isl_sample.c

   1 /*
   2  * Copyright 2008-2009 Katholieke Universiteit Leuven
   3  *
   4  * Use of this software is governed by the GNU LGPLv2.1 license
   5  *
   6  * Written by Sven Verdoolaege, K.U.Leuven, Departement
   7  * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
   8  */
   9
  10 #include "isl_sample.h"
  11 #include "isl_sample_piplib.h"
  12 #include <isl/vec.h>
  13 #include <isl/mat.h>
  14 #include <isl/seq.h>
  15 #include "isl_map_private.h"
  16 #include "isl_equalities.h"
  17 #include "isl_tab.h"
  18 #include "isl_basis_reduction.h"
  19 #include <isl_point_private.h>
  20
  21 static struct isl_vec *empty_sample(struct isl_basic_set *bset)
  22 {
  23         struct isl_vec *vec;
  24
  25         vec = isl_vec_alloc(bset->ctx, 0);
  26         isl_basic_set_free(bset);
  27         return vec;
  28 }
  29
  30 /* Construct a zero sample of the same dimension as bset.
  31  * As a special case, if bset is zero-dimensional, this
  32  * function creates a zero-dimensional sample point.
  33  */
  34 static struct isl_vec *zero_sample(struct isl_basic_set *bset)
  35 {
  36         unsigned dim;
  37         struct isl_vec *sample;
  38
  39         dim = isl_basic_set_total_dim(bset);
  40         sample = isl_vec_alloc(bset->ctx, 1 + dim);
  41         if (sample) {
  42                 isl_int_set_si(sample->el[0], 1);
  43                 isl_seq_clr(sample->el + 1, dim);
  44         }
  45         isl_basic_set_free(bset);
  46         return sample;
  47 }
  48
  49 static struct isl_vec *interval_sample(struct isl_basic_set *bset)
  50 {
  51         int i;
  52         isl_int t;
  53         struct isl_vec *sample;
  54
  55         bset = isl_basic_set_simplify(bset);
  56         if (!bset)
  57                 return NULL;
  58         if (isl_basic_set_fast_is_empty(bset))
  59                 return empty_sample(bset);
  60         if (bset->n_eq == 0 && bset->n_ineq == 0)
  61                 return zero_sample(bset);
  62
  63         sample = isl_vec_alloc(bset->ctx, 2);
  64         if (!sample)
  65                 goto error;
  66         if (!bset)
  67                 return NULL;
  68         isl_int_set_si(sample->block.data[0], 1);
  69
  70         if (bset->n_eq > 0) {
  71                 isl_assert(bset->ctx, bset->n_eq == 1, goto error);
  72                 isl_assert(bset->ctx, bset->n_ineq == 0, goto error);
  73                 if (isl_int_is_one(bset->eq[0][1]))
  74                         isl_int_neg(sample->el[1], bset->eq[0][0]);
  75                 else {
  76                         isl_assert(bset->ctx, isl_int_is_negone(bset->eq[0][1]),
  77                                    goto error);
  78                         isl_int_set(sample->el[1], bset->eq[0][0]);
  79                 }
  80                 isl_basic_set_free(bset);
  81                 return sample;
  82         }
  83
  84         isl_int_init(t);
  85         if (isl_int_is_one(bset->ineq[0][1]))
  86                 isl_int_neg(sample->block.data[1], bset->ineq[0][0]);
  87         else
  88                 isl_int_set(sample->block.data[1], bset->ineq[0][0]);
  89         for (i = 1; i < bset->n_ineq; ++i) {
  90                 isl_seq_inner_product(sample->block.data,
  91                                         bset->ineq[i], 2, &t);
  92                 if (isl_int_is_neg(t))
  93                         break;
  94         }
  95         isl_int_clear(t);
  96         if (i < bset->n_ineq) {
  97                 isl_vec_free(sample);
  98                 return empty_sample(bset);
  99         }
 100
 101         isl_basic_set_free(bset);
 102         return sample;
 103 error:
 104         isl_basic_set_free(bset);
 105         isl_vec_free(sample);
 106         return NULL;
 107 }
 108
 109 static struct isl_mat *independent_bounds(struct isl_basic_set *bset)
 110 {
 111         int i, j, n;
 112         struct isl_mat *dirs = NULL;
 113         struct isl_mat *bounds = NULL;
 114         unsigned dim;
 115
 116         if (!bset)
 117                 return NULL;
 118
 119         dim = isl_basic_set_n_dim(bset);
 120         bounds = isl_mat_alloc(bset->ctx, 1+dim, 1+dim);
 121         if (!bounds)
 122                 return NULL;
 123
 124         isl_int_set_si(bounds->row[0][0], 1);
 125         isl_seq_clr(bounds->row[0]+1, dim);
 126         bounds->n_row = 1;
 127
 128         if (bset->n_ineq == 0)
 129                 return bounds;
 130
 131         dirs = isl_mat_alloc(bset->ctx, dim, dim);
 132         if (!dirs) {
 133                 isl_mat_free(bounds);
 134                 return NULL;
 135         }
 136         isl_seq_cpy(dirs->row[0], bset->ineq[0]+1, dirs->n_col);
 137         isl_seq_cpy(bounds->row[1], bset->ineq[0], bounds->n_col);
 138         for (j = 1, n = 1; n < dim && j < bset->n_ineq; ++j) {
 139                 int pos;
 140
 141                 isl_seq_cpy(dirs->row[n], bset->ineq[j]+1, dirs->n_col);
 142
 143                 pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
 144                 if (pos < 0)
 145                         continue;
 146                 for (i = 0; i < n; ++i) {
 147                         int pos_i;
 148                         pos_i = isl_seq_first_non_zero(dirs->row[i], dirs->n_col);
 149                         if (pos_i < pos)
 150                                 continue;
 151                         if (pos_i > pos)
 152                                 break;
 153                         isl_seq_elim(dirs->row[n], dirs->row[i], pos,
 154                                         dirs->n_col, NULL);
 155                         pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
 156                         if (pos < 0)
 157                                 break;
 158                 }
 159                 if (pos < 0)
 160                         continue;
 161                 if (i < n) {
 162                         int k;
 163                         isl_int *t = dirs->row[n];
 164                         for (k = n; k > i; --k)
 165                                 dirs->row[k] = dirs->row[k-1];
 166                         dirs->row[i] = t;
 167                 }
 168                 ++n;
 169                 isl_seq_cpy(bounds->row[n], bset->ineq[j], bounds->n_col);
 170         }
 171         isl_mat_free(dirs);
 172         bounds->n_row = 1+n;
 173         return bounds;
 174 }
 175
 176 static void swap_inequality(struct isl_basic_set *bset, int a, int b)
 177 {
 178         isl_int *t = bset->ineq[a];
 179         bset->ineq[a] = bset->ineq[b];
 180         bset->ineq[b] = t;
 181 }
 182
 183 /* Skew into positive orthant and project out lineality space.
 184  *
 185  * We perform a unimodular transformation that turns a selected
 186  * maximal set of linearly independent bounds into constraints
 187  * on the first dimensions that impose that these first dimensions
 188  * are non-negative.  In particular, the constraint matrix is lower
 189  * triangular with positive entries on the diagonal and negative
 190  * entries below.
 191  * If "bset" has a lineality space then these constraints (and therefore
 192  * all constraints in bset) only involve the first dimensions.
 193  * The remaining dimensions then do not appear in any constraints and
 194  * we can select any value for them, say zero.  We therefore project
 195  * out this final dimensions and plug in the value zero later.  This
 196  * is accomplished by simply dropping the final columns of
 197  * the unimodular transformation.
 198  */
 199 static struct isl_basic_set *isl_basic_set_skew_to_positive_orthant(
 200         struct isl_basic_set *bset, struct isl_mat **T)
 201 {
 202         struct isl_mat *U = NULL;
 203         struct isl_mat *bounds = NULL;
 204         int i, j;
 205         unsigned old_dim, new_dim;
 206
 207         *T = NULL;
 208         if (!bset)
 209                 return NULL;
 210
 211         isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
 212         isl_assert(bset->ctx, bset->n_div == 0, goto error);
 213         isl_assert(bset->ctx, bset->n_eq == 0, goto error);
 214
 215         old_dim = isl_basic_set_n_dim(bset);
 216         /* Try to move (multiples of) unit rows up. */
 217         for (i = 0, j = 0; i < bset->n_ineq; ++i) {
 218                 int pos = isl_seq_first_non_zero(bset->ineq[i]+1, old_dim);
 219                 if (pos < 0)
 220                         continue;
 221                 if (isl_seq_first_non_zero(bset->ineq[i]+1+pos+1,
 222                                                 old_dim-pos-1) >= 0)
 223                         continue;
 224                 if (i != j)
 225                         swap_inequality(bset, i, j);
 226                 ++j;
 227         }
 228         bounds = independent_bounds(bset);
 229         if (!bounds)
 230                 goto error;
 231         new_dim = bounds->n_row - 1;
 232         bounds = isl_mat_left_hermite(bounds, 1, &U, NULL);
 233         if (!bounds)
 234                 goto error;
 235         U = isl_mat_drop_cols(U, 1 + new_dim, old_dim - new_dim);
 236         bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
 237         if (!bset)
 238                 goto error;
 239         *T = U;
 240         isl_mat_free(bounds);
 241         return bset;
 242 error:
 243         isl_mat_free(bounds);
 244         isl_mat_free(U);
 245         isl_basic_set_free(bset);
 246         return NULL;
 247 }
 248
 249 /* Find a sample integer point, if any, in bset, which is known
 250  * to have equalities.  If bset contains no integer points, then
 251  * return a zero-length vector.
 252  * We simply remove the known equalities, compute a sample
 253  * in the resulting bset, using the specified recurse function,
 254  * and then transform the sample back to the original space.
 255  */
 256 static struct isl_vec *sample_eq(struct isl_basic_set *bset,
 257         struct isl_vec *(*recurse)(struct isl_basic_set *))
 258 {
 259         struct isl_mat *T;
 260         struct isl_vec *sample;
 261
 262         if (!bset)
 263                 return NULL;
 264
 265         bset = isl_basic_set_remove_equalities(bset, &T, NULL);
 266         sample = recurse(bset);
 267         if (!sample || sample->size == 0)
 268                 isl_mat_free(T);
 269         else
 270                 sample = isl_mat_vec_product(T, sample);
 271         return sample;
 272 }
 273
 274 /* Return a matrix containing the equalities of the tableau
 275  * in constraint form.  The tableau is assumed to have
 276  * an associated bset that has been kept up-to-date.
 277  */
 278 static struct isl_mat *tab_equalities(struct isl_tab *tab)
 279 {
 280         int i, j;
 281         int n_eq;
 282         struct isl_mat *eq;
 283         struct isl_basic_set *bset;
 284
 285         if (!tab)
 286                 return NULL;
 287
 288         bset = isl_tab_peek_bset(tab);
 289         isl_assert(tab->mat->ctx, bset, return NULL);
 290
 291         n_eq = tab->n_var - tab->n_col + tab->n_dead;
 292         if (tab->empty || n_eq == 0)
 293                 return isl_mat_alloc(tab->mat->ctx, 0, tab->n_var);
 294         if (n_eq == tab->n_var)
 295                 return isl_mat_identity(tab->mat->ctx, tab->n_var);
 296
 297         eq = isl_mat_alloc(tab->mat->ctx, n_eq, tab->n_var);
 298         if (!eq)
 299                 return NULL;
 300         for (i = 0, j = 0; i < tab->n_con; ++i) {
 301                 if (tab->con[i].is_row)
 302                         continue;
 303                 if (tab->con[i].index >= 0 && tab->con[i].index >= tab->n_dead)
 304                         continue;
 305                 if (i < bset->n_eq)
 306                         isl_seq_cpy(eq->row[j], bset->eq[i] + 1, tab->n_var);
 307                 else
 308                         isl_seq_cpy(eq->row[j],
 309                                     bset->ineq[i - bset->n_eq] + 1, tab->n_var);
 310                 ++j;
 311         }
 312         isl_assert(bset->ctx, j == n_eq, goto error);
 313         return eq;
 314 error:
 315         isl_mat_free(eq);
 316         return NULL;
 317 }
 318
 319 /* Compute and return an initial basis for the bounded tableau "tab".
 320  *
 321  * If the tableau is either full-dimensional or zero-dimensional,
 322  * the we simply return an identity matrix.
 323  * Otherwise, we construct a basis whose first directions correspond
 324  * to equalities.
 325  */
 326 static struct isl_mat *initial_basis(struct isl_tab *tab)
 327 {
 328         int n_eq;
 329         struct isl_mat *eq;
 330         struct isl_mat *Q;
 331
 332         tab->n_unbounded = 0;
 333         tab->n_zero = n_eq = tab->n_var - tab->n_col + tab->n_dead;
 334         if (tab->empty || n_eq == 0 || n_eq == tab->n_var)
 335                 return isl_mat_identity(tab->mat->ctx, 1 + tab->n_var);
 336
 337         eq = tab_equalities(tab);
 338         eq = isl_mat_left_hermite(eq, 0, NULL, &Q);
 339         if (!eq)
 340                 return NULL;
 341         isl_mat_free(eq);
 342
 343         Q = isl_mat_lin_to_aff(Q);
 344         return Q;
 345 }
 346
 347 /* Given a tableau representing a set, find and return
 348  * an integer point in the set, if there is any.
 349  *
 350  * We perform a depth first search
 351  * for an integer point, by scanning all possible values in the range
 352  * attained by a basis vector, where an initial basis may have been set
 353  * by the calling function.  Otherwise an initial basis that exploits
 354  * the equalities in the tableau is created.
 355  * tab->n_zero is currently ignored and is clobbered by this function.
 356  *
 357  * The tableau is allowed to have unbounded direction, but then
 358  * the calling function needs to set an initial basis, with the
 359  * unbounded directions last and with tab->n_unbounded set
 360  * to the number of unbounded directions.
 361  * Furthermore, the calling functions needs to add shifted copies
 362  * of all constraints involving unbounded directions to ensure
 363  * that any feasible rational value in these directions can be rounded
 364  * up to yield a feasible integer value.
 365  * In particular, let B define the given basis x' = B x
 366  * and let T be the inverse of B, i.e., X = T x'.
 367  * Let a x + c >= 0 be a constraint of the set represented by the tableau,
 368  * or a T x' + c >= 0 in terms of the given basis.  Assume that
 369  * the bounded directions have an integer value, then we can safely
 370  * round up the values for the unbounded directions if we make sure
 371  * that x' not only satisfies the original constraint, but also
 372  * the constraint "a T x' + c + s >= 0" with s the sum of all
 373  * negative values in the last n_unbounded entries of "a T".
 374  * The calling function therefore needs to add the constraint
 375  * a x + c + s >= 0.  The current function then scans the first
 376  * directions for an integer value and once those have been found,
 377  * it can compute "T ceil(B x)" to yield an integer point in the set.
 378  * Note that during the search, the first rows of B may be changed
 379  * by a basis reduction, but the last n_unbounded rows of B remain
 380  * unaltered and are also not mixed into the first rows.
 381  *
 382  * The search is implemented iteratively.  "level" identifies the current
 383  * basis vector.  "init" is true if we want the first value at the current
 384  * level and false if we want the next value.
 385  *
 386  * The initial basis is the identity matrix.  If the range in some direction
 387  * contains more than one integer value, we perform basis reduction based
 388  * on the value of ctx->opt->gbr
 389  *      - ISL_GBR_NEVER:        never perform basis reduction
 390  *      - ISL_GBR_ONCE:         only perform basis reduction the first
 391  *                              time such a range is encountered
 392  *      - ISL_GBR_ALWAYS:       always perform basis reduction when
 393  *                              such a range is encountered
 394  *
 395  * When ctx->opt->gbr is set to ISL_GBR_ALWAYS, then we allow the basis
 396  * reduction computation to return early.  That is, as soon as it
 397  * finds a reasonable first direction.
 398  */
 399 struct isl_vec *isl_tab_sample(struct isl_tab *tab)
 400 {
 401         unsigned dim;
 402         unsigned gbr;
 403         struct isl_ctx *ctx;
 404         struct isl_vec *sample;
 405         struct isl_vec *min;
 406         struct isl_vec *max;
 407         enum isl_lp_result res;
 408         int level;
 409         int init;
 410         int reduced;
 411         struct isl_tab_undo **snap;
 412
 413         if (!tab)
 414                 return NULL;
 415         if (tab->empty)
 416                 return isl_vec_alloc(tab->mat->ctx, 0);
 417
 418         if (!tab->basis)
 419                 tab->basis = initial_basis(tab);
 420         if (!tab->basis)
 421                 return NULL;
 422         isl_assert(tab->mat->ctx, tab->basis->n_row == tab->n_var + 1,
 423                     return NULL);
 424         isl_assert(tab->mat->ctx, tab->basis->n_col == tab->n_var + 1,
 425                     return NULL);
 426
 427         ctx = tab->mat->ctx;
 428         dim = tab->n_var;
 429         gbr = ctx->opt->gbr;
 430
 431         if (tab->n_unbounded == tab->n_var) {
 432                 sample = isl_tab_get_sample_value(tab);
 433                 sample = isl_mat_vec_product(isl_mat_copy(tab->basis), sample);
 434                 sample = isl_vec_ceil(sample);
 435                 sample = isl_mat_vec_inverse_product(isl_mat_copy(tab->basis),
 436                                                         sample);
 437                 return sample;
 438         }
 439
 440         if (isl_tab_extend_cons(tab, dim + 1) < 0)
 441                 return NULL;
 442
 443         min = isl_vec_alloc(ctx, dim);
 444         max = isl_vec_alloc(ctx, dim);
 445         snap = isl_alloc_array(ctx, struct isl_tab_undo *, dim);
 446
 447         if (!min || !max || !snap)
 448                 goto error;
 449
 450         level = 0;
 451         init = 1;
 452         reduced = 0;
 453
 454         while (level >= 0) {
 455                 int empty = 0;
 456                 if (init) {
 457                         res = isl_tab_min(tab, tab->basis->row[1 + level],
 458                                     ctx->one, &min->el[level], NULL, 0);
 459                         if (res == isl_lp_empty)
 460                                 empty = 1;
 461                         isl_assert(ctx, res != isl_lp_unbounded, goto error);
 462                         if (res == isl_lp_error)
 463                                 goto error;
 464                         if (!empty && isl_tab_sample_is_integer(tab))
 465                                 break;
 466                         isl_seq_neg(tab->basis->row[1 + level] + 1,
 467                                     tab->basis->row[1 + level] + 1, dim);
 468                         res = isl_tab_min(tab, tab->basis->row[1 + level],
 469                                     ctx->one, &max->el[level], NULL, 0);
 470                         isl_seq_neg(tab->basis->row[1 + level] + 1,
 471                                     tab->basis->row[1 + level] + 1, dim);
 472                         isl_int_neg(max->el[level], max->el[level]);
 473                         if (res == isl_lp_empty)
 474                                 empty = 1;
 475                         isl_assert(ctx, res != isl_lp_unbounded, goto error);
 476                         if (res == isl_lp_error)
 477                                 goto error;
 478                         if (!empty && isl_tab_sample_is_integer(tab))
 479                                 break;
 480                         if (!empty && !reduced &&
 481                             ctx->opt->gbr != ISL_GBR_NEVER &&
 482                             isl_int_lt(min->el[level], max->el[level])) {
 483                                 unsigned gbr_only_first;
 484                                 if (ctx->opt->gbr == ISL_GBR_ONCE)
 485                                         ctx->opt->gbr = ISL_GBR_NEVER;
 486                                 tab->n_zero = level;
 487                                 gbr_only_first = ctx->opt->gbr_only_first;
 488                                 ctx->opt->gbr_only_first =
 489                                         ctx->opt->gbr == ISL_GBR_ALWAYS;
 490                                 tab = isl_tab_compute_reduced_basis(tab);
 491                                 ctx->opt->gbr_only_first = gbr_only_first;
 492                                 if (!tab || !tab->basis)
 493                                         goto error;
 494                                 reduced = 1;
 495                                 continue;
 496                         }
 497                         reduced = 0;
 498                         snap[level] = isl_tab_snap(tab);
 499                 } else
 500                         isl_int_add_ui(min->el[level], min->el[level], 1);
 501
 502                 if (empty || isl_int_gt(min->el[level], max->el[level])) {
 503                         level--;
 504                         init = 0;
 505                         if (level >= 0)
 506                                 if (isl_tab_rollback(tab, snap[level]) < 0)
 507                                         goto error;
 508                         continue;
 509                 }
 510                 isl_int_neg(tab->basis->row[1 + level][0], min->el[level]);
 511                 if (isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]) < 0)
 512                         goto error;
 513                 isl_int_set_si(tab->basis->row[1 + level][0], 0);
 514                 if (level + tab->n_unbounded < dim - 1) {
 515                         ++level;
 516                         init = 1;
 517                         continue;
 518                 }
 519                 break;
 520         }
 521
 522         if (level >= 0) {
 523                 sample = isl_tab_get_sample_value(tab);
 524                 if (!sample)
 525                         goto error;
 526                 if (tab->n_unbounded && !isl_int_is_one(sample->el[0])) {
 527                         sample = isl_mat_vec_product(isl_mat_copy(tab->basis),
 528                                                      sample);
 529                         sample = isl_vec_ceil(sample);
 530                         sample = isl_mat_vec_inverse_product(
 531                                         isl_mat_copy(tab->basis), sample);
 532                 }
 533         } else
 534                 sample = isl_vec_alloc(ctx, 0);
 535
 536         ctx->opt->gbr = gbr;
 537         isl_vec_free(min);
 538         isl_vec_free(max);
 539         free(snap);
 540         return sample;
 541 error:
 542         ctx->opt->gbr = gbr;
 543         isl_vec_free(min);
 544         isl_vec_free(max);
 545         free(snap);
 546         return NULL;
 547 }
 548
 549 /* Given a basic set that is known to be bounded, find and return
 550  * an integer point in the basic set, if there is any.
 551  *
 552  * After handling some trivial cases, we construct a tableau
 553  * and then use isl_tab_sample to find a sample, passing it
 554  * the identity matrix as initial basis.
 555  */
 556 static struct isl_vec *sample_bounded(struct isl_basic_set *bset)
 557 {
 558         unsigned dim;
 559         struct isl_ctx *ctx;
 560         struct isl_vec *sample;
 561         struct isl_tab *tab = NULL;
 562
 563         if (!bset)
 564                 return NULL;
 565
 566         if (isl_basic_set_fast_is_empty(bset))
 567                 return empty_sample(bset);
 568
 569         dim = isl_basic_set_total_dim(bset);
 570         if (dim == 0)
 571                 return zero_sample(bset);
 572         if (dim == 1)
 573                 return interval_sample(bset);
 574         if (bset->n_eq > 0)
 575                 return sample_eq(bset, sample_bounded);
 576
 577         ctx = bset->ctx;
 578
 579         tab = isl_tab_from_basic_set(bset);
 580         if (tab && tab->empty) {
 581                 isl_tab_free(tab);
 582                 ISL_F_SET(bset, ISL_BASIC_SET_EMPTY);
 583                 sample = isl_vec_alloc(bset->ctx, 0);
 584                 isl_basic_set_free(bset);
 585                 return sample;
 586         }
 587
 588         if (isl_tab_track_bset(tab, isl_basic_set_copy(bset)) < 0)
 589                 goto error;
 590         if (!ISL_F_ISSET(bset, ISL_BASIC_SET_NO_IMPLICIT))
 591                 if (isl_tab_detect_implicit_equalities(tab) < 0)
 592                         goto error;
 593
 594         sample = isl_tab_sample(tab);
 595         if (!sample)
 596                 goto error;
 597
 598         if (sample->size > 0) {
 599                 isl_vec_free(bset->sample);
 600                 bset->sample = isl_vec_copy(sample);
 601         }
 602
 603         isl_basic_set_free(bset);
 604         isl_tab_free(tab);
 605         return sample;
 606 error:
 607         isl_basic_set_free(bset);
 608         isl_tab_free(tab);
 609         return NULL;
 610 }
 611
 612 /* Given a basic set "bset" and a value "sample" for the first coordinates
 613  * of bset, plug in these values and drop the corresponding coordinates.
 614  *
 615  * We do this by computing the preimage of the transformation
 616  *
 617  *           [ 1 0 ]
 618  *      x =  [ s 0 ] x'
 619  *           [ 0 I ]
 620  *
 621  * where [1 s] is the sample value and I is the identity matrix of the
 622  * appropriate dimension.
 623  */
 624 static struct isl_basic_set *plug_in(struct isl_basic_set *bset,
 625         struct isl_vec *sample)
 626 {
 627         int i;
 628         unsigned total;
 629         struct isl_mat *T;
 630
 631         if (!bset || !sample)
 632                 goto error;
 633
 634         total = isl_basic_set_total_dim(bset);
 635         T = isl_mat_alloc(bset->ctx, 1 + total, 1 + total - (sample->size - 1));
 636         if (!T)
 637                 goto error;
 638
 639         for (i = 0; i < sample->size; ++i) {
 640                 isl_int_set(T->row[i][0], sample->el[i]);
 641                 isl_seq_clr(T->row[i] + 1, T->n_col - 1);
 642         }
 643         for (i = 0; i < T->n_col - 1; ++i) {
 644                 isl_seq_clr(T->row[sample->size + i], T->n_col);
 645                 isl_int_set_si(T->row[sample->size + i][1 + i], 1);
 646         }
 647         isl_vec_free(sample);
 648
 649         bset = isl_basic_set_preimage(bset, T);
 650         return bset;
 651 error:
 652         isl_basic_set_free(bset);
 653         isl_vec_free(sample);
 654         return NULL;
 655 }
 656
 657 /* Given a basic set "bset", return any (possibly non-integer) point
 658  * in the basic set.
 659  */
 660 static struct isl_vec *rational_sample(struct isl_basic_set *bset)
 661 {
 662         struct isl_tab *tab;
 663         struct isl_vec *sample;
 664
 665         if (!bset)
 666                 return NULL;
 667
 668         tab = isl_tab_from_basic_set(bset);
 669         sample = isl_tab_get_sample_value(tab);
 670         isl_tab_free(tab);
 671
 672         isl_basic_set_free(bset);
 673
 674         return sample;
 675 }
 676
 677 /* Given a linear cone "cone" and a rational point "vec",
 678  * construct a polyhedron with shifted copies of the constraints in "cone",
 679  * i.e., a polyhedron with "cone" as its recession cone, such that each
 680  * point x in this polyhedron is such that the unit box positioned at x
 681  * lies entirely inside the affine cone 'vec + cone'.
 682  * Any rational point in this polyhedron may therefore be rounded up
 683  * to yield an integer point that lies inside said affine cone.
 684  *
 685  * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
 686  * point "vec" by v/d.
 687  * Let b_i = <a_i, v>.  Then the affine cone 'vec + cone' is given
 688  * by <a_i, x> - b/d >= 0.
 689  * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
 690  * We prefer this polyhedron over the actual affine cone because it doesn't
 691  * require a scaling of the constraints.
 692  * If each of the vertices of the unit cube positioned at x lies inside
 693  * this polyhedron, then the whole unit cube at x lies inside the affine cone.
 694  * We therefore impose that x' = x + \sum e_i, for any selection of unit
 695  * vectors lies inside the polyhedron, i.e.,
 696  *
 697  *      <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
 698  *
 699  * The most stringent of these constraints is the one that selects
 700  * all negative a_i, so the polyhedron we are looking for has constraints
 701  *
 702  *      <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
 703  *
 704  * Note that if cone were known to have only non-negative rays
 705  * (which can be accomplished by a unimodular transformation),
 706  * then we would only have to check the points x' = x + e_i
 707  * and we only have to add the smallest negative a_i (if any)
 708  * instead of the sum of all negative a_i.
 709  */
 710 static struct isl_basic_set *shift_cone(struct isl_basic_set *cone,
 711         struct isl_vec *vec)
 712 {
 713         int i, j, k;
 714         unsigned total;
 715
 716         struct isl_basic_set *shift = NULL;
 717
 718         if (!cone || !vec)
 719                 goto error;
 720
 721         isl_assert(cone->ctx, cone->n_eq == 0, goto error);
 722
 723         total = isl_basic_set_total_dim(cone);
 724
 725         shift = isl_basic_set_alloc_dim(isl_basic_set_get_dim(cone),
 726                                         0, 0, cone->n_ineq);
 727
 728         for (i = 0; i < cone->n_ineq; ++i) {
 729                 k = isl_basic_set_alloc_inequality(shift);
 730                 if (k < 0)
 731                         goto error;
 732                 isl_seq_cpy(shift->ineq[k] + 1, cone->ineq[i] + 1, total);
 733                 isl_seq_inner_product(shift->ineq[k] + 1, vec->el + 1, total,
 734                                       &shift->ineq[k][0]);
 735                 isl_int_cdiv_q(shift->ineq[k][0],
 736                                shift->ineq[k][0], vec->el[0]);
 737                 isl_int_neg(shift->ineq[k][0], shift->ineq[k][0]);
 738                 for (j = 0; j < total; ++j) {
 739                         if (isl_int_is_nonneg(shift->ineq[k][1 + j]))
 740                                 continue;
 741                         isl_int_add(shift->ineq[k][0],
 742                                     shift->ineq[k][0], shift->ineq[k][1 + j]);
 743                 }
 744         }
 745
 746         isl_basic_set_free(cone);
 747         isl_vec_free(vec);
 748
 749         return isl_basic_set_finalize(shift);
 750 error:
 751         isl_basic_set_free(shift);
 752         isl_basic_set_free(cone);
 753         isl_vec_free(vec);
 754         return NULL;
 755 }
 756
 757 /* Given a rational point vec in a (transformed) basic set,
 758  * such that cone is the recession cone of the original basic set,
 759  * "round up" the rational point to an integer point.
 760  *
 761  * We first check if the rational point just happens to be integer.
 762  * If not, we transform the cone in the same way as the basic set,
 763  * pick a point x in this cone shifted to the rational point such that
 764  * the whole unit cube at x is also inside this affine cone.
 765  * Then we simply round up the coordinates of x and return the
 766  * resulting integer point.
 767  */
 768 static struct isl_vec *round_up_in_cone(struct isl_vec *vec,
 769         struct isl_basic_set *cone, struct isl_mat *U)
 770 {
 771         unsigned total;
 772
 773         if (!vec || !cone || !U)
 774                 goto error;
 775
 776         isl_assert(vec->ctx, vec->size != 0, goto error);
 777         if (isl_int_is_one(vec->el[0])) {
 778                 isl_mat_free(U);
 779                 isl_basic_set_free(cone);
 780                 return vec;
 781         }
 782
 783         total = isl_basic_set_total_dim(cone);
 784         cone = isl_basic_set_preimage(cone, U);
 785         cone = isl_basic_set_remove_dims(cone, isl_dim_set,
 786                                          0, total - (vec->size - 1));
 787
 788         cone = shift_cone(cone, vec);
 789
 790         vec = rational_sample(cone);
 791         vec = isl_vec_ceil(vec);
 792         return vec;
 793 error:
 794         isl_mat_free(U);
 795         isl_vec_free(vec);
 796         isl_basic_set_free(cone);
 797         return NULL;
 798 }
 799
 800 /* Concatenate two integer vectors, i.e., two vectors with denominator
 801  * (stored in element 0) equal to 1.
 802  */
 803 static struct isl_vec *vec_concat(struct isl_vec *vec1, struct isl_vec *vec2)
 804 {
 805         struct isl_vec *vec;
 806
 807         if (!vec1 || !vec2)
 808                 goto error;
 809         isl_assert(vec1->ctx, vec1->size > 0, goto error);
 810         isl_assert(vec2->ctx, vec2->size > 0, goto error);
 811         isl_assert(vec1->ctx, isl_int_is_one(vec1->el[0]), goto error);
 812         isl_assert(vec2->ctx, isl_int_is_one(vec2->el[0]), goto error);
 813
 814         vec = isl_vec_alloc(vec1->ctx, vec1->size + vec2->size - 1);
 815         if (!vec)
 816                 goto error;
 817
 818         isl_seq_cpy(vec->el, vec1->el, vec1->size);
 819         isl_seq_cpy(vec->el + vec1->size, vec2->el + 1, vec2->size - 1);
 820
 821         isl_vec_free(vec1);
 822         isl_vec_free(vec2);
 823
 824         return vec;
 825 error:
 826         isl_vec_free(vec1);
 827         isl_vec_free(vec2);
 828         return NULL;
 829 }
 830
 831 /* Give a basic set "bset" with recession cone "cone", compute and
 832  * return an integer point in bset, if any.
 833  *
 834  * If the recession cone is full-dimensional, then we know that
 835  * bset contains an infinite number of integer points and it is
 836  * fairly easy to pick one of them.
 837  * If the recession cone is not full-dimensional, then we first
 838  * transform bset such that the bounded directions appear as
 839  * the first dimensions of the transformed basic set.
 840  * We do this by using a unimodular transformation that transforms
 841  * the equalities in the recession cone to equalities on the first
 842  * dimensions.
 843  *
 844  * The transformed set is then projected onto its bounded dimensions.
 845  * Note that to compute this projection, we can simply drop all constraints
 846  * involving any of the unbounded dimensions since these constraints
 847  * cannot be combined to produce a constraint on the bounded dimensions.
 848  * To see this, assume that there is such a combination of constraints
 849  * that produces a constraint on the bounded dimensions.  This means
 850  * that some combination of the unbounded dimensions has both an upper
 851  * bound and a lower bound in terms of the bounded dimensions, but then
 852  * this combination would be a bounded direction too and would have been
 853  * transformed into a bounded dimensions.
 854  *
 855  * We then compute a sample value in the bounded dimensions.
 856  * If no such value can be found, then the original set did not contain
 857  * any integer points and we are done.
 858  * Otherwise, we plug in the value we found in the bounded dimensions,
 859  * project out these bounded dimensions and end up with a set with
 860  * a full-dimensional recession cone.
 861  * A sample point in this set is computed by "rounding up" any
 862  * rational point in the set.
 863  *
 864  * The sample points in the bounded and unbounded dimensions are
 865  * then combined into a single sample point and transformed back
 866  * to the original space.
 867  */
 868 __isl_give isl_vec *isl_basic_set_sample_with_cone(
 869         __isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone)
 870 {
 871         unsigned total;
 872         unsigned cone_dim;
 873         struct isl_mat *M, *U;
 874         struct isl_vec *sample;
 875         struct isl_vec *cone_sample;
 876         struct isl_ctx *ctx;
 877         struct isl_basic_set *bounded;
 878
 879         if (!bset || !cone)
 880                 goto error;
 881
 882         ctx = bset->ctx;
 883         total = isl_basic_set_total_dim(cone);
 884         cone_dim = total - cone->n_eq;
 885
 886         M = isl_mat_sub_alloc(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
 887         M = isl_mat_left_hermite(M, 0, &U, NULL);
 888         if (!M)
 889                 goto error;
 890         isl_mat_free(M);
 891
 892         U = isl_mat_lin_to_aff(U);
 893         bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
 894
 895         bounded = isl_basic_set_copy(bset);
 896         bounded = isl_basic_set_drop_constraints_involving(bounded,
 897                                                    total - cone_dim, cone_dim);
 898         bounded = isl_basic_set_drop_dims(bounded, total - cone_dim, cone_dim);
 899         sample = sample_bounded(bounded);
 900         if (!sample || sample->size == 0) {
 901                 isl_basic_set_free(bset);
 902                 isl_basic_set_free(cone);
 903                 isl_mat_free(U);
 904                 return sample;
 905         }
 906         bset = plug_in(bset, isl_vec_copy(sample));
 907         cone_sample = rational_sample(bset);
 908         cone_sample = round_up_in_cone(cone_sample, cone, isl_mat_copy(U));
 909         sample = vec_concat(sample, cone_sample);
 910         sample = isl_mat_vec_product(U, sample);
 911         return sample;
 912 error:
 913         isl_basic_set_free(cone);
 914         isl_basic_set_free(bset);
 915         return NULL;
 916 }
 917
 918 static void vec_sum_of_neg(struct isl_vec *v, isl_int *s)
 919 {
 920         int i;
 921
 922         isl_int_set_si(*s, 0);
 923
 924         for (i = 0; i < v->size; ++i)
 925                 if (isl_int_is_neg(v->el[i]))
 926                         isl_int_add(*s, *s, v->el[i]);
 927 }
 928
 929 /* Given a tableau "tab", a tableau "tab_cone" that corresponds
 930  * to the recession cone and the inverse of a new basis U = inv(B),
 931  * with the unbounded directions in B last,
 932  * add constraints to "tab" that ensure any rational value
 933  * in the unbounded directions can be rounded up to an integer value.
 934  *
 935  * The new basis is given by x' = B x, i.e., x = U x'.
 936  * For any rational value of the last tab->n_unbounded coordinates
 937  * in the update tableau, the value that is obtained by rounding
 938  * up this value should be contained in the original tableau.
 939  * For any constraint "a x + c >= 0", we therefore need to add
 940  * a constraint "a x + c + s >= 0", with s the sum of all negative
 941  * entries in the last elements of "a U".
 942  *
 943  * Since we are not interested in the first entries of any of the "a U",
 944  * we first drop the columns of U that correpond to bounded directions.
 945  */
 946 static int tab_shift_cone(struct isl_tab *tab,
 947         struct isl_tab *tab_cone, struct isl_mat *U)
 948 {
 949         int i;
 950         isl_int v;
 951         struct isl_basic_set *bset = NULL;
 952
 953         if (tab && tab->n_unbounded == 0) {
 954                 isl_mat_free(U);
 955                 return 0;
 956         }
 957         isl_int_init(v);
 958         if (!tab || !tab_cone || !U)
 959                 goto error;
 960         bset = isl_tab_peek_bset(tab_cone);
 961         U = isl_mat_drop_cols(U, 0, tab->n_var - tab->n_unbounded);
 962         for (i = 0; i < bset->n_ineq; ++i) {
 963                 int ok;
 964                 struct isl_vec *row = NULL;
 965                 if (isl_tab_is_equality(tab_cone, tab_cone->n_eq + i))
 966                         continue;
 967                 row = isl_vec_alloc(bset->ctx, tab_cone->n_var);
 968                 if (!row)
 969                         goto error;
 970                 isl_seq_cpy(row->el, bset->ineq[i] + 1, tab_cone->n_var);
 971                 row = isl_vec_mat_product(row, isl_mat_copy(U));
 972                 if (!row)
 973                         goto error;
 974                 vec_sum_of_neg(row, &v);
 975                 isl_vec_free(row);
 976                 if (isl_int_is_zero(v))
 977                         continue;
 978                 tab = isl_tab_extend(tab, 1);
 979                 isl_int_add(bset->ineq[i][0], bset->ineq[i][0], v);
 980                 ok = isl_tab_add_ineq(tab, bset->ineq[i]) >= 0;
 981                 isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], v);
 982                 if (!ok)
 983                         goto error;
 984         }
 985
 986         isl_mat_free(U);
 987         isl_int_clear(v);
 988         return 0;
 989 error:
 990         isl_mat_free(U);
 991         isl_int_clear(v);
 992         return -1;
 993 }
 994
 995 /* Compute and return an initial basis for the possibly
 996  * unbounded tableau "tab".  "tab_cone" is a tableau
 997  * for the corresponding recession cone.
 998  * Additionally, add constraints to "tab" that ensure
 999  * that any rational value for the unbounded directions
1000  * can be rounded up to an integer value.
1001  *
1002  * If the tableau is bounded, i.e., if the recession cone
1003  * is zero-dimensional, then we just use inital_basis.
1004  * Otherwise, we construct a basis whose first directions
1005  * correspond to equalities, followed by bounded directions,
1006  * i.e., equalities in the recession cone.
1007  * The remaining directions are then unbounded.
1008  */
1009 int isl_tab_set_initial_basis_with_cone(struct isl_tab *tab,
1010         struct isl_tab *tab_cone)
1011 {
1012         struct isl_mat *eq;
1013         struct isl_mat *cone_eq;
1014         struct isl_mat *U, *Q;
1015
1016         if (!tab || !tab_cone)
1017                 return -1;
1018
1019         if (tab_cone->n_col == tab_cone->n_dead) {
1020                 tab->basis = initial_basis(tab);
1021                 return tab->basis ? 0 : -1;
1022         }
1023
1024         eq = tab_equalities(tab);
1025         if (!eq)
1026                 return -1;
1027         tab->n_zero = eq->n_row;
1028         cone_eq = tab_equalities(tab_cone);
1029         eq = isl_mat_concat(eq, cone_eq);
1030         if (!eq)
1031                 return -1;
1032         tab->n_unbounded = tab->n_var - (eq->n_row - tab->n_zero);
1033         eq = isl_mat_left_hermite(eq, 0, &U, &Q);
1034         if (!eq)
1035                 return -1;
1036         isl_mat_free(eq);
1037         tab->basis = isl_mat_lin_to_aff(Q);
1038         if (tab_shift_cone(tab, tab_cone, U) < 0)
1039                 return -1;
1040         if (!tab->basis)
1041                 return -1;
1042         return 0;
1043 }
1044
1045 /* Compute and return a sample point in bset using generalized basis
1046  * reduction.  We first check if the input set has a non-trivial
1047  * recession cone.  If so, we perform some extra preprocessing in
1048  * sample_with_cone.  Otherwise, we directly perform generalized basis
1049  * reduction.
1050  */
1051 static struct isl_vec *gbr_sample(struct isl_basic_set *bset)
1052 {
1053         unsigned dim;
1054         struct isl_basic_set *cone;
1055
1056         dim = isl_basic_set_total_dim(bset);
1057
1058         cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
1059         if (!cone)
1060                 goto error;
1061
1062         if (cone->n_eq < dim)
1063                 return isl_basic_set_sample_with_cone(bset, cone);
1064
1065         isl_basic_set_free(cone);
1066         return sample_bounded(bset);
1067 error:
1068         isl_basic_set_free(bset);
1069         return NULL;
1070 }
1071
1072 static struct isl_vec *pip_sample(struct isl_basic_set *bset)
1073 {
1074         struct isl_mat *T;
1075         struct isl_ctx *ctx;
1076         struct isl_vec *sample;
1077
1078         bset = isl_basic_set_skew_to_positive_orthant(bset, &T);
1079         if (!bset)
1080                 return NULL;
1081
1082         ctx = bset->ctx;
1083         sample = isl_pip_basic_set_sample(bset);
1084
1085         if (sample && sample->size != 0)
1086                 sample = isl_mat_vec_product(T, sample);
1087         else
1088                 isl_mat_free(T);
1089
1090         return sample;
1091 }
1092
1093 static struct isl_vec *basic_set_sample(struct isl_basic_set *bset, int bounded)
1094 {
1095         struct isl_ctx *ctx;
1096         unsigned dim;
1097         if (!bset)
1098                 return NULL;
1099
1100         ctx = bset->ctx;
1101         if (isl_basic_set_fast_is_empty(bset))
1102                 return empty_sample(bset);
1103
1104         dim = isl_basic_set_n_dim(bset);
1105         isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
1106         isl_assert(ctx, bset->n_div == 0, goto error);
1107
1108         if (bset->sample && bset->sample->size == 1 + dim) {
1109                 int contains = isl_basic_set_contains(bset, bset->sample);
1110                 if (contains < 0)
1111                         goto error;
1112                 if (contains) {
1113                         struct isl_vec *sample = isl_vec_copy(bset->sample);
1114                         isl_basic_set_free(bset);
1115                         return sample;
1116                 }
1117         }
1118         isl_vec_free(bset->sample);
1119         bset->sample = NULL;
1120
1121         if (bset->n_eq > 0)
1122                 return sample_eq(bset, bounded ? isl_basic_set_sample_bounded
1123                                                : isl_basic_set_sample_vec);
1124         if (dim == 0)
1125                 return zero_sample(bset);
1126         if (dim == 1)
1127                 return interval_sample(bset);
1128
1129         switch (bset->ctx->opt->ilp_solver) {
1130         case ISL_ILP_PIP:
1131                 return pip_sample(bset);
1132         case ISL_ILP_GBR:
1133                 return bounded ? sample_bounded(bset) : gbr_sample(bset);
1134         }
1135         isl_assert(bset->ctx, 0, );
1136 error:
1137         isl_basic_set_free(bset);
1138         return NULL;
1139 }
1140
1141 __isl_give isl_vec *isl_basic_set_sample_vec(__isl_take isl_basic_set *bset)
1142 {
1143         return basic_set_sample(bset, 0);
1144 }
1145
1146 /* Compute an integer sample in "bset", where the caller guarantees
1147  * that "bset" is bounded.
1148  */
1149 struct isl_vec *isl_basic_set_sample_bounded(struct isl_basic_set *bset)
1150 {
1151         return basic_set_sample(bset, 1);
1152 }
1153
1154 __isl_give isl_basic_set *isl_basic_set_from_vec(__isl_take isl_vec *vec)
1155 {
1156         int i;
1157         int k;
1158         struct isl_basic_set *bset = NULL;
1159         struct isl_ctx *ctx;
1160         unsigned dim;
1161
1162         if (!vec)
1163                 return NULL;
1164         ctx = vec->ctx;
1165         isl_assert(ctx, vec->size != 0, goto error);
1166
1167         bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
1168         if (!bset)
1169                 goto error;
1170         dim = isl_basic_set_n_dim(bset);
1171         for (i = dim - 1; i >= 0; --i) {
1172                 k = isl_basic_set_alloc_equality(bset);
1173                 if (k < 0)
1174                         goto error;
1175                 isl_seq_clr(bset->eq[k], 1 + dim);
1176                 isl_int_neg(bset->eq[k][0], vec->el[1 + i]);
1177                 isl_int_set(bset->eq[k][1 + i], vec->el[0]);
1178         }
1179         bset->sample = vec;
1180
1181         return bset;
1182 error:
1183         isl_basic_set_free(bset);
1184         isl_vec_free(vec);
1185         return NULL;
1186 }
1187
1188 __isl_give isl_basic_map *isl_basic_map_sample(__isl_take isl_basic_map *bmap)
1189 {
1190         struct isl_basic_set *bset;
1191         struct isl_vec *sample_vec;
1192
1193         bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
1194         sample_vec = isl_basic_set_sample_vec(bset);
1195         if (!sample_vec)
1196                 goto error;
1197         if (sample_vec->size == 0) {
1198                 struct isl_basic_map *sample;
1199                 sample = isl_basic_map_empty_like(bmap);
1200                 isl_vec_free(sample_vec);
1201                 isl_basic_map_free(bmap);
1202                 return sample;
1203         }
1204         bset = isl_basic_set_from_vec(sample_vec);
1205         return isl_basic_map_overlying_set(bset, bmap);
1206 error:
1207         isl_basic_map_free(bmap);
1208         return NULL;
1209 }
1210
1211 __isl_give isl_basic_map *isl_map_sample(__isl_take isl_map *map)
1212 {
1213         int i;
1214         isl_basic_map *sample = NULL;
1215
1216         if (!map)
1217                 goto error;
1218
1219         for (i = 0; i < map->n; ++i) {
1220                 sample = isl_basic_map_sample(isl_basic_map_copy(map->p[i]));
1221                 if (!sample)
1222                         goto error;
1223                 if (!ISL_F_ISSET(sample, ISL_BASIC_MAP_EMPTY))
1224                         break;
1225                 isl_basic_map_free(sample);
1226         }
1227         if (i == map->n)
1228                 sample = isl_basic_map_empty_like_map(map);
1229         isl_map_free(map);
1230         return sample;
1231 error:
1232         isl_map_free(map);
1233         return NULL;
1234 }
1235
1236 __isl_give isl_basic_set *isl_set_sample(__isl_take isl_set *set)
1237 {
1238         return (isl_basic_set *) isl_map_sample((isl_map *)set);
1239 }
1240
1241 __isl_give isl_point *isl_basic_set_sample_point(__isl_take isl_basic_set *bset)
1242 {
1243         isl_vec *vec;
1244         isl_dim *dim;
1245
1246         dim = isl_basic_set_get_dim(bset);
1247         bset = isl_basic_set_underlying_set(bset);
1248         vec = isl_basic_set_sample_vec(bset);
1249
1250         return isl_point_alloc(dim, vec);
1251 }
1252
1253 __isl_give isl_point *isl_set_sample_point(__isl_take isl_set *set)
1254 {
1255         int i;
1256         isl_point *pnt;
1257
1258         if (!set)
1259                 return NULL;
1260
1261         for (i = 0; i < set->n; ++i) {
1262                 pnt = isl_basic_set_sample_point(isl_basic_set_copy(set->p[i]));
1263                 if (!pnt)
1264                         goto error;
1265                 if (!isl_point_is_void(pnt))
1266                         break;
1267                 isl_point_free(pnt);
1268         }
1269         if (i == set->n)
1270                 pnt = isl_point_void(isl_set_get_dim(set));
1271
1272         isl_set_free(set);
1273         return pnt;
1274 error:
1275         isl_set_free(set);
1276         return NULL;
1277 }