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 "bset" that lies outside of the equality
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 as a singleton set.
229 * If no point can be found, the empty set 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 yields another point
234 * inside the basic set.
236 * The caller of this function ensures that "bset" is bounded.
238 static struct isl_basic_set
*outside_point(struct isl_ctx
*ctx
,
239 struct isl_basic_set
*bset
, isl_int
*eq
, int up
)
241 struct isl_basic_set
*slice
= NULL
;
242 struct isl_vec
*sample
;
243 struct isl_basic_set
*point
;
247 dim
= isl_basic_set_n_dim(bset
);
248 sample
= isl_vec_alloc(ctx
, 1 + dim
);
251 isl_int_set_si(sample
->block
.data
[0], 1);
252 isl_seq_combine(sample
->block
.data
+ 1,
253 ctx
->one
, bset
->sample
->block
.data
+ 1,
254 up
? ctx
->one
: ctx
->negone
, eq
+ 1, dim
);
255 if (isl_basic_set_contains(bset
, sample
))
256 return isl_basic_set_from_vec(sample
);
257 isl_vec_free(sample
);
260 slice
= isl_basic_set_copy(bset
);
263 slice
= isl_basic_set_cow(slice
);
264 slice
= isl_basic_set_extend(slice
, 0, dim
, 0, 0, 1);
265 k
= isl_basic_set_alloc_inequality(slice
);
269 isl_seq_cpy(slice
->ineq
[k
], eq
, 1 + dim
);
271 isl_seq_neg(slice
->ineq
[k
], eq
, 1 + dim
);
272 isl_int_sub_ui(slice
->ineq
[k
][0], slice
->ineq
[k
][0], 1);
274 sample
= isl_basic_set_sample_bounded(slice
);
277 if (sample
->size
== 0) {
278 isl_vec_free(sample
);
279 point
= isl_basic_set_empty_like(bset
);
281 point
= isl_basic_set_from_vec(sample
);
285 isl_basic_set_free(slice
);
289 struct isl_basic_set
*isl_basic_set_recession_cone(struct isl_basic_set
*bset
)
293 bset
= isl_basic_set_cow(bset
);
296 isl_assert(bset
->ctx
, bset
->n_div
== 0, goto error
);
298 for (i
= 0; i
< bset
->n_eq
; ++i
)
299 isl_int_set_si(bset
->eq
[i
][0], 0);
301 for (i
= 0; i
< bset
->n_ineq
; ++i
)
302 isl_int_set_si(bset
->ineq
[i
][0], 0);
304 ISL_F_CLR(bset
, ISL_BASIC_SET_NO_IMPLICIT
);
305 return isl_basic_set_implicit_equalities(bset
);
307 isl_basic_set_free(bset
);
311 /* Extend an initial (under-)approximation of the affine hull of "bset"
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 "bset" is bounded.
318 static struct isl_basic_set
*extend_affine_hull(struct isl_basic_set
*bset
,
319 struct isl_basic_set
*hull
)
326 dim
= isl_basic_set_n_dim(bset
);
327 for (i
= 0; i
< dim
; ++i
) {
328 struct isl_basic_set
*point
;
329 for (j
= 0; j
< hull
->n_eq
; ++j
) {
330 point
= outside_point(ctx
, bset
, hull
->eq
[j
], 1);
333 if (!ISL_F_ISSET(point
, ISL_BASIC_SET_EMPTY
))
335 isl_basic_set_free(point
);
336 point
= outside_point(ctx
, bset
, hull
->eq
[j
], 0);
339 if (!ISL_F_ISSET(point
, ISL_BASIC_SET_EMPTY
))
341 isl_basic_set_free(point
);
343 bset
= isl_basic_set_extend_constraints(bset
, 1, 0);
344 k
= isl_basic_set_alloc_equality(bset
);
347 isl_seq_cpy(bset
->eq
[k
], hull
->eq
[j
],
348 1 + isl_basic_set_total_dim(hull
));
349 bset
= isl_basic_set_gauss(bset
, NULL
);
355 hull
= affine_hull(hull
, point
);
357 isl_basic_set_free(bset
);
361 isl_basic_set_free(bset
);
362 isl_basic_set_free(hull
);
366 /* Drop all constraints in bset that involve any of the dimensions
367 * first to first+n-1.
369 static struct isl_basic_set
*drop_constraints_involving
370 (struct isl_basic_set
*bset
, unsigned first
, unsigned n
)
377 bset
= isl_basic_set_cow(bset
);
379 for (i
= bset
->n_eq
- 1; i
>= 0; --i
) {
380 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + first
, n
) == -1)
382 isl_basic_set_drop_equality(bset
, i
);
385 for (i
= bset
->n_ineq
- 1; i
>= 0; --i
) {
386 if (isl_seq_first_non_zero(bset
->ineq
[i
] + 1 + first
, n
) == -1)
388 isl_basic_set_drop_inequality(bset
, i
);
394 /* Compute the affine hull of "bset", where "hull" is an initial approximation
395 * with only a single point of "bset" and "cone" is the recession cone
398 * We first compute a unimodular transformation that puts the unbounded
399 * directions in the last dimensions. In particular, we take a transformation
400 * that maps all equalities to equalities (in HNF) on the first dimensions.
401 * Let x be the original dimensions and y the transformed, with y_1 bounded
404 * [ y_1 ] [ y_1 ] [ Q_1 ]
405 * x = U [ y_2 ] [ y_2 ] = [ Q_2 ] x
407 * Let's call the input basic set S and the initial hull H.
408 * We compute S' = preimage(S, U) and H' = preimage(H, U)
409 * and drop the final dimensions including any constraints involving them.
410 * This results in sets S'' and H''.
411 * Then we extend H'' to the affine hull A'' of S''.
412 * Let F y_1 >= g be the constraint system of A''. In the transformed
413 * space the y_2 are unbounded, so we can add them back without any constraints,
417 * [ F 0 ] [ y_2 ] >= g
420 * [ F 0 ] [ Q_2 ] x >= g
424 * The affine hull in the original space is then obtained as
425 * A = preimage(A'', Q_1).
427 static struct isl_basic_set
*affine_hull_with_cone(struct isl_basic_set
*bset
,
428 struct isl_basic_set
*hull
, struct isl_basic_set
*cone
)
432 struct isl_mat
*M
, *U
, *Q
;
434 if (!bset
|| !hull
|| !cone
)
437 total
= isl_basic_set_total_dim(cone
);
438 cone_dim
= total
- cone
->n_eq
;
440 M
= isl_mat_sub_alloc(bset
->ctx
, cone
->eq
, 0, cone
->n_eq
, 1, total
);
441 M
= isl_mat_left_hermite(M
, 0, &U
, &Q
);
446 U
= isl_mat_lin_to_aff(U
);
447 bset
= isl_basic_set_preimage(bset
, isl_mat_copy(U
));
448 hull
= isl_basic_set_preimage(hull
, U
);
450 bset
= drop_constraints_involving(bset
, total
- cone_dim
, cone_dim
);
451 bset
= isl_basic_set_drop_dims(bset
, total
- cone_dim
, cone_dim
);
452 hull
= drop_constraints_involving(hull
, total
- cone_dim
, cone_dim
);
453 hull
= isl_basic_set_drop_dims(hull
, total
- cone_dim
, cone_dim
);
455 Q
= isl_mat_lin_to_aff(Q
);
456 Q
= isl_mat_drop_rows(Q
, 1 + total
- cone_dim
, cone_dim
);
458 if (bset
&& bset
->sample
)
459 bset
->sample
= isl_mat_vec_product(isl_mat_copy(Q
), bset
->sample
);
461 hull
= extend_affine_hull(bset
, hull
);
463 hull
= isl_basic_set_preimage(hull
, Q
);
465 isl_basic_set_free(cone
);
469 isl_basic_set_free(bset
);
470 isl_basic_set_free(hull
);
471 isl_basic_set_free(cone
);
475 /* Look for all equalities satisfied by the integer points in bset,
476 * which is assumed not to have any explicit equalities.
478 * The equalities are obtained by successively looking for
479 * a point that is affinely independent of the points found so far.
480 * In particular, for each equality satisfied by the points so far,
481 * we check if there is any point on a hyperplane parallel to the
482 * corresponding hyperplane shifted by at least one (in either direction).
484 * Before looking for any outside points, we first compute the recession
485 * cone. The directions of this recession cone will always be part
486 * of the affine hull, so there is no need for looking for any points
487 * in these directions.
488 * In particular, if the recession cone is full-dimensional, then
489 * the affine hull is simply the whole universe.
491 static struct isl_basic_set
*uset_affine_hull(struct isl_basic_set
*bset
)
493 struct isl_basic_set
*hull
= NULL
;
494 struct isl_vec
*sample
= NULL
;
495 struct isl_basic_set
*cone
;
497 if (isl_basic_set_is_empty(bset
))
500 sample
= isl_basic_set_sample_vec(isl_basic_set_copy(bset
));
503 if (sample
->size
== 0) {
504 isl_vec_free(sample
);
505 hull
= isl_basic_set_empty_like(bset
);
506 isl_basic_set_free(bset
);
509 if (sample
->size
== 1) {
510 isl_vec_free(sample
);
514 cone
= isl_basic_set_recession_cone(isl_basic_set_copy(bset
));
517 if (cone
->n_eq
== 0) {
518 isl_basic_set_free(cone
);
519 isl_vec_free(sample
);
520 hull
= isl_basic_set_universe_like(bset
);
521 isl_basic_set_free(bset
);
525 hull
= isl_basic_set_from_vec(sample
);
526 if (cone
->n_eq
< isl_basic_set_total_dim(cone
))
527 return affine_hull_with_cone(bset
, hull
, cone
);
529 isl_basic_set_free(cone
);
530 return extend_affine_hull(bset
, hull
);
532 isl_vec_free(sample
);
533 isl_basic_set_free(bset
);
534 isl_basic_set_free(hull
);
538 /* Look for all equalities satisfied by the integer points in bmap
539 * that are independent of the equalities already explicitly available
542 * We first remove all equalities already explicitly available,
543 * then look for additional equalities in the reduced space
544 * and then transform the result to the original space.
545 * The original equalities are _not_ added to this set. This is
546 * the responsibility of the calling function.
547 * The resulting basic set has all meaning about the dimensions removed.
548 * In particular, dimensions that correspond to existential variables
549 * in bmap and that are found to be fixed are not removed.
551 static struct isl_basic_set
*equalities_in_underlying_set(
552 struct isl_basic_map
*bmap
)
554 struct isl_mat
*T2
= NULL
;
555 struct isl_basic_set
*bset
= NULL
;
556 struct isl_basic_set
*hull
= NULL
;
558 bset
= isl_basic_map_underlying_set(bmap
);
559 bset
= isl_basic_set_remove_equalities(bset
, NULL
, &T2
);
563 hull
= uset_affine_hull(bset
);
565 hull
= isl_basic_set_preimage(hull
, T2
);
570 isl_basic_set_free(bset
);
571 isl_basic_set_free(hull
);
575 /* Detect and make explicit all equalities satisfied by the (integer)
578 struct isl_basic_map
*isl_basic_map_detect_equalities(
579 struct isl_basic_map
*bmap
)
582 struct isl_basic_set
*hull
= NULL
;
586 if (bmap
->n_ineq
== 0)
588 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
))
590 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_ALL_EQUALITIES
))
592 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
))
593 return isl_basic_map_implicit_equalities(bmap
);
595 hull
= equalities_in_underlying_set(isl_basic_map_copy(bmap
));
598 if (ISL_F_ISSET(hull
, ISL_BASIC_SET_EMPTY
)) {
599 isl_basic_set_free(hull
);
600 return isl_basic_map_set_to_empty(bmap
);
602 bmap
= isl_basic_map_extend_dim(bmap
, isl_dim_copy(bmap
->dim
), 0,
604 for (i
= 0; i
< hull
->n_eq
; ++i
) {
605 j
= isl_basic_map_alloc_equality(bmap
);
608 isl_seq_cpy(bmap
->eq
[j
], hull
->eq
[i
],
609 1 + isl_basic_set_total_dim(hull
));
611 isl_basic_set_free(hull
);
612 ISL_F_SET(bmap
, ISL_BASIC_MAP_NO_IMPLICIT
| ISL_BASIC_MAP_ALL_EQUALITIES
);
613 bmap
= isl_basic_map_simplify(bmap
);
614 return isl_basic_map_finalize(bmap
);
616 isl_basic_set_free(hull
);
617 isl_basic_map_free(bmap
);
621 __isl_give isl_basic_set
*isl_basic_set_detect_equalities(
622 __isl_take isl_basic_set
*bset
)
624 return (isl_basic_set
*)
625 isl_basic_map_detect_equalities((isl_basic_map
*)bset
);
628 struct isl_map
*isl_map_detect_equalities(struct isl_map
*map
)
630 struct isl_basic_map
*bmap
;
636 for (i
= 0; i
< map
->n
; ++i
) {
637 bmap
= isl_basic_map_copy(map
->p
[i
]);
638 bmap
= isl_basic_map_detect_equalities(bmap
);
641 isl_basic_map_free(map
->p
[i
]);
651 __isl_give isl_set
*isl_set_detect_equalities(__isl_take isl_set
*set
)
653 return (isl_set
*)isl_map_detect_equalities((isl_map
*)set
);
656 /* After computing the rational affine hull (by detecting the implicit
657 * equalities), we compute the additional equalities satisfied by
658 * the integer points (if any) and add the original equalities back in.
660 struct isl_basic_map
*isl_basic_map_affine_hull(struct isl_basic_map
*bmap
)
662 bmap
= isl_basic_map_detect_equalities(bmap
);
663 bmap
= isl_basic_map_cow(bmap
);
664 isl_basic_map_free_inequality(bmap
, bmap
->n_ineq
);
668 struct isl_basic_set
*isl_basic_set_affine_hull(struct isl_basic_set
*bset
)
670 return (struct isl_basic_set
*)
671 isl_basic_map_affine_hull((struct isl_basic_map
*)bset
);
674 struct isl_basic_map
*isl_map_affine_hull(struct isl_map
*map
)
677 struct isl_basic_map
*model
= NULL
;
678 struct isl_basic_map
*hull
= NULL
;
685 hull
= isl_basic_map_empty_like_map(map
);
690 map
= isl_map_detect_equalities(map
);
691 map
= isl_map_align_divs(map
);
694 model
= isl_basic_map_copy(map
->p
[0]);
695 set
= isl_map_underlying_set(map
);
696 set
= isl_set_cow(set
);
700 for (i
= 0; i
< set
->n
; ++i
) {
701 set
->p
[i
] = isl_basic_set_cow(set
->p
[i
]);
702 set
->p
[i
] = isl_basic_set_affine_hull(set
->p
[i
]);
703 set
->p
[i
] = isl_basic_set_gauss(set
->p
[i
], NULL
);
707 set
= isl_set_remove_empty_parts(set
);
709 hull
= isl_basic_map_empty_like(model
);
710 isl_basic_map_free(model
);
712 struct isl_basic_set
*bset
;
714 set
->p
[0] = affine_hull(set
->p
[0], set
->p
[--set
->n
]);
718 bset
= isl_basic_set_copy(set
->p
[0]);
719 hull
= isl_basic_map_overlying_set(bset
, model
);
722 hull
= isl_basic_map_simplify(hull
);
723 return isl_basic_map_finalize(hull
);
725 isl_basic_map_free(model
);
730 struct isl_basic_set
*isl_set_affine_hull(struct isl_set
*set
)
732 return (struct isl_basic_set
*)
733 isl_map_affine_hull((struct isl_map
*)set
);