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_vec *vec)
228 struct isl_basic_set *bset = NULL;
235 isl_assert(ctx, vec->size != 0, goto error);
237 bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
240 dim = isl_basic_set_n_dim(bset);
241 for (i = dim - 1; i >= 0; --i) {
242 k = isl_basic_set_alloc_equality(bset);
245 isl_seq_clr(bset->eq[k], 1 + dim);
246 isl_int_neg(bset->eq[k][0], vec->el[1 + i]);
247 isl_int_set(bset->eq[k][1 + i], vec->el[0]);
253 isl_basic_set_free(bset);
258 /* Find an integer point in "bset" that lies outside of the equality
260 * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
261 * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
262 * The point, if found, is returned as a singleton set.
263 * If no point can be found, the empty set is returned.
265 * Before solving an ILP problem, we first check if simply
266 * adding the normal of the constraint to one of the known
267 * integer points in the basic set yields another point
268 * inside the basic set.
270 static struct isl_basic_set *outside_point(struct isl_ctx *ctx,
271 struct isl_basic_set *bset, isl_int *eq, int up)
273 struct isl_basic_set *slice = NULL;
274 struct isl_vec *sample;
275 struct isl_basic_set *point;
279 dim = isl_basic_set_n_dim(bset);
280 sample = isl_vec_alloc(ctx, 1 + dim);
283 isl_int_set_si(sample->block.data[0], 1);
284 isl_seq_combine(sample->block.data + 1,
285 ctx->one, bset->sample->block.data + 1,
286 up ? ctx->one : ctx->negone, eq + 1, dim);
287 if (isl_basic_set_contains(bset, sample))
288 return isl_basic_set_from_vec(sample);
289 isl_vec_free(sample);
292 slice = isl_basic_set_copy(bset);
295 slice = isl_basic_set_cow(slice);
296 slice = isl_basic_set_extend(slice, 0, dim, 0, 0, 1);
297 k = isl_basic_set_alloc_inequality(slice);
301 isl_seq_cpy(slice->ineq[k], eq, 1 + dim);
303 isl_seq_neg(slice->ineq[k], eq, 1 + dim);
304 isl_int_sub_ui(slice->ineq[k][0], slice->ineq[k][0], 1);
306 sample = isl_basic_set_sample(slice);
309 if (sample->size == 0) {
310 isl_vec_free(sample);
311 point = isl_basic_set_empty_like(bset);
313 point = isl_basic_set_from_vec(sample);
317 isl_basic_set_free(slice);
321 static struct isl_basic_set *recession_cone(struct isl_basic_set *bset)
325 bset = isl_basic_set_cow(bset);
329 for (i = 0; i < bset->n_eq; ++i)
330 isl_int_set_si(bset->eq[i][0], 0);
332 for (i = 0; i < bset->n_ineq; ++i)
333 isl_int_set_si(bset->ineq[i][0], 0);
335 ISL_F_CLR(bset, ISL_BASIC_SET_NO_IMPLICIT);
336 return isl_basic_set_implicit_equalities(bset);
339 static struct isl_basic_set *shift(struct isl_basic_set *bset, isl_int *point)
344 bset = isl_basic_set_cow(bset);
348 dim = isl_basic_set_n_dim(bset);
349 for (i = 0; i < bset->n_eq; ++i) {
350 isl_seq_inner_product(bset->eq[i]+1, point+1, dim,
352 isl_int_neg(bset->eq[i][0], bset->eq[i][0]);
355 for (i = 0; i < bset->n_ineq; ++i) {
356 isl_seq_inner_product(bset->ineq[i]+1, point+1, dim,
358 isl_int_neg(bset->ineq[i][0], bset->ineq[i][0]);
364 /* Look for all equalities satisfied by the integer points in bset,
365 * which is assume not to have any explicit equalities.
367 * The equalities are obtained by successively looking for
368 * a point that is affinely independent of the points found so far.
369 * In particular, for each equality satisfied by the points so far,
370 * we check if there is any point on a hyperplane parallel to the
371 * corresponding hyperplane shifted by at least one (in either direction).
373 * Before looking for any outside points, we first remove the equalities
374 * that correspond to the affine hull of the recession cone.
375 * These equalities will never be equalities over the whols basic set.
377 static struct isl_basic_set *uset_affine_hull(struct isl_basic_set *bset)
380 struct isl_basic_set *hull = NULL;
381 struct isl_vec *sample;
385 if (isl_basic_set_is_empty(bset))
389 sample = isl_basic_set_sample(isl_basic_set_copy(bset));
392 if (sample->size == 0) {
393 isl_vec_free(sample);
394 hull = isl_basic_set_empty_like(bset);
395 isl_basic_set_free(bset);
398 hull = isl_basic_set_from_vec(sample);
400 if (hull->n_eq > 0) {
401 struct isl_basic_set *cone;
402 cone = recession_cone(isl_basic_set_copy(bset));
403 isl_basic_set_free_inequality(cone, cone->n_ineq);
404 cone = isl_basic_set_normalize_constraints(cone);
405 cone = shift(cone, bset->sample->block.data);
406 hull = affine_hull(hull, cone);
409 dim = isl_basic_set_n_dim(bset);
410 for (i = 0; i < dim; ++i) {
411 struct isl_basic_set *point;
412 for (j = 0; j < hull->n_eq; ++j) {
413 point = outside_point(ctx, bset, hull->eq[j], 1);
416 if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
418 isl_basic_set_free(point);
419 point = outside_point(ctx, bset, hull->eq[j], 0);
422 if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
424 isl_basic_set_free(point);
428 hull = affine_hull(hull, point);
430 isl_basic_set_free(bset);
434 isl_basic_set_free(bset);
435 isl_basic_set_free(hull);
439 /* Look for all equalities satisfied by the integer points in bmap
440 * that are independent of the equalities already explicitly available
443 * We first remove all equalities already explicitly available,
444 * then look for additional equalities in the reduced space
445 * and then transform the result to the original space.
446 * The original equalities are _not_ added to this set. This is
447 * the responsibility of the calling function.
448 * The resulting basic set has all meaning about the dimensions removed.
449 * In particular, dimensions that correspond to existential variables
450 * in bmap and that are found to be fixed are not removed.
452 static struct isl_basic_set *equalities_in_underlying_set(
453 struct isl_basic_map *bmap)
455 struct isl_mat *T2 = NULL;
456 struct isl_basic_set *bset = NULL;
457 struct isl_basic_set *hull = NULL;
461 bset = isl_basic_map_underlying_set(bmap);
462 bset = isl_basic_set_remove_equalities(bset, NULL, &T2);
466 hull = uset_affine_hull(bset);
468 hull = isl_basic_set_preimage(hull, T2);
472 isl_mat_free(ctx, T2);
473 isl_basic_set_free(bset);
474 isl_basic_set_free(hull);
478 /* Detect and make explicit all equalities satisfied by the (integer)
481 struct isl_basic_map *isl_basic_map_detect_equalities(
482 struct isl_basic_map *bmap)
485 struct isl_basic_set *hull = NULL;
489 if (bmap->n_ineq == 0)
491 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
493 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))
495 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
496 return isl_basic_map_implicit_equalities(bmap);
498 hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
501 if (ISL_F_ISSET(hull, ISL_BASIC_SET_EMPTY)) {
502 isl_basic_set_free(hull);
503 return isl_basic_map_set_to_empty(bmap);
505 bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim), 0,
507 for (i = 0; i < hull->n_eq; ++i) {
508 j = isl_basic_map_alloc_equality(bmap);
511 isl_seq_cpy(bmap->eq[j], hull->eq[i],
512 1 + isl_basic_set_total_dim(hull));
514 isl_basic_set_free(hull);
515 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);
516 bmap = isl_basic_map_simplify(bmap);
517 return isl_basic_map_finalize(bmap);
519 isl_basic_set_free(hull);
520 isl_basic_map_free(bmap);
524 struct isl_map *isl_map_detect_equalities(struct isl_map *map)
526 struct isl_basic_map *bmap;
532 for (i = 0; i < map->n; ++i) {
533 bmap = isl_basic_map_copy(map->p[i]);
534 bmap = isl_basic_map_detect_equalities(bmap);
537 isl_basic_map_free(map->p[i]);
547 /* After computing the rational affine hull (by detecting the implicit
548 * equalities), we compute the additional equalities satisfied by
549 * the integer points (if any) and add the original equalities back in.
551 struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
553 struct isl_basic_set *hull = NULL;
555 bmap = isl_basic_map_detect_equalities(bmap);
556 bmap = isl_basic_map_cow(bmap);
557 isl_basic_map_free_inequality(bmap, bmap->n_ineq);
561 struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
563 return (struct isl_basic_set *)
564 isl_basic_map_affine_hull((struct isl_basic_map *)bset);
567 struct isl_basic_map *isl_map_affine_hull(struct isl_map *map)
570 struct isl_basic_map *model = NULL;
571 struct isl_basic_map *hull = NULL;
578 hull = isl_basic_map_empty_like_map(map);
583 map = isl_map_detect_equalities(map);
584 map = isl_map_align_divs(map);
587 model = isl_basic_map_copy(map->p[0]);
588 set = isl_map_underlying_set(map);
589 set = isl_set_cow(set);
593 for (i = 0; i < set->n; ++i) {
594 set->p[i] = isl_basic_set_cow(set->p[i]);
595 set->p[i] = isl_basic_set_affine_hull(set->p[i]);
596 set->p[i] = isl_basic_set_gauss(set->p[i], NULL);
600 set = isl_set_remove_empty_parts(set);
602 hull = isl_basic_map_empty_like(model);
603 isl_basic_map_free(model);
605 struct isl_basic_set *bset;
607 set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
611 bset = isl_basic_set_copy(set->p[0]);
612 hull = isl_basic_map_overlying_set(bset, model);
615 hull = isl_basic_map_simplify(hull);
616 return isl_basic_map_finalize(hull);
618 isl_basic_map_free(model);
623 struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
625 return (struct isl_basic_set *)
626 isl_map_affine_hull((struct isl_map *)set);