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
  10 struct isl_basic_map *isl_basic_map_implicit_equalities(
  11                                                 struct isl_basic_map *bmap)
  12 {
  13         int i;
  14         int rational;
  15         isl_int opt;
  16         isl_int opt_denom;
  17         struct isl_ctx *ctx;
  18
  19         if (!bmap)
  20                 return bmap;
  21
  22         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
  23                 return bmap;
  24         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_IMPLICIT))
  25                 return bmap;
  26
  27         ctx = bmap->ctx;
  28         rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
  29         isl_int_init(opt);
  30         isl_int_init(opt_denom);
  31         if (!rational)
  32                 isl_int_set_si(opt_denom, 1);
  33         for (i = bmap->n_ineq - 1; i >= 0; --i) {
  34                 enum isl_lp_result res;
  35                 res = isl_solve_lp(bmap, 1, bmap->ineq[i]+1, ctx->one,
  36                                         &opt, rational ? &opt_denom : NULL);
  37                 if (res == isl_lp_unbounded)
  38                         continue;
  39                 if (res == isl_lp_error)
  40                         goto error;
  41                 if (res == isl_lp_empty) {
  42                         bmap = isl_basic_map_set_to_empty(bmap);
  43                         break;
  44                 }
  45                 if (!isl_int_is_one(opt_denom))
  46                         continue;
  47                 isl_int_add(opt, opt, bmap->ineq[i][0]);
  48                 if (isl_int_is_zero(opt))
  49                         isl_basic_map_inequality_to_equality(bmap, i);
  50         }
  51         isl_int_clear(opt_denom);
  52         isl_int_clear(opt);
  53
  54         ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
  55         return bmap;
  56 error:
  57         isl_int_clear(opt);
  58         isl_basic_map_free(bmap);
  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         return bset1;
 218 error:
 219         isl_basic_set_free(bset1);
 220         return NULL;
 221 }
 222
 223 static struct isl_basic_set *isl_basic_set_from_vec(struct isl_ctx *ctx,
 224         struct isl_vec *vec)
 225 {
 226         int i;
 227         int k;
 228         struct isl_basic_set *bset = NULL;
 229         unsigned dim;
 230
 231         if (!vec)
 232                 return NULL;
 233         isl_assert(ctx, vec->size != 0, goto error);
 234
 235         bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
 236         if (!bset)
 237                 goto error;
 238         dim = isl_basic_set_n_dim(bset);
 239         for (i = dim - 1; i >= 0; --i) {
 240                 k = isl_basic_set_alloc_equality(bset);
 241                 if (k < 0)
 242                         goto error;
 243                 isl_seq_clr(bset->eq[k], 1 + dim);
 244                 isl_int_neg(bset->eq[k][0], vec->block.data[1 + i]);
 245                 isl_int_set(bset->eq[k][1 + i], vec->block.data[0]);
 246         }
 247         isl_vec_free(ctx, vec);
 248
 249         return bset;
 250 error:
 251         isl_basic_set_free(bset);
 252         isl_vec_free(ctx, vec);
 253         return NULL;
 254 }
 255
 256 /* Find an integer point in "bset" that lies outside of the equality
 257  * "eq" e(x) = 0.
 258  * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
 259  * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
 260  * The point, if found, is returned as a singleton set.
 261  * If no point can be found, the empty set is returned.
 262  */
 263 static struct isl_basic_set *outside_point(struct isl_ctx *ctx,
 264         struct isl_basic_set *bset, isl_int *eq, int up)
 265 {
 266         struct isl_basic_set *slice = NULL;
 267         struct isl_vec *sample;
 268         struct isl_basic_set *point;
 269         unsigned dim;
 270         int k;
 271
 272         slice = isl_basic_set_copy(bset);
 273         if (!slice)
 274                 goto error;
 275         dim = isl_basic_set_n_dim(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 /* Look for all equalities satisfied by the integer points in bmap
 302  * that are independent of the equalities already explicitly available
 303  * in bmap.
 304  *
 305  * We first remove all equalities already explicitly available,
 306  * then look for additional equalities in the reduced space
 307  * and then transform the result to the original space.
 308  * The original equalities are _not_ added to this set.  This is
 309  * the responsibility of the calling function.
 310  * The resulting basic set has all meaning about the dimensions removed.
 311  * In particular, dimensions that correspond to existential variables
 312  * in bmap and that are found to be fixed are not removed.
 313  *
 314  * The additional equalities are obtained by successively looking for
 315  * a point that is affinely independent of the points found so far.
 316  * In particular, for each equality satisfied by the points so far,
 317  * we check if there is any point on a hyperplane parallel to the
 318  * corresponding hyperplane shifted by at least one (in either direction).
 319  */
 320 static struct isl_basic_set *equalities_in_underlying_set(
 321                                                 struct isl_basic_map *bmap)
 322 {
 323         int i, j;
 324         struct isl_mat *T2 = NULL;
 325         struct isl_basic_set *bset = NULL;
 326         struct isl_basic_set *hull = NULL;
 327         struct isl_vec *sample;
 328         struct isl_ctx *ctx;
 329         unsigned dim;
 330
 331         bset = isl_basic_map_underlying_set(bmap);
 332         bset = isl_basic_set_remove_equalities(bset, NULL, &T2);
 333         if (!bset)
 334                 goto error;
 335
 336         ctx = bset->ctx;
 337         sample = isl_basic_set_sample(isl_basic_set_copy(bset));
 338         if (!sample)
 339                 goto error;
 340         if (sample->size == 0) {
 341                 isl_vec_free(ctx, sample);
 342                 hull = isl_basic_set_empty_like(bset);
 343         } else
 344                 hull = isl_basic_set_from_vec(ctx, sample);
 345
 346         dim = isl_basic_set_n_dim(bset);
 347         for (i = 0; i < dim; ++i) {
 348                 struct isl_basic_set *point;
 349                 if (ISL_F_ISSET(hull, ISL_BASIC_SET_EMPTY))
 350                         break;
 351                 for (j = 0; j < hull->n_eq; ++j) {
 352                         point = outside_point(ctx, bset, hull->eq[j], 1);
 353                         if (!point)
 354                                 goto error;
 355                         if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
 356                                 break;
 357                         isl_basic_set_free(point);
 358                         point = outside_point(ctx, bset, hull->eq[j], 0);
 359                         if (!point)
 360                                 goto error;
 361                         if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
 362                                 break;
 363                         isl_basic_set_free(point);
 364                 }
 365                 if (j == hull->n_eq)
 366                         break;
 367                 hull = affine_hull(hull, point);
 368         }
 369         isl_basic_set_free(bset);
 370         if (T2)
 371                 hull = isl_basic_set_preimage(ctx, hull, T2);
 372
 373         return hull;
 374 error:
 375         isl_mat_free(ctx, T2);
 376         isl_basic_set_free(bset);
 377         isl_basic_set_free(hull);
 378         return NULL;
 379 }
 380
 381 /* Detect and make explicit all equalities satisfied by the (integer)
 382  * points in bmap.
 383  */
 384 struct isl_basic_map *isl_basic_map_detect_equalities(
 385                                                 struct isl_basic_map *bmap)
 386 {
 387         int i, j;
 388         struct isl_basic_set *hull = NULL;
 389
 390         if (!bmap)
 391                 return NULL;
 392         if (bmap->n_ineq == 0)
 393                 return bmap;
 394         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
 395                 return bmap;
 396         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))
 397                 return bmap;
 398         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 399                 return isl_basic_map_implicit_equalities(bmap);
 400
 401         hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
 402         if (!hull)
 403                 goto error;
 404         bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim), 0,
 405                                         hull->n_eq, 0);
 406         for (i = 0; i < hull->n_eq; ++i) {
 407                 j = isl_basic_map_alloc_equality(bmap);
 408                 if (j < 0)
 409                         goto error;
 410                 isl_seq_cpy(bmap->eq[j], hull->eq[i],
 411                                 1 + isl_basic_set_total_dim(hull));
 412         }
 413         isl_basic_set_free(hull);
 414         ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);
 415         bmap = isl_basic_map_simplify(bmap);
 416         return isl_basic_map_finalize(bmap);
 417 error:
 418         isl_basic_set_free(hull);
 419         isl_basic_map_free(bmap);
 420         return NULL;
 421 }
 422
 423 /* After computing the rational affine hull (by detecting the implicit
 424  * equalities), we compute the additional equalities satisfied by
 425  * the integer points (if any) and add the original equalities back in.
 426  */
 427 struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
 428 {
 429         struct isl_basic_set *hull = NULL;
 430
 431         bmap = isl_basic_map_implicit_equalities(bmap);
 432         if (!bmap)
 433                 return NULL;
 434         if (bmap->n_ineq == 0)
 435                 return bmap;
 436
 437         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) {
 438                 bmap = isl_basic_map_cow(bmap);
 439                 isl_basic_map_free_inequality(bmap, bmap->n_ineq);
 440                 return bmap;
 441         }
 442
 443         hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
 444         if (!hull)
 445                 goto error;
 446
 447         bmap = isl_basic_map_cow(bmap);
 448         if (!bmap)
 449                 goto error;
 450         isl_basic_map_free_inequality(bmap, bmap->n_ineq);
 451         bmap = isl_basic_map_intersect(bmap,
 452                                         isl_basic_map_overlying_set(hull,
 453                                                 isl_basic_map_copy(bmap)));
 454
 455         return isl_basic_map_finalize(bmap);
 456 error:
 457         isl_basic_set_free(hull);
 458         isl_basic_map_free(bmap);
 459         return NULL;
 460 }
 461
 462 struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
 463 {
 464         return (struct isl_basic_set *)
 465                 isl_basic_map_affine_hull((struct isl_basic_map *)bset);
 466 }
 467
 468 struct isl_basic_map *isl_map_affine_hull(struct isl_map *map)
 469 {
 470         int i;
 471         struct isl_basic_map *model = NULL;
 472         struct isl_basic_map *hull = NULL;
 473         struct isl_set *set;
 474
 475         if (!map)
 476                 return NULL;
 477
 478         if (map->n == 0) {
 479                 hull = isl_basic_map_empty_like_map(map);
 480                 isl_map_free(map);
 481                 return hull;
 482         }
 483
 484         map = isl_map_align_divs(map);
 485         model = isl_basic_map_copy(map->p[0]);
 486         set = isl_map_underlying_set(map);
 487         set = isl_set_cow(set);
 488         if (!set)
 489                 goto error;
 490
 491         for (i = 0; i < set->n; ++i) {
 492                 set->p[i] = isl_basic_set_cow(set->p[i]);
 493                 set->p[i] = isl_basic_set_affine_hull(set->p[i]);
 494                 set->p[i] = isl_basic_set_gauss(set->p[i], NULL);
 495                 if (!set->p[i])
 496                         goto error;
 497         }
 498         set = isl_set_remove_empty_parts(set);
 499         if (set->n == 0) {
 500                 hull = isl_basic_map_empty_like(model);
 501                 isl_basic_map_free(model);
 502         } else {
 503                 struct isl_basic_set *bset;
 504                 while (set->n > 1) {
 505                         set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
 506                         if (!set->p[0])
 507                                 goto error;
 508                 }
 509                 bset = isl_basic_set_copy(set->p[0]);
 510                 hull = isl_basic_map_overlying_set(bset, model);
 511         }
 512         isl_set_free(set);
 513         hull = isl_basic_map_simplify(hull);
 514         return isl_basic_map_finalize(hull);
 515 error:
 516         isl_basic_map_free(model);
 517         isl_set_free(set);
 518         return NULL;
 519 }
 520
 521 struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
 522 {
 523         return (struct isl_basic_set *)
 524                 isl_map_affine_hull((struct isl_map *)set);
 525 }