1253 lines · c
1/*2 * Copyright 2008-2009 Katholieke Universiteit Leuven3 * Copyright 2010 INRIA Saclay4 * Copyright 2012 Ecole Normale Superieure5 *6 * Use of this software is governed by the MIT license7 *8 * Written by Sven Verdoolaege, K.U.Leuven, Departement9 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium10 * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,11 * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France12 * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France13 */14 15#include <isl_ctx_private.h>16#include <isl_map_private.h>17#include <isl_seq.h>18#include <isl/set.h>19#include <isl/lp.h>20#include <isl/map.h>21#include "isl_equalities.h"22#include "isl_sample.h"23#include "isl_tab.h"24#include <isl_mat_private.h>25#include <isl_vec_private.h>26 27#include <bset_to_bmap.c>28#include <bset_from_bmap.c>29#include <set_to_map.c>30#include <set_from_map.c>31 32__isl_give isl_basic_map *isl_basic_map_implicit_equalities(33 __isl_take isl_basic_map *bmap)34{35 struct isl_tab *tab;36 37 if (!bmap)38 return bmap;39 40 bmap = isl_basic_map_gauss(bmap, NULL);41 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))42 return bmap;43 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_IMPLICIT))44 return bmap;45 if (bmap->n_ineq <= 1)46 return bmap;47 48 tab = isl_tab_from_basic_map(bmap, 0);49 if (isl_tab_detect_implicit_equalities(tab) < 0)50 goto error;51 bmap = isl_basic_map_update_from_tab(bmap, tab);52 isl_tab_free(tab);53 bmap = isl_basic_map_gauss(bmap, NULL);54 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);55 return bmap;56error:57 isl_tab_free(tab);58 isl_basic_map_free(bmap);59 return NULL;60}61 62__isl_give isl_basic_set *isl_basic_set_implicit_equalities(63 __isl_take isl_basic_set *bset)64{65 return bset_from_bmap(66 isl_basic_map_implicit_equalities(bset_to_bmap(bset)));67}68 69/* Make eq[row][col] of both bmaps equal so we can add the row70 * add the column to the common matrix.71 * Note that because of the echelon form, the columns of row row72 * after column col are zero.73 */74static void set_common_multiple(75 struct isl_basic_set *bset1, struct isl_basic_set *bset2,76 unsigned row, unsigned col)77{78 isl_int m, c;79 80 if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col]))81 return;82 83 isl_int_init(c);84 isl_int_init(m);85 isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]);86 isl_int_divexact(c, m, bset1->eq[row][col]);87 isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1);88 isl_int_divexact(c, m, bset2->eq[row][col]);89 isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1);90 isl_int_clear(c);91 isl_int_clear(m);92}93 94/* Delete a given equality, moving all the following equalities one up.95 */96static void delete_row(__isl_keep isl_basic_set *bset, unsigned row)97{98 isl_int *t;99 int r;100 101 t = bset->eq[row];102 bset->n_eq--;103 for (r = row; r < bset->n_eq; ++r)104 bset->eq[r] = bset->eq[r+1];105 bset->eq[bset->n_eq] = t;106}107 108/* Make first row entries in column col of bset1 identical to109 * those of bset2, using the fact that entry bset1->eq[row][col]=a110 * is non-zero. Initially, these elements of bset1 are all zero.111 * For each row i < row, we set112 * A[i] = a * A[i] + B[i][col] * A[row]113 * B[i] = a * B[i]114 * so that115 * A[i][col] = B[i][col] = a * old(B[i][col])116 */117static isl_stat construct_column(118 __isl_keep isl_basic_set *bset1, __isl_keep isl_basic_set *bset2,119 unsigned row, unsigned col)120{121 int r;122 isl_int a;123 isl_int b;124 isl_size total;125 126 total = isl_basic_set_dim(bset1, isl_dim_set);127 if (total < 0)128 return isl_stat_error;129 130 isl_int_init(a);131 isl_int_init(b);132 for (r = 0; r < row; ++r) {133 if (isl_int_is_zero(bset2->eq[r][col]))134 continue;135 isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]);136 isl_int_divexact(a, bset1->eq[row][col], b);137 isl_int_divexact(b, bset2->eq[r][col], b);138 isl_seq_combine(bset1->eq[r], a, bset1->eq[r],139 b, bset1->eq[row], 1 + total);140 isl_seq_scale(bset2->eq[r], bset2->eq[r], a, 1 + total);141 }142 isl_int_clear(a);143 isl_int_clear(b);144 delete_row(bset1, row);145 146 return isl_stat_ok;147}148 149/* Make first row entries in column col of bset1 identical to150 * those of bset2, using only these entries of the two matrices.151 * Let t be the last row with different entries.152 * For each row i < t, we set153 * A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]154 * B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]155 * so that156 * A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])157 */158static isl_bool transform_column(159 __isl_keep isl_basic_set *bset1, __isl_keep isl_basic_set *bset2,160 unsigned row, unsigned col)161{162 int i, t;163 isl_int a, b, g;164 isl_size total;165 166 for (t = row-1; t >= 0; --t)167 if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))168 break;169 if (t < 0)170 return isl_bool_false;171 172 total = isl_basic_set_dim(bset1, isl_dim_set);173 if (total < 0)174 return isl_bool_error;175 isl_int_init(a);176 isl_int_init(b);177 isl_int_init(g);178 isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]);179 for (i = 0; i < t; ++i) {180 isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]);181 isl_int_gcd(g, a, b);182 isl_int_divexact(a, a, g);183 isl_int_divexact(g, b, g);184 isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t],185 1 + total);186 isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],187 1 + total);188 }189 isl_int_clear(a);190 isl_int_clear(b);191 isl_int_clear(g);192 delete_row(bset1, t);193 delete_row(bset2, t);194 return isl_bool_true;195}196 197/* The implementation is based on Section 5.2 of Michael Karr,198 * "Affine Relationships Among Variables of a Program",199 * except that the echelon form we use starts from the last column200 * and that we are dealing with integer coefficients.201 */202static __isl_give isl_basic_set *affine_hull(203 __isl_take isl_basic_set *bset1, __isl_take isl_basic_set *bset2)204{205 isl_size dim;206 unsigned total;207 int col;208 int row;209 210 dim = isl_basic_set_dim(bset1, isl_dim_set);211 if (dim < 0 || !bset2)212 goto error;213 214 total = 1 + dim;215 216 row = 0;217 for (col = total-1; col >= 0; --col) {218 int is_zero1 = row >= bset1->n_eq ||219 isl_int_is_zero(bset1->eq[row][col]);220 int is_zero2 = row >= bset2->n_eq ||221 isl_int_is_zero(bset2->eq[row][col]);222 if (!is_zero1 && !is_zero2) {223 set_common_multiple(bset1, bset2, row, col);224 ++row;225 } else if (!is_zero1 && is_zero2) {226 if (construct_column(bset1, bset2, row, col) < 0)227 goto error;228 } else if (is_zero1 && !is_zero2) {229 if (construct_column(bset2, bset1, row, col) < 0)230 goto error;231 } else {232 isl_bool transform;233 234 transform = transform_column(bset1, bset2, row, col);235 if (transform < 0)236 goto error;237 if (transform)238 --row;239 }240 }241 isl_assert(bset1->ctx, row == bset1->n_eq, goto error);242 isl_basic_set_free(bset2);243 bset1 = isl_basic_set_normalize_constraints(bset1);244 return bset1;245error:246 isl_basic_set_free(bset1);247 isl_basic_set_free(bset2);248 return NULL;249}250 251/* Find an integer point in the set represented by "tab"252 * that lies outside of the equality "eq" e(x) = 0.253 * If "up" is true, look for a point satisfying e(x) - 1 >= 0.254 * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).255 * The point, if found, is returned.256 * If no point can be found, a zero-length vector is returned.257 *258 * Before solving an ILP problem, we first check if simply259 * adding the normal of the constraint to one of the known260 * integer points in the basic set represented by "tab"261 * yields another point inside the basic set.262 *263 * The caller of this function ensures that the tableau is bounded or264 * that tab->basis and tab->n_unbounded have been set appropriately.265 */266static __isl_give isl_vec *outside_point(struct isl_tab *tab, isl_int *eq,267 int up)268{269 struct isl_ctx *ctx;270 struct isl_vec *sample = NULL;271 struct isl_tab_undo *snap;272 unsigned dim;273 274 if (!tab)275 return NULL;276 ctx = tab->mat->ctx;277 278 dim = tab->n_var;279 sample = isl_vec_alloc(ctx, 1 + dim);280 if (!sample)281 return NULL;282 isl_int_set_si(sample->el[0], 1);283 isl_seq_combine(sample->el + 1,284 ctx->one, tab->bmap->sample->el + 1,285 up ? ctx->one : ctx->negone, eq + 1, dim);286 if (isl_basic_map_contains(tab->bmap, sample))287 return sample;288 isl_vec_free(sample);289 sample = NULL;290 291 snap = isl_tab_snap(tab);292 293 if (!up)294 isl_seq_neg(eq, eq, 1 + dim);295 isl_int_sub_ui(eq[0], eq[0], 1);296 297 if (isl_tab_extend_cons(tab, 1) < 0)298 goto error;299 if (isl_tab_add_ineq(tab, eq) < 0)300 goto error;301 302 sample = isl_tab_sample(tab);303 304 isl_int_add_ui(eq[0], eq[0], 1);305 if (!up)306 isl_seq_neg(eq, eq, 1 + dim);307 308 if (sample && isl_tab_rollback(tab, snap) < 0)309 goto error;310 311 return sample;312error:313 isl_vec_free(sample);314 return NULL;315}316 317__isl_give isl_basic_set *isl_basic_set_recession_cone(318 __isl_take isl_basic_set *bset)319{320 int i;321 isl_bool empty;322 323 empty = isl_basic_set_plain_is_empty(bset);324 if (empty < 0)325 return isl_basic_set_free(bset);326 if (empty)327 return bset;328 329 bset = isl_basic_set_cow(bset);330 if (isl_basic_set_check_no_locals(bset) < 0)331 return isl_basic_set_free(bset);332 333 for (i = 0; i < bset->n_eq; ++i)334 isl_int_set_si(bset->eq[i][0], 0);335 336 for (i = 0; i < bset->n_ineq; ++i)337 isl_int_set_si(bset->ineq[i][0], 0);338 339 ISL_F_CLR(bset, ISL_BASIC_SET_NO_IMPLICIT);340 return isl_basic_set_implicit_equalities(bset);341}342 343/* Move "sample" to a point that is one up (or down) from the original344 * point in dimension "pos".345 */346static void adjacent_point(__isl_keep isl_vec *sample, int pos, int up)347{348 if (up)349 isl_int_add_ui(sample->el[1 + pos], sample->el[1 + pos], 1);350 else351 isl_int_sub_ui(sample->el[1 + pos], sample->el[1 + pos], 1);352}353 354/* Check if any points that are adjacent to "sample" also belong to "bset".355 * If so, add them to "hull" and return the updated hull.356 *357 * Before checking whether and adjacent point belongs to "bset", we first358 * check whether it already belongs to "hull" as this test is typically359 * much cheaper.360 */361static __isl_give isl_basic_set *add_adjacent_points(362 __isl_take isl_basic_set *hull, __isl_take isl_vec *sample,363 __isl_keep isl_basic_set *bset)364{365 int i, up;366 isl_size dim;367 368 dim = isl_basic_set_dim(hull, isl_dim_set);369 if (!sample || dim < 0)370 goto error;371 372 for (i = 0; i < dim; ++i) {373 for (up = 0; up <= 1; ++up) {374 int contains;375 isl_basic_set *point;376 377 adjacent_point(sample, i, up);378 contains = isl_basic_set_contains(hull, sample);379 if (contains < 0)380 goto error;381 if (contains) {382 adjacent_point(sample, i, !up);383 continue;384 }385 contains = isl_basic_set_contains(bset, sample);386 if (contains < 0)387 goto error;388 if (contains) {389 point = isl_basic_set_from_vec(390 isl_vec_copy(sample));391 hull = affine_hull(hull, point);392 }393 adjacent_point(sample, i, !up);394 if (contains)395 break;396 }397 }398 399 isl_vec_free(sample);400 401 return hull;402error:403 isl_vec_free(sample);404 isl_basic_set_free(hull);405 return NULL;406}407 408/* Extend an initial (under-)approximation of the affine hull of basic409 * set represented by the tableau "tab"410 * by looking for points that do not satisfy one of the equalities411 * in the current approximation and adding them to that approximation412 * until no such points can be found any more.413 *414 * The caller of this function ensures that "tab" is bounded or415 * that tab->basis and tab->n_unbounded have been set appropriately.416 *417 * "bset" may be either NULL or the basic set represented by "tab".418 * If "bset" is not NULL, we check for any point we find if any419 * of its adjacent points also belong to "bset".420 */421static __isl_give isl_basic_set *extend_affine_hull(struct isl_tab *tab,422 __isl_take isl_basic_set *hull, __isl_keep isl_basic_set *bset)423{424 int i, j;425 unsigned dim;426 427 if (!tab || !hull)428 goto error;429 430 dim = tab->n_var;431 432 if (isl_tab_extend_cons(tab, 2 * dim + 1) < 0)433 goto error;434 435 for (i = 0; i < dim; ++i) {436 struct isl_vec *sample;437 struct isl_basic_set *point;438 for (j = 0; j < hull->n_eq; ++j) {439 sample = outside_point(tab, hull->eq[j], 1);440 if (!sample)441 goto error;442 if (sample->size > 0)443 break;444 isl_vec_free(sample);445 sample = outside_point(tab, hull->eq[j], 0);446 if (!sample)447 goto error;448 if (sample->size > 0)449 break;450 isl_vec_free(sample);451 452 if (isl_tab_add_eq(tab, hull->eq[j]) < 0)453 goto error;454 }455 if (j == hull->n_eq)456 break;457 if (tab->samples &&458 isl_tab_add_sample(tab, isl_vec_copy(sample)) < 0)459 hull = isl_basic_set_free(hull);460 if (bset)461 hull = add_adjacent_points(hull, isl_vec_copy(sample),462 bset);463 point = isl_basic_set_from_vec(sample);464 hull = affine_hull(hull, point);465 if (!hull)466 return NULL;467 }468 469 return hull;470error:471 isl_basic_set_free(hull);472 return NULL;473}474 475/* Construct an initial underapproximation of the hull of "bset"476 * from "sample" and any of its adjacent points that also belong to "bset".477 */478static __isl_give isl_basic_set *initialize_hull(__isl_keep isl_basic_set *bset,479 __isl_take isl_vec *sample)480{481 isl_basic_set *hull;482 483 hull = isl_basic_set_from_vec(isl_vec_copy(sample));484 hull = add_adjacent_points(hull, sample, bset);485 486 return hull;487}488 489/* Look for all equalities satisfied by the integer points in bset,490 * which is assumed to be bounded.491 *492 * The equalities are obtained by successively looking for493 * a point that is affinely independent of the points found so far.494 * In particular, for each equality satisfied by the points so far,495 * we check if there is any point on a hyperplane parallel to the496 * corresponding hyperplane shifted by at least one (in either direction).497 */498static __isl_give isl_basic_set *uset_affine_hull_bounded(499 __isl_take isl_basic_set *bset)500{501 struct isl_vec *sample = NULL;502 struct isl_basic_set *hull;503 struct isl_tab *tab = NULL;504 isl_size dim;505 506 if (isl_basic_set_plain_is_empty(bset))507 return bset;508 509 dim = isl_basic_set_dim(bset, isl_dim_set);510 if (dim < 0)511 return isl_basic_set_free(bset);512 513 if (bset->sample && bset->sample->size == 1 + dim) {514 int contains = isl_basic_set_contains(bset, bset->sample);515 if (contains < 0)516 goto error;517 if (contains) {518 if (dim == 0)519 return bset;520 sample = isl_vec_copy(bset->sample);521 } else {522 isl_vec_free(bset->sample);523 bset->sample = NULL;524 }525 }526 527 tab = isl_tab_from_basic_set(bset, 1);528 if (!tab)529 goto error;530 if (tab->empty) {531 isl_tab_free(tab);532 isl_vec_free(sample);533 return isl_basic_set_set_to_empty(bset);534 }535 536 if (!sample) {537 struct isl_tab_undo *snap;538 snap = isl_tab_snap(tab);539 sample = isl_tab_sample(tab);540 if (isl_tab_rollback(tab, snap) < 0)541 goto error;542 isl_vec_free(tab->bmap->sample);543 tab->bmap->sample = isl_vec_copy(sample);544 }545 546 if (!sample)547 goto error;548 if (sample->size == 0) {549 isl_tab_free(tab);550 isl_vec_free(sample);551 return isl_basic_set_set_to_empty(bset);552 }553 554 hull = initialize_hull(bset, sample);555 556 hull = extend_affine_hull(tab, hull, bset);557 isl_basic_set_free(bset);558 isl_tab_free(tab);559 560 return hull;561error:562 isl_vec_free(sample);563 isl_tab_free(tab);564 isl_basic_set_free(bset);565 return NULL;566}567 568/* Given an unbounded tableau and an integer point satisfying the tableau,569 * construct an initial affine hull containing the recession cone570 * shifted to the given point.571 *572 * The unbounded directions are taken from the last rows of the basis,573 * which is assumed to have been initialized appropriately.574 */575static __isl_give isl_basic_set *initial_hull(struct isl_tab *tab,576 __isl_take isl_vec *vec)577{578 int i;579 int k;580 struct isl_basic_set *bset = NULL;581 struct isl_ctx *ctx;582 isl_size dim;583 584 if (!vec || !tab)585 return NULL;586 ctx = vec->ctx;587 isl_assert(ctx, vec->size != 0, goto error);588 589 bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);590 dim = isl_basic_set_dim(bset, isl_dim_set);591 if (dim < 0)592 goto error;593 dim -= tab->n_unbounded;594 for (i = 0; i < dim; ++i) {595 k = isl_basic_set_alloc_equality(bset);596 if (k < 0)597 goto error;598 isl_seq_cpy(bset->eq[k] + 1, tab->basis->row[1 + i] + 1,599 vec->size - 1);600 isl_seq_inner_product(bset->eq[k] + 1, vec->el +1,601 vec->size - 1, &bset->eq[k][0]);602 isl_int_neg(bset->eq[k][0], bset->eq[k][0]);603 }604 bset->sample = vec;605 bset = isl_basic_set_gauss(bset, NULL);606 607 return bset;608error:609 isl_basic_set_free(bset);610 isl_vec_free(vec);611 return NULL;612}613 614/* Given a tableau of a set and a tableau of the corresponding615 * recession cone, detect and add all equalities to the tableau.616 * If the tableau is bounded, then we can simply keep the617 * tableau in its state after the return from extend_affine_hull.618 * However, if the tableau is unbounded, then619 * isl_tab_set_initial_basis_with_cone will add some additional620 * constraints to the tableau that have to be removed again.621 * In this case, we therefore rollback to the state before622 * any constraints were added and then add the equalities back in.623 */624struct isl_tab *isl_tab_detect_equalities(struct isl_tab *tab,625 struct isl_tab *tab_cone)626{627 int j;628 struct isl_vec *sample;629 struct isl_basic_set *hull = NULL;630 struct isl_tab_undo *snap;631 632 if (!tab || !tab_cone)633 goto error;634 635 snap = isl_tab_snap(tab);636 637 isl_mat_free(tab->basis);638 tab->basis = NULL;639 640 isl_assert(tab->mat->ctx, tab->bmap, goto error);641 isl_assert(tab->mat->ctx, tab->samples, goto error);642 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);643 isl_assert(tab->mat->ctx, tab->n_sample > tab->n_outside, goto error);644 645 if (isl_tab_set_initial_basis_with_cone(tab, tab_cone) < 0)646 goto error;647 648 sample = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);649 if (!sample)650 goto error;651 652 isl_seq_cpy(sample->el, tab->samples->row[tab->n_outside], sample->size);653 654 isl_vec_free(tab->bmap->sample);655 tab->bmap->sample = isl_vec_copy(sample);656 657 if (tab->n_unbounded == 0)658 hull = isl_basic_set_from_vec(isl_vec_copy(sample));659 else660 hull = initial_hull(tab, isl_vec_copy(sample));661 662 for (j = tab->n_outside + 1; j < tab->n_sample; ++j) {663 isl_seq_cpy(sample->el, tab->samples->row[j], sample->size);664 hull = affine_hull(hull,665 isl_basic_set_from_vec(isl_vec_copy(sample)));666 }667 668 isl_vec_free(sample);669 670 hull = extend_affine_hull(tab, hull, NULL);671 if (!hull)672 goto error;673 674 if (tab->n_unbounded == 0) {675 isl_basic_set_free(hull);676 return tab;677 }678 679 if (isl_tab_rollback(tab, snap) < 0)680 goto error;681 682 if (hull->n_eq > tab->n_zero) {683 for (j = 0; j < hull->n_eq; ++j) {684 isl_seq_normalize(tab->mat->ctx, hull->eq[j], 1 + tab->n_var);685 if (isl_tab_add_eq(tab, hull->eq[j]) < 0)686 goto error;687 }688 }689 690 isl_basic_set_free(hull);691 692 return tab;693error:694 isl_basic_set_free(hull);695 isl_tab_free(tab);696 return NULL;697}698 699/* Compute the affine hull of "bset", where "cone" is the recession cone700 * of "bset".701 *702 * We first compute a unimodular transformation that puts the unbounded703 * directions in the last dimensions. In particular, we take a transformation704 * that maps all equalities to equalities (in HNF) on the first dimensions.705 * Let x be the original dimensions and y the transformed, with y_1 bounded706 * and y_2 unbounded.707 *708 * [ y_1 ] [ y_1 ] [ Q_1 ]709 * x = U [ y_2 ] [ y_2 ] = [ Q_2 ] x710 *711 * Let's call the input basic set S. We compute S' = preimage(S, U)712 * and drop the final dimensions including any constraints involving them.713 * This results in set S''.714 * Then we compute the affine hull A'' of S''.715 * Let F y_1 >= g be the constraint system of A''. In the transformed716 * space the y_2 are unbounded, so we can add them back without any constraints,717 * resulting in718 *719 * [ y_1 ]720 * [ F 0 ] [ y_2 ] >= g721 * or722 * [ Q_1 ]723 * [ F 0 ] [ Q_2 ] x >= g724 * or725 * F Q_1 x >= g726 *727 * The affine hull in the original space is then obtained as728 * A = preimage(A'', Q_1).729 */730static __isl_give isl_basic_set *affine_hull_with_cone(731 __isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone)732{733 isl_size total;734 unsigned cone_dim;735 struct isl_basic_set *hull;736 struct isl_mat *M, *U, *Q;737 738 total = isl_basic_set_dim(cone, isl_dim_all);739 if (!bset || total < 0)740 goto error;741 742 cone_dim = total - cone->n_eq;743 744 M = isl_mat_sub_alloc6(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);745 M = isl_mat_left_hermite(M, 0, &U, &Q);746 if (!M)747 goto error;748 isl_mat_free(M);749 750 U = isl_mat_lin_to_aff(U);751 bset = isl_basic_set_preimage(bset, isl_mat_copy(U));752 753 bset = isl_basic_set_drop_constraints_involving(bset, total - cone_dim,754 cone_dim);755 bset = isl_basic_set_drop_dims(bset, total - cone_dim, cone_dim);756 757 Q = isl_mat_lin_to_aff(Q);758 Q = isl_mat_drop_rows(Q, 1 + total - cone_dim, cone_dim);759 760 if (bset && bset->sample && bset->sample->size == 1 + total)761 bset->sample = isl_mat_vec_product(isl_mat_copy(Q), bset->sample);762 763 hull = uset_affine_hull_bounded(bset);764 765 if (!hull) {766 isl_mat_free(Q);767 isl_mat_free(U);768 } else {769 struct isl_vec *sample = isl_vec_copy(hull->sample);770 U = isl_mat_drop_cols(U, 1 + total - cone_dim, cone_dim);771 if (sample && sample->size > 0)772 sample = isl_mat_vec_product(U, sample);773 else774 isl_mat_free(U);775 hull = isl_basic_set_preimage(hull, Q);776 if (hull) {777 isl_vec_free(hull->sample);778 hull->sample = sample;779 } else780 isl_vec_free(sample);781 }782 783 isl_basic_set_free(cone);784 785 return hull;786error:787 isl_basic_set_free(bset);788 isl_basic_set_free(cone);789 return NULL;790}791 792/* Look for all equalities satisfied by the integer points in bset,793 * which is assumed not to have any explicit equalities.794 *795 * The equalities are obtained by successively looking for796 * a point that is affinely independent of the points found so far.797 * In particular, for each equality satisfied by the points so far,798 * we check if there is any point on a hyperplane parallel to the799 * corresponding hyperplane shifted by at least one (in either direction).800 *801 * Before looking for any outside points, we first compute the recession802 * cone. The directions of this recession cone will always be part803 * of the affine hull, so there is no need for looking for any points804 * in these directions.805 * In particular, if the recession cone is full-dimensional, then806 * the affine hull is simply the whole universe.807 */808static __isl_give isl_basic_set *uset_affine_hull(809 __isl_take isl_basic_set *bset)810{811 struct isl_basic_set *cone;812 isl_size total;813 814 if (isl_basic_set_plain_is_empty(bset))815 return bset;816 817 cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));818 if (!cone)819 goto error;820 if (cone->n_eq == 0) {821 isl_space *space;822 space = isl_basic_set_get_space(bset);823 isl_basic_set_free(cone);824 isl_basic_set_free(bset);825 return isl_basic_set_universe(space);826 }827 828 total = isl_basic_set_dim(cone, isl_dim_all);829 if (total < 0)830 bset = isl_basic_set_free(bset);831 if (cone->n_eq < total)832 return affine_hull_with_cone(bset, cone);833 834 isl_basic_set_free(cone);835 return uset_affine_hull_bounded(bset);836error:837 isl_basic_set_free(bset);838 return NULL;839}840 841/* Look for all equalities satisfied by the integer points in bmap842 * that are independent of the equalities already explicitly available843 * in bmap.844 *845 * We first remove all equalities already explicitly available,846 * then look for additional equalities in the reduced space847 * and then transform the result to the original space.848 * The original equalities are _not_ added to this set. This is849 * the responsibility of the calling function.850 * The resulting basic set has all meaning about the dimensions removed.851 * In particular, dimensions that correspond to existential variables852 * in bmap and that are found to be fixed are not removed.853 */854static __isl_give isl_basic_set *equalities_in_underlying_set(855 __isl_take isl_basic_map *bmap)856{857 struct isl_mat *T1 = NULL;858 struct isl_mat *T2 = NULL;859 struct isl_basic_set *bset = NULL;860 struct isl_basic_set *hull = NULL;861 862 bset = isl_basic_map_underlying_set(bmap);863 if (!bset)864 return NULL;865 if (bset->n_eq)866 bset = isl_basic_set_remove_equalities(bset, &T1, &T2);867 if (!bset)868 goto error;869 870 hull = uset_affine_hull(bset);871 if (!T2)872 return hull;873 874 if (!hull) {875 isl_mat_free(T1);876 isl_mat_free(T2);877 } else {878 struct isl_vec *sample = isl_vec_copy(hull->sample);879 if (sample && sample->size > 0)880 sample = isl_mat_vec_product(T1, sample);881 else882 isl_mat_free(T1);883 hull = isl_basic_set_preimage(hull, T2);884 if (hull) {885 isl_vec_free(hull->sample);886 hull->sample = sample;887 } else888 isl_vec_free(sample);889 }890 891 return hull;892error:893 isl_mat_free(T1);894 isl_mat_free(T2);895 isl_basic_set_free(bset);896 isl_basic_set_free(hull);897 return NULL;898}899 900/* Detect and make explicit all equalities satisfied by the (integer)901 * points in bmap.902 */903__isl_give isl_basic_map *isl_basic_map_detect_equalities(904 __isl_take isl_basic_map *bmap)905{906 int i, j;907 isl_size total;908 struct isl_basic_set *hull = NULL;909 910 if (!bmap)911 return NULL;912 if (bmap->n_ineq == 0)913 return bmap;914 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))915 return bmap;916 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))917 return bmap;918 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))919 return isl_basic_map_implicit_equalities(bmap);920 921 hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));922 if (!hull)923 goto error;924 if (ISL_F_ISSET(hull, ISL_BASIC_SET_EMPTY)) {925 isl_basic_set_free(hull);926 return isl_basic_map_set_to_empty(bmap);927 }928 bmap = isl_basic_map_extend(bmap, 0, hull->n_eq, 0);929 total = isl_basic_set_dim(hull, isl_dim_all);930 if (total < 0)931 goto error;932 for (i = 0; i < hull->n_eq; ++i) {933 j = isl_basic_map_alloc_equality(bmap);934 if (j < 0)935 goto error;936 isl_seq_cpy(bmap->eq[j], hull->eq[i], 1 + total);937 }938 isl_vec_free(bmap->sample);939 bmap->sample = isl_vec_copy(hull->sample);940 isl_basic_set_free(hull);941 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);942 bmap = isl_basic_map_simplify(bmap);943 return isl_basic_map_finalize(bmap);944error:945 isl_basic_set_free(hull);946 isl_basic_map_free(bmap);947 return NULL;948}949 950__isl_give isl_basic_set *isl_basic_set_detect_equalities(951 __isl_take isl_basic_set *bset)952{953 return bset_from_bmap(954 isl_basic_map_detect_equalities(bset_to_bmap(bset)));955}956 957__isl_give isl_map *isl_map_detect_equalities(__isl_take isl_map *map)958{959 return isl_map_inline_foreach_basic_map(map,960 &isl_basic_map_detect_equalities);961}962 963__isl_give isl_set *isl_set_detect_equalities(__isl_take isl_set *set)964{965 return set_from_map(isl_map_detect_equalities(set_to_map(set)));966}967 968/* Return the superset of "bmap" described by the equalities969 * satisfied by "bmap" that are already known.970 */971__isl_give isl_basic_map *isl_basic_map_plain_affine_hull(972 __isl_take isl_basic_map *bmap)973{974 bmap = isl_basic_map_cow(bmap);975 if (bmap)976 isl_basic_map_free_inequality(bmap, bmap->n_ineq);977 bmap = isl_basic_map_finalize(bmap);978 return bmap;979}980 981/* Return the superset of "bset" described by the equalities982 * satisfied by "bset" that are already known.983 */984__isl_give isl_basic_set *isl_basic_set_plain_affine_hull(985 __isl_take isl_basic_set *bset)986{987 return isl_basic_map_plain_affine_hull(bset);988}989 990/* After computing the rational affine hull (by detecting the implicit991 * equalities), we compute the additional equalities satisfied by992 * the integer points (if any) and add the original equalities back in.993 */994__isl_give isl_basic_map *isl_basic_map_affine_hull(995 __isl_take isl_basic_map *bmap)996{997 bmap = isl_basic_map_detect_equalities(bmap);998 bmap = isl_basic_map_plain_affine_hull(bmap);999 return bmap;1000}1001 1002__isl_give isl_basic_set *isl_basic_set_affine_hull(1003 __isl_take isl_basic_set *bset)1004{1005 return bset_from_bmap(isl_basic_map_affine_hull(bset_to_bmap(bset)));1006}1007 1008/* Given a rational affine matrix "M", add stride constraints to "bmap"1009 * that ensure that1010 *1011 * M(x)1012 *1013 * is an integer vector. The variables x include all the variables1014 * of "bmap" except the unknown divs.1015 *1016 * If d is the common denominator of M, then we need to impose that1017 *1018 * d M(x) = 0 mod d1019 *1020 * or1021 *1022 * exists alpha : d M(x) = d alpha1023 *1024 * This function is similar to add_strides in isl_morph.c1025 */1026static __isl_give isl_basic_map *add_strides(__isl_take isl_basic_map *bmap,1027 __isl_keep isl_mat *M, int n_known)1028{1029 int i, div, k;1030 isl_int gcd;1031 1032 if (isl_int_is_one(M->row[0][0]))1033 return bmap;1034 1035 bmap = isl_basic_map_extend(bmap, M->n_row - 1, M->n_row - 1, 0);1036 1037 isl_int_init(gcd);1038 for (i = 1; i < M->n_row; ++i) {1039 isl_seq_gcd(M->row[i], M->n_col, &gcd);1040 if (isl_int_is_divisible_by(gcd, M->row[0][0]))1041 continue;1042 div = isl_basic_map_alloc_div(bmap);1043 if (div < 0)1044 goto error;1045 isl_int_set_si(bmap->div[div][0], 0);1046 k = isl_basic_map_alloc_equality(bmap);1047 if (k < 0)1048 goto error;1049 isl_seq_cpy(bmap->eq[k], M->row[i], M->n_col);1050 isl_seq_clr(bmap->eq[k] + M->n_col, bmap->n_div - n_known);1051 isl_int_set(bmap->eq[k][M->n_col - n_known + div],1052 M->row[0][0]);1053 }1054 isl_int_clear(gcd);1055 1056 return bmap;1057error:1058 isl_int_clear(gcd);1059 isl_basic_map_free(bmap);1060 return NULL;1061}1062 1063/* If there are any equalities that involve (multiple) unknown divs,1064 * then extract the stride information encoded by those equalities1065 * and make it explicitly available in "bmap".1066 *1067 * We first sort the divs so that the unknown divs appear last and1068 * then we count how many equalities involve these divs.1069 *1070 * Let these equalities be of the form1071 *1072 * A(x) + B y = 01073 *1074 * where y represents the unknown divs and x the remaining variables.1075 * Let [H 0] be the Hermite Normal Form of B, i.e.,1076 *1077 * B = [H 0] Q1078 *1079 * Then x is a solution of the equalities iff1080 *1081 * H^-1 A(x) (= - [I 0] Q y)1082 *1083 * is an integer vector. Let d be the common denominator of H^-1.1084 * We impose1085 *1086 * d H^-1 A(x) = d alpha1087 *1088 * in add_strides, with alpha fresh existentially quantified variables.1089 */1090static __isl_give isl_basic_map *isl_basic_map_make_strides_explicit(1091 __isl_take isl_basic_map *bmap)1092{1093 isl_bool known;1094 int n_known;1095 int n, n_col;1096 isl_size v_div;1097 isl_ctx *ctx;1098 isl_mat *A, *B, *M;1099 1100 known = isl_basic_map_divs_known(bmap);1101 if (known < 0)1102 return isl_basic_map_free(bmap);1103 if (known)1104 return bmap;1105 bmap = isl_basic_map_sort_divs(bmap);1106 bmap = isl_basic_map_gauss(bmap, NULL);1107 if (!bmap)1108 return NULL;1109 1110 for (n_known = 0; n_known < bmap->n_div; ++n_known)1111 if (isl_int_is_zero(bmap->div[n_known][0]))1112 break;1113 v_div = isl_basic_map_var_offset(bmap, isl_dim_div);1114 if (v_div < 0)1115 return isl_basic_map_free(bmap);1116 for (n = 0; n < bmap->n_eq; ++n)1117 if (isl_seq_first_non_zero(bmap->eq[n] + 1 + v_div + n_known,1118 bmap->n_div - n_known) == -1)1119 break;1120 if (n == 0)1121 return bmap;1122 ctx = isl_basic_map_get_ctx(bmap);1123 B = isl_mat_sub_alloc6(ctx, bmap->eq, 0, n, 0, 1 + v_div + n_known);1124 n_col = bmap->n_div - n_known;1125 A = isl_mat_sub_alloc6(ctx, bmap->eq, 0, n, 1 + v_div + n_known, n_col);1126 A = isl_mat_left_hermite(A, 0, NULL, NULL);1127 A = isl_mat_drop_cols(A, n, n_col - n);1128 A = isl_mat_lin_to_aff(A);1129 A = isl_mat_right_inverse(A);1130 B = isl_mat_insert_zero_rows(B, 0, 1);1131 B = isl_mat_set_element_si(B, 0, 0, 1);1132 M = isl_mat_product(A, B);1133 if (!M)1134 return isl_basic_map_free(bmap);1135 bmap = add_strides(bmap, M, n_known);1136 bmap = isl_basic_map_gauss(bmap, NULL);1137 isl_mat_free(M);1138 1139 return bmap;1140}1141 1142/* Compute the affine hull of each basic map in "map" separately1143 * and make all stride information explicit so that we can remove1144 * all unknown divs without losing this information.1145 * The result is also guaranteed to be gaussed.1146 *1147 * In simple cases where a div is determined by an equality,1148 * calling isl_basic_map_gauss is enough to make the stride information1149 * explicit, as it will derive an explicit representation for the div1150 * from the equality. If, however, the stride information1151 * is encoded through multiple unknown divs then we need to make1152 * some extra effort in isl_basic_map_make_strides_explicit.1153 */1154static __isl_give isl_map *isl_map_local_affine_hull(__isl_take isl_map *map)1155{1156 int i;1157 1158 map = isl_map_cow(map);1159 if (!map)1160 return NULL;1161 1162 for (i = 0; i < map->n; ++i) {1163 map->p[i] = isl_basic_map_affine_hull(map->p[i]);1164 map->p[i] = isl_basic_map_gauss(map->p[i], NULL);1165 map->p[i] = isl_basic_map_make_strides_explicit(map->p[i]);1166 if (!map->p[i])1167 return isl_map_free(map);1168 }1169 1170 return map;1171}1172 1173static __isl_give isl_set *isl_set_local_affine_hull(__isl_take isl_set *set)1174{1175 return isl_map_local_affine_hull(set);1176}1177 1178/* Return an empty basic map living in the same space as "map".1179 */1180static __isl_give isl_basic_map *replace_map_by_empty_basic_map(1181 __isl_take isl_map *map)1182{1183 isl_space *space;1184 1185 space = isl_map_get_space(map);1186 isl_map_free(map);1187 return isl_basic_map_empty(space);1188}1189 1190/* Compute the affine hull of "map".1191 *1192 * We first compute the affine hull of each basic map separately.1193 * Then we align the divs and recompute the affine hulls of the basic1194 * maps since some of them may now have extra divs.1195 * In order to avoid performing parametric integer programming to1196 * compute explicit expressions for the divs, possible leading to1197 * an explosion in the number of basic maps, we first drop all unknown1198 * divs before aligning the divs. Note that isl_map_local_affine_hull tries1199 * to make sure that all stride information is explicitly available1200 * in terms of known divs. This involves calling isl_basic_set_gauss,1201 * which is also needed because affine_hull assumes its input has been gaussed,1202 * while isl_map_affine_hull may be called on input that has not been gaussed,1203 * in particular from initial_facet_constraint.1204 * Similarly, align_divs may reorder some divs so that we need to1205 * gauss the result again.1206 * Finally, we combine the individual affine hulls into a single1207 * affine hull.1208 */1209__isl_give isl_basic_map *isl_map_affine_hull(__isl_take isl_map *map)1210{1211 struct isl_basic_map *model = NULL;1212 struct isl_basic_map *hull = NULL;1213 struct isl_set *set;1214 isl_basic_set *bset;1215 1216 map = isl_map_detect_equalities(map);1217 map = isl_map_local_affine_hull(map);1218 map = isl_map_remove_empty_parts(map);1219 map = isl_map_remove_unknown_divs(map);1220 map = isl_map_align_divs_internal(map);1221 1222 if (!map)1223 return NULL;1224 1225 if (map->n == 0)1226 return replace_map_by_empty_basic_map(map);1227 1228 model = isl_basic_map_copy(map->p[0]);1229 set = isl_map_underlying_set(map);1230 set = isl_set_cow(set);1231 set = isl_set_local_affine_hull(set);1232 if (!set)1233 goto error;1234 1235 while (set->n > 1)1236 set->p[0] = affine_hull(set->p[0], set->p[--set->n]);1237 1238 bset = isl_basic_set_copy(set->p[0]);1239 hull = isl_basic_map_overlying_set(bset, model);1240 isl_set_free(set);1241 hull = isl_basic_map_simplify(hull);1242 return isl_basic_map_finalize(hull);1243error:1244 isl_basic_map_free(model);1245 isl_set_free(set);1246 return NULL;1247}1248 1249__isl_give isl_basic_set *isl_set_affine_hull(__isl_take isl_set *set)1250{1251 return bset_from_bmap(isl_map_affine_hull(set_to_map(set)));1252}1253