG to a specific bipartite graph B
* - solve maximum matching on B
* - conver maximum matching to min vectex cover
* - convert min vertex cover to antichain on G
*/
int err;
unsigned long i,vg,found,added;
igraph_t b,gstar;
igraph_vector_t edges,*res = NULL;
igraph_vector_t c,s,t,todo,n,next,l,r;
igraph_eit_t eit;
igraph_es_t es;
igraph_integer_t from,to;
igraph_vit_t vit;
igraph_vs_t vs;
igraph_real_t value;
if(g == NULL)
return NULL;
/* before creating the bipartite graph, we need all relations
* between any two vertices : the transitive closure of g */
err = igraph_copy(&gstar,g);
if(err) return NULL;
err = ggen_transform_transitive_closure(&gstar);
if(err) goto error;
/* Bipartite convertion : let G = (S,C),
* we build B = (U,V,E) with
* - U = V = S (each vertex is present twice)
* - (u,v) \in E iff :
* - u \in U
* - v \in V
* - u < v in C (warning, this means that we take
* transitive closure into account, not just the
* original edges)
* We will also need two additional nodes further in the code.
*/
vg = igraph_vcount(g);
err = igraph_empty(&b,vg*2,1);
if(err) goto error;
/* id and id+vg will be a vertex in U and its copy in V,
* iterate over gstar edges to create edges in b
*/
err = igraph_vector_init(&edges,igraph_ecount(&gstar));
if(err) goto d_b;
igraph_vector_clear(&edges);
err = igraph_eit_create(&gstar,igraph_ess_all(IGRAPH_EDGEORDER_ID),&eit);
if(err) goto d_edges;
for(IGRAPH_EIT_RESET(eit); !IGRAPH_EIT_END(eit); IGRAPH_EIT_NEXT(eit))
{
err = igraph_edge(&gstar,IGRAPH_EIT_GET(eit),&from,&to);
if(err)
{
igraph_eit_destroy(&eit);
goto d_edges;
}
to += vg;
igraph_vector_push_back(&edges,(igraph_real_t)from);
igraph_vector_push_back(&edges,(igraph_real_t)to);
}
igraph_eit_destroy(&eit);
err = igraph_add_edges(&b,&edges,NULL);
if(err) goto d_edges;
/* maximum matching on b */
igraph_vector_clear(&edges);
err = bipartite_maximum_matching(&b,&edges);
if(err) goto d_edges;
/* Let M be the max matching, and N be E - M
* Define T as all unmatched vectices from U as well as all vertices
* reachable from those by going left-to-right along N and right-to-left along
* M.
* Define L = U - T, R = V \inter T
* C:= L + R
* C is a minimum vertex cover
*/
err = igraph_vector_init_seq(&n,0,igraph_ecount(&b)-1);
if(err) goto d_edges;
err = vector_diff(&n,&edges);
if(err) goto d_n;
err = igraph_vector_init(&c,vg);
if(err) goto d_n;
igraph_vector_clear(&c);
/* matched vertices : S */
err = igraph_vector_init(&s,vg);
if(err) goto d_c;
igraph_vector_clear(&s);
for(i = 0; i < igraph_vector_size(&edges); i++)
{
err = igraph_edge(&b,VECTOR(edges)[i],&from,&to);
if(err) goto d_s;
igraph_vector_push_back(&s,from);
}
/* we may have inserted the same vertex multiple times */
err = vector_uniq(&s);
if(err) goto d_s;
/* unmatched */
err = igraph_vector_init_seq(&t,0,vg-1);
if(err) goto d_s;
err = vector_diff(&t,&s);
if(err) goto d_t;
/* alternating paths
*/
err = igraph_vector_copy(&todo,&t);
if(err) goto d_t;
err = igraph_vector_init(&next,vg);
if(err) goto d_todo;
igraph_vector_clear(&next);
do {
vector_uniq(&todo);
added = 0;
for(i = 0; i < igraph_vector_size(&todo); i++)
{
if(VECTOR(todo)[i] < vg)
{
/* scan edges */
err = igraph_es_adj(&es,VECTOR(todo)[i],IGRAPH_OUT);
if(err) goto d_next;
err = igraph_eit_create(&b,es,&eit);
if(err)
{
igraph_es_destroy(&es);
goto d_next;
}
for(IGRAPH_EIT_RESET(eit); !IGRAPH_EIT_END(eit); IGRAPH_EIT_NEXT(eit))
{
if(igraph_vector_binsearch(&n,IGRAPH_EIT_GET(eit),NULL))
{
err = igraph_edge(&b,IGRAPH_EIT_GET(eit),&from,&to);
if(err)
{
igraph_eit_destroy(&eit);
igraph_es_destroy(&es);
goto d_next;
}
if(!igraph_vector_binsearch(&t,to,NULL))
{
igraph_vector_push_back(&next,to);
added = 1;
}
}
}
}
else
{
/* scan edges */
err = igraph_es_adj(&es,VECTOR(todo)[i],IGRAPH_IN);
if(err) goto d_next;
err = igraph_eit_create(&b,es,&eit);
if(err)
{
igraph_es_destroy(&es);
goto d_next;
}
for(IGRAPH_EIT_RESET(eit); !IGRAPH_EIT_END(eit); IGRAPH_EIT_NEXT(eit))
{
if(igraph_vector_binsearch(&edges,IGRAPH_EIT_GET(eit),NULL))
{
err = igraph_edge(&b,IGRAPH_EIT_GET(eit),&from,&to);
if(err)
{
igraph_eit_destroy(&eit);
igraph_es_destroy(&es);
goto d_next;
}
if(!igraph_vector_binsearch(&t,to,NULL))
{
igraph_vector_push_back(&next,from);
added = 1;
}
}
}
}
igraph_es_destroy(&es);
igraph_eit_destroy(&eit);
}
igraph_vector_append(&t,&todo);
igraph_vector_clear(&todo);
igraph_vector_append(&todo,&next);
igraph_vector_clear(&next);
} while(added);
err = igraph_vector_init_seq(&l,0,vg-1);
if(err) goto d_t;
err = vector_diff(&l,&t);
if(err) goto d_l;
err = igraph_vector_update(&c,&l);
if(err) goto d_l;
err = igraph_vector_init(&r,vg);
if(err) goto d_l;
igraph_vector_clear(&r);
/* compute V \inter T */
for(i = 0; i < igraph_vector_size(&t); i++)
{
if(VECTOR(t)[i] >= vg)
igraph_vector_push_back(&r,VECTOR(t)[i]);
}
igraph_vector_add_constant(&r,(igraph_real_t)-vg);
err = vector_union(&c,&r);
if(err) goto d_r;
/* our antichain is U - C */
res = malloc(sizeof(igraph_vector_t));
if(res == NULL) goto d_r;
err = igraph_vector_init_seq(res,0,vg-1);
if(err) goto f_res;
err = vector_diff(res,&c);
if(err) goto d_res;
goto ret;
d_res:
igraph_vector_destroy(res);
f_res:
free(res);
res = NULL;
ret:
d_r:
igraph_vector_destroy(&r);
d_l:
igraph_vector_destroy(&l);
d_next:
igraph_vector_destroy(&next);
d_todo:
igraph_vector_destroy(&todo);
d_t:
igraph_vector_destroy(&t);
d_s:
igraph_vector_destroy(&s);
d_c:
igraph_vector_destroy(&c);
d_n:
igraph_vector_destroy(&n);
d_edges:
igraph_vector_destroy(&edges);
d_b:
igraph_destroy(&b);
error:
igraph_destroy(&gstar);
return res;
}
/**
* \ingroup nongraph
* \function igraph_convex_hull
* \brief Determines the convex hull of a given set of points in the 2D plane
*
* </para><para>
* The convex hull is determined by the Graham scan algorithm.
* See the following reference for details:
*
* </para><para>
* Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford
* Stein. Introduction to Algorithms, Second Edition. MIT Press and
* McGraw-Hill, 2001. ISBN 0262032937. Pages 949-955 of section 33.3:
* Finding the convex hull.
*
* \param data vector containing the coordinates. The length of the
* vector must be even, since it contains X-Y coordinate pairs.
* \param resverts the vector containing the result, e.g. the vector of
* vertex indices used as the corners of the convex hull. Supply
* \c NULL here if you are only interested in the coordinates of
* the convex hull corners.
* \param rescoords the matrix containing the coordinates of the selected
* corner vertices. Supply \c NULL here if you are only interested in
* the vertex indices.
* \return Error code:
* \c IGRAPH_ENOMEM: not enough memory
*
* Time complexity: O(n log(n)) where n is the number of vertices
*/
int igraph_convex_hull(const igraph_matrix_t *data, igraph_vector_t *resverts,
igraph_matrix_t *rescoords) {
igraph_integer_t no_of_nodes;
long int i, pivot_idx=0, last_idx, before_last_idx, next_idx, j;
igraph_real_t* angles;
igraph_vector_t stack;
igraph_indheap_t order;
igraph_real_t px, py, cp;
no_of_nodes=igraph_matrix_nrow(data);
if (igraph_matrix_ncol(data) != 2) {
IGRAPH_ERROR("matrix must have 2 columns", IGRAPH_EINVAL);
}
if (no_of_nodes == 0) {
if (resverts != 0) {
IGRAPH_CHECK(igraph_vector_resize(resverts, 0));
}
if (rescoords != 0) {
IGRAPH_CHECK(igraph_matrix_resize(rescoords, 0, 2));
}
/**************************** this is an exit here *********/
return 0;
}
angles=igraph_Calloc(no_of_nodes, igraph_real_t);
if (!angles) IGRAPH_ERROR("not enough memory for angle array", IGRAPH_ENOMEM);
IGRAPH_FINALLY(free, angles);
IGRAPH_VECTOR_INIT_FINALLY(&stack, 0);
/* Search for the pivot vertex */
for (i=1; i<no_of_nodes; i++) {
if (MATRIX(*data, i, 1)<MATRIX(*data, pivot_idx, 1))
pivot_idx=i;
else if (MATRIX(*data, i, 1) == MATRIX(*data, pivot_idx, 1) &&
MATRIX(*data, i, 0) < MATRIX(*data, pivot_idx, 0))
pivot_idx=i;
}
px=MATRIX(*data, pivot_idx, 0);
py=MATRIX(*data, pivot_idx, 1);
/* Create angle array */
for (i=0; i<no_of_nodes; i++) {
if (i == pivot_idx) {
/* We can't calculate the angle of the pivot point with itself,
* so we use 10 here. This way, after sorting the angle vector,
* the pivot point will always be the first one, since the range
* of atan2 is -3.14..3.14 */
angles[i] = 10;
} else {
angles[i] = atan2(MATRIX(*data, i, 1)-py,
MATRIX(*data, i, 0)-px);
}
}
IGRAPH_CHECK(igraph_indheap_init_array(&order, angles, no_of_nodes));
IGRAPH_FINALLY(igraph_indheap_destroy, &order);
igraph_Free(angles);
IGRAPH_FINALLY_CLEAN(1);
if (no_of_nodes == 1) {
IGRAPH_CHECK(igraph_vector_push_back(&stack, 0));
igraph_indheap_delete_max(&order);
} else {
/* Do the trick */
IGRAPH_CHECK(igraph_vector_push_back(&stack, igraph_indheap_max_index(&order)-1));
igraph_indheap_delete_max(&order);
IGRAPH_CHECK(igraph_vector_push_back(&stack, igraph_indheap_max_index(&order)-1));
igraph_indheap_delete_max(&order);
j=2;
while (!igraph_indheap_empty(&order)) {
/* Determine whether we are at a left or right turn */
last_idx=VECTOR(stack)[j-1];
before_last_idx=VECTOR(stack)[j-2];
next_idx=(long)igraph_indheap_max_index(&order)-1;
igraph_indheap_delete_max(&order);
cp=(MATRIX(*data, last_idx, 0)-MATRIX(*data, before_last_idx, 0))*
(MATRIX(*data, next_idx, 1)-MATRIX(*data, before_last_idx, 1))-
(MATRIX(*data, next_idx, 0)-MATRIX(*data, before_last_idx, 0))*
(MATRIX(*data, last_idx, 1)-MATRIX(*data, before_last_idx, 1));
/*
printf("B L N cp: %d, %d, %d, %f [", before_last_idx, last_idx, next_idx, (float)cp);
for (k=0; k<j; k++) printf("%ld ", (long)VECTOR(stack)[k]);
printf("]\n");
*/
if (cp == 0) {
/* The last three points are collinear. Replace the last one in
* the stack to the newest one */
VECTOR(stack)[j-1]=next_idx;
} else if (cp < 0) {
/* We are turning into the right direction */
IGRAPH_CHECK(igraph_vector_push_back(&stack, next_idx));
j++;
} else {
/* No, skip back until we're okay */
while (cp >= 0 && j > 2) {
igraph_vector_pop_back(&stack);
j--;
last_idx=VECTOR(stack)[j-1];
before_last_idx=VECTOR(stack)[j-2];
cp=(MATRIX(*data, last_idx, 0)-MATRIX(*data, before_last_idx, 0))*
(MATRIX(*data, next_idx, 1)-MATRIX(*data, before_last_idx, 1))-
(MATRIX(*data, next_idx, 0)-MATRIX(*data, before_last_idx, 0))*
(MATRIX(*data, last_idx, 1)-MATRIX(*data, before_last_idx, 1));
}
IGRAPH_CHECK(igraph_vector_push_back(&stack, next_idx));
j++;
}
}
}
/* Create result vector */
if (resverts != 0) {
igraph_vector_clear(resverts);
IGRAPH_CHECK(igraph_vector_append(resverts, &stack));
}
if (rescoords != 0) {
igraph_matrix_select_rows(data, rescoords, &stack);
}
/* Free everything */
igraph_vector_destroy(&stack);
igraph_indheap_destroy(&order);
IGRAPH_FINALLY_CLEAN(2);
return 0;
}
/**
* \ingroup nongraph
* \function igraph_convex_hull
* \brief Determines the convex hull of a given set of points in the 2D plane
*
* </para><para>
* The convex hull is determined by the Graham scan algorithm.
* See the following reference for details:
*
* </para><para>
* Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford
* Stein. Introduction to Algorithms, Second Edition. MIT Press and
* McGraw-Hill, 2001. ISBN 0262032937. Pages 949-955 of section 33.3:
* Finding the convex hull.
*
* \param data vector containing the coordinates. The length of the
* vector must be even, since it contains X-Y coordinate pairs.
* \param resverts the vector containing the result, e.g. the vector of
* vertex indices used as the corners of the convex hull. Supply
* \c NULL here if you are only interested in the coordinates of
* the convex hull corners.
* \param rescoords the matrix containing the coordinates of the selected
* corner vertices. Supply \c NULL here if you are only interested in
* the vertex indices.
* \return Error code:
* \c IGRAPH_ENOMEM: not enough memory
*
* Time complexity: O(n log(n)) where n is the number of vertices
*
* \example examples/simple/igraph_convex_hull.c
*/
int igraph_convex_hull(const igraph_matrix_t *data, igraph_vector_t *resverts,
igraph_matrix_t *rescoords) {
igraph_integer_t no_of_nodes;
long int i, pivot_idx=0, last_idx, before_last_idx, next_idx, j;
igraph_vector_t angles, stack, order;
igraph_real_t px, py, cp;
no_of_nodes=(igraph_integer_t) igraph_matrix_nrow(data);
if (igraph_matrix_ncol(data) != 2) {
IGRAPH_ERROR("matrix must have 2 columns", IGRAPH_EINVAL);
}
if (no_of_nodes == 0) {
if (resverts != 0) {
IGRAPH_CHECK(igraph_vector_resize(resverts, 0));
}
if (rescoords != 0) {
IGRAPH_CHECK(igraph_matrix_resize(rescoords, 0, 2));
}
/**************************** this is an exit here *********/
return 0;
}
IGRAPH_VECTOR_INIT_FINALLY(&angles, no_of_nodes);
IGRAPH_VECTOR_INIT_FINALLY(&stack, 0);
/* Search for the pivot vertex */
for (i=1; i<no_of_nodes; i++) {
if (MATRIX(*data, i, 1)<MATRIX(*data, pivot_idx, 1))
pivot_idx=i;
else if (MATRIX(*data, i, 1) == MATRIX(*data, pivot_idx, 1) &&
MATRIX(*data, i, 0) < MATRIX(*data, pivot_idx, 0))
pivot_idx=i;
}
px=MATRIX(*data, pivot_idx, 0);
py=MATRIX(*data, pivot_idx, 1);
/* Create angle array */
for (i=0; i<no_of_nodes; i++) {
if (i == pivot_idx) {
/* We can't calculate the angle of the pivot point with itself,
* so we use 10 here. This way, after sorting the angle vector,
* the pivot point will always be the first one, since the range
* of atan2 is -3.14..3.14 */
VECTOR(angles)[i] = 10;
} else {
VECTOR(angles)[i] = atan2(MATRIX(*data, i, 1)-py, MATRIX(*data, i, 0)-px);
}
}
/* Sort points by angles */
IGRAPH_VECTOR_INIT_FINALLY(&order, no_of_nodes);
IGRAPH_CHECK(igraph_vector_qsort_ind(&angles, &order, 0));
/* Check if two points have the same angle. If so, keep only the point that
* is farthest from the pivot */
j = 0;
last_idx = (long int) VECTOR(order)[0];
pivot_idx = (long int) VECTOR(order)[no_of_nodes - 1];
for (i=1; i < no_of_nodes; i++) {
next_idx = (long int) VECTOR(order)[i];
if (VECTOR(angles)[last_idx] == VECTOR(angles)[next_idx]) {
/* Keep the vertex that is farther from the pivot, drop the one that is
* closer */
px = pow(MATRIX(*data, last_idx, 0) - MATRIX(*data, pivot_idx, 0), 2) +
pow(MATRIX(*data, last_idx, 1) - MATRIX(*data, pivot_idx, 1), 2);
py = pow(MATRIX(*data, next_idx, 0) - MATRIX(*data, pivot_idx, 0), 2) +
pow(MATRIX(*data, next_idx, 1) - MATRIX(*data, pivot_idx, 1), 2);
if (px > py) {
VECTOR(order)[i] = -1;
} else {
VECTOR(order)[j] = -1;
last_idx = next_idx;
j = i;
}
} else {
last_idx = next_idx;
j = i;
}
}
j=0;
last_idx=-1;
before_last_idx=-1;
while (!igraph_vector_empty(&order)) {
next_idx=(long int)VECTOR(order)[igraph_vector_size(&order) - 1];
if (next_idx < 0) {
/* This vertex should be skipped; was excluded in an earlier step */
igraph_vector_pop_back(&order);
continue;
}
/* Determine whether we are at a left or right turn */
if (j < 2) {
/* Pretend that we are turning into the right direction if we have less
* than two items in the stack */
cp=-1;
} else {
cp=(MATRIX(*data, last_idx, 0)-MATRIX(*data, before_last_idx, 0))*
(MATRIX(*data, next_idx, 1)-MATRIX(*data, before_last_idx, 1))-
(MATRIX(*data, next_idx, 0)-MATRIX(*data, before_last_idx, 0))*
(MATRIX(*data, last_idx, 1)-MATRIX(*data, before_last_idx, 1));
}
/*
printf("B L N cp: %ld, %ld, %ld, %f [", before_last_idx, last_idx, next_idx, (float)cp);
for (int k=0; k<j; k++) printf("%ld ", (long)VECTOR(stack)[k]);
printf("]\n");
*/
if (cp < 0) {
/* We are turning into the right direction */
igraph_vector_pop_back(&order);
IGRAPH_CHECK(igraph_vector_push_back(&stack, next_idx));
before_last_idx = last_idx;
last_idx = next_idx;
j++;
} else {
/* No, skip back and try again in the next iteration */
igraph_vector_pop_back(&stack);
j--;
last_idx = before_last_idx;
before_last_idx = (j >= 2) ? (long int) VECTOR(stack)[j-2] : -1;
}
}
/* Create result vector */
if (resverts != 0) {
igraph_vector_clear(resverts);
IGRAPH_CHECK(igraph_vector_append(resverts, &stack));
}
if (rescoords != 0) {
igraph_matrix_select_rows(data, rescoords, &stack);
}
/* Free everything */
igraph_vector_destroy(&order);
igraph_vector_destroy(&stack);
igraph_vector_destroy(&angles);
IGRAPH_FINALLY_CLEAN(3);
return 0;
}