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(bmap->ctx, tab);
  29         bmap = isl_basic_map_update_from_tab(bmap, tab);
  30         isl_tab_free(bmap->ctx, 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 /* Make eq[row][col] of both bmaps equal so we can add the row
  44  * add the column to the common matrix.
  45  * Note that because of the echelon form, the columns of row row
  46  * after column col are zero.
  47  */
  48 static void set_common_multiple(
  49         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
  50         unsigned row, unsigned col)
  51 {
  52         isl_int m, c;
  53
  54         if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col]))
  55                 return;
  56
  57         isl_int_init(c);
  58         isl_int_init(m);
  59         isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]);
  60         isl_int_divexact(c, m, bset1->eq[row][col]);
  61         isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1);
  62         isl_int_divexact(c, m, bset2->eq[row][col]);
  63         isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1);
  64         isl_int_clear(c);
  65         isl_int_clear(m);
  66 }
  67
  68 /* Delete a given equality, moving all the following equalities one up.
  69  */
  70 static void delete_row(struct isl_basic_set *bset, unsigned row)
  71 {
  72         isl_int *t;
  73         int r;
  74
  75         t = bset->eq[row];
  76         bset->n_eq--;
  77         for (r = row; r < bset->n_eq; ++r)
  78                 bset->eq[r] = bset->eq[r+1];
  79         bset->eq[bset->n_eq] = t;
  80 }
  81
  82 /* Make first row entries in column col of bset1 identical to
  83  * those of bset2, using the fact that entry bset1->eq[row][col]=a
  84  * is non-zero.  Initially, these elements of bset1 are all zero.
  85  * For each row i < row, we set
  86  *              A[i] = a * A[i] + B[i][col] * A[row]
  87  *              B[i] = a * B[i]
  88  * so that
  89  *              A[i][col] = B[i][col] = a * old(B[i][col])
  90  */
  91 static void construct_column(
  92         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
  93         unsigned row, unsigned col)
  94 {
  95         int r;
  96         isl_int a;
  97         isl_int b;
  98         unsigned total;
  99
 100         isl_int_init(a);
 101         isl_int_init(b);
 102         total = 1 + isl_basic_set_n_dim(bset1);
 103         for (r = 0; r < row; ++r) {
 104                 if (isl_int_is_zero(bset2->eq[r][col]))
 105                         continue;
 106                 isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]);
 107                 isl_int_divexact(a, bset1->eq[row][col], b);
 108                 isl_int_divexact(b, bset2->eq[r][col], b);
 109                 isl_seq_combine(bset1->eq[r], a, bset1->eq[r],
 110                                               b, bset1->eq[row], total);
 111                 isl_seq_scale(bset2->eq[r], bset2->eq[r], a, total);
 112         }
 113         isl_int_clear(a);
 114         isl_int_clear(b);
 115         delete_row(bset1, row);
 116 }
 117
 118 /* Make first row entries in column col of bset1 identical to
 119  * those of bset2, using only these entries of the two matrices.
 120  * Let t be the last row with different entries.
 121  * For each row i < t, we set
 122  *      A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
 123  *      B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
 124  * so that
 125  *      A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
 126  */
 127 static int transform_column(
 128         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
 129         unsigned row, unsigned col)
 130 {
 131         int i, t;
 132         isl_int a, b, g;
 133         unsigned total;
 134
 135         for (t = row-1; t >= 0; --t)
 136                 if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))
 137                         break;
 138         if (t < 0)
 139                 return 0;
 140
 141         total = 1 + isl_basic_set_n_dim(bset1);
 142         isl_int_init(a);
 143         isl_int_init(b);
 144         isl_int_init(g);
 145         isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]);
 146         for (i = 0; i < t; ++i) {
 147                 isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]);
 148                 isl_int_gcd(g, a, b);
 149                 isl_int_divexact(a, a, g);
 150                 isl_int_divexact(g, b, g);
 151                 isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t],
 152                                 total);
 153                 isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],
 154                                 total);
 155         }
 156         isl_int_clear(a);
 157         isl_int_clear(b);
 158         isl_int_clear(g);
 159         delete_row(bset1, t);
 160         delete_row(bset2, t);
 161         return 1;
 162 }
 163
 164 /* The implementation is based on Section 5.2 of Michael Karr,
 165  * "Affine Relationships Among Variables of a Program",
 166  * except that the echelon form we use starts from the last column
 167  * and that we are dealing with integer coefficients.
 168  */
 169 static struct isl_basic_set *affine_hull(
 170         struct isl_basic_set *bset1, struct isl_basic_set *bset2)
 171 {
 172         unsigned total;
 173         int col;
 174         int row;
 175
 176         total = 1 + isl_basic_set_n_dim(bset1);
 177
 178         row = 0;
 179         for (col = total-1; col >= 0; --col) {
 180                 int is_zero1 = row >= bset1->n_eq ||
 181                         isl_int_is_zero(bset1->eq[row][col]);
 182                 int is_zero2 = row >= bset2->n_eq ||
 183                         isl_int_is_zero(bset2->eq[row][col]);
 184                 if (!is_zero1 && !is_zero2) {
 185                         set_common_multiple(bset1, bset2, row, col);
 186                         ++row;
 187                 } else if (!is_zero1 && is_zero2) {
 188                         construct_column(bset1, bset2, row, col);
 189                 } else if (is_zero1 && !is_zero2) {
 190                         construct_column(bset2, bset1, row, col);
 191                 } else {
 192                         if (transform_column(bset1, bset2, row, col))
 193                                 --row;
 194                 }
 195         }
 196         isl_basic_set_free(bset2);
 197         isl_assert(ctx, row == bset1->n_eq, goto error);
 198         bset1 = isl_basic_set_normalize_constraints(bset1);
 199         return bset1;
 200 error:
 201         isl_basic_set_free(bset1);
 202         return NULL;
 203 }
 204
 205 static struct isl_basic_set *isl_basic_set_from_vec(struct isl_ctx *ctx,
 206         struct isl_vec *vec)
 207 {
 208         int i;
 209         int k;
 210         struct isl_basic_set *bset = NULL;
 211         unsigned dim;
 212
 213         if (!vec)
 214                 return NULL;
 215         isl_assert(ctx, vec->size != 0, goto error);
 216
 217         bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
 218         if (!bset)
 219                 goto error;
 220         dim = isl_basic_set_n_dim(bset);
 221         for (i = dim - 1; i >= 0; --i) {
 222                 k = isl_basic_set_alloc_equality(bset);
 223                 if (k < 0)
 224                         goto error;
 225                 isl_seq_clr(bset->eq[k], 1 + dim);
 226                 isl_int_neg(bset->eq[k][0], vec->block.data[1 + i]);
 227                 isl_int_set(bset->eq[k][1 + i], vec->block.data[0]);
 228         }
 229         isl_vec_free(ctx, vec);
 230
 231         return bset;
 232 error:
 233         isl_basic_set_free(bset);
 234         isl_vec_free(ctx, vec);
 235         return NULL;
 236 }
 237
 238 /* Find an integer point in "bset" that lies outside of the equality
 239  * "eq" e(x) = 0.
 240  * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
 241  * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
 242  * The point, if found, is returned as a singleton set.
 243  * If no point can be found, the empty set is returned.
 244  *
 245  * Before solving an ILP problem, we first check if simply
 246  * adding the normal of the constraint to one of the known
 247  * integer points in the basic set yields another point
 248  * inside the basic set.
 249  */
 250 static struct isl_basic_set *outside_point(struct isl_ctx *ctx,
 251         struct isl_basic_set *bset, isl_int *eq, int up)
 252 {
 253         struct isl_basic_set *slice = NULL;
 254         struct isl_vec *sample;
 255         struct isl_basic_set *point;
 256         unsigned dim;
 257         int k;
 258
 259         dim = isl_basic_set_n_dim(bset);
 260         sample = isl_vec_alloc(ctx, 1 + dim);
 261         if (!sample)
 262                 return NULL;
 263         isl_int_set_si(sample->block.data[0], 1);
 264         isl_seq_combine(sample->block.data + 1,
 265                 ctx->one, bset->sample->block.data + 1,
 266                 up ? ctx->one : ctx->negone, eq + 1, dim);
 267         if (isl_basic_set_contains(bset, sample))
 268                 return isl_basic_set_from_vec(ctx, sample);
 269         isl_vec_free(ctx, sample);
 270         sample = NULL;
 271
 272         slice = isl_basic_set_copy(bset);
 273         if (!slice)
 274                 goto error;
 275         slice = isl_basic_set_cow(slice);
 276         slice = isl_basic_set_extend(slice, 0, dim, 0, 0, 1);
 277         k = isl_basic_set_alloc_inequality(slice);
 278         if (k < 0)
 279                 goto error;
 280         if (up)
 281                 isl_seq_cpy(slice->ineq[k], eq, 1 + dim);
 282         else
 283                 isl_seq_neg(slice->ineq[k], eq, 1 + dim);
 284         isl_int_sub_ui(slice->ineq[k][0], slice->ineq[k][0], 1);
 285
 286         sample = isl_basic_set_sample(slice);
 287         if (!sample)
 288                 goto error;
 289         if (sample->size == 0) {
 290                 isl_vec_free(ctx, sample);
 291                 point = isl_basic_set_empty_like(bset);
 292         } else
 293                 point = isl_basic_set_from_vec(ctx, sample);
 294
 295         return point;
 296 error:
 297         isl_basic_set_free(slice);
 298         return NULL;
 299 }
 300
 301 static struct isl_basic_set *recession_cone(struct isl_basic_set *bset)
 302 {
 303         int i;
 304
 305         bset = isl_basic_set_cow(bset);
 306         if (!bset)
 307                 return NULL;
 308
 309         for (i = 0; i < bset->n_eq; ++i)
 310                 isl_int_set_si(bset->eq[i][0], 0);
 311
 312         for (i = 0; i < bset->n_ineq; ++i)
 313                 isl_int_set_si(bset->ineq[i][0], 0);
 314
 315         ISL_F_CLR(bset, ISL_BASIC_SET_NO_IMPLICIT);
 316         return isl_basic_set_implicit_equalities(bset);
 317 }
 318
 319 static struct isl_basic_set *shift(struct isl_basic_set *bset, isl_int *point)
 320 {
 321         int i;
 322         unsigned dim;
 323
 324         bset = isl_basic_set_cow(bset);
 325         if (!bset)
 326                 return NULL;
 327
 328         dim = isl_basic_set_n_dim(bset);
 329         for (i = 0; i < bset->n_eq; ++i) {
 330                 isl_seq_inner_product(bset->eq[i]+1, point+1, dim,
 331                                         &bset->eq[i][0]);
 332                 isl_int_neg(bset->eq[i][0], bset->eq[i][0]);
 333         }
 334
 335         for (i = 0; i < bset->n_ineq; ++i) {
 336                 isl_seq_inner_product(bset->ineq[i]+1, point+1, dim,
 337                                         &bset->ineq[i][0]);
 338                 isl_int_neg(bset->ineq[i][0], bset->ineq[i][0]);
 339         }
 340
 341         return bset;
 342 }
 343
 344 /* Look for all equalities satisfied by the integer points in bset,
 345  * which is assume not to have any explicit equalities.
 346  *
 347  * The equalities are obtained by successively looking for
 348  * a point that is affinely independent of the points found so far.
 349  * In particular, for each equality satisfied by the points so far,
 350  * we check if there is any point on a hyperplane parallel to the
 351  * corresponding hyperplane shifted by at least one (in either direction).
 352  *
 353  * Before looking for any outside points, we first remove the equalities
 354  * that correspond to the affine hull of the recession cone.
 355  * These equalities will never be equalities over the whols basic set.
 356  */
 357 static struct isl_basic_set *uset_affine_hull(struct isl_basic_set *bset)
 358 {
 359         int i, j;
 360         struct isl_basic_set *hull = NULL;
 361         struct isl_vec *sample;
 362         struct isl_ctx *ctx;
 363         unsigned dim;
 364
 365         if (isl_basic_set_is_empty(bset))
 366                 return bset;
 367
 368         ctx = bset->ctx;
 369         sample = isl_basic_set_sample(isl_basic_set_copy(bset));
 370         if (!sample)
 371                 goto error;
 372         if (sample->size == 0) {
 373                 isl_vec_free(ctx, sample);
 374                 hull = isl_basic_set_empty_like(bset);
 375                 isl_basic_set_free(bset);
 376                 return hull;
 377         } else
 378                 hull = isl_basic_set_from_vec(ctx, sample);
 379
 380         if (hull->n_eq > 0) {
 381                 struct isl_basic_set *cone;
 382                 cone = recession_cone(isl_basic_set_copy(bset));
 383                 isl_basic_set_free_inequality(cone, cone->n_ineq);
 384                 cone = isl_basic_set_normalize_constraints(cone);
 385                 cone = shift(cone, bset->sample->block.data);
 386                 hull = affine_hull(hull, cone);
 387         }
 388
 389         dim = isl_basic_set_n_dim(bset);
 390         for (i = 0; i < dim; ++i) {
 391                 struct isl_basic_set *point;
 392                 for (j = 0; j < hull->n_eq; ++j) {
 393                         point = outside_point(ctx, bset, hull->eq[j], 1);
 394                         if (!point)
 395                                 goto error;
 396                         if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
 397                                 break;
 398                         isl_basic_set_free(point);
 399                         point = outside_point(ctx, bset, hull->eq[j], 0);
 400                         if (!point)
 401                                 goto error;
 402                         if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
 403                                 break;
 404                         isl_basic_set_free(point);
 405                 }
 406                 if (j == hull->n_eq)
 407                         break;
 408                 hull = affine_hull(hull, point);
 409         }
 410         isl_basic_set_free(bset);
 411
 412         return hull;
 413 error:
 414         isl_basic_set_free(bset);
 415         isl_basic_set_free(hull);
 416         return NULL;
 417 }
 418
 419 /* Look for all equalities satisfied by the integer points in bmap
 420  * that are independent of the equalities already explicitly available
 421  * in bmap.
 422  *
 423  * We first remove all equalities already explicitly available,
 424  * then look for additional equalities in the reduced space
 425  * and then transform the result to the original space.
 426  * The original equalities are _not_ added to this set.  This is
 427  * the responsibility of the calling function.
 428  * The resulting basic set has all meaning about the dimensions removed.
 429  * In particular, dimensions that correspond to existential variables
 430  * in bmap and that are found to be fixed are not removed.
 431  */
 432 static struct isl_basic_set *equalities_in_underlying_set(
 433                                                 struct isl_basic_map *bmap)
 434 {
 435         struct isl_mat *T2 = NULL;
 436         struct isl_basic_set *bset = NULL;
 437         struct isl_basic_set *hull = NULL;
 438         struct isl_ctx *ctx;
 439
 440         ctx = bmap->ctx;
 441         bset = isl_basic_map_underlying_set(bmap);
 442         bset = isl_basic_set_remove_equalities(bset, NULL, &T2);
 443         if (!bset)
 444                 goto error;
 445
 446         hull = uset_affine_hull(bset);
 447         if (T2)
 448                 hull = isl_basic_set_preimage(hull, T2);
 449
 450         return hull;
 451 error:
 452         isl_mat_free(ctx, T2);
 453         isl_basic_set_free(bset);
 454         isl_basic_set_free(hull);
 455         return NULL;
 456 }
 457
 458 /* Detect and make explicit all equalities satisfied by the (integer)
 459  * points in bmap.
 460  */
 461 struct isl_basic_map *isl_basic_map_detect_equalities(
 462                                                 struct isl_basic_map *bmap)
 463 {
 464         int i, j;
 465         struct isl_basic_set *hull = NULL;
 466
 467         if (!bmap)
 468                 return NULL;
 469         if (bmap->n_ineq == 0)
 470                 return bmap;
 471         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
 472                 return bmap;
 473         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))
 474                 return bmap;
 475         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 476                 return isl_basic_map_implicit_equalities(bmap);
 477
 478         hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
 479         if (!hull)
 480                 goto error;
 481         bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim), 0,
 482                                         hull->n_eq, 0);
 483         for (i = 0; i < hull->n_eq; ++i) {
 484                 j = isl_basic_map_alloc_equality(bmap);
 485                 if (j < 0)
 486                         goto error;
 487                 isl_seq_cpy(bmap->eq[j], hull->eq[i],
 488                                 1 + isl_basic_set_total_dim(hull));
 489         }
 490         isl_basic_set_free(hull);
 491         ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);
 492         bmap = isl_basic_map_simplify(bmap);
 493         return isl_basic_map_finalize(bmap);
 494 error:
 495         isl_basic_set_free(hull);
 496         isl_basic_map_free(bmap);
 497         return NULL;
 498 }
 499
 500 struct isl_map *isl_map_detect_equalities(struct isl_map *map)
 501 {
 502         struct isl_basic_map *bmap;
 503         int i;
 504
 505         if (!map)
 506                 return NULL;
 507
 508         for (i = 0; i < map->n; ++i) {
 509                 bmap = isl_basic_map_copy(map->p[i]);
 510                 bmap = isl_basic_map_detect_equalities(bmap);
 511                 if (!bmap)
 512                         goto error;
 513                 isl_basic_map_free(map->p[i]);
 514                 map->p[i] = bmap;
 515         }
 516
 517         return map;
 518 error:
 519         isl_map_free(map);
 520         return NULL;
 521 }
 522
 523 /* After computing the rational affine hull (by detecting the implicit
 524  * equalities), we compute the additional equalities satisfied by
 525  * the integer points (if any) and add the original equalities back in.
 526  */
 527 struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
 528 {
 529         struct isl_basic_set *hull = NULL;
 530
 531         bmap = isl_basic_map_detect_equalities(bmap);
 532         bmap = isl_basic_map_cow(bmap);
 533         isl_basic_map_free_inequality(bmap, bmap->n_ineq);
 534         return bmap;
 535 }
 536
 537 struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
 538 {
 539         return (struct isl_basic_set *)
 540                 isl_basic_map_affine_hull((struct isl_basic_map *)bset);
 541 }
 542
 543 struct isl_basic_map *isl_map_affine_hull(struct isl_map *map)
 544 {
 545         int i;
 546         struct isl_basic_map *model = NULL;
 547         struct isl_basic_map *hull = NULL;
 548         struct isl_set *set;
 549
 550         if (!map)
 551                 return NULL;
 552
 553         if (map->n == 0) {
 554                 hull = isl_basic_map_empty_like_map(map);
 555                 isl_map_free(map);
 556                 return hull;
 557         }
 558
 559         map = isl_map_detect_equalities(map);
 560         map = isl_map_align_divs(map);
 561         if (!map)
 562                 return NULL;
 563         model = isl_basic_map_copy(map->p[0]);
 564         set = isl_map_underlying_set(map);
 565         set = isl_set_cow(set);
 566         if (!set)
 567                 goto error;
 568
 569         for (i = 0; i < set->n; ++i) {
 570                 set->p[i] = isl_basic_set_cow(set->p[i]);
 571                 set->p[i] = isl_basic_set_affine_hull(set->p[i]);
 572                 set->p[i] = isl_basic_set_gauss(set->p[i], NULL);
 573                 if (!set->p[i])
 574                         goto error;
 575         }
 576         set = isl_set_remove_empty_parts(set);
 577         if (set->n == 0) {
 578                 hull = isl_basic_map_empty_like(model);
 579                 isl_basic_map_free(model);
 580         } else {
 581                 struct isl_basic_set *bset;
 582                 while (set->n > 1) {
 583                         set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
 584                         if (!set->p[0])
 585                                 goto error;
 586                 }
 587                 bset = isl_basic_set_copy(set->p[0]);
 588                 hull = isl_basic_map_overlying_set(bset, model);
 589         }
 590         isl_set_free(set);
 591         hull = isl_basic_map_simplify(hull);
 592         return isl_basic_map_finalize(hull);
 593 error:
 594         isl_basic_map_free(model);
 595         isl_set_free(set);
 596         return NULL;
 597 }
 598
 599 struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
 600 {
 601         return (struct isl_basic_set *)
 602                 isl_map_affine_hull((struct isl_map *)set);
 603 }