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