/*

 * Copyright 2008-2009 Katholieke Universiteit Leuven

 *

 * Use of this software is governed by the MIT license

 *

 * Written by Sven Verdoolaege, K.U.Leuven, Departement

 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium

 */

#include <isl_ctx_private.h>

#include <isl_map_private.h>

#include "isl_sample.h"

#include <isl/vec.h>

#include <isl/mat.h>

#include <isl_seq.h>

#include "isl_equalities.h"

#include "isl_tab.h"

#include "isl_basis_reduction.h"

#include <isl_factorization.h>

#include <isl_point_private.h>

#include <isl_options_private.h>

#include <isl_vec_private.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_plain_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);

	if (!sample)

		goto error;

	if (!bset)

		return NULL;

	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;

}

/* 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;

}

/* Return a matrix containing the equalities of the tableau

 * in constraint form.  The tableau is assumed to have

 * an associated bset that has been kept up-to-date.

 */

static struct isl_mat *tab_equalities(struct isl_tab *tab)

{

	int i, j;

	int n_eq;

	struct isl_mat *eq;

	struct isl_basic_set *bset;

	if (!tab)

		return NULL;

	bset = isl_tab_peek_bset(tab);

	isl_assert(tab->mat->ctx, bset, return NULL);

	n_eq = tab->n_var - tab->n_col + tab->n_dead;

	if (tab->empty || n_eq == 0)

		return isl_mat_alloc(tab->mat->ctx, 0, tab->n_var);

	if (n_eq == tab->n_var)

		return isl_mat_identity(tab->mat->ctx, tab->n_var);

	eq = isl_mat_alloc(tab->mat->ctx, n_eq, tab->n_var);

	if (!eq)

		return NULL;

	for (i = 0, j = 0; i < tab->n_con; ++i) {

		if (tab->con[i].is_row)

			continue;

		if (tab->con[i].index >= 0 && tab->con[i].index >= tab->n_dead)

			continue;

		if (i < bset->n_eq)

			isl_seq_cpy(eq->row[j], bset->eq[i] + 1, tab->n_var);

		else

			isl_seq_cpy(eq->row[j],

				    bset->ineq[i - bset->n_eq] + 1, tab->n_var);

		++j;

	}

	isl_assert(bset->ctx, j == n_eq, goto error);

	return eq;

error:

	isl_mat_free(eq);

	return NULL;

}

/* Compute and return an initial basis for the bounded tableau "tab".

 *

 * If the tableau is either full-dimensional or zero-dimensional,

 * the we simply return an identity matrix.

 * Otherwise, we construct a basis whose first directions correspond

 * to equalities.

 */

static struct isl_mat *initial_basis(struct isl_tab *tab)

{

	int n_eq;

	struct isl_mat *eq;

	struct isl_mat *Q;

	tab->n_unbounded = 0;

	tab->n_zero = n_eq = tab->n_var - tab->n_col + tab->n_dead;

	if (tab->empty || n_eq == 0 || n_eq == tab->n_var)

		return isl_mat_identity(tab->mat->ctx, 1 + tab->n_var);

	eq = tab_equalities(tab);

	eq = isl_mat_left_hermite(eq, 0, NULL, &Q);

	if (!eq)

		return NULL;

	isl_mat_free(eq);

	Q = isl_mat_lin_to_aff(Q);

	return Q;

}

/* Compute the minimum of the current ("level") basis row over "tab"

 * and store the result in position "level" of "min".

 */

static enum isl_lp_result compute_min(isl_ctx *ctx, struct isl_tab *tab,

	__isl_keep isl_vec *min, int level)

{

	return isl_tab_min(tab, tab->basis->row[1 + level],

			    ctx->one, &min->el[level], NULL, 0);

}

/* Compute the maximum of the current ("level") basis row over "tab"

 * and store the result in position "level" of "max".

 */

static enum isl_lp_result compute_max(isl_ctx *ctx, struct isl_tab *tab,

	__isl_keep isl_vec *max, int level)

{

	enum isl_lp_result res;

	unsigned dim = tab->n_var;

	isl_seq_neg(tab->basis->row[1 + level] + 1,

		    tab->basis->row[1 + level] + 1, dim);

	res = isl_tab_min(tab, tab->basis->row[1 + level],

		    ctx->one, &max->el[level], NULL, 0);

	isl_seq_neg(tab->basis->row[1 + level] + 1,

		    tab->basis->row[1 + level] + 1, dim);

	isl_int_neg(max->el[level], max->el[level]);

	return res;

}

/* Perform a greedy search for an integer point in the set represented

 * by "tab", given that the minimal rational value (rounded up to the

 * nearest integer) at "level" is smaller than the maximal rational

 * value (rounded down to the nearest integer).

 *

 * Return 1 if we have found an integer point (if tab->n_unbounded > 0

 * then we may have only found integer values for the bounded dimensions

 * and it is the responsibility of the caller to extend this solution

 * to the unbounded dimensions).

 * Return 0 if greedy search did not result in a solution.

 * Return -1 if some error occurred.

 *

 * We assign a value half-way between the minimum and the maximum

 * to the current dimension and check if the minimal value of the

 * next dimension is still smaller than (or equal) to the maximal value.

 * We continue this process until either

 * - the minimal value (rounded up) is greater than the maximal value

 *	(rounded down).  In this case, greedy search has failed.

 * - we have exhausted all bounded dimensions, meaning that we have

 *	found a solution.

 * - the sample value of the tableau is integral.

 * - some error has occurred.

 */

static int greedy_search(isl_ctx *ctx, struct isl_tab *tab,

	__isl_keep isl_vec *min, __isl_keep isl_vec *max, int level)

{

	struct isl_tab_undo *snap;

	enum isl_lp_result res;

	snap = isl_tab_snap(tab);

	do {

		isl_int_add(tab->basis->row[1 + level][0],

			    min->el[level], max->el[level]);

		isl_int_fdiv_q_ui(tab->basis->row[1 + level][0],

			    tab->basis->row[1 + level][0], 2);

		isl_int_neg(tab->basis->row[1 + level][0],

			    tab->basis->row[1 + level][0]);

		if (isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]) < 0)

			return -1;

		isl_int_set_si(tab->basis->row[1 + level][0], 0);

		if (++level >= tab->n_var - tab->n_unbounded)

			return 1;

		if (isl_tab_sample_is_integer(tab))

			return 1;

		res = compute_min(ctx, tab, min, level);

		if (res == isl_lp_error)

			return -1;

		if (res != isl_lp_ok)

			isl_die(ctx, isl_error_internal,

				"expecting bounded rational solution",

				return -1);

		res = compute_max(ctx, tab, max, level);

		if (res == isl_lp_error)

			return -1;

		if (res != isl_lp_ok)

			isl_die(ctx, isl_error_internal,

				"expecting bounded rational solution",

				return -1);

	} while (isl_int_le(min->el[level], max->el[level]));

	if (isl_tab_rollback(tab, snap) < 0)

		return -1;

	return 0;

}

/* Given a tableau representing a set, find and return

 * an integer point in the set, if there is any.

 *

 * We perform a depth first search

 * for an integer point, by scanning all possible values in the range

 * attained by a basis vector, where an initial basis may have been set

 * by the calling function.  Otherwise an initial basis that exploits

 * the equalities in the tableau is created.

 * tab->n_zero is currently ignored and is clobbered by this function.

 *

 * The tableau is allowed to have unbounded direction, but then

 * the calling function needs to set an initial basis, with the

 * unbounded directions last and with tab->n_unbounded set

 * to the number of unbounded directions.

 * Furthermore, the calling functions needs to add shifted copies

 * of all constraints involving unbounded directions to ensure

 * that any feasible rational value in these directions can be rounded

 * up to yield a feasible integer value.

 * In particular, let B define the given basis x' = B x

 * and let T be the inverse of B, i.e., X = T x'.

 * Let a x + c >= 0 be a constraint of the set represented by the tableau,

 * or a T x' + c >= 0 in terms of the given basis.  Assume that

 * the bounded directions have an integer value, then we can safely

 * round up the values for the unbounded directions if we make sure

 * that x' not only satisfies the original constraint, but also

 * the constraint "a T x' + c + s >= 0" with s the sum of all

 * negative values in the last n_unbounded entries of "a T".

 * The calling function therefore needs to add the constraint

 * a x + c + s >= 0.  The current function then scans the first

 * directions for an integer value and once those have been found,

 * it can compute "T ceil(B x)" to yield an integer point in the set.

 * Note that during the search, the first rows of B may be changed

 * by a basis reduction, but the last n_unbounded rows of B remain

 * unaltered and are also not mixed into the first rows.

 *

 * The search is implemented iteratively.  "level" identifies the current

 * basis vector.  "init" is true if we want the first value at the current

 * level and false if we want the next value.

 *

 * At the start of each level, we first check if we can find a solution

 * using greedy search.  If not, we continue with the exhaustive search.

 *

 * The initial basis is the identity matrix.  If the range in some direction

 * contains more than one integer value, we perform basis reduction based

 * on the value of ctx->opt->gbr

 *	- ISL_GBR_NEVER:	never perform basis reduction

 *	- ISL_GBR_ONCE:		only perform basis reduction the first

 *				time such a range is encountered

 *	- ISL_GBR_ALWAYS:	always perform basis reduction when

 *				such a range is encountered

 *

 * When ctx->opt->gbr is set to ISL_GBR_ALWAYS, then we allow the basis

 * reduction computation to return early.  That is, as soon as it

 * finds a reasonable first direction.

 */ 

struct isl_vec *isl_tab_sample(struct isl_tab *tab)

{

	unsigned dim;

	unsigned gbr;

	struct isl_ctx *ctx;

	struct isl_vec *sample;

	struct isl_vec *min;

	struct isl_vec *max;

	enum isl_lp_result res;

	int level;

	int init;

	int reduced;

	struct isl_tab_undo **snap;

	if (!tab)

		return NULL;

	if (tab->empty)

		return isl_vec_alloc(tab->mat->ctx, 0);

	if (!tab->basis)

		tab->basis = initial_basis(tab);

	if (!tab->basis)

		return NULL;

	isl_assert(tab->mat->ctx, tab->basis->n_row == tab->n_var + 1,

		    return NULL);

	isl_assert(tab->mat->ctx, tab->basis->n_col == tab->n_var + 1,

		    return NULL);

	ctx = tab->mat->ctx;

	dim = tab->n_var;

	gbr = ctx->opt->gbr;

	if (tab->n_unbounded == tab->n_var) {

		sample = isl_tab_get_sample_value(tab);

		sample = isl_mat_vec_product(isl_mat_copy(tab->basis), sample);

		sample = isl_vec_ceil(sample);

		sample = isl_mat_vec_inverse_product(isl_mat_copy(tab->basis),

							sample);

		return sample;

	}

	if (isl_tab_extend_cons(tab, dim + 1) < 0)

		return NULL;

	min = isl_vec_alloc(ctx, dim);

	max = isl_vec_alloc(ctx, dim);

	snap = isl_alloc_array(ctx, struct isl_tab_undo *, dim);

	if (!min || !max || !snap)

		goto error;

	level = 0;

	init = 1;

	reduced = 0;

	while (level >= 0) {

		if (init) {

			int choice;

			res = compute_min(ctx, tab, min, level);

			if (res == isl_lp_error)

				goto error;

			if (res != isl_lp_ok)

				isl_die(ctx, isl_error_internal,

					"expecting bounded rational solution",

					goto error);

			if (isl_tab_sample_is_integer(tab))

				break;

			res = compute_max(ctx, tab, max, level);

			if (res == isl_lp_error)

				goto error;

			if (res != isl_lp_ok)

				isl_die(ctx, isl_error_internal,

					"expecting bounded rational solution",

					goto error);

			if (isl_tab_sample_is_integer(tab))

				break;

			choice = isl_int_lt(min->el[level], max->el[level]);

			if (choice) {

				int g;

				g = greedy_search(ctx, tab, min, max, level);

				if (g < 0)

					goto error;

				if (g)

					break;

			}

			if (!reduced && choice &&

			    ctx->opt->gbr != ISL_GBR_NEVER) {

				unsigned gbr_only_first;

				if (ctx->opt->gbr == ISL_GBR_ONCE)

					ctx->opt->gbr = ISL_GBR_NEVER;

				tab->n_zero = level;

				gbr_only_first = ctx->opt->gbr_only_first;

				ctx->opt->gbr_only_first =

					ctx->opt->gbr == ISL_GBR_ALWAYS;

				tab = isl_tab_compute_reduced_basis(tab);

				ctx->opt->gbr_only_first = gbr_only_first;

				if (!tab || !tab->basis)

					goto error;

				reduced = 1;

				continue;

			}

			reduced = 0;

			snap[level] = isl_tab_snap(tab);

		} else

			isl_int_add_ui(min->el[level], min->el[level], 1);

		if (isl_int_gt(min->el[level], max->el[level])) {

			level--;

			init = 0;

			if (level >= 0)

				if (isl_tab_rollback(tab, snap[level]) < 0)

					goto error;

			continue;

		}

		isl_int_neg(tab->basis->row[1 + level][0], min->el[level]);

		if (isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]) < 0)

			goto error;

		isl_int_set_si(tab->basis->row[1 + level][0], 0);

		if (level + tab->n_unbounded < dim - 1) {

			++level;

			init = 1;

			continue;

		}

		break;

	}

	if (level >= 0) {

		sample = isl_tab_get_sample_value(tab);

		if (!sample)

			goto error;

		if (tab->n_unbounded && !isl_int_is_one(sample->el[0])) {

			sample = isl_mat_vec_product(isl_mat_copy(tab->basis),

						     sample);

			sample = isl_vec_ceil(sample);

			sample = isl_mat_vec_inverse_product(

					isl_mat_copy(tab->basis), sample);

		}

	} else

		sample = isl_vec_alloc(ctx, 0);

	ctx->opt->gbr = gbr;

	isl_vec_free(min);

	isl_vec_free(max);

	free(snap);

	return sample;

error:

	ctx->opt->gbr = gbr;

	isl_vec_free(min);

	isl_vec_free(max);

	free(snap);

	return NULL;

}

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

/* Compute a sample point of the given basic set, based on the given,

 * non-trivial factorization.

 */

static __isl_give isl_vec *factored_sample(__isl_take isl_basic_set *bset,

	__isl_take isl_factorizer *f)

{

	int i, n;

	isl_vec *sample = NULL;

	isl_ctx *ctx;

	unsigned nparam;

	unsigned nvar;

	ctx = isl_basic_set_get_ctx(bset);

	if (!ctx)

		goto error;

	nparam = isl_basic_set_dim(bset, isl_dim_param);

	nvar = isl_basic_set_dim(bset, isl_dim_set);

	sample = isl_vec_alloc(ctx, 1 + isl_basic_set_total_dim(bset));

	if (!sample)

		goto error;

	isl_int_set_si(sample->el[0], 1);

	bset = isl_morph_basic_set(isl_morph_copy(f->morph), bset);

	for (i = 0, n = 0; i < f->n_group; ++i) {

		isl_basic_set *bset_i;

		isl_vec *sample_i;

		bset_i = isl_basic_set_copy(bset);

		bset_i = isl_basic_set_drop_constraints_involving(bset_i,

			    nparam + n + f->len[i], nvar - n - f->len[i]);

		bset_i = isl_basic_set_drop_constraints_involving(bset_i,

			    nparam, n);

		bset_i = isl_basic_set_drop(bset_i, isl_dim_set,

			    n + f->len[i], nvar - n - f->len[i]);

		bset_i = isl_basic_set_drop(bset_i, isl_dim_set, 0, n);

		sample_i = sample_bounded(bset_i);

		if (!sample_i)

			goto error;

		if (sample_i->size == 0) {

			isl_basic_set_free(bset);

			isl_factorizer_free(f);

			isl_vec_free(sample);

			return sample_i;

		}

		isl_seq_cpy(sample->el + 1 + nparam + n,

			    sample_i->el + 1, f->len[i]);

		isl_vec_free(sample_i);

		n += f->len[i];

	}

	f->morph = isl_morph_inverse(f->morph);

	sample = isl_morph_vec(isl_morph_copy(f->morph), sample);

	isl_basic_set_free(bset);

	isl_factorizer_free(f);

	return sample;

error:

	isl_basic_set_free(bset);

	isl_factorizer_free(f);

	isl_vec_free(sample);

	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 construct a tableau

 * and then use isl_tab_sample to find a sample, passing it

 * the identity matrix as initial basis.

 */ 

static struct isl_vec *sample_bounded(struct isl_basic_set *bset)

{

	unsigned dim;

	struct isl_ctx *ctx;

	struct isl_vec *sample;

	struct isl_tab *tab = NULL;

	isl_factorizer *f;

	if (!bset)

		return NULL;

	if (isl_basic_set_plain_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);

	f = isl_basic_set_factorizer(bset);

	if (!f)

		goto error;

	if (f->n_group != 0)

		return factored_sample(bset, f);

	isl_factorizer_free(f);

	ctx = bset->ctx;

	tab = isl_tab_from_basic_set(bset, 1);

	if (tab && tab->empty) {

		isl_tab_free(tab);

		ISL_F_SET(bset, ISL_BASIC_SET_EMPTY);

		sample = isl_vec_alloc(bset->ctx, 0);

		isl_basic_set_free(bset);

		return sample;

	}

	if (!ISL_F_ISSET(bset, ISL_BASIC_SET_NO_IMPLICIT))

		if (isl_tab_detect_implicit_equalities(tab) < 0)

			goto error;

	sample = isl_tab_sample(tab);

	if (!sample)

		goto error;

	if (sample->size > 0) {

		isl_vec_free(bset->sample);

		bset->sample = isl_vec_copy(sample);

	}

	isl_basic_set_free(bset);

	isl_tab_free(tab);

	return sample;

error:

	isl_basic_set_free(bset);

	isl_tab_free(tab);

	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, 0);

	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_space(isl_basic_set_get_space(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, isl_dim_set,

					 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;

}

/* 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