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 /* Make eq[row][col] of both bmaps equal so we can add the row
  37  * add the column to the common matrix.
  38  * Note that because of the echelon form, the columns of row row
  39  * after column col are zero.
  40  */
  41 static void set_common_multiple(
  42         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
  43         unsigned row, unsigned col)
  44 {
  45         isl_int m, c;
  46
  47         if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col]))
  48                 return;
  49
  50         isl_int_init(c);
  51         isl_int_init(m);
  52         isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]);
  53         isl_int_divexact(c, m, bset1->eq[row][col]);
  54         isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1);
  55         isl_int_divexact(c, m, bset2->eq[row][col]);
  56         isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1);
  57         isl_int_clear(c);
  58         isl_int_clear(m);
  59 }
  60
  61 /* Delete a given equality, moving all the following equalities one up.
  62  */
  63 static void delete_row(struct isl_basic_set *bset, unsigned row)
  64 {
  65         isl_int *t;
  66         int r;
  67
  68         t = bset->eq[row];
  69         bset->n_eq--;
  70         for (r = row; r < bset->n_eq; ++r)
  71                 bset->eq[r] = bset->eq[r+1];
  72         bset->eq[bset->n_eq] = t;
  73 }
  74
  75 /* Make first row entries in column col of bset1 identical to
  76  * those of bset2, using the fact that entry bset1->eq[row][col]=a
  77  * is non-zero.  Initially, these elements of bset1 are all zero.
  78  * For each row i < row, we set
  79  *              A[i] = a * A[i] + B[i][col] * A[row]
  80  *              B[i] = a * B[i]
  81  * so that
  82  *              A[i][col] = B[i][col] = a * old(B[i][col])
  83  */
  84 static void construct_column(
  85         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
  86         unsigned row, unsigned col)
  87 {
  88         int r;
  89         isl_int a;
  90         isl_int b;
  91         unsigned total;
  92
  93         isl_int_init(a);
  94         isl_int_init(b);
  95         total = 1 + isl_basic_set_n_dim(bset1);
  96         for (r = 0; r < row; ++r) {
  97                 if (isl_int_is_zero(bset2->eq[r][col]))
  98                         continue;
  99                 isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]);
 100                 isl_int_divexact(a, bset1->eq[row][col], b);
 101                 isl_int_divexact(b, bset2->eq[r][col], b);
 102                 isl_seq_combine(bset1->eq[r], a, bset1->eq[r],
 103                                               b, bset1->eq[row], total);
 104                 isl_seq_scale(bset2->eq[r], bset2->eq[r], a, total);
 105         }
 106         isl_int_clear(a);
 107         isl_int_clear(b);
 108         delete_row(bset1, row);
 109 }
 110
 111 /* Make first row entries in column col of bset1 identical to
 112  * those of bset2, using only these entries of the two matrices.
 113  * Let t be the last row with different entries.
 114  * For each row i < t, we set
 115  *      A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
 116  *      B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
 117  * so that
 118  *      A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
 119  */
 120 static int transform_column(
 121         struct isl_basic_set *bset1, struct isl_basic_set *bset2,
 122         unsigned row, unsigned col)
 123 {
 124         int i, t;
 125         isl_int a, b, g;
 126         unsigned total;
 127
 128         for (t = row-1; t >= 0; --t)
 129                 if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))
 130                         break;
 131         if (t < 0)
 132                 return 0;
 133
 134         total = 1 + isl_basic_set_n_dim(bset1);
 135         isl_int_init(a);
 136         isl_int_init(b);
 137         isl_int_init(g);
 138         isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]);
 139         for (i = 0; i < t; ++i) {
 140                 isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]);
 141                 isl_int_gcd(g, a, b);
 142                 isl_int_divexact(a, a, g);
 143                 isl_int_divexact(g, b, g);
 144                 isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t],
 145                                 total);
 146                 isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],
 147                                 total);
 148         }
 149         isl_int_clear(a);
 150         isl_int_clear(b);
 151         isl_int_clear(g);
 152         delete_row(bset1, t);
 153         delete_row(bset2, t);
 154         return 1;
 155 }
 156
 157 /* The implementation is based on Section 5.2 of Michael Karr,
 158  * "Affine Relationships Among Variables of a Program",
 159  * except that the echelon form we use starts from the last column
 160  * and that we are dealing with integer coefficients.
 161  */
 162 static struct isl_basic_set *affine_hull(
 163         struct isl_basic_set *bset1, struct isl_basic_set *bset2)
 164 {
 165         unsigned total;
 166         int col;
 167         int row;
 168
 169         total = 1 + isl_basic_set_n_dim(bset1);
 170
 171         row = 0;
 172         for (col = total-1; col >= 0; --col) {
 173                 int is_zero1 = row >= bset1->n_eq ||
 174                         isl_int_is_zero(bset1->eq[row][col]);
 175                 int is_zero2 = row >= bset2->n_eq ||
 176                         isl_int_is_zero(bset2->eq[row][col]);
 177                 if (!is_zero1 && !is_zero2) {
 178                         set_common_multiple(bset1, bset2, row, col);
 179                         ++row;
 180                 } else if (!is_zero1 && is_zero2) {
 181                         construct_column(bset1, bset2, row, col);
 182                 } else if (is_zero1 && !is_zero2) {
 183                         construct_column(bset2, bset1, row, col);
 184                 } else {
 185                         if (transform_column(bset1, bset2, row, col))
 186                                 --row;
 187                 }
 188         }
 189         isl_basic_set_free(bset2);
 190         isl_assert(ctx, row == bset1->n_eq, goto error);
 191         return bset1;
 192 error:
 193         isl_basic_set_free(bset1);
 194         return NULL;
 195 }
 196
 197 static struct isl_basic_set *isl_basic_set_from_vec(struct isl_ctx *ctx,
 198         struct isl_vec *vec)
 199 {
 200         int i;
 201         int k;
 202         struct isl_basic_set *bset = NULL;
 203         unsigned dim;
 204
 205         if (!vec)
 206                 return NULL;
 207         isl_assert(ctx, vec->size != 0, goto error);
 208
 209         bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
 210         if (!bset)
 211                 goto error;
 212         dim = isl_basic_set_n_dim(bset);
 213         for (i = dim - 1; i >= 0; --i) {
 214                 k = isl_basic_set_alloc_equality(bset);
 215                 if (k < 0)
 216                         goto error;
 217                 isl_seq_clr(bset->eq[k], 1 + dim);
 218                 isl_int_neg(bset->eq[k][0], vec->block.data[1 + i]);
 219                 isl_int_set(bset->eq[k][1 + i], vec->block.data[0]);
 220         }
 221         isl_vec_free(ctx, vec);
 222
 223         return bset;
 224 error:
 225         isl_basic_set_free(bset);
 226         isl_vec_free(ctx, vec);
 227         return NULL;
 228 }
 229
 230 /* Find an integer point in "bset" that lies outside of the equality
 231  * "eq" e(x) = 0.
 232  * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
 233  * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
 234  * The point, if found, is returned as a singleton set.
 235  * If no point can be found, the empty set is returned.
 236  *
 237  * Before solving an ILP problem, we first check if simply
 238  * adding the normal of the constraint to one of the known
 239  * integer points in the basic set yields another point
 240  * inside the basic set.
 241  */
 242 static struct isl_basic_set *outside_point(struct isl_ctx *ctx,
 243         struct isl_basic_set *bset, isl_int *eq, int up)
 244 {
 245         struct isl_basic_set *slice = NULL;
 246         struct isl_vec *sample;
 247         struct isl_basic_set *point;
 248         unsigned dim;
 249         int k;
 250
 251         dim = isl_basic_set_n_dim(bset);
 252         sample = isl_vec_alloc(ctx, 1 + dim);
 253         if (!sample)
 254                 return NULL;
 255         isl_int_set_si(sample->block.data[0], 1);
 256         isl_seq_combine(sample->block.data + 1,
 257                 ctx->one, bset->sample->block.data + 1,
 258                 up ? ctx->one : ctx->negone, eq + 1, dim);
 259         if (isl_basic_set_contains(bset, sample))
 260                 return isl_basic_set_from_vec(ctx, sample);
 261         isl_vec_free(ctx, sample);
 262         sample = NULL;
 263
 264         slice = isl_basic_set_copy(bset);
 265         if (!slice)
 266                 goto error;
 267         slice = isl_basic_set_cow(slice);
 268         slice = isl_basic_set_extend(slice, 0, dim, 0, 0, 1);
 269         k = isl_basic_set_alloc_inequality(slice);
 270         if (k < 0)
 271                 goto error;
 272         if (up)
 273                 isl_seq_cpy(slice->ineq[k], eq, 1 + dim);
 274         else
 275                 isl_seq_neg(slice->ineq[k], eq, 1 + dim);
 276         isl_int_sub_ui(slice->ineq[k][0], slice->ineq[k][0], 1);
 277
 278         sample = isl_basic_set_sample(slice);
 279         if (!sample)
 280                 goto error;
 281         if (sample->size == 0) {
 282                 isl_vec_free(ctx, sample);
 283                 point = isl_basic_set_empty_like(bset);
 284         } else
 285                 point = isl_basic_set_from_vec(ctx, sample);
 286
 287         return point;
 288 error:
 289         isl_basic_set_free(slice);
 290         return NULL;
 291 }
 292
 293 /* Look for all equalities satisfied by the integer points in bset,
 294  * which is assume not to have any explicit equalities.
 295  *
 296  * The equalities are obtained by successively looking for
 297  * a point that is affinely independent of the points found so far.
 298  * In particular, for each equality satisfied by the points so far,
 299  * we check if there is any point on a hyperplane parallel to the
 300  * corresponding hyperplane shifted by at least one (in either direction).
 301  */
 302 static struct isl_basic_set *uset_affine_hull(struct isl_basic_set *bset)
 303 {
 304         int i, j;
 305         struct isl_basic_set *hull = NULL;
 306         struct isl_vec *sample;
 307         struct isl_ctx *ctx;
 308         unsigned dim;
 309
 310         if (isl_basic_set_is_empty(bset))
 311                 return bset;
 312
 313         ctx = bset->ctx;
 314         sample = isl_basic_set_sample(isl_basic_set_copy(bset));
 315         if (!sample)
 316                 goto error;
 317         if (sample->size == 0) {
 318                 isl_vec_free(ctx, sample);
 319                 hull = isl_basic_set_empty_like(bset);
 320                 isl_basic_set_free(bset);
 321                 return hull;
 322         } else
 323                 hull = isl_basic_set_from_vec(ctx, sample);
 324
 325         dim = isl_basic_set_n_dim(bset);
 326         for (i = 0; i < dim; ++i) {
 327                 struct isl_basic_set *point;
 328                 for (j = 0; j < hull->n_eq; ++j) {
 329                         point = outside_point(ctx, bset, hull->eq[j], 1);
 330                         if (!point)
 331                                 goto error;
 332                         if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
 333                                 break;
 334                         isl_basic_set_free(point);
 335                         point = outside_point(ctx, bset, hull->eq[j], 0);
 336                         if (!point)
 337                                 goto error;
 338                         if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
 339                                 break;
 340                         isl_basic_set_free(point);
 341                 }
 342                 if (j == hull->n_eq)
 343                         break;
 344                 hull = affine_hull(hull, point);
 345         }
 346         isl_basic_set_free(bset);
 347
 348         return hull;
 349 error:
 350         isl_basic_set_free(bset);
 351         isl_basic_set_free(hull);
 352         return NULL;
 353 }
 354
 355 /* Look for all equalities satisfied by the integer points in bmap
 356  * that are independent of the equalities already explicitly available
 357  * in bmap.
 358  *
 359  * We first remove all equalities already explicitly available,
 360  * then look for additional equalities in the reduced space
 361  * and then transform the result to the original space.
 362  * The original equalities are _not_ added to this set.  This is
 363  * the responsibility of the calling function.
 364  * The resulting basic set has all meaning about the dimensions removed.
 365  * In particular, dimensions that correspond to existential variables
 366  * in bmap and that are found to be fixed are not removed.
 367  */
 368 static struct isl_basic_set *equalities_in_underlying_set(
 369                                                 struct isl_basic_map *bmap)
 370 {
 371         struct isl_mat *T2 = NULL;
 372         struct isl_basic_set *bset = NULL;
 373         struct isl_basic_set *hull = NULL;
 374         struct isl_ctx *ctx;
 375
 376         ctx = bmap->ctx;
 377         bset = isl_basic_map_underlying_set(bmap);
 378         bset = isl_basic_set_remove_equalities(bset, NULL, &T2);
 379         if (!bset)
 380                 goto error;
 381
 382         hull = uset_affine_hull(bset);
 383         if (T2)
 384                 hull = isl_basic_set_preimage(hull, T2);
 385
 386         return hull;
 387 error:
 388         isl_mat_free(ctx, T2);
 389         isl_basic_set_free(bset);
 390         isl_basic_set_free(hull);
 391         return NULL;
 392 }
 393
 394 /* Detect and make explicit all equalities satisfied by the (integer)
 395  * points in bmap.
 396  */
 397 struct isl_basic_map *isl_basic_map_detect_equalities(
 398                                                 struct isl_basic_map *bmap)
 399 {
 400         int i, j;
 401         struct isl_basic_set *hull = NULL;
 402
 403         if (!bmap)
 404                 return NULL;
 405         if (bmap->n_ineq == 0)
 406                 return bmap;
 407         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
 408                 return bmap;
 409         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))
 410                 return bmap;
 411         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 412                 return isl_basic_map_implicit_equalities(bmap);
 413
 414         hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
 415         if (!hull)
 416                 goto error;
 417         bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim), 0,
 418                                         hull->n_eq, 0);
 419         for (i = 0; i < hull->n_eq; ++i) {
 420                 j = isl_basic_map_alloc_equality(bmap);
 421                 if (j < 0)
 422                         goto error;
 423                 isl_seq_cpy(bmap->eq[j], hull->eq[i],
 424                                 1 + isl_basic_set_total_dim(hull));
 425         }
 426         isl_basic_set_free(hull);
 427         ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);
 428         bmap = isl_basic_map_simplify(bmap);
 429         return isl_basic_map_finalize(bmap);
 430 error:
 431         isl_basic_set_free(hull);
 432         isl_basic_map_free(bmap);
 433         return NULL;
 434 }
 435
 436 struct isl_map *isl_map_detect_equalities(struct isl_map *map)
 437 {
 438         struct isl_basic_map *bmap;
 439         int i;
 440
 441         if (!map)
 442                 return NULL;
 443
 444         for (i = 0; i < map->n; ++i) {
 445                 bmap = isl_basic_map_copy(map->p[i]);
 446                 bmap = isl_basic_map_detect_equalities(bmap);
 447                 if (!bmap)
 448                         goto error;
 449                 isl_basic_map_free(map->p[i]);
 450                 map->p[i] = bmap;
 451         }
 452
 453         return map;
 454 error:
 455         isl_map_free(map);
 456         return NULL;
 457 }
 458
 459 /* After computing the rational affine hull (by detecting the implicit
 460  * equalities), we compute the additional equalities satisfied by
 461  * the integer points (if any) and add the original equalities back in.
 462  */
 463 struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
 464 {
 465         struct isl_basic_set *hull = NULL;
 466
 467         bmap = isl_basic_map_detect_equalities(bmap);
 468         bmap = isl_basic_map_cow(bmap);
 469         isl_basic_map_free_inequality(bmap, bmap->n_ineq);
 470         return bmap;
 471 }
 472
 473 struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
 474 {
 475         return (struct isl_basic_set *)
 476                 isl_basic_map_affine_hull((struct isl_basic_map *)bset);
 477 }
 478
 479 struct isl_basic_map *isl_map_affine_hull(struct isl_map *map)
 480 {
 481         int i;
 482         struct isl_basic_map *model = NULL;
 483         struct isl_basic_map *hull = NULL;
 484         struct isl_set *set;
 485
 486         if (!map)
 487                 return NULL;
 488
 489         if (map->n == 0) {
 490                 hull = isl_basic_map_empty_like_map(map);
 491                 isl_map_free(map);
 492                 return hull;
 493         }
 494
 495         map = isl_map_detect_equalities(map);
 496         map = isl_map_align_divs(map);
 497         if (!map)
 498                 return NULL;
 499         model = isl_basic_map_copy(map->p[0]);
 500         set = isl_map_underlying_set(map);
 501         set = isl_set_cow(set);
 502         if (!set)
 503                 goto error;
 504
 505         for (i = 0; i < set->n; ++i) {
 506                 set->p[i] = isl_basic_set_cow(set->p[i]);
 507                 set->p[i] = isl_basic_set_affine_hull(set->p[i]);
 508                 set->p[i] = isl_basic_set_gauss(set->p[i], NULL);
 509                 if (!set->p[i])
 510                         goto error;
 511         }
 512         set = isl_set_remove_empty_parts(set);
 513         if (set->n == 0) {
 514                 hull = isl_basic_map_empty_like(model);
 515                 isl_basic_map_free(model);
 516         } else {
 517                 struct isl_basic_set *bset;
 518                 while (set->n > 1) {
 519                         set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
 520                         if (!set->p[0])
 521                                 goto error;
 522                 }
 523                 bset = isl_basic_set_copy(set->p[0]);
 524                 hull = isl_basic_map_overlying_set(bset, model);
 525         }
 526         isl_set_free(set);
 527         hull = isl_basic_map_simplify(hull);
 528         return isl_basic_map_finalize(hull);
 529 error:
 530         isl_basic_map_free(model);
 531         isl_set_free(set);
 532         return NULL;
 533 }
 534
 535 struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
 536 {
 537         return (struct isl_basic_set *)
 538                 isl_map_affine_hull((struct isl_map *)set);
 539 }