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 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
35 /* Make eq[row][col] of both bmaps equal so we can add the row
36 * add the column to the common matrix.
37 * Note that because of the echelon form, the columns of row row
38 * after column col are zero.
40 static void set_common_multiple(
41 struct isl_basic_set *bset1, struct isl_basic_set *bset2,
42 unsigned row, unsigned col)
46 if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col]))
51 isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]);
52 isl_int_divexact(c, m, bset1->eq[row][col]);
53 isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1);
54 isl_int_divexact(c, m, bset2->eq[row][col]);
55 isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1);
60 /* Delete a given equality, moving all the following equalities one up.
62 static void delete_row(struct isl_basic_set *bset, unsigned row)
69 for (r = row; r < bset->n_eq; ++r)
70 bset->eq[r] = bset->eq[r+1];
71 bset->eq[bset->n_eq] = t;
74 /* Make first row entries in column col of bset1 identical to
75 * those of bset2, using the fact that entry bset1->eq[row][col]=a
76 * is non-zero. Initially, these elements of bset1 are all zero.
77 * For each row i < row, we set
78 * A[i] = a * A[i] + B[i][col] * A[row]
81 * A[i][col] = B[i][col] = a * old(B[i][col])
83 static void construct_column(
84 struct isl_basic_set *bset1, struct isl_basic_set *bset2,
85 unsigned row, unsigned col)
94 total = 1 + isl_basic_set_n_dim(bset1);
95 for (r = 0; r < row; ++r) {
96 if (isl_int_is_zero(bset2->eq[r][col]))
98 isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]);
99 isl_int_divexact(a, bset1->eq[row][col], b);
100 isl_int_divexact(b, bset2->eq[r][col], b);
101 isl_seq_combine(bset1->eq[r], a, bset1->eq[r],
102 b, bset1->eq[row], total);
103 isl_seq_scale(bset2->eq[r], bset2->eq[r], a, total);
107 delete_row(bset1, row);
110 /* Make first row entries in column col of bset1 identical to
111 * those of bset2, using only these entries of the two matrices.
112 * Let t be the last row with different entries.
113 * For each row i < t, we set
114 * A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
115 * B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
117 * A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
119 static int transform_column(
120 struct isl_basic_set *bset1, struct isl_basic_set *bset2,
121 unsigned row, unsigned col)
127 for (t = row-1; t >= 0; --t)
128 if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))
133 total = 1 + isl_basic_set_n_dim(bset1);
137 isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]);
138 for (i = 0; i < t; ++i) {
139 isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]);
140 isl_int_gcd(g, a, b);
141 isl_int_divexact(a, a, g);
142 isl_int_divexact(g, b, g);
143 isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t],
145 isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],
151 delete_row(bset1, t);
152 delete_row(bset2, t);
156 /* The implementation is based on Section 5.2 of Michael Karr,
157 * "Affine Relationships Among Variables of a Program",
158 * except that the echelon form we use starts from the last column
159 * and that we are dealing with integer coefficients.
161 static struct isl_basic_set *affine_hull(
162 struct isl_basic_set *bset1, struct isl_basic_set *bset2)
168 total = 1 + isl_basic_set_n_dim(bset1);
171 for (col = total-1; col >= 0; --col) {
172 int is_zero1 = row >= bset1->n_eq ||
173 isl_int_is_zero(bset1->eq[row][col]);
174 int is_zero2 = row >= bset2->n_eq ||
175 isl_int_is_zero(bset2->eq[row][col]);
176 if (!is_zero1 && !is_zero2) {
177 set_common_multiple(bset1, bset2, row, col);
179 } else if (!is_zero1 && is_zero2) {
180 construct_column(bset1, bset2, row, col);
181 } else if (is_zero1 && !is_zero2) {
182 construct_column(bset2, bset1, row, col);
184 if (transform_column(bset1, bset2, row, col))
188 isl_basic_set_free(bset2);
189 isl_assert(ctx, row == bset1->n_eq, goto error);
192 isl_basic_set_free(bset1);
196 static struct isl_basic_set *isl_basic_set_from_vec(struct isl_ctx *ctx,
201 struct isl_basic_set *bset = NULL;
206 isl_assert(ctx, vec->size != 0, goto error);
208 bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
211 dim = isl_basic_set_n_dim(bset);
212 for (i = dim - 1; i >= 0; --i) {
213 k = isl_basic_set_alloc_equality(bset);
216 isl_seq_clr(bset->eq[k], 1 + dim);
217 isl_int_neg(bset->eq[k][0], vec->block.data[1 + i]);
218 isl_int_set(bset->eq[k][1 + i], vec->block.data[0]);
220 isl_vec_free(ctx, vec);
224 isl_basic_set_free(bset);
225 isl_vec_free(ctx, vec);
229 /* Find an integer point in "bset" that lies outside of the equality
231 * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
232 * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
233 * The point, if found, is returned as a singleton set.
234 * If no point can be found, the empty set is returned.
236 static struct isl_basic_set *outside_point(struct isl_ctx *ctx,
237 struct isl_basic_set *bset, isl_int *eq, int up)
239 struct isl_basic_set *slice = NULL;
240 struct isl_vec *sample;
241 struct isl_basic_set *point;
245 slice = isl_basic_set_copy(bset);
248 dim = isl_basic_set_n_dim(slice);
249 slice = isl_basic_set_cow(slice);
250 slice = isl_basic_set_extend(slice, 0, dim, 0, 0, 1);
251 k = isl_basic_set_alloc_inequality(slice);
255 isl_seq_cpy(slice->ineq[k], eq, 1 + dim);
257 isl_seq_neg(slice->ineq[k], eq, 1 + dim);
258 isl_int_sub_ui(slice->ineq[k][0], slice->ineq[k][0], 1);
260 sample = isl_basic_set_sample(slice);
263 if (sample->size == 0) {
264 isl_vec_free(ctx, sample);
265 point = isl_basic_set_empty_like(bset);
267 point = isl_basic_set_from_vec(ctx, sample);
271 isl_basic_set_free(slice);
275 /* Look for all equalities satisfied by the integer points in bmap
276 * that are independent of the equalities already explicitly available
279 * We first remove all equalities already explicitly available,
280 * then look for additional equalities in the reduced space
281 * and then transform the result to the original space.
282 * The original equalities are _not_ added to this set. This is
283 * the responsibility of the calling function.
284 * The resulting basic set has all meaning about the dimensions removed.
285 * In particular, dimensions that correspond to existential variables
286 * in bmap and that are found to be fixed are not removed.
288 * The additional equalities are obtained by successively looking for
289 * a point that is affinely independent of the points found so far.
290 * In particular, for each equality satisfied by the points so far,
291 * we check if there is any point on a hyperplane parallel to the
292 * corresponding hyperplane shifted by at least one (in either direction).
294 static struct isl_basic_set *equalities_in_underlying_set(
295 struct isl_basic_map *bmap)
298 struct isl_mat *T2 = NULL;
299 struct isl_basic_set *bset = NULL;
300 struct isl_basic_set *hull = NULL;
301 struct isl_vec *sample;
305 bset = isl_basic_map_underlying_set(bmap);
306 bset = isl_basic_set_remove_equalities(bset, NULL, &T2);
311 sample = isl_basic_set_sample(isl_basic_set_copy(bset));
314 if (sample->size == 0) {
315 isl_vec_free(ctx, sample);
316 hull = isl_basic_set_empty_like(bset);
318 hull = isl_basic_set_from_vec(ctx, sample);
320 dim = isl_basic_set_n_dim(bset);
321 for (i = 0; i < dim; ++i) {
322 struct isl_basic_set *point;
323 if (ISL_F_ISSET(hull, ISL_BASIC_SET_EMPTY))
325 for (j = 0; j < hull->n_eq; ++j) {
326 point = outside_point(ctx, bset, hull->eq[j], 1);
329 if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
331 isl_basic_set_free(point);
332 point = outside_point(ctx, bset, hull->eq[j], 0);
335 if (!ISL_F_ISSET(point, ISL_BASIC_SET_EMPTY))
337 isl_basic_set_free(point);
341 hull = affine_hull(hull, point);
343 isl_basic_set_free(bset);
345 hull = isl_basic_set_preimage(hull, T2);
349 isl_mat_free(ctx, T2);
350 isl_basic_set_free(bset);
351 isl_basic_set_free(hull);
355 /* Detect and make explicit all equalities satisfied by the (integer)
358 struct isl_basic_map *isl_basic_map_detect_equalities(
359 struct isl_basic_map *bmap)
362 struct isl_basic_set *hull = NULL;
366 if (bmap->n_ineq == 0)
368 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
370 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))
372 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
373 return isl_basic_map_implicit_equalities(bmap);
375 hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
378 bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim), 0,
380 for (i = 0; i < hull->n_eq; ++i) {
381 j = isl_basic_map_alloc_equality(bmap);
384 isl_seq_cpy(bmap->eq[j], hull->eq[i],
385 1 + isl_basic_set_total_dim(hull));
387 isl_basic_set_free(hull);
388 ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);
389 bmap = isl_basic_map_simplify(bmap);
390 return isl_basic_map_finalize(bmap);
392 isl_basic_set_free(hull);
393 isl_basic_map_free(bmap);
397 /* After computing the rational affine hull (by detecting the implicit
398 * equalities), we compute the additional equalities satisfied by
399 * the integer points (if any) and add the original equalities back in.
401 struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
403 struct isl_basic_set *hull = NULL;
405 bmap = isl_basic_map_implicit_equalities(bmap);
408 if (bmap->n_ineq == 0)
411 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) {
412 bmap = isl_basic_map_cow(bmap);
413 isl_basic_map_free_inequality(bmap, bmap->n_ineq);
417 hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
421 bmap = isl_basic_map_cow(bmap);
424 isl_basic_map_free_inequality(bmap, bmap->n_ineq);
425 bmap = isl_basic_map_intersect(bmap,
426 isl_basic_map_overlying_set(hull,
427 isl_basic_map_copy(bmap)));
429 return isl_basic_map_finalize(bmap);
431 isl_basic_set_free(hull);
432 isl_basic_map_free(bmap);
436 struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
438 return (struct isl_basic_set *)
439 isl_basic_map_affine_hull((struct isl_basic_map *)bset);
442 struct isl_basic_map *isl_map_affine_hull(struct isl_map *map)
445 struct isl_basic_map *model = NULL;
446 struct isl_basic_map *hull = NULL;
453 hull = isl_basic_map_empty_like_map(map);
458 map = isl_map_align_divs(map);
459 model = isl_basic_map_copy(map->p[0]);
460 set = isl_map_underlying_set(map);
461 set = isl_set_cow(set);
465 for (i = 0; i < set->n; ++i) {
466 set->p[i] = isl_basic_set_cow(set->p[i]);
467 set->p[i] = isl_basic_set_affine_hull(set->p[i]);
468 set->p[i] = isl_basic_set_gauss(set->p[i], NULL);
472 set = isl_set_remove_empty_parts(set);
474 hull = isl_basic_map_empty_like(model);
475 isl_basic_map_free(model);
477 struct isl_basic_set *bset;
479 set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
483 bset = isl_basic_set_copy(set->p[0]);
484 hull = isl_basic_map_overlying_set(bset, model);
487 hull = isl_basic_map_simplify(hull);
488 return isl_basic_map_finalize(hull);
490 isl_basic_map_free(model);
495 struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
497 return (struct isl_basic_set *)
498 isl_map_affine_hull((struct isl_map *)set);