add isl_set_sample_point
[isl.git] / isl_convex_hull.c
blob31b575b84a96706e82ecc3e17d1b4e638197a449
1 /*
2 * Copyright 2008-2009 Katholieke Universiteit Leuven
4 * Use of this software is governed by the GNU LGPLv2.1 license
6 * Written by Sven Verdoolaege, K.U.Leuven, Departement
7 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
8 */
10 #include "isl_lp.h"
11 #include "isl_map.h"
12 #include "isl_map_private.h"
13 #include "isl_mat.h"
14 #include "isl_set.h"
15 #include "isl_seq.h"
16 #include "isl_equalities.h"
17 #include "isl_tab.h"
19 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set);
21 static void swap_ineq(struct isl_basic_map *bmap, unsigned i, unsigned j)
23 isl_int *t;
25 if (i != j) {
26 t = bmap->ineq[i];
27 bmap->ineq[i] = bmap->ineq[j];
28 bmap->ineq[j] = t;
32 /* Return 1 if constraint c is redundant with respect to the constraints
33 * in bmap. If c is a lower [upper] bound in some variable and bmap
34 * does not have a lower [upper] bound in that variable, then c cannot
35 * be redundant and we do not need solve any lp.
37 int isl_basic_map_constraint_is_redundant(struct isl_basic_map **bmap,
38 isl_int *c, isl_int *opt_n, isl_int *opt_d)
40 enum isl_lp_result res;
41 unsigned total;
42 int i, j;
44 if (!bmap)
45 return -1;
47 total = isl_basic_map_total_dim(*bmap);
48 for (i = 0; i < total; ++i) {
49 int sign;
50 if (isl_int_is_zero(c[1+i]))
51 continue;
52 sign = isl_int_sgn(c[1+i]);
53 for (j = 0; j < (*bmap)->n_ineq; ++j)
54 if (sign == isl_int_sgn((*bmap)->ineq[j][1+i]))
55 break;
56 if (j == (*bmap)->n_ineq)
57 break;
59 if (i < total)
60 return 0;
62 res = isl_basic_map_solve_lp(*bmap, 0, c, (*bmap)->ctx->one,
63 opt_n, opt_d, NULL);
64 if (res == isl_lp_unbounded)
65 return 0;
66 if (res == isl_lp_error)
67 return -1;
68 if (res == isl_lp_empty) {
69 *bmap = isl_basic_map_set_to_empty(*bmap);
70 return 0;
72 return !isl_int_is_neg(*opt_n);
75 int isl_basic_set_constraint_is_redundant(struct isl_basic_set **bset,
76 isl_int *c, isl_int *opt_n, isl_int *opt_d)
78 return isl_basic_map_constraint_is_redundant(
79 (struct isl_basic_map **)bset, c, opt_n, opt_d);
82 /* Compute the convex hull of a basic map, by removing the redundant
83 * constraints. If the minimal value along the normal of a constraint
84 * is the same if the constraint is removed, then the constraint is redundant.
86 * Alternatively, we could have intersected the basic map with the
87 * corresponding equality and the checked if the dimension was that
88 * of a facet.
90 struct isl_basic_map *isl_basic_map_convex_hull(struct isl_basic_map *bmap)
92 struct isl_tab *tab;
94 if (!bmap)
95 return NULL;
97 bmap = isl_basic_map_gauss(bmap, NULL);
98 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
99 return bmap;
100 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_REDUNDANT))
101 return bmap;
102 if (bmap->n_ineq <= 1)
103 return bmap;
105 tab = isl_tab_from_basic_map(bmap);
106 tab = isl_tab_detect_implicit_equalities(tab);
107 if (isl_tab_detect_redundant(tab) < 0)
108 goto error;
109 bmap = isl_basic_map_update_from_tab(bmap, tab);
110 isl_tab_free(tab);
111 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
112 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
113 return bmap;
114 error:
115 isl_tab_free(tab);
116 isl_basic_map_free(bmap);
117 return NULL;
120 struct isl_basic_set *isl_basic_set_convex_hull(struct isl_basic_set *bset)
122 return (struct isl_basic_set *)
123 isl_basic_map_convex_hull((struct isl_basic_map *)bset);
126 /* Check if the set set is bound in the direction of the affine
127 * constraint c and if so, set the constant term such that the
128 * resulting constraint is a bounding constraint for the set.
130 static int uset_is_bound(struct isl_set *set, isl_int *c, unsigned len)
132 int first;
133 int j;
134 isl_int opt;
135 isl_int opt_denom;
137 isl_int_init(opt);
138 isl_int_init(opt_denom);
139 first = 1;
140 for (j = 0; j < set->n; ++j) {
141 enum isl_lp_result res;
143 if (ISL_F_ISSET(set->p[j], ISL_BASIC_SET_EMPTY))
144 continue;
146 res = isl_basic_set_solve_lp(set->p[j],
147 0, c, set->ctx->one, &opt, &opt_denom, NULL);
148 if (res == isl_lp_unbounded)
149 break;
150 if (res == isl_lp_error)
151 goto error;
152 if (res == isl_lp_empty) {
153 set->p[j] = isl_basic_set_set_to_empty(set->p[j]);
154 if (!set->p[j])
155 goto error;
156 continue;
158 if (first || isl_int_is_neg(opt)) {
159 if (!isl_int_is_one(opt_denom))
160 isl_seq_scale(c, c, opt_denom, len);
161 isl_int_sub(c[0], c[0], opt);
163 first = 0;
165 isl_int_clear(opt);
166 isl_int_clear(opt_denom);
167 return j >= set->n;
168 error:
169 isl_int_clear(opt);
170 isl_int_clear(opt_denom);
171 return -1;
174 /* Check if "c" is a direction that is independent of the previously found "n"
175 * bounds in "dirs".
176 * If so, add it to the list, with the negative of the lower bound
177 * in the constant position, i.e., such that c corresponds to a bounding
178 * hyperplane (but not necessarily a facet).
179 * Assumes set "set" is bounded.
181 static int is_independent_bound(struct isl_set *set, isl_int *c,
182 struct isl_mat *dirs, int n)
184 int is_bound;
185 int i = 0;
187 isl_seq_cpy(dirs->row[n]+1, c+1, dirs->n_col-1);
188 if (n != 0) {
189 int pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
190 if (pos < 0)
191 return 0;
192 for (i = 0; i < n; ++i) {
193 int pos_i;
194 pos_i = isl_seq_first_non_zero(dirs->row[i]+1, dirs->n_col-1);
195 if (pos_i < pos)
196 continue;
197 if (pos_i > pos)
198 break;
199 isl_seq_elim(dirs->row[n]+1, dirs->row[i]+1, pos,
200 dirs->n_col-1, NULL);
201 pos = isl_seq_first_non_zero(dirs->row[n]+1, dirs->n_col-1);
202 if (pos < 0)
203 return 0;
207 is_bound = uset_is_bound(set, dirs->row[n], dirs->n_col);
208 if (is_bound != 1)
209 return is_bound;
210 isl_seq_normalize(set->ctx, dirs->row[n], dirs->n_col);
211 if (i < n) {
212 int k;
213 isl_int *t = dirs->row[n];
214 for (k = n; k > i; --k)
215 dirs->row[k] = dirs->row[k-1];
216 dirs->row[i] = t;
218 return 1;
221 /* Compute and return a maximal set of linearly independent bounds
222 * on the set "set", based on the constraints of the basic sets
223 * in "set".
225 static struct isl_mat *independent_bounds(struct isl_set *set)
227 int i, j, n;
228 struct isl_mat *dirs = NULL;
229 unsigned dim = isl_set_n_dim(set);
231 dirs = isl_mat_alloc(set->ctx, dim, 1+dim);
232 if (!dirs)
233 goto error;
235 n = 0;
236 for (i = 0; n < dim && i < set->n; ++i) {
237 int f;
238 struct isl_basic_set *bset = set->p[i];
240 for (j = 0; n < dim && j < bset->n_eq; ++j) {
241 f = is_independent_bound(set, bset->eq[j], dirs, n);
242 if (f < 0)
243 goto error;
244 if (f)
245 ++n;
247 for (j = 0; n < dim && j < bset->n_ineq; ++j) {
248 f = is_independent_bound(set, bset->ineq[j], dirs, n);
249 if (f < 0)
250 goto error;
251 if (f)
252 ++n;
255 dirs->n_row = n;
256 return dirs;
257 error:
258 isl_mat_free(dirs);
259 return NULL;
262 struct isl_basic_set *isl_basic_set_set_rational(struct isl_basic_set *bset)
264 if (!bset)
265 return NULL;
267 if (ISL_F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
268 return bset;
270 bset = isl_basic_set_cow(bset);
271 if (!bset)
272 return NULL;
274 ISL_F_SET(bset, ISL_BASIC_MAP_RATIONAL);
276 return isl_basic_set_finalize(bset);
279 static struct isl_set *isl_set_set_rational(struct isl_set *set)
281 int i;
283 set = isl_set_cow(set);
284 if (!set)
285 return NULL;
286 for (i = 0; i < set->n; ++i) {
287 set->p[i] = isl_basic_set_set_rational(set->p[i]);
288 if (!set->p[i])
289 goto error;
291 return set;
292 error:
293 isl_set_free(set);
294 return NULL;
297 static struct isl_basic_set *isl_basic_set_add_equality(
298 struct isl_basic_set *bset, isl_int *c)
300 int i;
301 unsigned dim;
303 if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
304 return bset;
306 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
307 isl_assert(bset->ctx, bset->n_div == 0, goto error);
308 dim = isl_basic_set_n_dim(bset);
309 bset = isl_basic_set_cow(bset);
310 bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
311 i = isl_basic_set_alloc_equality(bset);
312 if (i < 0)
313 goto error;
314 isl_seq_cpy(bset->eq[i], c, 1 + dim);
315 return bset;
316 error:
317 isl_basic_set_free(bset);
318 return NULL;
321 static struct isl_set *isl_set_add_basic_set_equality(struct isl_set *set, isl_int *c)
323 int i;
325 set = isl_set_cow(set);
326 if (!set)
327 return NULL;
328 for (i = 0; i < set->n; ++i) {
329 set->p[i] = isl_basic_set_add_equality(set->p[i], c);
330 if (!set->p[i])
331 goto error;
333 return set;
334 error:
335 isl_set_free(set);
336 return NULL;
339 /* Given a union of basic sets, construct the constraints for wrapping
340 * a facet around one of its ridges.
341 * In particular, if each of n the d-dimensional basic sets i in "set"
342 * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
343 * and is defined by the constraints
344 * [ 1 ]
345 * A_i [ x ] >= 0
347 * then the resulting set is of dimension n*(1+d) and has as constraints
349 * [ a_i ]
350 * A_i [ x_i ] >= 0
352 * a_i >= 0
354 * \sum_i x_{i,1} = 1
356 static struct isl_basic_set *wrap_constraints(struct isl_set *set)
358 struct isl_basic_set *lp;
359 unsigned n_eq;
360 unsigned n_ineq;
361 int i, j, k;
362 unsigned dim, lp_dim;
364 if (!set)
365 return NULL;
367 dim = 1 + isl_set_n_dim(set);
368 n_eq = 1;
369 n_ineq = set->n;
370 for (i = 0; i < set->n; ++i) {
371 n_eq += set->p[i]->n_eq;
372 n_ineq += set->p[i]->n_ineq;
374 lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq);
375 if (!lp)
376 return NULL;
377 lp_dim = isl_basic_set_n_dim(lp);
378 k = isl_basic_set_alloc_equality(lp);
379 isl_int_set_si(lp->eq[k][0], -1);
380 for (i = 0; i < set->n; ++i) {
381 isl_int_set_si(lp->eq[k][1+dim*i], 0);
382 isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
383 isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
385 for (i = 0; i < set->n; ++i) {
386 k = isl_basic_set_alloc_inequality(lp);
387 isl_seq_clr(lp->ineq[k], 1+lp_dim);
388 isl_int_set_si(lp->ineq[k][1+dim*i], 1);
390 for (j = 0; j < set->p[i]->n_eq; ++j) {
391 k = isl_basic_set_alloc_equality(lp);
392 isl_seq_clr(lp->eq[k], 1+dim*i);
393 isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
394 isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
397 for (j = 0; j < set->p[i]->n_ineq; ++j) {
398 k = isl_basic_set_alloc_inequality(lp);
399 isl_seq_clr(lp->ineq[k], 1+dim*i);
400 isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
401 isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
404 return lp;
407 /* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
408 * of that facet, compute the other facet of the convex hull that contains
409 * the ridge.
411 * We first transform the set such that the facet constraint becomes
413 * x_1 >= 0
415 * I.e., the facet lies in
417 * x_1 = 0
419 * and on that facet, the constraint that defines the ridge is
421 * x_2 >= 0
423 * (This transformation is not strictly needed, all that is needed is
424 * that the ridge contains the origin.)
426 * Since the ridge contains the origin, the cone of the convex hull
427 * will be of the form
429 * x_1 >= 0
430 * x_2 >= a x_1
432 * with this second constraint defining the new facet.
433 * The constant a is obtained by settting x_1 in the cone of the
434 * convex hull to 1 and minimizing x_2.
435 * Now, each element in the cone of the convex hull is the sum
436 * of elements in the cones of the basic sets.
437 * If a_i is the dilation factor of basic set i, then the problem
438 * we need to solve is
440 * min \sum_i x_{i,2}
441 * st
442 * \sum_i x_{i,1} = 1
443 * a_i >= 0
444 * [ a_i ]
445 * A [ x_i ] >= 0
447 * with
448 * [ 1 ]
449 * A_i [ x_i ] >= 0
451 * the constraints of each (transformed) basic set.
452 * If a = n/d, then the constraint defining the new facet (in the transformed
453 * space) is
455 * -n x_1 + d x_2 >= 0
457 * In the original space, we need to take the same combination of the
458 * corresponding constraints "facet" and "ridge".
460 * If a = -infty = "-1/0", then we just return the original facet constraint.
461 * This means that the facet is unbounded, but has a bounded intersection
462 * with the union of sets.
464 isl_int *isl_set_wrap_facet(__isl_keep isl_set *set,
465 isl_int *facet, isl_int *ridge)
467 int i;
468 struct isl_mat *T = NULL;
469 struct isl_basic_set *lp = NULL;
470 struct isl_vec *obj;
471 enum isl_lp_result res;
472 isl_int num, den;
473 unsigned dim;
475 set = isl_set_copy(set);
476 set = isl_set_set_rational(set);
478 dim = 1 + isl_set_n_dim(set);
479 T = isl_mat_alloc(set->ctx, 3, dim);
480 if (!T)
481 goto error;
482 isl_int_set_si(T->row[0][0], 1);
483 isl_seq_clr(T->row[0]+1, dim - 1);
484 isl_seq_cpy(T->row[1], facet, dim);
485 isl_seq_cpy(T->row[2], ridge, dim);
486 T = isl_mat_right_inverse(T);
487 set = isl_set_preimage(set, T);
488 T = NULL;
489 if (!set)
490 goto error;
491 lp = wrap_constraints(set);
492 obj = isl_vec_alloc(set->ctx, 1 + dim*set->n);
493 if (!obj)
494 goto error;
495 isl_int_set_si(obj->block.data[0], 0);
496 for (i = 0; i < set->n; ++i) {
497 isl_seq_clr(obj->block.data + 1 + dim*i, 2);
498 isl_int_set_si(obj->block.data[1 + dim*i+2], 1);
499 isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3);
501 isl_int_init(num);
502 isl_int_init(den);
503 res = isl_basic_set_solve_lp(lp, 0,
504 obj->block.data, set->ctx->one, &num, &den, NULL);
505 if (res == isl_lp_ok) {
506 isl_int_neg(num, num);
507 isl_seq_combine(facet, num, facet, den, ridge, dim);
509 isl_int_clear(num);
510 isl_int_clear(den);
511 isl_vec_free(obj);
512 isl_basic_set_free(lp);
513 isl_set_free(set);
514 isl_assert(set->ctx, res == isl_lp_ok || res == isl_lp_unbounded,
515 return NULL);
516 return facet;
517 error:
518 isl_basic_set_free(lp);
519 isl_mat_free(T);
520 isl_set_free(set);
521 return NULL;
524 /* Drop rows in "rows" that are redundant with respect to earlier rows,
525 * assuming that "rows" is of full column rank.
527 * We compute the column echelon form. The non-redundant rows are
528 * those that are the first to contain a non-zero entry in a column.
529 * All the other rows can be removed.
531 static __isl_give isl_mat *drop_redundant_rows(__isl_take isl_mat *rows)
533 struct isl_mat *H = NULL;
534 int col;
535 int row;
536 int last_row;
538 if (!rows)
539 return NULL;
541 isl_assert(rows->ctx, rows->n_row >= rows->n_col, goto error);
543 if (rows->n_row == rows->n_col)
544 return rows;
546 H = isl_mat_left_hermite(isl_mat_copy(rows), 0, NULL, NULL);
547 if (!H)
548 goto error;
550 last_row = rows->n_row;
551 for (col = rows->n_col - 1; col >= 0; --col) {
552 for (row = col; row < last_row; ++row)
553 if (!isl_int_is_zero(H->row[row][col]))
554 break;
555 isl_assert(rows->ctx, row < last_row, goto error);
556 if (row + 1 < last_row) {
557 rows = isl_mat_drop_rows(rows, row + 1, last_row - (row + 1));
558 if (rows->n_row == rows->n_col)
559 break;
561 last_row = row;
564 isl_mat_free(H);
566 return rows;
567 error:
568 isl_mat_free(H);
569 isl_mat_free(rows);
570 return NULL;
573 /* Given a set of d linearly independent bounding constraints of the
574 * convex hull of "set", compute the constraint of a facet of "set".
576 * We first compute the intersection with the first bounding hyperplane
577 * and remove the component corresponding to this hyperplane from
578 * other bounds (in homogeneous space).
579 * We then wrap around one of the remaining bounding constraints
580 * and continue the process until all bounding constraints have been
581 * taken into account.
582 * The resulting linear combination of the bounding constraints will
583 * correspond to a facet of the convex hull.
585 static struct isl_mat *initial_facet_constraint(struct isl_set *set,
586 struct isl_mat *bounds)
588 struct isl_set *slice = NULL;
589 struct isl_basic_set *face = NULL;
590 struct isl_mat *m, *U, *Q;
591 int i;
592 unsigned dim = isl_set_n_dim(set);
594 isl_assert(set->ctx, set->n > 0, goto error);
595 isl_assert(set->ctx, bounds->n_row == dim, goto error);
597 while (bounds->n_row > 1) {
598 slice = isl_set_copy(set);
599 slice = isl_set_add_basic_set_equality(slice, bounds->row[0]);
600 face = isl_set_affine_hull(slice);
601 if (!face)
602 goto error;
603 if (face->n_eq == 1) {
604 isl_basic_set_free(face);
605 break;
607 m = isl_mat_alloc(set->ctx, 1 + face->n_eq, 1 + dim);
608 if (!m)
609 goto error;
610 isl_int_set_si(m->row[0][0], 1);
611 isl_seq_clr(m->row[0]+1, dim);
612 for (i = 0; i < face->n_eq; ++i)
613 isl_seq_cpy(m->row[1 + i], face->eq[i], 1 + dim);
614 U = isl_mat_right_inverse(m);
615 Q = isl_mat_right_inverse(isl_mat_copy(U));
616 U = isl_mat_drop_cols(U, 1 + face->n_eq, dim - face->n_eq);
617 Q = isl_mat_drop_rows(Q, 1 + face->n_eq, dim - face->n_eq);
618 U = isl_mat_drop_cols(U, 0, 1);
619 Q = isl_mat_drop_rows(Q, 0, 1);
620 bounds = isl_mat_product(bounds, U);
621 bounds = drop_redundant_rows(bounds);
622 bounds = isl_mat_product(bounds, Q);
623 isl_assert(set->ctx, bounds->n_row > 1, goto error);
624 if (!isl_set_wrap_facet(set, bounds->row[0],
625 bounds->row[bounds->n_row-1]))
626 goto error;
627 isl_basic_set_free(face);
628 bounds->n_row--;
630 return bounds;
631 error:
632 isl_basic_set_free(face);
633 isl_mat_free(bounds);
634 return NULL;
637 /* Given the bounding constraint "c" of a facet of the convex hull of "set",
638 * compute a hyperplane description of the facet, i.e., compute the facets
639 * of the facet.
641 * We compute an affine transformation that transforms the constraint
643 * [ 1 ]
644 * c [ x ] = 0
646 * to the constraint
648 * z_1 = 0
650 * by computing the right inverse U of a matrix that starts with the rows
652 * [ 1 0 ]
653 * [ c ]
655 * Then
656 * [ 1 ] [ 1 ]
657 * [ x ] = U [ z ]
658 * and
659 * [ 1 ] [ 1 ]
660 * [ z ] = Q [ x ]
662 * with Q = U^{-1}
663 * Since z_1 is zero, we can drop this variable as well as the corresponding
664 * column of U to obtain
666 * [ 1 ] [ 1 ]
667 * [ x ] = U' [ z' ]
668 * and
669 * [ 1 ] [ 1 ]
670 * [ z' ] = Q' [ x ]
672 * with Q' equal to Q, but without the corresponding row.
673 * After computing the facets of the facet in the z' space,
674 * we convert them back to the x space through Q.
676 static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c)
678 struct isl_mat *m, *U, *Q;
679 struct isl_basic_set *facet = NULL;
680 struct isl_ctx *ctx;
681 unsigned dim;
683 ctx = set->ctx;
684 set = isl_set_copy(set);
685 dim = isl_set_n_dim(set);
686 m = isl_mat_alloc(set->ctx, 2, 1 + dim);
687 if (!m)
688 goto error;
689 isl_int_set_si(m->row[0][0], 1);
690 isl_seq_clr(m->row[0]+1, dim);
691 isl_seq_cpy(m->row[1], c, 1+dim);
692 U = isl_mat_right_inverse(m);
693 Q = isl_mat_right_inverse(isl_mat_copy(U));
694 U = isl_mat_drop_cols(U, 1, 1);
695 Q = isl_mat_drop_rows(Q, 1, 1);
696 set = isl_set_preimage(set, U);
697 facet = uset_convex_hull_wrap_bounded(set);
698 facet = isl_basic_set_preimage(facet, Q);
699 isl_assert(ctx, facet->n_eq == 0, goto error);
700 return facet;
701 error:
702 isl_basic_set_free(facet);
703 isl_set_free(set);
704 return NULL;
707 /* Given an initial facet constraint, compute the remaining facets.
708 * We do this by running through all facets found so far and computing
709 * the adjacent facets through wrapping, adding those facets that we
710 * hadn't already found before.
712 * For each facet we have found so far, we first compute its facets
713 * in the resulting convex hull. That is, we compute the ridges
714 * of the resulting convex hull contained in the facet.
715 * We also compute the corresponding facet in the current approximation
716 * of the convex hull. There is no need to wrap around the ridges
717 * in this facet since that would result in a facet that is already
718 * present in the current approximation.
720 * This function can still be significantly optimized by checking which of
721 * the facets of the basic sets are also facets of the convex hull and
722 * using all the facets so far to help in constructing the facets of the
723 * facets
724 * and/or
725 * using the technique in section "3.1 Ridge Generation" of
726 * "Extended Convex Hull" by Fukuda et al.
728 static struct isl_basic_set *extend(struct isl_basic_set *hull,
729 struct isl_set *set)
731 int i, j, f;
732 int k;
733 struct isl_basic_set *facet = NULL;
734 struct isl_basic_set *hull_facet = NULL;
735 unsigned dim;
737 if (!hull)
738 return NULL;
740 isl_assert(set->ctx, set->n > 0, goto error);
742 dim = isl_set_n_dim(set);
744 for (i = 0; i < hull->n_ineq; ++i) {
745 facet = compute_facet(set, hull->ineq[i]);
746 facet = isl_basic_set_add_equality(facet, hull->ineq[i]);
747 facet = isl_basic_set_gauss(facet, NULL);
748 facet = isl_basic_set_normalize_constraints(facet);
749 hull_facet = isl_basic_set_copy(hull);
750 hull_facet = isl_basic_set_add_equality(hull_facet, hull->ineq[i]);
751 hull_facet = isl_basic_set_gauss(hull_facet, NULL);
752 hull_facet = isl_basic_set_normalize_constraints(hull_facet);
753 if (!facet)
754 goto error;
755 hull = isl_basic_set_cow(hull);
756 hull = isl_basic_set_extend_dim(hull,
757 isl_dim_copy(hull->dim), 0, 0, facet->n_ineq);
758 for (j = 0; j < facet->n_ineq; ++j) {
759 for (f = 0; f < hull_facet->n_ineq; ++f)
760 if (isl_seq_eq(facet->ineq[j],
761 hull_facet->ineq[f], 1 + dim))
762 break;
763 if (f < hull_facet->n_ineq)
764 continue;
765 k = isl_basic_set_alloc_inequality(hull);
766 if (k < 0)
767 goto error;
768 isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
769 if (!isl_set_wrap_facet(set, hull->ineq[k], facet->ineq[j]))
770 goto error;
772 isl_basic_set_free(hull_facet);
773 isl_basic_set_free(facet);
775 hull = isl_basic_set_simplify(hull);
776 hull = isl_basic_set_finalize(hull);
777 return hull;
778 error:
779 isl_basic_set_free(hull_facet);
780 isl_basic_set_free(facet);
781 isl_basic_set_free(hull);
782 return NULL;
785 /* Special case for computing the convex hull of a one dimensional set.
786 * We simply collect the lower and upper bounds of each basic set
787 * and the biggest of those.
789 static struct isl_basic_set *convex_hull_1d(struct isl_set *set)
791 struct isl_mat *c = NULL;
792 isl_int *lower = NULL;
793 isl_int *upper = NULL;
794 int i, j, k;
795 isl_int a, b;
796 struct isl_basic_set *hull;
798 for (i = 0; i < set->n; ++i) {
799 set->p[i] = isl_basic_set_simplify(set->p[i]);
800 if (!set->p[i])
801 goto error;
803 set = isl_set_remove_empty_parts(set);
804 if (!set)
805 goto error;
806 isl_assert(set->ctx, set->n > 0, goto error);
807 c = isl_mat_alloc(set->ctx, 2, 2);
808 if (!c)
809 goto error;
811 if (set->p[0]->n_eq > 0) {
812 isl_assert(set->ctx, set->p[0]->n_eq == 1, goto error);
813 lower = c->row[0];
814 upper = c->row[1];
815 if (isl_int_is_pos(set->p[0]->eq[0][1])) {
816 isl_seq_cpy(lower, set->p[0]->eq[0], 2);
817 isl_seq_neg(upper, set->p[0]->eq[0], 2);
818 } else {
819 isl_seq_neg(lower, set->p[0]->eq[0], 2);
820 isl_seq_cpy(upper, set->p[0]->eq[0], 2);
822 } else {
823 for (j = 0; j < set->p[0]->n_ineq; ++j) {
824 if (isl_int_is_pos(set->p[0]->ineq[j][1])) {
825 lower = c->row[0];
826 isl_seq_cpy(lower, set->p[0]->ineq[j], 2);
827 } else {
828 upper = c->row[1];
829 isl_seq_cpy(upper, set->p[0]->ineq[j], 2);
834 isl_int_init(a);
835 isl_int_init(b);
836 for (i = 0; i < set->n; ++i) {
837 struct isl_basic_set *bset = set->p[i];
838 int has_lower = 0;
839 int has_upper = 0;
841 for (j = 0; j < bset->n_eq; ++j) {
842 has_lower = 1;
843 has_upper = 1;
844 if (lower) {
845 isl_int_mul(a, lower[0], bset->eq[j][1]);
846 isl_int_mul(b, lower[1], bset->eq[j][0]);
847 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
848 isl_seq_cpy(lower, bset->eq[j], 2);
849 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
850 isl_seq_neg(lower, bset->eq[j], 2);
852 if (upper) {
853 isl_int_mul(a, upper[0], bset->eq[j][1]);
854 isl_int_mul(b, upper[1], bset->eq[j][0]);
855 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
856 isl_seq_neg(upper, bset->eq[j], 2);
857 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
858 isl_seq_cpy(upper, bset->eq[j], 2);
861 for (j = 0; j < bset->n_ineq; ++j) {
862 if (isl_int_is_pos(bset->ineq[j][1]))
863 has_lower = 1;
864 if (isl_int_is_neg(bset->ineq[j][1]))
865 has_upper = 1;
866 if (lower && isl_int_is_pos(bset->ineq[j][1])) {
867 isl_int_mul(a, lower[0], bset->ineq[j][1]);
868 isl_int_mul(b, lower[1], bset->ineq[j][0]);
869 if (isl_int_lt(a, b))
870 isl_seq_cpy(lower, bset->ineq[j], 2);
872 if (upper && isl_int_is_neg(bset->ineq[j][1])) {
873 isl_int_mul(a, upper[0], bset->ineq[j][1]);
874 isl_int_mul(b, upper[1], bset->ineq[j][0]);
875 if (isl_int_gt(a, b))
876 isl_seq_cpy(upper, bset->ineq[j], 2);
879 if (!has_lower)
880 lower = NULL;
881 if (!has_upper)
882 upper = NULL;
884 isl_int_clear(a);
885 isl_int_clear(b);
887 hull = isl_basic_set_alloc(set->ctx, 0, 1, 0, 0, 2);
888 hull = isl_basic_set_set_rational(hull);
889 if (!hull)
890 goto error;
891 if (lower) {
892 k = isl_basic_set_alloc_inequality(hull);
893 isl_seq_cpy(hull->ineq[k], lower, 2);
895 if (upper) {
896 k = isl_basic_set_alloc_inequality(hull);
897 isl_seq_cpy(hull->ineq[k], upper, 2);
899 hull = isl_basic_set_finalize(hull);
900 isl_set_free(set);
901 isl_mat_free(c);
902 return hull;
903 error:
904 isl_set_free(set);
905 isl_mat_free(c);
906 return NULL;
909 /* Project out final n dimensions using Fourier-Motzkin */
910 static struct isl_set *set_project_out(struct isl_ctx *ctx,
911 struct isl_set *set, unsigned n)
913 return isl_set_remove_dims(set, isl_set_n_dim(set) - n, n);
916 static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
918 struct isl_basic_set *convex_hull;
920 if (!set)
921 return NULL;
923 if (isl_set_is_empty(set))
924 convex_hull = isl_basic_set_empty(isl_dim_copy(set->dim));
925 else
926 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
927 isl_set_free(set);
928 return convex_hull;
931 /* Compute the convex hull of a pair of basic sets without any parameters or
932 * integer divisions using Fourier-Motzkin elimination.
933 * The convex hull is the set of all points that can be written as
934 * the sum of points from both basic sets (in homogeneous coordinates).
935 * We set up the constraints in a space with dimensions for each of
936 * the three sets and then project out the dimensions corresponding
937 * to the two original basic sets, retaining only those corresponding
938 * to the convex hull.
940 static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1,
941 struct isl_basic_set *bset2)
943 int i, j, k;
944 struct isl_basic_set *bset[2];
945 struct isl_basic_set *hull = NULL;
946 unsigned dim;
948 if (!bset1 || !bset2)
949 goto error;
951 dim = isl_basic_set_n_dim(bset1);
952 hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0,
953 1 + dim + bset1->n_eq + bset2->n_eq,
954 2 + bset1->n_ineq + bset2->n_ineq);
955 bset[0] = bset1;
956 bset[1] = bset2;
957 for (i = 0; i < 2; ++i) {
958 for (j = 0; j < bset[i]->n_eq; ++j) {
959 k = isl_basic_set_alloc_equality(hull);
960 if (k < 0)
961 goto error;
962 isl_seq_clr(hull->eq[k], (i+1) * (1+dim));
963 isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
964 isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j],
965 1+dim);
967 for (j = 0; j < bset[i]->n_ineq; ++j) {
968 k = isl_basic_set_alloc_inequality(hull);
969 if (k < 0)
970 goto error;
971 isl_seq_clr(hull->ineq[k], (i+1) * (1+dim));
972 isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
973 isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim),
974 bset[i]->ineq[j], 1+dim);
976 k = isl_basic_set_alloc_inequality(hull);
977 if (k < 0)
978 goto error;
979 isl_seq_clr(hull->ineq[k], 1+2+3*dim);
980 isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1);
982 for (j = 0; j < 1+dim; ++j) {
983 k = isl_basic_set_alloc_equality(hull);
984 if (k < 0)
985 goto error;
986 isl_seq_clr(hull->eq[k], 1+2+3*dim);
987 isl_int_set_si(hull->eq[k][j], -1);
988 isl_int_set_si(hull->eq[k][1+dim+j], 1);
989 isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1);
991 hull = isl_basic_set_set_rational(hull);
992 hull = isl_basic_set_remove_dims(hull, dim, 2*(1+dim));
993 hull = isl_basic_set_convex_hull(hull);
994 isl_basic_set_free(bset1);
995 isl_basic_set_free(bset2);
996 return hull;
997 error:
998 isl_basic_set_free(bset1);
999 isl_basic_set_free(bset2);
1000 isl_basic_set_free(hull);
1001 return NULL;
1004 static int isl_basic_set_is_bounded(struct isl_basic_set *bset)
1006 struct isl_tab *tab;
1007 int bounded;
1009 tab = isl_tab_from_recession_cone(bset);
1010 bounded = isl_tab_cone_is_bounded(tab);
1011 isl_tab_free(tab);
1012 return bounded;
1015 static int isl_set_is_bounded(struct isl_set *set)
1017 int i;
1019 for (i = 0; i < set->n; ++i) {
1020 int bounded = isl_basic_set_is_bounded(set->p[i]);
1021 if (!bounded || bounded < 0)
1022 return bounded;
1024 return 1;
1027 /* Compute the lineality space of the convex hull of bset1 and bset2.
1029 * We first compute the intersection of the recession cone of bset1
1030 * with the negative of the recession cone of bset2 and then compute
1031 * the linear hull of the resulting cone.
1033 static struct isl_basic_set *induced_lineality_space(
1034 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1036 int i, k;
1037 struct isl_basic_set *lin = NULL;
1038 unsigned dim;
1040 if (!bset1 || !bset2)
1041 goto error;
1043 dim = isl_basic_set_total_dim(bset1);
1044 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1), 0,
1045 bset1->n_eq + bset2->n_eq,
1046 bset1->n_ineq + bset2->n_ineq);
1047 lin = isl_basic_set_set_rational(lin);
1048 if (!lin)
1049 goto error;
1050 for (i = 0; i < bset1->n_eq; ++i) {
1051 k = isl_basic_set_alloc_equality(lin);
1052 if (k < 0)
1053 goto error;
1054 isl_int_set_si(lin->eq[k][0], 0);
1055 isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
1057 for (i = 0; i < bset1->n_ineq; ++i) {
1058 k = isl_basic_set_alloc_inequality(lin);
1059 if (k < 0)
1060 goto error;
1061 isl_int_set_si(lin->ineq[k][0], 0);
1062 isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
1064 for (i = 0; i < bset2->n_eq; ++i) {
1065 k = isl_basic_set_alloc_equality(lin);
1066 if (k < 0)
1067 goto error;
1068 isl_int_set_si(lin->eq[k][0], 0);
1069 isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
1071 for (i = 0; i < bset2->n_ineq; ++i) {
1072 k = isl_basic_set_alloc_inequality(lin);
1073 if (k < 0)
1074 goto error;
1075 isl_int_set_si(lin->ineq[k][0], 0);
1076 isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
1079 isl_basic_set_free(bset1);
1080 isl_basic_set_free(bset2);
1081 return isl_basic_set_affine_hull(lin);
1082 error:
1083 isl_basic_set_free(lin);
1084 isl_basic_set_free(bset1);
1085 isl_basic_set_free(bset2);
1086 return NULL;
1089 static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
1091 /* Given a set and a linear space "lin" of dimension n > 0,
1092 * project the linear space from the set, compute the convex hull
1093 * and then map the set back to the original space.
1095 * Let
1097 * M x = 0
1099 * describe the linear space. We first compute the Hermite normal
1100 * form H = M U of M = H Q, to obtain
1102 * H Q x = 0
1104 * The last n rows of H will be zero, so the last n variables of x' = Q x
1105 * are the one we want to project out. We do this by transforming each
1106 * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
1107 * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
1108 * we transform the hull back to the original space as A' Q_1 x >= b',
1109 * with Q_1 all but the last n rows of Q.
1111 static struct isl_basic_set *modulo_lineality(struct isl_set *set,
1112 struct isl_basic_set *lin)
1114 unsigned total = isl_basic_set_total_dim(lin);
1115 unsigned lin_dim;
1116 struct isl_basic_set *hull;
1117 struct isl_mat *M, *U, *Q;
1119 if (!set || !lin)
1120 goto error;
1121 lin_dim = total - lin->n_eq;
1122 M = isl_mat_sub_alloc(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
1123 M = isl_mat_left_hermite(M, 0, &U, &Q);
1124 if (!M)
1125 goto error;
1126 isl_mat_free(M);
1127 isl_basic_set_free(lin);
1129 Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim);
1131 U = isl_mat_lin_to_aff(U);
1132 Q = isl_mat_lin_to_aff(Q);
1134 set = isl_set_preimage(set, U);
1135 set = isl_set_remove_dims(set, total - lin_dim, lin_dim);
1136 hull = uset_convex_hull(set);
1137 hull = isl_basic_set_preimage(hull, Q);
1139 return hull;
1140 error:
1141 isl_basic_set_free(lin);
1142 isl_set_free(set);
1143 return NULL;
1146 /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
1147 * set up an LP for solving
1149 * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
1151 * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
1152 * The next \alpha{ij} correspond to the equalities and come in pairs.
1153 * The final \alpha{ij} correspond to the inequalities.
1155 static struct isl_basic_set *valid_direction_lp(
1156 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1158 struct isl_dim *dim;
1159 struct isl_basic_set *lp;
1160 unsigned d;
1161 int n;
1162 int i, j, k;
1164 if (!bset1 || !bset2)
1165 goto error;
1166 d = 1 + isl_basic_set_total_dim(bset1);
1167 n = 2 +
1168 2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
1169 dim = isl_dim_set_alloc(bset1->ctx, 0, n);
1170 lp = isl_basic_set_alloc_dim(dim, 0, d, n);
1171 if (!lp)
1172 goto error;
1173 for (i = 0; i < n; ++i) {
1174 k = isl_basic_set_alloc_inequality(lp);
1175 if (k < 0)
1176 goto error;
1177 isl_seq_clr(lp->ineq[k] + 1, n);
1178 isl_int_set_si(lp->ineq[k][0], -1);
1179 isl_int_set_si(lp->ineq[k][1 + i], 1);
1181 for (i = 0; i < d; ++i) {
1182 k = isl_basic_set_alloc_equality(lp);
1183 if (k < 0)
1184 goto error;
1185 n = 0;
1186 isl_int_set_si(lp->eq[k][n++], 0);
1187 /* positivity constraint 1 >= 0 */
1188 isl_int_set_si(lp->eq[k][n++], i == 0);
1189 for (j = 0; j < bset1->n_eq; ++j) {
1190 isl_int_set(lp->eq[k][n++], bset1->eq[j][i]);
1191 isl_int_neg(lp->eq[k][n++], bset1->eq[j][i]);
1193 for (j = 0; j < bset1->n_ineq; ++j)
1194 isl_int_set(lp->eq[k][n++], bset1->ineq[j][i]);
1195 /* positivity constraint 1 >= 0 */
1196 isl_int_set_si(lp->eq[k][n++], -(i == 0));
1197 for (j = 0; j < bset2->n_eq; ++j) {
1198 isl_int_neg(lp->eq[k][n++], bset2->eq[j][i]);
1199 isl_int_set(lp->eq[k][n++], bset2->eq[j][i]);
1201 for (j = 0; j < bset2->n_ineq; ++j)
1202 isl_int_neg(lp->eq[k][n++], bset2->ineq[j][i]);
1204 lp = isl_basic_set_gauss(lp, NULL);
1205 isl_basic_set_free(bset1);
1206 isl_basic_set_free(bset2);
1207 return lp;
1208 error:
1209 isl_basic_set_free(bset1);
1210 isl_basic_set_free(bset2);
1211 return NULL;
1214 /* Compute a vector s in the homogeneous space such that <s, r> > 0
1215 * for all rays in the homogeneous space of the two cones that correspond
1216 * to the input polyhedra bset1 and bset2.
1218 * We compute s as a vector that satisfies
1220 * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*)
1222 * with h_{ij} the normals of the facets of polyhedron i
1223 * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
1224 * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1.
1225 * We first set up an LP with as variables the \alpha{ij}.
1226 * In this formulateion, for each polyhedron i,
1227 * the first constraint is the positivity constraint, followed by pairs
1228 * of variables for the equalities, followed by variables for the inequalities.
1229 * We then simply pick a feasible solution and compute s using (*).
1231 * Note that we simply pick any valid direction and make no attempt
1232 * to pick a "good" or even the "best" valid direction.
1234 static struct isl_vec *valid_direction(
1235 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1237 struct isl_basic_set *lp;
1238 struct isl_tab *tab;
1239 struct isl_vec *sample = NULL;
1240 struct isl_vec *dir;
1241 unsigned d;
1242 int i;
1243 int n;
1245 if (!bset1 || !bset2)
1246 goto error;
1247 lp = valid_direction_lp(isl_basic_set_copy(bset1),
1248 isl_basic_set_copy(bset2));
1249 tab = isl_tab_from_basic_set(lp);
1250 sample = isl_tab_get_sample_value(tab);
1251 isl_tab_free(tab);
1252 isl_basic_set_free(lp);
1253 if (!sample)
1254 goto error;
1255 d = isl_basic_set_total_dim(bset1);
1256 dir = isl_vec_alloc(bset1->ctx, 1 + d);
1257 if (!dir)
1258 goto error;
1259 isl_seq_clr(dir->block.data + 1, dir->size - 1);
1260 n = 1;
1261 /* positivity constraint 1 >= 0 */
1262 isl_int_set(dir->block.data[0], sample->block.data[n++]);
1263 for (i = 0; i < bset1->n_eq; ++i) {
1264 isl_int_sub(sample->block.data[n],
1265 sample->block.data[n], sample->block.data[n+1]);
1266 isl_seq_combine(dir->block.data,
1267 bset1->ctx->one, dir->block.data,
1268 sample->block.data[n], bset1->eq[i], 1 + d);
1270 n += 2;
1272 for (i = 0; i < bset1->n_ineq; ++i)
1273 isl_seq_combine(dir->block.data,
1274 bset1->ctx->one, dir->block.data,
1275 sample->block.data[n++], bset1->ineq[i], 1 + d);
1276 isl_vec_free(sample);
1277 isl_seq_normalize(bset1->ctx, dir->block.data + 1, dir->size - 1);
1278 isl_basic_set_free(bset1);
1279 isl_basic_set_free(bset2);
1280 return dir;
1281 error:
1282 isl_vec_free(sample);
1283 isl_basic_set_free(bset1);
1284 isl_basic_set_free(bset2);
1285 return NULL;
1288 /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
1289 * compute b_i' + A_i' x' >= 0, with
1291 * [ b_i A_i ] [ y' ] [ y' ]
1292 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1294 * In particular, add the "positivity constraint" and then perform
1295 * the mapping.
1297 static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
1298 struct isl_mat *T)
1300 int k;
1302 if (!bset)
1303 goto error;
1304 bset = isl_basic_set_extend_constraints(bset, 0, 1);
1305 k = isl_basic_set_alloc_inequality(bset);
1306 if (k < 0)
1307 goto error;
1308 isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
1309 isl_int_set_si(bset->ineq[k][0], 1);
1310 bset = isl_basic_set_preimage(bset, T);
1311 return bset;
1312 error:
1313 isl_mat_free(T);
1314 isl_basic_set_free(bset);
1315 return NULL;
1318 /* Compute the convex hull of a pair of basic sets without any parameters or
1319 * integer divisions, where the convex hull is known to be pointed,
1320 * but the basic sets may be unbounded.
1322 * We turn this problem into the computation of a convex hull of a pair
1323 * _bounded_ polyhedra by "changing the direction of the homogeneous
1324 * dimension". This idea is due to Matthias Koeppe.
1326 * Consider the cones in homogeneous space that correspond to the
1327 * input polyhedra. The rays of these cones are also rays of the
1328 * polyhedra if the coordinate that corresponds to the homogeneous
1329 * dimension is zero. That is, if the inner product of the rays
1330 * with the homogeneous direction is zero.
1331 * The cones in the homogeneous space can also be considered to
1332 * correspond to other pairs of polyhedra by chosing a different
1333 * homogeneous direction. To ensure that both of these polyhedra
1334 * are bounded, we need to make sure that all rays of the cones
1335 * correspond to vertices and not to rays.
1336 * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
1337 * Then using s as a homogeneous direction, we obtain a pair of polytopes.
1338 * The vector s is computed in valid_direction.
1340 * Note that we need to consider _all_ rays of the cones and not just
1341 * the rays that correspond to rays in the polyhedra. If we were to
1342 * only consider those rays and turn them into vertices, then we
1343 * may inadvertently turn some vertices into rays.
1345 * The standard homogeneous direction is the unit vector in the 0th coordinate.
1346 * We therefore transform the two polyhedra such that the selected
1347 * direction is mapped onto this standard direction and then proceed
1348 * with the normal computation.
1349 * Let S be a non-singular square matrix with s as its first row,
1350 * then we want to map the polyhedra to the space
1352 * [ y' ] [ y ] [ y ] [ y' ]
1353 * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ]
1355 * We take S to be the unimodular completion of s to limit the growth
1356 * of the coefficients in the following computations.
1358 * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
1359 * We first move to the homogeneous dimension
1361 * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ]
1362 * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ]
1364 * Then we change directoin
1366 * [ b_i A_i ] [ y' ] [ y' ]
1367 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1369 * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
1370 * resulting in b' + A' x' >= 0, which we then convert back
1372 * [ y ] [ y ]
1373 * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0
1375 * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
1377 static struct isl_basic_set *convex_hull_pair_pointed(
1378 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1380 struct isl_ctx *ctx = NULL;
1381 struct isl_vec *dir = NULL;
1382 struct isl_mat *T = NULL;
1383 struct isl_mat *T2 = NULL;
1384 struct isl_basic_set *hull;
1385 struct isl_set *set;
1387 if (!bset1 || !bset2)
1388 goto error;
1389 ctx = bset1->ctx;
1390 dir = valid_direction(isl_basic_set_copy(bset1),
1391 isl_basic_set_copy(bset2));
1392 if (!dir)
1393 goto error;
1394 T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
1395 if (!T)
1396 goto error;
1397 isl_seq_cpy(T->row[0], dir->block.data, dir->size);
1398 T = isl_mat_unimodular_complete(T, 1);
1399 T2 = isl_mat_right_inverse(isl_mat_copy(T));
1401 bset1 = homogeneous_map(bset1, isl_mat_copy(T2));
1402 bset2 = homogeneous_map(bset2, T2);
1403 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1404 set = isl_set_add_basic_set(set, bset1);
1405 set = isl_set_add_basic_set(set, bset2);
1406 hull = uset_convex_hull(set);
1407 hull = isl_basic_set_preimage(hull, T);
1409 isl_vec_free(dir);
1411 return hull;
1412 error:
1413 isl_vec_free(dir);
1414 isl_basic_set_free(bset1);
1415 isl_basic_set_free(bset2);
1416 return NULL;
1419 /* Compute the convex hull of a pair of basic sets without any parameters or
1420 * integer divisions.
1422 * If the convex hull of the two basic sets would have a non-trivial
1423 * lineality space, we first project out this lineality space.
1425 static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
1426 struct isl_basic_set *bset2)
1428 struct isl_basic_set *lin;
1430 if (isl_basic_set_is_bounded(bset1) || isl_basic_set_is_bounded(bset2))
1431 return convex_hull_pair_pointed(bset1, bset2);
1433 lin = induced_lineality_space(isl_basic_set_copy(bset1),
1434 isl_basic_set_copy(bset2));
1435 if (!lin)
1436 goto error;
1437 if (isl_basic_set_is_universe(lin)) {
1438 isl_basic_set_free(bset1);
1439 isl_basic_set_free(bset2);
1440 return lin;
1442 if (lin->n_eq < isl_basic_set_total_dim(lin)) {
1443 struct isl_set *set;
1444 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1445 set = isl_set_add_basic_set(set, bset1);
1446 set = isl_set_add_basic_set(set, bset2);
1447 return modulo_lineality(set, lin);
1449 isl_basic_set_free(lin);
1451 return convex_hull_pair_pointed(bset1, bset2);
1452 error:
1453 isl_basic_set_free(bset1);
1454 isl_basic_set_free(bset2);
1455 return NULL;
1458 /* Compute the lineality space of a basic set.
1459 * We currently do not allow the basic set to have any divs.
1460 * We basically just drop the constants and turn every inequality
1461 * into an equality.
1463 struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset)
1465 int i, k;
1466 struct isl_basic_set *lin = NULL;
1467 unsigned dim;
1469 if (!bset)
1470 goto error;
1471 isl_assert(bset->ctx, bset->n_div == 0, goto error);
1472 dim = isl_basic_set_total_dim(bset);
1474 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset), 0, dim, 0);
1475 if (!lin)
1476 goto error;
1477 for (i = 0; i < bset->n_eq; ++i) {
1478 k = isl_basic_set_alloc_equality(lin);
1479 if (k < 0)
1480 goto error;
1481 isl_int_set_si(lin->eq[k][0], 0);
1482 isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
1484 lin = isl_basic_set_gauss(lin, NULL);
1485 if (!lin)
1486 goto error;
1487 for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
1488 k = isl_basic_set_alloc_equality(lin);
1489 if (k < 0)
1490 goto error;
1491 isl_int_set_si(lin->eq[k][0], 0);
1492 isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
1493 lin = isl_basic_set_gauss(lin, NULL);
1494 if (!lin)
1495 goto error;
1497 isl_basic_set_free(bset);
1498 return lin;
1499 error:
1500 isl_basic_set_free(lin);
1501 isl_basic_set_free(bset);
1502 return NULL;
1505 /* Compute the (linear) hull of the lineality spaces of the basic sets in the
1506 * "underlying" set "set".
1508 static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
1510 int i;
1511 struct isl_set *lin = NULL;
1513 if (!set)
1514 return NULL;
1515 if (set->n == 0) {
1516 struct isl_dim *dim = isl_set_get_dim(set);
1517 isl_set_free(set);
1518 return isl_basic_set_empty(dim);
1521 lin = isl_set_alloc_dim(isl_set_get_dim(set), set->n, 0);
1522 for (i = 0; i < set->n; ++i)
1523 lin = isl_set_add_basic_set(lin,
1524 isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i])));
1525 isl_set_free(set);
1526 return isl_set_affine_hull(lin);
1529 /* Compute the convex hull of a set without any parameters or
1530 * integer divisions.
1531 * In each step, we combined two basic sets until only one
1532 * basic set is left.
1533 * The input basic sets are assumed not to have a non-trivial
1534 * lineality space. If any of the intermediate results has
1535 * a non-trivial lineality space, it is projected out.
1537 static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
1539 struct isl_basic_set *convex_hull = NULL;
1541 convex_hull = isl_set_copy_basic_set(set);
1542 set = isl_set_drop_basic_set(set, convex_hull);
1543 if (!set)
1544 goto error;
1545 while (set->n > 0) {
1546 struct isl_basic_set *t;
1547 t = isl_set_copy_basic_set(set);
1548 if (!t)
1549 goto error;
1550 set = isl_set_drop_basic_set(set, t);
1551 if (!set)
1552 goto error;
1553 convex_hull = convex_hull_pair(convex_hull, t);
1554 if (set->n == 0)
1555 break;
1556 t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull));
1557 if (!t)
1558 goto error;
1559 if (isl_basic_set_is_universe(t)) {
1560 isl_basic_set_free(convex_hull);
1561 convex_hull = t;
1562 break;
1564 if (t->n_eq < isl_basic_set_total_dim(t)) {
1565 set = isl_set_add_basic_set(set, convex_hull);
1566 return modulo_lineality(set, t);
1568 isl_basic_set_free(t);
1570 isl_set_free(set);
1571 return convex_hull;
1572 error:
1573 isl_set_free(set);
1574 isl_basic_set_free(convex_hull);
1575 return NULL;
1578 /* Compute an initial hull for wrapping containing a single initial
1579 * facet by first computing bounds on the set and then using these
1580 * bounds to construct an initial facet.
1581 * This function is a remnant of an older implementation where the
1582 * bounds were also used to check whether the set was bounded.
1583 * Since this function will now only be called when we know the
1584 * set to be bounded, the initial facet should probably be constructed
1585 * by simply using the coordinate directions instead.
1587 static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
1588 struct isl_set *set)
1590 struct isl_mat *bounds = NULL;
1591 unsigned dim;
1592 int k;
1594 if (!hull)
1595 goto error;
1596 bounds = independent_bounds(set);
1597 if (!bounds)
1598 goto error;
1599 isl_assert(set->ctx, bounds->n_row == isl_set_n_dim(set), goto error);
1600 bounds = initial_facet_constraint(set, bounds);
1601 if (!bounds)
1602 goto error;
1603 k = isl_basic_set_alloc_inequality(hull);
1604 if (k < 0)
1605 goto error;
1606 dim = isl_set_n_dim(set);
1607 isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error);
1608 isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col);
1609 isl_mat_free(bounds);
1611 return hull;
1612 error:
1613 isl_basic_set_free(hull);
1614 isl_mat_free(bounds);
1615 return NULL;
1618 struct max_constraint {
1619 struct isl_mat *c;
1620 int count;
1621 int ineq;
1624 static int max_constraint_equal(const void *entry, const void *val)
1626 struct max_constraint *a = (struct max_constraint *)entry;
1627 isl_int *b = (isl_int *)val;
1629 return isl_seq_eq(a->c->row[0] + 1, b, a->c->n_col - 1);
1632 static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1633 isl_int *con, unsigned len, int n, int ineq)
1635 struct isl_hash_table_entry *entry;
1636 struct max_constraint *c;
1637 uint32_t c_hash;
1639 c_hash = isl_seq_get_hash(con + 1, len);
1640 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1641 con + 1, 0);
1642 if (!entry)
1643 return;
1644 c = entry->data;
1645 if (c->count < n) {
1646 isl_hash_table_remove(ctx, table, entry);
1647 return;
1649 c->count++;
1650 if (isl_int_gt(c->c->row[0][0], con[0]))
1651 return;
1652 if (isl_int_eq(c->c->row[0][0], con[0])) {
1653 if (ineq)
1654 c->ineq = ineq;
1655 return;
1657 c->c = isl_mat_cow(c->c);
1658 isl_int_set(c->c->row[0][0], con[0]);
1659 c->ineq = ineq;
1662 /* Check whether the constraint hash table "table" constains the constraint
1663 * "con".
1665 static int has_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1666 isl_int *con, unsigned len, int n)
1668 struct isl_hash_table_entry *entry;
1669 struct max_constraint *c;
1670 uint32_t c_hash;
1672 c_hash = isl_seq_get_hash(con + 1, len);
1673 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1674 con + 1, 0);
1675 if (!entry)
1676 return 0;
1677 c = entry->data;
1678 if (c->count < n)
1679 return 0;
1680 return isl_int_eq(c->c->row[0][0], con[0]);
1683 /* Check for inequality constraints of a basic set without equalities
1684 * such that the same or more stringent copies of the constraint appear
1685 * in all of the basic sets. Such constraints are necessarily facet
1686 * constraints of the convex hull.
1688 * If the resulting basic set is by chance identical to one of
1689 * the basic sets in "set", then we know that this basic set contains
1690 * all other basic sets and is therefore the convex hull of set.
1691 * In this case we set *is_hull to 1.
1693 static struct isl_basic_set *common_constraints(struct isl_basic_set *hull,
1694 struct isl_set *set, int *is_hull)
1696 int i, j, s, n;
1697 int min_constraints;
1698 int best;
1699 struct max_constraint *constraints = NULL;
1700 struct isl_hash_table *table = NULL;
1701 unsigned total;
1703 *is_hull = 0;
1705 for (i = 0; i < set->n; ++i)
1706 if (set->p[i]->n_eq == 0)
1707 break;
1708 if (i >= set->n)
1709 return hull;
1710 min_constraints = set->p[i]->n_ineq;
1711 best = i;
1712 for (i = best + 1; i < set->n; ++i) {
1713 if (set->p[i]->n_eq != 0)
1714 continue;
1715 if (set->p[i]->n_ineq >= min_constraints)
1716 continue;
1717 min_constraints = set->p[i]->n_ineq;
1718 best = i;
1720 constraints = isl_calloc_array(hull->ctx, struct max_constraint,
1721 min_constraints);
1722 if (!constraints)
1723 return hull;
1724 table = isl_alloc_type(hull->ctx, struct isl_hash_table);
1725 if (isl_hash_table_init(hull->ctx, table, min_constraints))
1726 goto error;
1728 total = isl_dim_total(set->dim);
1729 for (i = 0; i < set->p[best]->n_ineq; ++i) {
1730 constraints[i].c = isl_mat_sub_alloc(hull->ctx,
1731 set->p[best]->ineq + i, 0, 1, 0, 1 + total);
1732 if (!constraints[i].c)
1733 goto error;
1734 constraints[i].ineq = 1;
1736 for (i = 0; i < min_constraints; ++i) {
1737 struct isl_hash_table_entry *entry;
1738 uint32_t c_hash;
1739 c_hash = isl_seq_get_hash(constraints[i].c->row[0] + 1, total);
1740 entry = isl_hash_table_find(hull->ctx, table, c_hash,
1741 max_constraint_equal, constraints[i].c->row[0] + 1, 1);
1742 if (!entry)
1743 goto error;
1744 isl_assert(hull->ctx, !entry->data, goto error);
1745 entry->data = &constraints[i];
1748 n = 0;
1749 for (s = 0; s < set->n; ++s) {
1750 if (s == best)
1751 continue;
1753 for (i = 0; i < set->p[s]->n_eq; ++i) {
1754 isl_int *eq = set->p[s]->eq[i];
1755 for (j = 0; j < 2; ++j) {
1756 isl_seq_neg(eq, eq, 1 + total);
1757 update_constraint(hull->ctx, table,
1758 eq, total, n, 0);
1761 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1762 isl_int *ineq = set->p[s]->ineq[i];
1763 update_constraint(hull->ctx, table, ineq, total, n,
1764 set->p[s]->n_eq == 0);
1766 ++n;
1769 for (i = 0; i < min_constraints; ++i) {
1770 if (constraints[i].count < n)
1771 continue;
1772 if (!constraints[i].ineq)
1773 continue;
1774 j = isl_basic_set_alloc_inequality(hull);
1775 if (j < 0)
1776 goto error;
1777 isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total);
1780 for (s = 0; s < set->n; ++s) {
1781 if (set->p[s]->n_eq)
1782 continue;
1783 if (set->p[s]->n_ineq != hull->n_ineq)
1784 continue;
1785 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1786 isl_int *ineq = set->p[s]->ineq[i];
1787 if (!has_constraint(hull->ctx, table, ineq, total, n))
1788 break;
1790 if (i == set->p[s]->n_ineq)
1791 *is_hull = 1;
1794 isl_hash_table_clear(table);
1795 for (i = 0; i < min_constraints; ++i)
1796 isl_mat_free(constraints[i].c);
1797 free(constraints);
1798 free(table);
1799 return hull;
1800 error:
1801 isl_hash_table_clear(table);
1802 free(table);
1803 if (constraints)
1804 for (i = 0; i < min_constraints; ++i)
1805 isl_mat_free(constraints[i].c);
1806 free(constraints);
1807 return hull;
1810 /* Create a template for the convex hull of "set" and fill it up
1811 * obvious facet constraints, if any. If the result happens to
1812 * be the convex hull of "set" then *is_hull is set to 1.
1814 static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull)
1816 struct isl_basic_set *hull;
1817 unsigned n_ineq;
1818 int i;
1820 n_ineq = 1;
1821 for (i = 0; i < set->n; ++i) {
1822 n_ineq += set->p[i]->n_eq;
1823 n_ineq += set->p[i]->n_ineq;
1825 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
1826 hull = isl_basic_set_set_rational(hull);
1827 if (!hull)
1828 return NULL;
1829 return common_constraints(hull, set, is_hull);
1832 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
1834 struct isl_basic_set *hull;
1835 int is_hull;
1837 hull = proto_hull(set, &is_hull);
1838 if (hull && !is_hull) {
1839 if (hull->n_ineq == 0)
1840 hull = initial_hull(hull, set);
1841 hull = extend(hull, set);
1843 isl_set_free(set);
1845 return hull;
1848 /* Compute the convex hull of a set without any parameters or
1849 * integer divisions. Depending on whether the set is bounded,
1850 * we pass control to the wrapping based convex hull or
1851 * the Fourier-Motzkin elimination based convex hull.
1852 * We also handle a few special cases before checking the boundedness.
1854 static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
1856 struct isl_basic_set *convex_hull = NULL;
1857 struct isl_basic_set *lin;
1859 if (isl_set_n_dim(set) == 0)
1860 return convex_hull_0d(set);
1862 set = isl_set_coalesce(set);
1863 set = isl_set_set_rational(set);
1865 if (!set)
1866 goto error;
1867 if (!set)
1868 return NULL;
1869 if (set->n == 1) {
1870 convex_hull = isl_basic_set_copy(set->p[0]);
1871 isl_set_free(set);
1872 return convex_hull;
1874 if (isl_set_n_dim(set) == 1)
1875 return convex_hull_1d(set);
1877 if (isl_set_is_bounded(set))
1878 return uset_convex_hull_wrap(set);
1880 lin = uset_combined_lineality_space(isl_set_copy(set));
1881 if (!lin)
1882 goto error;
1883 if (isl_basic_set_is_universe(lin)) {
1884 isl_set_free(set);
1885 return lin;
1887 if (lin->n_eq < isl_basic_set_total_dim(lin))
1888 return modulo_lineality(set, lin);
1889 isl_basic_set_free(lin);
1891 return uset_convex_hull_unbounded(set);
1892 error:
1893 isl_set_free(set);
1894 isl_basic_set_free(convex_hull);
1895 return NULL;
1898 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1899 * without parameters or divs and where the convex hull of set is
1900 * known to be full-dimensional.
1902 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
1904 struct isl_basic_set *convex_hull = NULL;
1906 if (isl_set_n_dim(set) == 0) {
1907 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
1908 isl_set_free(set);
1909 convex_hull = isl_basic_set_set_rational(convex_hull);
1910 return convex_hull;
1913 set = isl_set_set_rational(set);
1915 if (!set)
1916 goto error;
1917 set = isl_set_coalesce(set);
1918 if (!set)
1919 goto error;
1920 if (set->n == 1) {
1921 convex_hull = isl_basic_set_copy(set->p[0]);
1922 isl_set_free(set);
1923 return convex_hull;
1925 if (isl_set_n_dim(set) == 1)
1926 return convex_hull_1d(set);
1928 return uset_convex_hull_wrap(set);
1929 error:
1930 isl_set_free(set);
1931 return NULL;
1934 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1935 * We first remove the equalities (transforming the set), compute the
1936 * convex hull of the transformed set and then add the equalities back
1937 * (after performing the inverse transformation.
1939 static struct isl_basic_set *modulo_affine_hull(struct isl_ctx *ctx,
1940 struct isl_set *set, struct isl_basic_set *affine_hull)
1942 struct isl_mat *T;
1943 struct isl_mat *T2;
1944 struct isl_basic_set *dummy;
1945 struct isl_basic_set *convex_hull;
1947 dummy = isl_basic_set_remove_equalities(
1948 isl_basic_set_copy(affine_hull), &T, &T2);
1949 if (!dummy)
1950 goto error;
1951 isl_basic_set_free(dummy);
1952 set = isl_set_preimage(set, T);
1953 convex_hull = uset_convex_hull(set);
1954 convex_hull = isl_basic_set_preimage(convex_hull, T2);
1955 convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
1956 return convex_hull;
1957 error:
1958 isl_basic_set_free(affine_hull);
1959 isl_set_free(set);
1960 return NULL;
1963 /* Compute the convex hull of a map.
1965 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1966 * specifically, the wrapping of facets to obtain new facets.
1968 struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
1970 struct isl_basic_set *bset;
1971 struct isl_basic_map *model = NULL;
1972 struct isl_basic_set *affine_hull = NULL;
1973 struct isl_basic_map *convex_hull = NULL;
1974 struct isl_set *set = NULL;
1975 struct isl_ctx *ctx;
1977 if (!map)
1978 goto error;
1980 ctx = map->ctx;
1981 if (map->n == 0) {
1982 convex_hull = isl_basic_map_empty_like_map(map);
1983 isl_map_free(map);
1984 return convex_hull;
1987 map = isl_map_detect_equalities(map);
1988 map = isl_map_align_divs(map);
1989 model = isl_basic_map_copy(map->p[0]);
1990 set = isl_map_underlying_set(map);
1991 if (!set)
1992 goto error;
1994 affine_hull = isl_set_affine_hull(isl_set_copy(set));
1995 if (!affine_hull)
1996 goto error;
1997 if (affine_hull->n_eq != 0)
1998 bset = modulo_affine_hull(ctx, set, affine_hull);
1999 else {
2000 isl_basic_set_free(affine_hull);
2001 bset = uset_convex_hull(set);
2004 convex_hull = isl_basic_map_overlying_set(bset, model);
2006 ISL_F_SET(convex_hull, ISL_BASIC_MAP_NO_IMPLICIT);
2007 ISL_F_SET(convex_hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2008 ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL);
2009 return convex_hull;
2010 error:
2011 isl_set_free(set);
2012 isl_basic_map_free(model);
2013 return NULL;
2016 struct isl_basic_set *isl_set_convex_hull(struct isl_set *set)
2018 return (struct isl_basic_set *)
2019 isl_map_convex_hull((struct isl_map *)set);
2022 struct sh_data_entry {
2023 struct isl_hash_table *table;
2024 struct isl_tab *tab;
2027 /* Holds the data needed during the simple hull computation.
2028 * In particular,
2029 * n the number of basic sets in the original set
2030 * hull_table a hash table of already computed constraints
2031 * in the simple hull
2032 * p for each basic set,
2033 * table a hash table of the constraints
2034 * tab the tableau corresponding to the basic set
2036 struct sh_data {
2037 struct isl_ctx *ctx;
2038 unsigned n;
2039 struct isl_hash_table *hull_table;
2040 struct sh_data_entry p[1];
2043 static void sh_data_free(struct sh_data *data)
2045 int i;
2047 if (!data)
2048 return;
2049 isl_hash_table_free(data->ctx, data->hull_table);
2050 for (i = 0; i < data->n; ++i) {
2051 isl_hash_table_free(data->ctx, data->p[i].table);
2052 isl_tab_free(data->p[i].tab);
2054 free(data);
2057 struct ineq_cmp_data {
2058 unsigned len;
2059 isl_int *p;
2062 static int has_ineq(const void *entry, const void *val)
2064 isl_int *row = (isl_int *)entry;
2065 struct ineq_cmp_data *v = (struct ineq_cmp_data *)val;
2067 return isl_seq_eq(row + 1, v->p + 1, v->len) ||
2068 isl_seq_is_neg(row + 1, v->p + 1, v->len);
2071 static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table,
2072 isl_int *ineq, unsigned len)
2074 uint32_t c_hash;
2075 struct ineq_cmp_data v;
2076 struct isl_hash_table_entry *entry;
2078 v.len = len;
2079 v.p = ineq;
2080 c_hash = isl_seq_get_hash(ineq + 1, len);
2081 entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1);
2082 if (!entry)
2083 return - 1;
2084 entry->data = ineq;
2085 return 0;
2088 /* Fill hash table "table" with the constraints of "bset".
2089 * Equalities are added as two inequalities.
2090 * The value in the hash table is a pointer to the (in)equality of "bset".
2092 static int hash_basic_set(struct isl_hash_table *table,
2093 struct isl_basic_set *bset)
2095 int i, j;
2096 unsigned dim = isl_basic_set_total_dim(bset);
2098 for (i = 0; i < bset->n_eq; ++i) {
2099 for (j = 0; j < 2; ++j) {
2100 isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim);
2101 if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0)
2102 return -1;
2105 for (i = 0; i < bset->n_ineq; ++i) {
2106 if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0)
2107 return -1;
2109 return 0;
2112 static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq)
2114 struct sh_data *data;
2115 int i;
2117 data = isl_calloc(set->ctx, struct sh_data,
2118 sizeof(struct sh_data) +
2119 (set->n - 1) * sizeof(struct sh_data_entry));
2120 if (!data)
2121 return NULL;
2122 data->ctx = set->ctx;
2123 data->n = set->n;
2124 data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq);
2125 if (!data->hull_table)
2126 goto error;
2127 for (i = 0; i < set->n; ++i) {
2128 data->p[i].table = isl_hash_table_alloc(set->ctx,
2129 2 * set->p[i]->n_eq + set->p[i]->n_ineq);
2130 if (!data->p[i].table)
2131 goto error;
2132 if (hash_basic_set(data->p[i].table, set->p[i]) < 0)
2133 goto error;
2135 return data;
2136 error:
2137 sh_data_free(data);
2138 return NULL;
2141 /* Check if inequality "ineq" is a bound for basic set "j" or if
2142 * it can be relaxed (by increasing the constant term) to become
2143 * a bound for that basic set. In the latter case, the constant
2144 * term is updated.
2145 * Return 1 if "ineq" is a bound
2146 * 0 if "ineq" may attain arbitrarily small values on basic set "j"
2147 * -1 if some error occurred
2149 static int is_bound(struct sh_data *data, struct isl_set *set, int j,
2150 isl_int *ineq)
2152 enum isl_lp_result res;
2153 isl_int opt;
2155 if (!data->p[j].tab) {
2156 data->p[j].tab = isl_tab_from_basic_set(set->p[j]);
2157 if (!data->p[j].tab)
2158 return -1;
2161 isl_int_init(opt);
2163 res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one,
2164 &opt, NULL, 0);
2165 if (res == isl_lp_ok && isl_int_is_neg(opt))
2166 isl_int_sub(ineq[0], ineq[0], opt);
2168 isl_int_clear(opt);
2170 return res == isl_lp_ok ? 1 :
2171 res == isl_lp_unbounded ? 0 : -1;
2174 /* Check if inequality "ineq" from basic set "i" can be relaxed to
2175 * become a bound on the whole set. If so, add the (relaxed) inequality
2176 * to "hull".
2178 * We first check if "hull" already contains a translate of the inequality.
2179 * If so, we are done.
2180 * Then, we check if any of the previous basic sets contains a translate
2181 * of the inequality. If so, then we have already considered this
2182 * inequality and we are done.
2183 * Otherwise, for each basic set other than "i", we check if the inequality
2184 * is a bound on the basic set.
2185 * For previous basic sets, we know that they do not contain a translate
2186 * of the inequality, so we directly call is_bound.
2187 * For following basic sets, we first check if a translate of the
2188 * inequality appears in its description and if so directly update
2189 * the inequality accordingly.
2191 static struct isl_basic_set *add_bound(struct isl_basic_set *hull,
2192 struct sh_data *data, struct isl_set *set, int i, isl_int *ineq)
2194 uint32_t c_hash;
2195 struct ineq_cmp_data v;
2196 struct isl_hash_table_entry *entry;
2197 int j, k;
2199 if (!hull)
2200 return NULL;
2202 v.len = isl_basic_set_total_dim(hull);
2203 v.p = ineq;
2204 c_hash = isl_seq_get_hash(ineq + 1, v.len);
2206 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2207 has_ineq, &v, 0);
2208 if (entry)
2209 return hull;
2211 for (j = 0; j < i; ++j) {
2212 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2213 c_hash, has_ineq, &v, 0);
2214 if (entry)
2215 break;
2217 if (j < i)
2218 return hull;
2220 k = isl_basic_set_alloc_inequality(hull);
2221 isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
2222 if (k < 0)
2223 goto error;
2225 for (j = 0; j < i; ++j) {
2226 int bound;
2227 bound = is_bound(data, set, j, hull->ineq[k]);
2228 if (bound < 0)
2229 goto error;
2230 if (!bound)
2231 break;
2233 if (j < i) {
2234 isl_basic_set_free_inequality(hull, 1);
2235 return hull;
2238 for (j = i + 1; j < set->n; ++j) {
2239 int bound, neg;
2240 isl_int *ineq_j;
2241 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2242 c_hash, has_ineq, &v, 0);
2243 if (entry) {
2244 ineq_j = entry->data;
2245 neg = isl_seq_is_neg(ineq_j + 1,
2246 hull->ineq[k] + 1, v.len);
2247 if (neg)
2248 isl_int_neg(ineq_j[0], ineq_j[0]);
2249 if (isl_int_gt(ineq_j[0], hull->ineq[k][0]))
2250 isl_int_set(hull->ineq[k][0], ineq_j[0]);
2251 if (neg)
2252 isl_int_neg(ineq_j[0], ineq_j[0]);
2253 continue;
2255 bound = is_bound(data, set, j, hull->ineq[k]);
2256 if (bound < 0)
2257 goto error;
2258 if (!bound)
2259 break;
2261 if (j < set->n) {
2262 isl_basic_set_free_inequality(hull, 1);
2263 return hull;
2266 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2267 has_ineq, &v, 1);
2268 if (!entry)
2269 goto error;
2270 entry->data = hull->ineq[k];
2272 return hull;
2273 error:
2274 isl_basic_set_free(hull);
2275 return NULL;
2278 /* Check if any inequality from basic set "i" can be relaxed to
2279 * become a bound on the whole set. If so, add the (relaxed) inequality
2280 * to "hull".
2282 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
2283 struct sh_data *data, struct isl_set *set, int i)
2285 int j, k;
2286 unsigned dim = isl_basic_set_total_dim(bset);
2288 for (j = 0; j < set->p[i]->n_eq; ++j) {
2289 for (k = 0; k < 2; ++k) {
2290 isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim);
2291 add_bound(bset, data, set, i, set->p[i]->eq[j]);
2294 for (j = 0; j < set->p[i]->n_ineq; ++j)
2295 add_bound(bset, data, set, i, set->p[i]->ineq[j]);
2296 return bset;
2299 /* Compute a superset of the convex hull of set that is described
2300 * by only translates of the constraints in the constituents of set.
2302 static struct isl_basic_set *uset_simple_hull(struct isl_set *set)
2304 struct sh_data *data = NULL;
2305 struct isl_basic_set *hull = NULL;
2306 unsigned n_ineq;
2307 int i;
2309 if (!set)
2310 return NULL;
2312 n_ineq = 0;
2313 for (i = 0; i < set->n; ++i) {
2314 if (!set->p[i])
2315 goto error;
2316 n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq;
2319 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
2320 if (!hull)
2321 goto error;
2323 data = sh_data_alloc(set, n_ineq);
2324 if (!data)
2325 goto error;
2327 for (i = 0; i < set->n; ++i)
2328 hull = add_bounds(hull, data, set, i);
2330 sh_data_free(data);
2331 isl_set_free(set);
2333 return hull;
2334 error:
2335 sh_data_free(data);
2336 isl_basic_set_free(hull);
2337 isl_set_free(set);
2338 return NULL;
2341 /* Compute a superset of the convex hull of map that is described
2342 * by only translates of the constraints in the constituents of map.
2344 struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
2346 struct isl_set *set = NULL;
2347 struct isl_basic_map *model = NULL;
2348 struct isl_basic_map *hull;
2349 struct isl_basic_map *affine_hull;
2350 struct isl_basic_set *bset = NULL;
2352 if (!map)
2353 return NULL;
2354 if (map->n == 0) {
2355 hull = isl_basic_map_empty_like_map(map);
2356 isl_map_free(map);
2357 return hull;
2359 if (map->n == 1) {
2360 hull = isl_basic_map_copy(map->p[0]);
2361 isl_map_free(map);
2362 return hull;
2365 map = isl_map_detect_equalities(map);
2366 affine_hull = isl_map_affine_hull(isl_map_copy(map));
2367 map = isl_map_align_divs(map);
2368 model = isl_basic_map_copy(map->p[0]);
2370 set = isl_map_underlying_set(map);
2372 bset = uset_simple_hull(set);
2374 hull = isl_basic_map_overlying_set(bset, model);
2376 hull = isl_basic_map_intersect(hull, affine_hull);
2377 hull = isl_basic_map_convex_hull(hull);
2378 ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT);
2379 ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2381 return hull;
2384 struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
2386 return (struct isl_basic_set *)
2387 isl_map_simple_hull((struct isl_map *)set);
2390 /* Given a set "set", return parametric bounds on the dimension "dim".
2392 static struct isl_basic_set *set_bounds(struct isl_set *set, int dim)
2394 unsigned set_dim = isl_set_dim(set, isl_dim_set);
2395 set = isl_set_copy(set);
2396 set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1));
2397 set = isl_set_eliminate_dims(set, 0, dim);
2398 return isl_set_convex_hull(set);
2401 /* Computes a "simple hull" and then check if each dimension in the
2402 * resulting hull is bounded by a symbolic constant. If not, the
2403 * hull is intersected with the corresponding bounds on the whole set.
2405 struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set)
2407 int i, j;
2408 struct isl_basic_set *hull;
2409 unsigned nparam, left;
2410 int removed_divs = 0;
2412 hull = isl_set_simple_hull(isl_set_copy(set));
2413 if (!hull)
2414 goto error;
2416 nparam = isl_basic_set_dim(hull, isl_dim_param);
2417 for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) {
2418 int lower = 0, upper = 0;
2419 struct isl_basic_set *bounds;
2421 left = isl_basic_set_total_dim(hull) - nparam - i - 1;
2422 for (j = 0; j < hull->n_eq; ++j) {
2423 if (isl_int_is_zero(hull->eq[j][1 + nparam + i]))
2424 continue;
2425 if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1,
2426 left) == -1)
2427 break;
2429 if (j < hull->n_eq)
2430 continue;
2432 for (j = 0; j < hull->n_ineq; ++j) {
2433 if (isl_int_is_zero(hull->ineq[j][1 + nparam + i]))
2434 continue;
2435 if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1,
2436 left) != -1 ||
2437 isl_seq_first_non_zero(hull->ineq[j]+1+nparam,
2438 i) != -1)
2439 continue;
2440 if (isl_int_is_pos(hull->ineq[j][1 + nparam + i]))
2441 lower = 1;
2442 else
2443 upper = 1;
2444 if (lower && upper)
2445 break;
2448 if (lower && upper)
2449 continue;
2451 if (!removed_divs) {
2452 set = isl_set_remove_divs(set);
2453 if (!set)
2454 goto error;
2455 removed_divs = 1;
2457 bounds = set_bounds(set, i);
2458 hull = isl_basic_set_intersect(hull, bounds);
2459 if (!hull)
2460 goto error;
2463 isl_set_free(set);
2464 return hull;
2465 error:
2466 isl_set_free(set);
2467 return NULL;