add rudimentary isl_morph
[isl.git] / isl_convex_hull.c
blob17212c36d5c94d000ac28547d59a0dc1e249293b
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 struct isl_basic_set *isl_basic_set_set_rational(struct isl_basic_set *bset)
176 if (!bset)
177 return NULL;
179 if (ISL_F_ISSET(bset, ISL_BASIC_MAP_RATIONAL))
180 return bset;
182 bset = isl_basic_set_cow(bset);
183 if (!bset)
184 return NULL;
186 ISL_F_SET(bset, ISL_BASIC_MAP_RATIONAL);
188 return isl_basic_set_finalize(bset);
191 static struct isl_set *isl_set_set_rational(struct isl_set *set)
193 int i;
195 set = isl_set_cow(set);
196 if (!set)
197 return NULL;
198 for (i = 0; i < set->n; ++i) {
199 set->p[i] = isl_basic_set_set_rational(set->p[i]);
200 if (!set->p[i])
201 goto error;
203 return set;
204 error:
205 isl_set_free(set);
206 return NULL;
209 static struct isl_basic_set *isl_basic_set_add_equality(
210 struct isl_basic_set *bset, isl_int *c)
212 int i;
213 unsigned dim;
215 if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
216 return bset;
218 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
219 isl_assert(bset->ctx, bset->n_div == 0, goto error);
220 dim = isl_basic_set_n_dim(bset);
221 bset = isl_basic_set_cow(bset);
222 bset = isl_basic_set_extend(bset, 0, dim, 0, 1, 0);
223 i = isl_basic_set_alloc_equality(bset);
224 if (i < 0)
225 goto error;
226 isl_seq_cpy(bset->eq[i], c, 1 + dim);
227 return bset;
228 error:
229 isl_basic_set_free(bset);
230 return NULL;
233 static struct isl_set *isl_set_add_basic_set_equality(struct isl_set *set, isl_int *c)
235 int i;
237 set = isl_set_cow(set);
238 if (!set)
239 return NULL;
240 for (i = 0; i < set->n; ++i) {
241 set->p[i] = isl_basic_set_add_equality(set->p[i], c);
242 if (!set->p[i])
243 goto error;
245 return set;
246 error:
247 isl_set_free(set);
248 return NULL;
251 /* Given a union of basic sets, construct the constraints for wrapping
252 * a facet around one of its ridges.
253 * In particular, if each of n the d-dimensional basic sets i in "set"
254 * contains the origin, satisfies the constraints x_1 >= 0 and x_2 >= 0
255 * and is defined by the constraints
256 * [ 1 ]
257 * A_i [ x ] >= 0
259 * then the resulting set is of dimension n*(1+d) and has as constraints
261 * [ a_i ]
262 * A_i [ x_i ] >= 0
264 * a_i >= 0
266 * \sum_i x_{i,1} = 1
268 static struct isl_basic_set *wrap_constraints(struct isl_set *set)
270 struct isl_basic_set *lp;
271 unsigned n_eq;
272 unsigned n_ineq;
273 int i, j, k;
274 unsigned dim, lp_dim;
276 if (!set)
277 return NULL;
279 dim = 1 + isl_set_n_dim(set);
280 n_eq = 1;
281 n_ineq = set->n;
282 for (i = 0; i < set->n; ++i) {
283 n_eq += set->p[i]->n_eq;
284 n_ineq += set->p[i]->n_ineq;
286 lp = isl_basic_set_alloc(set->ctx, 0, dim * set->n, 0, n_eq, n_ineq);
287 if (!lp)
288 return NULL;
289 lp_dim = isl_basic_set_n_dim(lp);
290 k = isl_basic_set_alloc_equality(lp);
291 isl_int_set_si(lp->eq[k][0], -1);
292 for (i = 0; i < set->n; ++i) {
293 isl_int_set_si(lp->eq[k][1+dim*i], 0);
294 isl_int_set_si(lp->eq[k][1+dim*i+1], 1);
295 isl_seq_clr(lp->eq[k]+1+dim*i+2, dim-2);
297 for (i = 0; i < set->n; ++i) {
298 k = isl_basic_set_alloc_inequality(lp);
299 isl_seq_clr(lp->ineq[k], 1+lp_dim);
300 isl_int_set_si(lp->ineq[k][1+dim*i], 1);
302 for (j = 0; j < set->p[i]->n_eq; ++j) {
303 k = isl_basic_set_alloc_equality(lp);
304 isl_seq_clr(lp->eq[k], 1+dim*i);
305 isl_seq_cpy(lp->eq[k]+1+dim*i, set->p[i]->eq[j], dim);
306 isl_seq_clr(lp->eq[k]+1+dim*(i+1), dim*(set->n-i-1));
309 for (j = 0; j < set->p[i]->n_ineq; ++j) {
310 k = isl_basic_set_alloc_inequality(lp);
311 isl_seq_clr(lp->ineq[k], 1+dim*i);
312 isl_seq_cpy(lp->ineq[k]+1+dim*i, set->p[i]->ineq[j], dim);
313 isl_seq_clr(lp->ineq[k]+1+dim*(i+1), dim*(set->n-i-1));
316 return lp;
319 /* Given a facet "facet" of the convex hull of "set" and a facet "ridge"
320 * of that facet, compute the other facet of the convex hull that contains
321 * the ridge.
323 * We first transform the set such that the facet constraint becomes
325 * x_1 >= 0
327 * I.e., the facet lies in
329 * x_1 = 0
331 * and on that facet, the constraint that defines the ridge is
333 * x_2 >= 0
335 * (This transformation is not strictly needed, all that is needed is
336 * that the ridge contains the origin.)
338 * Since the ridge contains the origin, the cone of the convex hull
339 * will be of the form
341 * x_1 >= 0
342 * x_2 >= a x_1
344 * with this second constraint defining the new facet.
345 * The constant a is obtained by settting x_1 in the cone of the
346 * convex hull to 1 and minimizing x_2.
347 * Now, each element in the cone of the convex hull is the sum
348 * of elements in the cones of the basic sets.
349 * If a_i is the dilation factor of basic set i, then the problem
350 * we need to solve is
352 * min \sum_i x_{i,2}
353 * st
354 * \sum_i x_{i,1} = 1
355 * a_i >= 0
356 * [ a_i ]
357 * A [ x_i ] >= 0
359 * with
360 * [ 1 ]
361 * A_i [ x_i ] >= 0
363 * the constraints of each (transformed) basic set.
364 * If a = n/d, then the constraint defining the new facet (in the transformed
365 * space) is
367 * -n x_1 + d x_2 >= 0
369 * In the original space, we need to take the same combination of the
370 * corresponding constraints "facet" and "ridge".
372 * If a = -infty = "-1/0", then we just return the original facet constraint.
373 * This means that the facet is unbounded, but has a bounded intersection
374 * with the union of sets.
376 isl_int *isl_set_wrap_facet(__isl_keep isl_set *set,
377 isl_int *facet, isl_int *ridge)
379 int i;
380 struct isl_mat *T = NULL;
381 struct isl_basic_set *lp = NULL;
382 struct isl_vec *obj;
383 enum isl_lp_result res;
384 isl_int num, den;
385 unsigned dim;
387 set = isl_set_copy(set);
388 set = isl_set_set_rational(set);
390 dim = 1 + isl_set_n_dim(set);
391 T = isl_mat_alloc(set->ctx, 3, dim);
392 if (!T)
393 goto error;
394 isl_int_set_si(T->row[0][0], 1);
395 isl_seq_clr(T->row[0]+1, dim - 1);
396 isl_seq_cpy(T->row[1], facet, dim);
397 isl_seq_cpy(T->row[2], ridge, dim);
398 T = isl_mat_right_inverse(T);
399 set = isl_set_preimage(set, T);
400 T = NULL;
401 if (!set)
402 goto error;
403 lp = wrap_constraints(set);
404 obj = isl_vec_alloc(set->ctx, 1 + dim*set->n);
405 if (!obj)
406 goto error;
407 isl_int_set_si(obj->block.data[0], 0);
408 for (i = 0; i < set->n; ++i) {
409 isl_seq_clr(obj->block.data + 1 + dim*i, 2);
410 isl_int_set_si(obj->block.data[1 + dim*i+2], 1);
411 isl_seq_clr(obj->block.data + 1 + dim*i+3, dim-3);
413 isl_int_init(num);
414 isl_int_init(den);
415 res = isl_basic_set_solve_lp(lp, 0,
416 obj->block.data, set->ctx->one, &num, &den, NULL);
417 if (res == isl_lp_ok) {
418 isl_int_neg(num, num);
419 isl_seq_combine(facet, num, facet, den, ridge, dim);
421 isl_int_clear(num);
422 isl_int_clear(den);
423 isl_vec_free(obj);
424 isl_basic_set_free(lp);
425 isl_set_free(set);
426 isl_assert(set->ctx, res == isl_lp_ok || res == isl_lp_unbounded,
427 return NULL);
428 return facet;
429 error:
430 isl_basic_set_free(lp);
431 isl_mat_free(T);
432 isl_set_free(set);
433 return NULL;
436 /* Compute the constraint of a facet of "set".
438 * We first compute the intersection with a bounding constraint
439 * that is orthogonal to one of the coordinate axes.
440 * If the affine hull of this intersection has only one equality,
441 * we have found a facet.
442 * Otherwise, we wrap the current bounding constraint around
443 * one of the equalities of the face (one that is not equal to
444 * the current bounding constraint).
445 * This process continues until we have found a facet.
446 * The dimension of the intersection increases by at least
447 * one on each iteration, so termination is guaranteed.
449 static __isl_give isl_mat *initial_facet_constraint(__isl_keep isl_set *set)
451 struct isl_set *slice = NULL;
452 struct isl_basic_set *face = NULL;
453 struct isl_mat *m, *U, *Q;
454 int i;
455 unsigned dim = isl_set_n_dim(set);
456 int is_bound;
457 isl_mat *bounds;
459 isl_assert(set->ctx, set->n > 0, goto error);
460 bounds = isl_mat_alloc(set->ctx, 1, 1 + dim);
461 if (!bounds)
462 return NULL;
464 isl_seq_clr(bounds->row[0], dim);
465 isl_int_set_si(bounds->row[0][1 + dim - 1], 1);
466 is_bound = uset_is_bound(set, bounds->row[0], 1 + dim);
467 isl_assert(set->ctx, is_bound == 1, goto error);
468 isl_seq_normalize(set->ctx, bounds->row[0], 1 + dim);
469 bounds->n_row = 1;
471 for (;;) {
472 slice = isl_set_copy(set);
473 slice = isl_set_add_basic_set_equality(slice, bounds->row[0]);
474 face = isl_set_affine_hull(slice);
475 if (!face)
476 goto error;
477 if (face->n_eq == 1) {
478 isl_basic_set_free(face);
479 break;
481 for (i = 0; i < face->n_eq; ++i)
482 if (!isl_seq_eq(bounds->row[0], face->eq[i], 1 + dim) &&
483 !isl_seq_is_neg(bounds->row[0],
484 face->eq[i], 1 + dim))
485 break;
486 isl_assert(set->ctx, i < face->n_eq, goto error);
487 if (!isl_set_wrap_facet(set, bounds->row[0], face->eq[i]))
488 goto error;
489 isl_seq_normalize(set->ctx, bounds->row[0], bounds->n_col);
490 isl_basic_set_free(face);
493 return bounds;
494 error:
495 isl_basic_set_free(face);
496 isl_mat_free(bounds);
497 return NULL;
500 /* Given the bounding constraint "c" of a facet of the convex hull of "set",
501 * compute a hyperplane description of the facet, i.e., compute the facets
502 * of the facet.
504 * We compute an affine transformation that transforms the constraint
506 * [ 1 ]
507 * c [ x ] = 0
509 * to the constraint
511 * z_1 = 0
513 * by computing the right inverse U of a matrix that starts with the rows
515 * [ 1 0 ]
516 * [ c ]
518 * Then
519 * [ 1 ] [ 1 ]
520 * [ x ] = U [ z ]
521 * and
522 * [ 1 ] [ 1 ]
523 * [ z ] = Q [ x ]
525 * with Q = U^{-1}
526 * Since z_1 is zero, we can drop this variable as well as the corresponding
527 * column of U to obtain
529 * [ 1 ] [ 1 ]
530 * [ x ] = U' [ z' ]
531 * and
532 * [ 1 ] [ 1 ]
533 * [ z' ] = Q' [ x ]
535 * with Q' equal to Q, but without the corresponding row.
536 * After computing the facets of the facet in the z' space,
537 * we convert them back to the x space through Q.
539 static struct isl_basic_set *compute_facet(struct isl_set *set, isl_int *c)
541 struct isl_mat *m, *U, *Q;
542 struct isl_basic_set *facet = NULL;
543 struct isl_ctx *ctx;
544 unsigned dim;
546 ctx = set->ctx;
547 set = isl_set_copy(set);
548 dim = isl_set_n_dim(set);
549 m = isl_mat_alloc(set->ctx, 2, 1 + dim);
550 if (!m)
551 goto error;
552 isl_int_set_si(m->row[0][0], 1);
553 isl_seq_clr(m->row[0]+1, dim);
554 isl_seq_cpy(m->row[1], c, 1+dim);
555 U = isl_mat_right_inverse(m);
556 Q = isl_mat_right_inverse(isl_mat_copy(U));
557 U = isl_mat_drop_cols(U, 1, 1);
558 Q = isl_mat_drop_rows(Q, 1, 1);
559 set = isl_set_preimage(set, U);
560 facet = uset_convex_hull_wrap_bounded(set);
561 facet = isl_basic_set_preimage(facet, Q);
562 isl_assert(ctx, facet->n_eq == 0, goto error);
563 return facet;
564 error:
565 isl_basic_set_free(facet);
566 isl_set_free(set);
567 return NULL;
570 /* Given an initial facet constraint, compute the remaining facets.
571 * We do this by running through all facets found so far and computing
572 * the adjacent facets through wrapping, adding those facets that we
573 * hadn't already found before.
575 * For each facet we have found so far, we first compute its facets
576 * in the resulting convex hull. That is, we compute the ridges
577 * of the resulting convex hull contained in the facet.
578 * We also compute the corresponding facet in the current approximation
579 * of the convex hull. There is no need to wrap around the ridges
580 * in this facet since that would result in a facet that is already
581 * present in the current approximation.
583 * This function can still be significantly optimized by checking which of
584 * the facets of the basic sets are also facets of the convex hull and
585 * using all the facets so far to help in constructing the facets of the
586 * facets
587 * and/or
588 * using the technique in section "3.1 Ridge Generation" of
589 * "Extended Convex Hull" by Fukuda et al.
591 static struct isl_basic_set *extend(struct isl_basic_set *hull,
592 struct isl_set *set)
594 int i, j, f;
595 int k;
596 struct isl_basic_set *facet = NULL;
597 struct isl_basic_set *hull_facet = NULL;
598 unsigned dim;
600 if (!hull)
601 return NULL;
603 isl_assert(set->ctx, set->n > 0, goto error);
605 dim = isl_set_n_dim(set);
607 for (i = 0; i < hull->n_ineq; ++i) {
608 facet = compute_facet(set, hull->ineq[i]);
609 facet = isl_basic_set_add_equality(facet, hull->ineq[i]);
610 facet = isl_basic_set_gauss(facet, NULL);
611 facet = isl_basic_set_normalize_constraints(facet);
612 hull_facet = isl_basic_set_copy(hull);
613 hull_facet = isl_basic_set_add_equality(hull_facet, hull->ineq[i]);
614 hull_facet = isl_basic_set_gauss(hull_facet, NULL);
615 hull_facet = isl_basic_set_normalize_constraints(hull_facet);
616 if (!facet)
617 goto error;
618 hull = isl_basic_set_cow(hull);
619 hull = isl_basic_set_extend_dim(hull,
620 isl_dim_copy(hull->dim), 0, 0, facet->n_ineq);
621 for (j = 0; j < facet->n_ineq; ++j) {
622 for (f = 0; f < hull_facet->n_ineq; ++f)
623 if (isl_seq_eq(facet->ineq[j],
624 hull_facet->ineq[f], 1 + dim))
625 break;
626 if (f < hull_facet->n_ineq)
627 continue;
628 k = isl_basic_set_alloc_inequality(hull);
629 if (k < 0)
630 goto error;
631 isl_seq_cpy(hull->ineq[k], hull->ineq[i], 1+dim);
632 if (!isl_set_wrap_facet(set, hull->ineq[k], facet->ineq[j]))
633 goto error;
635 isl_basic_set_free(hull_facet);
636 isl_basic_set_free(facet);
638 hull = isl_basic_set_simplify(hull);
639 hull = isl_basic_set_finalize(hull);
640 return hull;
641 error:
642 isl_basic_set_free(hull_facet);
643 isl_basic_set_free(facet);
644 isl_basic_set_free(hull);
645 return NULL;
648 /* Special case for computing the convex hull of a one dimensional set.
649 * We simply collect the lower and upper bounds of each basic set
650 * and the biggest of those.
652 static struct isl_basic_set *convex_hull_1d(struct isl_set *set)
654 struct isl_mat *c = NULL;
655 isl_int *lower = NULL;
656 isl_int *upper = NULL;
657 int i, j, k;
658 isl_int a, b;
659 struct isl_basic_set *hull;
661 for (i = 0; i < set->n; ++i) {
662 set->p[i] = isl_basic_set_simplify(set->p[i]);
663 if (!set->p[i])
664 goto error;
666 set = isl_set_remove_empty_parts(set);
667 if (!set)
668 goto error;
669 isl_assert(set->ctx, set->n > 0, goto error);
670 c = isl_mat_alloc(set->ctx, 2, 2);
671 if (!c)
672 goto error;
674 if (set->p[0]->n_eq > 0) {
675 isl_assert(set->ctx, set->p[0]->n_eq == 1, goto error);
676 lower = c->row[0];
677 upper = c->row[1];
678 if (isl_int_is_pos(set->p[0]->eq[0][1])) {
679 isl_seq_cpy(lower, set->p[0]->eq[0], 2);
680 isl_seq_neg(upper, set->p[0]->eq[0], 2);
681 } else {
682 isl_seq_neg(lower, set->p[0]->eq[0], 2);
683 isl_seq_cpy(upper, set->p[0]->eq[0], 2);
685 } else {
686 for (j = 0; j < set->p[0]->n_ineq; ++j) {
687 if (isl_int_is_pos(set->p[0]->ineq[j][1])) {
688 lower = c->row[0];
689 isl_seq_cpy(lower, set->p[0]->ineq[j], 2);
690 } else {
691 upper = c->row[1];
692 isl_seq_cpy(upper, set->p[0]->ineq[j], 2);
697 isl_int_init(a);
698 isl_int_init(b);
699 for (i = 0; i < set->n; ++i) {
700 struct isl_basic_set *bset = set->p[i];
701 int has_lower = 0;
702 int has_upper = 0;
704 for (j = 0; j < bset->n_eq; ++j) {
705 has_lower = 1;
706 has_upper = 1;
707 if (lower) {
708 isl_int_mul(a, lower[0], bset->eq[j][1]);
709 isl_int_mul(b, lower[1], bset->eq[j][0]);
710 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
711 isl_seq_cpy(lower, bset->eq[j], 2);
712 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
713 isl_seq_neg(lower, bset->eq[j], 2);
715 if (upper) {
716 isl_int_mul(a, upper[0], bset->eq[j][1]);
717 isl_int_mul(b, upper[1], bset->eq[j][0]);
718 if (isl_int_lt(a, b) && isl_int_is_pos(bset->eq[j][1]))
719 isl_seq_neg(upper, bset->eq[j], 2);
720 if (isl_int_gt(a, b) && isl_int_is_neg(bset->eq[j][1]))
721 isl_seq_cpy(upper, bset->eq[j], 2);
724 for (j = 0; j < bset->n_ineq; ++j) {
725 if (isl_int_is_pos(bset->ineq[j][1]))
726 has_lower = 1;
727 if (isl_int_is_neg(bset->ineq[j][1]))
728 has_upper = 1;
729 if (lower && isl_int_is_pos(bset->ineq[j][1])) {
730 isl_int_mul(a, lower[0], bset->ineq[j][1]);
731 isl_int_mul(b, lower[1], bset->ineq[j][0]);
732 if (isl_int_lt(a, b))
733 isl_seq_cpy(lower, bset->ineq[j], 2);
735 if (upper && isl_int_is_neg(bset->ineq[j][1])) {
736 isl_int_mul(a, upper[0], bset->ineq[j][1]);
737 isl_int_mul(b, upper[1], bset->ineq[j][0]);
738 if (isl_int_gt(a, b))
739 isl_seq_cpy(upper, bset->ineq[j], 2);
742 if (!has_lower)
743 lower = NULL;
744 if (!has_upper)
745 upper = NULL;
747 isl_int_clear(a);
748 isl_int_clear(b);
750 hull = isl_basic_set_alloc(set->ctx, 0, 1, 0, 0, 2);
751 hull = isl_basic_set_set_rational(hull);
752 if (!hull)
753 goto error;
754 if (lower) {
755 k = isl_basic_set_alloc_inequality(hull);
756 isl_seq_cpy(hull->ineq[k], lower, 2);
758 if (upper) {
759 k = isl_basic_set_alloc_inequality(hull);
760 isl_seq_cpy(hull->ineq[k], upper, 2);
762 hull = isl_basic_set_finalize(hull);
763 isl_set_free(set);
764 isl_mat_free(c);
765 return hull;
766 error:
767 isl_set_free(set);
768 isl_mat_free(c);
769 return NULL;
772 /* Project out final n dimensions using Fourier-Motzkin */
773 static struct isl_set *set_project_out(struct isl_ctx *ctx,
774 struct isl_set *set, unsigned n)
776 return isl_set_remove_dims(set, isl_set_n_dim(set) - n, n);
779 static struct isl_basic_set *convex_hull_0d(struct isl_set *set)
781 struct isl_basic_set *convex_hull;
783 if (!set)
784 return NULL;
786 if (isl_set_is_empty(set))
787 convex_hull = isl_basic_set_empty(isl_dim_copy(set->dim));
788 else
789 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
790 isl_set_free(set);
791 return convex_hull;
794 /* Compute the convex hull of a pair of basic sets without any parameters or
795 * integer divisions using Fourier-Motzkin elimination.
796 * The convex hull is the set of all points that can be written as
797 * the sum of points from both basic sets (in homogeneous coordinates).
798 * We set up the constraints in a space with dimensions for each of
799 * the three sets and then project out the dimensions corresponding
800 * to the two original basic sets, retaining only those corresponding
801 * to the convex hull.
803 static struct isl_basic_set *convex_hull_pair_elim(struct isl_basic_set *bset1,
804 struct isl_basic_set *bset2)
806 int i, j, k;
807 struct isl_basic_set *bset[2];
808 struct isl_basic_set *hull = NULL;
809 unsigned dim;
811 if (!bset1 || !bset2)
812 goto error;
814 dim = isl_basic_set_n_dim(bset1);
815 hull = isl_basic_set_alloc(bset1->ctx, 0, 2 + 3 * dim, 0,
816 1 + dim + bset1->n_eq + bset2->n_eq,
817 2 + bset1->n_ineq + bset2->n_ineq);
818 bset[0] = bset1;
819 bset[1] = bset2;
820 for (i = 0; i < 2; ++i) {
821 for (j = 0; j < bset[i]->n_eq; ++j) {
822 k = isl_basic_set_alloc_equality(hull);
823 if (k < 0)
824 goto error;
825 isl_seq_clr(hull->eq[k], (i+1) * (1+dim));
826 isl_seq_clr(hull->eq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
827 isl_seq_cpy(hull->eq[k]+(i+1)*(1+dim), bset[i]->eq[j],
828 1+dim);
830 for (j = 0; j < bset[i]->n_ineq; ++j) {
831 k = isl_basic_set_alloc_inequality(hull);
832 if (k < 0)
833 goto error;
834 isl_seq_clr(hull->ineq[k], (i+1) * (1+dim));
835 isl_seq_clr(hull->ineq[k]+(i+2)*(1+dim), (1-i)*(1+dim));
836 isl_seq_cpy(hull->ineq[k]+(i+1)*(1+dim),
837 bset[i]->ineq[j], 1+dim);
839 k = isl_basic_set_alloc_inequality(hull);
840 if (k < 0)
841 goto error;
842 isl_seq_clr(hull->ineq[k], 1+2+3*dim);
843 isl_int_set_si(hull->ineq[k][(i+1)*(1+dim)], 1);
845 for (j = 0; j < 1+dim; ++j) {
846 k = isl_basic_set_alloc_equality(hull);
847 if (k < 0)
848 goto error;
849 isl_seq_clr(hull->eq[k], 1+2+3*dim);
850 isl_int_set_si(hull->eq[k][j], -1);
851 isl_int_set_si(hull->eq[k][1+dim+j], 1);
852 isl_int_set_si(hull->eq[k][2*(1+dim)+j], 1);
854 hull = isl_basic_set_set_rational(hull);
855 hull = isl_basic_set_remove_dims(hull, dim, 2*(1+dim));
856 hull = isl_basic_set_convex_hull(hull);
857 isl_basic_set_free(bset1);
858 isl_basic_set_free(bset2);
859 return hull;
860 error:
861 isl_basic_set_free(bset1);
862 isl_basic_set_free(bset2);
863 isl_basic_set_free(hull);
864 return NULL;
867 static int isl_basic_set_is_bounded(struct isl_basic_set *bset)
869 struct isl_tab *tab;
870 int bounded;
872 if (!bset)
873 return -1;
874 if (isl_basic_set_fast_is_empty(bset))
875 return 1;
877 tab = isl_tab_from_recession_cone(bset);
878 bounded = isl_tab_cone_is_bounded(tab);
879 isl_tab_free(tab);
880 return bounded;
883 int isl_set_is_bounded(__isl_keep isl_set *set)
885 int i;
887 for (i = 0; i < set->n; ++i) {
888 int bounded = isl_basic_set_is_bounded(set->p[i]);
889 if (!bounded || bounded < 0)
890 return bounded;
892 return 1;
895 /* Compute the lineality space of the convex hull of bset1 and bset2.
897 * We first compute the intersection of the recession cone of bset1
898 * with the negative of the recession cone of bset2 and then compute
899 * the linear hull of the resulting cone.
901 static struct isl_basic_set *induced_lineality_space(
902 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
904 int i, k;
905 struct isl_basic_set *lin = NULL;
906 unsigned dim;
908 if (!bset1 || !bset2)
909 goto error;
911 dim = isl_basic_set_total_dim(bset1);
912 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1), 0,
913 bset1->n_eq + bset2->n_eq,
914 bset1->n_ineq + bset2->n_ineq);
915 lin = isl_basic_set_set_rational(lin);
916 if (!lin)
917 goto error;
918 for (i = 0; i < bset1->n_eq; ++i) {
919 k = isl_basic_set_alloc_equality(lin);
920 if (k < 0)
921 goto error;
922 isl_int_set_si(lin->eq[k][0], 0);
923 isl_seq_cpy(lin->eq[k] + 1, bset1->eq[i] + 1, dim);
925 for (i = 0; i < bset1->n_ineq; ++i) {
926 k = isl_basic_set_alloc_inequality(lin);
927 if (k < 0)
928 goto error;
929 isl_int_set_si(lin->ineq[k][0], 0);
930 isl_seq_cpy(lin->ineq[k] + 1, bset1->ineq[i] + 1, dim);
932 for (i = 0; i < bset2->n_eq; ++i) {
933 k = isl_basic_set_alloc_equality(lin);
934 if (k < 0)
935 goto error;
936 isl_int_set_si(lin->eq[k][0], 0);
937 isl_seq_neg(lin->eq[k] + 1, bset2->eq[i] + 1, dim);
939 for (i = 0; i < bset2->n_ineq; ++i) {
940 k = isl_basic_set_alloc_inequality(lin);
941 if (k < 0)
942 goto error;
943 isl_int_set_si(lin->ineq[k][0], 0);
944 isl_seq_neg(lin->ineq[k] + 1, bset2->ineq[i] + 1, dim);
947 isl_basic_set_free(bset1);
948 isl_basic_set_free(bset2);
949 return isl_basic_set_affine_hull(lin);
950 error:
951 isl_basic_set_free(lin);
952 isl_basic_set_free(bset1);
953 isl_basic_set_free(bset2);
954 return NULL;
957 static struct isl_basic_set *uset_convex_hull(struct isl_set *set);
959 /* Given a set and a linear space "lin" of dimension n > 0,
960 * project the linear space from the set, compute the convex hull
961 * and then map the set back to the original space.
963 * Let
965 * M x = 0
967 * describe the linear space. We first compute the Hermite normal
968 * form H = M U of M = H Q, to obtain
970 * H Q x = 0
972 * The last n rows of H will be zero, so the last n variables of x' = Q x
973 * are the one we want to project out. We do this by transforming each
974 * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
975 * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
976 * we transform the hull back to the original space as A' Q_1 x >= b',
977 * with Q_1 all but the last n rows of Q.
979 static struct isl_basic_set *modulo_lineality(struct isl_set *set,
980 struct isl_basic_set *lin)
982 unsigned total = isl_basic_set_total_dim(lin);
983 unsigned lin_dim;
984 struct isl_basic_set *hull;
985 struct isl_mat *M, *U, *Q;
987 if (!set || !lin)
988 goto error;
989 lin_dim = total - lin->n_eq;
990 M = isl_mat_sub_alloc(set->ctx, lin->eq, 0, lin->n_eq, 1, total);
991 M = isl_mat_left_hermite(M, 0, &U, &Q);
992 if (!M)
993 goto error;
994 isl_mat_free(M);
995 isl_basic_set_free(lin);
997 Q = isl_mat_drop_rows(Q, Q->n_row - lin_dim, lin_dim);
999 U = isl_mat_lin_to_aff(U);
1000 Q = isl_mat_lin_to_aff(Q);
1002 set = isl_set_preimage(set, U);
1003 set = isl_set_remove_dims(set, total - lin_dim, lin_dim);
1004 hull = uset_convex_hull(set);
1005 hull = isl_basic_set_preimage(hull, Q);
1007 return hull;
1008 error:
1009 isl_basic_set_free(lin);
1010 isl_set_free(set);
1011 return NULL;
1014 /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
1015 * set up an LP for solving
1017 * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
1019 * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
1020 * The next \alpha{ij} correspond to the equalities and come in pairs.
1021 * The final \alpha{ij} correspond to the inequalities.
1023 static struct isl_basic_set *valid_direction_lp(
1024 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1026 struct isl_dim *dim;
1027 struct isl_basic_set *lp;
1028 unsigned d;
1029 int n;
1030 int i, j, k;
1032 if (!bset1 || !bset2)
1033 goto error;
1034 d = 1 + isl_basic_set_total_dim(bset1);
1035 n = 2 +
1036 2 * bset1->n_eq + bset1->n_ineq + 2 * bset2->n_eq + bset2->n_ineq;
1037 dim = isl_dim_set_alloc(bset1->ctx, 0, n);
1038 lp = isl_basic_set_alloc_dim(dim, 0, d, n);
1039 if (!lp)
1040 goto error;
1041 for (i = 0; i < n; ++i) {
1042 k = isl_basic_set_alloc_inequality(lp);
1043 if (k < 0)
1044 goto error;
1045 isl_seq_clr(lp->ineq[k] + 1, n);
1046 isl_int_set_si(lp->ineq[k][0], -1);
1047 isl_int_set_si(lp->ineq[k][1 + i], 1);
1049 for (i = 0; i < d; ++i) {
1050 k = isl_basic_set_alloc_equality(lp);
1051 if (k < 0)
1052 goto error;
1053 n = 0;
1054 isl_int_set_si(lp->eq[k][n++], 0);
1055 /* positivity constraint 1 >= 0 */
1056 isl_int_set_si(lp->eq[k][n++], i == 0);
1057 for (j = 0; j < bset1->n_eq; ++j) {
1058 isl_int_set(lp->eq[k][n++], bset1->eq[j][i]);
1059 isl_int_neg(lp->eq[k][n++], bset1->eq[j][i]);
1061 for (j = 0; j < bset1->n_ineq; ++j)
1062 isl_int_set(lp->eq[k][n++], bset1->ineq[j][i]);
1063 /* positivity constraint 1 >= 0 */
1064 isl_int_set_si(lp->eq[k][n++], -(i == 0));
1065 for (j = 0; j < bset2->n_eq; ++j) {
1066 isl_int_neg(lp->eq[k][n++], bset2->eq[j][i]);
1067 isl_int_set(lp->eq[k][n++], bset2->eq[j][i]);
1069 for (j = 0; j < bset2->n_ineq; ++j)
1070 isl_int_neg(lp->eq[k][n++], bset2->ineq[j][i]);
1072 lp = isl_basic_set_gauss(lp, NULL);
1073 isl_basic_set_free(bset1);
1074 isl_basic_set_free(bset2);
1075 return lp;
1076 error:
1077 isl_basic_set_free(bset1);
1078 isl_basic_set_free(bset2);
1079 return NULL;
1082 /* Compute a vector s in the homogeneous space such that <s, r> > 0
1083 * for all rays in the homogeneous space of the two cones that correspond
1084 * to the input polyhedra bset1 and bset2.
1086 * We compute s as a vector that satisfies
1088 * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*)
1090 * with h_{ij} the normals of the facets of polyhedron i
1091 * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
1092 * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1.
1093 * We first set up an LP with as variables the \alpha{ij}.
1094 * In this formulation, for each polyhedron i,
1095 * the first constraint is the positivity constraint, followed by pairs
1096 * of variables for the equalities, followed by variables for the inequalities.
1097 * We then simply pick a feasible solution and compute s using (*).
1099 * Note that we simply pick any valid direction and make no attempt
1100 * to pick a "good" or even the "best" valid direction.
1102 static struct isl_vec *valid_direction(
1103 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1105 struct isl_basic_set *lp;
1106 struct isl_tab *tab;
1107 struct isl_vec *sample = NULL;
1108 struct isl_vec *dir;
1109 unsigned d;
1110 int i;
1111 int n;
1113 if (!bset1 || !bset2)
1114 goto error;
1115 lp = valid_direction_lp(isl_basic_set_copy(bset1),
1116 isl_basic_set_copy(bset2));
1117 tab = isl_tab_from_basic_set(lp);
1118 sample = isl_tab_get_sample_value(tab);
1119 isl_tab_free(tab);
1120 isl_basic_set_free(lp);
1121 if (!sample)
1122 goto error;
1123 d = isl_basic_set_total_dim(bset1);
1124 dir = isl_vec_alloc(bset1->ctx, 1 + d);
1125 if (!dir)
1126 goto error;
1127 isl_seq_clr(dir->block.data + 1, dir->size - 1);
1128 n = 1;
1129 /* positivity constraint 1 >= 0 */
1130 isl_int_set(dir->block.data[0], sample->block.data[n++]);
1131 for (i = 0; i < bset1->n_eq; ++i) {
1132 isl_int_sub(sample->block.data[n],
1133 sample->block.data[n], sample->block.data[n+1]);
1134 isl_seq_combine(dir->block.data,
1135 bset1->ctx->one, dir->block.data,
1136 sample->block.data[n], bset1->eq[i], 1 + d);
1138 n += 2;
1140 for (i = 0; i < bset1->n_ineq; ++i)
1141 isl_seq_combine(dir->block.data,
1142 bset1->ctx->one, dir->block.data,
1143 sample->block.data[n++], bset1->ineq[i], 1 + d);
1144 isl_vec_free(sample);
1145 isl_seq_normalize(bset1->ctx, dir->el, dir->size);
1146 isl_basic_set_free(bset1);
1147 isl_basic_set_free(bset2);
1148 return dir;
1149 error:
1150 isl_vec_free(sample);
1151 isl_basic_set_free(bset1);
1152 isl_basic_set_free(bset2);
1153 return NULL;
1156 /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
1157 * compute b_i' + A_i' x' >= 0, with
1159 * [ b_i A_i ] [ y' ] [ y' ]
1160 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1162 * In particular, add the "positivity constraint" and then perform
1163 * the mapping.
1165 static struct isl_basic_set *homogeneous_map(struct isl_basic_set *bset,
1166 struct isl_mat *T)
1168 int k;
1170 if (!bset)
1171 goto error;
1172 bset = isl_basic_set_extend_constraints(bset, 0, 1);
1173 k = isl_basic_set_alloc_inequality(bset);
1174 if (k < 0)
1175 goto error;
1176 isl_seq_clr(bset->ineq[k] + 1, isl_basic_set_total_dim(bset));
1177 isl_int_set_si(bset->ineq[k][0], 1);
1178 bset = isl_basic_set_preimage(bset, T);
1179 return bset;
1180 error:
1181 isl_mat_free(T);
1182 isl_basic_set_free(bset);
1183 return NULL;
1186 /* Compute the convex hull of a pair of basic sets without any parameters or
1187 * integer divisions, where the convex hull is known to be pointed,
1188 * but the basic sets may be unbounded.
1190 * We turn this problem into the computation of a convex hull of a pair
1191 * _bounded_ polyhedra by "changing the direction of the homogeneous
1192 * dimension". This idea is due to Matthias Koeppe.
1194 * Consider the cones in homogeneous space that correspond to the
1195 * input polyhedra. The rays of these cones are also rays of the
1196 * polyhedra if the coordinate that corresponds to the homogeneous
1197 * dimension is zero. That is, if the inner product of the rays
1198 * with the homogeneous direction is zero.
1199 * The cones in the homogeneous space can also be considered to
1200 * correspond to other pairs of polyhedra by chosing a different
1201 * homogeneous direction. To ensure that both of these polyhedra
1202 * are bounded, we need to make sure that all rays of the cones
1203 * correspond to vertices and not to rays.
1204 * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
1205 * Then using s as a homogeneous direction, we obtain a pair of polytopes.
1206 * The vector s is computed in valid_direction.
1208 * Note that we need to consider _all_ rays of the cones and not just
1209 * the rays that correspond to rays in the polyhedra. If we were to
1210 * only consider those rays and turn them into vertices, then we
1211 * may inadvertently turn some vertices into rays.
1213 * The standard homogeneous direction is the unit vector in the 0th coordinate.
1214 * We therefore transform the two polyhedra such that the selected
1215 * direction is mapped onto this standard direction and then proceed
1216 * with the normal computation.
1217 * Let S be a non-singular square matrix with s as its first row,
1218 * then we want to map the polyhedra to the space
1220 * [ y' ] [ y ] [ y ] [ y' ]
1221 * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ]
1223 * We take S to be the unimodular completion of s to limit the growth
1224 * of the coefficients in the following computations.
1226 * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
1227 * We first move to the homogeneous dimension
1229 * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ]
1230 * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ]
1232 * Then we change directoin
1234 * [ b_i A_i ] [ y' ] [ y' ]
1235 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1237 * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
1238 * resulting in b' + A' x' >= 0, which we then convert back
1240 * [ y ] [ y ]
1241 * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0
1243 * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
1245 static struct isl_basic_set *convex_hull_pair_pointed(
1246 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
1248 struct isl_ctx *ctx = NULL;
1249 struct isl_vec *dir = NULL;
1250 struct isl_mat *T = NULL;
1251 struct isl_mat *T2 = NULL;
1252 struct isl_basic_set *hull;
1253 struct isl_set *set;
1255 if (!bset1 || !bset2)
1256 goto error;
1257 ctx = bset1->ctx;
1258 dir = valid_direction(isl_basic_set_copy(bset1),
1259 isl_basic_set_copy(bset2));
1260 if (!dir)
1261 goto error;
1262 T = isl_mat_alloc(bset1->ctx, dir->size, dir->size);
1263 if (!T)
1264 goto error;
1265 isl_seq_cpy(T->row[0], dir->block.data, dir->size);
1266 T = isl_mat_unimodular_complete(T, 1);
1267 T2 = isl_mat_right_inverse(isl_mat_copy(T));
1269 bset1 = homogeneous_map(bset1, isl_mat_copy(T2));
1270 bset2 = homogeneous_map(bset2, T2);
1271 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1272 set = isl_set_add_basic_set(set, bset1);
1273 set = isl_set_add_basic_set(set, bset2);
1274 hull = uset_convex_hull(set);
1275 hull = isl_basic_set_preimage(hull, T);
1277 isl_vec_free(dir);
1279 return hull;
1280 error:
1281 isl_vec_free(dir);
1282 isl_basic_set_free(bset1);
1283 isl_basic_set_free(bset2);
1284 return NULL;
1287 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set);
1288 static struct isl_basic_set *modulo_affine_hull(
1289 struct isl_set *set, struct isl_basic_set *affine_hull);
1291 /* Compute the convex hull of a pair of basic sets without any parameters or
1292 * integer divisions.
1294 * This function is called from uset_convex_hull_unbounded, which
1295 * means that the complete convex hull is unbounded. Some pairs
1296 * of basic sets may still be bounded, though.
1297 * They may even lie inside a lower dimensional space, in which
1298 * case they need to be handled inside their affine hull since
1299 * the main algorithm assumes that the result is full-dimensional.
1301 * If the convex hull of the two basic sets would have a non-trivial
1302 * lineality space, we first project out this lineality space.
1304 static struct isl_basic_set *convex_hull_pair(struct isl_basic_set *bset1,
1305 struct isl_basic_set *bset2)
1307 isl_basic_set *lin, *aff;
1308 int bounded1, bounded2;
1310 aff = isl_set_affine_hull(isl_basic_set_union(isl_basic_set_copy(bset1),
1311 isl_basic_set_copy(bset2)));
1312 if (!aff)
1313 goto error;
1314 if (aff->n_eq != 0)
1315 return modulo_affine_hull(isl_basic_set_union(bset1, bset2), aff);
1316 isl_basic_set_free(aff);
1318 bounded1 = isl_basic_set_is_bounded(bset1);
1319 bounded2 = isl_basic_set_is_bounded(bset2);
1321 if (bounded1 < 0 || bounded2 < 0)
1322 goto error;
1324 if (bounded1 && bounded2)
1325 uset_convex_hull_wrap(isl_basic_set_union(bset1, bset2));
1327 if (bounded1 || bounded2)
1328 return convex_hull_pair_pointed(bset1, bset2);
1330 lin = induced_lineality_space(isl_basic_set_copy(bset1),
1331 isl_basic_set_copy(bset2));
1332 if (!lin)
1333 goto error;
1334 if (isl_basic_set_is_universe(lin)) {
1335 isl_basic_set_free(bset1);
1336 isl_basic_set_free(bset2);
1337 return lin;
1339 if (lin->n_eq < isl_basic_set_total_dim(lin)) {
1340 struct isl_set *set;
1341 set = isl_set_alloc_dim(isl_basic_set_get_dim(bset1), 2, 0);
1342 set = isl_set_add_basic_set(set, bset1);
1343 set = isl_set_add_basic_set(set, bset2);
1344 return modulo_lineality(set, lin);
1346 isl_basic_set_free(lin);
1348 return convex_hull_pair_pointed(bset1, bset2);
1349 error:
1350 isl_basic_set_free(bset1);
1351 isl_basic_set_free(bset2);
1352 return NULL;
1355 /* Compute the lineality space of a basic set.
1356 * We currently do not allow the basic set to have any divs.
1357 * We basically just drop the constants and turn every inequality
1358 * into an equality.
1360 struct isl_basic_set *isl_basic_set_lineality_space(struct isl_basic_set *bset)
1362 int i, k;
1363 struct isl_basic_set *lin = NULL;
1364 unsigned dim;
1366 if (!bset)
1367 goto error;
1368 isl_assert(bset->ctx, bset->n_div == 0, goto error);
1369 dim = isl_basic_set_total_dim(bset);
1371 lin = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset), 0, dim, 0);
1372 if (!lin)
1373 goto error;
1374 for (i = 0; i < bset->n_eq; ++i) {
1375 k = isl_basic_set_alloc_equality(lin);
1376 if (k < 0)
1377 goto error;
1378 isl_int_set_si(lin->eq[k][0], 0);
1379 isl_seq_cpy(lin->eq[k] + 1, bset->eq[i] + 1, dim);
1381 lin = isl_basic_set_gauss(lin, NULL);
1382 if (!lin)
1383 goto error;
1384 for (i = 0; i < bset->n_ineq && lin->n_eq < dim; ++i) {
1385 k = isl_basic_set_alloc_equality(lin);
1386 if (k < 0)
1387 goto error;
1388 isl_int_set_si(lin->eq[k][0], 0);
1389 isl_seq_cpy(lin->eq[k] + 1, bset->ineq[i] + 1, dim);
1390 lin = isl_basic_set_gauss(lin, NULL);
1391 if (!lin)
1392 goto error;
1394 isl_basic_set_free(bset);
1395 return lin;
1396 error:
1397 isl_basic_set_free(lin);
1398 isl_basic_set_free(bset);
1399 return NULL;
1402 /* Compute the (linear) hull of the lineality spaces of the basic sets in the
1403 * "underlying" set "set".
1405 static struct isl_basic_set *uset_combined_lineality_space(struct isl_set *set)
1407 int i;
1408 struct isl_set *lin = NULL;
1410 if (!set)
1411 return NULL;
1412 if (set->n == 0) {
1413 struct isl_dim *dim = isl_set_get_dim(set);
1414 isl_set_free(set);
1415 return isl_basic_set_empty(dim);
1418 lin = isl_set_alloc_dim(isl_set_get_dim(set), set->n, 0);
1419 for (i = 0; i < set->n; ++i)
1420 lin = isl_set_add_basic_set(lin,
1421 isl_basic_set_lineality_space(isl_basic_set_copy(set->p[i])));
1422 isl_set_free(set);
1423 return isl_set_affine_hull(lin);
1426 /* Compute the convex hull of a set without any parameters or
1427 * integer divisions.
1428 * In each step, we combined two basic sets until only one
1429 * basic set is left.
1430 * The input basic sets are assumed not to have a non-trivial
1431 * lineality space. If any of the intermediate results has
1432 * a non-trivial lineality space, it is projected out.
1434 static struct isl_basic_set *uset_convex_hull_unbounded(struct isl_set *set)
1436 struct isl_basic_set *convex_hull = NULL;
1438 convex_hull = isl_set_copy_basic_set(set);
1439 set = isl_set_drop_basic_set(set, convex_hull);
1440 if (!set)
1441 goto error;
1442 while (set->n > 0) {
1443 struct isl_basic_set *t;
1444 t = isl_set_copy_basic_set(set);
1445 if (!t)
1446 goto error;
1447 set = isl_set_drop_basic_set(set, t);
1448 if (!set)
1449 goto error;
1450 convex_hull = convex_hull_pair(convex_hull, t);
1451 if (set->n == 0)
1452 break;
1453 t = isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull));
1454 if (!t)
1455 goto error;
1456 if (isl_basic_set_is_universe(t)) {
1457 isl_basic_set_free(convex_hull);
1458 convex_hull = t;
1459 break;
1461 if (t->n_eq < isl_basic_set_total_dim(t)) {
1462 set = isl_set_add_basic_set(set, convex_hull);
1463 return modulo_lineality(set, t);
1465 isl_basic_set_free(t);
1467 isl_set_free(set);
1468 return convex_hull;
1469 error:
1470 isl_set_free(set);
1471 isl_basic_set_free(convex_hull);
1472 return NULL;
1475 /* Compute an initial hull for wrapping containing a single initial
1476 * facet.
1477 * This function assumes that the given set is bounded.
1479 static struct isl_basic_set *initial_hull(struct isl_basic_set *hull,
1480 struct isl_set *set)
1482 struct isl_mat *bounds = NULL;
1483 unsigned dim;
1484 int k;
1486 if (!hull)
1487 goto error;
1488 bounds = initial_facet_constraint(set);
1489 if (!bounds)
1490 goto error;
1491 k = isl_basic_set_alloc_inequality(hull);
1492 if (k < 0)
1493 goto error;
1494 dim = isl_set_n_dim(set);
1495 isl_assert(set->ctx, 1 + dim == bounds->n_col, goto error);
1496 isl_seq_cpy(hull->ineq[k], bounds->row[0], bounds->n_col);
1497 isl_mat_free(bounds);
1499 return hull;
1500 error:
1501 isl_basic_set_free(hull);
1502 isl_mat_free(bounds);
1503 return NULL;
1506 struct max_constraint {
1507 struct isl_mat *c;
1508 int count;
1509 int ineq;
1512 static int max_constraint_equal(const void *entry, const void *val)
1514 struct max_constraint *a = (struct max_constraint *)entry;
1515 isl_int *b = (isl_int *)val;
1517 return isl_seq_eq(a->c->row[0] + 1, b, a->c->n_col - 1);
1520 static void update_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1521 isl_int *con, unsigned len, int n, int ineq)
1523 struct isl_hash_table_entry *entry;
1524 struct max_constraint *c;
1525 uint32_t c_hash;
1527 c_hash = isl_seq_get_hash(con + 1, len);
1528 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1529 con + 1, 0);
1530 if (!entry)
1531 return;
1532 c = entry->data;
1533 if (c->count < n) {
1534 isl_hash_table_remove(ctx, table, entry);
1535 return;
1537 c->count++;
1538 if (isl_int_gt(c->c->row[0][0], con[0]))
1539 return;
1540 if (isl_int_eq(c->c->row[0][0], con[0])) {
1541 if (ineq)
1542 c->ineq = ineq;
1543 return;
1545 c->c = isl_mat_cow(c->c);
1546 isl_int_set(c->c->row[0][0], con[0]);
1547 c->ineq = ineq;
1550 /* Check whether the constraint hash table "table" constains the constraint
1551 * "con".
1553 static int has_constraint(struct isl_ctx *ctx, struct isl_hash_table *table,
1554 isl_int *con, unsigned len, int n)
1556 struct isl_hash_table_entry *entry;
1557 struct max_constraint *c;
1558 uint32_t c_hash;
1560 c_hash = isl_seq_get_hash(con + 1, len);
1561 entry = isl_hash_table_find(ctx, table, c_hash, max_constraint_equal,
1562 con + 1, 0);
1563 if (!entry)
1564 return 0;
1565 c = entry->data;
1566 if (c->count < n)
1567 return 0;
1568 return isl_int_eq(c->c->row[0][0], con[0]);
1571 /* Check for inequality constraints of a basic set without equalities
1572 * such that the same or more stringent copies of the constraint appear
1573 * in all of the basic sets. Such constraints are necessarily facet
1574 * constraints of the convex hull.
1576 * If the resulting basic set is by chance identical to one of
1577 * the basic sets in "set", then we know that this basic set contains
1578 * all other basic sets and is therefore the convex hull of set.
1579 * In this case we set *is_hull to 1.
1581 static struct isl_basic_set *common_constraints(struct isl_basic_set *hull,
1582 struct isl_set *set, int *is_hull)
1584 int i, j, s, n;
1585 int min_constraints;
1586 int best;
1587 struct max_constraint *constraints = NULL;
1588 struct isl_hash_table *table = NULL;
1589 unsigned total;
1591 *is_hull = 0;
1593 for (i = 0; i < set->n; ++i)
1594 if (set->p[i]->n_eq == 0)
1595 break;
1596 if (i >= set->n)
1597 return hull;
1598 min_constraints = set->p[i]->n_ineq;
1599 best = i;
1600 for (i = best + 1; i < set->n; ++i) {
1601 if (set->p[i]->n_eq != 0)
1602 continue;
1603 if (set->p[i]->n_ineq >= min_constraints)
1604 continue;
1605 min_constraints = set->p[i]->n_ineq;
1606 best = i;
1608 constraints = isl_calloc_array(hull->ctx, struct max_constraint,
1609 min_constraints);
1610 if (!constraints)
1611 return hull;
1612 table = isl_alloc_type(hull->ctx, struct isl_hash_table);
1613 if (isl_hash_table_init(hull->ctx, table, min_constraints))
1614 goto error;
1616 total = isl_dim_total(set->dim);
1617 for (i = 0; i < set->p[best]->n_ineq; ++i) {
1618 constraints[i].c = isl_mat_sub_alloc(hull->ctx,
1619 set->p[best]->ineq + i, 0, 1, 0, 1 + total);
1620 if (!constraints[i].c)
1621 goto error;
1622 constraints[i].ineq = 1;
1624 for (i = 0; i < min_constraints; ++i) {
1625 struct isl_hash_table_entry *entry;
1626 uint32_t c_hash;
1627 c_hash = isl_seq_get_hash(constraints[i].c->row[0] + 1, total);
1628 entry = isl_hash_table_find(hull->ctx, table, c_hash,
1629 max_constraint_equal, constraints[i].c->row[0] + 1, 1);
1630 if (!entry)
1631 goto error;
1632 isl_assert(hull->ctx, !entry->data, goto error);
1633 entry->data = &constraints[i];
1636 n = 0;
1637 for (s = 0; s < set->n; ++s) {
1638 if (s == best)
1639 continue;
1641 for (i = 0; i < set->p[s]->n_eq; ++i) {
1642 isl_int *eq = set->p[s]->eq[i];
1643 for (j = 0; j < 2; ++j) {
1644 isl_seq_neg(eq, eq, 1 + total);
1645 update_constraint(hull->ctx, table,
1646 eq, total, n, 0);
1649 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1650 isl_int *ineq = set->p[s]->ineq[i];
1651 update_constraint(hull->ctx, table, ineq, total, n,
1652 set->p[s]->n_eq == 0);
1654 ++n;
1657 for (i = 0; i < min_constraints; ++i) {
1658 if (constraints[i].count < n)
1659 continue;
1660 if (!constraints[i].ineq)
1661 continue;
1662 j = isl_basic_set_alloc_inequality(hull);
1663 if (j < 0)
1664 goto error;
1665 isl_seq_cpy(hull->ineq[j], constraints[i].c->row[0], 1 + total);
1668 for (s = 0; s < set->n; ++s) {
1669 if (set->p[s]->n_eq)
1670 continue;
1671 if (set->p[s]->n_ineq != hull->n_ineq)
1672 continue;
1673 for (i = 0; i < set->p[s]->n_ineq; ++i) {
1674 isl_int *ineq = set->p[s]->ineq[i];
1675 if (!has_constraint(hull->ctx, table, ineq, total, n))
1676 break;
1678 if (i == set->p[s]->n_ineq)
1679 *is_hull = 1;
1682 isl_hash_table_clear(table);
1683 for (i = 0; i < min_constraints; ++i)
1684 isl_mat_free(constraints[i].c);
1685 free(constraints);
1686 free(table);
1687 return hull;
1688 error:
1689 isl_hash_table_clear(table);
1690 free(table);
1691 if (constraints)
1692 for (i = 0; i < min_constraints; ++i)
1693 isl_mat_free(constraints[i].c);
1694 free(constraints);
1695 return hull;
1698 /* Create a template for the convex hull of "set" and fill it up
1699 * obvious facet constraints, if any. If the result happens to
1700 * be the convex hull of "set" then *is_hull is set to 1.
1702 static struct isl_basic_set *proto_hull(struct isl_set *set, int *is_hull)
1704 struct isl_basic_set *hull;
1705 unsigned n_ineq;
1706 int i;
1708 n_ineq = 1;
1709 for (i = 0; i < set->n; ++i) {
1710 n_ineq += set->p[i]->n_eq;
1711 n_ineq += set->p[i]->n_ineq;
1713 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
1714 hull = isl_basic_set_set_rational(hull);
1715 if (!hull)
1716 return NULL;
1717 return common_constraints(hull, set, is_hull);
1720 static struct isl_basic_set *uset_convex_hull_wrap(struct isl_set *set)
1722 struct isl_basic_set *hull;
1723 int is_hull;
1725 hull = proto_hull(set, &is_hull);
1726 if (hull && !is_hull) {
1727 if (hull->n_ineq == 0)
1728 hull = initial_hull(hull, set);
1729 hull = extend(hull, set);
1731 isl_set_free(set);
1733 return hull;
1736 /* Compute the convex hull of a set without any parameters or
1737 * integer divisions. Depending on whether the set is bounded,
1738 * we pass control to the wrapping based convex hull or
1739 * the Fourier-Motzkin elimination based convex hull.
1740 * We also handle a few special cases before checking the boundedness.
1742 static struct isl_basic_set *uset_convex_hull(struct isl_set *set)
1744 struct isl_basic_set *convex_hull = NULL;
1745 struct isl_basic_set *lin;
1747 if (isl_set_n_dim(set) == 0)
1748 return convex_hull_0d(set);
1750 set = isl_set_coalesce(set);
1751 set = isl_set_set_rational(set);
1753 if (!set)
1754 goto error;
1755 if (!set)
1756 return NULL;
1757 if (set->n == 1) {
1758 convex_hull = isl_basic_set_copy(set->p[0]);
1759 isl_set_free(set);
1760 return convex_hull;
1762 if (isl_set_n_dim(set) == 1)
1763 return convex_hull_1d(set);
1765 if (isl_set_is_bounded(set))
1766 return uset_convex_hull_wrap(set);
1768 lin = uset_combined_lineality_space(isl_set_copy(set));
1769 if (!lin)
1770 goto error;
1771 if (isl_basic_set_is_universe(lin)) {
1772 isl_set_free(set);
1773 return lin;
1775 if (lin->n_eq < isl_basic_set_total_dim(lin))
1776 return modulo_lineality(set, lin);
1777 isl_basic_set_free(lin);
1779 return uset_convex_hull_unbounded(set);
1780 error:
1781 isl_set_free(set);
1782 isl_basic_set_free(convex_hull);
1783 return NULL;
1786 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1787 * without parameters or divs and where the convex hull of set is
1788 * known to be full-dimensional.
1790 static struct isl_basic_set *uset_convex_hull_wrap_bounded(struct isl_set *set)
1792 struct isl_basic_set *convex_hull = NULL;
1794 if (isl_set_n_dim(set) == 0) {
1795 convex_hull = isl_basic_set_universe(isl_dim_copy(set->dim));
1796 isl_set_free(set);
1797 convex_hull = isl_basic_set_set_rational(convex_hull);
1798 return convex_hull;
1801 set = isl_set_set_rational(set);
1803 if (!set)
1804 goto error;
1805 set = isl_set_coalesce(set);
1806 if (!set)
1807 goto error;
1808 if (set->n == 1) {
1809 convex_hull = isl_basic_set_copy(set->p[0]);
1810 isl_set_free(set);
1811 return convex_hull;
1813 if (isl_set_n_dim(set) == 1)
1814 return convex_hull_1d(set);
1816 return uset_convex_hull_wrap(set);
1817 error:
1818 isl_set_free(set);
1819 return NULL;
1822 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1823 * We first remove the equalities (transforming the set), compute the
1824 * convex hull of the transformed set and then add the equalities back
1825 * (after performing the inverse transformation.
1827 static struct isl_basic_set *modulo_affine_hull(
1828 struct isl_set *set, struct isl_basic_set *affine_hull)
1830 struct isl_mat *T;
1831 struct isl_mat *T2;
1832 struct isl_basic_set *dummy;
1833 struct isl_basic_set *convex_hull;
1835 dummy = isl_basic_set_remove_equalities(
1836 isl_basic_set_copy(affine_hull), &T, &T2);
1837 if (!dummy)
1838 goto error;
1839 isl_basic_set_free(dummy);
1840 set = isl_set_preimage(set, T);
1841 convex_hull = uset_convex_hull(set);
1842 convex_hull = isl_basic_set_preimage(convex_hull, T2);
1843 convex_hull = isl_basic_set_intersect(convex_hull, affine_hull);
1844 return convex_hull;
1845 error:
1846 isl_basic_set_free(affine_hull);
1847 isl_set_free(set);
1848 return NULL;
1851 /* Compute the convex hull of a map.
1853 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1854 * specifically, the wrapping of facets to obtain new facets.
1856 struct isl_basic_map *isl_map_convex_hull(struct isl_map *map)
1858 struct isl_basic_set *bset;
1859 struct isl_basic_map *model = NULL;
1860 struct isl_basic_set *affine_hull = NULL;
1861 struct isl_basic_map *convex_hull = NULL;
1862 struct isl_set *set = NULL;
1863 struct isl_ctx *ctx;
1865 if (!map)
1866 goto error;
1868 ctx = map->ctx;
1869 if (map->n == 0) {
1870 convex_hull = isl_basic_map_empty_like_map(map);
1871 isl_map_free(map);
1872 return convex_hull;
1875 map = isl_map_detect_equalities(map);
1876 map = isl_map_align_divs(map);
1877 model = isl_basic_map_copy(map->p[0]);
1878 set = isl_map_underlying_set(map);
1879 if (!set)
1880 goto error;
1882 affine_hull = isl_set_affine_hull(isl_set_copy(set));
1883 if (!affine_hull)
1884 goto error;
1885 if (affine_hull->n_eq != 0)
1886 bset = modulo_affine_hull(set, affine_hull);
1887 else {
1888 isl_basic_set_free(affine_hull);
1889 bset = uset_convex_hull(set);
1892 convex_hull = isl_basic_map_overlying_set(bset, model);
1894 ISL_F_SET(convex_hull, ISL_BASIC_MAP_NO_IMPLICIT);
1895 ISL_F_SET(convex_hull, ISL_BASIC_MAP_ALL_EQUALITIES);
1896 ISL_F_CLR(convex_hull, ISL_BASIC_MAP_RATIONAL);
1897 return convex_hull;
1898 error:
1899 isl_set_free(set);
1900 isl_basic_map_free(model);
1901 return NULL;
1904 struct isl_basic_set *isl_set_convex_hull(struct isl_set *set)
1906 return (struct isl_basic_set *)
1907 isl_map_convex_hull((struct isl_map *)set);
1910 struct sh_data_entry {
1911 struct isl_hash_table *table;
1912 struct isl_tab *tab;
1915 /* Holds the data needed during the simple hull computation.
1916 * In particular,
1917 * n the number of basic sets in the original set
1918 * hull_table a hash table of already computed constraints
1919 * in the simple hull
1920 * p for each basic set,
1921 * table a hash table of the constraints
1922 * tab the tableau corresponding to the basic set
1924 struct sh_data {
1925 struct isl_ctx *ctx;
1926 unsigned n;
1927 struct isl_hash_table *hull_table;
1928 struct sh_data_entry p[1];
1931 static void sh_data_free(struct sh_data *data)
1933 int i;
1935 if (!data)
1936 return;
1937 isl_hash_table_free(data->ctx, data->hull_table);
1938 for (i = 0; i < data->n; ++i) {
1939 isl_hash_table_free(data->ctx, data->p[i].table);
1940 isl_tab_free(data->p[i].tab);
1942 free(data);
1945 struct ineq_cmp_data {
1946 unsigned len;
1947 isl_int *p;
1950 static int has_ineq(const void *entry, const void *val)
1952 isl_int *row = (isl_int *)entry;
1953 struct ineq_cmp_data *v = (struct ineq_cmp_data *)val;
1955 return isl_seq_eq(row + 1, v->p + 1, v->len) ||
1956 isl_seq_is_neg(row + 1, v->p + 1, v->len);
1959 static int hash_ineq(struct isl_ctx *ctx, struct isl_hash_table *table,
1960 isl_int *ineq, unsigned len)
1962 uint32_t c_hash;
1963 struct ineq_cmp_data v;
1964 struct isl_hash_table_entry *entry;
1966 v.len = len;
1967 v.p = ineq;
1968 c_hash = isl_seq_get_hash(ineq + 1, len);
1969 entry = isl_hash_table_find(ctx, table, c_hash, has_ineq, &v, 1);
1970 if (!entry)
1971 return - 1;
1972 entry->data = ineq;
1973 return 0;
1976 /* Fill hash table "table" with the constraints of "bset".
1977 * Equalities are added as two inequalities.
1978 * The value in the hash table is a pointer to the (in)equality of "bset".
1980 static int hash_basic_set(struct isl_hash_table *table,
1981 struct isl_basic_set *bset)
1983 int i, j;
1984 unsigned dim = isl_basic_set_total_dim(bset);
1986 for (i = 0; i < bset->n_eq; ++i) {
1987 for (j = 0; j < 2; ++j) {
1988 isl_seq_neg(bset->eq[i], bset->eq[i], 1 + dim);
1989 if (hash_ineq(bset->ctx, table, bset->eq[i], dim) < 0)
1990 return -1;
1993 for (i = 0; i < bset->n_ineq; ++i) {
1994 if (hash_ineq(bset->ctx, table, bset->ineq[i], dim) < 0)
1995 return -1;
1997 return 0;
2000 static struct sh_data *sh_data_alloc(struct isl_set *set, unsigned n_ineq)
2002 struct sh_data *data;
2003 int i;
2005 data = isl_calloc(set->ctx, struct sh_data,
2006 sizeof(struct sh_data) +
2007 (set->n - 1) * sizeof(struct sh_data_entry));
2008 if (!data)
2009 return NULL;
2010 data->ctx = set->ctx;
2011 data->n = set->n;
2012 data->hull_table = isl_hash_table_alloc(set->ctx, n_ineq);
2013 if (!data->hull_table)
2014 goto error;
2015 for (i = 0; i < set->n; ++i) {
2016 data->p[i].table = isl_hash_table_alloc(set->ctx,
2017 2 * set->p[i]->n_eq + set->p[i]->n_ineq);
2018 if (!data->p[i].table)
2019 goto error;
2020 if (hash_basic_set(data->p[i].table, set->p[i]) < 0)
2021 goto error;
2023 return data;
2024 error:
2025 sh_data_free(data);
2026 return NULL;
2029 /* Check if inequality "ineq" is a bound for basic set "j" or if
2030 * it can be relaxed (by increasing the constant term) to become
2031 * a bound for that basic set. In the latter case, the constant
2032 * term is updated.
2033 * Return 1 if "ineq" is a bound
2034 * 0 if "ineq" may attain arbitrarily small values on basic set "j"
2035 * -1 if some error occurred
2037 static int is_bound(struct sh_data *data, struct isl_set *set, int j,
2038 isl_int *ineq)
2040 enum isl_lp_result res;
2041 isl_int opt;
2043 if (!data->p[j].tab) {
2044 data->p[j].tab = isl_tab_from_basic_set(set->p[j]);
2045 if (!data->p[j].tab)
2046 return -1;
2049 isl_int_init(opt);
2051 res = isl_tab_min(data->p[j].tab, ineq, data->ctx->one,
2052 &opt, NULL, 0);
2053 if (res == isl_lp_ok && isl_int_is_neg(opt))
2054 isl_int_sub(ineq[0], ineq[0], opt);
2056 isl_int_clear(opt);
2058 return (res == isl_lp_ok || res == isl_lp_empty) ? 1 :
2059 res == isl_lp_unbounded ? 0 : -1;
2062 /* Check if inequality "ineq" from basic set "i" can be relaxed to
2063 * become a bound on the whole set. If so, add the (relaxed) inequality
2064 * to "hull".
2066 * We first check if "hull" already contains a translate of the inequality.
2067 * If so, we are done.
2068 * Then, we check if any of the previous basic sets contains a translate
2069 * of the inequality. If so, then we have already considered this
2070 * inequality and we are done.
2071 * Otherwise, for each basic set other than "i", we check if the inequality
2072 * is a bound on the basic set.
2073 * For previous basic sets, we know that they do not contain a translate
2074 * of the inequality, so we directly call is_bound.
2075 * For following basic sets, we first check if a translate of the
2076 * inequality appears in its description and if so directly update
2077 * the inequality accordingly.
2079 static struct isl_basic_set *add_bound(struct isl_basic_set *hull,
2080 struct sh_data *data, struct isl_set *set, int i, isl_int *ineq)
2082 uint32_t c_hash;
2083 struct ineq_cmp_data v;
2084 struct isl_hash_table_entry *entry;
2085 int j, k;
2087 if (!hull)
2088 return NULL;
2090 v.len = isl_basic_set_total_dim(hull);
2091 v.p = ineq;
2092 c_hash = isl_seq_get_hash(ineq + 1, v.len);
2094 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2095 has_ineq, &v, 0);
2096 if (entry)
2097 return hull;
2099 for (j = 0; j < i; ++j) {
2100 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2101 c_hash, has_ineq, &v, 0);
2102 if (entry)
2103 break;
2105 if (j < i)
2106 return hull;
2108 k = isl_basic_set_alloc_inequality(hull);
2109 isl_seq_cpy(hull->ineq[k], ineq, 1 + v.len);
2110 if (k < 0)
2111 goto error;
2113 for (j = 0; j < i; ++j) {
2114 int bound;
2115 bound = is_bound(data, set, j, hull->ineq[k]);
2116 if (bound < 0)
2117 goto error;
2118 if (!bound)
2119 break;
2121 if (j < i) {
2122 isl_basic_set_free_inequality(hull, 1);
2123 return hull;
2126 for (j = i + 1; j < set->n; ++j) {
2127 int bound, neg;
2128 isl_int *ineq_j;
2129 entry = isl_hash_table_find(hull->ctx, data->p[j].table,
2130 c_hash, has_ineq, &v, 0);
2131 if (entry) {
2132 ineq_j = entry->data;
2133 neg = isl_seq_is_neg(ineq_j + 1,
2134 hull->ineq[k] + 1, v.len);
2135 if (neg)
2136 isl_int_neg(ineq_j[0], ineq_j[0]);
2137 if (isl_int_gt(ineq_j[0], hull->ineq[k][0]))
2138 isl_int_set(hull->ineq[k][0], ineq_j[0]);
2139 if (neg)
2140 isl_int_neg(ineq_j[0], ineq_j[0]);
2141 continue;
2143 bound = is_bound(data, set, j, hull->ineq[k]);
2144 if (bound < 0)
2145 goto error;
2146 if (!bound)
2147 break;
2149 if (j < set->n) {
2150 isl_basic_set_free_inequality(hull, 1);
2151 return hull;
2154 entry = isl_hash_table_find(hull->ctx, data->hull_table, c_hash,
2155 has_ineq, &v, 1);
2156 if (!entry)
2157 goto error;
2158 entry->data = hull->ineq[k];
2160 return hull;
2161 error:
2162 isl_basic_set_free(hull);
2163 return NULL;
2166 /* Check if any inequality from basic set "i" can be relaxed to
2167 * become a bound on the whole set. If so, add the (relaxed) inequality
2168 * to "hull".
2170 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
2171 struct sh_data *data, struct isl_set *set, int i)
2173 int j, k;
2174 unsigned dim = isl_basic_set_total_dim(bset);
2176 for (j = 0; j < set->p[i]->n_eq; ++j) {
2177 for (k = 0; k < 2; ++k) {
2178 isl_seq_neg(set->p[i]->eq[j], set->p[i]->eq[j], 1+dim);
2179 bset = add_bound(bset, data, set, i, set->p[i]->eq[j]);
2182 for (j = 0; j < set->p[i]->n_ineq; ++j)
2183 bset = add_bound(bset, data, set, i, set->p[i]->ineq[j]);
2184 return bset;
2187 /* Compute a superset of the convex hull of set that is described
2188 * by only translates of the constraints in the constituents of set.
2190 static struct isl_basic_set *uset_simple_hull(struct isl_set *set)
2192 struct sh_data *data = NULL;
2193 struct isl_basic_set *hull = NULL;
2194 unsigned n_ineq;
2195 int i;
2197 if (!set)
2198 return NULL;
2200 n_ineq = 0;
2201 for (i = 0; i < set->n; ++i) {
2202 if (!set->p[i])
2203 goto error;
2204 n_ineq += 2 * set->p[i]->n_eq + set->p[i]->n_ineq;
2207 hull = isl_basic_set_alloc_dim(isl_dim_copy(set->dim), 0, 0, n_ineq);
2208 if (!hull)
2209 goto error;
2211 data = sh_data_alloc(set, n_ineq);
2212 if (!data)
2213 goto error;
2215 for (i = 0; i < set->n; ++i)
2216 hull = add_bounds(hull, data, set, i);
2218 sh_data_free(data);
2219 isl_set_free(set);
2221 return hull;
2222 error:
2223 sh_data_free(data);
2224 isl_basic_set_free(hull);
2225 isl_set_free(set);
2226 return NULL;
2229 /* Compute a superset of the convex hull of map that is described
2230 * by only translates of the constraints in the constituents of map.
2232 struct isl_basic_map *isl_map_simple_hull(struct isl_map *map)
2234 struct isl_set *set = NULL;
2235 struct isl_basic_map *model = NULL;
2236 struct isl_basic_map *hull;
2237 struct isl_basic_map *affine_hull;
2238 struct isl_basic_set *bset = NULL;
2240 if (!map)
2241 return NULL;
2242 if (map->n == 0) {
2243 hull = isl_basic_map_empty_like_map(map);
2244 isl_map_free(map);
2245 return hull;
2247 if (map->n == 1) {
2248 hull = isl_basic_map_copy(map->p[0]);
2249 isl_map_free(map);
2250 return hull;
2253 map = isl_map_detect_equalities(map);
2254 affine_hull = isl_map_affine_hull(isl_map_copy(map));
2255 map = isl_map_align_divs(map);
2256 model = isl_basic_map_copy(map->p[0]);
2258 set = isl_map_underlying_set(map);
2260 bset = uset_simple_hull(set);
2262 hull = isl_basic_map_overlying_set(bset, model);
2264 hull = isl_basic_map_intersect(hull, affine_hull);
2265 hull = isl_basic_map_convex_hull(hull);
2266 ISL_F_SET(hull, ISL_BASIC_MAP_NO_IMPLICIT);
2267 ISL_F_SET(hull, ISL_BASIC_MAP_ALL_EQUALITIES);
2269 return hull;
2272 struct isl_basic_set *isl_set_simple_hull(struct isl_set *set)
2274 return (struct isl_basic_set *)
2275 isl_map_simple_hull((struct isl_map *)set);
2278 /* Given a set "set", return parametric bounds on the dimension "dim".
2280 static struct isl_basic_set *set_bounds(struct isl_set *set, int dim)
2282 unsigned set_dim = isl_set_dim(set, isl_dim_set);
2283 set = isl_set_copy(set);
2284 set = isl_set_eliminate_dims(set, dim + 1, set_dim - (dim + 1));
2285 set = isl_set_eliminate_dims(set, 0, dim);
2286 return isl_set_convex_hull(set);
2289 /* Computes a "simple hull" and then check if each dimension in the
2290 * resulting hull is bounded by a symbolic constant. If not, the
2291 * hull is intersected with the corresponding bounds on the whole set.
2293 struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set)
2295 int i, j;
2296 struct isl_basic_set *hull;
2297 unsigned nparam, left;
2298 int removed_divs = 0;
2300 hull = isl_set_simple_hull(isl_set_copy(set));
2301 if (!hull)
2302 goto error;
2304 nparam = isl_basic_set_dim(hull, isl_dim_param);
2305 for (i = 0; i < isl_basic_set_dim(hull, isl_dim_set); ++i) {
2306 int lower = 0, upper = 0;
2307 struct isl_basic_set *bounds;
2309 left = isl_basic_set_total_dim(hull) - nparam - i - 1;
2310 for (j = 0; j < hull->n_eq; ++j) {
2311 if (isl_int_is_zero(hull->eq[j][1 + nparam + i]))
2312 continue;
2313 if (isl_seq_first_non_zero(hull->eq[j]+1+nparam+i+1,
2314 left) == -1)
2315 break;
2317 if (j < hull->n_eq)
2318 continue;
2320 for (j = 0; j < hull->n_ineq; ++j) {
2321 if (isl_int_is_zero(hull->ineq[j][1 + nparam + i]))
2322 continue;
2323 if (isl_seq_first_non_zero(hull->ineq[j]+1+nparam+i+1,
2324 left) != -1 ||
2325 isl_seq_first_non_zero(hull->ineq[j]+1+nparam,
2326 i) != -1)
2327 continue;
2328 if (isl_int_is_pos(hull->ineq[j][1 + nparam + i]))
2329 lower = 1;
2330 else
2331 upper = 1;
2332 if (lower && upper)
2333 break;
2336 if (lower && upper)
2337 continue;
2339 if (!removed_divs) {
2340 set = isl_set_remove_divs(set);
2341 if (!set)
2342 goto error;
2343 removed_divs = 1;
2345 bounds = set_bounds(set, i);
2346 hull = isl_basic_set_intersect(hull, bounds);
2347 if (!hull)
2348 goto error;
2351 isl_set_free(set);
2352 return hull;
2353 error:
2354 isl_set_free(set);
2355 return NULL;