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_equalities(bmap
->ctx
, tab
);
29 bmap
= isl_basic_map_update_from_tab(bmap
, tab
);
30 isl_tab_free(bmap
->ctx
, 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(ctx
, row
== bset1
->n_eq
, goto error
);
217 bset1
= isl_basic_set_normalize_constraints(bset1
);
220 isl_basic_set_free(bset1
);
224 static struct isl_basic_set
*isl_basic_set_from_vec(struct isl_ctx
*ctx
,
229 struct isl_basic_set
*bset
= NULL
;
234 isl_assert(ctx
, vec
->size
!= 0, goto error
);
236 bset
= isl_basic_set_alloc(ctx
, 0, vec
->size
- 1, 0, vec
->size
- 1, 0);
239 dim
= isl_basic_set_n_dim(bset
);
240 for (i
= dim
- 1; i
>= 0; --i
) {
241 k
= isl_basic_set_alloc_equality(bset
);
244 isl_seq_clr(bset
->eq
[k
], 1 + dim
);
245 isl_int_neg(bset
->eq
[k
][0], vec
->block
.data
[1 + i
]);
246 isl_int_set(bset
->eq
[k
][1 + i
], vec
->block
.data
[0]);
248 isl_vec_free(ctx
, vec
);
252 isl_basic_set_free(bset
);
253 isl_vec_free(ctx
, vec
);
257 /* Find an integer point in "bset" that lies outside of the equality
259 * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
260 * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
261 * The point, if found, is returned as a singleton set.
262 * If no point can be found, the empty set is returned.
264 * Before solving an ILP problem, we first check if simply
265 * adding the normal of the constraint to one of the known
266 * integer points in the basic set yields another point
267 * inside the basic set.
269 static struct isl_basic_set
*outside_point(struct isl_ctx
*ctx
,
270 struct isl_basic_set
*bset
, isl_int
*eq
, int up
)
272 struct isl_basic_set
*slice
= NULL
;
273 struct isl_vec
*sample
;
274 struct isl_basic_set
*point
;
278 dim
= isl_basic_set_n_dim(bset
);
279 sample
= isl_vec_alloc(ctx
, 1 + dim
);
282 isl_int_set_si(sample
->block
.data
[0], 1);
283 isl_seq_combine(sample
->block
.data
+ 1,
284 ctx
->one
, bset
->sample
->block
.data
+ 1,
285 up
? ctx
->one
: ctx
->negone
, eq
+ 1, dim
);
286 if (isl_basic_set_contains(bset
, sample
))
287 return isl_basic_set_from_vec(ctx
, sample
);
288 isl_vec_free(ctx
, sample
);
291 slice
= isl_basic_set_copy(bset
);
294 slice
= isl_basic_set_cow(slice
);
295 slice
= isl_basic_set_extend(slice
, 0, dim
, 0, 0, 1);
296 k
= isl_basic_set_alloc_inequality(slice
);
300 isl_seq_cpy(slice
->ineq
[k
], eq
, 1 + dim
);
302 isl_seq_neg(slice
->ineq
[k
], eq
, 1 + dim
);
303 isl_int_sub_ui(slice
->ineq
[k
][0], slice
->ineq
[k
][0], 1);
305 sample
= isl_basic_set_sample(slice
);
308 if (sample
->size
== 0) {
309 isl_vec_free(ctx
, sample
);
310 point
= isl_basic_set_empty_like(bset
);
312 point
= isl_basic_set_from_vec(ctx
, sample
);
316 isl_basic_set_free(slice
);
320 static struct isl_basic_set
*recession_cone(struct isl_basic_set
*bset
)
324 bset
= isl_basic_set_cow(bset
);
328 for (i
= 0; i
< bset
->n_eq
; ++i
)
329 isl_int_set_si(bset
->eq
[i
][0], 0);
331 for (i
= 0; i
< bset
->n_ineq
; ++i
)
332 isl_int_set_si(bset
->ineq
[i
][0], 0);
334 ISL_F_CLR(bset
, ISL_BASIC_SET_NO_IMPLICIT
);
335 return isl_basic_set_implicit_equalities(bset
);
338 static struct isl_basic_set
*shift(struct isl_basic_set
*bset
, isl_int
*point
)
343 bset
= isl_basic_set_cow(bset
);
347 dim
= isl_basic_set_n_dim(bset
);
348 for (i
= 0; i
< bset
->n_eq
; ++i
) {
349 isl_seq_inner_product(bset
->eq
[i
]+1, point
+1, dim
,
351 isl_int_neg(bset
->eq
[i
][0], bset
->eq
[i
][0]);
354 for (i
= 0; i
< bset
->n_ineq
; ++i
) {
355 isl_seq_inner_product(bset
->ineq
[i
]+1, point
+1, dim
,
357 isl_int_neg(bset
->ineq
[i
][0], bset
->ineq
[i
][0]);
363 /* Look for all equalities satisfied by the integer points in bset,
364 * which is assume not to have any explicit equalities.
366 * The equalities are obtained by successively looking for
367 * a point that is affinely independent of the points found so far.
368 * In particular, for each equality satisfied by the points so far,
369 * we check if there is any point on a hyperplane parallel to the
370 * corresponding hyperplane shifted by at least one (in either direction).
372 * Before looking for any outside points, we first remove the equalities
373 * that correspond to the affine hull of the recession cone.
374 * These equalities will never be equalities over the whols basic set.
376 static struct isl_basic_set
*uset_affine_hull(struct isl_basic_set
*bset
)
379 struct isl_basic_set
*hull
= NULL
;
380 struct isl_vec
*sample
;
384 if (isl_basic_set_is_empty(bset
))
388 sample
= isl_basic_set_sample(isl_basic_set_copy(bset
));
391 if (sample
->size
== 0) {
392 isl_vec_free(ctx
, sample
);
393 hull
= isl_basic_set_empty_like(bset
);
394 isl_basic_set_free(bset
);
397 hull
= isl_basic_set_from_vec(ctx
, sample
);
399 if (hull
->n_eq
> 0) {
400 struct isl_basic_set
*cone
;
401 cone
= recession_cone(isl_basic_set_copy(bset
));
402 isl_basic_set_free_inequality(cone
, cone
->n_ineq
);
403 cone
= isl_basic_set_normalize_constraints(cone
);
404 cone
= shift(cone
, bset
->sample
->block
.data
);
405 hull
= affine_hull(hull
, cone
);
408 dim
= isl_basic_set_n_dim(bset
);
409 for (i
= 0; i
< dim
; ++i
) {
410 struct isl_basic_set
*point
;
411 for (j
= 0; j
< hull
->n_eq
; ++j
) {
412 point
= outside_point(ctx
, bset
, hull
->eq
[j
], 1);
415 if (!ISL_F_ISSET(point
, ISL_BASIC_SET_EMPTY
))
417 isl_basic_set_free(point
);
418 point
= outside_point(ctx
, bset
, hull
->eq
[j
], 0);
421 if (!ISL_F_ISSET(point
, ISL_BASIC_SET_EMPTY
))
423 isl_basic_set_free(point
);
427 hull
= affine_hull(hull
, point
);
429 isl_basic_set_free(bset
);
433 isl_basic_set_free(bset
);
434 isl_basic_set_free(hull
);
438 /* Look for all equalities satisfied by the integer points in bmap
439 * that are independent of the equalities already explicitly available
442 * We first remove all equalities already explicitly available,
443 * then look for additional equalities in the reduced space
444 * and then transform the result to the original space.
445 * The original equalities are _not_ added to this set. This is
446 * the responsibility of the calling function.
447 * The resulting basic set has all meaning about the dimensions removed.
448 * In particular, dimensions that correspond to existential variables
449 * in bmap and that are found to be fixed are not removed.
451 static struct isl_basic_set
*equalities_in_underlying_set(
452 struct isl_basic_map
*bmap
)
454 struct isl_mat
*T2
= NULL
;
455 struct isl_basic_set
*bset
= NULL
;
456 struct isl_basic_set
*hull
= NULL
;
460 bset
= isl_basic_map_underlying_set(bmap
);
461 bset
= isl_basic_set_remove_equalities(bset
, NULL
, &T2
);
465 hull
= uset_affine_hull(bset
);
467 hull
= isl_basic_set_preimage(hull
, T2
);
471 isl_mat_free(ctx
, T2
);
472 isl_basic_set_free(bset
);
473 isl_basic_set_free(hull
);
477 /* Detect and make explicit all equalities satisfied by the (integer)
480 struct isl_basic_map
*isl_basic_map_detect_equalities(
481 struct isl_basic_map
*bmap
)
484 struct isl_basic_set
*hull
= NULL
;
488 if (bmap
->n_ineq
== 0)
490 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
))
492 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_ALL_EQUALITIES
))
494 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
))
495 return isl_basic_map_implicit_equalities(bmap
);
497 hull
= equalities_in_underlying_set(isl_basic_map_copy(bmap
));
500 bmap
= isl_basic_map_extend_dim(bmap
, isl_dim_copy(bmap
->dim
), 0,
502 for (i
= 0; i
< hull
->n_eq
; ++i
) {
503 j
= isl_basic_map_alloc_equality(bmap
);
506 isl_seq_cpy(bmap
->eq
[j
], hull
->eq
[i
],
507 1 + isl_basic_set_total_dim(hull
));
509 isl_basic_set_free(hull
);
510 ISL_F_SET(bmap
, ISL_BASIC_MAP_NO_IMPLICIT
| ISL_BASIC_MAP_ALL_EQUALITIES
);
511 bmap
= isl_basic_map_simplify(bmap
);
512 return isl_basic_map_finalize(bmap
);
514 isl_basic_set_free(hull
);
515 isl_basic_map_free(bmap
);
519 struct isl_map
*isl_map_detect_equalities(struct isl_map
*map
)
521 struct isl_basic_map
*bmap
;
527 for (i
= 0; i
< map
->n
; ++i
) {
528 bmap
= isl_basic_map_copy(map
->p
[i
]);
529 bmap
= isl_basic_map_detect_equalities(bmap
);
532 isl_basic_map_free(map
->p
[i
]);
542 /* After computing the rational affine hull (by detecting the implicit
543 * equalities), we compute the additional equalities satisfied by
544 * the integer points (if any) and add the original equalities back in.
546 struct isl_basic_map
*isl_basic_map_affine_hull(struct isl_basic_map
*bmap
)
548 struct isl_basic_set
*hull
= NULL
;
550 bmap
= isl_basic_map_detect_equalities(bmap
);
551 bmap
= isl_basic_map_cow(bmap
);
552 isl_basic_map_free_inequality(bmap
, bmap
->n_ineq
);
556 struct isl_basic_set
*isl_basic_set_affine_hull(struct isl_basic_set
*bset
)
558 return (struct isl_basic_set
*)
559 isl_basic_map_affine_hull((struct isl_basic_map
*)bset
);
562 struct isl_basic_map
*isl_map_affine_hull(struct isl_map
*map
)
565 struct isl_basic_map
*model
= NULL
;
566 struct isl_basic_map
*hull
= NULL
;
573 hull
= isl_basic_map_empty_like_map(map
);
578 map
= isl_map_detect_equalities(map
);
579 map
= isl_map_align_divs(map
);
582 model
= isl_basic_map_copy(map
->p
[0]);
583 set
= isl_map_underlying_set(map
);
584 set
= isl_set_cow(set
);
588 for (i
= 0; i
< set
->n
; ++i
) {
589 set
->p
[i
] = isl_basic_set_cow(set
->p
[i
]);
590 set
->p
[i
] = isl_basic_set_affine_hull(set
->p
[i
]);
591 set
->p
[i
] = isl_basic_set_gauss(set
->p
[i
], NULL
);
595 set
= isl_set_remove_empty_parts(set
);
597 hull
= isl_basic_map_empty_like(model
);
598 isl_basic_map_free(model
);
600 struct isl_basic_set
*bset
;
602 set
->p
[0] = affine_hull(set
->p
[0], set
->p
[--set
->n
]);
606 bset
= isl_basic_set_copy(set
->p
[0]);
607 hull
= isl_basic_map_overlying_set(bset
, model
);
610 hull
= isl_basic_map_simplify(hull
);
611 return isl_basic_map_finalize(hull
);
613 isl_basic_map_free(model
);
618 struct isl_basic_set
*isl_set_affine_hull(struct isl_set
*set
)
620 return (struct isl_basic_set
*)
621 isl_map_affine_hull((struct isl_map
*)set
);