isl_affine_hull.c

   1 #include "isl_ctx.h"
   2 #include "isl_seq.h"
   3 #include "isl_set.h"
   4 #include "isl_lp.h"
   5 #include "isl_map.h"
   6 #include "isl_map_private.h"
   7 #include "isl_equalities.h"
   8 #include "isl_sample.h"
   9 #include "isl_tab.h"
  10
  11 struct isl_basic_map *isl_basic_map_implicit_equalities(
  12                                                 struct isl_basic_map *bmap)
  13 {
  14         struct isl_tab *tab;
  15
  16         if (!bmap)
  17                 return bmap;
  18
  19         bmap = isl_basic_map_gauss(bmap, NULL);
  20         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
  21                 return bmap;
  22         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_IMPLICIT))
  23                 return bmap;
  24         if (bmap->n_ineq <= 1)
  25                 return bmap;
  26
  27         tab = isl_tab_from_basic_map(bmap);
  28         tab = isl_tab_detect_implicit_equalities(tab);
  29         bmap = isl_basic_map_update_from_tab(bmap, tab);
  30         isl_tab_free(tab);
  31         bmap = isl_basic_map_gauss(bmap, NULL);
  32         ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
  33         return bmap;
  34 }
  35
  36 struct isl_basic_set *isl_basic_set_implicit_equalities(
  37                                                 struct isl_basic_set *bset)
  38 {
  39         return (struct isl_basic_set *)
  40                 isl_basic_map_implicit_equalities((struct isl_basic_map*)bset);
  41 }
  42
  43 struct isl_map *isl_map_implicit_equalities(struct isl_map *map)
  44 {
  45         int i;
  46
  47         if (!map)
  48                 return map;
  49
  50         for (i = 0; i < map->n; ++i) {
  51                 map->p[i] = isl_basic_map_implicit_equalities(map->p[i]);
  52                 if (!map->p[i])
  53                         goto error;
  54         }
  55
  56         return map;
  57 error:
  58         isl_map_free(map);
  59         return NULL;
  60 }
  61
  62 /* Make eq[row][col] of both bmaps equal so we can add the row
  63  * add the column to the common matrix.
  64  * Note that because of the echelon form, the columns of row row
  65  * after column col are zero.
  66  */
  67 static void set_common_multiple(
  68         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
  69         unsigned row, unsigned col)
  70 {
  71         isl_int m, c;
  72
  73         if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col]))
  74                 return;
  75
  76         isl_int_init(c);
  77         isl_int_init(m);
  78         isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]);
  79         isl_int_divexact(c, m, bset1->eq[row][col]);
  80         isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1);
  81         isl_int_divexact(c, m, bset2->eq[row][col]);
  82         isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1);
  83         isl_int_clear(c);
  84         isl_int_clear(m);
  85 }
  86
  87 /* Delete a given equality, moving all the following equalities one up.
  88  */
  89 static void delete_row(struct isl_basic_set *bset, unsigned row)
  90 {
  91         isl_int *t;
  92         int r;
  93
  94         t = bset->eq[row];
  95         bset->n_eq--;
  96         for (r = row; r < bset->n_eq; ++r)
  97                 bset->eq[r] = bset->eq[r+1];
  98         bset->eq[bset->n_eq] = t;
  99 }
 100
 101 /* Make first row entries in column col of bset1 identical to
 102  * those of bset2, using the fact that entry bset1->eq[row][col]=a
 103  * is non-zero.  Initially, these elements of bset1 are all zero.
 104  * For each row i < row, we set
 105  *              A[i] = a * A[i] + B[i][col] * A[row]
 106  *              B[i] = a * B[i]
 107  * so that
 108  *              A[i][col] = B[i][col] = a * old(B[i][col])
 109  */
 110 static void construct_column(
 111         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
 112         unsigned row, unsigned col)
 113 {
 114         int r;
 115         isl_int a;
 116         isl_int b;
 117         unsigned total;
 118
 119         isl_int_init(a);
 120         isl_int_init(b);
 121         total = 1 + isl_basic_set_n_dim(bset1);
 122         for (r = 0; r < row; ++r) {
 123                 if (isl_int_is_zero(bset2->eq[r][col]))
 124                         continue;
 125                 isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]);
 126                 isl_int_divexact(a, bset1->eq[row][col], b);
 127                 isl_int_divexact(b, bset2->eq[r][col], b);
 128                 isl_seq_combine(bset1->eq[r], a, bset1->eq[r],
 129                                               b, bset1->eq[row], total);
 130                 isl_seq_scale(bset2->eq[r], bset2->eq[r], a, total);
 131         }
 132         isl_int_clear(a);
 133         isl_int_clear(b);
 134         delete_row(bset1, row);
 135 }
 136
 137 /* Make first row entries in column col of bset1 identical to
 138  * those of bset2, using only these entries of the two matrices.
 139  * Let t be the last row with different entries.
 140  * For each row i < t, we set
 141  *      A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
 142  *      B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
 143  * so that
 144  *      A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
 145  */
 146 static int transform_column(
 147         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
 148         unsigned row, unsigned col)
 149 {
 150         int i, t;
 151         isl_int a, b, g;
 152         unsigned total;
 153
 154         for (t = row-1; t >= 0; --t)
 155                 if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))
 156                         break;
 157         if (t < 0)
 158                 return 0;
 159
 160         total = 1 + isl_basic_set_n_dim(bset1);
 161         isl_int_init(a);
 162         isl_int_init(b);
 163         isl_int_init(g);
 164         isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]);
 165         for (i = 0; i < t; ++i) {
 166                 isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]);
 167                 isl_int_gcd(g, a, b);
 168                 isl_int_divexact(a, a, g);
 169                 isl_int_divexact(g, b, g);
 170                 isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t],
 171                                 total);
 172                 isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],
 173                                 total);
 174         }
 175         isl_int_clear(a);
 176         isl_int_clear(b);
 177         isl_int_clear(g);
 178         delete_row(bset1, t);
 179         delete_row(bset2, t);
 180         return 1;
 181 }
 182
 183 /* The implementation is based on Section 5.2 of Michael Karr,
 184  * "Affine Relationships Among Variables of a Program",
 185  * except that the echelon form we use starts from the last column
 186  * and that we are dealing with integer coefficients.
 187  */
 188 static struct isl_basic_set *affine_hull(
 189         struct isl_basic_set *bset1, struct isl_basic_set *bset2)
 190 {
 191         unsigned total;
 192         int col;
 193         int row;
 194
 195         total = 1 + isl_basic_set_n_dim(bset1);
 196
 197         row = 0;
 198         for (col = total-1; col >= 0; --col) {
 199                 int is_zero1 = row >= bset1->n_eq ||
 200                         isl_int_is_zero(bset1->eq[row][col]);
 201                 int is_zero2 = row >= bset2->n_eq ||
 202                         isl_int_is_zero(bset2->eq[row][col]);
 203                 if (!is_zero1 && !is_zero2) {
 204                         set_common_multiple(bset1, bset2, row, col);
 205                         ++row;
 206                 } else if (!is_zero1 && is_zero2) {
 207                         construct_column(bset1, bset2, row, col);
 208                 } else if (is_zero1 && !is_zero2) {
 209                         construct_column(bset2, bset1, row, col);
 210                 } else {
 211                         if (transform_column(bset1, bset2, row, col))
 212                                 --row;
 213                 }
 214         }
 215         isl_basic_set_free(bset2);
 216         isl_assert(bset1->ctx, row == bset1->n_eq, goto error);
 217         bset1 = isl_basic_set_normalize_constraints(bset1);
 218         return bset1;
 219 error:
 220         isl_basic_set_free(bset1);
 221         return NULL;
 222 }
 223
 224 /* Find an integer point in the set represented by "tab"
 225  * that lies outside of the equality "eq" e(x) = 0.
 226  * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
 227  * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
 228  * The point, if found, is returned.
 229  * If no point can be found, a zero-length vector is returned.
 230  *
 231  * Before solving an ILP problem, we first check if simply
 232  * adding the normal of the constraint to one of the known
 233  * integer points in the basic set represented by "tab"
 234  * yields another point inside the basic set.
 235  *
 236  * The caller of this function ensures that the tableau is bounded.
 237  */
 238 static struct isl_vec *outside_point(struct isl_tab *tab, isl_int *eq, int up)
 239 {
 240         struct isl_ctx *ctx;
 241         struct isl_vec *sample;
 242         struct isl_tab_undo *snap;
 243         unsigned dim;
 244         int k;
 245
 246         if (!tab)
 247                 return NULL;
 248         ctx = tab->mat->ctx;
 249
 250         dim = tab->n_var;
 251         sample = isl_vec_alloc(ctx, 1 + dim);
 252         if (!sample)
 253                 return NULL;
 254         isl_int_set_si(sample->el[0], 1);
 255         isl_seq_combine(sample->el + 1,
 256                 ctx->one, tab->bset->sample->el + 1,
 257                 up ? ctx->one : ctx->negone, eq + 1, dim);
 258         if (isl_basic_set_contains(tab->bset, sample))
 259                 return sample;
 260         isl_vec_free(sample);
 261         sample = NULL;
 262
 263         snap = isl_tab_snap(tab);
 264
 265         if (!up)
 266                 isl_seq_neg(eq, eq, 1 + dim);
 267         isl_int_sub_ui(eq[0], eq[0], 1);
 268
 269         if (isl_tab_extend_cons(tab, 1) < 0)
 270                 goto error;
 271         tab = isl_tab_add_ineq(tab, eq);
 272
 273         sample = isl_tab_sample(tab);
 274
 275         isl_int_add_ui(eq[0], eq[0], 1);
 276         if (!up)
 277                 isl_seq_neg(eq, eq, 1 + dim);
 278
 279         if (isl_tab_rollback(tab, snap) < 0)
 280                 goto error;
 281
 282         return sample;
 283 error:
 284         isl_vec_free(sample);
 285         return NULL;
 286 }
 287
 288 struct isl_basic_set *isl_basic_set_recession_cone(struct isl_basic_set *bset)
 289 {
 290         int i;
 291
 292         bset = isl_basic_set_cow(bset);
 293         if (!bset)
 294                 return NULL;
 295         isl_assert(bset->ctx, bset->n_div == 0, goto error);
 296
 297         for (i = 0; i < bset->n_eq; ++i)
 298                 isl_int_set_si(bset->eq[i][0], 0);
 299
 300         for (i = 0; i < bset->n_ineq; ++i)
 301                 isl_int_set_si(bset->ineq[i][0], 0);
 302
 303         ISL_F_CLR(bset, ISL_BASIC_SET_NO_IMPLICIT);
 304         return isl_basic_set_implicit_equalities(bset);
 305 error:
 306         isl_basic_set_free(bset);
 307         return NULL;
 308 }
 309
 310 /* Extend an initial (under-)approximation of the affine hull of basic
 311  * set represented by the tableau "tab"
 312  * by looking for points that do not satisfy one of the equalities
 313  * in the current approximation and adding them to that approximation
 314  * until no such points can be found any more.
 315  *
 316  * The caller of this function ensures that "tab" is bounded.
 317  */
 318 static struct isl_basic_set *extend_affine_hull(struct isl_tab *tab,
 319         struct isl_basic_set *hull)
 320 {
 321         int i, j, k;
 322         unsigned dim;
 323
 324         if (!tab || !hull)
 325                 goto error;
 326
 327         dim = tab->n_var;
 328
 329         if (isl_tab_extend_cons(tab, 2 * dim + 1) < 0)
 330                 goto error;
 331
 332         for (i = 0; i < dim; ++i) {
 333                 struct isl_vec *sample;
 334                 struct isl_basic_set *point;
 335                 for (j = 0; j < hull->n_eq; ++j) {
 336                         sample = outside_point(tab, hull->eq[j], 1);
 337                         if (!sample)
 338                                 goto error;
 339                         if (sample->size > 0)
 340                                 break;
 341                         isl_vec_free(sample);
 342                         sample = outside_point(tab, hull->eq[j], 0);
 343                         if (!sample)
 344                                 goto error;
 345                         if (sample->size > 0)
 346                                 break;
 347                         isl_vec_free(sample);
 348
 349                         tab = isl_tab_add_eq(tab, hull->eq[j]);
 350                         if (!tab)
 351                                 goto error;
 352                 }
 353                 if (j == hull->n_eq)
 354                         break;
 355                 point = isl_basic_set_from_vec(sample);
 356                 hull = affine_hull(hull, point);
 357         }
 358
 359         return hull;
 360 error:
 361         isl_basic_set_free(hull);
 362         return NULL;
 363 }
 364
 365 /* Drop all constraints in bset that involve any of the dimensions
 366  * first to first+n-1.
 367  */
 368 static struct isl_basic_set *drop_constraints_involving
 369         (struct isl_basic_set *bset, unsigned first, unsigned n)
 370 {
 371         int i;
 372
 373         if (!bset)
 374                 return NULL;
 375
 376         bset = isl_basic_set_cow(bset);
 377
 378         for (i = bset->n_eq - 1; i >= 0; --i) {
 379                 if (isl_seq_first_non_zero(bset->eq[i] + 1 + first, n) == -1)
 380                         continue;
 381                 isl_basic_set_drop_equality(bset, i);
 382         }
 383
 384         for (i = bset->n_ineq - 1; i >= 0; --i) {
 385                 if (isl_seq_first_non_zero(bset->ineq[i] + 1 + first, n) == -1)
 386                         continue;
 387                 isl_basic_set_drop_inequality(bset, i);
 388         }
 389
 390         return bset;
 391 }
 392
 393 /* Look for all equalities satisfied by the integer points in bset,
 394  * which is assumed to be bounded.
 395  *
 396  * The equalities are obtained by successively looking for
 397  * a point that is affinely independent of the points found so far.
 398  * In particular, for each equality satisfied by the points so far,
 399  * we check if there is any point on a hyperplane parallel to the
 400  * corresponding hyperplane shifted by at least one (in either direction).
 401  */
 402 static struct isl_basic_set *uset_affine_hull_bounded(struct isl_basic_set *bset)
 403 {
 404         struct isl_vec *sample = NULL;
 405         struct isl_basic_set *hull;
 406         struct isl_tab *tab = NULL;
 407         unsigned dim;
 408
 409         if (isl_basic_set_fast_is_empty(bset))
 410                 return bset;
 411
 412         dim = isl_basic_set_n_dim(bset);
 413
 414         if (bset->sample && bset->sample->size == 1 + dim) {
 415                 int contains = isl_basic_set_contains(bset, bset->sample);
 416                 if (contains < 0)
 417                         goto error;
 418                 if (contains) {
 419                         if (dim == 0)
 420                                 return bset;
 421                         sample = isl_vec_copy(bset->sample);
 422                 } else {
 423                         isl_vec_free(bset->sample);
 424                         bset->sample = NULL;
 425                 }
 426         }
 427
 428         tab = isl_tab_from_basic_set(bset);
 429         if (!tab)
 430                 goto error;
 431         tab->bset = isl_basic_set_copy(bset);
 432
 433         if (!sample) {
 434                 struct isl_tab_undo *snap;
 435                 snap = isl_tab_snap(tab);
 436                 sample = isl_tab_sample(tab);
 437                 if (isl_tab_rollback(tab, snap) < 0)
 438                         goto error;
 439                 isl_vec_free(tab->bset->sample);
 440                 tab->bset->sample = isl_vec_copy(sample);
 441         }
 442
 443         if (!sample)
 444                 goto error;
 445         if (sample->size == 0) {
 446                 isl_tab_free(tab);
 447                 isl_vec_free(sample);
 448                 return isl_basic_set_set_to_empty(bset);
 449         }
 450
 451         hull = isl_basic_set_from_vec(sample);
 452
 453         isl_basic_set_free(bset);
 454         hull = extend_affine_hull(tab, hull);
 455         isl_tab_free(tab);
 456
 457         return hull;
 458 error:
 459         isl_vec_free(sample);
 460         isl_tab_free(tab);
 461         isl_basic_set_free(bset);
 462         return NULL;
 463 }
 464
 465 /* Compute the affine hull of "bset", where "cone" is the recession cone
 466  * of "bset".
 467  *
 468  * We first compute a unimodular transformation that puts the unbounded
 469  * directions in the last dimensions.  In particular, we take a transformation
 470  * that maps all equalities to equalities (in HNF) on the first dimensions.
 471  * Let x be the original dimensions and y the transformed, with y_1 bounded
 472  * and y_2 unbounded.
 473  *
 474  *             [ y_1 ]                  [ y_1 ]   [ Q_1 ]
 475  *      x = U  [ y_2 ]                  [ y_2 ] = [ Q_2 ] x
 476  *
 477  * Let's call the input basic set S.  We compute S' = preimage(S, U)
 478  * and drop the final dimensions including any constraints involving them.
 479  * This results in set S''.
 480  * Then we compute the affine hull A'' of S''.
 481  * Let F y_1 >= g be the constraint system of A''.  In the transformed
 482  * space the y_2 are unbounded, so we can add them back without any constraints,
 483  * resulting in
 484  *
 485  *                      [ y_1 ]
 486  *              [ F 0 ] [ y_2 ] >= g
 487  * or
 488  *                      [ Q_1 ]
 489  *              [ F 0 ] [ Q_2 ] x >= g
 490  * or
 491  *              F Q_1 x >= g
 492  *
 493  * The affine hull in the original space is then obtained as
 494  * A = preimage(A'', Q_1).
 495  */
 496 static struct isl_basic_set *affine_hull_with_cone(struct isl_basic_set *bset,
 497         struct isl_basic_set *cone)
 498 {
 499         unsigned total;
 500         unsigned cone_dim;
 501         struct isl_basic_set *hull;
 502         struct isl_mat *M, *U, *Q;
 503
 504         if (!bset || !cone)
 505                 goto error;
 506
 507         total = isl_basic_set_total_dim(cone);
 508         cone_dim = total - cone->n_eq;
 509
 510         M = isl_mat_sub_alloc(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
 511         M = isl_mat_left_hermite(M, 0, &U, &Q);
 512         if (!M)
 513                 goto error;
 514         isl_mat_free(M);
 515
 516         U = isl_mat_lin_to_aff(U);
 517         bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
 518
 519         bset = drop_constraints_involving(bset, total - cone_dim, cone_dim);
 520         bset = isl_basic_set_drop_dims(bset, total - cone_dim, cone_dim);
 521
 522         Q = isl_mat_lin_to_aff(Q);
 523         Q = isl_mat_drop_rows(Q, 1 + total - cone_dim, cone_dim);
 524
 525         if (bset && bset->sample && bset->sample->size == 1 + total)
 526                 bset->sample = isl_mat_vec_product(isl_mat_copy(Q), bset->sample);
 527
 528         hull = uset_affine_hull_bounded(bset);
 529
 530         if (!hull)
 531                 isl_mat_free(U);
 532         else {
 533                 struct isl_vec *sample = isl_vec_copy(hull->sample);
 534                 U = isl_mat_drop_cols(U, 1 + total - cone_dim, cone_dim);
 535                 if (sample && sample->size > 0)
 536                         sample = isl_mat_vec_product(U, sample);
 537                 else
 538                         isl_mat_free(U);
 539                 hull = isl_basic_set_preimage(hull, Q);
 540                 isl_vec_free(hull->sample);
 541                 hull->sample = sample;
 542         }
 543
 544         isl_basic_set_free(cone);
 545
 546         return hull;
 547 error:
 548         isl_basic_set_free(bset);
 549         isl_basic_set_free(cone);
 550         return NULL;
 551 }
 552
 553 /* Look for all equalities satisfied by the integer points in bset,
 554  * which is assumed not to have any explicit equalities.
 555  *
 556  * The equalities are obtained by successively looking for
 557  * a point that is affinely independent of the points found so far.
 558  * In particular, for each equality satisfied by the points so far,
 559  * we check if there is any point on a hyperplane parallel to the
 560  * corresponding hyperplane shifted by at least one (in either direction).
 561  *
 562  * Before looking for any outside points, we first compute the recession
 563  * cone.  The directions of this recession cone will always be part
 564  * of the affine hull, so there is no need for looking for any points
 565  * in these directions.
 566  * In particular, if the recession cone is full-dimensional, then
 567  * the affine hull is simply the whole universe.
 568  */
 569 static struct isl_basic_set *uset_affine_hull(struct isl_basic_set *bset)
 570 {
 571         struct isl_basic_set *cone;
 572
 573         if (isl_basic_set_fast_is_empty(bset))
 574                 return bset;
 575
 576         cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
 577         if (!cone)
 578                 goto error;
 579         if (cone->n_eq == 0) {
 580                 struct isl_basic_set *hull;
 581                 isl_basic_set_free(cone);
 582                 hull = isl_basic_set_universe_like(bset);
 583                 isl_basic_set_free(bset);
 584                 return hull;
 585         }
 586
 587         if (cone->n_eq < isl_basic_set_total_dim(cone))
 588                 return affine_hull_with_cone(bset, cone);
 589
 590         isl_basic_set_free(cone);
 591         return uset_affine_hull_bounded(bset);
 592 error:
 593         isl_basic_set_free(bset);
 594         return NULL;
 595 }
 596
 597 /* Look for all equalities satisfied by the integer points in bmap
 598  * that are independent of the equalities already explicitly available
 599  * in bmap.
 600  *
 601  * We first remove all equalities already explicitly available,
 602  * then look for additional equalities in the reduced space
 603  * and then transform the result to the original space.
 604  * The original equalities are _not_ added to this set.  This is
 605  * the responsibility of the calling function.
 606  * The resulting basic set has all meaning about the dimensions removed.
 607  * In particular, dimensions that correspond to existential variables
 608  * in bmap and that are found to be fixed are not removed.
 609  */
 610 static struct isl_basic_set *equalities_in_underlying_set(
 611                                                 struct isl_basic_map *bmap)
 612 {
 613         struct isl_mat *T1 = NULL;
 614         struct isl_mat *T2 = NULL;
 615         struct isl_basic_set *bset = NULL;
 616         struct isl_basic_set *hull = NULL;
 617
 618         bset = isl_basic_map_underlying_set(bmap);
 619         if (!bset)
 620                 return NULL;
 621         if (bset->n_eq)
 622                 bset = isl_basic_set_remove_equalities(bset, &T1, &T2);
 623         if (!bset)
 624                 goto error;
 625
 626         hull = uset_affine_hull(bset);
 627         if (!T2)
 628                 return hull;
 629
 630         if (!hull)
 631                 isl_mat_free(T1);
 632         else {
 633                 struct isl_vec *sample = isl_vec_copy(hull->sample);
 634                 if (sample && sample->size > 0)
 635                         sample = isl_mat_vec_product(T1, sample);
 636                 else
 637                         isl_mat_free(T1);
 638                 hull = isl_basic_set_preimage(hull, T2);
 639                 isl_vec_free(hull->sample);
 640                 hull->sample = sample;
 641         }
 642
 643         return hull;
 644 error:
 645         isl_mat_free(T2);
 646         isl_basic_set_free(bset);
 647         isl_basic_set_free(hull);
 648         return NULL;
 649 }
 650
 651 /* Detect and make explicit all equalities satisfied by the (integer)
 652  * points in bmap.
 653  */
 654 struct isl_basic_map *isl_basic_map_detect_equalities(
 655                                                 struct isl_basic_map *bmap)
 656 {
 657         int i, j;
 658         struct isl_basic_set *hull = NULL;
 659
 660         if (!bmap)
 661                 return NULL;
 662         if (bmap->n_ineq == 0)
 663                 return bmap;
 664         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
 665                 return bmap;
 666         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))
 667                 return bmap;
 668         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 669                 return isl_basic_map_implicit_equalities(bmap);
 670
 671         hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
 672         if (!hull)
 673                 goto error;
 674         if (ISL_F_ISSET(hull, ISL_BASIC_SET_EMPTY)) {
 675                 isl_basic_set_free(hull);
 676                 return isl_basic_map_set_to_empty(bmap);
 677         }
 678         bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim), 0,
 679                                         hull->n_eq, 0);
 680         for (i = 0; i < hull->n_eq; ++i) {
 681                 j = isl_basic_map_alloc_equality(bmap);
 682                 if (j < 0)
 683                         goto error;
 684                 isl_seq_cpy(bmap->eq[j], hull->eq[i],
 685                                 1 + isl_basic_set_total_dim(hull));
 686         }
 687         isl_vec_free(bmap->sample);
 688         bmap->sample = isl_vec_copy(hull->sample);
 689         isl_basic_set_free(hull);
 690         ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);
 691         bmap = isl_basic_map_simplify(bmap);
 692         return isl_basic_map_finalize(bmap);
 693 error:
 694         isl_basic_set_free(hull);
 695         isl_basic_map_free(bmap);
 696         return NULL;
 697 }
 698
 699 __isl_give isl_basic_set *isl_basic_set_detect_equalities(
 700                                                 __isl_take isl_basic_set *bset)
 701 {
 702         return (isl_basic_set *)
 703                 isl_basic_map_detect_equalities((isl_basic_map *)bset);
 704 }
 705
 706 struct isl_map *isl_map_detect_equalities(struct isl_map *map)
 707 {
 708         struct isl_basic_map *bmap;
 709         int i;
 710
 711         if (!map)
 712                 return NULL;
 713
 714         for (i = 0; i < map->n; ++i) {
 715                 bmap = isl_basic_map_copy(map->p[i]);
 716                 bmap = isl_basic_map_detect_equalities(bmap);
 717                 if (!bmap)
 718                         goto error;
 719                 isl_basic_map_free(map->p[i]);
 720                 map->p[i] = bmap;
 721         }
 722
 723         return map;
 724 error:
 725         isl_map_free(map);
 726         return NULL;
 727 }
 728
 729 __isl_give isl_set *isl_set_detect_equalities(__isl_take isl_set *set)
 730 {
 731         return (isl_set *)isl_map_detect_equalities((isl_map *)set);
 732 }
 733
 734 /* After computing the rational affine hull (by detecting the implicit
 735  * equalities), we compute the additional equalities satisfied by
 736  * the integer points (if any) and add the original equalities back in.
 737  */
 738 struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
 739 {
 740         bmap = isl_basic_map_detect_equalities(bmap);
 741         bmap = isl_basic_map_cow(bmap);
 742         isl_basic_map_free_inequality(bmap, bmap->n_ineq);
 743         return bmap;
 744 }
 745
 746 struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
 747 {
 748         return (struct isl_basic_set *)
 749                 isl_basic_map_affine_hull((struct isl_basic_map *)bset);
 750 }
 751
 752 struct isl_basic_map *isl_map_affine_hull(struct isl_map *map)
 753 {
 754         int i;
 755         struct isl_basic_map *model = NULL;
 756         struct isl_basic_map *hull = NULL;
 757         struct isl_set *set;
 758
 759         if (!map)
 760                 return NULL;
 761
 762         if (map->n == 0) {
 763                 hull = isl_basic_map_empty_like_map(map);
 764                 isl_map_free(map);
 765                 return hull;
 766         }
 767
 768         map = isl_map_detect_equalities(map);
 769         map = isl_map_align_divs(map);
 770         if (!map)
 771                 return NULL;
 772         model = isl_basic_map_copy(map->p[0]);
 773         set = isl_map_underlying_set(map);
 774         set = isl_set_cow(set);
 775         if (!set)
 776                 goto error;
 777
 778         for (i = 0; i < set->n; ++i) {
 779                 set->p[i] = isl_basic_set_cow(set->p[i]);
 780                 set->p[i] = isl_basic_set_affine_hull(set->p[i]);
 781                 set->p[i] = isl_basic_set_gauss(set->p[i], NULL);
 782                 if (!set->p[i])
 783                         goto error;
 784         }
 785         set = isl_set_remove_empty_parts(set);
 786         if (set->n == 0) {
 787                 hull = isl_basic_map_empty_like(model);
 788                 isl_basic_map_free(model);
 789         } else {
 790                 struct isl_basic_set *bset;
 791                 while (set->n > 1) {
 792                         set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
 793                         if (!set->p[0])
 794                                 goto error;
 795                 }
 796                 bset = isl_basic_set_copy(set->p[0]);
 797                 hull = isl_basic_map_overlying_set(bset, model);
 798         }
 799         isl_set_free(set);
 800         hull = isl_basic_map_simplify(hull);
 801         return isl_basic_map_finalize(hull);
 802 error:
 803         isl_basic_map_free(model);
 804         isl_set_free(set);
 805         return NULL;
 806 }
 807
 808 struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
 809 {
 810         return (struct isl_basic_set *)
 811                 isl_map_affine_hull((struct isl_map *)set);
 812 }