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
12 #include "isl_map_private.h"
16 #include "isl_equalities.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
)
27 bmap
->ineq
[i
] = bmap
->ineq
[j
];
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
;
47 total
= isl_basic_map_total_dim(*bmap
);
48 for (i
= 0; i
< total
; ++i
) {
50 if (isl_int_is_zero(c
[1+i
]))
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
]))
56 if (j
== (*bmap
)->n_ineq
)
62 res
= isl_basic_map_solve_lp(*bmap
, 0, c
, (*bmap
)->ctx
->one
,
64 if (res
== isl_lp_unbounded
)
66 if (res
== isl_lp_error
)
68 if (res
== isl_lp_empty
) {
69 *bmap
= isl_basic_map_set_to_empty(*bmap
);
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
90 struct isl_basic_map
*isl_basic_map_convex_hull(struct isl_basic_map
*bmap
)
97 bmap
= isl_basic_map_gauss(bmap
, NULL
);
98 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
))
100 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_NO_REDUNDANT
))
102 if (bmap
->n_ineq
<= 1)
105 tab
= isl_tab_from_basic_map(bmap
);
106 tab
= isl_tab_detect_implicit_equalities(tab
);
107 if (isl_tab_detect_redundant(tab
) < 0)
109 bmap
= isl_basic_map_update_from_tab(bmap
, tab
);
111 ISL_F_SET(bmap
, ISL_BASIC_MAP_NO_IMPLICIT
);
112 ISL_F_SET(bmap
, ISL_BASIC_MAP_NO_REDUNDANT
);
116 isl_basic_map_free(bmap
);
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
)
138 isl_int_init(opt_denom
);
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
))
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
)
150 if (res
== isl_lp_error
)
152 if (res
== isl_lp_empty
) {
153 set
->p
[j
] = isl_basic_set_set_to_empty(set
->p
[j
]);
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
);
166 isl_int_clear(opt_denom
);
170 isl_int_clear(opt_denom
);
174 /* Check if "c" is a direction that is independent of the previously found "n"
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
)
187 isl_seq_cpy(dirs
->row
[n
]+1, c
+1, dirs
->n_col
-1);
189 int pos
= isl_seq_first_non_zero(dirs
->row
[n
]+1, dirs
->n_col
-1);
192 for (i
= 0; i
< n
; ++i
) {
194 pos_i
= isl_seq_first_non_zero(dirs
->row
[i
]+1, dirs
->n_col
-1);
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);
207 is_bound
= uset_is_bound(set
, dirs
->row
[n
], dirs
->n_col
);
210 isl_seq_normalize(set
->ctx
, dirs
->row
[n
], dirs
->n_col
);
213 isl_int
*t
= dirs
->row
[n
];
214 for (k
= n
; k
> i
; --k
)
215 dirs
->row
[k
] = dirs
->row
[k
-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
225 static struct isl_mat
*independent_bounds(struct isl_set
*set
)
228 struct isl_mat
*dirs
= NULL
;
229 unsigned dim
= isl_set_n_dim(set
);
231 dirs
= isl_mat_alloc(set
->ctx
, dim
, 1+dim
);
236 for (i
= 0; n
< dim
&& i
< set
->n
; ++i
) {
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
);
247 for (j
= 0; n
< dim
&& j
< bset
->n_ineq
; ++j
) {
248 f
= is_independent_bound(set
, bset
->ineq
[j
], dirs
, n
);
262 struct isl_basic_set
*isl_basic_set_set_rational(struct isl_basic_set
*bset
)
267 if (ISL_F_ISSET(bset
, ISL_BASIC_MAP_RATIONAL
))
270 bset
= isl_basic_set_cow(bset
);
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
)
283 set
= isl_set_cow(set
);
286 for (i
= 0; i
< set
->n
; ++i
) {
287 set
->p
[i
] = isl_basic_set_set_rational(set
->p
[i
]);
297 static struct isl_basic_set
*isl_basic_set_add_equality(
298 struct isl_basic_set
*bset
, isl_int
*c
)
303 if (ISL_F_ISSET(bset
, ISL_BASIC_SET_EMPTY
))
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
);
314 isl_seq_cpy(bset
->eq
[i
], c
, 1 + dim
);
317 isl_basic_set_free(bset
);
321 static struct isl_set
*isl_set_add_basic_set_equality(struct isl_set
*set
, isl_int
*c
)
325 set
= isl_set_cow(set
);
328 for (i
= 0; i
< set
->n
; ++i
) {
329 set
->p
[i
] = isl_basic_set_add_equality(set
->p
[i
], c
);
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
347 * then the resulting set is of dimension n*(1+d) and has as constraints
356 static struct isl_basic_set
*wrap_constraints(struct isl_set
*set
)
358 struct isl_basic_set
*lp
;
362 unsigned dim
, lp_dim
;
367 dim
= 1 + isl_set_n_dim(set
);
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
);
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));
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
411 * We first transform the set such that the facet constraint becomes
415 * I.e., the facet lies in
419 * and on that facet, the constraint that defines the ridge is
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
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
451 * the constraints of each (transformed) basic set.
452 * If a = n/d, then the constraint defining the new facet (in the transformed
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
)
468 struct isl_mat
*T
= NULL
;
469 struct isl_basic_set
*lp
= NULL
;
471 enum isl_lp_result res
;
475 set
= isl_set_copy(set
);
477 dim
= 1 + isl_set_n_dim(set
);
478 T
= isl_mat_alloc(set
->ctx
, 3, dim
);
481 isl_int_set_si(T
->row
[0][0], 1);
482 isl_seq_clr(T
->row
[0]+1, dim
- 1);
483 isl_seq_cpy(T
->row
[1], facet
, dim
);
484 isl_seq_cpy(T
->row
[2], ridge
, dim
);
485 T
= isl_mat_right_inverse(T
);
486 set
= isl_set_preimage(set
, T
);
490 lp
= wrap_constraints(set
);
491 obj
= isl_vec_alloc(set
->ctx
, 1 + dim
*set
->n
);
494 isl_int_set_si(obj
->block
.data
[0], 0);
495 for (i
= 0; i
< set
->n
; ++i
) {
496 isl_seq_clr(obj
->block
.data
+ 1 + dim
*i
, 2);
497 isl_int_set_si(obj
->block
.data
[1 + dim
*i
+2], 1);
498 isl_seq_clr(obj
->block
.data
+ 1 + dim
*i
+3, dim
-3);
502 res
= isl_basic_set_solve_lp(lp
, 0,
503 obj
->block
.data
, set
->ctx
->one
, &num
, &den
, NULL
);
504 if (res
== isl_lp_ok
) {
505 isl_int_neg(num
, num
);
506 isl_seq_combine(facet
, num
, facet
, den
, ridge
, dim
);
511 isl_basic_set_free(lp
);
513 isl_assert(set
->ctx
, res
== isl_lp_ok
|| res
== isl_lp_unbounded
,
517 isl_basic_set_free(lp
);
523 /* Drop rows in "rows" that are redundant with respect to earlier rows,
524 * assuming that "rows" is of full column rank.
526 * We compute the column echelon form. The non-redundant rows are
527 * those that are the first to contain a non-zero entry in a column.
528 * All the other rows can be removed.
530 static __isl_give isl_mat
*drop_redundant_rows(__isl_take isl_mat
*rows
)
532 struct isl_mat
*H
= NULL
;
540 isl_assert(rows
->ctx
, rows
->n_row
>= rows
->n_col
, goto error
);
542 if (rows
->n_row
== rows
->n_col
)
545 H
= isl_mat_left_hermite(isl_mat_copy(rows
), 0, NULL
, NULL
);
549 last_row
= rows
->n_row
;
550 for (col
= rows
->n_col
- 1; col
>= 0; --col
) {
551 for (row
= col
; row
< last_row
; ++row
)
552 if (!isl_int_is_zero(H
->row
[row
][col
]))
554 isl_assert(rows
->ctx
, row
< last_row
, goto error
);
555 if (row
+ 1 < last_row
) {
556 rows
= isl_mat_drop_rows(rows
, row
+ 1, last_row
- (row
+ 1));
557 if (rows
->n_row
== rows
->n_col
)
572 /* Given a set of d linearly independent bounding constraints of the
573 * convex hull of "set", compute the constraint of a facet of "set".
575 * We first compute the intersection with the first bounding hyperplane
576 * and remove the component corresponding to this hyperplane from
577 * other bounds (in homogeneous space).
578 * We then wrap around one of the remaining bounding constraints
579 * and continue the process until all bounding constraints have been
580 * taken into account.
581 * The resulting linear combination of the bounding constraints will
582 * correspond to a facet of the convex hull.
584 static struct isl_mat
*initial_facet_constraint(struct isl_set
*set
,
585 struct isl_mat
*bounds
)
587 struct isl_set
*slice
= NULL
;
588 struct isl_basic_set
*face
= NULL
;
589 struct isl_mat
*m
, *U
, *Q
;
591 unsigned dim
= isl_set_n_dim(set
);
593 isl_assert(set
->ctx
, set
->n
> 0, goto error
);
594 isl_assert(set
->ctx
, bounds
->n_row
== dim
, goto error
);
596 while (bounds
->n_row
> 1) {
597 slice
= isl_set_copy(set
);
598 slice
= isl_set_add_basic_set_equality(slice
, bounds
->row
[0]);
599 face
= isl_set_affine_hull(slice
);
602 if (face
->n_eq
== 1) {
603 isl_basic_set_free(face
);
606 m
= isl_mat_alloc(set
->ctx
, 1 + face
->n_eq
, 1 + dim
);
609 isl_int_set_si(m
->row
[0][0], 1);
610 isl_seq_clr(m
->row
[0]+1, dim
);
611 for (i
= 0; i
< face
->n_eq
; ++i
)
612 isl_seq_cpy(m
->row
[1 + i
], face
->eq
[i
], 1 + dim
);
613 U
= isl_mat_right_inverse(m
);
614 Q
= isl_mat_right_inverse(isl_mat_copy(U
));
615 U
= isl_mat_drop_cols(U
, 1 + face
->n_eq
, dim
- face
->n_eq
);
616 Q
= isl_mat_drop_rows(Q
, 1 + face
->n_eq
, dim
- face
->n_eq
);
617 U
= isl_mat_drop_cols(U
, 0, 1);
618 Q
= isl_mat_drop_rows(Q
, 0, 1);
619 bounds
= isl_mat_product(bounds
, U
);
620 bounds
= drop_redundant_rows(bounds
);
621 bounds
= isl_mat_product(bounds
, Q
);
622 isl_assert(set
->ctx
, bounds
->n_row
> 1, goto error
);
623 if (!isl_set_wrap_facet(set
, bounds
->row
[0],
624 bounds
->row
[bounds
->n_row
-1]))
626 isl_basic_set_free(face
);
631 isl_basic_set_free(face
);
632 isl_mat_free(bounds
);
636 /* Given the bounding constraint "c" of a facet of the convex hull of "set",
637 * compute a hyperplane description of the facet, i.e., compute the facets
640 * We compute an affine transformation that transforms the constraint
649 * by computing the right inverse U of a matrix that starts with the rows
662 * Since z_1 is zero, we can drop this variable as well as the corresponding
663 * column of U to obtain
671 * with Q' equal to Q, but without the corresponding row.
672 * After computing the facets of the facet in the z' space,
673 * we convert them back to the x space through Q.
675 static struct isl_basic_set
*compute_facet(struct isl_set
*set
, isl_int
*c
)
677 struct isl_mat
*m
, *U
, *Q
;
678 struct isl_basic_set
*facet
= NULL
;
683 set
= isl_set_copy(set
);
684 dim
= isl_set_n_dim(set
);
685 m
= isl_mat_alloc(set
->ctx
, 2, 1 + dim
);
688 isl_int_set_si(m
->row
[0][0], 1);
689 isl_seq_clr(m
->row
[0]+1, dim
);
690 isl_seq_cpy(m
->row
[1], c
, 1+dim
);
691 U
= isl_mat_right_inverse(m
);
692 Q
= isl_mat_right_inverse(isl_mat_copy(U
));
693 U
= isl_mat_drop_cols(U
, 1, 1);
694 Q
= isl_mat_drop_rows(Q
, 1, 1);
695 set
= isl_set_preimage(set
, U
);
696 facet
= uset_convex_hull_wrap_bounded(set
);
697 facet
= isl_basic_set_preimage(facet
, Q
);
698 isl_assert(ctx
, facet
->n_eq
== 0, goto error
);
701 isl_basic_set_free(facet
);
706 /* Given an initial facet constraint, compute the remaining facets.
707 * We do this by running through all facets found so far and computing
708 * the adjacent facets through wrapping, adding those facets that we
709 * hadn't already found before.
711 * For each facet we have found so far, we first compute its facets
712 * in the resulting convex hull. That is, we compute the ridges
713 * of the resulting convex hull contained in the facet.
714 * We also compute the corresponding facet in the current approximation
715 * of the convex hull. There is no need to wrap around the ridges
716 * in this facet since that would result in a facet that is already
717 * present in the current approximation.
719 * This function can still be significantly optimized by checking which of
720 * the facets of the basic sets are also facets of the convex hull and
721 * using all the facets so far to help in constructing the facets of the
724 * using the technique in section "3.1 Ridge Generation" of
725 * "Extended Convex Hull" by Fukuda et al.
727 static struct isl_basic_set
*extend(struct isl_basic_set
*hull
,
732 struct isl_basic_set
*facet
= NULL
;
733 struct isl_basic_set
*hull_facet
= NULL
;
739 isl_assert(set
->ctx
, set
->n
> 0, goto error
);
741 dim
= isl_set_n_dim(set
);
743 for (i
= 0; i
< hull
->n_ineq
; ++i
) {
744 facet
= compute_facet(set
, hull
->ineq
[i
]);
745 facet
= isl_basic_set_add_equality(facet
, hull
->ineq
[i
]);
746 facet
= isl_basic_set_gauss(facet
, NULL
);
747 facet
= isl_basic_set_normalize_constraints(facet
);
748 hull_facet
= isl_basic_set_copy(hull
);
749 hull_facet
= isl_basic_set_add_equality(hull_facet
, hull
->ineq
[i
]);
750 hull_facet
= isl_basic_set_gauss(hull_facet
, NULL
);
751 hull_facet
= isl_basic_set_normalize_constraints(hull_facet
);
754 hull
= isl_basic_set_cow(hull
);
755 hull
= isl_basic_set_extend_dim(hull
,
756 isl_dim_copy(hull
->dim
), 0, 0, facet
->n_ineq
);
757 for (j
= 0; j
< facet
->n_ineq
; ++j
) {
758 for (f
= 0; f
< hull_facet
->n_ineq
; ++f
)
759 if (isl_seq_eq(facet
->ineq
[j
],
760 hull_facet
->ineq
[f
], 1 + dim
))
762 if (f
< hull_facet
->n_ineq
)
764 k
= isl_basic_set_alloc_inequality(hull
);
767 isl_seq_cpy(hull
->ineq
[k
], hull
->ineq
[i
], 1+dim
);
768 if (!isl_set_wrap_facet(set
, hull
->ineq
[k
], facet
->ineq
[j
]))
771 isl_basic_set_free(hull_facet
);
772 isl_basic_set_free(facet
);
774 hull
= isl_basic_set_simplify(hull
);
775 hull
= isl_basic_set_finalize(hull
);
778 isl_basic_set_free(hull_facet
);
779 isl_basic_set_free(facet
);
780 isl_basic_set_free(hull
);
784 /* Special case for computing the convex hull of a one dimensional set.
785 * We simply collect the lower and upper bounds of each basic set
786 * and the biggest of those.
788 static struct isl_basic_set
*convex_hull_1d(struct isl_set
*set
)
790 struct isl_mat
*c
= NULL
;
791 isl_int
*lower
= NULL
;
792 isl_int
*upper
= NULL
;
795 struct isl_basic_set
*hull
;
797 for (i
= 0; i
< set
->n
; ++i
) {
798 set
->p
[i
] = isl_basic_set_simplify(set
->p
[i
]);
802 set
= isl_set_remove_empty_parts(set
);
805 isl_assert(set
->ctx
, set
->n
> 0, goto error
);
806 c
= isl_mat_alloc(set
->ctx
, 2, 2);
810 if (set
->p
[0]->n_eq
> 0) {
811 isl_assert(set
->ctx
, set
->p
[0]->n_eq
== 1, goto error
);
814 if (isl_int_is_pos(set
->p
[0]->eq
[0][1])) {
815 isl_seq_cpy(lower
, set
->p
[0]->eq
[0], 2);
816 isl_seq_neg(upper
, set
->p
[0]->eq
[0], 2);
818 isl_seq_neg(lower
, set
->p
[0]->eq
[0], 2);
819 isl_seq_cpy(upper
, set
->p
[0]->eq
[0], 2);
822 for (j
= 0; j
< set
->p
[0]->n_ineq
; ++j
) {
823 if (isl_int_is_pos(set
->p
[0]->ineq
[j
][1])) {
825 isl_seq_cpy(lower
, set
->p
[0]->ineq
[j
], 2);
828 isl_seq_cpy(upper
, set
->p
[0]->ineq
[j
], 2);
835 for (i
= 0; i
< set
->n
; ++i
) {
836 struct isl_basic_set
*bset
= set
->p
[i
];
840 for (j
= 0; j
< bset
->n_eq
; ++j
) {
844 isl_int_mul(a
, lower
[0], bset
->eq
[j
][1]);
845 isl_int_mul(b
, lower
[1], bset
->eq
[j
][0]);
846 if (isl_int_lt(a
, b
) && isl_int_is_pos(bset
->eq
[j
][1]))
847 isl_seq_cpy(lower
, bset
->eq
[j
], 2);
848 if (isl_int_gt(a
, b
) && isl_int_is_neg(bset
->eq
[j
][1]))
849 isl_seq_neg(lower
, bset
->eq
[j
], 2);
852 isl_int_mul(a
, upper
[0], bset
->eq
[j
][1]);
853 isl_int_mul(b
, upper
[1], bset
->eq
[j
][0]);
854 if (isl_int_lt(a
, b
) && isl_int_is_pos(bset
->eq
[j
][1]))
855 isl_seq_neg(upper
, bset
->eq
[j
], 2);
856 if (isl_int_gt(a
, b
) && isl_int_is_neg(bset
->eq
[j
][1]))
857 isl_seq_cpy(upper
, bset
->eq
[j
], 2);
860 for (j
= 0; j
< bset
->n_ineq
; ++j
) {
861 if (isl_int_is_pos(bset
->ineq
[j
][1]))
863 if (isl_int_is_neg(bset
->ineq
[j
][1]))
865 if (lower
&& isl_int_is_pos(bset
->ineq
[j
][1])) {
866 isl_int_mul(a
, lower
[0], bset
->ineq
[j
][1]);
867 isl_int_mul(b
, lower
[1], bset
->ineq
[j
][0]);
868 if (isl_int_lt(a
, b
))
869 isl_seq_cpy(lower
, bset
->ineq
[j
], 2);
871 if (upper
&& isl_int_is_neg(bset
->ineq
[j
][1])) {
872 isl_int_mul(a
, upper
[0], bset
->ineq
[j
][1]);
873 isl_int_mul(b
, upper
[1], bset
->ineq
[j
][0]);
874 if (isl_int_gt(a
, b
))
875 isl_seq_cpy(upper
, bset
->ineq
[j
], 2);
886 hull
= isl_basic_set_alloc(set
->ctx
, 0, 1, 0, 0, 2);
887 hull
= isl_basic_set_set_rational(hull
);
891 k
= isl_basic_set_alloc_inequality(hull
);
892 isl_seq_cpy(hull
->ineq
[k
], lower
, 2);
895 k
= isl_basic_set_alloc_inequality(hull
);
896 isl_seq_cpy(hull
->ineq
[k
], upper
, 2);
898 hull
= isl_basic_set_finalize(hull
);
908 /* Project out final n dimensions using Fourier-Motzkin */
909 static struct isl_set
*set_project_out(struct isl_ctx
*ctx
,
910 struct isl_set
*set
, unsigned n
)
912 return isl_set_remove_dims(set
, isl_set_n_dim(set
) - n
, n
);
915 static struct isl_basic_set
*convex_hull_0d(struct isl_set
*set
)
917 struct isl_basic_set
*convex_hull
;
922 if (isl_set_is_empty(set
))
923 convex_hull
= isl_basic_set_empty(isl_dim_copy(set
->dim
));
925 convex_hull
= isl_basic_set_universe(isl_dim_copy(set
->dim
));
930 /* Compute the convex hull of a pair of basic sets without any parameters or
931 * integer divisions using Fourier-Motzkin elimination.
932 * The convex hull is the set of all points that can be written as
933 * the sum of points from both basic sets (in homogeneous coordinates).
934 * We set up the constraints in a space with dimensions for each of
935 * the three sets and then project out the dimensions corresponding
936 * to the two original basic sets, retaining only those corresponding
937 * to the convex hull.
939 static struct isl_basic_set
*convex_hull_pair_elim(struct isl_basic_set
*bset1
,
940 struct isl_basic_set
*bset2
)
943 struct isl_basic_set
*bset
[2];
944 struct isl_basic_set
*hull
= NULL
;
947 if (!bset1
|| !bset2
)
950 dim
= isl_basic_set_n_dim(bset1
);
951 hull
= isl_basic_set_alloc(bset1
->ctx
, 0, 2 + 3 * dim
, 0,
952 1 + dim
+ bset1
->n_eq
+ bset2
->n_eq
,
953 2 + bset1
->n_ineq
+ bset2
->n_ineq
);
956 for (i
= 0; i
< 2; ++i
) {
957 for (j
= 0; j
< bset
[i
]->n_eq
; ++j
) {
958 k
= isl_basic_set_alloc_equality(hull
);
961 isl_seq_clr(hull
->eq
[k
], (i
+1) * (1+dim
));
962 isl_seq_clr(hull
->eq
[k
]+(i
+2)*(1+dim
), (1-i
)*(1+dim
));
963 isl_seq_cpy(hull
->eq
[k
]+(i
+1)*(1+dim
), bset
[i
]->eq
[j
],
966 for (j
= 0; j
< bset
[i
]->n_ineq
; ++j
) {
967 k
= isl_basic_set_alloc_inequality(hull
);
970 isl_seq_clr(hull
->ineq
[k
], (i
+1) * (1+dim
));
971 isl_seq_clr(hull
->ineq
[k
]+(i
+2)*(1+dim
), (1-i
)*(1+dim
));
972 isl_seq_cpy(hull
->ineq
[k
]+(i
+1)*(1+dim
),
973 bset
[i
]->ineq
[j
], 1+dim
);
975 k
= isl_basic_set_alloc_inequality(hull
);
978 isl_seq_clr(hull
->ineq
[k
], 1+2+3*dim
);
979 isl_int_set_si(hull
->ineq
[k
][(i
+1)*(1+dim
)], 1);
981 for (j
= 0; j
< 1+dim
; ++j
) {
982 k
= isl_basic_set_alloc_equality(hull
);
985 isl_seq_clr(hull
->eq
[k
], 1+2+3*dim
);
986 isl_int_set_si(hull
->eq
[k
][j
], -1);
987 isl_int_set_si(hull
->eq
[k
][1+dim
+j
], 1);
988 isl_int_set_si(hull
->eq
[k
][2*(1+dim
)+j
], 1);
990 hull
= isl_basic_set_set_rational(hull
);
991 hull
= isl_basic_set_remove_dims(hull
, dim
, 2*(1+dim
));
992 hull
= isl_basic_set_convex_hull(hull
);
993 isl_basic_set_free(bset1
);
994 isl_basic_set_free(bset2
);
997 isl_basic_set_free(bset1
);
998 isl_basic_set_free(bset2
);
999 isl_basic_set_free(hull
);
1003 static int isl_basic_set_is_bounded(struct isl_basic_set
*bset
)
1005 struct isl_tab
*tab
;
1008 tab
= isl_tab_from_recession_cone(bset
);
1009 bounded
= isl_tab_cone_is_bounded(tab
);
1014 static int isl_set_is_bounded(struct isl_set
*set
)
1018 for (i
= 0; i
< set
->n
; ++i
) {
1019 int bounded
= isl_basic_set_is_bounded(set
->p
[i
]);
1020 if (!bounded
|| bounded
< 0)
1026 /* Compute the lineality space of the convex hull of bset1 and bset2.
1028 * We first compute the intersection of the recession cone of bset1
1029 * with the negative of the recession cone of bset2 and then compute
1030 * the linear hull of the resulting cone.
1032 static struct isl_basic_set
*induced_lineality_space(
1033 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
)
1036 struct isl_basic_set
*lin
= NULL
;
1039 if (!bset1
|| !bset2
)
1042 dim
= isl_basic_set_total_dim(bset1
);
1043 lin
= isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset1
), 0,
1044 bset1
->n_eq
+ bset2
->n_eq
,
1045 bset1
->n_ineq
+ bset2
->n_ineq
);
1046 lin
= isl_basic_set_set_rational(lin
);
1049 for (i
= 0; i
< bset1
->n_eq
; ++i
) {
1050 k
= isl_basic_set_alloc_equality(lin
);
1053 isl_int_set_si(lin
->eq
[k
][0], 0);
1054 isl_seq_cpy(lin
->eq
[k
] + 1, bset1
->eq
[i
] + 1, dim
);
1056 for (i
= 0; i
< bset1
->n_ineq
; ++i
) {
1057 k
= isl_basic_set_alloc_inequality(lin
);
1060 isl_int_set_si(lin
->ineq
[k
][0], 0);
1061 isl_seq_cpy(lin
->ineq
[k
] + 1, bset1
->ineq
[i
] + 1, dim
);
1063 for (i
= 0; i
< bset2
->n_eq
; ++i
) {
1064 k
= isl_basic_set_alloc_equality(lin
);
1067 isl_int_set_si(lin
->eq
[k
][0], 0);
1068 isl_seq_neg(lin
->eq
[k
] + 1, bset2
->eq
[i
] + 1, dim
);
1070 for (i
= 0; i
< bset2
->n_ineq
; ++i
) {
1071 k
= isl_basic_set_alloc_inequality(lin
);
1074 isl_int_set_si(lin
->ineq
[k
][0], 0);
1075 isl_seq_neg(lin
->ineq
[k
] + 1, bset2
->ineq
[i
] + 1, dim
);
1078 isl_basic_set_free(bset1
);
1079 isl_basic_set_free(bset2
);
1080 return isl_basic_set_affine_hull(lin
);
1082 isl_basic_set_free(lin
);
1083 isl_basic_set_free(bset1
);
1084 isl_basic_set_free(bset2
);
1088 static struct isl_basic_set
*uset_convex_hull(struct isl_set
*set
);
1090 /* Given a set and a linear space "lin" of dimension n > 0,
1091 * project the linear space from the set, compute the convex hull
1092 * and then map the set back to the original space.
1098 * describe the linear space. We first compute the Hermite normal
1099 * form H = M U of M = H Q, to obtain
1103 * The last n rows of H will be zero, so the last n variables of x' = Q x
1104 * are the one we want to project out. We do this by transforming each
1105 * basic set A x >= b to A U x' >= b and then removing the last n dimensions.
1106 * After computing the convex hull in x'_1, i.e., A' x'_1 >= b',
1107 * we transform the hull back to the original space as A' Q_1 x >= b',
1108 * with Q_1 all but the last n rows of Q.
1110 static struct isl_basic_set
*modulo_lineality(struct isl_set
*set
,
1111 struct isl_basic_set
*lin
)
1113 unsigned total
= isl_basic_set_total_dim(lin
);
1115 struct isl_basic_set
*hull
;
1116 struct isl_mat
*M
, *U
, *Q
;
1120 lin_dim
= total
- lin
->n_eq
;
1121 M
= isl_mat_sub_alloc(set
->ctx
, lin
->eq
, 0, lin
->n_eq
, 1, total
);
1122 M
= isl_mat_left_hermite(M
, 0, &U
, &Q
);
1126 isl_basic_set_free(lin
);
1128 Q
= isl_mat_drop_rows(Q
, Q
->n_row
- lin_dim
, lin_dim
);
1130 U
= isl_mat_lin_to_aff(U
);
1131 Q
= isl_mat_lin_to_aff(Q
);
1133 set
= isl_set_preimage(set
, U
);
1134 set
= isl_set_remove_dims(set
, total
- lin_dim
, lin_dim
);
1135 hull
= uset_convex_hull(set
);
1136 hull
= isl_basic_set_preimage(hull
, Q
);
1140 isl_basic_set_free(lin
);
1145 /* Given two polyhedra with as constraints h_{ij} x >= 0 in homegeneous space,
1146 * set up an LP for solving
1148 * \sum_j \alpha_{1j} h_{1j} = \sum_j \alpha_{2j} h_{2j}
1150 * \alpha{i0} corresponds to the (implicit) positivity constraint 1 >= 0
1151 * The next \alpha{ij} correspond to the equalities and come in pairs.
1152 * The final \alpha{ij} correspond to the inequalities.
1154 static struct isl_basic_set
*valid_direction_lp(
1155 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
)
1157 struct isl_dim
*dim
;
1158 struct isl_basic_set
*lp
;
1163 if (!bset1
|| !bset2
)
1165 d
= 1 + isl_basic_set_total_dim(bset1
);
1167 2 * bset1
->n_eq
+ bset1
->n_ineq
+ 2 * bset2
->n_eq
+ bset2
->n_ineq
;
1168 dim
= isl_dim_set_alloc(bset1
->ctx
, 0, n
);
1169 lp
= isl_basic_set_alloc_dim(dim
, 0, d
, n
);
1172 for (i
= 0; i
< n
; ++i
) {
1173 k
= isl_basic_set_alloc_inequality(lp
);
1176 isl_seq_clr(lp
->ineq
[k
] + 1, n
);
1177 isl_int_set_si(lp
->ineq
[k
][0], -1);
1178 isl_int_set_si(lp
->ineq
[k
][1 + i
], 1);
1180 for (i
= 0; i
< d
; ++i
) {
1181 k
= isl_basic_set_alloc_equality(lp
);
1185 isl_int_set_si(lp
->eq
[k
][n
++], 0);
1186 /* positivity constraint 1 >= 0 */
1187 isl_int_set_si(lp
->eq
[k
][n
++], i
== 0);
1188 for (j
= 0; j
< bset1
->n_eq
; ++j
) {
1189 isl_int_set(lp
->eq
[k
][n
++], bset1
->eq
[j
][i
]);
1190 isl_int_neg(lp
->eq
[k
][n
++], bset1
->eq
[j
][i
]);
1192 for (j
= 0; j
< bset1
->n_ineq
; ++j
)
1193 isl_int_set(lp
->eq
[k
][n
++], bset1
->ineq
[j
][i
]);
1194 /* positivity constraint 1 >= 0 */
1195 isl_int_set_si(lp
->eq
[k
][n
++], -(i
== 0));
1196 for (j
= 0; j
< bset2
->n_eq
; ++j
) {
1197 isl_int_neg(lp
->eq
[k
][n
++], bset2
->eq
[j
][i
]);
1198 isl_int_set(lp
->eq
[k
][n
++], bset2
->eq
[j
][i
]);
1200 for (j
= 0; j
< bset2
->n_ineq
; ++j
)
1201 isl_int_neg(lp
->eq
[k
][n
++], bset2
->ineq
[j
][i
]);
1203 lp
= isl_basic_set_gauss(lp
, NULL
);
1204 isl_basic_set_free(bset1
);
1205 isl_basic_set_free(bset2
);
1208 isl_basic_set_free(bset1
);
1209 isl_basic_set_free(bset2
);
1213 /* Compute a vector s in the homogeneous space such that <s, r> > 0
1214 * for all rays in the homogeneous space of the two cones that correspond
1215 * to the input polyhedra bset1 and bset2.
1217 * We compute s as a vector that satisfies
1219 * s = \sum_j \alpha_{ij} h_{ij} for i = 1,2 (*)
1221 * with h_{ij} the normals of the facets of polyhedron i
1222 * (including the "positivity constraint" 1 >= 0) and \alpha_{ij}
1223 * strictly positive numbers. For simplicity we impose \alpha_{ij} >= 1.
1224 * We first set up an LP with as variables the \alpha{ij}.
1225 * In this formulateion, for each polyhedron i,
1226 * the first constraint is the positivity constraint, followed by pairs
1227 * of variables for the equalities, followed by variables for the inequalities.
1228 * We then simply pick a feasible solution and compute s using (*).
1230 * Note that we simply pick any valid direction and make no attempt
1231 * to pick a "good" or even the "best" valid direction.
1233 static struct isl_vec
*valid_direction(
1234 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
)
1236 struct isl_basic_set
*lp
;
1237 struct isl_tab
*tab
;
1238 struct isl_vec
*sample
= NULL
;
1239 struct isl_vec
*dir
;
1244 if (!bset1
|| !bset2
)
1246 lp
= valid_direction_lp(isl_basic_set_copy(bset1
),
1247 isl_basic_set_copy(bset2
));
1248 tab
= isl_tab_from_basic_set(lp
);
1249 sample
= isl_tab_get_sample_value(tab
);
1251 isl_basic_set_free(lp
);
1254 d
= isl_basic_set_total_dim(bset1
);
1255 dir
= isl_vec_alloc(bset1
->ctx
, 1 + d
);
1258 isl_seq_clr(dir
->block
.data
+ 1, dir
->size
- 1);
1260 /* positivity constraint 1 >= 0 */
1261 isl_int_set(dir
->block
.data
[0], sample
->block
.data
[n
++]);
1262 for (i
= 0; i
< bset1
->n_eq
; ++i
) {
1263 isl_int_sub(sample
->block
.data
[n
],
1264 sample
->block
.data
[n
], sample
->block
.data
[n
+1]);
1265 isl_seq_combine(dir
->block
.data
,
1266 bset1
->ctx
->one
, dir
->block
.data
,
1267 sample
->block
.data
[n
], bset1
->eq
[i
], 1 + d
);
1271 for (i
= 0; i
< bset1
->n_ineq
; ++i
)
1272 isl_seq_combine(dir
->block
.data
,
1273 bset1
->ctx
->one
, dir
->block
.data
,
1274 sample
->block
.data
[n
++], bset1
->ineq
[i
], 1 + d
);
1275 isl_vec_free(sample
);
1276 isl_seq_normalize(bset1
->ctx
, dir
->block
.data
+ 1, dir
->size
- 1);
1277 isl_basic_set_free(bset1
);
1278 isl_basic_set_free(bset2
);
1281 isl_vec_free(sample
);
1282 isl_basic_set_free(bset1
);
1283 isl_basic_set_free(bset2
);
1287 /* Given a polyhedron b_i + A_i x >= 0 and a map T = S^{-1},
1288 * compute b_i' + A_i' x' >= 0, with
1290 * [ b_i A_i ] [ y' ] [ y' ]
1291 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1293 * In particular, add the "positivity constraint" and then perform
1296 static struct isl_basic_set
*homogeneous_map(struct isl_basic_set
*bset
,
1303 bset
= isl_basic_set_extend_constraints(bset
, 0, 1);
1304 k
= isl_basic_set_alloc_inequality(bset
);
1307 isl_seq_clr(bset
->ineq
[k
] + 1, isl_basic_set_total_dim(bset
));
1308 isl_int_set_si(bset
->ineq
[k
][0], 1);
1309 bset
= isl_basic_set_preimage(bset
, T
);
1313 isl_basic_set_free(bset
);
1317 /* Compute the convex hull of a pair of basic sets without any parameters or
1318 * integer divisions, where the convex hull is known to be pointed,
1319 * but the basic sets may be unbounded.
1321 * We turn this problem into the computation of a convex hull of a pair
1322 * _bounded_ polyhedra by "changing the direction of the homogeneous
1323 * dimension". This idea is due to Matthias Koeppe.
1325 * Consider the cones in homogeneous space that correspond to the
1326 * input polyhedra. The rays of these cones are also rays of the
1327 * polyhedra if the coordinate that corresponds to the homogeneous
1328 * dimension is zero. That is, if the inner product of the rays
1329 * with the homogeneous direction is zero.
1330 * The cones in the homogeneous space can also be considered to
1331 * correspond to other pairs of polyhedra by chosing a different
1332 * homogeneous direction. To ensure that both of these polyhedra
1333 * are bounded, we need to make sure that all rays of the cones
1334 * correspond to vertices and not to rays.
1335 * Let s be a direction such that <s, r> > 0 for all rays r of both cones.
1336 * Then using s as a homogeneous direction, we obtain a pair of polytopes.
1337 * The vector s is computed in valid_direction.
1339 * Note that we need to consider _all_ rays of the cones and not just
1340 * the rays that correspond to rays in the polyhedra. If we were to
1341 * only consider those rays and turn them into vertices, then we
1342 * may inadvertently turn some vertices into rays.
1344 * The standard homogeneous direction is the unit vector in the 0th coordinate.
1345 * We therefore transform the two polyhedra such that the selected
1346 * direction is mapped onto this standard direction and then proceed
1347 * with the normal computation.
1348 * Let S be a non-singular square matrix with s as its first row,
1349 * then we want to map the polyhedra to the space
1351 * [ y' ] [ y ] [ y ] [ y' ]
1352 * [ x' ] = S [ x ] i.e., [ x ] = S^{-1} [ x' ]
1354 * We take S to be the unimodular completion of s to limit the growth
1355 * of the coefficients in the following computations.
1357 * Let b_i + A_i x >= 0 be the constraints of polyhedron i.
1358 * We first move to the homogeneous dimension
1360 * b_i y + A_i x >= 0 [ b_i A_i ] [ y ] [ 0 ]
1361 * y >= 0 or [ 1 0 ] [ x ] >= [ 0 ]
1363 * Then we change directoin
1365 * [ b_i A_i ] [ y' ] [ y' ]
1366 * [ 1 0 ] S^{-1} [ x' ] >= 0 or [ b_i' A_i' ] [ x' ] >= 0
1368 * Then we compute the convex hull of the polytopes b_i' + A_i' x' >= 0
1369 * resulting in b' + A' x' >= 0, which we then convert back
1372 * [ b' A' ] S [ x ] >= 0 or [ b A ] [ x ] >= 0
1374 * The polyhedron b + A x >= 0 is then the convex hull of the input polyhedra.
1376 static struct isl_basic_set
*convex_hull_pair_pointed(
1377 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
)
1379 struct isl_ctx
*ctx
= NULL
;
1380 struct isl_vec
*dir
= NULL
;
1381 struct isl_mat
*T
= NULL
;
1382 struct isl_mat
*T2
= NULL
;
1383 struct isl_basic_set
*hull
;
1384 struct isl_set
*set
;
1386 if (!bset1
|| !bset2
)
1389 dir
= valid_direction(isl_basic_set_copy(bset1
),
1390 isl_basic_set_copy(bset2
));
1393 T
= isl_mat_alloc(bset1
->ctx
, dir
->size
, dir
->size
);
1396 isl_seq_cpy(T
->row
[0], dir
->block
.data
, dir
->size
);
1397 T
= isl_mat_unimodular_complete(T
, 1);
1398 T2
= isl_mat_right_inverse(isl_mat_copy(T
));
1400 bset1
= homogeneous_map(bset1
, isl_mat_copy(T2
));
1401 bset2
= homogeneous_map(bset2
, T2
);
1402 set
= isl_set_alloc_dim(isl_basic_set_get_dim(bset1
), 2, 0);
1403 set
= isl_set_add_basic_set(set
, bset1
);
1404 set
= isl_set_add_basic_set(set
, bset2
);
1405 hull
= uset_convex_hull(set
);
1406 hull
= isl_basic_set_preimage(hull
, T
);
1413 isl_basic_set_free(bset1
);
1414 isl_basic_set_free(bset2
);
1418 /* Compute the convex hull of a pair of basic sets without any parameters or
1419 * integer divisions.
1421 * If the convex hull of the two basic sets would have a non-trivial
1422 * lineality space, we first project out this lineality space.
1424 static struct isl_basic_set
*convex_hull_pair(struct isl_basic_set
*bset1
,
1425 struct isl_basic_set
*bset2
)
1427 struct isl_basic_set
*lin
;
1429 if (isl_basic_set_is_bounded(bset1
) || isl_basic_set_is_bounded(bset2
))
1430 return convex_hull_pair_pointed(bset1
, bset2
);
1432 lin
= induced_lineality_space(isl_basic_set_copy(bset1
),
1433 isl_basic_set_copy(bset2
));
1436 if (isl_basic_set_is_universe(lin
)) {
1437 isl_basic_set_free(bset1
);
1438 isl_basic_set_free(bset2
);
1441 if (lin
->n_eq
< isl_basic_set_total_dim(lin
)) {
1442 struct isl_set
*set
;
1443 set
= isl_set_alloc_dim(isl_basic_set_get_dim(bset1
), 2, 0);
1444 set
= isl_set_add_basic_set(set
, bset1
);
1445 set
= isl_set_add_basic_set(set
, bset2
);
1446 return modulo_lineality(set
, lin
);
1448 isl_basic_set_free(lin
);
1450 return convex_hull_pair_pointed(bset1
, bset2
);
1452 isl_basic_set_free(bset1
);
1453 isl_basic_set_free(bset2
);
1457 /* Compute the lineality space of a basic set.
1458 * We currently do not allow the basic set to have any divs.
1459 * We basically just drop the constants and turn every inequality
1462 struct isl_basic_set
*isl_basic_set_lineality_space(struct isl_basic_set
*bset
)
1465 struct isl_basic_set
*lin
= NULL
;
1470 isl_assert(bset
->ctx
, bset
->n_div
== 0, goto error
);
1471 dim
= isl_basic_set_total_dim(bset
);
1473 lin
= isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset
), 0, dim
, 0);
1476 for (i
= 0; i
< bset
->n_eq
; ++i
) {
1477 k
= isl_basic_set_alloc_equality(lin
);
1480 isl_int_set_si(lin
->eq
[k
][0], 0);
1481 isl_seq_cpy(lin
->eq
[k
] + 1, bset
->eq
[i
] + 1, dim
);
1483 lin
= isl_basic_set_gauss(lin
, NULL
);
1486 for (i
= 0; i
< bset
->n_ineq
&& lin
->n_eq
< dim
; ++i
) {
1487 k
= isl_basic_set_alloc_equality(lin
);
1490 isl_int_set_si(lin
->eq
[k
][0], 0);
1491 isl_seq_cpy(lin
->eq
[k
] + 1, bset
->ineq
[i
] + 1, dim
);
1492 lin
= isl_basic_set_gauss(lin
, NULL
);
1496 isl_basic_set_free(bset
);
1499 isl_basic_set_free(lin
);
1500 isl_basic_set_free(bset
);
1504 /* Compute the (linear) hull of the lineality spaces of the basic sets in the
1505 * "underlying" set "set".
1507 static struct isl_basic_set
*uset_combined_lineality_space(struct isl_set
*set
)
1510 struct isl_set
*lin
= NULL
;
1515 struct isl_dim
*dim
= isl_set_get_dim(set
);
1517 return isl_basic_set_empty(dim
);
1520 lin
= isl_set_alloc_dim(isl_set_get_dim(set
), set
->n
, 0);
1521 for (i
= 0; i
< set
->n
; ++i
)
1522 lin
= isl_set_add_basic_set(lin
,
1523 isl_basic_set_lineality_space(isl_basic_set_copy(set
->p
[i
])));
1525 return isl_set_affine_hull(lin
);
1528 /* Compute the convex hull of a set without any parameters or
1529 * integer divisions.
1530 * In each step, we combined two basic sets until only one
1531 * basic set is left.
1532 * The input basic sets are assumed not to have a non-trivial
1533 * lineality space. If any of the intermediate results has
1534 * a non-trivial lineality space, it is projected out.
1536 static struct isl_basic_set
*uset_convex_hull_unbounded(struct isl_set
*set
)
1538 struct isl_basic_set
*convex_hull
= NULL
;
1540 convex_hull
= isl_set_copy_basic_set(set
);
1541 set
= isl_set_drop_basic_set(set
, convex_hull
);
1544 while (set
->n
> 0) {
1545 struct isl_basic_set
*t
;
1546 t
= isl_set_copy_basic_set(set
);
1549 set
= isl_set_drop_basic_set(set
, t
);
1552 convex_hull
= convex_hull_pair(convex_hull
, t
);
1555 t
= isl_basic_set_lineality_space(isl_basic_set_copy(convex_hull
));
1558 if (isl_basic_set_is_universe(t
)) {
1559 isl_basic_set_free(convex_hull
);
1563 if (t
->n_eq
< isl_basic_set_total_dim(t
)) {
1564 set
= isl_set_add_basic_set(set
, convex_hull
);
1565 return modulo_lineality(set
, t
);
1567 isl_basic_set_free(t
);
1573 isl_basic_set_free(convex_hull
);
1577 /* Compute an initial hull for wrapping containing a single initial
1578 * facet by first computing bounds on the set and then using these
1579 * bounds to construct an initial facet.
1580 * This function is a remnant of an older implementation where the
1581 * bounds were also used to check whether the set was bounded.
1582 * Since this function will now only be called when we know the
1583 * set to be bounded, the initial facet should probably be constructed
1584 * by simply using the coordinate directions instead.
1586 static struct isl_basic_set
*initial_hull(struct isl_basic_set
*hull
,
1587 struct isl_set
*set
)
1589 struct isl_mat
*bounds
= NULL
;
1595 bounds
= independent_bounds(set
);
1598 isl_assert(set
->ctx
, bounds
->n_row
== isl_set_n_dim(set
), goto error
);
1599 bounds
= initial_facet_constraint(set
, bounds
);
1602 k
= isl_basic_set_alloc_inequality(hull
);
1605 dim
= isl_set_n_dim(set
);
1606 isl_assert(set
->ctx
, 1 + dim
== bounds
->n_col
, goto error
);
1607 isl_seq_cpy(hull
->ineq
[k
], bounds
->row
[0], bounds
->n_col
);
1608 isl_mat_free(bounds
);
1612 isl_basic_set_free(hull
);
1613 isl_mat_free(bounds
);
1617 struct max_constraint
{
1623 static int max_constraint_equal(const void *entry
, const void *val
)
1625 struct max_constraint
*a
= (struct max_constraint
*)entry
;
1626 isl_int
*b
= (isl_int
*)val
;
1628 return isl_seq_eq(a
->c
->row
[0] + 1, b
, a
->c
->n_col
- 1);
1631 static void update_constraint(struct isl_ctx
*ctx
, struct isl_hash_table
*table
,
1632 isl_int
*con
, unsigned len
, int n
, int ineq
)
1634 struct isl_hash_table_entry
*entry
;
1635 struct max_constraint
*c
;
1638 c_hash
= isl_seq_get_hash(con
+ 1, len
);
1639 entry
= isl_hash_table_find(ctx
, table
, c_hash
, max_constraint_equal
,
1645 isl_hash_table_remove(ctx
, table
, entry
);
1649 if (isl_int_gt(c
->c
->row
[0][0], con
[0]))
1651 if (isl_int_eq(c
->c
->row
[0][0], con
[0])) {
1656 c
->c
= isl_mat_cow(c
->c
);
1657 isl_int_set(c
->c
->row
[0][0], con
[0]);
1661 /* Check whether the constraint hash table "table" constains the constraint
1664 static int has_constraint(struct isl_ctx
*ctx
, struct isl_hash_table
*table
,
1665 isl_int
*con
, unsigned len
, int n
)
1667 struct isl_hash_table_entry
*entry
;
1668 struct max_constraint
*c
;
1671 c_hash
= isl_seq_get_hash(con
+ 1, len
);
1672 entry
= isl_hash_table_find(ctx
, table
, c_hash
, max_constraint_equal
,
1679 return isl_int_eq(c
->c
->row
[0][0], con
[0]);
1682 /* Check for inequality constraints of a basic set without equalities
1683 * such that the same or more stringent copies of the constraint appear
1684 * in all of the basic sets. Such constraints are necessarily facet
1685 * constraints of the convex hull.
1687 * If the resulting basic set is by chance identical to one of
1688 * the basic sets in "set", then we know that this basic set contains
1689 * all other basic sets and is therefore the convex hull of set.
1690 * In this case we set *is_hull to 1.
1692 static struct isl_basic_set
*common_constraints(struct isl_basic_set
*hull
,
1693 struct isl_set
*set
, int *is_hull
)
1696 int min_constraints
;
1698 struct max_constraint
*constraints
= NULL
;
1699 struct isl_hash_table
*table
= NULL
;
1704 for (i
= 0; i
< set
->n
; ++i
)
1705 if (set
->p
[i
]->n_eq
== 0)
1709 min_constraints
= set
->p
[i
]->n_ineq
;
1711 for (i
= best
+ 1; i
< set
->n
; ++i
) {
1712 if (set
->p
[i
]->n_eq
!= 0)
1714 if (set
->p
[i
]->n_ineq
>= min_constraints
)
1716 min_constraints
= set
->p
[i
]->n_ineq
;
1719 constraints
= isl_calloc_array(hull
->ctx
, struct max_constraint
,
1723 table
= isl_alloc_type(hull
->ctx
, struct isl_hash_table
);
1724 if (isl_hash_table_init(hull
->ctx
, table
, min_constraints
))
1727 total
= isl_dim_total(set
->dim
);
1728 for (i
= 0; i
< set
->p
[best
]->n_ineq
; ++i
) {
1729 constraints
[i
].c
= isl_mat_sub_alloc(hull
->ctx
,
1730 set
->p
[best
]->ineq
+ i
, 0, 1, 0, 1 + total
);
1731 if (!constraints
[i
].c
)
1733 constraints
[i
].ineq
= 1;
1735 for (i
= 0; i
< min_constraints
; ++i
) {
1736 struct isl_hash_table_entry
*entry
;
1738 c_hash
= isl_seq_get_hash(constraints
[i
].c
->row
[0] + 1, total
);
1739 entry
= isl_hash_table_find(hull
->ctx
, table
, c_hash
,
1740 max_constraint_equal
, constraints
[i
].c
->row
[0] + 1, 1);
1743 isl_assert(hull
->ctx
, !entry
->data
, goto error
);
1744 entry
->data
= &constraints
[i
];
1748 for (s
= 0; s
< set
->n
; ++s
) {
1752 for (i
= 0; i
< set
->p
[s
]->n_eq
; ++i
) {
1753 isl_int
*eq
= set
->p
[s
]->eq
[i
];
1754 for (j
= 0; j
< 2; ++j
) {
1755 isl_seq_neg(eq
, eq
, 1 + total
);
1756 update_constraint(hull
->ctx
, table
,
1760 for (i
= 0; i
< set
->p
[s
]->n_ineq
; ++i
) {
1761 isl_int
*ineq
= set
->p
[s
]->ineq
[i
];
1762 update_constraint(hull
->ctx
, table
, ineq
, total
, n
,
1763 set
->p
[s
]->n_eq
== 0);
1768 for (i
= 0; i
< min_constraints
; ++i
) {
1769 if (constraints
[i
].count
< n
)
1771 if (!constraints
[i
].ineq
)
1773 j
= isl_basic_set_alloc_inequality(hull
);
1776 isl_seq_cpy(hull
->ineq
[j
], constraints
[i
].c
->row
[0], 1 + total
);
1779 for (s
= 0; s
< set
->n
; ++s
) {
1780 if (set
->p
[s
]->n_eq
)
1782 if (set
->p
[s
]->n_ineq
!= hull
->n_ineq
)
1784 for (i
= 0; i
< set
->p
[s
]->n_ineq
; ++i
) {
1785 isl_int
*ineq
= set
->p
[s
]->ineq
[i
];
1786 if (!has_constraint(hull
->ctx
, table
, ineq
, total
, n
))
1789 if (i
== set
->p
[s
]->n_ineq
)
1793 isl_hash_table_clear(table
);
1794 for (i
= 0; i
< min_constraints
; ++i
)
1795 isl_mat_free(constraints
[i
].c
);
1800 isl_hash_table_clear(table
);
1803 for (i
= 0; i
< min_constraints
; ++i
)
1804 isl_mat_free(constraints
[i
].c
);
1809 /* Create a template for the convex hull of "set" and fill it up
1810 * obvious facet constraints, if any. If the result happens to
1811 * be the convex hull of "set" then *is_hull is set to 1.
1813 static struct isl_basic_set
*proto_hull(struct isl_set
*set
, int *is_hull
)
1815 struct isl_basic_set
*hull
;
1820 for (i
= 0; i
< set
->n
; ++i
) {
1821 n_ineq
+= set
->p
[i
]->n_eq
;
1822 n_ineq
+= set
->p
[i
]->n_ineq
;
1824 hull
= isl_basic_set_alloc_dim(isl_dim_copy(set
->dim
), 0, 0, n_ineq
);
1825 hull
= isl_basic_set_set_rational(hull
);
1828 return common_constraints(hull
, set
, is_hull
);
1831 static struct isl_basic_set
*uset_convex_hull_wrap(struct isl_set
*set
)
1833 struct isl_basic_set
*hull
;
1836 hull
= proto_hull(set
, &is_hull
);
1837 if (hull
&& !is_hull
) {
1838 if (hull
->n_ineq
== 0)
1839 hull
= initial_hull(hull
, set
);
1840 hull
= extend(hull
, set
);
1847 /* Compute the convex hull of a set without any parameters or
1848 * integer divisions. Depending on whether the set is bounded,
1849 * we pass control to the wrapping based convex hull or
1850 * the Fourier-Motzkin elimination based convex hull.
1851 * We also handle a few special cases before checking the boundedness.
1853 static struct isl_basic_set
*uset_convex_hull(struct isl_set
*set
)
1855 struct isl_basic_set
*convex_hull
= NULL
;
1856 struct isl_basic_set
*lin
;
1858 if (isl_set_n_dim(set
) == 0)
1859 return convex_hull_0d(set
);
1861 set
= isl_set_coalesce(set
);
1862 set
= isl_set_set_rational(set
);
1869 convex_hull
= isl_basic_set_copy(set
->p
[0]);
1873 if (isl_set_n_dim(set
) == 1)
1874 return convex_hull_1d(set
);
1876 if (isl_set_is_bounded(set
))
1877 return uset_convex_hull_wrap(set
);
1879 lin
= uset_combined_lineality_space(isl_set_copy(set
));
1882 if (isl_basic_set_is_universe(lin
)) {
1886 if (lin
->n_eq
< isl_basic_set_total_dim(lin
))
1887 return modulo_lineality(set
, lin
);
1888 isl_basic_set_free(lin
);
1890 return uset_convex_hull_unbounded(set
);
1893 isl_basic_set_free(convex_hull
);
1897 /* This is the core procedure, where "set" is a "pure" set, i.e.,
1898 * without parameters or divs and where the convex hull of set is
1899 * known to be full-dimensional.
1901 static struct isl_basic_set
*uset_convex_hull_wrap_bounded(struct isl_set
*set
)
1903 struct isl_basic_set
*convex_hull
= NULL
;
1905 if (isl_set_n_dim(set
) == 0) {
1906 convex_hull
= isl_basic_set_universe(isl_dim_copy(set
->dim
));
1908 convex_hull
= isl_basic_set_set_rational(convex_hull
);
1912 set
= isl_set_set_rational(set
);
1916 set
= isl_set_coalesce(set
);
1920 convex_hull
= isl_basic_set_copy(set
->p
[0]);
1924 if (isl_set_n_dim(set
) == 1)
1925 return convex_hull_1d(set
);
1927 return uset_convex_hull_wrap(set
);
1933 /* Compute the convex hull of set "set" with affine hull "affine_hull",
1934 * We first remove the equalities (transforming the set), compute the
1935 * convex hull of the transformed set and then add the equalities back
1936 * (after performing the inverse transformation.
1938 static struct isl_basic_set
*modulo_affine_hull(struct isl_ctx
*ctx
,
1939 struct isl_set
*set
, struct isl_basic_set
*affine_hull
)
1943 struct isl_basic_set
*dummy
;
1944 struct isl_basic_set
*convex_hull
;
1946 dummy
= isl_basic_set_remove_equalities(
1947 isl_basic_set_copy(affine_hull
), &T
, &T2
);
1950 isl_basic_set_free(dummy
);
1951 set
= isl_set_preimage(set
, T
);
1952 convex_hull
= uset_convex_hull(set
);
1953 convex_hull
= isl_basic_set_preimage(convex_hull
, T2
);
1954 convex_hull
= isl_basic_set_intersect(convex_hull
, affine_hull
);
1957 isl_basic_set_free(affine_hull
);
1962 /* Compute the convex hull of a map.
1964 * The implementation was inspired by "Extended Convex Hull" by Fukuda et al.,
1965 * specifically, the wrapping of facets to obtain new facets.
1967 struct isl_basic_map
*isl_map_convex_hull(struct isl_map
*map
)
1969 struct isl_basic_set
*bset
;
1970 struct isl_basic_map
*model
= NULL
;
1971 struct isl_basic_set
*affine_hull
= NULL
;
1972 struct isl_basic_map
*convex_hull
= NULL
;
1973 struct isl_set
*set
= NULL
;
1974 struct isl_ctx
*ctx
;
1981 convex_hull
= isl_basic_map_empty_like_map(map
);
1986 map
= isl_map_detect_equalities(map
);
1987 map
= isl_map_align_divs(map
);
1988 model
= isl_basic_map_copy(map
->p
[0]);
1989 set
= isl_map_underlying_set(map
);
1993 affine_hull
= isl_set_affine_hull(isl_set_copy(set
));
1996 if (affine_hull
->n_eq
!= 0)
1997 bset
= modulo_affine_hull(ctx
, set
, affine_hull
);
1999 isl_basic_set_free(affine_hull
);
2000 bset
= uset_convex_hull(set
);
2003 convex_hull
= isl_basic_map_overlying_set(bset
, model
);
2005 ISL_F_SET(convex_hull
, ISL_BASIC_MAP_NO_IMPLICIT
);
2006 ISL_F_SET(convex_hull
, ISL_BASIC_MAP_ALL_EQUALITIES
);
2007 ISL_F_CLR(convex_hull
, ISL_BASIC_MAP_RATIONAL
);
2011 isl_basic_map_free(model
);
2015 struct isl_basic_set
*isl_set_convex_hull(struct isl_set
*set
)
2017 return (struct isl_basic_set
*)
2018 isl_map_convex_hull((struct isl_map
*)set
);
2021 struct sh_data_entry
{
2022 struct isl_hash_table
*table
;
2023 struct isl_tab
*tab
;
2026 /* Holds the data needed during the simple hull computation.
2028 * n the number of basic sets in the original set
2029 * hull_table a hash table of already computed constraints
2030 * in the simple hull
2031 * p for each basic set,
2032 * table a hash table of the constraints
2033 * tab the tableau corresponding to the basic set
2036 struct isl_ctx
*ctx
;
2038 struct isl_hash_table
*hull_table
;
2039 struct sh_data_entry p
[1];
2042 static void sh_data_free(struct sh_data
*data
)
2048 isl_hash_table_free(data
->ctx
, data
->hull_table
);
2049 for (i
= 0; i
< data
->n
; ++i
) {
2050 isl_hash_table_free(data
->ctx
, data
->p
[i
].table
);
2051 isl_tab_free(data
->p
[i
].tab
);
2056 struct ineq_cmp_data
{
2061 static int has_ineq(const void *entry
, const void *val
)
2063 isl_int
*row
= (isl_int
*)entry
;
2064 struct ineq_cmp_data
*v
= (struct ineq_cmp_data
*)val
;
2066 return isl_seq_eq(row
+ 1, v
->p
+ 1, v
->len
) ||
2067 isl_seq_is_neg(row
+ 1, v
->p
+ 1, v
->len
);
2070 static int hash_ineq(struct isl_ctx
*ctx
, struct isl_hash_table
*table
,
2071 isl_int
*ineq
, unsigned len
)
2074 struct ineq_cmp_data v
;
2075 struct isl_hash_table_entry
*entry
;
2079 c_hash
= isl_seq_get_hash(ineq
+ 1, len
);
2080 entry
= isl_hash_table_find(ctx
, table
, c_hash
, has_ineq
, &v
, 1);
2087 /* Fill hash table "table" with the constraints of "bset".
2088 * Equalities are added as two inequalities.
2089 * The value in the hash table is a pointer to the (in)equality of "bset".
2091 static int hash_basic_set(struct isl_hash_table
*table
,
2092 struct isl_basic_set
*bset
)
2095 unsigned dim
= isl_basic_set_total_dim(bset
);
2097 for (i
= 0; i
< bset
->n_eq
; ++i
) {
2098 for (j
= 0; j
< 2; ++j
) {
2099 isl_seq_neg(bset
->eq
[i
], bset
->eq
[i
], 1 + dim
);
2100 if (hash_ineq(bset
->ctx
, table
, bset
->eq
[i
], dim
) < 0)
2104 for (i
= 0; i
< bset
->n_ineq
; ++i
) {
2105 if (hash_ineq(bset
->ctx
, table
, bset
->ineq
[i
], dim
) < 0)
2111 static struct sh_data
*sh_data_alloc(struct isl_set
*set
, unsigned n_ineq
)
2113 struct sh_data
*data
;
2116 data
= isl_calloc(set
->ctx
, struct sh_data
,
2117 sizeof(struct sh_data
) +
2118 (set
->n
- 1) * sizeof(struct sh_data_entry
));
2121 data
->ctx
= set
->ctx
;
2123 data
->hull_table
= isl_hash_table_alloc(set
->ctx
, n_ineq
);
2124 if (!data
->hull_table
)
2126 for (i
= 0; i
< set
->n
; ++i
) {
2127 data
->p
[i
].table
= isl_hash_table_alloc(set
->ctx
,
2128 2 * set
->p
[i
]->n_eq
+ set
->p
[i
]->n_ineq
);
2129 if (!data
->p
[i
].table
)
2131 if (hash_basic_set(data
->p
[i
].table
, set
->p
[i
]) < 0)
2140 /* Check if inequality "ineq" is a bound for basic set "j" or if
2141 * it can be relaxed (by increasing the constant term) to become
2142 * a bound for that basic set. In the latter case, the constant
2144 * Return 1 if "ineq" is a bound
2145 * 0 if "ineq" may attain arbitrarily small values on basic set "j"
2146 * -1 if some error occurred
2148 static int is_bound(struct sh_data
*data
, struct isl_set
*set
, int j
,
2151 enum isl_lp_result res
;
2154 if (!data
->p
[j
].tab
) {
2155 data
->p
[j
].tab
= isl_tab_from_basic_set(set
->p
[j
]);
2156 if (!data
->p
[j
].tab
)
2162 res
= isl_tab_min(data
->p
[j
].tab
, ineq
, data
->ctx
->one
,
2164 if (res
== isl_lp_ok
&& isl_int_is_neg(opt
))
2165 isl_int_sub(ineq
[0], ineq
[0], opt
);
2169 return res
== isl_lp_ok
? 1 :
2170 res
== isl_lp_unbounded
? 0 : -1;
2173 /* Check if inequality "ineq" from basic set "i" can be relaxed to
2174 * become a bound on the whole set. If so, add the (relaxed) inequality
2177 * We first check if "hull" already contains a translate of the inequality.
2178 * If so, we are done.
2179 * Then, we check if any of the previous basic sets contains a translate
2180 * of the inequality. If so, then we have already considered this
2181 * inequality and we are done.
2182 * Otherwise, for each basic set other than "i", we check if the inequality
2183 * is a bound on the basic set.
2184 * For previous basic sets, we know that they do not contain a translate
2185 * of the inequality, so we directly call is_bound.
2186 * For following basic sets, we first check if a translate of the
2187 * inequality appears in its description and if so directly update
2188 * the inequality accordingly.
2190 static struct isl_basic_set
*add_bound(struct isl_basic_set
*hull
,
2191 struct sh_data
*data
, struct isl_set
*set
, int i
, isl_int
*ineq
)
2194 struct ineq_cmp_data v
;
2195 struct isl_hash_table_entry
*entry
;
2201 v
.len
= isl_basic_set_total_dim(hull
);
2203 c_hash
= isl_seq_get_hash(ineq
+ 1, v
.len
);
2205 entry
= isl_hash_table_find(hull
->ctx
, data
->hull_table
, c_hash
,
2210 for (j
= 0; j
< i
; ++j
) {
2211 entry
= isl_hash_table_find(hull
->ctx
, data
->p
[j
].table
,
2212 c_hash
, has_ineq
, &v
, 0);
2219 k
= isl_basic_set_alloc_inequality(hull
);
2220 isl_seq_cpy(hull
->ineq
[k
], ineq
, 1 + v
.len
);
2224 for (j
= 0; j
< i
; ++j
) {
2226 bound
= is_bound(data
, set
, j
, hull
->ineq
[k
]);
2233 isl_basic_set_free_inequality(hull
, 1);
2237 for (j
= i
+ 1; j
< set
->n
; ++j
) {
2240 entry
= isl_hash_table_find(hull
->ctx
, data
->p
[j
].table
,
2241 c_hash
, has_ineq
, &v
, 0);
2243 ineq_j
= entry
->data
;
2244 neg
= isl_seq_is_neg(ineq_j
+ 1,
2245 hull
->ineq
[k
] + 1, v
.len
);
2247 isl_int_neg(ineq_j
[0], ineq_j
[0]);
2248 if (isl_int_gt(ineq_j
[0], hull
->ineq
[k
][0]))
2249 isl_int_set(hull
->ineq
[k
][0], ineq_j
[0]);
2251 isl_int_neg(ineq_j
[0], ineq_j
[0]);
2254 bound
= is_bound(data
, set
, j
, hull
->ineq
[k
]);
2261 isl_basic_set_free_inequality(hull
, 1);
2265 entry
= isl_hash_table_find(hull
->ctx
, data
->hull_table
, c_hash
,
2269 entry
->data
= hull
->ineq
[k
];
2273 isl_basic_set_free(hull
);
2277 /* Check if any inequality from basic set "i" can be relaxed to
2278 * become a bound on the whole set. If so, add the (relaxed) inequality
2281 static struct isl_basic_set
*add_bounds(struct isl_basic_set
*bset
,
2282 struct sh_data
*data
, struct isl_set
*set
, int i
)
2285 unsigned dim
= isl_basic_set_total_dim(bset
);
2287 for (j
= 0; j
< set
->p
[i
]->n_eq
; ++j
) {
2288 for (k
= 0; k
< 2; ++k
) {
2289 isl_seq_neg(set
->p
[i
]->eq
[j
], set
->p
[i
]->eq
[j
], 1+dim
);
2290 add_bound(bset
, data
, set
, i
, set
->p
[i
]->eq
[j
]);
2293 for (j
= 0; j
< set
->p
[i
]->n_ineq
; ++j
)
2294 add_bound(bset
, data
, set
, i
, set
->p
[i
]->ineq
[j
]);
2298 /* Compute a superset of the convex hull of set that is described
2299 * by only translates of the constraints in the constituents of set.
2301 static struct isl_basic_set
*uset_simple_hull(struct isl_set
*set
)
2303 struct sh_data
*data
= NULL
;
2304 struct isl_basic_set
*hull
= NULL
;
2312 for (i
= 0; i
< set
->n
; ++i
) {
2315 n_ineq
+= 2 * set
->p
[i
]->n_eq
+ set
->p
[i
]->n_ineq
;
2318 hull
= isl_basic_set_alloc_dim(isl_dim_copy(set
->dim
), 0, 0, n_ineq
);
2322 data
= sh_data_alloc(set
, n_ineq
);
2326 for (i
= 0; i
< set
->n
; ++i
)
2327 hull
= add_bounds(hull
, data
, set
, i
);
2335 isl_basic_set_free(hull
);
2340 /* Compute a superset of the convex hull of map that is described
2341 * by only translates of the constraints in the constituents of map.
2343 struct isl_basic_map
*isl_map_simple_hull(struct isl_map
*map
)
2345 struct isl_set
*set
= NULL
;
2346 struct isl_basic_map
*model
= NULL
;
2347 struct isl_basic_map
*hull
;
2348 struct isl_basic_map
*affine_hull
;
2349 struct isl_basic_set
*bset
= NULL
;
2354 hull
= isl_basic_map_empty_like_map(map
);
2359 hull
= isl_basic_map_copy(map
->p
[0]);
2364 map
= isl_map_detect_equalities(map
);
2365 affine_hull
= isl_map_affine_hull(isl_map_copy(map
));
2366 map
= isl_map_align_divs(map
);
2367 model
= isl_basic_map_copy(map
->p
[0]);
2369 set
= isl_map_underlying_set(map
);
2371 bset
= uset_simple_hull(set
);
2373 hull
= isl_basic_map_overlying_set(bset
, model
);
2375 hull
= isl_basic_map_intersect(hull
, affine_hull
);
2376 hull
= isl_basic_map_convex_hull(hull
);
2377 ISL_F_SET(hull
, ISL_BASIC_MAP_NO_IMPLICIT
);
2378 ISL_F_SET(hull
, ISL_BASIC_MAP_ALL_EQUALITIES
);
2383 struct isl_basic_set
*isl_set_simple_hull(struct isl_set
*set
)
2385 return (struct isl_basic_set
*)
2386 isl_map_simple_hull((struct isl_map
*)set
);
2389 /* Given a set "set", return parametric bounds on the dimension "dim".
2391 static struct isl_basic_set
*set_bounds(struct isl_set
*set
, int dim
)
2393 unsigned set_dim
= isl_set_dim(set
, isl_dim_set
);
2394 set
= isl_set_copy(set
);
2395 set
= isl_set_eliminate_dims(set
, dim
+ 1, set_dim
- (dim
+ 1));
2396 set
= isl_set_eliminate_dims(set
, 0, dim
);
2397 return isl_set_convex_hull(set
);
2400 /* Computes a "simple hull" and then check if each dimension in the
2401 * resulting hull is bounded by a symbolic constant. If not, the
2402 * hull is intersected with the corresponding bounds on the whole set.
2404 struct isl_basic_set
*isl_set_bounded_simple_hull(struct isl_set
*set
)
2407 struct isl_basic_set
*hull
;
2408 unsigned nparam
, left
;
2409 int removed_divs
= 0;
2411 hull
= isl_set_simple_hull(isl_set_copy(set
));
2415 nparam
= isl_basic_set_dim(hull
, isl_dim_param
);
2416 for (i
= 0; i
< isl_basic_set_dim(hull
, isl_dim_set
); ++i
) {
2417 int lower
= 0, upper
= 0;
2418 struct isl_basic_set
*bounds
;
2420 left
= isl_basic_set_total_dim(hull
) - nparam
- i
- 1;
2421 for (j
= 0; j
< hull
->n_eq
; ++j
) {
2422 if (isl_int_is_zero(hull
->eq
[j
][1 + nparam
+ i
]))
2424 if (isl_seq_first_non_zero(hull
->eq
[j
]+1+nparam
+i
+1,
2431 for (j
= 0; j
< hull
->n_ineq
; ++j
) {
2432 if (isl_int_is_zero(hull
->ineq
[j
][1 + nparam
+ i
]))
2434 if (isl_seq_first_non_zero(hull
->ineq
[j
]+1+nparam
+i
+1,
2436 isl_seq_first_non_zero(hull
->ineq
[j
]+1+nparam
,
2439 if (isl_int_is_pos(hull
->ineq
[j
][1 + nparam
+ i
]))
2450 if (!removed_divs
) {
2451 set
= isl_set_remove_divs(set
);
2456 bounds
= set_bounds(set
, i
);
2457 hull
= isl_basic_set_intersect(hull
, bounds
);