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 /* Make eq[row][col] of both bmaps equal so we can add the row
44 * add the column to the common matrix.
45 * Note that because of the echelon form, the columns of row row
46 * after column col are zero.
48 static void set_common_multiple(
49 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
,
50 unsigned row
, unsigned col
)
54 if (isl_int_eq(bset1
->eq
[row
][col
], bset2
->eq
[row
][col
]))
59 isl_int_lcm(m
, bset1
->eq
[row
][col
], bset2
->eq
[row
][col
]);
60 isl_int_divexact(c
, m
, bset1
->eq
[row
][col
]);
61 isl_seq_scale(bset1
->eq
[row
], bset1
->eq
[row
], c
, col
+1);
62 isl_int_divexact(c
, m
, bset2
->eq
[row
][col
]);
63 isl_seq_scale(bset2
->eq
[row
], bset2
->eq
[row
], c
, col
+1);
68 /* Delete a given equality, moving all the following equalities one up.
70 static void delete_row(struct isl_basic_set
*bset
, unsigned row
)
77 for (r
= row
; r
< bset
->n_eq
; ++r
)
78 bset
->eq
[r
] = bset
->eq
[r
+1];
79 bset
->eq
[bset
->n_eq
] = t
;
82 /* Make first row entries in column col of bset1 identical to
83 * those of bset2, using the fact that entry bset1->eq[row][col]=a
84 * is non-zero. Initially, these elements of bset1 are all zero.
85 * For each row i < row, we set
86 * A[i] = a * A[i] + B[i][col] * A[row]
89 * A[i][col] = B[i][col] = a * old(B[i][col])
91 static void construct_column(
92 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
,
93 unsigned row
, unsigned col
)
102 total
= 1 + isl_basic_set_n_dim(bset1
);
103 for (r
= 0; r
< row
; ++r
) {
104 if (isl_int_is_zero(bset2
->eq
[r
][col
]))
106 isl_int_gcd(b
, bset2
->eq
[r
][col
], bset1
->eq
[row
][col
]);
107 isl_int_divexact(a
, bset1
->eq
[row
][col
], b
);
108 isl_int_divexact(b
, bset2
->eq
[r
][col
], b
);
109 isl_seq_combine(bset1
->eq
[r
], a
, bset1
->eq
[r
],
110 b
, bset1
->eq
[row
], total
);
111 isl_seq_scale(bset2
->eq
[r
], bset2
->eq
[r
], a
, total
);
115 delete_row(bset1
, row
);
118 /* Make first row entries in column col of bset1 identical to
119 * those of bset2, using only these entries of the two matrices.
120 * Let t be the last row with different entries.
121 * For each row i < t, we set
122 * A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
123 * B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
125 * A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
127 static int transform_column(
128 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
,
129 unsigned row
, unsigned col
)
135 for (t
= row
-1; t
>= 0; --t
)
136 if (isl_int_ne(bset1
->eq
[t
][col
], bset2
->eq
[t
][col
]))
141 total
= 1 + isl_basic_set_n_dim(bset1
);
145 isl_int_sub(b
, bset1
->eq
[t
][col
], bset2
->eq
[t
][col
]);
146 for (i
= 0; i
< t
; ++i
) {
147 isl_int_sub(a
, bset2
->eq
[i
][col
], bset1
->eq
[i
][col
]);
148 isl_int_gcd(g
, a
, b
);
149 isl_int_divexact(a
, a
, g
);
150 isl_int_divexact(g
, b
, g
);
151 isl_seq_combine(bset1
->eq
[i
], g
, bset1
->eq
[i
], a
, bset1
->eq
[t
],
153 isl_seq_combine(bset2
->eq
[i
], g
, bset2
->eq
[i
], a
, bset2
->eq
[t
],
159 delete_row(bset1
, t
);
160 delete_row(bset2
, t
);
164 /* The implementation is based on Section 5.2 of Michael Karr,
165 * "Affine Relationships Among Variables of a Program",
166 * except that the echelon form we use starts from the last column
167 * and that we are dealing with integer coefficients.
169 static struct isl_basic_set
*affine_hull(
170 struct isl_basic_set
*bset1
, struct isl_basic_set
*bset2
)
176 total
= 1 + isl_basic_set_n_dim(bset1
);
179 for (col
= total
-1; col
>= 0; --col
) {
180 int is_zero1
= row
>= bset1
->n_eq
||
181 isl_int_is_zero(bset1
->eq
[row
][col
]);
182 int is_zero2
= row
>= bset2
->n_eq
||
183 isl_int_is_zero(bset2
->eq
[row
][col
]);
184 if (!is_zero1
&& !is_zero2
) {
185 set_common_multiple(bset1
, bset2
, row
, col
);
187 } else if (!is_zero1
&& is_zero2
) {
188 construct_column(bset1
, bset2
, row
, col
);
189 } else if (is_zero1
&& !is_zero2
) {
190 construct_column(bset2
, bset1
, row
, col
);
192 if (transform_column(bset1
, bset2
, row
, col
))
196 isl_basic_set_free(bset2
);
197 isl_assert(ctx
, row
== bset1
->n_eq
, goto error
);
198 bset1
= isl_basic_set_normalize_constraints(bset1
);
201 isl_basic_set_free(bset1
);
205 static struct isl_basic_set
*isl_basic_set_from_vec(struct isl_ctx
*ctx
,
210 struct isl_basic_set
*bset
= NULL
;
215 isl_assert(ctx
, vec
->size
!= 0, goto error
);
217 bset
= isl_basic_set_alloc(ctx
, 0, vec
->size
- 1, 0, vec
->size
- 1, 0);
220 dim
= isl_basic_set_n_dim(bset
);
221 for (i
= dim
- 1; i
>= 0; --i
) {
222 k
= isl_basic_set_alloc_equality(bset
);
225 isl_seq_clr(bset
->eq
[k
], 1 + dim
);
226 isl_int_neg(bset
->eq
[k
][0], vec
->block
.data
[1 + i
]);
227 isl_int_set(bset
->eq
[k
][1 + i
], vec
->block
.data
[0]);
229 isl_vec_free(ctx
, vec
);
233 isl_basic_set_free(bset
);
234 isl_vec_free(ctx
, vec
);
238 /* Find an integer point in "bset" that lies outside of the equality
240 * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
241 * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
242 * The point, if found, is returned as a singleton set.
243 * If no point can be found, the empty set is returned.
245 * Before solving an ILP problem, we first check if simply
246 * adding the normal of the constraint to one of the known
247 * integer points in the basic set yields another point
248 * inside the basic set.
250 static struct isl_basic_set
*outside_point(struct isl_ctx
*ctx
,
251 struct isl_basic_set
*bset
, isl_int
*eq
, int up
)
253 struct isl_basic_set
*slice
= NULL
;
254 struct isl_vec
*sample
;
255 struct isl_basic_set
*point
;
259 dim
= isl_basic_set_n_dim(bset
);
260 sample
= isl_vec_alloc(ctx
, 1 + dim
);
263 isl_int_set_si(sample
->block
.data
[0], 1);
264 isl_seq_combine(sample
->block
.data
+ 1,
265 ctx
->one
, bset
->sample
->block
.data
+ 1,
266 up
? ctx
->one
: ctx
->negone
, eq
+ 1, dim
);
267 if (isl_basic_set_contains(bset
, sample
))
268 return isl_basic_set_from_vec(ctx
, sample
);
269 isl_vec_free(ctx
, sample
);
272 slice
= isl_basic_set_copy(bset
);
275 slice
= isl_basic_set_cow(slice
);
276 slice
= isl_basic_set_extend(slice
, 0, dim
, 0, 0, 1);
277 k
= isl_basic_set_alloc_inequality(slice
);
281 isl_seq_cpy(slice
->ineq
[k
], eq
, 1 + dim
);
283 isl_seq_neg(slice
->ineq
[k
], eq
, 1 + dim
);
284 isl_int_sub_ui(slice
->ineq
[k
][0], slice
->ineq
[k
][0], 1);
286 sample
= isl_basic_set_sample(slice
);
289 if (sample
->size
== 0) {
290 isl_vec_free(ctx
, sample
);
291 point
= isl_basic_set_empty_like(bset
);
293 point
= isl_basic_set_from_vec(ctx
, sample
);
297 isl_basic_set_free(slice
);
301 static struct isl_basic_set
*recession_cone(struct isl_basic_set
*bset
)
305 bset
= isl_basic_set_cow(bset
);
309 for (i
= 0; i
< bset
->n_eq
; ++i
)
310 isl_int_set_si(bset
->eq
[i
][0], 0);
312 for (i
= 0; i
< bset
->n_ineq
; ++i
)
313 isl_int_set_si(bset
->ineq
[i
][0], 0);
315 ISL_F_CLR(bset
, ISL_BASIC_SET_NO_IMPLICIT
);
316 return isl_basic_set_implicit_equalities(bset
);
319 static struct isl_basic_set
*shift(struct isl_basic_set
*bset
, isl_int
*point
)
324 bset
= isl_basic_set_cow(bset
);
328 dim
= isl_basic_set_n_dim(bset
);
329 for (i
= 0; i
< bset
->n_eq
; ++i
) {
330 isl_seq_inner_product(bset
->eq
[i
]+1, point
+1, dim
,
332 isl_int_neg(bset
->eq
[i
][0], bset
->eq
[i
][0]);
335 for (i
= 0; i
< bset
->n_ineq
; ++i
) {
336 isl_seq_inner_product(bset
->ineq
[i
]+1, point
+1, dim
,
338 isl_int_neg(bset
->ineq
[i
][0], bset
->ineq
[i
][0]);
344 /* Look for all equalities satisfied by the integer points in bset,
345 * which is assume not to have any explicit equalities.
347 * The equalities are obtained by successively looking for
348 * a point that is affinely independent of the points found so far.
349 * In particular, for each equality satisfied by the points so far,
350 * we check if there is any point on a hyperplane parallel to the
351 * corresponding hyperplane shifted by at least one (in either direction).
353 * Before looking for any outside points, we first remove the equalities
354 * that correspond to the affine hull of the recession cone.
355 * These equalities will never be equalities over the whols basic set.
357 static struct isl_basic_set
*uset_affine_hull(struct isl_basic_set
*bset
)
360 struct isl_basic_set
*hull
= NULL
;
361 struct isl_vec
*sample
;
365 if (isl_basic_set_is_empty(bset
))
369 sample
= isl_basic_set_sample(isl_basic_set_copy(bset
));
372 if (sample
->size
== 0) {
373 isl_vec_free(ctx
, sample
);
374 hull
= isl_basic_set_empty_like(bset
);
375 isl_basic_set_free(bset
);
378 hull
= isl_basic_set_from_vec(ctx
, sample
);
380 if (hull
->n_eq
> 0) {
381 struct isl_basic_set
*cone
;
382 cone
= recession_cone(isl_basic_set_copy(bset
));
383 isl_basic_set_free_inequality(cone
, cone
->n_ineq
);
384 cone
= isl_basic_set_normalize_constraints(cone
);
385 cone
= shift(cone
, bset
->sample
->block
.data
);
386 hull
= affine_hull(hull
, cone
);
389 dim
= isl_basic_set_n_dim(bset
);
390 for (i
= 0; i
< dim
; ++i
) {
391 struct isl_basic_set
*point
;
392 for (j
= 0; j
< hull
->n_eq
; ++j
) {
393 point
= outside_point(ctx
, bset
, hull
->eq
[j
], 1);
396 if (!ISL_F_ISSET(point
, ISL_BASIC_SET_EMPTY
))
398 isl_basic_set_free(point
);
399 point
= outside_point(ctx
, bset
, hull
->eq
[j
], 0);
402 if (!ISL_F_ISSET(point
, ISL_BASIC_SET_EMPTY
))
404 isl_basic_set_free(point
);
408 hull
= affine_hull(hull
, point
);
410 isl_basic_set_free(bset
);
414 isl_basic_set_free(bset
);
415 isl_basic_set_free(hull
);
419 /* Look for all equalities satisfied by the integer points in bmap
420 * that are independent of the equalities already explicitly available
423 * We first remove all equalities already explicitly available,
424 * then look for additional equalities in the reduced space
425 * and then transform the result to the original space.
426 * The original equalities are _not_ added to this set. This is
427 * the responsibility of the calling function.
428 * The resulting basic set has all meaning about the dimensions removed.
429 * In particular, dimensions that correspond to existential variables
430 * in bmap and that are found to be fixed are not removed.
432 static struct isl_basic_set
*equalities_in_underlying_set(
433 struct isl_basic_map
*bmap
)
435 struct isl_mat
*T2
= NULL
;
436 struct isl_basic_set
*bset
= NULL
;
437 struct isl_basic_set
*hull
= NULL
;
441 bset
= isl_basic_map_underlying_set(bmap
);
442 bset
= isl_basic_set_remove_equalities(bset
, NULL
, &T2
);
446 hull
= uset_affine_hull(bset
);
448 hull
= isl_basic_set_preimage(hull
, T2
);
452 isl_mat_free(ctx
, T2
);
453 isl_basic_set_free(bset
);
454 isl_basic_set_free(hull
);
458 /* Detect and make explicit all equalities satisfied by the (integer)
461 struct isl_basic_map
*isl_basic_map_detect_equalities(
462 struct isl_basic_map
*bmap
)
465 struct isl_basic_set
*hull
= NULL
;
469 if (bmap
->n_ineq
== 0)
471 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
))
473 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_ALL_EQUALITIES
))
475 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
))
476 return isl_basic_map_implicit_equalities(bmap
);
478 hull
= equalities_in_underlying_set(isl_basic_map_copy(bmap
));
481 bmap
= isl_basic_map_extend_dim(bmap
, isl_dim_copy(bmap
->dim
), 0,
483 for (i
= 0; i
< hull
->n_eq
; ++i
) {
484 j
= isl_basic_map_alloc_equality(bmap
);
487 isl_seq_cpy(bmap
->eq
[j
], hull
->eq
[i
],
488 1 + isl_basic_set_total_dim(hull
));
490 isl_basic_set_free(hull
);
491 ISL_F_SET(bmap
, ISL_BASIC_MAP_NO_IMPLICIT
| ISL_BASIC_MAP_ALL_EQUALITIES
);
492 bmap
= isl_basic_map_simplify(bmap
);
493 return isl_basic_map_finalize(bmap
);
495 isl_basic_set_free(hull
);
496 isl_basic_map_free(bmap
);
500 struct isl_map
*isl_map_detect_equalities(struct isl_map
*map
)
502 struct isl_basic_map
*bmap
;
508 for (i
= 0; i
< map
->n
; ++i
) {
509 bmap
= isl_basic_map_copy(map
->p
[i
]);
510 bmap
= isl_basic_map_detect_equalities(bmap
);
513 isl_basic_map_free(map
->p
[i
]);
523 /* After computing the rational affine hull (by detecting the implicit
524 * equalities), we compute the additional equalities satisfied by
525 * the integer points (if any) and add the original equalities back in.
527 struct isl_basic_map
*isl_basic_map_affine_hull(struct isl_basic_map
*bmap
)
529 struct isl_basic_set
*hull
= NULL
;
531 bmap
= isl_basic_map_detect_equalities(bmap
);
532 bmap
= isl_basic_map_cow(bmap
);
533 isl_basic_map_free_inequality(bmap
, bmap
->n_ineq
);
537 struct isl_basic_set
*isl_basic_set_affine_hull(struct isl_basic_set
*bset
)
539 return (struct isl_basic_set
*)
540 isl_basic_map_affine_hull((struct isl_basic_map
*)bset
);
543 struct isl_basic_map
*isl_map_affine_hull(struct isl_map
*map
)
546 struct isl_basic_map
*model
= NULL
;
547 struct isl_basic_map
*hull
= NULL
;
554 hull
= isl_basic_map_empty_like_map(map
);
559 map
= isl_map_detect_equalities(map
);
560 map
= isl_map_align_divs(map
);
563 model
= isl_basic_map_copy(map
->p
[0]);
564 set
= isl_map_underlying_set(map
);
565 set
= isl_set_cow(set
);
569 for (i
= 0; i
< set
->n
; ++i
) {
570 set
->p
[i
] = isl_basic_set_cow(set
->p
[i
]);
571 set
->p
[i
] = isl_basic_set_affine_hull(set
->p
[i
]);
572 set
->p
[i
] = isl_basic_set_gauss(set
->p
[i
], NULL
);
576 set
= isl_set_remove_empty_parts(set
);
578 hull
= isl_basic_map_empty_like(model
);
579 isl_basic_map_free(model
);
581 struct isl_basic_set
*bset
;
583 set
->p
[0] = affine_hull(set
->p
[0], set
->p
[--set
->n
]);
587 bset
= isl_basic_set_copy(set
->p
[0]);
588 hull
= isl_basic_map_overlying_set(bset
, model
);
591 hull
= isl_basic_map_simplify(hull
);
592 return isl_basic_map_finalize(hull
);
594 isl_basic_map_free(model
);
599 struct isl_basic_set
*isl_set_affine_hull(struct isl_set
*set
)
601 return (struct isl_basic_set
*)
602 isl_map_affine_hull((struct isl_map
*)set
);