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         ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
  32         return bmap;
  33 }
  34
  35 /* Make eq[row][col] of both bmaps equal so we can add the row
  36  * add the column to the common matrix.
  37  * Note that because of the echelon form, the columns of row row
  38  * after column col are zero.
  39  */
  40 static void set_common_multiple(
  41         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
  42         unsigned row, unsigned col)
  43 {
  44         isl_int m, c;
  45
  46         if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col]))
  47                 return;
  48
  49         isl_int_init(c);
  50         isl_int_init(m);
  51         isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]);
  52         isl_int_divexact(c, m, bset1->eq[row][col]);
  53         isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1);
  54         isl_int_divexact(c, m, bset2->eq[row][col]);
  55         isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1);
  56         isl_int_clear(c);
  57         isl_int_clear(m);
  58 }
  59
  60 /* Delete a given equality, moving all the following equalities one up.
  61  */
  62 static void delete_row(struct isl_basic_set *bset, unsigned row)
  63 {
  64         isl_int *t;
  65         int r;
  66
  67         t = bset->eq[row];
  68         bset->n_eq--;
  69         for (r = row; r < bset->n_eq; ++r)
  70                 bset->eq[r] = bset->eq[r+1];
  71         bset->eq[bset->n_eq] = t;
  72 }
  73
  74 /* Make first row entries in column col of bset1 identical to
  75  * those of bset2, using the fact that entry bset1->eq[row][col]=a
  76  * is non-zero.  Initially, these elements of bset1 are all zero.
  77  * For each row i < row, we set
  78  *              A[i] = a * A[i] + B[i][col] * A[row]
  79  *              B[i] = a * B[i]
  80  * so that
  81  *              A[i][col] = B[i][col] = a * old(B[i][col])
  82  */
  83 static void construct_column(
  84         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
  85         unsigned row, unsigned col)
  86 {
  87         int r;
  88         isl_int a;
  89         isl_int b;
  90         unsigned total;
  91
  92         isl_int_init(a);
  93         isl_int_init(b);
  94         total = 1 + isl_basic_set_n_dim(bset1);
  95         for (r = 0; r < row; ++r) {
  96                 if (isl_int_is_zero(bset2->eq[r][col]))
  97                         continue;
  98                 isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]);
  99                 isl_int_divexact(a, bset1->eq[row][col], b);
 100                 isl_int_divexact(b, bset2->eq[r][col], b);
 101                 isl_seq_combine(bset1->eq[r], a, bset1->eq[r],
 102                                               b, bset1->eq[row], total);
 103                 isl_seq_scale(bset2->eq[r], bset2->eq[r], a, total);
 104         }
 105         isl_int_clear(a);
 106         isl_int_clear(b);
 107         delete_row(bset1, row);
 108 }
 109
 110 /* Make first row entries in column col of bset1 identical to
 111  * those of bset2, using only these entries of the two matrices.
 112  * Let t be the last row with different entries.
 113  * For each row i < t, we set
 114  *      A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
 115  *      B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
 116  * so that
 117  *      A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
 118  */
 119 static int transform_column(
 120         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
 121         unsigned row, unsigned col)
 122 {
 123         int i, t;
 124         isl_int a, b, g;
 125         unsigned total;
 126
 127         for (t = row-1; t >= 0; --t)
 128                 if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))
 129                         break;
 130         if (t < 0)
 131                 return 0;
 132
 133         total = 1 + isl_basic_set_n_dim(bset1);
 134         isl_int_init(a);
 135         isl_int_init(b);
 136         isl_int_init(g);
 137         isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]);
 138         for (i = 0; i < t; ++i) {
 139                 isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]);
 140                 isl_int_gcd(g, a, b);
 141                 isl_int_divexact(a, a, g);
 142                 isl_int_divexact(g, b, g);
 143                 isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t],
 144                                 total);
 145                 isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],
 146                                 total);
 147         }
 148         isl_int_clear(a);
 149         isl_int_clear(b);
 150         isl_int_clear(g);
 151         delete_row(bset1, t);
 152         delete_row(bset2, t);
 153         return 1;
 154 }
 155
 156 /* The implementation is based on Section 5.2 of Michael Karr,
 157  * "Affine Relationships Among Variables of a Program",
 158  * except that the echelon form we use starts from the last column
 159  * and that we are dealing with integer coefficients.
 160  */
 161 static struct isl_basic_set *affine_hull(
 162         struct isl_basic_set *bset1, struct isl_basic_set *bset2)
 163 {
 164         unsigned total;
 165         int col;
 166         int row;
 167
 168         total = 1 + isl_basic_set_n_dim(bset1);
 169
 170         row = 0;
 171         for (col = total-1; col >= 0; --col) {
 172                 int is_zero1 = row >= bset1->n_eq ||
 173                         isl_int_is_zero(bset1->eq[row][col]);
 174                 int is_zero2 = row >= bset2->n_eq ||
 175                         isl_int_is_zero(bset2->eq[row][col]);
 176                 if (!is_zero1 && !is_zero2) {
 177                         set_common_multiple(bset1, bset2, row, col);
 178                         ++row;
 179                 } else if (!is_zero1 && is_zero2) {
 180                         construct_column(bset1, bset2, row, col);
 181                 } else if (is_zero1 && !is_zero2) {
 182                         construct_column(bset2, bset1, row, col);
 183                 } else {
 184                         if (transform_column(bset1, bset2, row, col))
 185                                 --row;
 186                 }
 187         }
 188         isl_basic_set_free(bset2);
 189         isl_assert(ctx, row == bset1->n_eq, goto error);
 190         return bset1;
 191 error:
 192         isl_basic_set_free(bset1);
 193         return NULL;
 194 }
 195
 196 static struct isl_basic_set *isl_basic_set_from_vec(struct isl_ctx *ctx,
 197         struct isl_vec *vec)
 198 {
 199         int i;
 200         int k;
 201         struct isl_basic_set *bset = NULL;
 202         unsigned dim;
 203
 204         if (!vec)
 205                 return NULL;
 206         isl_assert(ctx, vec->size != 0, goto error);
 207
 208         bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
 209         if (!bset)
 210                 goto error;
 211         dim = isl_basic_set_n_dim(bset);
 212         for (i = dim - 1; i >= 0; --i) {
 213                 k = isl_basic_set_alloc_equality(bset);
 214                 if (k < 0)
 215                         goto error;
 216                 isl_seq_clr(bset->eq[k], 1 + dim);
 217                 isl_int_neg(bset->eq[k][0], vec->block.data[1 + i]);
 218                 isl_int_set(bset->eq[k][1 + i], vec->block.data[0]);
 219         }
 220         isl_vec_free(ctx, vec);
 221
 222         return bset;
 223 error:
 224         isl_basic_set_free(bset);
 225         isl_vec_free(ctx, vec);
 226         return NULL;
 227 }
 228
 229 /* Find an integer point in "bset" that lies outside of the equality
 230  * "eq" e(x) = 0.
 231  * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
 232  * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
 233  * The point, if found, is returned as a singleton set.
 234  * If no point can be found, the empty set is returned.
 235  *
 236  * Before solving an ILP problem, we first check if simply
 237  * adding the normal of the constraint to one of the known
 238  * integer points in the basic set yields another point
 239  * inside the basic set.
 240  */
 241 static struct isl_basic_set *outside_point(struct isl_ctx *ctx,
 242         struct isl_basic_set *bset, isl_int *eq, int up)
 243 {
 244         struct isl_basic_set *slice = NULL;
 245         struct isl_vec *sample;
 246         struct isl_basic_set *point;
 247         unsigned dim;
 248         int k;
 249
 250         dim = isl_basic_set_n_dim(bset);
 251         sample = isl_vec_alloc(ctx, 1 + dim);
 252         if (!sample)
 253                 return NULL;
 254         isl_int_set_si(sample->block.data[0], 1);
 255         isl_seq_combine(sample->block.data + 1,
 256                 ctx->one, bset->sample->block.data + 1,
 257                 up ? ctx->one : ctx->negone, eq + 1, dim);
 258         if (isl_basic_set_contains(bset, sample))
 259                 return isl_basic_set_from_vec(ctx, sample);
 260         isl_vec_free(ctx, sample);
 261         sample = NULL;
 262
 263         slice = isl_basic_set_copy(bset);
 264         if (!slice)
 265                 goto error;
 266         slice = isl_basic_set_cow(slice);
 267         slice = isl_basic_set_extend(slice, 0, dim, 0, 0, 1);
 268         k = isl_basic_set_alloc_inequality(slice);
 269         if (k < 0)
 270                 goto error;
 271         if (up)
 272                 isl_seq_cpy(slice->ineq[k], eq, 1 + dim);
 273         else
 274                 isl_seq_neg(slice->ineq[k], eq, 1 + dim);
 275         isl_int_sub_ui(slice->ineq[k][0], slice->ineq[k][0], 1);
 276
 277         sample = isl_basic_set_sample(slice);
 278         if (!sample)
 279                 goto error;
 280         if (sample->size == 0) {
 281                 isl_vec_free(ctx, sample);
 282                 point = isl_basic_set_empty_like(bset);
 283         } else
 284                 point = isl_basic_set_from_vec(ctx, sample);
 285
 286         return point;
 287 error:
 288         isl_basic_set_free(slice);
 289         return NULL;
 290 }
 291
 292 /* Look for all equalities satisfied by the integer points in bset,
 293  * which is assume not to have any explicit equalities.
 294  *
 295  * The equalities are obtained by successively looking for
 296  * a point that is affinely independent of the points found so far.
 297  * In particular, for each equality satisfied by the points so far,
 298  * we check if there is any point on a hyperplane parallel to the
 299  * corresponding hyperplane shifted by at least one (in either direction).
 300  */
 301 static struct isl_basic_set *uset_affine_hull(struct isl_basic_set *bset)
 302 {
 303         int i, j;
 304         struct isl_basic_set *hull = NULL;
 305         struct isl_vec *sample;
 306         struct isl_ctx *ctx;
 307         unsigned dim;
 308
 309         if (isl_basic_set_is_empty(bset))
 310                 return bset;
 311
 312         ctx = bset->ctx;
 313         sample = isl_basic_set_sample(isl_basic_set_copy(bset));
 314         if (!sample)
 315                 goto error;
 316         if (sample->size == 0) {
 317                 isl_vec_free(ctx, sample);
 318                 hull = isl_basic_set_empty_like(bset);
 319                 isl_basic_set_free(bset);
 320                 return hull;
 321         } else
 322                 hull = isl_basic_set_from_vec(ctx, sample);
 323
 324         dim = isl_basic_set_n_dim(bset);
 325         for (i = 0; i < dim; ++i) {
 326                 struct isl_basic_set *point;
 327                 for (j = 0; j < hull->n_eq; ++j) {
 328                         point = outside_point(ctx, bset, hull->eq[j], 1);
 329                         if (!point)
 330                                 goto error;
 331                         if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
 332                                 break;
 333                         isl_basic_set_free(point);
 334                         point = outside_point(ctx, bset, hull->eq[j], 0);
 335                         if (!point)
 336                                 goto error;
 337                         if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
 338                                 break;
 339                         isl_basic_set_free(point);
 340                 }
 341                 if (j == hull->n_eq)
 342                         break;
 343                 hull = affine_hull(hull, point);
 344         }
 345         isl_basic_set_free(bset);
 346
 347         return hull;
 348 error:
 349         isl_basic_set_free(bset);
 350         isl_basic_set_free(hull);
 351         return NULL;
 352 }
 353
 354 /* Look for all equalities satisfied by the integer points in bmap
 355  * that are independent of the equalities already explicitly available
 356  * in bmap.
 357  *
 358  * We first remove all equalities already explicitly available,
 359  * then look for additional equalities in the reduced space
 360  * and then transform the result to the original space.
 361  * The original equalities are _not_ added to this set.  This is
 362  * the responsibility of the calling function.
 363  * The resulting basic set has all meaning about the dimensions removed.
 364  * In particular, dimensions that correspond to existential variables
 365  * in bmap and that are found to be fixed are not removed.
 366  */
 367 static struct isl_basic_set *equalities_in_underlying_set(
 368                                                 struct isl_basic_map *bmap)
 369 {
 370         struct isl_mat *T2 = NULL;
 371         struct isl_basic_set *bset = NULL;
 372         struct isl_basic_set *hull = NULL;
 373         struct isl_ctx *ctx;
 374
 375         ctx = bmap->ctx;
 376         bset = isl_basic_map_underlying_set(bmap);
 377         bset = isl_basic_set_remove_equalities(bset, NULL, &T2);
 378         if (!bset)
 379                 goto error;
 380
 381         hull = uset_affine_hull(bset);
 382         if (T2)
 383                 hull = isl_basic_set_preimage(hull, T2);
 384
 385         return hull;
 386 error:
 387         isl_mat_free(ctx, T2);
 388         isl_basic_set_free(bset);
 389         isl_basic_set_free(hull);
 390         return NULL;
 391 }
 392
 393 /* Detect and make explicit all equalities satisfied by the (integer)
 394  * points in bmap.
 395  */
 396 struct isl_basic_map *isl_basic_map_detect_equalities(
 397                                                 struct isl_basic_map *bmap)
 398 {
 399         int i, j;
 400         struct isl_basic_set *hull = NULL;
 401
 402         if (!bmap)
 403                 return NULL;
 404         if (bmap->n_ineq == 0)
 405                 return bmap;
 406         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
 407                 return bmap;
 408         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))
 409                 return bmap;
 410         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 411                 return isl_basic_map_implicit_equalities(bmap);
 412
 413         hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
 414         if (!hull)
 415                 goto error;
 416         bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim), 0,
 417                                         hull->n_eq, 0);
 418         for (i = 0; i < hull->n_eq; ++i) {
 419                 j = isl_basic_map_alloc_equality(bmap);
 420                 if (j < 0)
 421                         goto error;
 422                 isl_seq_cpy(bmap->eq[j], hull->eq[i],
 423                                 1 + isl_basic_set_total_dim(hull));
 424         }
 425         isl_basic_set_free(hull);
 426         ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);
 427         bmap = isl_basic_map_simplify(bmap);
 428         return isl_basic_map_finalize(bmap);
 429 error:
 430         isl_basic_set_free(hull);
 431         isl_basic_map_free(bmap);
 432         return NULL;
 433 }
 434
 435 struct isl_map *isl_map_detect_equalities(struct isl_map *map)
 436 {
 437         struct isl_basic_map *bmap;
 438         int i;
 439
 440         if (!map)
 441                 return NULL;
 442
 443         for (i = 0; i < map->n; ++i) {
 444                 bmap = isl_basic_map_copy(map->p[i]);
 445                 bmap = isl_basic_map_detect_equalities(bmap);
 446                 if (!bmap)
 447                         goto error;
 448                 isl_basic_map_free(map->p[i]);
 449                 map->p[i] = bmap;
 450         }
 451
 452         return map;
 453 error:
 454         isl_map_free(map);
 455         return NULL;
 456 }
 457
 458 /* After computing the rational affine hull (by detecting the implicit
 459  * equalities), we compute the additional equalities satisfied by
 460  * the integer points (if any) and add the original equalities back in.
 461  */
 462 struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
 463 {
 464         struct isl_basic_set *hull = NULL;
 465
 466         bmap = isl_basic_map_detect_equalities(bmap);
 467         bmap = isl_basic_map_cow(bmap);
 468         isl_basic_map_free_inequality(bmap, bmap->n_ineq);
 469         return bmap;
 470 }
 471
 472 struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
 473 {
 474         return (struct isl_basic_set *)
 475                 isl_basic_map_affine_hull((struct isl_basic_map *)bset);
 476 }
 477
 478 struct isl_basic_map *isl_map_affine_hull(struct isl_map *map)
 479 {
 480         int i;
 481         struct isl_basic_map *model = NULL;
 482         struct isl_basic_map *hull = NULL;
 483         struct isl_set *set;
 484
 485         if (!map)
 486                 return NULL;
 487
 488         if (map->n == 0) {
 489                 hull = isl_basic_map_empty_like_map(map);
 490                 isl_map_free(map);
 491                 return hull;
 492         }
 493
 494         map = isl_map_detect_equalities(map);
 495         map = isl_map_align_divs(map);
 496         if (!map)
 497                 return NULL;
 498         model = isl_basic_map_copy(map->p[0]);
 499         set = isl_map_underlying_set(map);
 500         set = isl_set_cow(set);
 501         if (!set)
 502                 goto error;
 503
 504         for (i = 0; i < set->n; ++i) {
 505                 set->p[i] = isl_basic_set_cow(set->p[i]);
 506                 set->p[i] = isl_basic_set_affine_hull(set->p[i]);
 507                 set->p[i] = isl_basic_set_gauss(set->p[i], NULL);
 508                 if (!set->p[i])
 509                         goto error;
 510         }
 511         set = isl_set_remove_empty_parts(set);
 512         if (set->n == 0) {
 513                 hull = isl_basic_map_empty_like(model);
 514                 isl_basic_map_free(model);
 515         } else {
 516                 struct isl_basic_set *bset;
 517                 while (set->n > 1) {
 518                         set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
 519                         if (!set->p[0])
 520                                 goto error;
 521                 }
 522                 bset = isl_basic_set_copy(set->p[0]);
 523                 hull = isl_basic_map_overlying_set(bset, model);
 524         }
 525         isl_set_free(set);
 526         hull = isl_basic_map_simplify(hull);
 527         return isl_basic_map_finalize(hull);
 528 error:
 529         isl_basic_map_free(model);
 530         isl_set_free(set);
 531         return NULL;
 532 }
 533
 534 struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
 535 {
 536         return (struct isl_basic_set *)
 537                 isl_map_affine_hull((struct isl_map *)set);
 538 }