isl_tab_pip.c: fix typos in comments

[isl.git] / isl_sample.c

/*
 * 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".
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 */
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".
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 */
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_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);

        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(isl_basic_set_get_ctx(bset), 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
 * 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.
 */
__isl_give isl_vec *isl_basic_set_sample_with_cone(
        __isl_take isl_basic_set *bset, __isl_take 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 = isl_basic_set_get_ctx(bset);
        total = isl_basic_set_total_dim(cone);
        cone_dim = total - cone->n_eq;

        M = isl_mat_sub_alloc6(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 = isl_basic_set_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;
}

static void vec_sum_of_neg(struct isl_vec *v, isl_int *s)
{
        int i;

        isl_int_set_si(*s, 0);

        for (i = 0; i < v->size; ++i)
                if (isl_int_is_neg(v->el[i]))
                        isl_int_add(*s, *s, v->el[i]);
}

/* Given a tableau "tab", a tableau "tab_cone" that corresponds
 * to the recession cone and the inverse of a new basis U = inv(B),
 * with the unbounded directions in B last,
 * add constraints to "tab" that ensure any rational value
 * in the unbounded directions can be rounded up to an integer value.
 *
 * The new basis is given by x' = B x, i.e., x = U x'.
 * For any rational value of the last tab->n_unbounded coordinates
 * in the update tableau, the value that is obtained by rounding
 * up this value should be contained in the original tableau.
 * For any constraint "a x + c >= 0", we therefore need to add
 * a constraint "a x + c + s >= 0", with s the sum of all negative
 * entries in the last elements of "a U".
 *
 * Since we are not interested in the first entries of any of the "a U",
 * we first drop the columns of U that correpond to bounded directions.
 */
static int tab_shift_cone(struct isl_tab *tab,
        struct isl_tab *tab_cone, struct isl_mat *U)
{
        int i;
        isl_int v;
        struct isl_basic_set *bset = NULL;

        if (tab && tab->n_unbounded == 0) {
                isl_mat_free(U);
                return 0;
        }
        isl_int_init(v);
        if (!tab || !tab_cone || !U)
                goto error;
        bset = isl_tab_peek_bset(tab_cone);
        U = isl_mat_drop_cols(U, 0, tab->n_var - tab->n_unbounded);
        for (i = 0; i < bset->n_ineq; ++i) {
                int ok;
                struct isl_vec *row = NULL;
                if (isl_tab_is_equality(tab_cone, tab_cone->n_eq + i))
                        continue;
                row = isl_vec_alloc(bset->ctx, tab_cone->n_var);
                if (!row)
                        goto error;
                isl_seq_cpy(row->el, bset->ineq[i] + 1, tab_cone->n_var);
                row = isl_vec_mat_product(row, isl_mat_copy(U));
                if (!row)
                        goto error;
                vec_sum_of_neg(row, &v);
                isl_vec_free(row);
                if (isl_int_is_zero(v))
                        continue;
                if (isl_tab_extend_cons(tab, 1) < 0)
                        goto error;
                isl_int_add(bset->ineq[i][0], bset->ineq[i][0], v);
                ok = isl_tab_add_ineq(tab, bset->ineq[i]) >= 0;
                isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], v);
                if (!ok)
                        goto error;
        }

        isl_mat_free(U);
        isl_int_clear(v);
        return 0;
error:
        isl_mat_free(U);
        isl_int_clear(v);
        return -1;
}

/* Compute and return an initial basis for the possibly
 * unbounded tableau "tab".  "tab_cone" is a tableau
 * for the corresponding recession cone.
 * Additionally, add constraints to "tab" that ensure
 * that any rational value for the unbounded directions
 * can be rounded up to an integer value.
 *
 * If the tableau is bounded, i.e., if the recession cone
 * is zero-dimensional, then we just use inital_basis.
 * Otherwise, we construct a basis whose first directions
 * correspond to equalities, followed by bounded directions,
 * i.e., equalities in the recession cone.
 * The remaining directions are then unbounded.
 */
int isl_tab_set_initial_basis_with_cone(struct isl_tab *tab,
        struct isl_tab *tab_cone)
{
        struct isl_mat *eq;
        struct isl_mat *cone_eq;
        struct isl_mat *U, *Q;

        if (!tab || !tab_cone)
                return -1;

        if (tab_cone->n_col == tab_cone->n_dead) {
                tab->basis = initial_basis(tab);
                return tab->basis ? 0 : -1;
        }

        eq = tab_equalities(tab);
        if (!eq)
                return -1;
        tab->n_zero = eq->n_row;
        cone_eq = tab_equalities(tab_cone);
        eq = isl_mat_concat(eq, cone_eq);
        if (!eq)
                return -1;
        tab->n_unbounded = tab->n_var - (eq->n_row - tab->n_zero);
        eq = isl_mat_left_hermite(eq, 0, &U, &Q);
        if (!eq)
                return -1;
        isl_mat_free(eq);
        tab->basis = isl_mat_lin_to_aff(Q);
        if (tab_shift_cone(tab, tab_cone, U) < 0)
                return -1;
        if (!tab->basis)
                return -1;
        return 0;
}

/* 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)
                goto error;

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

        isl_basic_set_free(cone);
        return sample_bounded(bset);
error:
        isl_basic_set_free(bset);
        return NULL;
}

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_plain_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, bounded ? isl_basic_set_sample_bounded
                                               : isl_basic_set_sample_vec);
        if (dim == 0)
                return zero_sample(bset);
        if (dim == 1)
                return interval_sample(bset);

        return bounded ? sample_bounded(bset) : gbr_sample(bset);
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]);
        }
        bset->sample = 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) {
                isl_vec_free(sample_vec);
                return isl_basic_map_set_to_empty(bmap);
        }
        isl_vec_free(bmap->sample);
        bmap->sample = isl_vec_copy(sample_vec);
        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_set *isl_basic_set_sample(__isl_take isl_basic_set *bset)
{
        return isl_basic_map_sample(bset);
}

__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(isl_map_get_space(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);
}

__isl_give isl_point *isl_basic_set_sample_point(__isl_take isl_basic_set *bset)
{
        isl_vec *vec;
        isl_space *dim;

        dim = isl_basic_set_get_space(bset);
        bset = isl_basic_set_underlying_set(bset);
        vec = isl_basic_set_sample_vec(bset);

        return isl_point_alloc(dim, vec);
}

__isl_give isl_point *isl_set_sample_point(__isl_take isl_set *set)
{
        int i;
        isl_point *pnt;

        if (!set)
                return NULL;

        for (i = 0; i < set->n; ++i) {
                pnt = isl_basic_set_sample_point(isl_basic_set_copy(set->p[i]));
                if (!pnt)
                        goto error;
                if (!isl_point_is_void(pnt))
                        break;
                isl_point_free(pnt);
        }
        if (i == set->n)
                pnt = isl_point_void(isl_set_get_space(set));

        isl_set_free(set);
        return pnt;
error:
        isl_set_free(set);
        return NULL;
}