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1/*2 * Copyright 2008-2009 Katholieke Universiteit Leuven3 *4 * Use of this software is governed by the MIT license5 *6 * Written by Sven Verdoolaege, K.U.Leuven, Departement7 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium8 */9 10#include <isl_ctx_private.h>11#include <isl_map_private.h>12#include "isl_sample.h"13#include <isl/vec.h>14#include <isl/mat.h>15#include <isl_seq.h>16#include "isl_equalities.h"17#include "isl_tab.h"18#include "isl_basis_reduction.h"19#include <isl_factorization.h>20#include <isl_point_private.h>21#include <isl_options_private.h>22#include <isl_vec_private.h>23 24#include <bset_from_bmap.c>25#include <set_to_map.c>26 27static __isl_give isl_vec *isl_basic_set_sample_bounded(28 __isl_take isl_basic_set *bset);29 30static __isl_give isl_vec *empty_sample(__isl_take isl_basic_set *bset)31{32 struct isl_vec *vec;33 34 vec = isl_vec_alloc(bset->ctx, 0);35 isl_basic_set_free(bset);36 return vec;37}38 39/* Construct a zero sample of the same dimension as bset.40 * As a special case, if bset is zero-dimensional, this41 * function creates a zero-dimensional sample point.42 */43static __isl_give isl_vec *zero_sample(__isl_take isl_basic_set *bset)44{45 isl_size dim;46 struct isl_vec *sample;47 48 dim = isl_basic_set_dim(bset, isl_dim_all);49 if (dim < 0)50 goto error;51 sample = isl_vec_alloc(bset->ctx, 1 + dim);52 if (sample) {53 isl_int_set_si(sample->el[0], 1);54 isl_seq_clr(sample->el + 1, dim);55 }56 isl_basic_set_free(bset);57 return sample;58error:59 isl_basic_set_free(bset);60 return NULL;61}62 63static __isl_give isl_vec *interval_sample(__isl_take isl_basic_set *bset)64{65 int i;66 isl_int t;67 struct isl_vec *sample;68 69 bset = isl_basic_set_simplify(bset);70 if (!bset)71 return NULL;72 if (isl_basic_set_plain_is_empty(bset))73 return empty_sample(bset);74 if (bset->n_eq == 0 && bset->n_ineq == 0)75 return zero_sample(bset);76 77 sample = isl_vec_alloc(bset->ctx, 2);78 if (!sample)79 goto error;80 if (!bset)81 return NULL;82 isl_int_set_si(sample->block.data[0], 1);83 84 if (bset->n_eq > 0) {85 isl_assert(bset->ctx, bset->n_eq == 1, goto error);86 isl_assert(bset->ctx, bset->n_ineq == 0, goto error);87 if (isl_int_is_one(bset->eq[0][1]))88 isl_int_neg(sample->el[1], bset->eq[0][0]);89 else {90 isl_assert(bset->ctx, isl_int_is_negone(bset->eq[0][1]),91 goto error);92 isl_int_set(sample->el[1], bset->eq[0][0]);93 }94 isl_basic_set_free(bset);95 return sample;96 }97 98 isl_int_init(t);99 if (isl_int_is_one(bset->ineq[0][1]))100 isl_int_neg(sample->block.data[1], bset->ineq[0][0]);101 else102 isl_int_set(sample->block.data[1], bset->ineq[0][0]);103 for (i = 1; i < bset->n_ineq; ++i) {104 isl_seq_inner_product(sample->block.data,105 bset->ineq[i], 2, &t);106 if (isl_int_is_neg(t))107 break;108 }109 isl_int_clear(t);110 if (i < bset->n_ineq) {111 isl_vec_free(sample);112 return empty_sample(bset);113 }114 115 isl_basic_set_free(bset);116 return sample;117error:118 isl_basic_set_free(bset);119 isl_vec_free(sample);120 return NULL;121}122 123/* Find a sample integer point, if any, in bset, which is known124 * to have equalities. If bset contains no integer points, then125 * return a zero-length vector.126 * We simply remove the known equalities, compute a sample127 * in the resulting bset, using the specified recurse function,128 * and then transform the sample back to the original space.129 */130static __isl_give isl_vec *sample_eq(__isl_take isl_basic_set *bset,131 __isl_give isl_vec *(*recurse)(__isl_take isl_basic_set *))132{133 struct isl_mat *T;134 struct isl_vec *sample;135 136 if (!bset)137 return NULL;138 139 bset = isl_basic_set_remove_equalities(bset, &T, NULL);140 sample = recurse(bset);141 if (!sample || sample->size == 0)142 isl_mat_free(T);143 else144 sample = isl_mat_vec_product(T, sample);145 return sample;146}147 148/* Return a matrix containing the equalities of the tableau149 * in constraint form. The tableau is assumed to have150 * an associated bset that has been kept up-to-date.151 */152static struct isl_mat *tab_equalities(struct isl_tab *tab)153{154 int i, j;155 int n_eq;156 struct isl_mat *eq;157 struct isl_basic_set *bset;158 159 if (!tab)160 return NULL;161 162 bset = isl_tab_peek_bset(tab);163 isl_assert(tab->mat->ctx, bset, return NULL);164 165 n_eq = tab->n_var - tab->n_col + tab->n_dead;166 if (tab->empty || n_eq == 0)167 return isl_mat_alloc(tab->mat->ctx, 0, tab->n_var);168 if (n_eq == tab->n_var)169 return isl_mat_identity(tab->mat->ctx, tab->n_var);170 171 eq = isl_mat_alloc(tab->mat->ctx, n_eq, tab->n_var);172 if (!eq)173 return NULL;174 for (i = 0, j = 0; i < tab->n_con; ++i) {175 if (tab->con[i].is_row)176 continue;177 if (tab->con[i].index >= 0 && tab->con[i].index >= tab->n_dead)178 continue;179 if (i < bset->n_eq)180 isl_seq_cpy(eq->row[j], bset->eq[i] + 1, tab->n_var);181 else182 isl_seq_cpy(eq->row[j],183 bset->ineq[i - bset->n_eq] + 1, tab->n_var);184 ++j;185 }186 isl_assert(bset->ctx, j == n_eq, goto error);187 return eq;188error:189 isl_mat_free(eq);190 return NULL;191}192 193/* Compute and return an initial basis for the bounded tableau "tab".194 *195 * If the tableau is either full-dimensional or zero-dimensional,196 * the we simply return an identity matrix.197 * Otherwise, we construct a basis whose first directions correspond198 * to equalities.199 */200static struct isl_mat *initial_basis(struct isl_tab *tab)201{202 int n_eq;203 struct isl_mat *eq;204 struct isl_mat *Q;205 206 tab->n_unbounded = 0;207 tab->n_zero = n_eq = tab->n_var - tab->n_col + tab->n_dead;208 if (tab->empty || n_eq == 0 || n_eq == tab->n_var)209 return isl_mat_identity(tab->mat->ctx, 1 + tab->n_var);210 211 eq = tab_equalities(tab);212 eq = isl_mat_left_hermite(eq, 0, NULL, &Q);213 if (!eq)214 return NULL;215 isl_mat_free(eq);216 217 Q = isl_mat_lin_to_aff(Q);218 return Q;219}220 221/* Compute the minimum of the current ("level") basis row over "tab"222 * and store the result in position "level" of "min".223 *224 * This function assumes that at least one more row and at least225 * one more element in the constraint array are available in the tableau.226 */227static enum isl_lp_result compute_min(isl_ctx *ctx, struct isl_tab *tab,228 __isl_keep isl_vec *min, int level)229{230 return isl_tab_min(tab, tab->basis->row[1 + level],231 ctx->one, &min->el[level], NULL, 0);232}233 234/* Compute the maximum of the current ("level") basis row over "tab"235 * and store the result in position "level" of "max".236 *237 * This function assumes that at least one more row and at least238 * one more element in the constraint array are available in the tableau.239 */240static enum isl_lp_result compute_max(isl_ctx *ctx, struct isl_tab *tab,241 __isl_keep isl_vec *max, int level)242{243 enum isl_lp_result res;244 unsigned dim = tab->n_var;245 246 isl_seq_neg(tab->basis->row[1 + level] + 1,247 tab->basis->row[1 + level] + 1, dim);248 res = isl_tab_min(tab, tab->basis->row[1 + level],249 ctx->one, &max->el[level], NULL, 0);250 isl_seq_neg(tab->basis->row[1 + level] + 1,251 tab->basis->row[1 + level] + 1, dim);252 isl_int_neg(max->el[level], max->el[level]);253 254 return res;255}256 257/* Perform a greedy search for an integer point in the set represented258 * by "tab", given that the minimal rational value (rounded up to the259 * nearest integer) at "level" is smaller than the maximal rational260 * value (rounded down to the nearest integer).261 *262 * Return 1 if we have found an integer point (if tab->n_unbounded > 0263 * then we may have only found integer values for the bounded dimensions264 * and it is the responsibility of the caller to extend this solution265 * to the unbounded dimensions).266 * Return 0 if greedy search did not result in a solution.267 * Return -1 if some error occurred.268 *269 * We assign a value half-way between the minimum and the maximum270 * to the current dimension and check if the minimal value of the271 * next dimension is still smaller than (or equal) to the maximal value.272 * We continue this process until either273 * - the minimal value (rounded up) is greater than the maximal value274 * (rounded down). In this case, greedy search has failed.275 * - we have exhausted all bounded dimensions, meaning that we have276 * found a solution.277 * - the sample value of the tableau is integral.278 * - some error has occurred.279 */280static int greedy_search(isl_ctx *ctx, struct isl_tab *tab,281 __isl_keep isl_vec *min, __isl_keep isl_vec *max, int level)282{283 struct isl_tab_undo *snap;284 enum isl_lp_result res;285 286 snap = isl_tab_snap(tab);287 288 do {289 isl_int_add(tab->basis->row[1 + level][0],290 min->el[level], max->el[level]);291 isl_int_fdiv_q_ui(tab->basis->row[1 + level][0],292 tab->basis->row[1 + level][0], 2);293 isl_int_neg(tab->basis->row[1 + level][0],294 tab->basis->row[1 + level][0]);295 if (isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]) < 0)296 return -1;297 isl_int_set_si(tab->basis->row[1 + level][0], 0);298 299 if (++level >= tab->n_var - tab->n_unbounded)300 return 1;301 if (isl_tab_sample_is_integer(tab))302 return 1;303 304 res = compute_min(ctx, tab, min, level);305 if (res == isl_lp_error)306 return -1;307 if (res != isl_lp_ok)308 isl_die(ctx, isl_error_internal,309 "expecting bounded rational solution",310 return -1);311 res = compute_max(ctx, tab, max, level);312 if (res == isl_lp_error)313 return -1;314 if (res != isl_lp_ok)315 isl_die(ctx, isl_error_internal,316 "expecting bounded rational solution",317 return -1);318 } while (isl_int_le(min->el[level], max->el[level]));319 320 if (isl_tab_rollback(tab, snap) < 0)321 return -1;322 323 return 0;324}325 326/* Given a tableau representing a set, find and return327 * an integer point in the set, if there is any.328 *329 * We perform a depth first search330 * for an integer point, by scanning all possible values in the range331 * attained by a basis vector, where an initial basis may have been set332 * by the calling function. Otherwise an initial basis that exploits333 * the equalities in the tableau is created.334 * tab->n_zero is currently ignored and is clobbered by this function.335 *336 * The tableau is allowed to have unbounded direction, but then337 * the calling function needs to set an initial basis, with the338 * unbounded directions last and with tab->n_unbounded set339 * to the number of unbounded directions.340 * Furthermore, the calling functions needs to add shifted copies341 * of all constraints involving unbounded directions to ensure342 * that any feasible rational value in these directions can be rounded343 * up to yield a feasible integer value.344 * In particular, let B define the given basis x' = B x345 * and let T be the inverse of B, i.e., X = T x'.346 * Let a x + c >= 0 be a constraint of the set represented by the tableau,347 * or a T x' + c >= 0 in terms of the given basis. Assume that348 * the bounded directions have an integer value, then we can safely349 * round up the values for the unbounded directions if we make sure350 * that x' not only satisfies the original constraint, but also351 * the constraint "a T x' + c + s >= 0" with s the sum of all352 * negative values in the last n_unbounded entries of "a T".353 * The calling function therefore needs to add the constraint354 * a x + c + s >= 0. The current function then scans the first355 * directions for an integer value and once those have been found,356 * it can compute "T ceil(B x)" to yield an integer point in the set.357 * Note that during the search, the first rows of B may be changed358 * by a basis reduction, but the last n_unbounded rows of B remain359 * unaltered and are also not mixed into the first rows.360 *361 * The search is implemented iteratively. "level" identifies the current362 * basis vector. "init" is true if we want the first value at the current363 * level and false if we want the next value.364 *365 * At the start of each level, we first check if we can find a solution366 * using greedy search. If not, we continue with the exhaustive search.367 *368 * The initial basis is the identity matrix. If the range in some direction369 * contains more than one integer value, we perform basis reduction based370 * on the value of ctx->opt->gbr371 * - ISL_GBR_NEVER: never perform basis reduction372 * - ISL_GBR_ONCE: only perform basis reduction the first373 * time such a range is encountered374 * - ISL_GBR_ALWAYS: always perform basis reduction when375 * such a range is encountered376 *377 * When ctx->opt->gbr is set to ISL_GBR_ALWAYS, then we allow the basis378 * reduction computation to return early. That is, as soon as it379 * finds a reasonable first direction.380 */ 381__isl_give isl_vec *isl_tab_sample(struct isl_tab *tab)382{383 unsigned dim;384 unsigned gbr;385 struct isl_ctx *ctx;386 struct isl_vec *sample;387 struct isl_vec *min;388 struct isl_vec *max;389 enum isl_lp_result res;390 int level;391 int init;392 int reduced;393 struct isl_tab_undo **snap;394 395 if (!tab)396 return NULL;397 if (tab->empty)398 return isl_vec_alloc(tab->mat->ctx, 0);399 400 if (!tab->basis)401 tab->basis = initial_basis(tab);402 if (!tab->basis)403 return NULL;404 isl_assert(tab->mat->ctx, tab->basis->n_row == tab->n_var + 1,405 return NULL);406 isl_assert(tab->mat->ctx, tab->basis->n_col == tab->n_var + 1,407 return NULL);408 409 ctx = tab->mat->ctx;410 dim = tab->n_var;411 gbr = ctx->opt->gbr;412 413 if (tab->n_unbounded == tab->n_var) {414 sample = isl_tab_get_sample_value(tab);415 sample = isl_mat_vec_product(isl_mat_copy(tab->basis), sample);416 sample = isl_vec_ceil(sample);417 sample = isl_mat_vec_inverse_product(isl_mat_copy(tab->basis),418 sample);419 return sample;420 }421 422 if (isl_tab_extend_cons(tab, dim + 1) < 0)423 return NULL;424 425 min = isl_vec_alloc(ctx, dim);426 max = isl_vec_alloc(ctx, dim);427 snap = isl_alloc_array(ctx, struct isl_tab_undo *, dim);428 429 if (!min || !max || !snap)430 goto error;431 432 level = 0;433 init = 1;434 reduced = 0;435 436 while (level >= 0) {437 if (init) {438 int choice;439 440 res = compute_min(ctx, tab, min, level);441 if (res == isl_lp_error)442 goto error;443 if (res != isl_lp_ok)444 isl_die(ctx, isl_error_internal,445 "expecting bounded rational solution",446 goto error);447 if (isl_tab_sample_is_integer(tab))448 break;449 res = compute_max(ctx, tab, max, level);450 if (res == isl_lp_error)451 goto error;452 if (res != isl_lp_ok)453 isl_die(ctx, isl_error_internal,454 "expecting bounded rational solution",455 goto error);456 if (isl_tab_sample_is_integer(tab))457 break;458 choice = isl_int_lt(min->el[level], max->el[level]);459 if (choice) {460 int g;461 g = greedy_search(ctx, tab, min, max, level);462 if (g < 0)463 goto error;464 if (g)465 break;466 }467 if (!reduced && choice &&468 ctx->opt->gbr != ISL_GBR_NEVER) {469 unsigned gbr_only_first;470 if (ctx->opt->gbr == ISL_GBR_ONCE)471 ctx->opt->gbr = ISL_GBR_NEVER;472 tab->n_zero = level;473 gbr_only_first = ctx->opt->gbr_only_first;474 ctx->opt->gbr_only_first =475 ctx->opt->gbr == ISL_GBR_ALWAYS;476 tab = isl_tab_compute_reduced_basis(tab);477 ctx->opt->gbr_only_first = gbr_only_first;478 if (!tab || !tab->basis)479 goto error;480 reduced = 1;481 continue;482 }483 reduced = 0;484 snap[level] = isl_tab_snap(tab);485 } else486 isl_int_add_ui(min->el[level], min->el[level], 1);487 488 if (isl_int_gt(min->el[level], max->el[level])) {489 level--;490 init = 0;491 if (level >= 0)492 if (isl_tab_rollback(tab, snap[level]) < 0)493 goto error;494 continue;495 }496 isl_int_neg(tab->basis->row[1 + level][0], min->el[level]);497 if (isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]) < 0)498 goto error;499 isl_int_set_si(tab->basis->row[1 + level][0], 0);500 if (level + tab->n_unbounded < dim - 1) {501 ++level;502 init = 1;503 continue;504 }505 break;506 }507 508 if (level >= 0) {509 sample = isl_tab_get_sample_value(tab);510 if (!sample)511 goto error;512 if (tab->n_unbounded && !isl_int_is_one(sample->el[0])) {513 sample = isl_mat_vec_product(isl_mat_copy(tab->basis),514 sample);515 sample = isl_vec_ceil(sample);516 sample = isl_mat_vec_inverse_product(517 isl_mat_copy(tab->basis), sample);518 }519 } else520 sample = isl_vec_alloc(ctx, 0);521 522 ctx->opt->gbr = gbr;523 isl_vec_free(min);524 isl_vec_free(max);525 free(snap);526 return sample;527error:528 ctx->opt->gbr = gbr;529 isl_vec_free(min);530 isl_vec_free(max);531 free(snap);532 return NULL;533}534 535static __isl_give isl_vec *sample_bounded(__isl_take isl_basic_set *bset);536 537/* Internal data for factored_sample.538 * "sample" collects the sample and may get reset to a zero-length vector539 * signaling the absence of a sample vector.540 * "pos" is the position of the contribution of the next factor.541 */542struct isl_factored_sample_data {543 isl_vec *sample;544 int pos;545};546 547/* isl_factorizer_every_factor_basic_set callback that extends548 * the sample in data->sample with the contribution549 * of the factor "bset".550 * If "bset" turns out to be empty, then the product is empty too and551 * no further factors need to be considered.552 */553static isl_bool factor_sample(__isl_keep isl_basic_set *bset, void *user)554{555 struct isl_factored_sample_data *data = user;556 isl_vec *sample;557 isl_size n;558 559 n = isl_basic_set_dim(bset, isl_dim_set);560 if (n < 0)561 return isl_bool_error;562 563 sample = sample_bounded(isl_basic_set_copy(bset));564 if (!sample)565 return isl_bool_error;566 if (sample->size == 0) {567 isl_vec_free(data->sample);568 data->sample = sample;569 return isl_bool_false;570 }571 isl_seq_cpy(data->sample->el + data->pos, sample->el + 1, n);572 isl_vec_free(sample);573 data->pos += n;574 575 return isl_bool_true;576}577 578/* Compute a sample point of the given basic set, based on the given,579 * non-trivial factorization.580 */581static __isl_give isl_vec *factored_sample(__isl_take isl_basic_set *bset,582 __isl_take isl_factorizer *f)583{584 struct isl_factored_sample_data data = { NULL };585 isl_ctx *ctx;586 isl_size total;587 isl_bool every;588 589 ctx = isl_basic_set_get_ctx(bset);590 total = isl_basic_set_dim(bset, isl_dim_all);591 if (!ctx || total < 0)592 goto error;593 594 data.sample = isl_vec_alloc(ctx, 1 + total);595 if (!data.sample)596 goto error;597 isl_int_set_si(data.sample->el[0], 1);598 data.pos = 1;599 600 every = isl_factorizer_every_factor_basic_set(f, &factor_sample, &data);601 if (every < 0) {602 data.sample = isl_vec_free(data.sample);603 } else if (every) {604 isl_morph *morph;605 606 morph = isl_morph_inverse(isl_morph_copy(f->morph));607 data.sample = isl_morph_vec(morph, data.sample);608 }609 610 isl_basic_set_free(bset);611 isl_factorizer_free(f);612 return data.sample;613error:614 isl_basic_set_free(bset);615 isl_factorizer_free(f);616 isl_vec_free(data.sample);617 return NULL;618}619 620/* Given a basic set that is known to be bounded, find and return621 * an integer point in the basic set, if there is any.622 *623 * After handling some trivial cases, we construct a tableau624 * and then use isl_tab_sample to find a sample, passing it625 * the identity matrix as initial basis.626 */ 627static __isl_give isl_vec *sample_bounded(__isl_take isl_basic_set *bset)628{629 isl_size dim;630 struct isl_vec *sample;631 struct isl_tab *tab = NULL;632 isl_factorizer *f;633 634 if (!bset)635 return NULL;636 637 if (isl_basic_set_plain_is_empty(bset))638 return empty_sample(bset);639 640 dim = isl_basic_set_dim(bset, isl_dim_all);641 if (dim < 0)642 bset = isl_basic_set_free(bset);643 if (dim == 0)644 return zero_sample(bset);645 if (dim == 1)646 return interval_sample(bset);647 if (bset->n_eq > 0)648 return sample_eq(bset, sample_bounded);649 650 f = isl_basic_set_factorizer(bset);651 if (!f)652 goto error;653 if (f->n_group != 0)654 return factored_sample(bset, f);655 isl_factorizer_free(f);656 657 tab = isl_tab_from_basic_set(bset, 1);658 if (tab && tab->empty) {659 isl_tab_free(tab);660 ISL_F_SET(bset, ISL_BASIC_SET_EMPTY);661 sample = isl_vec_alloc(isl_basic_set_get_ctx(bset), 0);662 isl_basic_set_free(bset);663 return sample;664 }665 666 if (!ISL_F_ISSET(bset, ISL_BASIC_SET_NO_IMPLICIT))667 if (isl_tab_detect_implicit_equalities(tab) < 0)668 goto error;669 670 sample = isl_tab_sample(tab);671 if (!sample)672 goto error;673 674 if (sample->size > 0) {675 isl_vec_free(bset->sample);676 bset->sample = isl_vec_copy(sample);677 }678 679 isl_basic_set_free(bset);680 isl_tab_free(tab);681 return sample;682error:683 isl_basic_set_free(bset);684 isl_tab_free(tab);685 return NULL;686}687 688/* Given a basic set "bset" and a value "sample" for the first coordinates689 * of bset, plug in these values and drop the corresponding coordinates.690 *691 * We do this by computing the preimage of the transformation692 *693 * [ 1 0 ]694 * x = [ s 0 ] x'695 * [ 0 I ]696 *697 * where [1 s] is the sample value and I is the identity matrix of the698 * appropriate dimension.699 */700static __isl_give isl_basic_set *plug_in(__isl_take isl_basic_set *bset,701 __isl_take isl_vec *sample)702{703 int i;704 isl_size total;705 struct isl_mat *T;706 707 total = isl_basic_set_dim(bset, isl_dim_all);708 if (total < 0 || !sample)709 goto error;710 711 T = isl_mat_alloc(bset->ctx, 1 + total, 1 + total - (sample->size - 1));712 if (!T)713 goto error;714 715 for (i = 0; i < sample->size; ++i) {716 isl_int_set(T->row[i][0], sample->el[i]);717 isl_seq_clr(T->row[i] + 1, T->n_col - 1);718 }719 for (i = 0; i < T->n_col - 1; ++i) {720 isl_seq_clr(T->row[sample->size + i], T->n_col);721 isl_int_set_si(T->row[sample->size + i][1 + i], 1);722 }723 isl_vec_free(sample);724 725 bset = isl_basic_set_preimage(bset, T);726 return bset;727error:728 isl_basic_set_free(bset);729 isl_vec_free(sample);730 return NULL;731}732 733/* Given a basic set "bset", return any (possibly non-integer) point734 * in the basic set.735 */736static __isl_give isl_vec *rational_sample(__isl_take isl_basic_set *bset)737{738 struct isl_tab *tab;739 struct isl_vec *sample;740 741 if (!bset)742 return NULL;743 744 tab = isl_tab_from_basic_set(bset, 0);745 sample = isl_tab_get_sample_value(tab);746 isl_tab_free(tab);747 748 isl_basic_set_free(bset);749 750 return sample;751}752 753/* Given a linear cone "cone" and a rational point "vec",754 * construct a polyhedron with shifted copies of the constraints in "cone",755 * i.e., a polyhedron with "cone" as its recession cone, such that each756 * point x in this polyhedron is such that the unit box positioned at x757 * lies entirely inside the affine cone 'vec + cone'.758 * Any rational point in this polyhedron may therefore be rounded up759 * to yield an integer point that lies inside said affine cone.760 *761 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational762 * point "vec" by v/d.763 * Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given764 * by <a_i, x> - b/d >= 0.765 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.766 * We prefer this polyhedron over the actual affine cone because it doesn't767 * require a scaling of the constraints.768 * If each of the vertices of the unit cube positioned at x lies inside769 * this polyhedron, then the whole unit cube at x lies inside the affine cone.770 * We therefore impose that x' = x + \sum e_i, for any selection of unit771 * vectors lies inside the polyhedron, i.e.,772 *773 * <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0774 *775 * The most stringent of these constraints is the one that selects776 * all negative a_i, so the polyhedron we are looking for has constraints777 *778 * <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0779 *780 * Note that if cone were known to have only non-negative rays781 * (which can be accomplished by a unimodular transformation),782 * then we would only have to check the points x' = x + e_i783 * and we only have to add the smallest negative a_i (if any)784 * instead of the sum of all negative a_i.785 */786static __isl_give isl_basic_set *shift_cone(__isl_take isl_basic_set *cone,787 __isl_take isl_vec *vec)788{789 int i, j, k;790 isl_size total;791 792 struct isl_basic_set *shift = NULL;793 794 total = isl_basic_set_dim(cone, isl_dim_all);795 if (total < 0 || !vec)796 goto error;797 798 isl_assert(cone->ctx, cone->n_eq == 0, goto error);799 800 shift = isl_basic_set_alloc_space(isl_basic_set_get_space(cone),801 0, 0, cone->n_ineq);802 803 for (i = 0; i < cone->n_ineq; ++i) {804 k = isl_basic_set_alloc_inequality(shift);805 if (k < 0)806 goto error;807 isl_seq_cpy(shift->ineq[k] + 1, cone->ineq[i] + 1, total);808 isl_seq_inner_product(shift->ineq[k] + 1, vec->el + 1, total,809 &shift->ineq[k][0]);810 isl_int_cdiv_q(shift->ineq[k][0],811 shift->ineq[k][0], vec->el[0]);812 isl_int_neg(shift->ineq[k][0], shift->ineq[k][0]);813 for (j = 0; j < total; ++j) {814 if (isl_int_is_nonneg(shift->ineq[k][1 + j]))815 continue;816 isl_int_add(shift->ineq[k][0],817 shift->ineq[k][0], shift->ineq[k][1 + j]);818 }819 }820 821 isl_basic_set_free(cone);822 isl_vec_free(vec);823 824 return isl_basic_set_finalize(shift);825error:826 isl_basic_set_free(shift);827 isl_basic_set_free(cone);828 isl_vec_free(vec);829 return NULL;830}831 832/* Given a rational point vec in a (transformed) basic set,833 * such that cone is the recession cone of the original basic set,834 * "round up" the rational point to an integer point.835 *836 * We first check if the rational point just happens to be integer.837 * If not, we transform the cone in the same way as the basic set,838 * pick a point x in this cone shifted to the rational point such that839 * the whole unit cube at x is also inside this affine cone.840 * Then we simply round up the coordinates of x and return the841 * resulting integer point.842 */843static __isl_give isl_vec *round_up_in_cone(__isl_take isl_vec *vec,844 __isl_take isl_basic_set *cone, __isl_take isl_mat *U)845{846 isl_size total;847 848 if (!vec || !cone || !U)849 goto error;850 851 isl_assert(vec->ctx, vec->size != 0, goto error);852 if (isl_int_is_one(vec->el[0])) {853 isl_mat_free(U);854 isl_basic_set_free(cone);855 return vec;856 }857 858 total = isl_basic_set_dim(cone, isl_dim_all);859 if (total < 0)860 goto error;861 cone = isl_basic_set_preimage(cone, U);862 cone = isl_basic_set_remove_dims(cone, isl_dim_set,863 0, total - (vec->size - 1));864 865 cone = shift_cone(cone, vec);866 867 vec = rational_sample(cone);868 vec = isl_vec_ceil(vec);869 return vec;870error:871 isl_mat_free(U);872 isl_vec_free(vec);873 isl_basic_set_free(cone);874 return NULL;875}876 877/* Concatenate two integer vectors, i.e., two vectors with denominator878 * (stored in element 0) equal to 1.879 */880static __isl_give isl_vec *vec_concat(__isl_take isl_vec *vec1,881 __isl_take isl_vec *vec2)882{883 struct isl_vec *vec;884 885 if (!vec1 || !vec2)886 goto error;887 isl_assert(vec1->ctx, vec1->size > 0, goto error);888 isl_assert(vec2->ctx, vec2->size > 0, goto error);889 isl_assert(vec1->ctx, isl_int_is_one(vec1->el[0]), goto error);890 isl_assert(vec2->ctx, isl_int_is_one(vec2->el[0]), goto error);891 892 vec = isl_vec_alloc(vec1->ctx, vec1->size + vec2->size - 1);893 if (!vec)894 goto error;895 896 isl_seq_cpy(vec->el, vec1->el, vec1->size);897 isl_seq_cpy(vec->el + vec1->size, vec2->el + 1, vec2->size - 1);898 899 isl_vec_free(vec1);900 isl_vec_free(vec2);901 902 return vec;903error:904 isl_vec_free(vec1);905 isl_vec_free(vec2);906 return NULL;907}908 909/* Give a basic set "bset" with recession cone "cone", compute and910 * return an integer point in bset, if any.911 *912 * If the recession cone is full-dimensional, then we know that913 * bset contains an infinite number of integer points and it is914 * fairly easy to pick one of them.915 * If the recession cone is not full-dimensional, then we first916 * transform bset such that the bounded directions appear as917 * the first dimensions of the transformed basic set.918 * We do this by using a unimodular transformation that transforms919 * the equalities in the recession cone to equalities on the first920 * dimensions.921 *922 * The transformed set is then projected onto its bounded dimensions.923 * Note that to compute this projection, we can simply drop all constraints924 * involving any of the unbounded dimensions since these constraints925 * cannot be combined to produce a constraint on the bounded dimensions.926 * To see this, assume that there is such a combination of constraints927 * that produces a constraint on the bounded dimensions. This means928 * that some combination of the unbounded dimensions has both an upper929 * bound and a lower bound in terms of the bounded dimensions, but then930 * this combination would be a bounded direction too and would have been931 * transformed into a bounded dimensions.932 *933 * We then compute a sample value in the bounded dimensions.934 * If no such value can be found, then the original set did not contain935 * any integer points and we are done.936 * Otherwise, we plug in the value we found in the bounded dimensions,937 * project out these bounded dimensions and end up with a set with938 * a full-dimensional recession cone.939 * A sample point in this set is computed by "rounding up" any940 * rational point in the set.941 *942 * The sample points in the bounded and unbounded dimensions are943 * then combined into a single sample point and transformed back944 * to the original space.945 */946__isl_give isl_vec *isl_basic_set_sample_with_cone(947 __isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone)948{949 isl_size total;950 unsigned cone_dim;951 struct isl_mat *M, *U;952 struct isl_vec *sample;953 struct isl_vec *cone_sample;954 struct isl_ctx *ctx;955 struct isl_basic_set *bounded;956 957 total = isl_basic_set_dim(cone, isl_dim_all);958 if (!bset || total < 0)959 goto error;960 961 ctx = isl_basic_set_get_ctx(bset);962 cone_dim = total - cone->n_eq;963 964 M = isl_mat_sub_alloc6(ctx, cone->eq, 0, cone->n_eq, 1, total);965 M = isl_mat_left_hermite(M, 0, &U, NULL);966 if (!M)967 goto error;968 isl_mat_free(M);969 970 U = isl_mat_lin_to_aff(U);971 bset = isl_basic_set_preimage(bset, isl_mat_copy(U));972 973 bounded = isl_basic_set_copy(bset);974 bounded = isl_basic_set_drop_constraints_involving(bounded,975 total - cone_dim, cone_dim);976 bounded = isl_basic_set_drop_dims(bounded, total - cone_dim, cone_dim);977 sample = sample_bounded(bounded);978 if (!sample || sample->size == 0) {979 isl_basic_set_free(bset);980 isl_basic_set_free(cone);981 isl_mat_free(U);982 return sample;983 }984 bset = plug_in(bset, isl_vec_copy(sample));985 cone_sample = rational_sample(bset);986 cone_sample = round_up_in_cone(cone_sample, cone, isl_mat_copy(U));987 sample = vec_concat(sample, cone_sample);988 sample = isl_mat_vec_product(U, sample);989 return sample;990error:991 isl_basic_set_free(cone);992 isl_basic_set_free(bset);993 return NULL;994}995 996static void vec_sum_of_neg(__isl_keep isl_vec *v, isl_int *s)997{998 int i;999 1000 isl_int_set_si(*s, 0);1001 1002 for (i = 0; i < v->size; ++i)1003 if (isl_int_is_neg(v->el[i]))1004 isl_int_add(*s, *s, v->el[i]);1005}1006 1007/* Given a tableau "tab", a tableau "tab_cone" that corresponds1008 * to the recession cone and the inverse of a new basis U = inv(B),1009 * with the unbounded directions in B last,1010 * add constraints to "tab" that ensure any rational value1011 * in the unbounded directions can be rounded up to an integer value.1012 *1013 * The new basis is given by x' = B x, i.e., x = U x'.1014 * For any rational value of the last tab->n_unbounded coordinates1015 * in the update tableau, the value that is obtained by rounding1016 * up this value should be contained in the original tableau.1017 * For any constraint "a x + c >= 0", we therefore need to add1018 * a constraint "a x + c + s >= 0", with s the sum of all negative1019 * entries in the last elements of "a U".1020 *1021 * Since we are not interested in the first entries of any of the "a U",1022 * we first drop the columns of U that correpond to bounded directions.1023 */1024static int tab_shift_cone(struct isl_tab *tab,1025 struct isl_tab *tab_cone, struct isl_mat *U)1026{1027 int i;1028 isl_int v;1029 struct isl_basic_set *bset = NULL;1030 1031 if (tab && tab->n_unbounded == 0) {1032 isl_mat_free(U);1033 return 0;1034 }1035 isl_int_init(v);1036 if (!tab || !tab_cone || !U)1037 goto error;1038 bset = isl_tab_peek_bset(tab_cone);1039 U = isl_mat_drop_cols(U, 0, tab->n_var - tab->n_unbounded);1040 for (i = 0; i < bset->n_ineq; ++i) {1041 int ok;1042 struct isl_vec *row = NULL;1043 if (isl_tab_is_equality(tab_cone, tab_cone->n_eq + i))1044 continue;1045 row = isl_vec_alloc(bset->ctx, tab_cone->n_var);1046 if (!row)1047 goto error;1048 isl_seq_cpy(row->el, bset->ineq[i] + 1, tab_cone->n_var);1049 row = isl_vec_mat_product(row, isl_mat_copy(U));1050 if (!row)1051 goto error;1052 vec_sum_of_neg(row, &v);1053 isl_vec_free(row);1054 if (isl_int_is_zero(v))1055 continue;1056 if (isl_tab_extend_cons(tab, 1) < 0)1057 goto error;1058 isl_int_add(bset->ineq[i][0], bset->ineq[i][0], v);1059 ok = isl_tab_add_ineq(tab, bset->ineq[i]) >= 0;1060 isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], v);1061 if (!ok)1062 goto error;1063 }1064 1065 isl_mat_free(U);1066 isl_int_clear(v);1067 return 0;1068error:1069 isl_mat_free(U);1070 isl_int_clear(v);1071 return -1;1072}1073 1074/* Compute and return an initial basis for the possibly1075 * unbounded tableau "tab". "tab_cone" is a tableau1076 * for the corresponding recession cone.1077 * Additionally, add constraints to "tab" that ensure1078 * that any rational value for the unbounded directions1079 * can be rounded up to an integer value.1080 *1081 * If the tableau is bounded, i.e., if the recession cone1082 * is zero-dimensional, then we just use inital_basis.1083 * Otherwise, we construct a basis whose first directions1084 * correspond to equalities, followed by bounded directions,1085 * i.e., equalities in the recession cone.1086 * The remaining directions are then unbounded.1087 */1088int isl_tab_set_initial_basis_with_cone(struct isl_tab *tab,1089 struct isl_tab *tab_cone)1090{1091 struct isl_mat *eq;1092 struct isl_mat *cone_eq;1093 struct isl_mat *U, *Q;1094 1095 if (!tab || !tab_cone)1096 return -1;1097 1098 if (tab_cone->n_col == tab_cone->n_dead) {1099 tab->basis = initial_basis(tab);1100 return tab->basis ? 0 : -1;1101 }1102 1103 eq = tab_equalities(tab);1104 if (!eq)1105 return -1;1106 tab->n_zero = eq->n_row;1107 cone_eq = tab_equalities(tab_cone);1108 eq = isl_mat_concat(eq, cone_eq);1109 if (!eq)1110 return -1;1111 tab->n_unbounded = tab->n_var - (eq->n_row - tab->n_zero);1112 eq = isl_mat_left_hermite(eq, 0, &U, &Q);1113 if (!eq)1114 return -1;1115 isl_mat_free(eq);1116 tab->basis = isl_mat_lin_to_aff(Q);1117 if (tab_shift_cone(tab, tab_cone, U) < 0)1118 return -1;1119 if (!tab->basis)1120 return -1;1121 return 0;1122}1123 1124/* Compute and return a sample point in bset using generalized basis1125 * reduction. We first check if the input set has a non-trivial1126 * recession cone. If so, we perform some extra preprocessing in1127 * sample_with_cone. Otherwise, we directly perform generalized basis1128 * reduction.1129 */1130static __isl_give isl_vec *gbr_sample(__isl_take isl_basic_set *bset)1131{1132 isl_size dim;1133 struct isl_basic_set *cone;1134 1135 dim = isl_basic_set_dim(bset, isl_dim_all);1136 if (dim < 0)1137 goto error;1138 1139 cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));1140 if (!cone)1141 goto error;1142 1143 if (cone->n_eq < dim)1144 return isl_basic_set_sample_with_cone(bset, cone);1145 1146 isl_basic_set_free(cone);1147 return sample_bounded(bset);1148error:1149 isl_basic_set_free(bset);1150 return NULL;1151}1152 1153static __isl_give isl_vec *basic_set_sample(__isl_take isl_basic_set *bset,1154 int bounded)1155{1156 isl_size dim;1157 if (!bset)1158 return NULL;1159 1160 if (isl_basic_set_plain_is_empty(bset))1161 return empty_sample(bset);1162 1163 dim = isl_basic_set_dim(bset, isl_dim_set);1164 if (dim < 0 ||1165 isl_basic_set_check_no_params(bset) < 0 ||1166 isl_basic_set_check_no_locals(bset) < 0)1167 goto error;1168 1169 if (bset->sample && bset->sample->size == 1 + dim) {1170 int contains = isl_basic_set_contains(bset, bset->sample);1171 if (contains < 0)1172 goto error;1173 if (contains) {1174 struct isl_vec *sample = isl_vec_copy(bset->sample);1175 isl_basic_set_free(bset);1176 return sample;1177 }1178 }1179 isl_vec_free(bset->sample);1180 bset->sample = NULL;1181 1182 if (bset->n_eq > 0)1183 return sample_eq(bset, bounded ? isl_basic_set_sample_bounded1184 : isl_basic_set_sample_vec);1185 if (dim == 0)1186 return zero_sample(bset);1187 if (dim == 1)1188 return interval_sample(bset);1189 1190 return bounded ? sample_bounded(bset) : gbr_sample(bset);1191error:1192 isl_basic_set_free(bset);1193 return NULL;1194}1195 1196__isl_give isl_vec *isl_basic_set_sample_vec(__isl_take isl_basic_set *bset)1197{1198 return basic_set_sample(bset, 0);1199}1200 1201/* Compute an integer sample in "bset", where the caller guarantees1202 * that "bset" is bounded.1203 */1204__isl_give isl_vec *isl_basic_set_sample_bounded(__isl_take isl_basic_set *bset)1205{1206 return basic_set_sample(bset, 1);1207}1208 1209__isl_give isl_basic_set *isl_basic_set_from_vec(__isl_take isl_vec *vec)1210{1211 int i;1212 int k;1213 struct isl_basic_set *bset = NULL;1214 struct isl_ctx *ctx;1215 isl_size dim;1216 1217 if (!vec)1218 return NULL;1219 ctx = vec->ctx;1220 isl_assert(ctx, vec->size != 0, goto error);1221 1222 bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);1223 dim = isl_basic_set_dim(bset, isl_dim_set);1224 if (dim < 0)1225 goto error;1226 for (i = dim - 1; i >= 0; --i) {1227 k = isl_basic_set_alloc_equality(bset);1228 if (k < 0)1229 goto error;1230 isl_seq_clr(bset->eq[k], 1 + dim);1231 isl_int_neg(bset->eq[k][0], vec->el[1 + i]);1232 isl_int_set(bset->eq[k][1 + i], vec->el[0]);1233 }1234 bset->sample = vec;1235 1236 return bset;1237error:1238 isl_basic_set_free(bset);1239 isl_vec_free(vec);1240 return NULL;1241}1242 1243__isl_give isl_basic_map *isl_basic_map_sample(__isl_take isl_basic_map *bmap)1244{1245 struct isl_basic_set *bset;1246 struct isl_vec *sample_vec;1247 1248 bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));1249 sample_vec = isl_basic_set_sample_vec(bset);1250 if (!sample_vec)1251 goto error;1252 if (sample_vec->size == 0) {1253 isl_vec_free(sample_vec);1254 return isl_basic_map_set_to_empty(bmap);1255 }1256 isl_vec_free(bmap->sample);1257 bmap->sample = isl_vec_copy(sample_vec);1258 bset = isl_basic_set_from_vec(sample_vec);1259 return isl_basic_map_overlying_set(bset, bmap);1260error:1261 isl_basic_map_free(bmap);1262 return NULL;1263}1264 1265__isl_give isl_basic_set *isl_basic_set_sample(__isl_take isl_basic_set *bset)1266{1267 return isl_basic_map_sample(bset);1268}1269 1270__isl_give isl_basic_map *isl_map_sample(__isl_take isl_map *map)1271{1272 int i;1273 isl_basic_map *sample = NULL;1274 1275 if (!map)1276 goto error;1277 1278 for (i = 0; i < map->n; ++i) {1279 sample = isl_basic_map_sample(isl_basic_map_copy(map->p[i]));1280 if (!sample)1281 goto error;1282 if (!ISL_F_ISSET(sample, ISL_BASIC_MAP_EMPTY))1283 break;1284 isl_basic_map_free(sample);1285 }1286 if (i == map->n)1287 sample = isl_basic_map_empty(isl_map_get_space(map));1288 isl_map_free(map);1289 return sample;1290error:1291 isl_map_free(map);1292 return NULL;1293}1294 1295__isl_give isl_basic_set *isl_set_sample(__isl_take isl_set *set)1296{1297 return bset_from_bmap(isl_map_sample(set_to_map(set)));1298}1299 1300__isl_give isl_point *isl_basic_set_sample_point(__isl_take isl_basic_set *bset)1301{1302 isl_vec *vec;1303 isl_space *space;1304 1305 space = isl_basic_set_get_space(bset);1306 bset = isl_basic_set_underlying_set(bset);1307 vec = isl_basic_set_sample_vec(bset);1308 1309 return isl_point_alloc(space, vec);1310}1311 1312__isl_give isl_point *isl_set_sample_point(__isl_take isl_set *set)1313{1314 int i;1315 isl_point *pnt;1316 1317 if (!set)1318 return NULL;1319 1320 for (i = 0; i < set->n; ++i) {1321 pnt = isl_basic_set_sample_point(isl_basic_set_copy(set->p[i]));1322 if (!pnt)1323 goto error;1324 if (!isl_point_is_void(pnt))1325 break;1326 isl_point_free(pnt);1327 }1328 if (i == set->n)1329 pnt = isl_point_void(isl_set_get_space(set));1330 1331 isl_set_free(set);1332 return pnt;1333error:1334 isl_set_free(set);1335 return NULL;1336}1337