1/* 2 * Copyright 2008-2009 Katholieke Universiteit Leuven 3 * Copyright 2010 INRIA Saclay 4 * Copyright 2012 Ecole Normale Superieure 5 * 6 * Use of this software is governed by the MIT license 7 * 8 * Written by Sven Verdoolaege, K.U.Leuven, Departement 9 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium 10 * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite, 11 * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 12 * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France 13 */ 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 row 70 * add the column to the common matrix. 71 * Note that because of the echelon form, the columns of row row 72 * 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 to 109 * those of bset2, using the fact that entry bset1->eq[row][col]=a 110 * is non-zero. Initially, these elements of bset1 are all zero. 111 * For each row i < row, we set 112 * A[i] = a * A[i] + B[i][col] * A[row] 113 * B[i] = a * B[i] 114 * so that 115 * 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 to 150 * 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 set 153 * 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 that 156 * 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 column 200 * 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 simply 259 * adding the normal of the constraint to one of the known 260 * 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 or 264 * 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 original 344 * 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 else 351 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 first 358 * check whether it already belongs to "hull" as this test is typically 359 * 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 basic 409 * set represented by the tableau "tab" 410 * by looking for points that do not satisfy one of the equalities 411 * in the current approximation and adding them to that approximation 412 * until no such points can be found any more. 413 * 414 * The caller of this function ensures that "tab" is bounded or 415 * 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 any 419 * 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 for 493 * 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 the 496 * 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 cone 570 * 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 corresponding 615 * recession cone, detect and add all equalities to the tableau. 616 * If the tableau is bounded, then we can simply keep the 617 * tableau in its state after the return from extend_affine_hull. 618 * However, if the tableau is unbounded, then 619 * isl_tab_set_initial_basis_with_cone will add some additional 620 * constraints to the tableau that have to be removed again. 621 * In this case, we therefore rollback to the state before 622 * 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 else 660 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 cone 700 * of "bset". 701 * 702 * We first compute a unimodular transformation that puts the unbounded 703 * directions in the last dimensions. In particular, we take a transformation 704 * 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 bounded 706 * and y_2 unbounded. 707 * 708 * [ y_1 ] [ y_1 ] [ Q_1 ] 709 * x = U [ y_2 ] [ y_2 ] = [ Q_2 ] x 710 * 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 transformed 716 * space the y_2 are unbounded, so we can add them back without any constraints, 717 * resulting in 718 * 719 * [ y_1 ] 720 * [ F 0 ] [ y_2 ] >= g 721 * or 722 * [ Q_1 ] 723 * [ F 0 ] [ Q_2 ] x >= g 724 * or 725 * F Q_1 x >= g 726 * 727 * The affine hull in the original space is then obtained as 728 * 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 else 774 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 } else 780 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 for 796 * 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 the 799 * corresponding hyperplane shifted by at least one (in either direction). 800 * 801 * Before looking for any outside points, we first compute the recession 802 * cone. The directions of this recession cone will always be part 803 * of the affine hull, so there is no need for looking for any points 804 * in these directions. 805 * In particular, if the recession cone is full-dimensional, then 806 * 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 bmap 842 * that are independent of the equalities already explicitly available 843 * in bmap. 844 * 845 * We first remove all equalities already explicitly available, 846 * then look for additional equalities in the reduced space 847 * and then transform the result to the original space. 848 * The original equalities are _not_ added to this set. This is 849 * 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 variables 852 * 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 else 882 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 } else 888 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 equalities 969 * 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 equalities 982 * 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 implicit 991 * equalities), we compute the additional equalities satisfied by 992 * 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 that 1010 * 1011 * M(x) 1012 * 1013 * is an integer vector. The variables x include all the variables 1014 * of "bmap" except the unknown divs. 1015 * 1016 * If d is the common denominator of M, then we need to impose that 1017 * 1018 * d M(x) = 0 mod d 1019 * 1020 * or 1021 * 1022 * exists alpha : d M(x) = d alpha 1023 * 1024 * This function is similar to add_strides in isl_morph.c 1025 */ 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 equalities 1065 * and make it explicitly available in "bmap". 1066 * 1067 * We first sort the divs so that the unknown divs appear last and 1068 * then we count how many equalities involve these divs. 1069 * 1070 * Let these equalities be of the form 1071 * 1072 * A(x) + B y = 0 1073 * 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] Q 1078 * 1079 * Then x is a solution of the equalities iff 1080 * 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 impose 1085 * 1086 * d H^-1 A(x) = d alpha 1087 * 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" separately 1143 * and make all stride information explicit so that we can remove 1144 * 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 information 1149 * explicit, as it will derive an explicit representation for the div 1150 * from the equality. If, however, the stride information 1151 * is encoded through multiple unknown divs then we need to make 1152 * 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 basic 1194 * maps since some of them may now have extra divs. 1195 * In order to avoid performing parametric integer programming to 1196 * compute explicit expressions for the divs, possible leading to 1197 * an explosion in the number of basic maps, we first drop all unknown 1198 * divs before aligning the divs. Note that isl_map_local_affine_hull tries 1199 * to make sure that all stride information is explicitly available 1200 * 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 to 1205 * gauss the result again. 1206 * Finally, we combine the individual affine hulls into a single 1207 * 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