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_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(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 static struct isl_basic_set *isl_basic_set_from_vec(struct isl_vec *vec)
 225 {
 226         int i;
 227         int k;
 228         struct isl_basic_set *bset = NULL;
 229         struct isl_ctx *ctx;
 230         unsigned dim;
 231
 232         if (!vec)
 233                 return NULL;
 234         ctx = vec->ctx;
 235         isl_assert(ctx, vec->size != 0, goto error);
 236
 237         bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
 238         if (!bset)
 239                 goto error;
 240         dim = isl_basic_set_n_dim(bset);
 241         for (i = dim - 1; i >= 0; --i) {
 242                 k = isl_basic_set_alloc_equality(bset);
 243                 if (k < 0)
 244                         goto error;
 245                 isl_seq_clr(bset->eq[k], 1 + dim);
 246                 isl_int_neg(bset->eq[k][0], vec->el[1 + i]);
 247                 isl_int_set(bset->eq[k][1 + i], vec->el[0]);
 248         }
 249         isl_vec_free(vec);
 250
 251         return bset;
 252 error:
 253         isl_basic_set_free(bset);
 254         isl_vec_free(vec);
 255         return NULL;
 256 }
 257
 258 /* Find an integer point in "bset" that lies outside of the equality
 259  * "eq" e(x) = 0.
 260  * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
 261  * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
 262  * The point, if found, is returned as a singleton set.
 263  * If no point can be found, the empty set is returned.
 264  *
 265  * Before solving an ILP problem, we first check if simply
 266  * adding the normal of the constraint to one of the known
 267  * integer points in the basic set yields another point
 268  * inside the basic set.
 269  */
 270 static struct isl_basic_set *outside_point(struct isl_ctx *ctx,
 271         struct isl_basic_set *bset, isl_int *eq, int up)
 272 {
 273         struct isl_basic_set *slice = NULL;
 274         struct isl_vec *sample;
 275         struct isl_basic_set *point;
 276         unsigned dim;
 277         int k;
 278
 279         dim = isl_basic_set_n_dim(bset);
 280         sample = isl_vec_alloc(ctx, 1 + dim);
 281         if (!sample)
 282                 return NULL;
 283         isl_int_set_si(sample->block.data[0], 1);
 284         isl_seq_combine(sample->block.data + 1,
 285                 ctx->one, bset->sample->block.data + 1,
 286                 up ? ctx->one : ctx->negone, eq + 1, dim);
 287         if (isl_basic_set_contains(bset, sample))
 288                 return isl_basic_set_from_vec(sample);
 289         isl_vec_free(sample);
 290         sample = NULL;
 291
 292         slice = isl_basic_set_copy(bset);
 293         if (!slice)
 294                 goto error;
 295         slice = isl_basic_set_cow(slice);
 296         slice = isl_basic_set_extend(slice, 0, dim, 0, 0, 1);
 297         k = isl_basic_set_alloc_inequality(slice);
 298         if (k < 0)
 299                 goto error;
 300         if (up)
 301                 isl_seq_cpy(slice->ineq[k], eq, 1 + dim);
 302         else
 303                 isl_seq_neg(slice->ineq[k], eq, 1 + dim);
 304         isl_int_sub_ui(slice->ineq[k][0], slice->ineq[k][0], 1);
 305
 306         sample = isl_basic_set_sample(slice);
 307         if (!sample)
 308                 goto error;
 309         if (sample->size == 0) {
 310                 isl_vec_free(sample);
 311                 point = isl_basic_set_empty_like(bset);
 312         } else
 313                 point = isl_basic_set_from_vec(sample);
 314
 315         return point;
 316 error:
 317         isl_basic_set_free(slice);
 318         return NULL;
 319 }
 320
 321 struct isl_basic_set *isl_basic_set_recession_cone(struct isl_basic_set *bset)
 322 {
 323         int i;
 324
 325         bset = isl_basic_set_cow(bset);
 326         if (!bset)
 327                 return NULL;
 328         isl_assert(bset->ctx, bset->n_div == 0, goto error);
 329
 330         for (i = 0; i < bset->n_eq; ++i)
 331                 isl_int_set_si(bset->eq[i][0], 0);
 332
 333         for (i = 0; i < bset->n_ineq; ++i)
 334                 isl_int_set_si(bset->ineq[i][0], 0);
 335
 336         ISL_F_CLR(bset, ISL_BASIC_SET_NO_IMPLICIT);
 337         return isl_basic_set_implicit_equalities(bset);
 338 error:
 339         isl_basic_set_free(bset);
 340         return NULL;
 341 }
 342
 343 static struct isl_basic_set *shift(struct isl_basic_set *bset, isl_int *point)
 344 {
 345         int i;
 346         unsigned dim;
 347
 348         bset = isl_basic_set_cow(bset);
 349         if (!bset)
 350                 return NULL;
 351
 352         dim = isl_basic_set_n_dim(bset);
 353         for (i = 0; i < bset->n_eq; ++i) {
 354                 isl_seq_inner_product(bset->eq[i]+1, point+1, dim,
 355                                         &bset->eq[i][0]);
 356                 isl_int_neg(bset->eq[i][0], bset->eq[i][0]);
 357         }
 358
 359         for (i = 0; i < bset->n_ineq; ++i) {
 360                 isl_seq_inner_product(bset->ineq[i]+1, point+1, dim,
 361                                         &bset->ineq[i][0]);
 362                 isl_int_neg(bset->ineq[i][0], bset->ineq[i][0]);
 363         }
 364
 365         return bset;
 366 }
 367
 368 /* Look for all equalities satisfied by the integer points in bset,
 369  * which is assume not to have any explicit equalities.
 370  *
 371  * The equalities are obtained by successively looking for
 372  * a point that is affinely independent of the points found so far.
 373  * In particular, for each equality satisfied by the points so far,
 374  * we check if there is any point on a hyperplane parallel to the
 375  * corresponding hyperplane shifted by at least one (in either direction).
 376  *
 377  * Before looking for any outside points, we first remove the equalities
 378  * that correspond to the affine hull of the recession cone.
 379  * These equalities will never be equalities over the whols basic set.
 380  */
 381 static struct isl_basic_set *uset_affine_hull(struct isl_basic_set *bset)
 382 {
 383         int i, j;
 384         struct isl_basic_set *hull = NULL;
 385         struct isl_vec *sample;
 386         struct isl_ctx *ctx;
 387         unsigned dim;
 388
 389         if (isl_basic_set_is_empty(bset))
 390                 return bset;
 391
 392         ctx = bset->ctx;
 393         sample = isl_basic_set_sample(isl_basic_set_copy(bset));
 394         if (!sample)
 395                 goto error;
 396         if (sample->size == 0) {
 397                 isl_vec_free(sample);
 398                 hull = isl_basic_set_empty_like(bset);
 399                 isl_basic_set_free(bset);
 400                 return hull;
 401         } else
 402                 hull = isl_basic_set_from_vec(sample);
 403
 404         if (hull->n_eq > 0) {
 405                 struct isl_basic_set *cone;
 406                 cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
 407                 isl_basic_set_free_inequality(cone, cone->n_ineq);
 408                 cone = isl_basic_set_normalize_constraints(cone);
 409                 cone = shift(cone, bset->sample->block.data);
 410                 hull = affine_hull(hull, cone);
 411         }
 412
 413         dim = isl_basic_set_n_dim(bset);
 414         for (i = 0; i < dim; ++i) {
 415                 struct isl_basic_set *point;
 416                 for (j = 0; j < hull->n_eq; ++j) {
 417                         point = outside_point(ctx, bset, hull->eq[j], 1);
 418                         if (!point)
 419                                 goto error;
 420                         if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
 421                                 break;
 422                         isl_basic_set_free(point);
 423                         point = outside_point(ctx, bset, hull->eq[j], 0);
 424                         if (!point)
 425                                 goto error;
 426                         if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
 427                                 break;
 428                         isl_basic_set_free(point);
 429                 }
 430                 if (j == hull->n_eq)
 431                         break;
 432                 hull = affine_hull(hull, point);
 433         }
 434         isl_basic_set_free(bset);
 435
 436         return hull;
 437 error:
 438         isl_basic_set_free(bset);
 439         isl_basic_set_free(hull);
 440         return NULL;
 441 }
 442
 443 /* Look for all equalities satisfied by the integer points in bmap
 444  * that are independent of the equalities already explicitly available
 445  * in bmap.
 446  *
 447  * We first remove all equalities already explicitly available,
 448  * then look for additional equalities in the reduced space
 449  * and then transform the result to the original space.
 450  * The original equalities are _not_ added to this set.  This is
 451  * the responsibility of the calling function.
 452  * The resulting basic set has all meaning about the dimensions removed.
 453  * In particular, dimensions that correspond to existential variables
 454  * in bmap and that are found to be fixed are not removed.
 455  */
 456 static struct isl_basic_set *equalities_in_underlying_set(
 457                                                 struct isl_basic_map *bmap)
 458 {
 459         struct isl_mat *T2 = NULL;
 460         struct isl_basic_set *bset = NULL;
 461         struct isl_basic_set *hull = NULL;
 462
 463         bset = isl_basic_map_underlying_set(bmap);
 464         bset = isl_basic_set_remove_equalities(bset, NULL, &T2);
 465         if (!bset)
 466                 goto error;
 467
 468         hull = uset_affine_hull(bset);
 469         if (T2)
 470                 hull = isl_basic_set_preimage(hull, T2);
 471
 472         return hull;
 473 error:
 474         isl_mat_free(T2);
 475         isl_basic_set_free(bset);
 476         isl_basic_set_free(hull);
 477         return NULL;
 478 }
 479
 480 /* Detect and make explicit all equalities satisfied by the (integer)
 481  * points in bmap.
 482  */
 483 struct isl_basic_map *isl_basic_map_detect_equalities(
 484                                                 struct isl_basic_map *bmap)
 485 {
 486         int i, j;
 487         struct isl_basic_set *hull = NULL;
 488
 489         if (!bmap)
 490                 return NULL;
 491         if (bmap->n_ineq == 0)
 492                 return bmap;
 493         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
 494                 return bmap;
 495         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))
 496                 return bmap;
 497         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 498                 return isl_basic_map_implicit_equalities(bmap);
 499
 500         hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
 501         if (!hull)
 502                 goto error;
 503         if (ISL_F_ISSET(hull, ISL_BASIC_SET_EMPTY)) {
 504                 isl_basic_set_free(hull);
 505                 return isl_basic_map_set_to_empty(bmap);
 506         }
 507         bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim), 0,
 508                                         hull->n_eq, 0);
 509         for (i = 0; i < hull->n_eq; ++i) {
 510                 j = isl_basic_map_alloc_equality(bmap);
 511                 if (j < 0)
 512                         goto error;
 513                 isl_seq_cpy(bmap->eq[j], hull->eq[i],
 514                                 1 + isl_basic_set_total_dim(hull));
 515         }
 516         isl_basic_set_free(hull);
 517         ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);
 518         bmap = isl_basic_map_simplify(bmap);
 519         return isl_basic_map_finalize(bmap);
 520 error:
 521         isl_basic_set_free(hull);
 522         isl_basic_map_free(bmap);
 523         return NULL;
 524 }
 525
 526 struct isl_map *isl_map_detect_equalities(struct isl_map *map)
 527 {
 528         struct isl_basic_map *bmap;
 529         int i;
 530
 531         if (!map)
 532                 return NULL;
 533
 534         for (i = 0; i < map->n; ++i) {
 535                 bmap = isl_basic_map_copy(map->p[i]);
 536                 bmap = isl_basic_map_detect_equalities(bmap);
 537                 if (!bmap)
 538                         goto error;
 539                 isl_basic_map_free(map->p[i]);
 540                 map->p[i] = bmap;
 541         }
 542
 543         return map;
 544 error:
 545         isl_map_free(map);
 546         return NULL;
 547 }
 548
 549 /* After computing the rational affine hull (by detecting the implicit
 550  * equalities), we compute the additional equalities satisfied by
 551  * the integer points (if any) and add the original equalities back in.
 552  */
 553 struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
 554 {
 555         struct isl_basic_set *hull = NULL;
 556
 557         bmap = isl_basic_map_detect_equalities(bmap);
 558         bmap = isl_basic_map_cow(bmap);
 559         isl_basic_map_free_inequality(bmap, bmap->n_ineq);
 560         return bmap;
 561 }
 562
 563 struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
 564 {
 565         return (struct isl_basic_set *)
 566                 isl_basic_map_affine_hull((struct isl_basic_map *)bset);
 567 }
 568
 569 struct isl_basic_map *isl_map_affine_hull(struct isl_map *map)
 570 {
 571         int i;
 572         struct isl_basic_map *model = NULL;
 573         struct isl_basic_map *hull = NULL;
 574         struct isl_set *set;
 575
 576         if (!map)
 577                 return NULL;
 578
 579         if (map->n == 0) {
 580                 hull = isl_basic_map_empty_like_map(map);
 581                 isl_map_free(map);
 582                 return hull;
 583         }
 584
 585         map = isl_map_detect_equalities(map);
 586         map = isl_map_align_divs(map);
 587         if (!map)
 588                 return NULL;
 589         model = isl_basic_map_copy(map->p[0]);
 590         set = isl_map_underlying_set(map);
 591         set = isl_set_cow(set);
 592         if (!set)
 593                 goto error;
 594
 595         for (i = 0; i < set->n; ++i) {
 596                 set->p[i] = isl_basic_set_cow(set->p[i]);
 597                 set->p[i] = isl_basic_set_affine_hull(set->p[i]);
 598                 set->p[i] = isl_basic_set_gauss(set->p[i], NULL);
 599                 if (!set->p[i])
 600                         goto error;
 601         }
 602         set = isl_set_remove_empty_parts(set);
 603         if (set->n == 0) {
 604                 hull = isl_basic_map_empty_like(model);
 605                 isl_basic_map_free(model);
 606         } else {
 607                 struct isl_basic_set *bset;
 608                 while (set->n > 1) {
 609                         set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
 610                         if (!set->p[0])
 611                                 goto error;
 612                 }
 613                 bset = isl_basic_set_copy(set->p[0]);
 614                 hull = isl_basic_map_overlying_set(bset, model);
 615         }
 616         isl_set_free(set);
 617         hull = isl_basic_map_simplify(hull);
 618         return isl_basic_map_finalize(hull);
 619 error:
 620         isl_basic_map_free(model);
 621         isl_set_free(set);
 622         return NULL;
 623 }
 624
 625 struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
 626 {
 627         return (struct isl_basic_set *)
 628                 isl_map_affine_hull((struct isl_map *)set);
 629 }