X-Git-Url: http://review.tizen.org/git/?a=blobdiff_plain;f=isl_convex_hull.c;h=0bab5fa5b67b92c183ad68dbe2d9e6c2bfba4d06;hb=a3474ffbe9847cd6df8936df50f0d0f351d86afe;hp=50795617df2718a8022ff3cf9f3aedd8a7124607;hpb=7c083539dd3121d783fc1d3d8a241826b7e6dbdd;p=platform%2Fupstream%2Fisl.git

diff --git a/isl_convex_hull.c b/isl_convex_hull.c
index 5079561..0bab5fa 100644
--- a/isl_convex_hull.c
+++ b/isl_convex_hull.c
@@ -50,7 +50,8 @@ int isl_basic_map_constraint_is_redundant(struct isl_basic_map **bmap,
 	if (i < total)
 		return 0;
 
-	res = isl_solve_lp(*bmap, 0, c, (*bmap)->ctx->one, opt_n, opt_d);
+	res = isl_basic_map_solve_lp(*bmap, 0, c, (*bmap)->ctx->one,
+					opt_n, opt_d, NULL);
 	if (res == isl_lp_unbounded)
 		return 0;
 	if (res == isl_lp_error)
@@ -93,10 +94,10 @@ struct isl_basic_map *isl_basic_map_convex_hull(struct isl_basic_map *bmap)
 		return bmap;
 
 	tab = isl_tab_from_basic_map(bmap);
-	tab = isl_tab_detect_equalities(bmap->ctx, tab);
-	tab = isl_tab_detect_redundant(bmap->ctx, tab);
+	tab = isl_tab_detect_equalities(tab);
+	tab = isl_tab_detect_redundant(tab);
 	bmap = isl_basic_map_update_from_tab(bmap, tab);
-	isl_tab_free(bmap->ctx, tab);
+	isl_tab_free(tab);
 	ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
 	ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
 	return bmap;
@@ -112,8 +113,7 @@ struct isl_basic_set *isl_basic_set_convex_hull(struct isl_basic_set *bset)
  * constraint c and if so, set the constant term such that the
  * resulting constraint is a bounding constraint for the set.
  */
-static int uset_is_bound(struct isl_ctx *ctx, struct isl_set *set,
-	isl_int *c, unsigned len)
+static int uset_is_bound(struct isl_set *set, isl_int *c, unsigned len)
 {
 	int first;
 	int j;
@@ -129,8 +129,8 @@ static int uset_is_bound(struct isl_ctx *ctx, struct isl_set *set,
 		if (ISL_F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY))
 			continue;
 
-		res = isl_solve_lp((struct isl_basic_map*)set->p[j],
-				0, c, ctx->one, &opt, &opt_denom);
+		res = isl_basic_set_solve_lp(set->p[j],
+				0, c, set->ctx->one, &opt, &opt_denom, NULL);
 		if (res == isl_lp_unbounded)
 			break;
 		if (res == isl_lp_error)
@@ -163,8 +163,7 @@ error:
  * hyperplane (but not necessarily a facet).
  * Assumes set "set" is bounded.
  */
-static int is_independent_bound(struct isl_ctx *ctx,
-	struct isl_set *set, isl_int *c,
+static int is_independent_bound(struct isl_set *set, isl_int *c,
 	struct isl_mat *dirs, int n)
 {
 	int is_bound;
@@ -190,7 +189,7 @@ static int is_independent_bound(struct isl_ctx *ctx,
 		}
 	}
 
-	is_bound = uset_is_bound(ctx, set, dirs->row[n], dirs->n_col);
+	is_bound = uset_is_bound(set, dirs->row[n], dirs->n_col);
 	if (is_bound != 1)
 		return is_bound;
 	if (i < n) {
@@ -207,14 +206,13 @@ static int is_independent_bound(struct isl_ctx *ctx,
  * on the set "set", based on the constraints of the basic sets
  * in "set".
  */
-static struct isl_mat *independent_bounds(struct isl_ctx *ctx,
-	struct isl_set *set)
+static struct isl_mat *independent_bounds(struct isl_set *set)
 {
 	int i, j, n;
 	struct isl_mat *dirs = NULL;
 	unsigned dim = isl_set_n_dim(set);
 
-	dirs = isl_mat_alloc(ctx, dim, 1+dim);
+	dirs = isl_mat_alloc(set->ctx, dim, 1+dim);
 	if (!dirs)
 		goto error;
 
@@ -224,16 +222,14 @@ static struct isl_mat *independent_bounds(struct isl_ctx *ctx,
 		struct isl_basic_set *bset = set->p[i];
 
 		for (j = 0; n < dim && j < bset->n_eq; ++j) {
-			f = is_independent_bound(ctx, set, bset->eq[j],
-						dirs, n);
+			f = is_independent_bound(set, bset->eq[j], dirs, n);
 			if (f < 0)
 				goto error;
 			if (f)
 				++n;
 		}
 		for (j = 0; n < dim && j < bset->n_ineq; ++j) {
-			f = is_independent_bound(ctx, set, bset->ineq[j],
-						dirs, n);
+			f = is_independent_bound(set, bset->ineq[j], dirs, n);
 			if (f < 0)
 				goto error;
 			if (f)
@@ -243,12 +239,11 @@ static struct isl_mat *independent_bounds(struct isl_ctx *ctx,
 	dirs->n_row = n;
 	return dirs;
 error:
-	isl_mat_free(ctx, dirs);
+	isl_mat_free(dirs);
 	return NULL;
 }
 
-static struct isl_basic_set *isl_basic_set_set_rational(
-	struct isl_basic_set *bset)
+struct isl_basic_set *isl_basic_set_set_rational(struct isl_basic_set *bset)
 {
 	if (!bset)
 		return NULL;
@@ -283,7 +278,7 @@ error:
 	return NULL;
 }
 
-static struct isl_basic_set *isl_basic_set_add_equality(struct isl_ctx *ctx,
+static struct isl_basic_set *isl_basic_set_add_equality(
 	struct isl_basic_set *bset, isl_int *c)
 {
 	int i;
@@ -308,8 +303,7 @@ error:
 	return NULL;
 }
 
-static struct isl_set *isl_set_add_equality(struct isl_ctx *ctx,
-	struct isl_set *set, isl_int *c)
+static struct isl_set *isl_set_add_equality(struct isl_set *set, isl_int *c)
 {
 	int i;
 
@@ -317,7 +311,7 @@ static struct isl_set *isl_set_add_equality(struct isl_ctx *ctx,
 	if (!set)
 		return NULL;
 	for (i = 0; i < set->n; ++i) {
-		set->p[i] = isl_basic_set_add_equality(ctx, set->p[i], c);
+		set->p[i] = isl_basic_set_add_equality(set->p[i], c);
 		if (!set->p[i])
 			goto error;
 	}
@@ -471,7 +465,7 @@ static isl_int *wrap_facet(struct isl_set *set, isl_int *facet, isl_int *ridge)
 	isl_seq_clr(T->row[0]+1, dim - 1);
 	isl_seq_cpy(T->row[1], facet, dim);
 	isl_seq_cpy(T->row[2], ridge, dim);
-	T = isl_mat_right_inverse(set->ctx, T);
+	T = isl_mat_right_inverse(T);
 	set = isl_set_preimage(set, T);
 	T = NULL;
 	if (!set)
@@ -488,22 +482,22 @@ static isl_int *wrap_facet(struct isl_set *set, isl_int *facet, isl_int *ridge)
 	}
 	isl_int_init(num);
 	isl_int_init(den);
-	res = isl_solve_lp((struct isl_basic_map *)lp, 0,
-				    obj->block.data, set->ctx->one, &num, &den);
+	res = isl_basic_set_solve_lp(lp, 0,
+			    obj->block.data, set->ctx->one, &num, &den, NULL);
 	if (res == isl_lp_ok) {
 		isl_int_neg(num, num);
 		isl_seq_combine(facet, num, facet, den, ridge, dim);
 	}
 	isl_int_clear(num);
 	isl_int_clear(den);
-	isl_vec_free(set->ctx, obj);
+	isl_vec_free(obj);
 	isl_basic_set_free(lp);
 	isl_set_free(set);
 	isl_assert(set->ctx, res == isl_lp_ok, return NULL);
 	return facet;
 error:
 	isl_basic_set_free(lp);
-	isl_mat_free(set->ctx, T);
+	isl_mat_free(T);
 	isl_set_free(set);
 	return NULL;
 }
@@ -520,8 +514,8 @@ error:
  * The resulting linear combination of the bounding constraints will
  * correspond to a facet of the convex hull.
  */
-static struct isl_mat *initial_facet_constraint(struct isl_ctx *ctx,
-	struct isl_set *set, struct isl_mat *bounds)
+static struct isl_mat *initial_facet_constraint(struct isl_set *set,
+	struct isl_mat *bounds)
 {
 	struct isl_set *slice = NULL;
 	struct isl_basic_set *face = NULL;
@@ -534,7 +528,7 @@ static struct isl_mat *initial_facet_constraint(struct isl_ctx *ctx,
 
 	while (bounds->n_row > 1) {
 		slice = isl_set_copy(set);
-		slice = isl_set_add_equality(ctx, slice, bounds->row[0]);
+		slice = isl_set_add_equality(slice, bounds->row[0]);
 		face = isl_set_affine_hull(slice);
 		if (!face)
 			goto error;
@@ -542,23 +536,21 @@ static struct isl_mat *initial_facet_constraint(struct isl_ctx *ctx,
 			isl_basic_set_free(face);
 			break;
 		}
-		m = isl_mat_alloc(ctx, 1 + face->n_eq, 1 + dim);
+		m = isl_mat_alloc(set->ctx, 1 + face->n_eq, 1 + dim);
 		if (!m)
 			goto error;
 		isl_int_set_si(m->row[0][0], 1);
 		isl_seq_clr(m->row[0]+1, dim);
 		for (i = 0; i < face->n_eq; ++i)
 			isl_seq_cpy(m->row[1 + i], face->eq[i], 1 + dim);
-		U = isl_mat_right_inverse(ctx, m);
-		Q = isl_mat_right_inverse(ctx, isl_mat_copy(ctx, U));
-		U = isl_mat_drop_cols(ctx, U, 1 + face->n_eq,
-						dim - face->n_eq);
-		Q = isl_mat_drop_rows(ctx, Q, 1 + face->n_eq,
-						dim - face->n_eq);
-		U = isl_mat_drop_cols(ctx, U, 0, 1);
-		Q = isl_mat_drop_rows(ctx, Q, 0, 1);
-		bounds = isl_mat_product(ctx, bounds, U);
-		bounds = isl_mat_product(ctx, bounds, Q);
+		U = isl_mat_right_inverse(m);
+		Q = isl_mat_right_inverse(isl_mat_copy(U));
+		U = isl_mat_drop_cols(U, 1 + face->n_eq, dim - face->n_eq);
+		Q = isl_mat_drop_rows(Q, 1 + face->n_eq, dim - face->n_eq);
+		U = isl_mat_drop_cols(U, 0, 1);
+		Q = isl_mat_drop_rows(Q, 0, 1);
+		bounds = isl_mat_product(bounds, U);
+		bounds = isl_mat_product(bounds, Q);
 		while (isl_seq_first_non_zero(bounds->row[bounds->n_row-1],
 					      bounds->n_col) == -1) {
 			bounds->n_row--;
@@ -573,7 +565,7 @@ static struct isl_mat *initial_facet_constraint(struct isl_ctx *ctx,
 	return bounds;
 error:
 	isl_basic_set_free(face);
-	isl_mat_free(ctx, bounds);
+	isl_mat_free(bounds);
 	return NULL;
 }
 
@@ -632,10 +624,10 @@ static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c)
 	isl_int_set_si(m->row[0][0], 1);
 	isl_seq_clr(m->row[0]+1, dim);
 	isl_seq_cpy(m->row[1], c, 1+dim);
-	U = isl_mat_right_inverse(set->ctx, m);
-	Q = isl_mat_right_inverse(set->ctx, isl_mat_copy(set->ctx, U));
-	U = isl_mat_drop_cols(set->ctx, U, 1, 1);
-	Q = isl_mat_drop_rows(set->ctx, Q, 1, 1);
+	U = isl_mat_right_inverse(m);
+	Q = isl_mat_right_inverse(isl_mat_copy(U));
+	U = isl_mat_drop_cols(U, 1, 1);
+	Q = isl_mat_drop_rows(Q, 1, 1);
 	set = isl_set_preimage(set, U);
 	facet = uset_convex_hull_wrap_bounded(set);
 	facet = isl_basic_set_preimage(facet, Q);
@@ -684,11 +676,11 @@ static struct isl_basic_set *extend(struct isl_basic_set *hull,
 
 	for (i = 0; i < hull->n_ineq; ++i) {
 		facet = compute_facet(set, hull->ineq[i]);
-		facet = isl_basic_set_add_equality(facet->ctx, facet, hull->ineq[i]);
+		facet = isl_basic_set_add_equality(facet, hull->ineq[i]);
 		facet = isl_basic_set_gauss(facet, NULL);
 		facet = isl_basic_set_normalize_constraints(facet);
 		hull_facet = isl_basic_set_copy(hull);
-		hull_facet = isl_basic_set_add_equality(hull_facet->ctx, hull_facet, hull->ineq[i]);
+		hull_facet = isl_basic_set_add_equality(hull_facet, hull->ineq[i]);
 		hull_facet = isl_basic_set_gauss(hull_facet, NULL);
 		hull_facet = isl_basic_set_normalize_constraints(hull_facet);
 		if (!facet)
@@ -727,8 +719,7 @@ error:
  * We simply collect the lower and upper bounds of each basic set
  * and the biggest of those.
  */
-static struct isl_basic_set *convex_hull_1d(struct isl_ctx *ctx,
-	struct isl_set *set)
+static struct isl_basic_set *convex_hull_1d(struct isl_set *set)
 {
 	struct isl_mat *c = NULL;
 	isl_int *lower = NULL;
@@ -745,13 +736,13 @@ static struct isl_basic_set *convex_hull_1d(struct isl_ctx *ctx,
 	set = isl_set_remove_empty_parts(set);
 	if (!set)
 		goto error;
-	isl_assert(ctx, set->n > 0, goto error);
-	c = isl_mat_alloc(ctx, 2, 2);
+	isl_assert(set->ctx, set->n > 0, goto error);
+	c = isl_mat_alloc(set->ctx, 2, 2);
 	if (!c)
 		goto error;
 
 	if (set->p[0]->n_eq > 0) {
-		isl_assert(ctx, set->p[0]->n_eq == 1, goto error);
+		isl_assert(set->ctx, set->p[0]->n_eq == 1, goto error);
 		lower = c->row[0];
 		upper = c->row[1];
 		if (isl_int_is_pos(set->p[0]->eq[0][1])) {
@@ -826,7 +817,7 @@ static struct isl_basic_set *convex_hull_1d(struct isl_ctx *ctx,
 	isl_int_clear(a);
 	isl_int_clear(b);
 
-	hull = isl_basic_set_alloc(ctx, 0, 1, 0, 0, 2);
+	hull = isl_basic_set_alloc(set->ctx, 0, 1, 0, 0, 2);
 	hull = isl_basic_set_set_rational(hull);
 	if (!hull)
 		goto error;
@@ -840,11 +831,11 @@ static struct isl_basic_set *convex_hull_1d(struct isl_ctx *ctx,
 	}
 	hull = isl_basic_set_finalize(hull);
 	isl_set_free(set);
-	isl_mat_free(ctx, c);
+	isl_mat_free(c);
 	return hull;
 error:
 	isl_set_free(set);
-	isl_mat_free(ctx, c);
+	isl_mat_free(c);
 	return NULL;
 }
 
@@ -879,7 +870,7 @@ static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
  * to the two original basic sets, retaining only those corresponding
  * to the convex hull.
  */
-static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
+static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1,
 	struct isl_basic_set *bset2)
 {
 	int i, j, k;
@@ -943,12 +934,540 @@ error:
 	return NULL;
 }
 
+static int isl_basic_set_is_bounded(struct isl_basic_set *bset)
+{
+	struct isl_tab *tab;
+	int bounded;
+
+	tab = isl_tab_from_recession_cone((struct isl_basic_map *)bset);
+	bounded = isl_tab_cone_is_bounded(tab);
+	isl_tab_free(tab);
+	return bounded;
+}
+
+static int isl_set_is_bounded(struct isl_set *set)
+{
+	int i;
+
+	for (i = 0; i < set->n; ++i) {
+		int bounded = isl_basic_set_is_bounded(set->p[i]);
+		if (!bounded || bounded < 0)
+			return bounded;
+	}
+	return 1;
+}
+
+/* Compute the lineality space of the convex hull of bset1 and bset2.
+ *
+ * We first compute the intersection of the recession cone of bset1
+ * with the negative of the recession cone of bset2 and then compute
+ * the linear hull of the resulting cone.
+ */
+static struct isl_basic_set *induced_lineality_space(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	int i, k;
+	struct isl_basic_set *lin = NULL;
+	unsigned dim;
+
+	if (!bset1 || !bset2)
+		goto error;
+
+	dim = isl_basic_set_total_dim(bset1);
+	lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1), 0,
+					bset1->n_eq + bset2->n_eq,
+					bset1->n_ineq + bset2->n_ineq);
+	lin = isl_basic_set_set_rational(lin);
+	if (!lin)
+		goto error;
+	for (i = 0; i < bset1->n_eq; ++i) {
+		k = isl_basic_set_alloc_equality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->eq[k][0], 0);
+		isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
+	}
+	for (i = 0; i < bset1->n_ineq; ++i) {
+		k = isl_basic_set_alloc_inequality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->ineq[k][0], 0);
+		isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
+	}
+	for (i = 0; i < bset2->n_eq; ++i) {
+		k = isl_basic_set_alloc_equality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->eq[k][0], 0);
+		isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
+	}
+	for (i = 0; i < bset2->n_ineq; ++i) {
+		k = isl_basic_set_alloc_inequality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->ineq[k][0], 0);
+		isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
+	}
+
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return isl_basic_set_affine_hull(lin);
+error:
+	isl_basic_set_free(lin);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return NULL;
+}
+
+static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
+
+/* Given a set and a linear space "lin" of dimension n > 0,
+ * project the linear space from the set, compute the convex hull
+ * and then map the set back to the original space.
+ *
+ * Let
+ *
+ *	M x = 0
+ *
+ * describe the linear space.  We first compute the Hermite normal
+ * form H = M U of M = H Q, to obtain
+ *
+ *	H Q x = 0
+ *
+ * The last n rows of H will be zero, so the last n variables of x' = Q x
+ * are the one we want to project out.  We do this by transforming each
+ * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
+ * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
+ * we transform the hull back to the original space as A' Q_1 x >= b',
+ * with Q_1 all but the last n rows of Q.
+ */
+static struct isl_basic_set *modulo_lineality(struct isl_set *set,
+	struct isl_basic_set *lin)
+{
+	unsigned total = isl_basic_set_total_dim(lin);
+	unsigned lin_dim;
+	struct isl_basic_set *hull;
+	struct isl_mat *M, *U, *Q;
+
+	if (!set || !lin)
+		goto error;
+	lin_dim = total - lin->n_eq;
+	M = isl_mat_sub_alloc(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
+	M = isl_mat_left_hermite(M, 0, &U, &Q);
+	if (!M)
+		goto error;
+	isl_mat_free(M);
+	isl_basic_set_free(lin);
+
+	Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim);
+
+	U = isl_mat_lin_to_aff(U);
+	Q = isl_mat_lin_to_aff(Q);
+
+	set = isl_set_preimage(set, U);
+	set = isl_set_remove_dims(set, total - lin_dim, lin_dim);
+	hull = uset_convex_hull(set);
+	hull = isl_basic_set_preimage(hull, Q);
+
+	return hull;
+error:
+	isl_basic_set_free(lin);
+	isl_set_free(set);
+	return NULL;
+}
+
+/* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
+ * set up an LP for solving
+ *
+ *	\sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
+ *
+ * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
+ * The next \alpha{ij} correspond to the equalities and come in pairs.
+ * The final \alpha{ij} correspond to the inequalities.
+ */
+static struct isl_basic_set *valid_direction_lp(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	struct isl_dim *dim;
+	struct isl_basic_set *lp;
+	unsigned d;
+	int n;
+	int i, j, k;
+
+	if (!bset1 || !bset2)
+		goto error;
+	d = 1 + isl_basic_set_total_dim(bset1);
+	n = 2 +
+	    2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
+	dim = isl_dim_set_alloc(bset1->ctx, 0, n);
+	lp = isl_basic_set_alloc_dim(dim, 0, d, n);
+	if (!lp)
+		goto error;
+	for (i = 0; i < n; ++i) {
+		k = isl_basic_set_alloc_inequality(lp);
+		if (k < 0)
+			goto error;
+		isl_seq_clr(lp->ineq[k] + 1, n);
+		isl_int_set_si(lp->ineq[k][0], -1);
+		isl_int_set_si(lp->ineq[k][1 + i], 1);
+	}
+	for (i = 0; i < d; ++i) {
+		k = isl_basic_set_alloc_equality(lp);
+		if (k < 0)
+			goto error;
+		n = 0;
+		isl_int_set_si(lp->eq[k][n++], 0);
+		/* positivity constraint 1 >= 0 */
+		isl_int_set_si(lp->eq[k][n++], i == 0);
+		for (j = 0; j < bset1->n_eq; ++j) {
+			isl_int_set(lp->eq[k][n++], bset1->eq[j][i]);
+			isl_int_neg(lp->eq[k][n++], bset1->eq[j][i]);
+		}
+		for (j = 0; j < bset1->n_ineq; ++j)
+			isl_int_set(lp->eq[k][n++], bset1->ineq[j][i]);
+		/* positivity constraint 1 >= 0 */
+		isl_int_set_si(lp->eq[k][n++], -(i == 0));
+		for (j = 0; j < bset2->n_eq; ++j) {
+			isl_int_neg(lp->eq[k][n++], bset2->eq[j][i]);
+			isl_int_set(lp->eq[k][n++], bset2->eq[j][i]);
+		}
+		for (j = 0; j < bset2->n_ineq; ++j)
+			isl_int_neg(lp->eq[k][n++], bset2->ineq[j][i]);
+	}
+	lp = isl_basic_set_gauss(lp, NULL);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return lp;
+error:
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return NULL;
+}
+
+/* Compute a vector s in the homogeneous space such that <s, r> > 0
+ * for all rays in the homogeneous space of the two cones that correspond
+ * to the input polyhedra bset1 and bset2.
+ *
+ * We compute s as a vector that satisfies
+ *
+ *	s = \sum_j \alpha_{ij} h_{ij}	for i = 1,2			(*)
+ *
+ * with h_{ij} the normals of the facets of polyhedron i
+ * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
+ * strictly positive numbers.  For simplicity we impose \alpha_{ij} >= 1.
+ * We first set up an LP with as variables the \alpha{ij}.
+ * In this formulateion, for each polyhedron i,
+ * the first constraint is the positivity constraint, followed by pairs
+ * of variables for the equalities, followed by variables for the inequalities.
+ * We then simply pick a feasible solution and compute s using (*).
+ *
+ * Note that we simply pick any valid direction and make no attempt
+ * to pick a "good" or even the "best" valid direction.
+ */
+static struct isl_vec *valid_direction(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	struct isl_basic_set *lp;
+	struct isl_tab *tab;
+	struct isl_vec *sample = NULL;
+	struct isl_vec *dir;
+	unsigned d;
+	int i;
+	int n;
+
+	if (!bset1 || !bset2)
+		goto error;
+	lp = valid_direction_lp(isl_basic_set_copy(bset1),
+				isl_basic_set_copy(bset2));
+	tab = isl_tab_from_basic_set(lp);
+	sample = isl_tab_get_sample_value(tab);
+	isl_tab_free(tab);
+	isl_basic_set_free(lp);
+	if (!sample)
+		goto error;
+	d = isl_basic_set_total_dim(bset1);
+	dir = isl_vec_alloc(bset1->ctx, 1 + d);
+	if (!dir)
+		goto error;
+	isl_seq_clr(dir->block.data + 1, dir->size - 1);
+	n = 1;
+	/* positivity constraint 1 >= 0 */
+	isl_int_set(dir->block.data[0], sample->block.data[n++]);
+	for (i = 0; i < bset1->n_eq; ++i) {
+		isl_int_sub(sample->block.data[n],
+			    sample->block.data[n], sample->block.data[n+1]);
+		isl_seq_combine(dir->block.data,
+				bset1->ctx->one, dir->block.data,
+				sample->block.data[n], bset1->eq[i], 1 + d);
+
+		n += 2;
+	}
+	for (i = 0; i < bset1->n_ineq; ++i)
+		isl_seq_combine(dir->block.data,
+				bset1->ctx->one, dir->block.data,
+				sample->block.data[n++], bset1->ineq[i], 1 + d);
+	isl_vec_free(sample);
+	isl_seq_normalize(bset1->ctx, dir->block.data + 1, dir->size - 1);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return dir;
+error:
+	isl_vec_free(sample);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return NULL;
+}
+
+/* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
+ * compute b_i' + A_i' x' >= 0, with
+ *
+ *	[ b_i A_i ]        [ y' ]		              [ y' ]
+ *	[  1   0  ] S^{-1} [ x' ] >= 0	or	[ b_i' A_i' ] [ x' ] >= 0
+ *
+ * In particular, add the "positivity constraint" and then perform
+ * the mapping.
+ */
+static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
+	struct isl_mat *T)
+{
+	int k;
+
+	if (!bset)
+		goto error;
+	bset = isl_basic_set_extend_constraints(bset, 0, 1);
+	k = isl_basic_set_alloc_inequality(bset);
+	if (k < 0)
+		goto error;
+	isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
+	isl_int_set_si(bset->ineq[k][0], 1);
+	bset = isl_basic_set_preimage(bset, T);
+	return bset;
+error:
+	isl_mat_free(T);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Compute the convex hull of a pair of basic sets without any parameters or
+ * integer divisions, where the convex hull is known to be pointed,
+ * but the basic sets may be unbounded.
+ *
+ * We turn this problem into the computation of a convex hull of a pair
+ * _bounded_ polyhedra by "changing the direction of the homogeneous
+ * dimension".  This idea is due to Matthias Koeppe.
+ *
+ * Consider the cones in homogeneous space that correspond to the
+ * input polyhedra.  The rays of these cones are also rays of the
+ * polyhedra if the coordinate that corresponds to the homogeneous
+ * dimension is zero.  That is, if the inner product of the rays
+ * with the homogeneous direction is zero.
+ * The cones in the homogeneous space can also be considered to
+ * correspond to other pairs of polyhedra by chosing a different
+ * homogeneous direction.  To ensure that both of these polyhedra
+ * are bounded, we need to make sure that all rays of the cones
+ * correspond to vertices and not to rays.
+ * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
+ * Then using s as a homogeneous direction, we obtain a pair of polytopes.
+ * The vector s is computed in valid_direction.
+ *
+ * Note that we need to consider _all_ rays of the cones and not just
+ * the rays that correspond to rays in the polyhedra.  If we were to
+ * only consider those rays and turn them into vertices, then we
+ * may inadvertently turn some vertices into rays.
+ *
+ * The standard homogeneous direction is the unit vector in the 0th coordinate.
+ * We therefore transform the two polyhedra such that the selected
+ * direction is mapped onto this standard direction and then proceed
+ * with the normal computation.
+ * Let S be a non-singular square matrix with s as its first row,
+ * then we want to map the polyhedra to the space
+ *
+ *	[ y' ]     [ y ]		[ y ]          [ y' ]
+ *	[ x' ] = S [ x ]	i.e.,	[ x ] = S^{-1} [ x' ]
+ *
+ * We take S to be the unimodular completion of s to limit the growth
+ * of the coefficients in the following computations.
+ *
+ * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
+ * We first move to the homogeneous dimension
+ *
+ *	b_i y + A_i x >= 0		[ b_i A_i ] [ y ]    [ 0 ]
+ *	    y         >= 0	or	[  1   0  ] [ x ] >= [ 0 ]
+ *
+ * Then we change directoin
+ *
+ *	[ b_i A_i ]        [ y' ]		              [ y' ]
+ *	[  1   0  ] S^{-1} [ x' ] >= 0	or	[ b_i' A_i' ] [ x' ] >= 0
+ *
+ * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
+ * resulting in b' + A' x' >= 0, which we then convert back
+ *
+ *	            [ y ]		        [ y ]
+ *	[ b' A' ] S [ x ] >= 0	or	[ b A ] [ x ] >= 0
+ *
+ * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
+ */
+static struct isl_basic_set *convex_hull_pair_pointed(
+	struct isl_basic_set *bset1, struct isl_basic_set *bset2)
+{
+	struct isl_ctx *ctx = NULL;
+	struct isl_vec *dir = NULL;
+	struct isl_mat *T = NULL;
+	struct isl_mat *T2 = NULL;
+	struct isl_basic_set *hull;
+	struct isl_set *set;
+
+	if (!bset1 || !bset2)
+		goto error;
+	ctx = bset1->ctx;
+	dir = valid_direction(isl_basic_set_copy(bset1),
+				isl_basic_set_copy(bset2));
+	if (!dir)
+		goto error;
+	T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
+	if (!T)
+		goto error;
+	isl_seq_cpy(T->row[0], dir->block.data, dir->size);
+	T = isl_mat_unimodular_complete(T, 1);
+	T2 = isl_mat_right_inverse(isl_mat_copy(T));
+
+	bset1 = homogeneous_map(bset1, isl_mat_copy(T2));
+	bset2 = homogeneous_map(bset2, T2);
+	set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
+	set = isl_set_add(set, bset1);
+	set = isl_set_add(set, bset2);
+	hull = uset_convex_hull(set);
+	hull = isl_basic_set_preimage(hull, T);
+	 
+	isl_vec_free(dir);
+
+	return hull;
+error:
+	isl_vec_free(dir);
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return NULL;
+}
+
+/* Compute the convex hull of a pair of basic sets without any parameters or
+ * integer divisions.
+ *
+ * If the convex hull of the two basic sets would have a non-trivial
+ * lineality space, we first project out this lineality space.
+ */
+static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
+	struct isl_basic_set *bset2)
+{
+	struct isl_basic_set *lin;
+
+	if (isl_basic_set_is_bounded(bset1) || isl_basic_set_is_bounded(bset2))
+		return convex_hull_pair_pointed(bset1, bset2);
+
+	lin = induced_lineality_space(isl_basic_set_copy(bset1),
+				      isl_basic_set_copy(bset2));
+	if (!lin)
+		goto error;
+	if (isl_basic_set_is_universe(lin)) {
+		isl_basic_set_free(bset1);
+		isl_basic_set_free(bset2);
+		return lin;
+	}
+	if (lin->n_eq < isl_basic_set_total_dim(lin)) {
+		struct isl_set *set;
+		set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
+		set = isl_set_add(set, bset1);
+		set = isl_set_add(set, bset2);
+		return modulo_lineality(set, lin);
+	}
+	isl_basic_set_free(lin);
+
+	return convex_hull_pair_pointed(bset1, bset2);
+error:
+	isl_basic_set_free(bset1);
+	isl_basic_set_free(bset2);
+	return NULL;
+}
+
+/* Compute the lineality space of a basic set.
+ * We currently do not allow the basic set to have any divs.
+ * We basically just drop the constants and turn every inequality
+ * into an equality.
+ */
+struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset)
+{
+	int i, k;
+	struct isl_basic_set *lin = NULL;
+	unsigned dim;
+
+	if (!bset)
+		goto error;
+	isl_assert(bset->ctx, bset->n_div == 0, goto error);
+	dim = isl_basic_set_total_dim(bset);
+
+	lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset), 0, dim, 0);
+	if (!lin)
+		goto error;
+	for (i = 0; i < bset->n_eq; ++i) {
+		k = isl_basic_set_alloc_equality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->eq[k][0], 0);
+		isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
+	}
+	lin = isl_basic_set_gauss(lin, NULL);
+	if (!lin)
+		goto error;
+	for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
+		k = isl_basic_set_alloc_equality(lin);
+		if (k < 0)
+			goto error;
+		isl_int_set_si(lin->eq[k][0], 0);
+		isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
+		lin = isl_basic_set_gauss(lin, NULL);
+		if (!lin)
+			goto error;
+	}
+	isl_basic_set_free(bset);
+	return lin;
+error:
+	isl_basic_set_free(lin);
+	isl_basic_set_free(bset);
+	return NULL;
+}
+
+/* Compute the (linear) hull of the lineality spaces of the basic sets in the
+ * "underlying" set "set".
+ */
+static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
+{
+	int i;
+	struct isl_set *lin = NULL;
+
+	if (!set)
+		return NULL;
+	if (set->n == 0) {
+		struct isl_dim *dim = isl_set_get_dim(set);
+		isl_set_free(set);
+		return isl_basic_set_empty(dim);
+	}
+
+	lin = isl_set_alloc_dim(isl_set_get_dim(set), set->n, 0);
+	for (i = 0; i < set->n; ++i)
+		lin = isl_set_add(lin,
+		    isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i])));
+	isl_set_free(set);
+	return isl_set_affine_hull(lin);
+}
+
 /* Compute the convex hull of a set without any parameters or
- * integer divisions using Fourier-Motzkin elimination.
+ * integer divisions.
  * In each step, we combined two basic sets until only one
  * basic set is left.
+ * The input basic sets are assumed not to have a non-trivial
+ * lineality space.  If any of the intermediate results has
+ * a non-trivial lineality space, it is projected out.
  */
-static struct isl_basic_set *uset_convex_hull_elim(struct isl_set *set)
+static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
 {
 	struct isl_basic_set *convex_hull = NULL;
 
@@ -965,6 +1484,21 @@ static struct isl_basic_set *uset_convex_hull_elim(struct isl_set *set)
 		if (!set)
 			goto error;
 		convex_hull = convex_hull_pair(convex_hull, t);
+		if (set->n == 0)
+			break;
+		t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull));
+		if (!t)
+			goto error;
+		if (isl_basic_set_is_universe(t)) {
+			isl_basic_set_free(convex_hull);
+			convex_hull = t;
+			break;
+		}
+		if (t->n_eq < isl_basic_set_total_dim(t)) {
+			set = isl_set_add(set, convex_hull);
+			return modulo_lineality(set, t);
+		}
+		isl_basic_set_free(t);
 	}
 	isl_set_free(set);
 	return convex_hull;
@@ -992,11 +1526,11 @@ static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
 
 	if (!hull)
 		goto error;
-	bounds = independent_bounds(set->ctx, set);
+	bounds = independent_bounds(set);
 	if (!bounds)
 		goto error;
 	isl_assert(set->ctx, bounds->n_row == isl_set_n_dim(set), goto error);
-	bounds = initial_facet_constraint(set->ctx, set, bounds);
+	bounds = initial_facet_constraint(set, bounds);
 	if (!bounds)
 		goto error;
 	k = isl_basic_set_alloc_inequality(hull);
@@ -1005,12 +1539,12 @@ static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
 	dim = isl_set_n_dim(set);
 	isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error);
 	isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col);
-	isl_mat_free(set->ctx, bounds);
+	isl_mat_free(bounds);
 
 	return hull;
 error:
 	isl_basic_set_free(hull);
-	isl_mat_free(set->ctx, bounds);
+	isl_mat_free(bounds);
 	return NULL;
 }
 
@@ -1053,7 +1587,7 @@ static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
 			c->ineq = ineq;
 		return;
 	}
-	c->c = isl_mat_cow(ctx, c->c);
+	c->c = isl_mat_cow(c->c);
 	isl_int_set(c->c->row[0][0], con[0]);
 	c->ineq = ineq;
 }
@@ -1193,7 +1727,7 @@ static struct isl_basic_set *common_constraints(struct isl_basic_set *hull,
 
 	isl_hash_table_clear(table);
 	for (i = 0; i < min_constraints; ++i)
-		isl_mat_free(hull->ctx, constraints[i].c);
+		isl_mat_free(constraints[i].c);
 	free(constraints);
 	free(table);
 	return hull;
@@ -1202,7 +1736,7 @@ error:
 	free(table);
 	if (constraints)
 		for (i = 0; i < min_constraints; ++i)
-			isl_mat_free(hull->ctx, constraints[i].c);
+			isl_mat_free(constraints[i].c);
 	free(constraints);
 	return hull;
 }
@@ -1245,29 +1779,6 @@ static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
 	return hull;
 }
 
-static int isl_basic_set_is_bounded(struct isl_basic_set *bset)
-{
-	struct isl_tab *tab;
-	int bounded;
-
-	tab = isl_tab_from_recession_cone((struct isl_basic_map *)bset);
-	bounded = isl_tab_cone_is_bounded(bset->ctx, tab);
-	isl_tab_free(bset->ctx, tab);
-	return bounded;
-}
-
-static int isl_set_is_bounded(struct isl_set *set)
-{
-	int i;
-
-	for (i = 0; i < set->n; ++i) {
-		int bounded = isl_basic_set_is_bounded(set->p[i]);
-		if (!bounded || bounded < 0)
-			return bounded;
-	}
-	return 1;
-}
-
 /* Compute the convex hull of a set without any parameters or
  * integer divisions.  Depending on whether the set is bounded,
  * we pass control to the wrapping based convex hull or
@@ -1278,6 +1789,7 @@ static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
 {
 	int i;
 	struct isl_basic_set *convex_hull = NULL;
+	struct isl_basic_set *lin;
 
 	if (isl_set_n_dim(set) == 0)
 		return convex_hull_0d(set);
@@ -1295,12 +1807,23 @@ static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
 		return convex_hull;
 	}
 	if (isl_set_n_dim(set) == 1)
-		return convex_hull_1d(set->ctx, set);
+		return convex_hull_1d(set);
 
-	if (!isl_set_is_bounded(set))
-		return uset_convex_hull_elim(set);
+	if (isl_set_is_bounded(set))
+		return uset_convex_hull_wrap(set);
 
-	return uset_convex_hull_wrap(set);
+	lin = uset_combined_lineality_space(isl_set_copy(set));
+	if (!lin)
+		goto error;
+	if (isl_basic_set_is_universe(lin)) {
+		isl_set_free(set);
+		return lin;
+	}
+	if (lin->n_eq < isl_basic_set_total_dim(lin))
+		return modulo_lineality(set, lin);
+	isl_basic_set_free(lin);
+
+	return uset_convex_hull_unbounded(set);
 error:
 	isl_set_free(set);
 	isl_basic_set_free(convex_hull);
@@ -1327,7 +1850,7 @@ static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
 
 	if (!set)
 		goto error;
-	set = isl_set_normalize(set);
+	set = isl_set_coalesce(set);
 	if (!set)
 		goto error;
 	if (set->n == 1) {
@@ -1336,7 +1859,7 @@ static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
 		return convex_hull;
 	}
 	if (isl_set_n_dim(set) == 1)
-		return convex_hull_1d(set->ctx, set);
+		return convex_hull_1d(set);
 
 	return uset_convex_hull_wrap(set);
 error:
@@ -1462,7 +1985,7 @@ static void sh_data_free(struct sh_data *data)
 	isl_hash_table_free(data->ctx, data->hull_table);
 	for (i = 0; i < data->n; ++i) {
 		isl_hash_table_free(data->ctx, data->p[i].table);
-		isl_tab_free(data->ctx, data->p[i].tab);
+		isl_tab_free(data->p[i].tab);
 	}
 	free(data);
 }
@@ -1572,8 +2095,8 @@ static int is_bound(struct sh_data *data, struct isl_set *set, int j,
 
 	isl_int_init(opt);
 
-	res = isl_tab_min(data->ctx, data->p[j].tab, ineq, data->ctx->one,
-				&opt, NULL);
+	res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one,
+				&opt, NULL, 0);
 	if (res == isl_lp_ok && isl_int_is_neg(opt))
 		isl_int_sub(ineq[0], ineq[0], opt);