isl_map_piplib.c: extremum_on: detect equalities first

[isl.git] / isl_affine_hull.c

#include "isl_ctx.h"
#include "isl_seq.h"
#include "isl_set.h"
#include "isl_lp.h"
#include "isl_map.h"
#include "isl_map_private.h"
#include "isl_equalities.h"
#include "isl_sample.h"
#include "isl_tab.h"

struct isl_basic_map *isl_basic_map_implicit_equalities(
                                                struct isl_basic_map *bmap)
{
        struct isl_tab *tab;

        if (!bmap)
                return bmap;

        bmap = isl_basic_map_gauss(bmap, NULL);
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
                return bmap;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_IMPLICIT))
                return bmap;
        if (bmap->n_ineq <= 1)
                return bmap;

        tab = isl_tab_from_basic_map(bmap);
        tab = isl_tab_detect_equalities(bmap->ctx, tab);
        bmap = isl_basic_map_update_from_tab(bmap, tab);
        isl_tab_free(bmap->ctx, tab);
        bmap = isl_basic_map_gauss(bmap, NULL);
        ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
        return bmap;
}

struct isl_basic_set *isl_basic_set_implicit_equalities(
                                                struct isl_basic_set *bset)
{
        return (struct isl_basic_set *)
                isl_basic_map_implicit_equalities((struct isl_basic_map*)bset);
}

struct isl_map *isl_map_implicit_equalities(struct isl_map *map)
{
        int i;

        if (!map)
                return map;

        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_implicit_equalities(map->p[i]);
                if (!map->p[i])
                        goto error;
        }

        return map;
error:
        isl_map_free(map);
        return NULL;
}

/* Make eq[row][col] of both bmaps equal so we can add the row
 * add the column to the common matrix.
 * Note that because of the echelon form, the columns of row row
 * after column col are zero.
 */
static void set_common_multiple(
        struct isl_basic_set *bset1, struct isl_basic_set *bset2,
        unsigned row, unsigned col)
{
        isl_int m, c;

        if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col]))
                return;

        isl_int_init(c);
        isl_int_init(m);
        isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]);
        isl_int_divexact(c, m, bset1->eq[row][col]);
        isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1);
        isl_int_divexact(c, m, bset2->eq[row][col]);
        isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1);
        isl_int_clear(c);
        isl_int_clear(m);
}

/* Delete a given equality, moving all the following equalities one up.
 */
static void delete_row(struct isl_basic_set *bset, unsigned row)
{
        isl_int *t;
        int r;

        t = bset->eq[row];
        bset->n_eq--;
        for (r = row; r < bset->n_eq; ++r)
                bset->eq[r] = bset->eq[r+1];
        bset->eq[bset->n_eq] = t;
}

/* Make first row entries in column col of bset1 identical to
 * those of bset2, using the fact that entry bset1->eq[row][col]=a
 * is non-zero.  Initially, these elements of bset1 are all zero.
 * For each row i < row, we set
 *              A[i] = a * A[i] + B[i][col] * A[row]
 *              B[i] = a * B[i]
 * so that
 *              A[i][col] = B[i][col] = a * old(B[i][col])
 */
static void construct_column(
        struct isl_basic_set *bset1, struct isl_basic_set *bset2,
        unsigned row, unsigned col)
{
        int r;
        isl_int a;
        isl_int b;
        unsigned total;

        isl_int_init(a);
        isl_int_init(b);
        total = 1 + isl_basic_set_n_dim(bset1);
        for (r = 0; r < row; ++r) {
                if (isl_int_is_zero(bset2->eq[r][col]))
                        continue;
                isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]);
                isl_int_divexact(a, bset1->eq[row][col], b);
                isl_int_divexact(b, bset2->eq[r][col], b);
                isl_seq_combine(bset1->eq[r], a, bset1->eq[r],
                                              b, bset1->eq[row], total);
                isl_seq_scale(bset2->eq[r], bset2->eq[r], a, total);
        }
        isl_int_clear(a);
        isl_int_clear(b);
        delete_row(bset1, row);
}

/* Make first row entries in column col of bset1 identical to
 * those of bset2, using only these entries of the two matrices.
 * Let t be the last row with different entries.
 * For each row i < t, we set
 *      A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
 *      B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
 * so that
 *      A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
 */
static int transform_column(
        struct isl_basic_set *bset1, struct isl_basic_set *bset2,
        unsigned row, unsigned col)
{
        int i, t;
        isl_int a, b, g;
        unsigned total;

        for (t = row-1; t >= 0; --t)
                if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))
                        break;
        if (t < 0)
                return 0;

        total = 1 + isl_basic_set_n_dim(bset1);
        isl_int_init(a);
        isl_int_init(b);
        isl_int_init(g);
        isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]);
        for (i = 0; i < t; ++i) {
                isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]);
                isl_int_gcd(g, a, b);
                isl_int_divexact(a, a, g);
                isl_int_divexact(g, b, g);
                isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t],
                                total);
                isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],
                                total);
        }
        isl_int_clear(a);
        isl_int_clear(b);
        isl_int_clear(g);
        delete_row(bset1, t);
        delete_row(bset2, t);
        return 1;
}

/* The implementation is based on Section 5.2 of Michael Karr,
 * "Affine Relationships Among Variables of a Program",
 * except that the echelon form we use starts from the last column
 * and that we are dealing with integer coefficients.
 */
static struct isl_basic_set *affine_hull(
        struct isl_basic_set *bset1, struct isl_basic_set *bset2)
{
        unsigned total;
        int col;
        int row;

        total = 1 + isl_basic_set_n_dim(bset1);

        row = 0;
        for (col = total-1; col >= 0; --col) {
                int is_zero1 = row >= bset1->n_eq ||
                        isl_int_is_zero(bset1->eq[row][col]);
                int is_zero2 = row >= bset2->n_eq ||
                        isl_int_is_zero(bset2->eq[row][col]);
                if (!is_zero1 && !is_zero2) {
                        set_common_multiple(bset1, bset2, row, col);
                        ++row;
                } else if (!is_zero1 && is_zero2) {
                        construct_column(bset1, bset2, row, col);
                } else if (is_zero1 && !is_zero2) {
                        construct_column(bset2, bset1, row, col);
                } else {
                        if (transform_column(bset1, bset2, row, col))
                                --row;
                }
        }
        isl_basic_set_free(bset2);
        isl_assert(ctx, row == bset1->n_eq, goto error);
        bset1 = isl_basic_set_normalize_constraints(bset1);
        return bset1;
error:
        isl_basic_set_free(bset1);
        return NULL;
}

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

        if (!vec)
                return NULL;
        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->block.data[1 + i]);
                isl_int_set(bset->eq[k][1 + i], vec->block.data[0]);
        }
        isl_vec_free(ctx, vec);

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

/* Find an integer point in "bset" that lies outside of the equality
 * "eq" e(x) = 0.
 * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
 * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
 * The point, if found, is returned as a singleton set.
 * If no point can be found, the empty set is returned.
 *
 * Before solving an ILP problem, we first check if simply
 * adding the normal of the constraint to one of the known
 * integer points in the basic set yields another point
 * inside the basic set.
 */
static struct isl_basic_set *outside_point(struct isl_ctx *ctx,
        struct isl_basic_set *bset, isl_int *eq, int up)
{
        struct isl_basic_set *slice = NULL;
        struct isl_vec *sample;
        struct isl_basic_set *point;
        unsigned dim;
        int k;

        dim = isl_basic_set_n_dim(bset);
        sample = isl_vec_alloc(ctx, 1 + dim);
        if (!sample)
                return NULL;
        isl_int_set_si(sample->block.data[0], 1);
        isl_seq_combine(sample->block.data + 1,
                ctx->one, bset->sample->block.data + 1,
                up ? ctx->one : ctx->negone, eq + 1, dim);
        if (isl_basic_set_contains(bset, sample))
                return isl_basic_set_from_vec(ctx, sample);
        isl_vec_free(ctx, sample);
        sample = NULL;

        slice = isl_basic_set_copy(bset);
        if (!slice)
                goto error;
        slice = isl_basic_set_cow(slice);
        slice = isl_basic_set_extend(slice, 0, dim, 0, 0, 1);
        k = isl_basic_set_alloc_inequality(slice);
        if (k < 0)
                goto error;
        if (up)
                isl_seq_cpy(slice->ineq[k], eq, 1 + dim);
        else
                isl_seq_neg(slice->ineq[k], eq, 1 + dim);
        isl_int_sub_ui(slice->ineq[k][0], slice->ineq[k][0], 1);

        sample = isl_basic_set_sample(slice);
        if (!sample)
                goto error;
        if (sample->size == 0) {
                isl_vec_free(ctx, sample);
                point = isl_basic_set_empty_like(bset);
        } else
                point = isl_basic_set_from_vec(ctx, sample);

        return point;
error:
        isl_basic_set_free(slice);
        return NULL;
}

static struct isl_basic_set *recession_cone(struct isl_basic_set *bset)
{
        int i;

        bset = isl_basic_set_cow(bset);
        if (!bset)
                return NULL;

        for (i = 0; i < bset->n_eq; ++i)
                isl_int_set_si(bset->eq[i][0], 0);

        for (i = 0; i < bset->n_ineq; ++i)
                isl_int_set_si(bset->ineq[i][0], 0);

        ISL_F_CLR(bset, ISL_BASIC_SET_NO_IMPLICIT);
        return isl_basic_set_implicit_equalities(bset);
}

static struct isl_basic_set *shift(struct isl_basic_set *bset, isl_int *point)
{
        int i;
        unsigned dim;

        bset = isl_basic_set_cow(bset);
        if (!bset)
                return NULL;

        dim = isl_basic_set_n_dim(bset);
        for (i = 0; i < bset->n_eq; ++i) {
                isl_seq_inner_product(bset->eq[i]+1, point+1, dim,
                                        &bset->eq[i][0]);
                isl_int_neg(bset->eq[i][0], bset->eq[i][0]);
        }

        for (i = 0; i < bset->n_ineq; ++i) {
                isl_seq_inner_product(bset->ineq[i]+1, point+1, dim,
                                        &bset->ineq[i][0]);
                isl_int_neg(bset->ineq[i][0], bset->ineq[i][0]);
        }

        return bset;
}

/* Look for all equalities satisfied by the integer points in bset,
 * which is assume not to have any explicit equalities.
 *
 * The equalities are obtained by successively looking for
 * a point that is affinely independent of the points found so far.
 * In particular, for each equality satisfied by the points so far,
 * we check if there is any point on a hyperplane parallel to the
 * corresponding hyperplane shifted by at least one (in either direction).
 *
 * Before looking for any outside points, we first remove the equalities
 * that correspond to the affine hull of the recession cone.
 * These equalities will never be equalities over the whols basic set.
 */
static struct isl_basic_set *uset_affine_hull(struct isl_basic_set *bset)
{
        int i, j;
        struct isl_basic_set *hull = NULL;
        struct isl_vec *sample;
        struct isl_ctx *ctx;
        unsigned dim;

        if (isl_basic_set_is_empty(bset))
                return bset;

        ctx = bset->ctx;
        sample = isl_basic_set_sample(isl_basic_set_copy(bset));
        if (!sample)
                goto error;
        if (sample->size == 0) {
                isl_vec_free(ctx, sample);
                hull = isl_basic_set_empty_like(bset);
                isl_basic_set_free(bset);
                return hull;
        } else
                hull = isl_basic_set_from_vec(ctx, sample);

        if (hull->n_eq > 0) {
                struct isl_basic_set *cone;
                cone = recession_cone(isl_basic_set_copy(bset));
                isl_basic_set_free_inequality(cone, cone->n_ineq);
                cone = isl_basic_set_normalize_constraints(cone);
                cone = shift(cone, bset->sample->block.data);
                hull = affine_hull(hull, cone);
        }

        dim = isl_basic_set_n_dim(bset);
        for (i = 0; i < dim; ++i) {
                struct isl_basic_set *point;
                for (j = 0; j < hull->n_eq; ++j) {
                        point = outside_point(ctx, bset, hull->eq[j], 1);
                        if (!point)
                                goto error;
                        if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
                                break;
                        isl_basic_set_free(point);
                        point = outside_point(ctx, bset, hull->eq[j], 0);
                        if (!point)
                                goto error;
                        if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
                                break;
                        isl_basic_set_free(point);
                }
                if (j == hull->n_eq)
                        break;
                hull = affine_hull(hull, point);
        }
        isl_basic_set_free(bset);

        return hull;
error:
        isl_basic_set_free(bset);
        isl_basic_set_free(hull);
        return NULL;
}

/* Look for all equalities satisfied by the integer points in bmap
 * that are independent of the equalities already explicitly available
 * in bmap.
 *
 * We first remove all equalities already explicitly available,
 * then look for additional equalities in the reduced space
 * and then transform the result to the original space.
 * The original equalities are _not_ added to this set.  This is
 * the responsibility of the calling function.
 * The resulting basic set has all meaning about the dimensions removed.
 * In particular, dimensions that correspond to existential variables
 * in bmap and that are found to be fixed are not removed.
 */
static struct isl_basic_set *equalities_in_underlying_set(
                                                struct isl_basic_map *bmap)
{
        struct isl_mat *T2 = NULL;
        struct isl_basic_set *bset = NULL;
        struct isl_basic_set *hull = NULL;
        struct isl_ctx *ctx;

        ctx = bmap->ctx;
        bset = isl_basic_map_underlying_set(bmap);
        bset = isl_basic_set_remove_equalities(bset, NULL, &T2);
        if (!bset)
                goto error;

        hull = uset_affine_hull(bset);
        if (T2)
                hull = isl_basic_set_preimage(hull, T2);

        return hull;
error:
        isl_mat_free(ctx, T2);
        isl_basic_set_free(bset);
        isl_basic_set_free(hull);
        return NULL;
}

/* Detect and make explicit all equalities satisfied by the (integer)
 * points in bmap.
 */
struct isl_basic_map *isl_basic_map_detect_equalities(
                                                struct isl_basic_map *bmap)
{
        int i, j;
        struct isl_basic_set *hull = NULL;

        if (!bmap)
                return NULL;
        if (bmap->n_ineq == 0)
                return bmap;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
                return bmap;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))
                return bmap;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
                return isl_basic_map_implicit_equalities(bmap);

        hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
        if (!hull)
                goto error;
        if (ISL_F_ISSET(hull, ISL_BASIC_SET_EMPTY)) {
                isl_basic_set_free(hull);
                return isl_basic_map_set_to_empty(bmap);
        }
        bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim), 0,
                                        hull->n_eq, 0);
        for (i = 0; i < hull->n_eq; ++i) {
                j = isl_basic_map_alloc_equality(bmap);
                if (j < 0)
                        goto error;
                isl_seq_cpy(bmap->eq[j], hull->eq[i],
                                1 + isl_basic_set_total_dim(hull));
        }
        isl_basic_set_free(hull);
        ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);
        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_set_free(hull);
        isl_basic_map_free(bmap);
        return NULL;
}

struct isl_map *isl_map_detect_equalities(struct isl_map *map)
{
        struct isl_basic_map *bmap;
        int i;

        if (!map)
                return NULL;

        for (i = 0; i < map->n; ++i) {
                bmap = isl_basic_map_copy(map->p[i]);
                bmap = isl_basic_map_detect_equalities(bmap);
                if (!bmap)
                        goto error;
                isl_basic_map_free(map->p[i]);
                map->p[i] = bmap;
        }

        return map;
error:
        isl_map_free(map);
        return NULL;
}

/* After computing the rational affine hull (by detecting the implicit
 * equalities), we compute the additional equalities satisfied by
 * the integer points (if any) and add the original equalities back in.
 */
struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
{
        struct isl_basic_set *hull = NULL;

        bmap = isl_basic_map_detect_equalities(bmap);
        bmap = isl_basic_map_cow(bmap);
        isl_basic_map_free_inequality(bmap, bmap->n_ineq);
        return bmap;
}

struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
{
        return (struct isl_basic_set *)
                isl_basic_map_affine_hull((struct isl_basic_map *)bset);
}

struct isl_basic_map *isl_map_affine_hull(struct isl_map *map)
{
        int i;
        struct isl_basic_map *model = NULL;
        struct isl_basic_map *hull = NULL;
        struct isl_set *set;

        if (!map)
                return NULL;

        if (map->n == 0) {
                hull = isl_basic_map_empty_like_map(map);
                isl_map_free(map);
                return hull;
        }

        map = isl_map_detect_equalities(map);
        map = isl_map_align_divs(map);
        if (!map)
                return NULL;
        model = isl_basic_map_copy(map->p[0]);
        set = isl_map_underlying_set(map);
        set = isl_set_cow(set);
        if (!set)
                goto error;

        for (i = 0; i < set->n; ++i) {
                set->p[i] = isl_basic_set_cow(set->p[i]);
                set->p[i] = isl_basic_set_affine_hull(set->p[i]);
                set->p[i] = isl_basic_set_gauss(set->p[i], NULL);
                if (!set->p[i])
                        goto error;
        }
        set = isl_set_remove_empty_parts(set);
        if (set->n == 0) {
                hull = isl_basic_map_empty_like(model);
                isl_basic_map_free(model);
        } else {
                struct isl_basic_set *bset;
                while (set->n > 1) {
                        set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
                        if (!set->p[0])
                                goto error;
                }
                bset = isl_basic_set_copy(set->p[0]);
                hull = isl_basic_map_overlying_set(bset, model);
        }
        isl_set_free(set);
        hull = isl_basic_map_simplify(hull);
        return isl_basic_map_finalize(hull);
error:
        isl_basic_map_free(model);
        isl_set_free(set);
        return NULL;
}

struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
{
        return (struct isl_basic_set *)
                isl_map_affine_hull((struct isl_map *)set);
}