isl_affine_hull.c

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/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2010      INRIA Saclay
 * Copyright 2012      Ecole Normale Superieure
 *
 * 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
 * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
 * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
 * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
 */
 
#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include <isl_seq.h>
#include <isl/set.h>
#include <isl/lp.h>
#include <isl/map.h>
#include "isl_equalities.h"
#include "isl_sample.h"
#include "isl_tab.h"
#include <isl_mat_private.h>
#include <isl_vec_private.h>
 
#include <bset_to_bmap.c>
#include <bset_from_bmap.c>
#include <set_to_map.c>
#include <set_from_map.c>
 
__isl_give isl_basic_map *isl_basic_map_implicit_equalities(
  __isl_take 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, 0);
  if (isl_tab_detect_implicit_equalities(tab) < 0)
    goto error;
  bmap = isl_basic_map_update_from_tab(bmap, tab);
  isl_tab_free(tab);
  bmap = isl_basic_map_gauss(bmap, NULL);
  ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
  return bmap;
error:
  isl_tab_free(tab);
  isl_basic_map_free(bmap);
  return NULL;
}
 
__isl_give isl_basic_set *isl_basic_set_implicit_equalities(
  __isl_take isl_basic_set *bset)
{
  return bset_from_bmap(
    isl_basic_map_implicit_equalities(bset_to_bmap(bset)));
}
 
/* 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(__isl_keep 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 isl_stat construct_column(
  __isl_keep isl_basic_set *bset1, __isl_keep isl_basic_set *bset2,
  unsigned row, unsigned col)
{
  int r;
  isl_int a;
  isl_int b;
  isl_size total;
 
  total = isl_basic_set_dim(bset1, isl_dim_set);
  if (total < 0)
    return isl_stat_error;
 
  isl_int_init(a);
  isl_int_init(b);
  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], 1 + total);
    isl_seq_scale(bset2->eq[r], bset2->eq[r], a, 1 + total);
  }
  isl_int_clear(a);
  isl_int_clear(b);
  delete_row(bset1, row);
 
  return isl_stat_ok;
}
 
/* 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 isl_bool transform_column(
  __isl_keep isl_basic_set *bset1, __isl_keep isl_basic_set *bset2,
  unsigned row, unsigned col)
{
  int i, t;
  isl_int a, b, g;
  isl_size 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 isl_bool_false;
 
  total = isl_basic_set_dim(bset1, isl_dim_set);
  if (total < 0)
    return isl_bool_error;
  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],
        1 + total);
    isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],
        1 + total);
  }
  isl_int_clear(a);
  isl_int_clear(b);
  isl_int_clear(g);
  delete_row(bset1, t);
  delete_row(bset2, t);
  return isl_bool_true;
}
 
/* 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 __isl_give isl_basic_set *affine_hull(
  __isl_take isl_basic_set *bset1, __isl_take isl_basic_set *bset2)
{
  isl_size dim;
  unsigned total;
  int col;
  int row;
 
  dim = isl_basic_set_dim(bset1, isl_dim_set);
  if (dim < 0 || !bset2)
    goto error;
 
  total = 1 + dim;
 
  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) {
      if (construct_column(bset1, bset2, row, col) < 0)
        goto error;
    } else if (is_zero1 && !is_zero2) {
      if (construct_column(bset2, bset1, row, col) < 0)
        goto error;
    } else {
      isl_bool transform;
 
      transform = transform_column(bset1, bset2, row, col);
      if (transform < 0)
        goto error;
      if (transform)
        --row;
    }
  }
  isl_assert(bset1->ctx, row == bset1->n_eq, goto error);
  isl_basic_set_free(bset2);
  bset1 = isl_basic_set_normalize_constraints(bset1);
  return bset1;
error:
  isl_basic_set_free(bset1);
  isl_basic_set_free(bset2);
  return NULL;
}
 
/* Find an integer point in the set represented by "tab"
 * 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.
 * If no point can be found, a zero-length vector 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 represented by "tab"
 * yields another point inside the basic set.
 *
 * The caller of this function ensures that the tableau is bounded or
 * that tab->basis and tab->n_unbounded have been set appropriately.
 */
static __isl_give isl_vec *outside_point(struct isl_tab *tab, isl_int *eq,
  int up)
{
  struct isl_ctx *ctx;
  struct isl_vec *sample = NULL;
  struct isl_tab_undo *snap;
  unsigned dim;
 
  if (!tab)
    return NULL;
  ctx = tab->mat->ctx;
 
  dim = tab->n_var;
  sample = isl_vec_alloc(ctx, 1 + dim);
  if (!sample)
    return NULL;
  isl_int_set_si(sample->el[0], 1);
  isl_seq_combine(sample->el + 1,
    ctx->one, tab->bmap->sample->el + 1,
    up ? ctx->one : ctx->negone, eq + 1, dim);
  if (isl_basic_map_contains(tab->bmap, sample))
    return sample;
  isl_vec_free(sample);
  sample = NULL;
 
  snap = isl_tab_snap(tab);
 
  if (!up)
    isl_seq_neg(eq, eq, 1 + dim);
  isl_int_sub_ui(eq[0], eq[0], 1);
 
  if (isl_tab_extend_cons(tab, 1) < 0)
    goto error;
  if (isl_tab_add_ineq(tab, eq) < 0)
    goto error;
 
  sample = isl_tab_sample(tab);
 
  isl_int_add_ui(eq[0], eq[0], 1);
  if (!up)
    isl_seq_neg(eq, eq, 1 + dim);
 
  if (sample && isl_tab_rollback(tab, snap) < 0)
    goto error;
 
  return sample;
error:
  isl_vec_free(sample);
  return NULL;
}
 
__isl_give isl_basic_set *isl_basic_set_recession_cone(
  __isl_take isl_basic_set *bset)
{
  int i;
  isl_bool empty;
 
  empty = isl_basic_set_plain_is_empty(bset);
  if (empty < 0)
    return isl_basic_set_free(bset);
  if (empty)
    return bset;
 
  bset = isl_basic_set_cow(bset);
  if (isl_basic_set_check_no_locals(bset) < 0)
    return isl_basic_set_free(bset);
 
  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);
}
 
/* Move "sample" to a point that is one up (or down) from the original
 * point in dimension "pos".
 */
static void adjacent_point(__isl_keep isl_vec *sample, int pos, int up)
{
  if (up)
    isl_int_add_ui(sample->el[1 + pos], sample->el[1 + pos], 1);
  else
    isl_int_sub_ui(sample->el[1 + pos], sample->el[1 + pos], 1);
}
 
/* Check if any points that are adjacent to "sample" also belong to "bset".
 * If so, add them to "hull" and return the updated hull.
 *
 * Before checking whether and adjacent point belongs to "bset", we first
 * check whether it already belongs to "hull" as this test is typically
 * much cheaper.
 */
static __isl_give isl_basic_set *add_adjacent_points(
  __isl_take isl_basic_set *hull, __isl_take isl_vec *sample,
  __isl_keep isl_basic_set *bset)
{
  int i, up;
  isl_size dim;
 
  dim = isl_basic_set_dim(hull, isl_dim_set);
  if (!sample || dim < 0)
    goto error;
 
  for (i = 0; i < dim; ++i) {
    for (up = 0; up <= 1; ++up) {
      int contains;
      isl_basic_set *point;
 
      adjacent_point(sample, i, up);
      contains = isl_basic_set_contains(hull, sample);
      if (contains < 0)
        goto error;
      if (contains) {
        adjacent_point(sample, i, !up);
        continue;
      }
      contains = isl_basic_set_contains(bset, sample);
      if (contains < 0)
        goto error;
      if (contains) {
        point = isl_basic_set_from_vec(
              isl_vec_copy(sample));
        hull = affine_hull(hull, point);
      }
      adjacent_point(sample, i, !up);
      if (contains)
        break;
    }
  }
 
  isl_vec_free(sample);
 
  return hull;
error:
  isl_vec_free(sample);
  isl_basic_set_free(hull);
  return NULL;
}
 
/* Extend an initial (under-)approximation of the affine hull of basic
 * set represented by the tableau "tab"
 * by looking for points that do not satisfy one of the equalities
 * in the current approximation and adding them to that approximation
 * until no such points can be found any more.
 *
 * The caller of this function ensures that "tab" is bounded or
 * that tab->basis and tab->n_unbounded have been set appropriately.
 *
 * "bset" may be either NULL or the basic set represented by "tab".
 * If "bset" is not NULL, we check for any point we find if any
 * of its adjacent points also belong to "bset".
 */
static __isl_give isl_basic_set *extend_affine_hull(struct isl_tab *tab,
  __isl_take isl_basic_set *hull, __isl_keep isl_basic_set *bset)
{
  int i, j;
  unsigned dim;
 
  if (!tab || !hull)
    goto error;
 
  dim = tab->n_var;
 
  if (isl_tab_extend_cons(tab, 2 * dim + 1) < 0)
    goto error;
 
  for (i = 0; i < dim; ++i) {
    struct isl_vec *sample;
    struct isl_basic_set *point;
    for (j = 0; j < hull->n_eq; ++j) {
      sample = outside_point(tab, hull->eq[j], 1);
      if (!sample)
        goto error;
      if (sample->size > 0)
        break;
      isl_vec_free(sample);
      sample = outside_point(tab, hull->eq[j], 0);
      if (!sample)
        goto error;
      if (sample->size > 0)
        break;
      isl_vec_free(sample);
 
      if (isl_tab_add_eq(tab, hull->eq[j]) < 0)
        goto error;
    }
    if (j == hull->n_eq)
      break;
    if (tab->samples &&
        isl_tab_add_sample(tab, isl_vec_copy(sample)) < 0)
      hull = isl_basic_set_free(hull);
    if (bset)
      hull = add_adjacent_points(hull, isl_vec_copy(sample),
                bset);
    point = isl_basic_set_from_vec(sample);
    hull = affine_hull(hull, point);
    if (!hull)
      return NULL;
  }
 
  return hull;
error:
  isl_basic_set_free(hull);
  return NULL;
}
 
/* Construct an initial underapproximation of the hull of "bset"
 * from "sample" and any of its adjacent points that also belong to "bset".
 */
static __isl_give isl_basic_set *initialize_hull(__isl_keep isl_basic_set *bset,
  __isl_take isl_vec *sample)
{
  isl_basic_set *hull;
 
  hull = isl_basic_set_from_vec(isl_vec_copy(sample));
  hull = add_adjacent_points(hull, sample, bset);
 
  return hull;
}
 
/* Look for all equalities satisfied by the integer points in bset,
 * which is assumed to be bounded.
 *
 * 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).
 */
static __isl_give isl_basic_set *uset_affine_hull_bounded(
  __isl_take isl_basic_set *bset)
{
  struct isl_vec *sample = NULL;
  struct isl_basic_set *hull;
  struct isl_tab *tab = NULL;
  isl_size dim;
 
  if (isl_basic_set_plain_is_empty(bset))
    return bset;
 
  dim = isl_basic_set_dim(bset, isl_dim_set);
  if (dim < 0)
    return isl_basic_set_free(bset);
 
  if (bset->sample && bset->sample->size == 1 + dim) {
    int contains = isl_basic_set_contains(bset, bset->sample);
    if (contains < 0)
      goto error;
    if (contains) {
      if (dim == 0)
        return bset;
      sample = isl_vec_copy(bset->sample);
    } else {
      isl_vec_free(bset->sample);
      bset->sample = NULL;
    }
  }
 
  tab = isl_tab_from_basic_set(bset, 1);
  if (!tab)
    goto error;
  if (tab->empty) {
    isl_tab_free(tab);
    isl_vec_free(sample);
    return isl_basic_set_set_to_empty(bset);
  }
 
  if (!sample) {
    struct isl_tab_undo *snap;
    snap = isl_tab_snap(tab);
    sample = isl_tab_sample(tab);
    if (isl_tab_rollback(tab, snap) < 0)
      goto error;
    isl_vec_free(tab->bmap->sample);
    tab->bmap->sample = isl_vec_copy(sample);
  }
 
  if (!sample)
    goto error;
  if (sample->size == 0) {
    isl_tab_free(tab);
    isl_vec_free(sample);
    return isl_basic_set_set_to_empty(bset);
  }
 
  hull = initialize_hull(bset, sample);
 
  hull = extend_affine_hull(tab, hull, bset);
  isl_basic_set_free(bset);
  isl_tab_free(tab);
 
  return hull;
error:
  isl_vec_free(sample);
  isl_tab_free(tab);
  isl_basic_set_free(bset);
  return NULL;
}
 
/* Given an unbounded tableau and an integer point satisfying the tableau,
 * construct an initial affine hull containing the recession cone
 * shifted to the given point.
 *
 * The unbounded directions are taken from the last rows of the basis,
 * which is assumed to have been initialized appropriately.
 */
static __isl_give isl_basic_set *initial_hull(struct isl_tab *tab,
  __isl_take isl_vec *vec)
{
  int i;
  int k;
  struct isl_basic_set *bset = NULL;
  struct isl_ctx *ctx;
  isl_size dim;
 
  if (!vec || !tab)
    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);
  dim = isl_basic_set_dim(bset, isl_dim_set);
  if (dim < 0)
    goto error;
  dim -= tab->n_unbounded;
  for (i = 0; i < dim; ++i) {
    k = isl_basic_set_alloc_equality(bset);
    if (k < 0)
      goto error;
    isl_seq_cpy(bset->eq[k] + 1, tab->basis->row[1 + i] + 1,
          vec->size - 1);
    isl_seq_inner_product(bset->eq[k] + 1, vec->el +1,
              vec->size - 1, &bset->eq[k][0]);
    isl_int_neg(bset->eq[k][0], bset->eq[k][0]);
  }
  bset->sample = vec;
  bset = isl_basic_set_gauss(bset, NULL);
 
  return bset;
error:
  isl_basic_set_free(bset);
  isl_vec_free(vec);
  return NULL;
}
 
/* Given a tableau of a set and a tableau of the corresponding
 * recession cone, detect and add all equalities to the tableau.
 * If the tableau is bounded, then we can simply keep the
 * tableau in its state after the return from extend_affine_hull.
 * However, if the tableau is unbounded, then
 * isl_tab_set_initial_basis_with_cone will add some additional
 * constraints to the tableau that have to be removed again.
 * In this case, we therefore rollback to the state before
 * any constraints were added and then add the equalities back in.
 */
struct isl_tab *isl_tab_detect_equalities(struct isl_tab *tab,
  struct isl_tab *tab_cone)
{
  int j;
  struct isl_vec *sample;
  struct isl_basic_set *hull = NULL;
  struct isl_tab_undo *snap;
 
  if (!tab || !tab_cone)
    goto error;
 
  snap = isl_tab_snap(tab);
 
  isl_mat_free(tab->basis);
  tab->basis = NULL;
 
  isl_assert(tab->mat->ctx, tab->bmap, goto error);
  isl_assert(tab->mat->ctx, tab->samples, goto error);
  isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);
  isl_assert(tab->mat->ctx, tab->n_sample > tab->n_outside, goto error);
 
  if (isl_tab_set_initial_basis_with_cone(tab, tab_cone) < 0)
    goto error;
 
  sample = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
  if (!sample)
    goto error;
 
  isl_seq_cpy(sample->el, tab->samples->row[tab->n_outside], sample->size);
 
  isl_vec_free(tab->bmap->sample);
  tab->bmap->sample = isl_vec_copy(sample);
 
  if (tab->n_unbounded == 0)
    hull = isl_basic_set_from_vec(isl_vec_copy(sample));
  else
    hull = initial_hull(tab, isl_vec_copy(sample));
 
  for (j = tab->n_outside + 1; j < tab->n_sample; ++j) {
    isl_seq_cpy(sample->el, tab->samples->row[j], sample->size);
    hull = affine_hull(hull,
        isl_basic_set_from_vec(isl_vec_copy(sample)));
  }
 
  isl_vec_free(sample);
 
  hull = extend_affine_hull(tab, hull, NULL);
  if (!hull)
    goto error;
 
  if (tab->n_unbounded == 0) {
    isl_basic_set_free(hull);
    return tab;
  }
 
  if (isl_tab_rollback(tab, snap) < 0)
    goto error;
 
  if (hull->n_eq > tab->n_zero) {
    for (j = 0; j < hull->n_eq; ++j) {
      isl_seq_normalize(tab->mat->ctx, hull->eq[j], 1 + tab->n_var);
      if (isl_tab_add_eq(tab, hull->eq[j]) < 0)
        goto error;
    }
  }
 
  isl_basic_set_free(hull);
 
  return tab;
error:
  isl_basic_set_free(hull);
  isl_tab_free(tab);
  return NULL;
}
 
/* Compute the affine hull of "bset", where "cone" is the recession cone
 * of "bset".
 *
 * We first compute a unimodular transformation that puts the unbounded
 * directions in the last dimensions.  In particular, we take a transformation
 * that maps all equalities to equalities (in HNF) on the first dimensions.
 * Let x be the original dimensions and y the transformed, with y_1 bounded
 * and y_2 unbounded.
 *
 *         [ y_1 ]      [ y_1 ]   [ Q_1 ]
 *  x = U  [ y_2 ]      [ y_2 ] = [ Q_2 ] x
 *
 * Let's call the input basic set S.  We compute S' = preimage(S, U)
 * and drop the final dimensions including any constraints involving them.
 * This results in set S''.
 * Then we compute the affine hull A'' of S''.
 * Let F y_1 >= g be the constraint system of A''.  In the transformed
 * space the y_2 are unbounded, so we can add them back without any constraints,
 * resulting in
 *
 *            [ y_1 ]
 *    [ F 0 ] [ y_2 ] >= g
 * or
 *            [ Q_1 ]
 *    [ F 0 ] [ Q_2 ] x >= g
 * or
 *    F Q_1 x >= g
 *
 * The affine hull in the original space is then obtained as
 * A = preimage(A'', Q_1).
 */
static __isl_give isl_basic_set *affine_hull_with_cone(
  __isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone)
{
  isl_size total;
  unsigned cone_dim;
  struct isl_basic_set *hull;
  struct isl_mat *M, *U, *Q;
 
  total = isl_basic_set_dim(cone, isl_dim_all);
  if (!bset || total < 0)
    goto error;
 
  cone_dim = total - cone->n_eq;
 
  M = isl_mat_sub_alloc6(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
  M = isl_mat_left_hermite(M, 0, &U, &Q);
  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));
 
  bset = isl_basic_set_drop_constraints_involving(bset, total - cone_dim,
              cone_dim);
  bset = isl_basic_set_drop_dims(bset, total - cone_dim, cone_dim);
 
  Q = isl_mat_lin_to_aff(Q);
  Q = isl_mat_drop_rows(Q, 1 + total - cone_dim, cone_dim);
 
  if (bset && bset->sample && bset->sample->size == 1 + total)
    bset->sample = isl_mat_vec_product(isl_mat_copy(Q), bset->sample);
 
  hull = uset_affine_hull_bounded(bset);
 
  if (!hull) {
    isl_mat_free(Q);
    isl_mat_free(U);
  } else {
    struct isl_vec *sample = isl_vec_copy(hull->sample);
    U = isl_mat_drop_cols(U, 1 + total - cone_dim, cone_dim);
    if (sample && sample->size > 0)
      sample = isl_mat_vec_product(U, sample);
    else
      isl_mat_free(U);
    hull = isl_basic_set_preimage(hull, Q);
    if (hull) {
      isl_vec_free(hull->sample);
      hull->sample = sample;
    } else
      isl_vec_free(sample);
  }
 
  isl_basic_set_free(cone);
 
  return hull;
error:
  isl_basic_set_free(bset);
  isl_basic_set_free(cone);
  return NULL;
}
 
/* Look for all equalities satisfied by the integer points in bset,
 * which is assumed 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 compute the recession
 * cone.  The directions of this recession cone will always be part
 * of the affine hull, so there is no need for looking for any points
 * in these directions.
 * In particular, if the recession cone is full-dimensional, then
 * the affine hull is simply the whole universe.
 */
static __isl_give isl_basic_set *uset_affine_hull(
  __isl_take isl_basic_set *bset)
{
  struct isl_basic_set *cone;
  isl_size total;
 
  if (isl_basic_set_plain_is_empty(bset))
    return bset;
 
  cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
  if (!cone)
    goto error;
  if (cone->n_eq == 0) {
    isl_space *space;
    space = isl_basic_set_get_space(bset);
    isl_basic_set_free(cone);
    isl_basic_set_free(bset);
    return isl_basic_set_universe(space);
  }
 
  total = isl_basic_set_dim(cone, isl_dim_all);
  if (total < 0)
    bset = isl_basic_set_free(bset);
  if (cone->n_eq < total)
    return affine_hull_with_cone(bset, cone);
 
  isl_basic_set_free(cone);
  return uset_affine_hull_bounded(bset);
error:
  isl_basic_set_free(bset);
  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 __isl_give isl_basic_set *equalities_in_underlying_set(
  __isl_take isl_basic_map *bmap)
{
  struct isl_mat *T1 = NULL;
  struct isl_mat *T2 = NULL;
  struct isl_basic_set *bset = NULL;
  struct isl_basic_set *hull = NULL;
 
  bset = isl_basic_map_underlying_set(bmap);
  if (!bset)
    return NULL;
  if (bset->n_eq)
    bset = isl_basic_set_remove_equalities(bset, &T1, &T2);
  if (!bset)
    goto error;
 
  hull = uset_affine_hull(bset);
  if (!T2)
    return hull;
 
  if (!hull) {
    isl_mat_free(T1);
    isl_mat_free(T2);
  } else {
    struct isl_vec *sample = isl_vec_copy(hull->sample);
    if (sample && sample->size > 0)
      sample = isl_mat_vec_product(T1, sample);
    else
      isl_mat_free(T1);
    hull = isl_basic_set_preimage(hull, T2);
    if (hull) {
      isl_vec_free(hull->sample);
      hull->sample = sample;
    } else
      isl_vec_free(sample);
  }
 
  return hull;
error:
  isl_mat_free(T1);
  isl_mat_free(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.
 */
__isl_give isl_basic_map *isl_basic_map_detect_equalities(
  __isl_take isl_basic_map *bmap)
{
  int i, j;
  isl_size total;
  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(bmap, 0, hull->n_eq, 0);
  total = isl_basic_set_dim(hull, isl_dim_all);
  if (total < 0)
    goto error;
  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 + total);
  }
  isl_vec_free(bmap->sample);
  bmap->sample = isl_vec_copy(hull->sample);
  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;
}
 
__isl_give isl_basic_set *isl_basic_set_detect_equalities(
            __isl_take isl_basic_set *bset)
{
  return bset_from_bmap(
    isl_basic_map_detect_equalities(bset_to_bmap(bset)));
}
 
__isl_give isl_map *isl_map_detect_equalities(__isl_take isl_map *map)
{
  return isl_map_inline_foreach_basic_map(map,
              &isl_basic_map_detect_equalities);
}
 
__isl_give isl_set *isl_set_detect_equalities(__isl_take isl_set *set)
{
  return set_from_map(isl_map_detect_equalities(set_to_map(set)));
}
 
/* Return the superset of "bmap" described by the equalities
 * satisfied by "bmap" that are already known.
 */
__isl_give isl_basic_map *isl_basic_map_plain_affine_hull(
  __isl_take isl_basic_map *bmap)
{
  bmap = isl_basic_map_cow(bmap);
  if (bmap)
    isl_basic_map_free_inequality(bmap, bmap->n_ineq);
  bmap = isl_basic_map_finalize(bmap);
  return bmap;
}
 
/* Return the superset of "bset" described by the equalities
 * satisfied by "bset" that are already known.
 */
__isl_give isl_basic_set *isl_basic_set_plain_affine_hull(
  __isl_take isl_basic_set *bset)
{
  return isl_basic_map_plain_affine_hull(bset);
}
 
/* 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.
 */
__isl_give isl_basic_map *isl_basic_map_affine_hull(
  __isl_take isl_basic_map *bmap)
{
  bmap = isl_basic_map_detect_equalities(bmap);
  bmap = isl_basic_map_plain_affine_hull(bmap);
  return bmap;
}
 
__isl_give isl_basic_set *isl_basic_set_affine_hull(
  __isl_take isl_basic_set *bset)
{
  return bset_from_bmap(isl_basic_map_affine_hull(bset_to_bmap(bset)));
}
 
/* Given a rational affine matrix "M", add stride constraints to "bmap"
 * that ensure that
 *
 *    M(x)
 *
 * is an integer vector.  The variables x include all the variables
 * of "bmap" except the unknown divs.
 *
 * If d is the common denominator of M, then we need to impose that
 *
 *    d M(x) = 0  mod d
 *
 * or
 *
 *    exists alpha : d M(x) = d alpha
 *
 * This function is similar to add_strides in isl_morph.c
 */
static __isl_give isl_basic_map *add_strides(__isl_take isl_basic_map *bmap,
  __isl_keep isl_mat *M, int n_known)
{
  int i, div, k;
  isl_int gcd;
 
  if (isl_int_is_one(M->row[0][0]))
    return bmap;
 
  bmap = isl_basic_map_extend(bmap, M->n_row - 1, M->n_row - 1, 0);
 
  isl_int_init(gcd);
  for (i = 1; i < M->n_row; ++i) {
    isl_seq_gcd(M->row[i], M->n_col, &gcd);
    if (isl_int_is_divisible_by(gcd, M->row[0][0]))
      continue;
    div = isl_basic_map_alloc_div(bmap);
    if (div < 0)
      goto error;
    isl_int_set_si(bmap->div[div][0], 0);
    k = isl_basic_map_alloc_equality(bmap);
    if (k < 0)
      goto error;
    isl_seq_cpy(bmap->eq[k], M->row[i], M->n_col);
    isl_seq_clr(bmap->eq[k] + M->n_col, bmap->n_div - n_known);
    isl_int_set(bmap->eq[k][M->n_col - n_known + div],
          M->row[0][0]);
  }
  isl_int_clear(gcd);
 
  return bmap;
error:
  isl_int_clear(gcd);
  isl_basic_map_free(bmap);
  return NULL;
}
 
/* If there are any equalities that involve (multiple) unknown divs,
 * then extract the stride information encoded by those equalities
 * and make it explicitly available in "bmap".
 *
 * We first sort the divs so that the unknown divs appear last and
 * then we count how many equalities involve these divs.
 *
 * Let these equalities be of the form
 *
 *    A(x) + B y = 0
 *
 * where y represents the unknown divs and x the remaining variables.
 * Let [H 0] be the Hermite Normal Form of B, i.e.,
 *
 *    B = [H 0] Q
 *
 * Then x is a solution of the equalities iff
 *
 *    H^-1 A(x) (= - [I 0] Q y)
 *
 * is an integer vector.  Let d be the common denominator of H^-1.
 * We impose
 *
 *    d H^-1 A(x) = d alpha
 *
 * in add_strides, with alpha fresh existentially quantified variables.
 */
static __isl_give isl_basic_map *isl_basic_map_make_strides_explicit(
  __isl_take isl_basic_map *bmap)
{
  isl_bool known;
  int n_known;
  int n, n_col;
  isl_size v_div;
  isl_ctx *ctx;
  isl_mat *A, *B, *M;
 
  known = isl_basic_map_divs_known(bmap);
  if (known < 0)
    return isl_basic_map_free(bmap);
  if (known)
    return bmap;
  bmap = isl_basic_map_sort_divs(bmap);
  bmap = isl_basic_map_gauss(bmap, NULL);
  if (!bmap)
    return NULL;
 
  for (n_known = 0; n_known < bmap->n_div; ++n_known)
    if (isl_int_is_zero(bmap->div[n_known][0]))
      break;
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  if (v_div < 0)
    return isl_basic_map_free(bmap);
  for (n = 0; n < bmap->n_eq; ++n)
    if (isl_seq_first_non_zero(bmap->eq[n] + 1 + v_div + n_known,
              bmap->n_div - n_known) == -1)
      break;
  if (n == 0)
    return bmap;
  ctx = isl_basic_map_get_ctx(bmap);
  B = isl_mat_sub_alloc6(ctx, bmap->eq, 0, n, 0, 1 + v_div + n_known);
  n_col = bmap->n_div - n_known;
  A = isl_mat_sub_alloc6(ctx, bmap->eq, 0, n, 1 + v_div + n_known, n_col);
  A = isl_mat_left_hermite(A, 0, NULL, NULL);
  A = isl_mat_drop_cols(A, n, n_col - n);
  A = isl_mat_lin_to_aff(A);
  A = isl_mat_right_inverse(A);
  B = isl_mat_insert_zero_rows(B, 0, 1);
  B = isl_mat_set_element_si(B, 0, 0, 1);
  M = isl_mat_product(A, B);
  if (!M)
    return isl_basic_map_free(bmap);
  bmap = add_strides(bmap, M, n_known);
  bmap = isl_basic_map_gauss(bmap, NULL);
  isl_mat_free(M);
 
  return bmap;
}
 
/* Compute the affine hull of each basic map in "map" separately
 * and make all stride information explicit so that we can remove
 * all unknown divs without losing this information.
 * The result is also guaranteed to be gaussed.
 *
 * In simple cases where a div is determined by an equality,
 * calling isl_basic_map_gauss is enough to make the stride information
 * explicit, as it will derive an explicit representation for the div
 * from the equality.  If, however, the stride information
 * is encoded through multiple unknown divs then we need to make
 * some extra effort in isl_basic_map_make_strides_explicit.
 */
static __isl_give isl_map *isl_map_local_affine_hull(__isl_take isl_map *map)
{
  int i;
 
  map = isl_map_cow(map);
  if (!map)
    return NULL;
 
  for (i = 0; i < map->n; ++i) {
    map->p[i] = isl_basic_map_affine_hull(map->p[i]);
    map->p[i] = isl_basic_map_gauss(map->p[i], NULL);
    map->p[i] = isl_basic_map_make_strides_explicit(map->p[i]);
    if (!map->p[i])
      return isl_map_free(map);
  }
 
  return map;
}
 
static __isl_give isl_set *isl_set_local_affine_hull(__isl_take isl_set *set)
{
  return isl_map_local_affine_hull(set);
}
 
/* Return an empty basic map living in the same space as "map".
 */
static __isl_give isl_basic_map *replace_map_by_empty_basic_map(
  __isl_take isl_map *map)
{
  isl_space *space;
 
  space = isl_map_get_space(map);
  isl_map_free(map);
  return isl_basic_map_empty(space);
}
 
/* Compute the affine hull of "map".
 *
 * We first compute the affine hull of each basic map separately.
 * Then we align the divs and recompute the affine hulls of the basic
 * maps since some of them may now have extra divs.
 * In order to avoid performing parametric integer programming to
 * compute explicit expressions for the divs, possible leading to
 * an explosion in the number of basic maps, we first drop all unknown
 * divs before aligning the divs.  Note that isl_map_local_affine_hull tries
 * to make sure that all stride information is explicitly available
 * in terms of known divs.  This involves calling isl_basic_set_gauss,
 * which is also needed because affine_hull assumes its input has been gaussed,
 * while isl_map_affine_hull may be called on input that has not been gaussed,
 * in particular from initial_facet_constraint.
 * Similarly, align_divs may reorder some divs so that we need to
 * gauss the result again.
 * Finally, we combine the individual affine hulls into a single
 * affine hull.
 */
__isl_give isl_basic_map *isl_map_affine_hull(__isl_take isl_map *map)
{
  struct isl_basic_map *model = NULL;
  struct isl_basic_map *hull = NULL;
  struct isl_set *set;
  isl_basic_set *bset;
 
  map = isl_map_detect_equalities(map);
  map = isl_map_local_affine_hull(map);
  map = isl_map_remove_empty_parts(map);
  map = isl_map_remove_unknown_divs(map);
  map = isl_map_align_divs_internal(map);
 
  if (!map)
    return NULL;
 
  if (map->n == 0)
    return replace_map_by_empty_basic_map(map);
 
  model = isl_basic_map_copy(map->p[0]);
  set = isl_map_underlying_set(map);
  set = isl_set_cow(set);
  set = isl_set_local_affine_hull(set);
  if (!set)
    goto error;
 
  while (set->n > 1)
    set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
 
  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;
}
 
__isl_give isl_basic_set *isl_set_affine_hull(__isl_take isl_set *set)
{
  return bset_from_bmap(isl_map_affine_hull(set_to_map(set)));
}