path: root/isl_sample.c

#include "isl_sample.h"
#include "isl_sample_piplib.h"
#include "isl_vec.h"
#include "isl_mat.h"
#include "isl_seq.h"
#include "isl_map_private.h"
#include "isl_equalities.h"
#include "isl_tab.h"
#include "isl_basis_reduction.h"

static struct isl_vec *empty_sample(struct isl_basic_set *bset)
{
	struct isl_vec *vec;

	vec = isl_vec_alloc(bset->ctx, 0);
	isl_basic_set_free(bset);
	return vec;
}

/* Construct a zero sample of the same dimension as bset.
 * As a special case, if bset is zero-dimensional, this
 * function creates a zero-dimensional sample point.
 */
static struct isl_vec *zero_sample(struct isl_basic_set *bset)
{
	unsigned dim;
	struct isl_vec *sample;

	dim = isl_basic_set_total_dim(bset);
	sample = isl_vec_alloc(bset->ctx, 1 + dim);
	if (sample) {
		isl_int_set_si(sample->el[0], 1);
		isl_seq_clr(sample->el + 1, dim);
	}
	isl_basic_set_free(bset);
	return sample;
}

static struct isl_vec *interval_sample(struct isl_basic_set *bset)
{
	int i;
	isl_int t;
	struct isl_vec *sample;

	bset = isl_basic_set_simplify(bset);
	if (!bset)
		return NULL;
	if (isl_basic_set_fast_is_empty(bset))
		return empty_sample(bset);
	if (bset->n_eq == 0 && bset->n_ineq == 0)
		return zero_sample(bset);

	sample = isl_vec_alloc(bset->ctx, 2);
	isl_int_set_si(sample->block.data[0], 1);

	if (bset->n_eq > 0) {
		isl_assert(bset->ctx, bset->n_eq == 1, goto error);
		isl_assert(bset->ctx, bset->n_ineq == 0, goto error);
		if (isl_int_is_one(bset->eq[0][1]))
			isl_int_neg(sample->el[1], bset->eq[0][0]);
		else {
			isl_assert(bset->ctx, isl_int_is_negone(bset->eq[0][1]),
				   goto error);
			isl_int_set(sample->el[1], bset->eq[0][0]);
		}
		isl_basic_set_free(bset);
		return sample;
	}

	isl_int_init(t);
	if (isl_int_is_one(bset->ineq[0][1]))
		isl_int_neg(sample->block.data[1], bset->ineq[0][0]);
	else
		isl_int_set(sample->block.data[1], bset->ineq[0][0]);
	for (i = 1; i < bset->n_ineq; ++i) {
		isl_seq_inner_product(sample->block.data,
					bset->ineq[i], 2, &t);
		if (isl_int_is_neg(t))
			break;
	}
	isl_int_clear(t);
	if (i < bset->n_ineq) {
		isl_vec_free(sample);
		return empty_sample(bset);
	}

	isl_basic_set_free(bset);
	return sample;
error:
	isl_basic_set_free(bset);
	isl_vec_free(sample);
	return NULL;
}

static struct isl_mat *independent_bounds(struct isl_basic_set *bset)
{
	int i, j, n;
	struct isl_mat *dirs = NULL;
	struct isl_mat *bounds = NULL;
	unsigned dim;

	if (!bset)
		return NULL;

	dim = isl_basic_set_n_dim(bset);
	bounds = isl_mat_alloc(bset->ctx, 1+dim, 1+dim);
	if (!bounds)
		return NULL;

	isl_int_set_si(bounds->row[0][0], 1);
	isl_seq_clr(bounds->row[0]+1, dim);
	bounds->n_row = 1;

	if (bset->n_ineq == 0)
		return bounds;

	dirs = isl_mat_alloc(bset->ctx, dim, dim);
	if (!dirs) {
		isl_mat_free(bounds);
		return NULL;
	}
	isl_seq_cpy(dirs->row[0], bset->ineq[0]+1, dirs->n_col);
	isl_seq_cpy(bounds->row[1], bset->ineq[0], bounds->n_col);
	for (j = 1, n = 1; n < dim && j < bset->n_ineq; ++j) {
		int pos;

		isl_seq_cpy(dirs->row[n], bset->ineq[j]+1, dirs->n_col);

		pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
		if (pos < 0)
			continue;
		for (i = 0; i < n; ++i) {
			int pos_i;
			pos_i = isl_seq_first_non_zero(dirs->row[i], dirs->n_col);
			if (pos_i < pos)
				continue;
			if (pos_i > pos)
				break;
			isl_seq_elim(dirs->row[n], dirs->row[i], pos,
					dirs->n_col, NULL);
			pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
			if (pos < 0)
				break;
		}
		if (pos < 0)
			continue;
		if (i < n) {
			int k;
			isl_int *t = dirs->row[n];
			for (k = n; k > i; --k)
				dirs->row[k] = dirs->row[k-1];
			dirs->row[i] = t;
		}
		++n;
		isl_seq_cpy(bounds->row[n], bset->ineq[j], bounds->n_col);
	}
	isl_mat_free(dirs);
	bounds->n_row = 1+n;
	return bounds;
}

static void swap_inequality(struct isl_basic_set *bset, int a, int b)
{
	isl_int *t = bset->ineq[a];
	bset->ineq[a] = bset->ineq[b];
	bset->ineq[b] = t;
}

/* Skew into positive orthant and project out lineality space.
 *
 * We perform a unimodular transformation that turns a selected
 * maximal set of linearly independent bounds into constraints
 * on the first dimensions that impose that these first dimensions
 * are non-negative.  In particular, the constraint matrix is lower
 * triangular with positive entries on the diagonal and negative
 * entries below.
 * If "bset" has a lineality space then these constraints (and therefore
 * all constraints in bset) only involve the first dimensions.
 * The remaining dimensions then do not appear in any constraints and
 * we can select any value for them, say zero.  We therefore project
 * out this final dimensions and plug in the value zero later.  This
 * is accomplished by simply dropping the final columns of
 * the unimodular transformation.
 */
static struct isl_basic_set *isl_basic_set_skew_to_positive_orthant(
	struct isl_basic_set *bset, struct isl_mat **T)
{
	struct isl_mat *U = NULL;
	struct isl_mat *bounds = NULL;
	int i, j;
	unsigned old_dim, new_dim;

	*T = NULL;
	if (!bset)
		return NULL;

	isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
	isl_assert(bset->ctx, bset->n_div == 0, goto error);
	isl_assert(bset->ctx, bset->n_eq == 0, goto error);
	
	old_dim = isl_basic_set_n_dim(bset);
	/* Try to move (multiples of) unit rows up. */
	for (i = 0, j = 0; i < bset->n_ineq; ++i) {
		int pos = isl_seq_first_non_zero(bset->ineq[i]+1, old_dim);
		if (pos < 0)
			continue;
		if (isl_seq_first_non_zero(bset->ineq[i]+1+pos+1,
						old_dim-pos-1) >= 0)
			continue;
		if (i != j)
			swap_inequality(bset, i, j);
		++j;
	}
	bounds = independent_bounds(bset);
	if (!bounds)
		goto error;
	new_dim = bounds->n_row - 1;
	bounds = isl_mat_left_hermite(bounds, 1, &U, NULL);
	if (!bounds)
		goto error;
	U = isl_mat_drop_cols(U, 1 + new_dim, old_dim - new_dim);
	bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
	if (!bset)
		goto error;
	*T = U;
	isl_mat_free(bounds);
	return bset;
error:
	isl_mat_free(bounds);
	isl_mat_free(U);
	isl_basic_set_free(bset);
	return NULL;
}

/* Find a sample integer point, if any, in bset, which is known
 * to have equalities.  If bset contains no integer points, then
 * return a zero-length vector.
 * We simply remove the known equalities, compute a sample
 * in the resulting bset, using the specified recurse function,
 * and then transform the sample back to the original space.
 */
static struct isl_vec *sample_eq(struct isl_basic_set *bset,
	struct isl_vec *(*recurse)(struct isl_basic_set *))
{
	struct isl_mat *T;
	struct isl_vec *sample;

	if (!bset)
		return NULL;

	bset = isl_basic_set_remove_equalities(bset, &T, NULL);
	sample = recurse(bset);
	if (!sample || sample->size == 0)
		isl_mat_free(T);
	else
		sample = isl_mat_vec_product(T, sample);
	return sample;
}

/* Given a basic set "bset" and an affine function "f"/"denom",
 * check if bset is bounded and non-empty and if so, return the minimal
 * and maximal value attained by the affine function in "min" and "max".
 * The minimal value is rounded up to the nearest integer, while the
 * maximal value is rounded down.
 * The return value indicates whether the set was empty or unbounded.
 *
 * If we happen to find an integer point while looking for the minimal
 * or maximal value, then we record that value in "bset" and return early.
 */
static enum isl_lp_result basic_set_range(struct isl_basic_set *bset,
	isl_int *f, isl_int denom, isl_int *min, isl_int *max)
{
	unsigned dim;
	struct isl_tab *tab;
	enum isl_lp_result res;

	if (!bset)
		return isl_lp_error;
	if (isl_basic_set_fast_is_empty(bset))
		return isl_lp_empty;

	tab = isl_tab_from_basic_set(bset);
	res = isl_tab_min(tab, f, denom, min, NULL, 0);
	if (res != isl_lp_ok)
		goto done;

	if (isl_tab_sample_is_integer(tab)) {
		isl_vec_free(bset->sample);
		bset->sample = isl_tab_get_sample_value(tab);
		if (!bset->sample)
			goto error;
		isl_int_set(*max, *min);
		goto done;
	}

	dim = isl_basic_set_total_dim(bset);
	isl_seq_neg(f, f, 1 + dim);
	res = isl_tab_min(tab, f, denom, max, NULL, 0);
	isl_seq_neg(f, f, 1 + dim);
	isl_int_neg(*max, *max);

	if (isl_tab_sample_is_integer(tab)) {
		isl_vec_free(bset->sample);
		bset->sample = isl_tab_get_sample_value(tab);
		if (!bset->sample)
			goto error;
	}

done:
	isl_tab_free(tab);
	return res;
error:
	isl_tab_free(tab);
	return isl_lp_error;
}

/* Perform a basis reduction on "bset" and return the inverse of
 * the new basis, i.e., an affine mapping from the new coordinates to the old,
 * in *T.
 */
static struct isl_basic_set *basic_set_reduced(struct isl_basic_set *bset,
	struct isl_mat **T)
{
	unsigned gbr_only_first;

	*T = NULL;
	if (!bset)
		return NULL;

	gbr_only_first = bset->ctx->gbr_only_first;
	bset->ctx->gbr_only_first = bset->ctx->gbr == ISL_GBR_ONCE;
	*T = isl_basic_set_reduced_basis(bset);
	bset->ctx->gbr_only_first = gbr_only_first;

	*T = isl_mat_lin_to_aff(*T);
	*T = isl_mat_right_inverse(*T);

	bset = isl_basic_set_preimage(bset, isl_mat_copy(*T));
	if (!bset)
		goto error;

	return bset;
error:
	isl_mat_free(*T);
	*T = NULL;
	return NULL;
}

static struct isl_vec *sample_bounded(struct isl_basic_set *bset);

/* Given a basic set "bset" whose first coordinate ranges between
 * "min" and "max", step through all values from min to max, until
 * the slice of bset with the first coordinate fixed to one of these
 * values contains an integer point.  If such a point is found, return it.
 * If none of the slices contains any integer point, then bset itself
 * doesn't contain any integer point and an empty sample is returned.
 */
static struct isl_vec *sample_scan(struct isl_basic_set *bset,
	isl_int min, isl_int max)
{
	unsigned total;
	struct isl_basic_set *slice = NULL;
	struct isl_vec *sample = NULL;
	isl_int tmp;

	total = isl_basic_set_total_dim(bset);

	isl_int_init(tmp);
	for (isl_int_set(tmp, min); isl_int_le(tmp, max);
	     isl_int_add_ui(tmp, tmp, 1)) {
		int k;

		slice = isl_basic_set_copy(bset);
		slice = isl_basic_set_cow(slice);
		slice = isl_basic_set_extend_constraints(slice, 1, 0);
		k = isl_basic_set_alloc_equality(slice);
		if (k < 0)
			goto error;
		isl_int_set(slice->eq[k][0], tmp);
		isl_int_set_si(slice->eq[k][1], -1);
		isl_seq_clr(slice->eq[k] + 2, total - 1);
		slice = isl_basic_set_simplify(slice);
		sample = sample_bounded(slice);
		slice = NULL;
		if (!sample)
			goto error;
		if (sample->size > 0)
			break;
		isl_vec_free(sample);
		sample = NULL;
	}
	if (!sample)
		sample = empty_sample(bset);
	else
		isl_basic_set_free(bset);
	isl_int_clear(tmp);
	return sample;
error:
	isl_basic_set_free(bset);
	isl_basic_set_free(slice);
	isl_int_clear(tmp);
	return NULL;
}

/* Given a basic set that is known to be bounded, find and return
 * an integer point in the basic set, if there is any.
 *
 * After handling some trivial cases, we check the range of the
 * first coordinate.  If this coordinate can only attain one integer
 * value, we are happy.  Otherwise, we perform basis reduction and
 * determine the new range.
 *
 * Then we step through all possible values in the range in sample_scan.
 *
 * If any basis reduction was performed, the sample value found, if any,
 * is transformed back to the original space.
 */ 
static struct isl_vec *sample_bounded(struct isl_basic_set *bset)
{
	unsigned dim;
	unsigned gbr;
	struct isl_ctx *ctx;
	struct isl_vec *sample;
	struct isl_vec *obj = NULL;
	struct isl_mat *T = NULL;
	isl_int min, max;
	enum isl_lp_result res;

	if (!bset)
		return NULL;

	if (isl_basic_set_fast_is_empty(bset))
		return empty_sample(bset);

	dim = isl_basic_set_total_dim(bset);
	if (dim == 0)
		return zero_sample(bset);
	if (dim == 1)
		return interval_sample(bset);
	if (bset->n_eq > 0)
		return sample_eq(bset, sample_bounded);

	ctx = bset->ctx;
	gbr = ctx->gbr;

	isl_int_init(min);
	isl_int_init(max);
	obj = isl_vec_alloc(bset->ctx, 1 + dim);
	if (!obj)
		goto error;
	isl_seq_clr(obj->el, 1+ dim);
	isl_int_set_si(obj->el[1], 1);

	res = basic_set_range(bset, obj->el, bset->ctx->one, &min, &max);
	if (res == isl_lp_error)
		goto error;
	isl_assert(bset->ctx, res != isl_lp_unbounded, goto error);
	if (bset->sample) {
		sample = isl_vec_copy(bset->sample);
		isl_basic_set_free(bset);
		goto out;
	}
	if (res == isl_lp_empty || isl_int_lt(max, min)) {
		sample = empty_sample(bset);
		goto out;
	}

	if (ctx->gbr != ISL_GBR_NEVER && isl_int_ne(min, max)) {
		if (ctx->gbr == ISL_GBR_ONCE)
			ctx->gbr = ISL_GBR_NEVER;

		bset = basic_set_reduced(bset, &T);
		if (!bset)
			goto error;

		res = basic_set_range(bset, obj->el, bset->ctx->one, &min, &max);
		if (res == isl_lp_error)
			goto error;
		isl_assert(bset->ctx, res != isl_lp_unbounded, goto error);
		if (bset->sample) {
			sample = isl_vec_copy(bset->sample);
			isl_basic_set_free(bset);
			goto out;
		}
		if (res == isl_lp_empty || isl_int_lt(max, min)) {
			sample = empty_sample(bset);
			goto out;
		}
	}

	sample = sample_scan(bset, min, max);
out:
	if (T) {
		if (!sample || sample->size == 0)
			isl_mat_free(T);
		else
			sample = isl_mat_vec_product(T, sample);
	}
	isl_vec_free(obj);
	isl_int_clear(min);
	isl_int_clear(max);
	ctx->gbr = gbr;
	return sample;
error:
	ctx->gbr = gbr;
	isl_mat_free(T);
	isl_basic_set_free(bset);
	isl_vec_free(obj);
	isl_int_clear(min);
	isl_int_clear(max);
	return NULL;
}

/* Given a basic set "bset" and a value "sample" for the first coordinates
 * of bset, plug in these values and drop the corresponding coordinates.
 *
 * We do this by computing the preimage of the transformation
 *
 *	     [ 1 0 ]
 *	x =  [ s 0 ] x'
 *	     [ 0 I ]
 *
 * where [1 s] is the sample value and I is the identity matrix of the
 * appropriate dimension.
 */
static struct isl_basic_set *plug_in(struct isl_basic_set *bset,
	struct isl_vec *sample)
{
	int i;
	unsigned total;
	struct isl_mat *T;

	if (!bset || !sample)
		goto error;

	total = isl_basic_set_total_dim(bset);
	T = isl_mat_alloc(bset->ctx, 1 + total, 1 + total - (sample->size - 1));
	if (!T)
		goto error;

	for (i = 0; i < sample->size; ++i) {
		isl_int_set(T->row[i][0], sample->el[i]);
		isl_seq_clr(T->row[i] + 1, T->n_col - 1);
	}
	for (i = 0; i < T->n_col - 1; ++i) {
		isl_seq_clr(T->row[sample->size + i], T->n_col);
		isl_int_set_si(T->row[sample->size + i][1 + i], 1);
	}
	isl_vec_free(sample);

	bset = isl_basic_set_preimage(bset, T);
	return bset;
error:
	isl_basic_set_free(bset);
	isl_vec_free(sample);
	return NULL;
}

/* Given a basic set "bset", return any (possibly non-integer) point
 * in the basic set.
 */
static struct isl_vec *rational_sample(struct isl_basic_set *bset)
{
	struct isl_tab *tab;
	struct isl_vec *sample;

	if (!bset)
		return NULL;

	tab = isl_tab_from_basic_set(bset);
	sample = isl_tab_get_sample_value(tab);
	isl_tab_free(tab);

	isl_basic_set_free(bset);

	return sample;
}

/* Given a linear cone "cone" and a rational point "vec",
 * construct a polyhedron with shifted copies of the constraints in "cone",
 * i.e., a polyhedron with "cone" as its recession cone, such that each
 * point x in this polyhedron is such that the unit box positioned at x
 * lies entirely inside the affine cone 'vec + cone'.
 * Any rational point in this polyhedron may therefore be rounded up
 * to yield an integer point that lies inside said affine cone.
 *
 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
 * point "vec" by v/d.
 * Let b_i = <a_i, v>.  Then the affine cone 'vec + cone' is given
 * by <a_i, x> - b/d >= 0.
 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
 * We prefer this polyhedron over the actual affine cone because it doesn't
 * require a scaling of the constraints.
 * If each of the vertices of the unit cube positioned at x lies inside
 * this polyhedron, then the whole unit cube at x lies inside the affine cone.
 * We therefore impose that x' = x + \sum e_i, for any selection of unit
 * vectors lies inside the polyhedron, i.e.,
 *
 *	<a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
 *
 * The most stringent of these constraints is the one that selects
 * all negative a_i, so the polyhedron we are looking for has constraints
 *
 *	<a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
 *
 * Note that if cone were known to have only non-negative rays
 * (which can be accomplished by a unimodular transformation),
 * then we would only have to check the points x' = x + e_i
 * and we only have to add the smallest negative a_i (if any)
 * instead of the sum of all negative a_i.
 */
static struct isl_basic_set *shift_cone(struct isl_basic_set *cone,
	struct isl_vec *vec)
{
	int i, j, k;
	unsigned total;

	struct isl_basic_set *shift = NULL;

	if (!cone || !vec)
		goto error;

	isl_assert(cone->ctx, cone->n_eq == 0, goto error);

	total = isl_basic_set_total_dim(cone);

	shift = isl_basic_set_alloc_dim(isl_basic_set_get_dim(cone),
					0, 0, cone->n_ineq);

	for (i = 0; i < cone->n_ineq; ++i) {
		k = isl_basic_set_alloc_inequality(shift);
		if (k < 0)
			goto error;
		isl_seq_cpy(shift->ineq[k] + 1, cone->ineq[i] + 1, total);
		isl_seq_inner_product(shift->ineq[k] + 1, vec->el + 1, total,
				      &shift->ineq[k][0]);
		isl_int_cdiv_q(shift->ineq[k][0],
			       shift->ineq[k][0], vec->el[0]);
		isl_int_neg(shift->ineq[k][0], shift->ineq[k][0]);
		for (j = 0; j < total; ++j) {
			if (isl_int_is_nonneg(shift->ineq[k][1 + j]))
				continue;
			isl_int_add(shift->ineq[k][0],
				    shift->ineq[k][0], shift->ineq[k][1 + j]);
		}
	}

	isl_basic_set_free(cone);
	isl_vec_free(vec);

	return isl_basic_set_finalize(shift);
error:
	isl_basic_set_free(shift);
	isl_basic_set_free(cone);
	isl_vec_free(vec);
	return NULL;
}

/* Given a rational point vec in a (transformed) basic set,
 * such that cone is the recession cone of the original basic set,
 * "round up" the rational point to an integer point.
 *
 * We first check if the rational point just happens to be integer.
 * If not, we transform the cone in the same way as the basic set,
 * pick a point x in this cone shifted to the rational point such that
 * the whole unit cube at x is also inside this affine cone.
 * Then we simply round up the coordinates of x and return the
 * resulting integer point.
 */
static struct isl_vec *round_up_in_cone(struct isl_vec *vec,
	struct isl_basic_set *cone, struct isl_mat *U)
{
	unsigned total;

	if (!vec || !cone || !U)
		goto error;

	isl_assert(vec->ctx, vec->size != 0, goto error);
	if (isl_int_is_one(vec->el[0])) {
		isl_mat_free(U);
		isl_basic_set_free(cone);
		return vec;
	}

	total = isl_basic_set_total_dim(cone);
	cone = isl_basic_set_preimage(cone, U);
	cone = isl_basic_set_remove_dims(cone, 0, total - (vec->size - 1));

	cone = shift_cone(cone, vec);

	vec = rational_sample(cone);
	vec = isl_vec_ceil(vec);
	return vec;
error:
	isl_mat_free(U);
	isl_vec_free(vec);
	isl_basic_set_free(cone);
	return NULL;
}

/* Concatenate two integer vectors, i.e., two vectors with denominator
 * (stored in element 0) equal to 1.
 */
static struct isl_vec *vec_concat(struct isl_vec *vec1, struct isl_vec *vec2)
{
	struct isl_vec *vec;

	if (!vec1 || !vec2)
		goto error;
	isl_assert(vec1->ctx, vec1->size > 0, goto error);
	isl_assert(vec2->ctx, vec2->size > 0, goto error);
	isl_assert(vec1->ctx, isl_int_is_one(vec1->el[0]), goto error);
	isl_assert(vec2->ctx, isl_int_is_one(vec2->el[0]), goto error);

	vec = isl_vec_alloc(vec1->ctx, vec1->size + vec2->size - 1);
	if (!vec)
		goto error;

	isl_seq_cpy(vec->el, vec1->el, vec1->size);
	isl_seq_cpy(vec->el + vec1->size, vec2->el + 1, vec2->size - 1);

	isl_vec_free(vec1);
	isl_vec_free(vec2);

	return vec;
error:
	isl_vec_free(vec1);
	isl_vec_free(vec2);
	return NULL;
}

/* Drop all constraints in bset that involve any of the dimensions
 * first to first+n-1.
 */
static struct isl_basic_set *drop_constraints_involving
	(struct isl_basic_set *bset, unsigned first, unsigned n)
{
	int i;

	if (!bset)
		return NULL;

	bset = isl_basic_set_cow(bset);

	for (i = bset->n_ineq - 1; i >= 0; --i) {
		if (isl_seq_first_non_zero(bset->ineq[i] + 1 + first, n) == -1)
			continue;
		isl_basic_set_drop_inequality(bset, i);
	}

	return bset;
}

/* Give a basic set "bset" with recession cone "cone", compute and
 * return an integer point in bset, if any.
 *
 * If the recession cone is full-dimensional, then we know that
 * bset contains an infinite number of integer points and it is
 * fairly easy to pick one of them.
 * If the recession cone is not full-dimensional, then we first
 * transform bset such that the bounded directions appear as
 * the first dimensions of the transformed basic set.
 * We do this by using a unimodular transformation that transforms
 * the equalities in the recession cone to equalities on the first
 * dimensions.
 *
 * The transformed set is then projected onto its bounded dimensions.
 * Note that to compute this projection, we can simply drop all constraints
 * involving any of the unbounded dimensions since these constraints
 * cannot be combined to produce a constraint on the bounded dimensions.
 * To see this, assume that there is such a combination of constraints
 * that produces a constraint on the bounded dimensions.  This means
 * that some combination of the unbounded dimensions has both an upper
 * bound and a lower bound in terms of the bounded dimensions, but then
 * this combination would be a bounded direction too and would have been
 * transformed into a bounded dimensions.
 *
 * We then compute a sample value in the bounded dimensions.
 * If no such value can be found, then the original set did not contain
 * any integer points and we are done.
 * Otherwise, we plug in the value we found in the bounded dimensions,
 * project out these bounded dimensions and end up with a set with
 * a full-dimensional recession cone.
 * A sample point in this set is computed by "rounding up" any
 * rational point in the set.
 *
 * The sample points in the bounded and unbounded dimensions are
 * then combined into a single sample point and transformed back
 * to the original space.
 */
static struct isl_vec *sample_with_cone(struct isl_basic_set *bset,
	struct isl_basic_set *cone)
{
	unsigned total;
	unsigned cone_dim;
	struct isl_mat *M, *U;
	struct isl_vec *sample;
	struct isl_vec *cone_sample;
	struct isl_ctx *ctx;
	struct isl_basic_set *bounded;

	if (!bset || !cone)
		goto error;

	ctx = bset->ctx;
	total = isl_basic_set_total_dim(cone);
	cone_dim = total - cone->n_eq;

	M = isl_mat_sub_alloc(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
	M = isl_mat_left_hermite(M, 0, &U, NULL);
	if (!M)
		goto error;
	isl_mat_free(M);

	U = isl_mat_lin_to_aff(U);
	bset = isl_basic_set_preimage(bset, isl_mat_copy(U));

	bounded = isl_basic_set_copy(bset);
	bounded = drop_constraints_involving(bounded, total - cone_dim, cone_dim);
	bounded = isl_basic_set_drop_dims(bounded, total - cone_dim, cone_dim);
	sample = sample_bounded(bounded);
	if (!sample || sample->size == 0) {
		isl_basic_set_free(bset);
		isl_basic_set_free(cone);
		isl_mat_free(U);
		return sample;
	}
	bset = plug_in(bset, isl_vec_copy(sample));
	cone_sample = rational_sample(bset);
	cone_sample = round_up_in_cone(cone_sample, cone, isl_mat_copy(U));
	sample = vec_concat(sample, cone_sample);
	sample = isl_mat_vec_product(U, sample);
	return sample;
error:
	isl_basic_set_free(cone);
	isl_basic_set_free(bset);
	return NULL;
}

/* Compute and return a sample point in bset using generalized basis
 * reduction.  We first check if the input set has a non-trivial
 * recession cone.  If so, we perform some extra preprocessing in
 * sample_with_cone.  Otherwise, we directly perform generalized basis
 * reduction.
 */
static struct isl_vec *gbr_sample(struct isl_basic_set *bset)
{
	unsigned dim;
	struct isl_basic_set *cone;

	dim = isl_basic_set_total_dim(bset);

	cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));

	if (cone->n_eq < dim)
		return sample_with_cone(bset, cone);

	isl_basic_set_free(cone);
	return sample_bounded(bset);
}

static struct isl_vec *pip_sample(struct isl_basic_set *bset)
{
	struct isl_mat *T;
	struct isl_ctx *ctx;
	struct isl_vec *sample;

	bset = isl_basic_set_skew_to_positive_orthant(bset, &T);
	if (!bset)
		return NULL;

	ctx = bset->ctx;
	sample = isl_pip_basic_set_sample(bset);

	if (sample && sample->size != 0)
		sample = isl_mat_vec_product(T, sample);
	else
		isl_mat_free(T);

	return sample;
}

static struct isl_vec *basic_set_sample(struct isl_basic_set *bset, int bounded)
{
	struct isl_ctx *ctx;
	unsigned dim;
	if (!bset)
		return NULL;

	ctx = bset->ctx;
	if (isl_basic_set_fast_is_empty(bset))
		return empty_sample(bset);

	dim = isl_basic_set_n_dim(bset);
	isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
	isl_assert(ctx, bset->n_div == 0, goto error);

	if (bset->sample && bset->sample->size == 1 + dim) {
		int contains = isl_basic_set_contains(bset, bset->sample);
		if (contains < 0)
			goto error;
		if (contains) {
			struct isl_vec *sample = isl_vec_copy(bset->sample);
			isl_basic_set_free(bset);
			return sample;
		}
	}
	isl_vec_free(bset->sample);
	bset->sample = NULL;

	if (bset->n_eq > 0)
		return sample_eq(bset, isl_basic_set_sample_vec);
	if (dim == 0)
		return zero_sample(bset);
	if (dim == 1)
		return interval_sample(bset);

	switch (bset->ctx->ilp_solver) {
	case ISL_ILP_PIP:
		return pip_sample(bset);
	case ISL_ILP_GBR:
		return bounded ? sample_bounded(bset) : gbr_sample(bset);
	}
	isl_assert(bset->ctx, 0, );
error:
	isl_basic_set_free(bset);
	return NULL;
}

__isl_give isl_vec *isl_basic_set_sample_vec(__isl_take isl_basic_set *bset)
{
	return basic_set_sample(bset, 0);
}

/* Compute an integer sample in "bset", where the caller guarantees
 * that "bset" is bounded.
 */
struct isl_vec *isl_basic_set_sample_bounded(struct isl_basic_set *bset)
{
	return basic_set_sample(bset, 1);
}

__isl_give isl_basic_set *isl_basic_set_from_vec(__isl_take isl_vec *vec)
{
	int i;
	int k;
	struct isl_basic_set *bset = NULL;
	struct isl_ctx *ctx;
	unsigned dim;

	if (!vec)
		return NULL;
	ctx = vec->ctx;
	isl_assert(ctx, vec->size != 0, goto error);

	bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
	if (!bset)
		goto error;
	dim = isl_basic_set_n_dim(bset);
	for (i = dim - 1; i >= 0; --i) {
		k = isl_basic_set_alloc_equality(bset);
		if (k < 0)
			goto error;
		isl_seq_clr(bset->eq[k], 1 + dim);
		isl_int_neg(bset->eq[k][0], vec->el[1 + i]);
		isl_int_set(bset->eq[k][1 + i], vec->el[0]);
	}
	isl_vec_free(vec);

	return bset;
error:
	isl_basic_set_free(bset);
	isl_vec_free(vec);
	return NULL;
}

__isl_give isl_basic_map *isl_basic_map_sample(__isl_take isl_basic_map *bmap)
{
	struct isl_basic_set *bset;
	struct isl_vec *sample_vec;

	bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
	sample_vec = isl_basic_set_sample_vec(bset);
	if (!sample_vec)
		goto error;
	if (sample_vec->size == 0) {
		struct isl_basic_map *sample;
		sample = isl_basic_map_empty_like(bmap);
		isl_vec_free(sample_vec);
		isl_basic_map_free(bmap);
		return sample;
	}
	bset = isl_basic_set_from_vec(sample_vec);
	return isl_basic_map_overlying_set(bset, bmap);
error:
	isl_basic_map_free(bmap);
	return NULL;
}

__isl_give isl_basic_map *isl_map_sample(__isl_take isl_map *map)
{
	int i;
	isl_basic_map *sample = NULL;

	if (!map)
		goto error;

	for (i = 0; i < map->n; ++i) {
		sample = isl_basic_map_sample(isl_basic_map_copy(map->p[i]));
		if (!sample)
			goto error;
		if (!ISL_F_ISSET(sample, ISL_BASIC_MAP_EMPTY))
			break;
		isl_basic_map_free(sample);
	}
	if (i == map->n)
		sample = isl_basic_map_empty_like_map(map);
	isl_map_free(map);
	return sample;
error:
	isl_map_free(map);
	return NULL;
}

__isl_give isl_basic_set *isl_set_sample(__isl_take isl_set *set)
{
	return (isl_basic_set *) isl_map_sample((isl_map *)set);
}