6 #include "isl_map_private.h"
7 #include "isl_equalities.h"
8 #include "isl_sample.h"
11 struct isl_basic_map
*isl_basic_map_implicit_equalities(
12 struct isl_basic_map
*bmap
)
19 bmap
= isl_basic_map_gauss(bmap
, NULL
);
20 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
))
22 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_NO_IMPLICIT
))
24 if (bmap
->n_ineq
<= 1)
27 tab
= isl_tab_from_basic_map(bmap
);
28 tab
= isl_tab_detect_implicit_equalities(tab
);
29 bmap
= isl_basic_map_update_from_tab(bmap
, tab
);
31 bmap
= isl_basic_map_gauss(bmap
, NULL
);
32 ISL_F_SET(bmap
, ISL_BASIC_MAP_NO_IMPLICIT
);
36 struct isl_basic_set
*isl_basic_set_implicit_equalities(
37 struct isl_basic_set
*bset
)
39 return (struct isl_basic_set
*)
40 isl_basic_map_implicit_equalities((struct isl_basic_map
*)bset
);
43 struct isl_map
*isl_map_implicit_equalities(struct isl_map
*map
)
50 for (i
= 0; i
< map
->n
; ++i
) {
51 map
->p
[i
] = isl_basic_map_implicit_equalities(map
->p
[i
]);
62 /* Make eq[row][col] of both bmaps equal so we can add the row
63 * add the column to the common matrix.
64 * Note that because of the echelon form, the columns of row row
65 * after column col are zero.
67 static void set_common_multiple(
68 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
,
69 unsigned row
, unsigned col
)
73 if (isl_int_eq(bset1
->eq
[row
][col
], bset2
->eq
[row
][col
]))
78 isl_int_lcm(m
, bset1
->eq
[row
][col
], bset2
->eq
[row
][col
]);
79 isl_int_divexact(c
, m
, bset1
->eq
[row
][col
]);
80 isl_seq_scale(bset1
->eq
[row
], bset1
->eq
[row
], c
, col
+1);
81 isl_int_divexact(c
, m
, bset2
->eq
[row
][col
]);
82 isl_seq_scale(bset2
->eq
[row
], bset2
->eq
[row
], c
, col
+1);
87 /* Delete a given equality, moving all the following equalities one up.
89 static void delete_row(struct isl_basic_set
*bset
, unsigned row
)
96 for (r
= row
; r
< bset
->n_eq
; ++r
)
97 bset
->eq
[r
] = bset
->eq
[r
+1];
98 bset
->eq
[bset
->n_eq
] = t
;
101 /* Make first row entries in column col of bset1 identical to
102 * those of bset2, using the fact that entry bset1->eq[row][col]=a
103 * is non-zero. Initially, these elements of bset1 are all zero.
104 * For each row i < row, we set
105 * A[i] = a * A[i] + B[i][col] * A[row]
108 * A[i][col] = B[i][col] = a * old(B[i][col])
110 static void construct_column(
111 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
,
112 unsigned row
, unsigned col
)
121 total
= 1 + isl_basic_set_n_dim(bset1
);
122 for (r
= 0; r
< row
; ++r
) {
123 if (isl_int_is_zero(bset2
->eq
[r
][col
]))
125 isl_int_gcd(b
, bset2
->eq
[r
][col
], bset1
->eq
[row
][col
]);
126 isl_int_divexact(a
, bset1
->eq
[row
][col
], b
);
127 isl_int_divexact(b
, bset2
->eq
[r
][col
], b
);
128 isl_seq_combine(bset1
->eq
[r
], a
, bset1
->eq
[r
],
129 b
, bset1
->eq
[row
], total
);
130 isl_seq_scale(bset2
->eq
[r
], bset2
->eq
[r
], a
, total
);
134 delete_row(bset1
, row
);
137 /* Make first row entries in column col of bset1 identical to
138 * those of bset2, using only these entries of the two matrices.
139 * Let t be the last row with different entries.
140 * For each row i < t, we set
141 * A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
142 * B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
144 * A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
146 static int transform_column(
147 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
,
148 unsigned row
, unsigned col
)
154 for (t
= row
-1; t
>= 0; --t
)
155 if (isl_int_ne(bset1
->eq
[t
][col
], bset2
->eq
[t
][col
]))
160 total
= 1 + isl_basic_set_n_dim(bset1
);
164 isl_int_sub(b
, bset1
->eq
[t
][col
], bset2
->eq
[t
][col
]);
165 for (i
= 0; i
< t
; ++i
) {
166 isl_int_sub(a
, bset2
->eq
[i
][col
], bset1
->eq
[i
][col
]);
167 isl_int_gcd(g
, a
, b
);
168 isl_int_divexact(a
, a
, g
);
169 isl_int_divexact(g
, b
, g
);
170 isl_seq_combine(bset1
->eq
[i
], g
, bset1
->eq
[i
], a
, bset1
->eq
[t
],
172 isl_seq_combine(bset2
->eq
[i
], g
, bset2
->eq
[i
], a
, bset2
->eq
[t
],
178 delete_row(bset1
, t
);
179 delete_row(bset2
, t
);
183 /* The implementation is based on Section 5.2 of Michael Karr,
184 * "Affine Relationships Among Variables of a Program",
185 * except that the echelon form we use starts from the last column
186 * and that we are dealing with integer coefficients.
188 static struct isl_basic_set
*affine_hull(
189 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
)
195 total
= 1 + isl_basic_set_n_dim(bset1
);
198 for (col
= total
-1; col
>= 0; --col
) {
199 int is_zero1
= row
>= bset1
->n_eq
||
200 isl_int_is_zero(bset1
->eq
[row
][col
]);
201 int is_zero2
= row
>= bset2
->n_eq
||
202 isl_int_is_zero(bset2
->eq
[row
][col
]);
203 if (!is_zero1
&& !is_zero2
) {
204 set_common_multiple(bset1
, bset2
, row
, col
);
206 } else if (!is_zero1
&& is_zero2
) {
207 construct_column(bset1
, bset2
, row
, col
);
208 } else if (is_zero1
&& !is_zero2
) {
209 construct_column(bset2
, bset1
, row
, col
);
211 if (transform_column(bset1
, bset2
, row
, col
))
215 isl_basic_set_free(bset2
);
216 isl_assert(bset1
->ctx
, row
== bset1
->n_eq
, goto error
);
217 bset1
= isl_basic_set_normalize_constraints(bset1
);
220 isl_basic_set_free(bset1
);
224 /* Find an integer point in the set represented by "tab"
225 * that lies outside of the equality "eq" e(x) = 0.
226 * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
227 * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
228 * The point, if found, is returned.
229 * If no point can be found, a zero-length vector is returned.
231 * Before solving an ILP problem, we first check if simply
232 * adding the normal of the constraint to one of the known
233 * integer points in the basic set represented by "tab"
234 * yields another point inside the basic set.
236 * The caller of this function ensures that the tableau is bounded.
238 static struct isl_vec
*outside_point(struct isl_tab
*tab
, isl_int
*eq
, int up
)
241 struct isl_vec
*sample
;
242 struct isl_tab_undo
*snap
;
251 sample
= isl_vec_alloc(ctx
, 1 + dim
);
254 isl_int_set_si(sample
->el
[0], 1);
255 isl_seq_combine(sample
->el
+ 1,
256 ctx
->one
, tab
->bset
->sample
->el
+ 1,
257 up
? ctx
->one
: ctx
->negone
, eq
+ 1, dim
);
258 if (isl_basic_set_contains(tab
->bset
, sample
))
260 isl_vec_free(sample
);
263 snap
= isl_tab_snap(tab
);
266 isl_seq_neg(eq
, eq
, 1 + dim
);
267 isl_int_sub_ui(eq
[0], eq
[0], 1);
269 if (isl_tab_extend_cons(tab
, 1) < 0)
271 tab
= isl_tab_add_ineq(tab
, eq
);
273 sample
= isl_tab_sample(tab
);
275 isl_int_add_ui(eq
[0], eq
[0], 1);
277 isl_seq_neg(eq
, eq
, 1 + dim
);
279 if (isl_tab_rollback(tab
, snap
) < 0)
284 isl_vec_free(sample
);
288 struct isl_basic_set
*isl_basic_set_recession_cone(struct isl_basic_set
*bset
)
292 bset
= isl_basic_set_cow(bset
);
295 isl_assert(bset
->ctx
, bset
->n_div
== 0, goto error
);
297 for (i
= 0; i
< bset
->n_eq
; ++i
)
298 isl_int_set_si(bset
->eq
[i
][0], 0);
300 for (i
= 0; i
< bset
->n_ineq
; ++i
)
301 isl_int_set_si(bset
->ineq
[i
][0], 0);
303 ISL_F_CLR(bset
, ISL_BASIC_SET_NO_IMPLICIT
);
304 return isl_basic_set_implicit_equalities(bset
);
306 isl_basic_set_free(bset
);
310 /* Extend an initial (under-)approximation of the affine hull of basic
311 * set represented by the tableau "tab"
312 * by looking for points that do not satisfy one of the equalities
313 * in the current approximation and adding them to that approximation
314 * until no such points can be found any more.
316 * The caller of this function ensures that "tab" is bounded.
318 static struct isl_basic_set
*extend_affine_hull(struct isl_tab
*tab
,
319 struct isl_basic_set
*hull
)
329 if (isl_tab_extend_cons(tab
, 2 * dim
+ 1) < 0)
332 for (i
= 0; i
< dim
; ++i
) {
333 struct isl_vec
*sample
;
334 struct isl_basic_set
*point
;
335 for (j
= 0; j
< hull
->n_eq
; ++j
) {
336 sample
= outside_point(tab
, hull
->eq
[j
], 1);
339 if (sample
->size
> 0)
341 isl_vec_free(sample
);
342 sample
= outside_point(tab
, hull
->eq
[j
], 0);
345 if (sample
->size
> 0)
347 isl_vec_free(sample
);
349 tab
= isl_tab_add_eq(tab
, hull
->eq
[j
]);
355 point
= isl_basic_set_from_vec(sample
);
356 hull
= affine_hull(hull
, point
);
361 isl_basic_set_free(hull
);
365 /* Drop all constraints in bset that involve any of the dimensions
366 * first to first+n-1.
368 static struct isl_basic_set
*drop_constraints_involving
369 (struct isl_basic_set
*bset
, unsigned first
, unsigned n
)
376 bset
= isl_basic_set_cow(bset
);
378 for (i
= bset
->n_eq
- 1; i
>= 0; --i
) {
379 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + first
, n
) == -1)
381 isl_basic_set_drop_equality(bset
, i
);
384 for (i
= bset
->n_ineq
- 1; i
>= 0; --i
) {
385 if (isl_seq_first_non_zero(bset
->ineq
[i
] + 1 + first
, n
) == -1)
387 isl_basic_set_drop_inequality(bset
, i
);
393 /* Look for all equalities satisfied by the integer points in bset,
394 * which is assumed to be bounded.
396 * The equalities are obtained by successively looking for
397 * a point that is affinely independent of the points found so far.
398 * In particular, for each equality satisfied by the points so far,
399 * we check if there is any point on a hyperplane parallel to the
400 * corresponding hyperplane shifted by at least one (in either direction).
402 static struct isl_basic_set
*uset_affine_hull_bounded(struct isl_basic_set
*bset
)
404 struct isl_vec
*sample
= NULL
;
405 struct isl_basic_set
*hull
;
406 struct isl_tab
*tab
= NULL
;
409 if (isl_basic_set_fast_is_empty(bset
))
412 dim
= isl_basic_set_n_dim(bset
);
414 if (bset
->sample
&& bset
->sample
->size
== 1 + dim
) {
415 int contains
= isl_basic_set_contains(bset
, bset
->sample
);
421 sample
= isl_vec_copy(bset
->sample
);
423 isl_vec_free(bset
->sample
);
428 tab
= isl_tab_from_basic_set(bset
);
431 tab
->bset
= isl_basic_set_copy(bset
);
434 struct isl_tab_undo
*snap
;
435 snap
= isl_tab_snap(tab
);
436 sample
= isl_tab_sample(tab
);
437 if (isl_tab_rollback(tab
, snap
) < 0)
439 isl_vec_free(tab
->bset
->sample
);
440 tab
->bset
->sample
= isl_vec_copy(sample
);
445 if (sample
->size
== 0) {
447 isl_vec_free(sample
);
448 return isl_basic_set_set_to_empty(bset
);
451 hull
= isl_basic_set_from_vec(sample
);
453 isl_basic_set_free(bset
);
454 hull
= extend_affine_hull(tab
, hull
);
459 isl_vec_free(sample
);
461 isl_basic_set_free(bset
);
465 /* Compute the affine hull of "bset", where "cone" is the recession cone
468 * We first compute a unimodular transformation that puts the unbounded
469 * directions in the last dimensions. In particular, we take a transformation
470 * that maps all equalities to equalities (in HNF) on the first dimensions.
471 * Let x be the original dimensions and y the transformed, with y_1 bounded
474 * [ y_1 ] [ y_1 ] [ Q_1 ]
475 * x = U [ y_2 ] [ y_2 ] = [ Q_2 ] x
477 * Let's call the input basic set S. We compute S' = preimage(S, U)
478 * and drop the final dimensions including any constraints involving them.
479 * This results in set S''.
480 * Then we compute the affine hull A'' of S''.
481 * Let F y_1 >= g be the constraint system of A''. In the transformed
482 * space the y_2 are unbounded, so we can add them back without any constraints,
486 * [ F 0 ] [ y_2 ] >= g
489 * [ F 0 ] [ Q_2 ] x >= g
493 * The affine hull in the original space is then obtained as
494 * A = preimage(A'', Q_1).
496 static struct isl_basic_set
*affine_hull_with_cone(struct isl_basic_set
*bset
,
497 struct isl_basic_set
*cone
)
501 struct isl_basic_set
*hull
;
502 struct isl_mat
*M
, *U
, *Q
;
507 total
= isl_basic_set_total_dim(cone
);
508 cone_dim
= total
- cone
->n_eq
;
510 M
= isl_mat_sub_alloc(bset
->ctx
, cone
->eq
, 0, cone
->n_eq
, 1, total
);
511 M
= isl_mat_left_hermite(M
, 0, &U
, &Q
);
516 U
= isl_mat_lin_to_aff(U
);
517 bset
= isl_basic_set_preimage(bset
, isl_mat_copy(U
));
519 bset
= drop_constraints_involving(bset
, total
- cone_dim
, cone_dim
);
520 bset
= isl_basic_set_drop_dims(bset
, total
- cone_dim
, cone_dim
);
522 Q
= isl_mat_lin_to_aff(Q
);
523 Q
= isl_mat_drop_rows(Q
, 1 + total
- cone_dim
, cone_dim
);
525 if (bset
&& bset
->sample
&& bset
->sample
->size
== 1 + total
)
526 bset
->sample
= isl_mat_vec_product(isl_mat_copy(Q
), bset
->sample
);
528 hull
= uset_affine_hull_bounded(bset
);
533 struct isl_vec
*sample
= isl_vec_copy(hull
->sample
);
534 U
= isl_mat_drop_cols(U
, 1 + total
- cone_dim
, cone_dim
);
535 if (sample
&& sample
->size
> 0)
536 sample
= isl_mat_vec_product(U
, sample
);
539 hull
= isl_basic_set_preimage(hull
, Q
);
540 isl_vec_free(hull
->sample
);
541 hull
->sample
= sample
;
544 isl_basic_set_free(cone
);
548 isl_basic_set_free(bset
);
549 isl_basic_set_free(cone
);
553 /* Look for all equalities satisfied by the integer points in bset,
554 * which is assumed not to have any explicit equalities.
556 * The equalities are obtained by successively looking for
557 * a point that is affinely independent of the points found so far.
558 * In particular, for each equality satisfied by the points so far,
559 * we check if there is any point on a hyperplane parallel to the
560 * corresponding hyperplane shifted by at least one (in either direction).
562 * Before looking for any outside points, we first compute the recession
563 * cone. The directions of this recession cone will always be part
564 * of the affine hull, so there is no need for looking for any points
565 * in these directions.
566 * In particular, if the recession cone is full-dimensional, then
567 * the affine hull is simply the whole universe.
569 static struct isl_basic_set
*uset_affine_hull(struct isl_basic_set
*bset
)
571 struct isl_basic_set
*cone
;
573 if (isl_basic_set_fast_is_empty(bset
))
576 cone
= isl_basic_set_recession_cone(isl_basic_set_copy(bset
));
579 if (cone
->n_eq
== 0) {
580 struct isl_basic_set
*hull
;
581 isl_basic_set_free(cone
);
582 hull
= isl_basic_set_universe_like(bset
);
583 isl_basic_set_free(bset
);
587 if (cone
->n_eq
< isl_basic_set_total_dim(cone
))
588 return affine_hull_with_cone(bset
, cone
);
590 isl_basic_set_free(cone
);
591 return uset_affine_hull_bounded(bset
);
593 isl_basic_set_free(bset
);
597 /* Look for all equalities satisfied by the integer points in bmap
598 * that are independent of the equalities already explicitly available
601 * We first remove all equalities already explicitly available,
602 * then look for additional equalities in the reduced space
603 * and then transform the result to the original space.
604 * The original equalities are _not_ added to this set. This is
605 * the responsibility of the calling function.
606 * The resulting basic set has all meaning about the dimensions removed.
607 * In particular, dimensions that correspond to existential variables
608 * in bmap and that are found to be fixed are not removed.
610 static struct isl_basic_set
*equalities_in_underlying_set(
611 struct isl_basic_map
*bmap
)
613 struct isl_mat
*T1
= NULL
;
614 struct isl_mat
*T2
= NULL
;
615 struct isl_basic_set
*bset
= NULL
;
616 struct isl_basic_set
*hull
= NULL
;
618 bset
= isl_basic_map_underlying_set(bmap
);
622 bset
= isl_basic_set_remove_equalities(bset
, &T1
, &T2
);
626 hull
= uset_affine_hull(bset
);
633 struct isl_vec
*sample
= isl_vec_copy(hull
->sample
);
634 if (sample
&& sample
->size
> 0)
635 sample
= isl_mat_vec_product(T1
, sample
);
638 hull
= isl_basic_set_preimage(hull
, T2
);
639 isl_vec_free(hull
->sample
);
640 hull
->sample
= sample
;
646 isl_basic_set_free(bset
);
647 isl_basic_set_free(hull
);
651 /* Detect and make explicit all equalities satisfied by the (integer)
654 struct isl_basic_map
*isl_basic_map_detect_equalities(
655 struct isl_basic_map
*bmap
)
658 struct isl_basic_set
*hull
= NULL
;
662 if (bmap
->n_ineq
== 0)
664 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
))
666 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_ALL_EQUALITIES
))
668 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
))
669 return isl_basic_map_implicit_equalities(bmap
);
671 hull
= equalities_in_underlying_set(isl_basic_map_copy(bmap
));
674 if (ISL_F_ISSET(hull
, ISL_BASIC_SET_EMPTY
)) {
675 isl_basic_set_free(hull
);
676 return isl_basic_map_set_to_empty(bmap
);
678 bmap
= isl_basic_map_extend_dim(bmap
, isl_dim_copy(bmap
->dim
), 0,
680 for (i
= 0; i
< hull
->n_eq
; ++i
) {
681 j
= isl_basic_map_alloc_equality(bmap
);
684 isl_seq_cpy(bmap
->eq
[j
], hull
->eq
[i
],
685 1 + isl_basic_set_total_dim(hull
));
687 isl_vec_free(bmap
->sample
);
688 bmap
->sample
= isl_vec_copy(hull
->sample
);
689 isl_basic_set_free(hull
);
690 ISL_F_SET(bmap
, ISL_BASIC_MAP_NO_IMPLICIT
| ISL_BASIC_MAP_ALL_EQUALITIES
);
691 bmap
= isl_basic_map_simplify(bmap
);
692 return isl_basic_map_finalize(bmap
);
694 isl_basic_set_free(hull
);
695 isl_basic_map_free(bmap
);
699 __isl_give isl_basic_set
*isl_basic_set_detect_equalities(
700 __isl_take isl_basic_set
*bset
)
702 return (isl_basic_set
*)
703 isl_basic_map_detect_equalities((isl_basic_map
*)bset
);
706 struct isl_map
*isl_map_detect_equalities(struct isl_map
*map
)
708 struct isl_basic_map
*bmap
;
714 for (i
= 0; i
< map
->n
; ++i
) {
715 bmap
= isl_basic_map_copy(map
->p
[i
]);
716 bmap
= isl_basic_map_detect_equalities(bmap
);
719 isl_basic_map_free(map
->p
[i
]);
729 __isl_give isl_set
*isl_set_detect_equalities(__isl_take isl_set
*set
)
731 return (isl_set
*)isl_map_detect_equalities((isl_map
*)set
);
734 /* After computing the rational affine hull (by detecting the implicit
735 * equalities), we compute the additional equalities satisfied by
736 * the integer points (if any) and add the original equalities back in.
738 struct isl_basic_map
*isl_basic_map_affine_hull(struct isl_basic_map
*bmap
)
740 bmap
= isl_basic_map_detect_equalities(bmap
);
741 bmap
= isl_basic_map_cow(bmap
);
742 isl_basic_map_free_inequality(bmap
, bmap
->n_ineq
);
746 struct isl_basic_set
*isl_basic_set_affine_hull(struct isl_basic_set
*bset
)
748 return (struct isl_basic_set
*)
749 isl_basic_map_affine_hull((struct isl_basic_map
*)bset
);
752 struct isl_basic_map
*isl_map_affine_hull(struct isl_map
*map
)
755 struct isl_basic_map
*model
= NULL
;
756 struct isl_basic_map
*hull
= NULL
;
763 hull
= isl_basic_map_empty_like_map(map
);
768 map
= isl_map_detect_equalities(map
);
769 map
= isl_map_align_divs(map
);
772 model
= isl_basic_map_copy(map
->p
[0]);
773 set
= isl_map_underlying_set(map
);
774 set
= isl_set_cow(set
);
778 for (i
= 0; i
< set
->n
; ++i
) {
779 set
->p
[i
] = isl_basic_set_cow(set
->p
[i
]);
780 set
->p
[i
] = isl_basic_set_affine_hull(set
->p
[i
]);
781 set
->p
[i
] = isl_basic_set_gauss(set
->p
[i
], NULL
);
785 set
= isl_set_remove_empty_parts(set
);
787 hull
= isl_basic_map_empty_like(model
);
788 isl_basic_map_free(model
);
790 struct isl_basic_set
*bset
;
792 set
->p
[0] = affine_hull(set
->p
[0], set
->p
[--set
->n
]);
796 bset
= isl_basic_set_copy(set
->p
[0]);
797 hull
= isl_basic_map_overlying_set(bset
, model
);
800 hull
= isl_basic_map_simplify(hull
);
801 return isl_basic_map_finalize(hull
);
803 isl_basic_map_free(model
);
808 struct isl_basic_set
*isl_set_affine_hull(struct isl_set
*set
)
810 return (struct isl_basic_set
*)
811 isl_map_affine_hull((struct isl_map
*)set
);