isl 0.27
[isl.git] / isl_transitive_closure.c
blobc3da35ca27238595c3b79ff21f3fd90db4d768ed
1 /*
2 * Copyright 2010 INRIA Saclay
4 * Use of this software is governed by the MIT license
6 * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
7 * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
8 * 91893 Orsay, France
9 */
11 #include <isl_ctx_private.h>
12 #include <isl_map_private.h>
13 #include <isl/map.h>
14 #include <isl_seq.h>
15 #include <isl_space_private.h>
16 #include <isl_lp_private.h>
17 #include <isl/union_map.h>
18 #include <isl_mat_private.h>
19 #include <isl_vec_private.h>
20 #include <isl_options_private.h>
21 #include <isl_tarjan.h>
23 isl_bool isl_map_is_transitively_closed(__isl_keep isl_map *map)
25 isl_map *map2;
26 isl_bool closed;
28 map2 = isl_map_apply_range(isl_map_copy(map), isl_map_copy(map));
29 closed = isl_map_is_subset(map2, map);
30 isl_map_free(map2);
32 return closed;
35 isl_bool isl_union_map_is_transitively_closed(__isl_keep isl_union_map *umap)
37 isl_union_map *umap2;
38 isl_bool closed;
40 umap2 = isl_union_map_apply_range(isl_union_map_copy(umap),
41 isl_union_map_copy(umap));
42 closed = isl_union_map_is_subset(umap2, umap);
43 isl_union_map_free(umap2);
45 return closed;
48 /* Given a map that represents a path with the length of the path
49 * encoded as the difference between the last output coordindate
50 * and the last input coordinate, set this length to either
51 * exactly "length" (if "exactly" is set) or at least "length"
52 * (if "exactly" is not set).
54 static __isl_give isl_map *set_path_length(__isl_take isl_map *map,
55 int exactly, int length)
57 isl_space *space;
58 struct isl_basic_map *bmap;
59 isl_size d;
60 isl_size nparam;
61 isl_size total;
62 int k;
63 isl_int *c;
65 if (!map)
66 return NULL;
68 space = isl_map_get_space(map);
69 d = isl_space_dim(space, isl_dim_in);
70 nparam = isl_space_dim(space, isl_dim_param);
71 total = isl_space_dim(space, isl_dim_all);
72 if (d < 0 || nparam < 0 || total < 0)
73 space = isl_space_free(space);
74 bmap = isl_basic_map_alloc_space(space, 0, 1, 1);
75 if (exactly) {
76 k = isl_basic_map_alloc_equality(bmap);
77 if (k < 0)
78 goto error;
79 c = bmap->eq[k];
80 } else {
81 k = isl_basic_map_alloc_inequality(bmap);
82 if (k < 0)
83 goto error;
84 c = bmap->ineq[k];
86 isl_seq_clr(c, 1 + total);
87 isl_int_set_si(c[0], -length);
88 isl_int_set_si(c[1 + nparam + d - 1], -1);
89 isl_int_set_si(c[1 + nparam + d + d - 1], 1);
91 bmap = isl_basic_map_finalize(bmap);
92 map = isl_map_intersect(map, isl_map_from_basic_map(bmap));
94 return map;
95 error:
96 isl_basic_map_free(bmap);
97 isl_map_free(map);
98 return NULL;
101 /* Check whether the overapproximation of the power of "map" is exactly
102 * the power of "map". Let R be "map" and A_k the overapproximation.
103 * The approximation is exact if
105 * A_1 = R
106 * A_k = A_{k-1} \circ R k >= 2
108 * Since A_k is known to be an overapproximation, we only need to check
110 * A_1 \subset R
111 * A_k \subset A_{k-1} \circ R k >= 2
113 * In practice, "app" has an extra input and output coordinate
114 * to encode the length of the path. So, we first need to add
115 * this coordinate to "map" and set the length of the path to
116 * one.
118 static isl_bool check_power_exactness(__isl_take isl_map *map,
119 __isl_take isl_map *app)
121 isl_bool exact;
122 isl_map *app_1;
123 isl_map *app_2;
125 map = isl_map_add_dims(map, isl_dim_in, 1);
126 map = isl_map_add_dims(map, isl_dim_out, 1);
127 map = set_path_length(map, 1, 1);
129 app_1 = set_path_length(isl_map_copy(app), 1, 1);
131 exact = isl_map_is_subset(app_1, map);
132 isl_map_free(app_1);
134 if (!exact || exact < 0) {
135 isl_map_free(app);
136 isl_map_free(map);
137 return exact;
140 app_1 = set_path_length(isl_map_copy(app), 0, 1);
141 app_2 = set_path_length(app, 0, 2);
142 app_1 = isl_map_apply_range(map, app_1);
144 exact = isl_map_is_subset(app_2, app_1);
146 isl_map_free(app_1);
147 isl_map_free(app_2);
149 return exact;
152 /* Check whether the overapproximation of the power of "map" is exactly
153 * the power of "map", possibly after projecting out the power (if "project"
154 * is set).
156 * If "project" is set and if "steps" can only result in acyclic paths,
157 * then we check
159 * A = R \cup (A \circ R)
161 * where A is the overapproximation with the power projected out, i.e.,
162 * an overapproximation of the transitive closure.
163 * More specifically, since A is known to be an overapproximation, we check
165 * A \subset R \cup (A \circ R)
167 * Otherwise, we check if the power is exact.
169 * Note that "app" has an extra input and output coordinate to encode
170 * the length of the part. If we are only interested in the transitive
171 * closure, then we can simply project out these coordinates first.
173 static isl_bool check_exactness(__isl_take isl_map *map,
174 __isl_take isl_map *app, int project)
176 isl_map *test;
177 isl_bool exact;
178 isl_size d;
180 if (!project)
181 return check_power_exactness(map, app);
183 d = isl_map_dim(map, isl_dim_in);
184 if (d < 0)
185 app = isl_map_free(app);
186 app = set_path_length(app, 0, 1);
187 app = isl_map_project_out(app, isl_dim_in, d, 1);
188 app = isl_map_project_out(app, isl_dim_out, d, 1);
190 app = isl_map_reset_space(app, isl_map_get_space(map));
192 test = isl_map_apply_range(isl_map_copy(map), isl_map_copy(app));
193 test = isl_map_union(test, isl_map_copy(map));
195 exact = isl_map_is_subset(app, test);
197 isl_map_free(app);
198 isl_map_free(test);
200 isl_map_free(map);
202 return exact;
206 * The transitive closure implementation is based on the paper
207 * "Computing the Transitive Closure of a Union of Affine Integer
208 * Tuple Relations" by Anna Beletska, Denis Barthou, Wlodzimierz Bielecki and
209 * Albert Cohen.
212 /* Given a set of n offsets v_i (the rows of "steps"), construct a relation
213 * of the given dimension specification (Z^{n+1} -> Z^{n+1})
214 * that maps an element x to any element that can be reached
215 * by taking a non-negative number of steps along any of
216 * the extended offsets v'_i = [v_i 1].
217 * That is, construct
219 * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v'_i }
221 * For any element in this relation, the number of steps taken
222 * is equal to the difference in the final coordinates.
224 static __isl_give isl_map *path_along_steps(__isl_take isl_space *space,
225 __isl_keep isl_mat *steps)
227 int i, j, k;
228 struct isl_basic_map *path = NULL;
229 isl_size d;
230 unsigned n;
231 isl_size nparam;
232 isl_size total;
234 d = isl_space_dim(space, isl_dim_in);
235 nparam = isl_space_dim(space, isl_dim_param);
236 if (d < 0 || nparam < 0 || !steps)
237 goto error;
239 n = steps->n_row;
241 path = isl_basic_map_alloc_space(isl_space_copy(space), n, d, n);
243 for (i = 0; i < n; ++i) {
244 k = isl_basic_map_alloc_div(path);
245 if (k < 0)
246 goto error;
247 isl_assert(steps->ctx, i == k, goto error);
248 isl_int_set_si(path->div[k][0], 0);
251 total = isl_basic_map_dim(path, isl_dim_all);
252 if (total < 0)
253 goto error;
254 for (i = 0; i < d; ++i) {
255 k = isl_basic_map_alloc_equality(path);
256 if (k < 0)
257 goto error;
258 isl_seq_clr(path->eq[k], 1 + total);
259 isl_int_set_si(path->eq[k][1 + nparam + i], 1);
260 isl_int_set_si(path->eq[k][1 + nparam + d + i], -1);
261 if (i == d - 1)
262 for (j = 0; j < n; ++j)
263 isl_int_set_si(path->eq[k][1 + nparam + 2 * d + j], 1);
264 else
265 for (j = 0; j < n; ++j)
266 isl_int_set(path->eq[k][1 + nparam + 2 * d + j],
267 steps->row[j][i]);
270 for (i = 0; i < n; ++i) {
271 k = isl_basic_map_alloc_inequality(path);
272 if (k < 0)
273 goto error;
274 isl_seq_clr(path->ineq[k], 1 + total);
275 isl_int_set_si(path->ineq[k][1 + nparam + 2 * d + i], 1);
278 isl_space_free(space);
280 path = isl_basic_map_simplify(path);
281 path = isl_basic_map_finalize(path);
282 return isl_map_from_basic_map(path);
283 error:
284 isl_space_free(space);
285 isl_basic_map_free(path);
286 return NULL;
289 #define IMPURE 0
290 #define PURE_PARAM 1
291 #define PURE_VAR 2
292 #define MIXED 3
294 /* Check whether the parametric constant term of constraint c is never
295 * positive in "bset".
297 static isl_bool parametric_constant_never_positive(
298 __isl_keep isl_basic_set *bset, isl_int *c, int *div_purity)
300 isl_size d;
301 isl_size n_div;
302 isl_size nparam;
303 isl_size total;
304 int i;
305 int k;
306 isl_bool empty;
308 n_div = isl_basic_set_dim(bset, isl_dim_div);
309 d = isl_basic_set_dim(bset, isl_dim_set);
310 nparam = isl_basic_set_dim(bset, isl_dim_param);
311 total = isl_basic_set_dim(bset, isl_dim_all);
312 if (n_div < 0 || d < 0 || nparam < 0 || total < 0)
313 return isl_bool_error;
315 bset = isl_basic_set_copy(bset);
316 bset = isl_basic_set_cow(bset);
317 bset = isl_basic_set_extend_constraints(bset, 0, 1);
318 k = isl_basic_set_alloc_inequality(bset);
319 if (k < 0)
320 goto error;
321 isl_seq_clr(bset->ineq[k], 1 + total);
322 isl_seq_cpy(bset->ineq[k], c, 1 + nparam);
323 for (i = 0; i < n_div; ++i) {
324 if (div_purity[i] != PURE_PARAM)
325 continue;
326 isl_int_set(bset->ineq[k][1 + nparam + d + i],
327 c[1 + nparam + d + i]);
329 isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);
330 empty = isl_basic_set_is_empty(bset);
331 isl_basic_set_free(bset);
333 return empty;
334 error:
335 isl_basic_set_free(bset);
336 return isl_bool_error;
339 /* Return PURE_PARAM if only the coefficients of the parameters are non-zero.
340 * Return PURE_VAR if only the coefficients of the set variables are non-zero.
341 * Return MIXED if only the coefficients of the parameters and the set
342 * variables are non-zero and if moreover the parametric constant
343 * can never attain positive values.
344 * Return IMPURE otherwise.
346 static int purity(__isl_keep isl_basic_set *bset, isl_int *c, int *div_purity,
347 int eq)
349 isl_size d;
350 isl_size n_div;
351 isl_size nparam;
352 isl_bool empty;
353 int i;
354 int p = 0, v = 0;
356 n_div = isl_basic_set_dim(bset, isl_dim_div);
357 d = isl_basic_set_dim(bset, isl_dim_set);
358 nparam = isl_basic_set_dim(bset, isl_dim_param);
359 if (n_div < 0 || d < 0 || nparam < 0)
360 return -1;
362 for (i = 0; i < n_div; ++i) {
363 if (isl_int_is_zero(c[1 + nparam + d + i]))
364 continue;
365 switch (div_purity[i]) {
366 case PURE_PARAM: p = 1; break;
367 case PURE_VAR: v = 1; break;
368 default: return IMPURE;
371 if (!p && !isl_seq_any_non_zero(c + 1, nparam))
372 return PURE_VAR;
373 if (!v && !isl_seq_any_non_zero(c + 1 + nparam, d))
374 return PURE_PARAM;
376 empty = parametric_constant_never_positive(bset, c, div_purity);
377 if (eq && empty >= 0 && !empty) {
378 isl_seq_neg(c, c, 1 + nparam + d + n_div);
379 empty = parametric_constant_never_positive(bset, c, div_purity);
382 return empty < 0 ? -1 : empty ? MIXED : IMPURE;
385 /* Return an array of integers indicating the type of each div in bset.
386 * If the div is (recursively) defined in terms of only the parameters,
387 * then the type is PURE_PARAM.
388 * If the div is (recursively) defined in terms of only the set variables,
389 * then the type is PURE_VAR.
390 * Otherwise, the type is IMPURE.
392 static __isl_give int *get_div_purity(__isl_keep isl_basic_set *bset)
394 int i, j;
395 int *div_purity;
396 isl_size d;
397 isl_size n_div;
398 isl_size nparam;
400 n_div = isl_basic_set_dim(bset, isl_dim_div);
401 d = isl_basic_set_dim(bset, isl_dim_set);
402 nparam = isl_basic_set_dim(bset, isl_dim_param);
403 if (n_div < 0 || d < 0 || nparam < 0)
404 return NULL;
406 div_purity = isl_alloc_array(bset->ctx, int, n_div);
407 if (n_div && !div_purity)
408 return NULL;
410 for (i = 0; i < bset->n_div; ++i) {
411 int p = 0, v = 0;
412 if (isl_int_is_zero(bset->div[i][0])) {
413 div_purity[i] = IMPURE;
414 continue;
416 if (isl_seq_any_non_zero(bset->div[i] + 2, nparam))
417 p = 1;
418 if (isl_seq_any_non_zero(bset->div[i] + 2 + nparam, d))
419 v = 1;
420 for (j = 0; j < i; ++j) {
421 if (isl_int_is_zero(bset->div[i][2 + nparam + d + j]))
422 continue;
423 switch (div_purity[j]) {
424 case PURE_PARAM: p = 1; break;
425 case PURE_VAR: v = 1; break;
426 default: p = v = 1; break;
429 div_purity[i] = v ? p ? IMPURE : PURE_VAR : PURE_PARAM;
432 return div_purity;
435 /* Given a path with the as yet unconstrained length at div position "pos",
436 * check if setting the length to zero results in only the identity
437 * mapping.
439 static isl_bool empty_path_is_identity(__isl_keep isl_basic_map *path,
440 unsigned pos)
442 isl_basic_map *test = NULL;
443 isl_basic_map *id = NULL;
444 isl_bool is_id;
446 test = isl_basic_map_copy(path);
447 test = isl_basic_map_fix_si(test, isl_dim_div, pos, 0);
448 id = isl_basic_map_identity(isl_basic_map_get_space(path));
449 is_id = isl_basic_map_is_equal(test, id);
450 isl_basic_map_free(test);
451 isl_basic_map_free(id);
452 return is_id;
455 /* If any of the constraints is found to be impure then this function
456 * sets *impurity to 1.
458 * If impurity is NULL then we are dealing with a non-parametric set
459 * and so the constraints are obviously PURE_VAR.
461 static __isl_give isl_basic_map *add_delta_constraints(
462 __isl_take isl_basic_map *path,
463 __isl_keep isl_basic_set *delta, unsigned off, unsigned nparam,
464 unsigned d, int *div_purity, int eq, int *impurity)
466 int i, k;
467 int n = eq ? delta->n_eq : delta->n_ineq;
468 isl_int **delta_c = eq ? delta->eq : delta->ineq;
469 isl_size n_div, total;
471 n_div = isl_basic_set_dim(delta, isl_dim_div);
472 total = isl_basic_map_dim(path, isl_dim_all);
473 if (n_div < 0 || total < 0)
474 return isl_basic_map_free(path);
476 for (i = 0; i < n; ++i) {
477 isl_int *path_c;
478 int p = PURE_VAR;
479 if (impurity)
480 p = purity(delta, delta_c[i], div_purity, eq);
481 if (p < 0)
482 goto error;
483 if (p != PURE_VAR && p != PURE_PARAM && !*impurity)
484 *impurity = 1;
485 if (p == IMPURE)
486 continue;
487 if (eq && p != MIXED) {
488 k = isl_basic_map_alloc_equality(path);
489 if (k < 0)
490 goto error;
491 path_c = path->eq[k];
492 } else {
493 k = isl_basic_map_alloc_inequality(path);
494 if (k < 0)
495 goto error;
496 path_c = path->ineq[k];
498 isl_seq_clr(path_c, 1 + total);
499 if (p == PURE_VAR) {
500 isl_seq_cpy(path_c + off,
501 delta_c[i] + 1 + nparam, d);
502 isl_int_set(path_c[off + d], delta_c[i][0]);
503 } else if (p == PURE_PARAM) {
504 isl_seq_cpy(path_c, delta_c[i], 1 + nparam);
505 } else {
506 isl_seq_cpy(path_c + off,
507 delta_c[i] + 1 + nparam, d);
508 isl_seq_cpy(path_c, delta_c[i], 1 + nparam);
510 isl_seq_cpy(path_c + off - n_div,
511 delta_c[i] + 1 + nparam + d, n_div);
514 return path;
515 error:
516 isl_basic_map_free(path);
517 return NULL;
520 /* Given a set of offsets "delta", construct a relation of the
521 * given dimension specification (Z^{n+1} -> Z^{n+1}) that
522 * is an overapproximation of the relation that
523 * maps an element x to any element that can be reached
524 * by taking a non-negative number of steps along any of
525 * the elements in "delta".
526 * That is, construct an approximation of
528 * { [x] -> [y] : exists f \in \delta, k \in Z :
529 * y = x + k [f, 1] and k >= 0 }
531 * For any element in this relation, the number of steps taken
532 * is equal to the difference in the final coordinates.
534 * In particular, let delta be defined as
536 * \delta = [p] -> { [x] : A x + a >= 0 and B p + b >= 0 and
537 * C x + C'p + c >= 0 and
538 * D x + D'p + d >= 0 }
540 * where the constraints C x + C'p + c >= 0 are such that the parametric
541 * constant term of each constraint j, "C_j x + C'_j p + c_j",
542 * can never attain positive values, then the relation is constructed as
544 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
545 * A f + k a >= 0 and B p + b >= 0 and
546 * C f + C'p + c >= 0 and k >= 1 }
547 * union { [x] -> [x] }
549 * If the zero-length paths happen to correspond exactly to the identity
550 * mapping, then we return
552 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
553 * A f + k a >= 0 and B p + b >= 0 and
554 * C f + C'p + c >= 0 and k >= 0 }
556 * instead.
558 * Existentially quantified variables in \delta are handled by
559 * classifying them as independent of the parameters, purely
560 * parameter dependent and others. Constraints containing
561 * any of the other existentially quantified variables are removed.
562 * This is safe, but leads to an additional overapproximation.
564 * If there are any impure constraints, then we also eliminate
565 * the parameters from \delta, resulting in a set
567 * \delta' = { [x] : E x + e >= 0 }
569 * and add the constraints
571 * E f + k e >= 0
573 * to the constructed relation.
575 static __isl_give isl_map *path_along_delta(__isl_take isl_space *space,
576 __isl_take isl_basic_set *delta)
578 isl_basic_map *path = NULL;
579 isl_size d;
580 isl_size n_div;
581 isl_size nparam;
582 isl_size total;
583 unsigned off;
584 int i, k;
585 isl_bool is_id;
586 int *div_purity = NULL;
587 int impurity = 0;
589 n_div = isl_basic_set_dim(delta, isl_dim_div);
590 d = isl_basic_set_dim(delta, isl_dim_set);
591 nparam = isl_basic_set_dim(delta, isl_dim_param);
592 if (n_div < 0 || d < 0 || nparam < 0)
593 goto error;
594 path = isl_basic_map_alloc_space(isl_space_copy(space), n_div + d + 1,
595 d + 1 + delta->n_eq, delta->n_eq + delta->n_ineq + 1);
596 off = 1 + nparam + 2 * (d + 1) + n_div;
598 for (i = 0; i < n_div + d + 1; ++i) {
599 k = isl_basic_map_alloc_div(path);
600 if (k < 0)
601 goto error;
602 isl_int_set_si(path->div[k][0], 0);
605 total = isl_basic_map_dim(path, isl_dim_all);
606 if (total < 0)
607 goto error;
608 for (i = 0; i < d + 1; ++i) {
609 k = isl_basic_map_alloc_equality(path);
610 if (k < 0)
611 goto error;
612 isl_seq_clr(path->eq[k], 1 + total);
613 isl_int_set_si(path->eq[k][1 + nparam + i], 1);
614 isl_int_set_si(path->eq[k][1 + nparam + d + 1 + i], -1);
615 isl_int_set_si(path->eq[k][off + i], 1);
618 div_purity = get_div_purity(delta);
619 if (n_div && !div_purity)
620 goto error;
622 path = add_delta_constraints(path, delta, off, nparam, d,
623 div_purity, 1, &impurity);
624 path = add_delta_constraints(path, delta, off, nparam, d,
625 div_purity, 0, &impurity);
626 if (impurity) {
627 isl_space *space = isl_basic_set_get_space(delta);
628 delta = isl_basic_set_project_out(delta,
629 isl_dim_param, 0, nparam);
630 delta = isl_basic_set_add_dims(delta, isl_dim_param, nparam);
631 delta = isl_basic_set_reset_space(delta, space);
632 if (!delta)
633 goto error;
634 path = isl_basic_map_extend_constraints(path, delta->n_eq,
635 delta->n_ineq + 1);
636 path = add_delta_constraints(path, delta, off, nparam, d,
637 NULL, 1, NULL);
638 path = add_delta_constraints(path, delta, off, nparam, d,
639 NULL, 0, NULL);
640 path = isl_basic_map_gauss(path, NULL);
643 is_id = empty_path_is_identity(path, n_div + d);
644 if (is_id < 0)
645 goto error;
647 k = isl_basic_map_alloc_inequality(path);
648 if (k < 0)
649 goto error;
650 isl_seq_clr(path->ineq[k], 1 + total);
651 if (!is_id)
652 isl_int_set_si(path->ineq[k][0], -1);
653 isl_int_set_si(path->ineq[k][off + d], 1);
655 free(div_purity);
656 isl_basic_set_free(delta);
657 path = isl_basic_map_finalize(path);
658 if (is_id) {
659 isl_space_free(space);
660 return isl_map_from_basic_map(path);
662 return isl_basic_map_union(path, isl_basic_map_identity(space));
663 error:
664 free(div_purity);
665 isl_space_free(space);
666 isl_basic_set_free(delta);
667 isl_basic_map_free(path);
668 return NULL;
671 /* Given a dimension specification Z^{n+1} -> Z^{n+1} and a parameter "param",
672 * construct a map that equates the parameter to the difference
673 * in the final coordinates and imposes that this difference is positive.
674 * That is, construct
676 * { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
678 static __isl_give isl_map *equate_parameter_to_length(
679 __isl_take isl_space *space, unsigned param)
681 struct isl_basic_map *bmap;
682 isl_size d;
683 isl_size nparam;
684 isl_size total;
685 int k;
687 d = isl_space_dim(space, isl_dim_in);
688 nparam = isl_space_dim(space, isl_dim_param);
689 total = isl_space_dim(space, isl_dim_all);
690 if (d < 0 || nparam < 0 || total < 0)
691 space = isl_space_free(space);
692 bmap = isl_basic_map_alloc_space(space, 0, 1, 1);
693 k = isl_basic_map_alloc_equality(bmap);
694 if (k < 0)
695 goto error;
696 isl_seq_clr(bmap->eq[k], 1 + total);
697 isl_int_set_si(bmap->eq[k][1 + param], -1);
698 isl_int_set_si(bmap->eq[k][1 + nparam + d - 1], -1);
699 isl_int_set_si(bmap->eq[k][1 + nparam + d + d - 1], 1);
701 k = isl_basic_map_alloc_inequality(bmap);
702 if (k < 0)
703 goto error;
704 isl_seq_clr(bmap->ineq[k], 1 + total);
705 isl_int_set_si(bmap->ineq[k][1 + param], 1);
706 isl_int_set_si(bmap->ineq[k][0], -1);
708 bmap = isl_basic_map_finalize(bmap);
709 return isl_map_from_basic_map(bmap);
710 error:
711 isl_basic_map_free(bmap);
712 return NULL;
715 /* Check whether "path" is acyclic, where the last coordinates of domain
716 * and range of path encode the number of steps taken.
717 * That is, check whether
719 * { d | d = y - x and (x,y) in path }
721 * does not contain any element with positive last coordinate (positive length)
722 * and zero remaining coordinates (cycle).
724 static isl_bool is_acyclic(__isl_take isl_map *path)
726 int i;
727 isl_bool acyclic;
728 isl_size dim;
729 struct isl_set *delta;
731 delta = isl_map_deltas(path);
732 dim = isl_set_dim(delta, isl_dim_set);
733 if (dim < 0)
734 delta = isl_set_free(delta);
735 for (i = 0; i < dim; ++i) {
736 if (i == dim -1)
737 delta = isl_set_lower_bound_si(delta, isl_dim_set, i, 1);
738 else
739 delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
742 acyclic = isl_set_is_empty(delta);
743 isl_set_free(delta);
745 return acyclic;
748 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
749 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
750 * construct a map that is an overapproximation of the map
751 * that takes an element from the space D \times Z to another
752 * element from the same space, such that the first n coordinates of the
753 * difference between them is a sum of differences between images
754 * and pre-images in one of the R_i and such that the last coordinate
755 * is equal to the number of steps taken.
756 * That is, let
758 * \Delta_i = { y - x | (x, y) in R_i }
760 * then the constructed map is an overapproximation of
762 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
763 * d = (\sum_i k_i \delta_i, \sum_i k_i) }
765 * The elements of the singleton \Delta_i's are collected as the
766 * rows of the steps matrix. For all these \Delta_i's together,
767 * a single path is constructed.
768 * For each of the other \Delta_i's, we compute an overapproximation
769 * of the paths along elements of \Delta_i.
770 * Since each of these paths performs an addition, composition is
771 * symmetric and we can simply compose all resulting paths in any order.
773 static __isl_give isl_map *construct_extended_path(__isl_take isl_space *space,
774 __isl_keep isl_map *map, int *project)
776 struct isl_mat *steps = NULL;
777 struct isl_map *path = NULL;
778 isl_size d;
779 int i, j, n;
781 d = isl_map_dim(map, isl_dim_in);
782 if (d < 0)
783 goto error;
785 path = isl_map_identity(isl_space_copy(space));
787 steps = isl_mat_alloc(map->ctx, map->n, d);
788 if (!steps)
789 goto error;
791 n = 0;
792 for (i = 0; i < map->n; ++i) {
793 struct isl_basic_set *delta;
795 delta = isl_basic_map_deltas(isl_basic_map_copy(map->p[i]));
797 for (j = 0; j < d; ++j) {
798 isl_bool fixed;
800 fixed = isl_basic_set_plain_dim_is_fixed(delta, j,
801 &steps->row[n][j]);
802 if (fixed < 0) {
803 isl_basic_set_free(delta);
804 goto error;
806 if (!fixed)
807 break;
811 if (j < d) {
812 path = isl_map_apply_range(path,
813 path_along_delta(isl_space_copy(space), delta));
814 path = isl_map_coalesce(path);
815 } else {
816 isl_basic_set_free(delta);
817 ++n;
821 if (n > 0) {
822 steps->n_row = n;
823 path = isl_map_apply_range(path,
824 path_along_steps(isl_space_copy(space), steps));
827 if (project && *project) {
828 *project = is_acyclic(isl_map_copy(path));
829 if (*project < 0)
830 goto error;
833 isl_space_free(space);
834 isl_mat_free(steps);
835 return path;
836 error:
837 isl_space_free(space);
838 isl_mat_free(steps);
839 isl_map_free(path);
840 return NULL;
843 static isl_bool isl_set_overlaps(__isl_keep isl_set *set1,
844 __isl_keep isl_set *set2)
846 return isl_bool_not(isl_set_is_disjoint(set1, set2));
849 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
850 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
851 * construct a map that is an overapproximation of the map
852 * that takes an element from the dom R \times Z to an
853 * element from ran R \times Z, such that the first n coordinates of the
854 * difference between them is a sum of differences between images
855 * and pre-images in one of the R_i and such that the last coordinate
856 * is equal to the number of steps taken.
857 * That is, let
859 * \Delta_i = { y - x | (x, y) in R_i }
861 * then the constructed map is an overapproximation of
863 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
864 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
865 * x in dom R and x + d in ran R and
866 * \sum_i k_i >= 1 }
868 static __isl_give isl_map *construct_component(__isl_take isl_space *space,
869 __isl_keep isl_map *map, isl_bool *exact, int project)
871 struct isl_set *domain = NULL;
872 struct isl_set *range = NULL;
873 struct isl_map *app = NULL;
874 struct isl_map *path = NULL;
875 isl_bool overlaps;
876 int check;
878 domain = isl_map_domain(isl_map_copy(map));
879 domain = isl_set_coalesce(domain);
880 range = isl_map_range(isl_map_copy(map));
881 range = isl_set_coalesce(range);
882 overlaps = isl_set_overlaps(domain, range);
883 if (overlaps < 0 || !overlaps) {
884 isl_set_free(domain);
885 isl_set_free(range);
886 isl_space_free(space);
888 if (overlaps < 0)
889 map = NULL;
890 map = isl_map_copy(map);
891 map = isl_map_add_dims(map, isl_dim_in, 1);
892 map = isl_map_add_dims(map, isl_dim_out, 1);
893 map = set_path_length(map, 1, 1);
894 return map;
896 app = isl_map_from_domain_and_range(domain, range);
897 app = isl_map_add_dims(app, isl_dim_in, 1);
898 app = isl_map_add_dims(app, isl_dim_out, 1);
900 check = exact && *exact == isl_bool_true;
901 path = construct_extended_path(isl_space_copy(space), map,
902 check ? &project : NULL);
903 app = isl_map_intersect(app, path);
905 if (check &&
906 (*exact = check_exactness(isl_map_copy(map), isl_map_copy(app),
907 project)) < 0)
908 goto error;
910 isl_space_free(space);
911 app = set_path_length(app, 0, 1);
912 return app;
913 error:
914 isl_space_free(space);
915 isl_map_free(app);
916 return NULL;
919 /* Call construct_component and, if "project" is set, project out
920 * the final coordinates.
922 static __isl_give isl_map *construct_projected_component(
923 __isl_take isl_space *space,
924 __isl_keep isl_map *map, isl_bool *exact, int project)
926 isl_map *app;
927 unsigned d;
929 if (!space)
930 return NULL;
931 d = isl_space_dim(space, isl_dim_in);
933 app = construct_component(space, map, exact, project);
934 if (project) {
935 app = isl_map_project_out(app, isl_dim_in, d - 1, 1);
936 app = isl_map_project_out(app, isl_dim_out, d - 1, 1);
938 return app;
941 /* Compute an extended version, i.e., with path lengths, of
942 * an overapproximation of the transitive closure of "bmap"
943 * with path lengths greater than or equal to zero and with
944 * domain and range equal to "dom".
946 static __isl_give isl_map *q_closure(__isl_take isl_space *space,
947 __isl_take isl_set *dom, __isl_keep isl_basic_map *bmap,
948 isl_bool *exact)
950 int project = 1;
951 isl_map *path;
952 isl_map *map;
953 isl_map *app;
955 dom = isl_set_add_dims(dom, isl_dim_set, 1);
956 app = isl_map_from_domain_and_range(dom, isl_set_copy(dom));
957 map = isl_map_from_basic_map(isl_basic_map_copy(bmap));
958 path = construct_extended_path(space, map, &project);
959 app = isl_map_intersect(app, path);
961 if ((*exact = check_exactness(map, isl_map_copy(app), project)) < 0)
962 goto error;
964 return app;
965 error:
966 isl_map_free(app);
967 return NULL;
970 /* Check whether qc has any elements of length at least one
971 * with domain and/or range outside of dom and ran.
973 static isl_bool has_spurious_elements(__isl_keep isl_map *qc,
974 __isl_keep isl_set *dom, __isl_keep isl_set *ran)
976 isl_set *s;
977 isl_bool subset;
978 isl_size d;
980 d = isl_map_dim(qc, isl_dim_in);
981 if (d < 0 || !dom || !ran)
982 return isl_bool_error;
984 qc = isl_map_copy(qc);
985 qc = set_path_length(qc, 0, 1);
986 qc = isl_map_project_out(qc, isl_dim_in, d - 1, 1);
987 qc = isl_map_project_out(qc, isl_dim_out, d - 1, 1);
989 s = isl_map_domain(isl_map_copy(qc));
990 subset = isl_set_is_subset(s, dom);
991 isl_set_free(s);
992 if (subset < 0)
993 goto error;
994 if (!subset) {
995 isl_map_free(qc);
996 return isl_bool_true;
999 s = isl_map_range(qc);
1000 subset = isl_set_is_subset(s, ran);
1001 isl_set_free(s);
1003 return isl_bool_not(subset);
1004 error:
1005 isl_map_free(qc);
1006 return isl_bool_error;
1009 #define LEFT 2
1010 #define RIGHT 1
1012 /* For each basic map in "map", except i, check whether it combines
1013 * with the transitive closure that is reflexive on C combines
1014 * to the left and to the right.
1016 * In particular, if
1018 * dom map_j \subseteq C
1020 * then right[j] is set to 1. Otherwise, if
1022 * ran map_i \cap dom map_j = \emptyset
1024 * then right[j] is set to 0. Otherwise, composing to the right
1025 * is impossible.
1027 * Similar, for composing to the left, we have if
1029 * ran map_j \subseteq C
1031 * then left[j] is set to 1. Otherwise, if
1033 * dom map_i \cap ran map_j = \emptyset
1035 * then left[j] is set to 0. Otherwise, composing to the left
1036 * is impossible.
1038 * The return value is or'd with LEFT if composing to the left
1039 * is possible and with RIGHT if composing to the right is possible.
1041 static int composability(__isl_keep isl_set *C, int i,
1042 isl_set **dom, isl_set **ran, int *left, int *right,
1043 __isl_keep isl_map *map)
1045 int j;
1046 int ok;
1048 ok = LEFT | RIGHT;
1049 for (j = 0; j < map->n && ok; ++j) {
1050 isl_bool overlaps, subset;
1051 if (j == i)
1052 continue;
1054 if (ok & RIGHT) {
1055 if (!dom[j])
1056 dom[j] = isl_set_from_basic_set(
1057 isl_basic_map_domain(
1058 isl_basic_map_copy(map->p[j])));
1059 if (!dom[j])
1060 return -1;
1061 overlaps = isl_set_overlaps(ran[i], dom[j]);
1062 if (overlaps < 0)
1063 return -1;
1064 if (!overlaps)
1065 right[j] = 0;
1066 else {
1067 subset = isl_set_is_subset(dom[j], C);
1068 if (subset < 0)
1069 return -1;
1070 if (subset)
1071 right[j] = 1;
1072 else
1073 ok &= ~RIGHT;
1077 if (ok & LEFT) {
1078 if (!ran[j])
1079 ran[j] = isl_set_from_basic_set(
1080 isl_basic_map_range(
1081 isl_basic_map_copy(map->p[j])));
1082 if (!ran[j])
1083 return -1;
1084 overlaps = isl_set_overlaps(dom[i], ran[j]);
1085 if (overlaps < 0)
1086 return -1;
1087 if (!overlaps)
1088 left[j] = 0;
1089 else {
1090 subset = isl_set_is_subset(ran[j], C);
1091 if (subset < 0)
1092 return -1;
1093 if (subset)
1094 left[j] = 1;
1095 else
1096 ok &= ~LEFT;
1101 return ok;
1104 static __isl_give isl_map *anonymize(__isl_take isl_map *map)
1106 map = isl_map_reset(map, isl_dim_in);
1107 map = isl_map_reset(map, isl_dim_out);
1108 return map;
1111 /* Return a map that is a union of the basic maps in "map", except i,
1112 * composed to left and right with qc based on the entries of "left"
1113 * and "right".
1115 static __isl_give isl_map *compose(__isl_keep isl_map *map, int i,
1116 __isl_take isl_map *qc, int *left, int *right)
1118 int j;
1119 isl_map *comp;
1121 comp = isl_map_empty(isl_map_get_space(map));
1122 for (j = 0; j < map->n; ++j) {
1123 isl_map *map_j;
1125 if (j == i)
1126 continue;
1128 map_j = isl_map_from_basic_map(isl_basic_map_copy(map->p[j]));
1129 map_j = anonymize(map_j);
1130 if (left && left[j])
1131 map_j = isl_map_apply_range(map_j, isl_map_copy(qc));
1132 if (right && right[j])
1133 map_j = isl_map_apply_range(isl_map_copy(qc), map_j);
1134 comp = isl_map_union(comp, map_j);
1137 comp = isl_map_compute_divs(comp);
1138 comp = isl_map_coalesce(comp);
1140 isl_map_free(qc);
1142 return comp;
1145 /* Compute the transitive closure of "map" incrementally by
1146 * computing
1148 * map_i^+ \cup qc^+
1150 * or
1152 * map_i^+ \cup ((id \cup map_i^) \circ qc^+)
1154 * or
1156 * map_i^+ \cup (qc^+ \circ (id \cup map_i^))
1158 * depending on whether left or right are NULL.
1160 static __isl_give isl_map *compute_incremental(
1161 __isl_take isl_space *space, __isl_keep isl_map *map,
1162 int i, __isl_take isl_map *qc, int *left, int *right, isl_bool *exact)
1164 isl_map *map_i;
1165 isl_map *tc;
1166 isl_map *rtc = NULL;
1168 if (!map)
1169 goto error;
1170 isl_assert(map->ctx, left || right, goto error);
1172 map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
1173 tc = construct_projected_component(isl_space_copy(space), map_i,
1174 exact, 1);
1175 isl_map_free(map_i);
1177 if (*exact)
1178 qc = isl_map_transitive_closure(qc, exact);
1180 if (!*exact) {
1181 isl_space_free(space);
1182 isl_map_free(tc);
1183 isl_map_free(qc);
1184 return isl_map_universe(isl_map_get_space(map));
1187 if (!left || !right)
1188 rtc = isl_map_union(isl_map_copy(tc),
1189 isl_map_identity(isl_map_get_space(tc)));
1190 if (!right)
1191 qc = isl_map_apply_range(rtc, qc);
1192 if (!left)
1193 qc = isl_map_apply_range(qc, rtc);
1194 qc = isl_map_union(tc, qc);
1196 isl_space_free(space);
1198 return qc;
1199 error:
1200 isl_space_free(space);
1201 isl_map_free(qc);
1202 return NULL;
1205 /* Given a map "map", try to find a basic map such that
1206 * map^+ can be computed as
1208 * map^+ = map_i^+ \cup
1209 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1211 * with C the simple hull of the domain and range of the input map.
1212 * map_i^ \cup Id_C is computed by allowing the path lengths to be zero
1213 * and by intersecting domain and range with C.
1214 * Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
1215 * Also, we only use the incremental computation if all the transitive
1216 * closures are exact and if the number of basic maps in the union,
1217 * after computing the integer divisions, is smaller than the number
1218 * of basic maps in the input map.
1220 static isl_bool incremental_on_entire_domain(__isl_keep isl_space *space,
1221 __isl_keep isl_map *map,
1222 isl_set **dom, isl_set **ran, int *left, int *right,
1223 __isl_give isl_map **res)
1225 int i;
1226 isl_set *C;
1227 isl_size d;
1229 *res = NULL;
1231 d = isl_map_dim(map, isl_dim_in);
1232 if (d < 0)
1233 return isl_bool_error;
1235 C = isl_set_union(isl_map_domain(isl_map_copy(map)),
1236 isl_map_range(isl_map_copy(map)));
1237 C = isl_set_from_basic_set(isl_set_simple_hull(C));
1238 if (!C)
1239 return isl_bool_error;
1240 if (C->n != 1) {
1241 isl_set_free(C);
1242 return isl_bool_false;
1245 for (i = 0; i < map->n; ++i) {
1246 isl_map *qc;
1247 isl_bool exact_i;
1248 isl_bool spurious;
1249 int j;
1250 dom[i] = isl_set_from_basic_set(isl_basic_map_domain(
1251 isl_basic_map_copy(map->p[i])));
1252 ran[i] = isl_set_from_basic_set(isl_basic_map_range(
1253 isl_basic_map_copy(map->p[i])));
1254 qc = q_closure(isl_space_copy(space), isl_set_copy(C),
1255 map->p[i], &exact_i);
1256 if (!qc)
1257 goto error;
1258 if (!exact_i) {
1259 isl_map_free(qc);
1260 continue;
1262 spurious = has_spurious_elements(qc, dom[i], ran[i]);
1263 if (spurious) {
1264 isl_map_free(qc);
1265 if (spurious < 0)
1266 goto error;
1267 continue;
1269 qc = isl_map_project_out(qc, isl_dim_in, d, 1);
1270 qc = isl_map_project_out(qc, isl_dim_out, d, 1);
1271 qc = isl_map_compute_divs(qc);
1272 for (j = 0; j < map->n; ++j)
1273 left[j] = right[j] = 1;
1274 qc = compose(map, i, qc, left, right);
1275 if (!qc)
1276 goto error;
1277 if (qc->n >= map->n) {
1278 isl_map_free(qc);
1279 continue;
1281 *res = compute_incremental(isl_space_copy(space), map, i, qc,
1282 left, right, &exact_i);
1283 if (!*res)
1284 goto error;
1285 if (exact_i)
1286 break;
1287 isl_map_free(*res);
1288 *res = NULL;
1291 isl_set_free(C);
1293 return isl_bool_ok(*res != NULL);
1294 error:
1295 isl_set_free(C);
1296 return isl_bool_error;
1299 /* Try and compute the transitive closure of "map" as
1301 * map^+ = map_i^+ \cup
1302 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1304 * with C either the simple hull of the domain and range of the entire
1305 * map or the simple hull of domain and range of map_i.
1307 static __isl_give isl_map *incremental_closure(__isl_take isl_space *space,
1308 __isl_keep isl_map *map, isl_bool *exact, int project)
1310 int i;
1311 isl_set **dom = NULL;
1312 isl_set **ran = NULL;
1313 int *left = NULL;
1314 int *right = NULL;
1315 isl_set *C;
1316 isl_size d;
1317 isl_map *res = NULL;
1319 if (!project)
1320 return construct_projected_component(space, map, exact,
1321 project);
1323 if (!map)
1324 goto error;
1325 if (map->n <= 1)
1326 return construct_projected_component(space, map, exact,
1327 project);
1329 d = isl_map_dim(map, isl_dim_in);
1330 if (d < 0)
1331 goto error;
1333 dom = isl_calloc_array(map->ctx, isl_set *, map->n);
1334 ran = isl_calloc_array(map->ctx, isl_set *, map->n);
1335 left = isl_calloc_array(map->ctx, int, map->n);
1336 right = isl_calloc_array(map->ctx, int, map->n);
1337 if (!ran || !dom || !left || !right)
1338 goto error;
1340 if (incremental_on_entire_domain(space, map, dom, ran, left, right,
1341 &res) < 0)
1342 goto error;
1344 for (i = 0; !res && i < map->n; ++i) {
1345 isl_map *qc;
1346 int comp;
1347 isl_bool exact_i, spurious;
1348 if (!dom[i])
1349 dom[i] = isl_set_from_basic_set(
1350 isl_basic_map_domain(
1351 isl_basic_map_copy(map->p[i])));
1352 if (!dom[i])
1353 goto error;
1354 if (!ran[i])
1355 ran[i] = isl_set_from_basic_set(
1356 isl_basic_map_range(
1357 isl_basic_map_copy(map->p[i])));
1358 if (!ran[i])
1359 goto error;
1360 C = isl_set_union(isl_set_copy(dom[i]),
1361 isl_set_copy(ran[i]));
1362 C = isl_set_from_basic_set(isl_set_simple_hull(C));
1363 if (!C)
1364 goto error;
1365 if (C->n != 1) {
1366 isl_set_free(C);
1367 continue;
1369 comp = composability(C, i, dom, ran, left, right, map);
1370 if (!comp || comp < 0) {
1371 isl_set_free(C);
1372 if (comp < 0)
1373 goto error;
1374 continue;
1376 qc = q_closure(isl_space_copy(space), C, map->p[i], &exact_i);
1377 if (!qc)
1378 goto error;
1379 if (!exact_i) {
1380 isl_map_free(qc);
1381 continue;
1383 spurious = has_spurious_elements(qc, dom[i], ran[i]);
1384 if (spurious) {
1385 isl_map_free(qc);
1386 if (spurious < 0)
1387 goto error;
1388 continue;
1390 qc = isl_map_project_out(qc, isl_dim_in, d, 1);
1391 qc = isl_map_project_out(qc, isl_dim_out, d, 1);
1392 qc = isl_map_compute_divs(qc);
1393 qc = compose(map, i, qc, (comp & LEFT) ? left : NULL,
1394 (comp & RIGHT) ? right : NULL);
1395 if (!qc)
1396 goto error;
1397 if (qc->n >= map->n) {
1398 isl_map_free(qc);
1399 continue;
1401 res = compute_incremental(isl_space_copy(space), map, i, qc,
1402 (comp & LEFT) ? left : NULL,
1403 (comp & RIGHT) ? right : NULL, &exact_i);
1404 if (!res)
1405 goto error;
1406 if (exact_i)
1407 break;
1408 isl_map_free(res);
1409 res = NULL;
1412 for (i = 0; i < map->n; ++i) {
1413 isl_set_free(dom[i]);
1414 isl_set_free(ran[i]);
1416 free(dom);
1417 free(ran);
1418 free(left);
1419 free(right);
1421 if (res) {
1422 isl_space_free(space);
1423 return res;
1426 return construct_projected_component(space, map, exact, project);
1427 error:
1428 if (dom)
1429 for (i = 0; i < map->n; ++i)
1430 isl_set_free(dom[i]);
1431 free(dom);
1432 if (ran)
1433 for (i = 0; i < map->n; ++i)
1434 isl_set_free(ran[i]);
1435 free(ran);
1436 free(left);
1437 free(right);
1438 isl_space_free(space);
1439 return NULL;
1442 /* Given an array of sets "set", add "dom" at position "pos"
1443 * and search for elements at earlier positions that overlap with "dom".
1444 * If any can be found, then merge all of them, together with "dom", into
1445 * a single set and assign the union to the first in the array,
1446 * which becomes the new group leader for all groups involved in the merge.
1447 * During the search, we only consider group leaders, i.e., those with
1448 * group[i] = i, as the other sets have already been combined
1449 * with one of the group leaders.
1451 static int merge(isl_set **set, int *group, __isl_take isl_set *dom, int pos)
1453 int i;
1455 group[pos] = pos;
1456 set[pos] = isl_set_copy(dom);
1458 for (i = pos - 1; i >= 0; --i) {
1459 isl_bool o;
1461 if (group[i] != i)
1462 continue;
1464 o = isl_set_overlaps(set[i], dom);
1465 if (o < 0)
1466 goto error;
1467 if (!o)
1468 continue;
1470 set[i] = isl_set_union(set[i], set[group[pos]]);
1471 set[group[pos]] = NULL;
1472 if (!set[i])
1473 goto error;
1474 group[group[pos]] = i;
1475 group[pos] = i;
1478 isl_set_free(dom);
1479 return 0;
1480 error:
1481 isl_set_free(dom);
1482 return -1;
1485 /* Construct a map [x] -> [x+1], with parameters prescribed by "space".
1487 static __isl_give isl_map *increment(__isl_take isl_space *space)
1489 int k;
1490 isl_basic_map *bmap;
1491 isl_size total;
1493 space = isl_space_set_from_params(space);
1494 space = isl_space_add_dims(space, isl_dim_set, 1);
1495 space = isl_space_map_from_set(space);
1496 bmap = isl_basic_map_alloc_space(space, 0, 1, 0);
1497 total = isl_basic_map_dim(bmap, isl_dim_all);
1498 k = isl_basic_map_alloc_equality(bmap);
1499 if (total < 0 || k < 0)
1500 goto error;
1501 isl_seq_clr(bmap->eq[k], 1 + total);
1502 isl_int_set_si(bmap->eq[k][0], 1);
1503 isl_int_set_si(bmap->eq[k][isl_basic_map_offset(bmap, isl_dim_in)], 1);
1504 isl_int_set_si(bmap->eq[k][isl_basic_map_offset(bmap, isl_dim_out)], -1);
1505 return isl_map_from_basic_map(bmap);
1506 error:
1507 isl_basic_map_free(bmap);
1508 return NULL;
1511 /* Replace each entry in the n by n grid of maps by the cross product
1512 * with the relation { [i] -> [i + 1] }.
1514 static isl_stat add_length(__isl_keep isl_map *map, isl_map ***grid, int n)
1516 int i, j;
1517 isl_space *space;
1518 isl_map *step;
1520 space = isl_space_params(isl_map_get_space(map));
1521 step = increment(space);
1523 if (!step)
1524 return isl_stat_error;
1526 for (i = 0; i < n; ++i)
1527 for (j = 0; j < n; ++j)
1528 grid[i][j] = isl_map_product(grid[i][j],
1529 isl_map_copy(step));
1531 isl_map_free(step);
1533 return isl_stat_ok;
1536 /* The core of the Floyd-Warshall algorithm.
1537 * Updates the given n x x matrix of relations in place.
1539 * The algorithm iterates over all vertices. In each step, the whole
1540 * matrix is updated to include all paths that go to the current vertex,
1541 * possibly stay there a while (including passing through earlier vertices)
1542 * and then come back. At the start of each iteration, the diagonal
1543 * element corresponding to the current vertex is replaced by its
1544 * transitive closure to account for all indirect paths that stay
1545 * in the current vertex.
1547 static void floyd_warshall_iterate(isl_map ***grid, int n, isl_bool *exact)
1549 int r, p, q;
1551 for (r = 0; r < n; ++r) {
1552 isl_bool r_exact;
1553 int check = exact && *exact == isl_bool_true;
1554 grid[r][r] = isl_map_transitive_closure(grid[r][r],
1555 check ? &r_exact : NULL);
1556 if (check && !r_exact)
1557 *exact = isl_bool_false;
1559 for (p = 0; p < n; ++p)
1560 for (q = 0; q < n; ++q) {
1561 isl_map *loop;
1562 if (p == r && q == r)
1563 continue;
1564 loop = isl_map_apply_range(
1565 isl_map_copy(grid[p][r]),
1566 isl_map_copy(grid[r][q]));
1567 grid[p][q] = isl_map_union(grid[p][q], loop);
1568 loop = isl_map_apply_range(
1569 isl_map_copy(grid[p][r]),
1570 isl_map_apply_range(
1571 isl_map_copy(grid[r][r]),
1572 isl_map_copy(grid[r][q])));
1573 grid[p][q] = isl_map_union(grid[p][q], loop);
1574 grid[p][q] = isl_map_coalesce(grid[p][q]);
1579 /* Given a partition of the domains and ranges of the basic maps in "map",
1580 * apply the Floyd-Warshall algorithm with the elements in the partition
1581 * as vertices.
1583 * In particular, there are "n" elements in the partition and "group" is
1584 * an array of length 2 * map->n with entries in [0,n-1].
1586 * We first construct a matrix of relations based on the partition information,
1587 * apply Floyd-Warshall on this matrix of relations and then take the
1588 * union of all entries in the matrix as the final result.
1590 * If we are actually computing the power instead of the transitive closure,
1591 * i.e., when "project" is not set, then the result should have the
1592 * path lengths encoded as the difference between an extra pair of
1593 * coordinates. We therefore apply the nested transitive closures
1594 * to relations that include these lengths. In particular, we replace
1595 * the input relation by the cross product with the unit length relation
1596 * { [i] -> [i + 1] }.
1598 static __isl_give isl_map *floyd_warshall_with_groups(
1599 __isl_take isl_space *space, __isl_keep isl_map *map,
1600 isl_bool *exact, int project, int *group, int n)
1602 int i, j, k;
1603 isl_map ***grid = NULL;
1604 isl_map *app;
1606 if (!map)
1607 goto error;
1609 if (n == 1) {
1610 free(group);
1611 return incremental_closure(space, map, exact, project);
1614 grid = isl_calloc_array(map->ctx, isl_map **, n);
1615 if (!grid)
1616 goto error;
1617 for (i = 0; i < n; ++i) {
1618 grid[i] = isl_calloc_array(map->ctx, isl_map *, n);
1619 if (!grid[i])
1620 goto error;
1621 for (j = 0; j < n; ++j)
1622 grid[i][j] = isl_map_empty(isl_map_get_space(map));
1625 for (k = 0; k < map->n; ++k) {
1626 i = group[2 * k];
1627 j = group[2 * k + 1];
1628 grid[i][j] = isl_map_union(grid[i][j],
1629 isl_map_from_basic_map(
1630 isl_basic_map_copy(map->p[k])));
1633 if (!project && add_length(map, grid, n) < 0)
1634 goto error;
1636 floyd_warshall_iterate(grid, n, exact);
1638 app = isl_map_empty(isl_map_get_space(grid[0][0]));
1640 for (i = 0; i < n; ++i) {
1641 for (j = 0; j < n; ++j)
1642 app = isl_map_union(app, grid[i][j]);
1643 free(grid[i]);
1645 free(grid);
1647 free(group);
1648 isl_space_free(space);
1650 return app;
1651 error:
1652 if (grid)
1653 for (i = 0; i < n; ++i) {
1654 if (!grid[i])
1655 continue;
1656 for (j = 0; j < n; ++j)
1657 isl_map_free(grid[i][j]);
1658 free(grid[i]);
1660 free(grid);
1661 free(group);
1662 isl_space_free(space);
1663 return NULL;
1666 /* Partition the domains and ranges of the n basic relations in list
1667 * into disjoint cells.
1669 * To find the partition, we simply consider all of the domains
1670 * and ranges in turn and combine those that overlap.
1671 * "set" contains the partition elements and "group" indicates
1672 * to which partition element a given domain or range belongs.
1673 * The domain of basic map i corresponds to element 2 * i in these arrays,
1674 * while the domain corresponds to element 2 * i + 1.
1675 * During the construction group[k] is either equal to k,
1676 * in which case set[k] contains the union of all the domains and
1677 * ranges in the corresponding group, or is equal to some l < k,
1678 * with l another domain or range in the same group.
1680 static int *setup_groups(isl_ctx *ctx, __isl_keep isl_basic_map **list, int n,
1681 isl_set ***set, int *n_group)
1683 int i;
1684 int *group = NULL;
1685 int g;
1687 *set = isl_calloc_array(ctx, isl_set *, 2 * n);
1688 group = isl_alloc_array(ctx, int, 2 * n);
1690 if (!*set || !group)
1691 goto error;
1693 for (i = 0; i < n; ++i) {
1694 isl_set *dom;
1695 dom = isl_set_from_basic_set(isl_basic_map_domain(
1696 isl_basic_map_copy(list[i])));
1697 if (merge(*set, group, dom, 2 * i) < 0)
1698 goto error;
1699 dom = isl_set_from_basic_set(isl_basic_map_range(
1700 isl_basic_map_copy(list[i])));
1701 if (merge(*set, group, dom, 2 * i + 1) < 0)
1702 goto error;
1705 g = 0;
1706 for (i = 0; i < 2 * n; ++i)
1707 if (group[i] == i) {
1708 if (g != i) {
1709 (*set)[g] = (*set)[i];
1710 (*set)[i] = NULL;
1712 group[i] = g++;
1713 } else
1714 group[i] = group[group[i]];
1716 *n_group = g;
1718 return group;
1719 error:
1720 if (*set) {
1721 for (i = 0; i < 2 * n; ++i)
1722 isl_set_free((*set)[i]);
1723 free(*set);
1724 *set = NULL;
1726 free(group);
1727 return NULL;
1730 /* Check if the domains and ranges of the basic maps in "map" can
1731 * be partitioned, and if so, apply Floyd-Warshall on the elements
1732 * of the partition. Note that we also apply this algorithm
1733 * if we want to compute the power, i.e., when "project" is not set.
1734 * However, the results are unlikely to be exact since the recursive
1735 * calls inside the Floyd-Warshall algorithm typically result in
1736 * non-linear path lengths quite quickly.
1738 static __isl_give isl_map *floyd_warshall(__isl_take isl_space *space,
1739 __isl_keep isl_map *map, isl_bool *exact, int project)
1741 int i;
1742 isl_set **set = NULL;
1743 int *group = NULL;
1744 int n;
1746 if (!map)
1747 goto error;
1748 if (map->n <= 1)
1749 return incremental_closure(space, map, exact, project);
1751 group = setup_groups(map->ctx, map->p, map->n, &set, &n);
1752 if (!group)
1753 goto error;
1755 for (i = 0; i < 2 * map->n; ++i)
1756 isl_set_free(set[i]);
1758 free(set);
1760 return floyd_warshall_with_groups(space, map, exact, project, group, n);
1761 error:
1762 isl_space_free(space);
1763 return NULL;
1766 /* Structure for representing the nodes of the graph of which
1767 * strongly connected components are being computed.
1769 * list contains the actual nodes
1770 * check_closed is set if we may have used the fact that
1771 * a pair of basic maps can be interchanged
1773 struct isl_tc_follows_data {
1774 isl_basic_map **list;
1775 int check_closed;
1778 /* Check whether in the computation of the transitive closure
1779 * "list[i]" (R_1) should follow (or be part of the same component as)
1780 * "list[j]" (R_2).
1782 * That is check whether
1784 * R_1 \circ R_2
1786 * is a subset of
1788 * R_2 \circ R_1
1790 * If so, then there is no reason for R_1 to immediately follow R_2
1791 * in any path.
1793 * *check_closed is set if the subset relation holds while
1794 * R_1 \circ R_2 is not empty.
1796 static isl_bool basic_map_follows(int i, int j, void *user)
1798 struct isl_tc_follows_data *data = user;
1799 struct isl_map *map12 = NULL;
1800 struct isl_map *map21 = NULL;
1801 isl_bool applies, subset;
1803 applies = isl_basic_map_applies_range(data->list[j], data->list[i]);
1804 if (applies < 0)
1805 return isl_bool_error;
1806 if (!applies)
1807 return isl_bool_false;
1809 map21 = isl_map_from_basic_map(
1810 isl_basic_map_apply_range(
1811 isl_basic_map_copy(data->list[j]),
1812 isl_basic_map_copy(data->list[i])));
1813 subset = isl_map_is_empty(map21);
1814 if (subset < 0)
1815 goto error;
1816 if (subset) {
1817 isl_map_free(map21);
1818 return isl_bool_false;
1821 if (!isl_basic_map_is_transformation(data->list[i]) ||
1822 !isl_basic_map_is_transformation(data->list[j])) {
1823 isl_map_free(map21);
1824 return isl_bool_true;
1827 map12 = isl_map_from_basic_map(
1828 isl_basic_map_apply_range(
1829 isl_basic_map_copy(data->list[i]),
1830 isl_basic_map_copy(data->list[j])));
1832 subset = isl_map_is_subset(map21, map12);
1834 isl_map_free(map12);
1835 isl_map_free(map21);
1837 if (subset)
1838 data->check_closed = 1;
1840 return isl_bool_not(subset);
1841 error:
1842 isl_map_free(map21);
1843 return isl_bool_error;
1846 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
1847 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
1848 * construct a map that is an overapproximation of the map
1849 * that takes an element from the dom R \times Z to an
1850 * element from ran R \times Z, such that the first n coordinates of the
1851 * difference between them is a sum of differences between images
1852 * and pre-images in one of the R_i and such that the last coordinate
1853 * is equal to the number of steps taken.
1854 * If "project" is set, then these final coordinates are not included,
1855 * i.e., a relation of type Z^n -> Z^n is returned.
1856 * That is, let
1858 * \Delta_i = { y - x | (x, y) in R_i }
1860 * then the constructed map is an overapproximation of
1862 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1863 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
1864 * x in dom R and x + d in ran R }
1866 * or
1868 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1869 * d = (\sum_i k_i \delta_i) and
1870 * x in dom R and x + d in ran R }
1872 * if "project" is set.
1874 * We first split the map into strongly connected components, perform
1875 * the above on each component and then join the results in the correct
1876 * order, at each join also taking in the union of both arguments
1877 * to allow for paths that do not go through one of the two arguments.
1879 static __isl_give isl_map *construct_power_components(
1880 __isl_take isl_space *space, __isl_keep isl_map *map, isl_bool *exact,
1881 int project)
1883 int i, n, c;
1884 struct isl_map *path = NULL;
1885 struct isl_tc_follows_data data;
1886 struct isl_tarjan_graph *g = NULL;
1887 isl_bool *orig_exact;
1888 isl_bool local_exact;
1890 if (!map)
1891 goto error;
1892 if (map->n <= 1)
1893 return floyd_warshall(space, map, exact, project);
1895 data.list = map->p;
1896 data.check_closed = 0;
1897 g = isl_tarjan_graph_init(map->ctx, map->n, &basic_map_follows, &data);
1898 if (!g)
1899 goto error;
1901 orig_exact = exact;
1902 if (data.check_closed && !exact)
1903 exact = &local_exact;
1905 c = 0;
1906 i = 0;
1907 n = map->n;
1908 if (project)
1909 path = isl_map_empty(isl_map_get_space(map));
1910 else
1911 path = isl_map_empty(isl_space_copy(space));
1912 path = anonymize(path);
1913 while (n) {
1914 struct isl_map *comp;
1915 isl_map *path_comp, *path_comb;
1916 comp = isl_map_alloc_space(isl_map_get_space(map), n, 0);
1917 while (g->order[i] != -1) {
1918 comp = isl_map_add_basic_map(comp,
1919 isl_basic_map_copy(map->p[g->order[i]]));
1920 --n;
1921 ++i;
1923 path_comp = floyd_warshall(isl_space_copy(space),
1924 comp, exact, project);
1925 path_comp = anonymize(path_comp);
1926 path_comb = isl_map_apply_range(isl_map_copy(path),
1927 isl_map_copy(path_comp));
1928 path = isl_map_union(path, path_comp);
1929 path = isl_map_union(path, path_comb);
1930 isl_map_free(comp);
1931 ++i;
1932 ++c;
1935 if (c > 1 && data.check_closed && !*exact) {
1936 isl_bool closed;
1938 closed = isl_map_is_transitively_closed(path);
1939 if (closed < 0)
1940 goto error;
1941 if (!closed) {
1942 isl_tarjan_graph_free(g);
1943 isl_map_free(path);
1944 return floyd_warshall(space, map, orig_exact, project);
1948 isl_tarjan_graph_free(g);
1949 isl_space_free(space);
1951 return path;
1952 error:
1953 isl_tarjan_graph_free(g);
1954 isl_space_free(space);
1955 isl_map_free(path);
1956 return NULL;
1959 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
1960 * construct a map that is an overapproximation of the map
1961 * that takes an element from the space D to another
1962 * element from the same space, such that the difference between
1963 * them is a strictly positive sum of differences between images
1964 * and pre-images in one of the R_i.
1965 * The number of differences in the sum is equated to parameter "param".
1966 * That is, let
1968 * \Delta_i = { y - x | (x, y) in R_i }
1970 * then the constructed map is an overapproximation of
1972 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1973 * d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
1974 * or
1976 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1977 * d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
1979 * if "project" is set.
1981 * If "project" is not set, then
1982 * we construct an extended mapping with an extra coordinate
1983 * that indicates the number of steps taken. In particular,
1984 * the difference in the last coordinate is equal to the number
1985 * of steps taken to move from a domain element to the corresponding
1986 * image element(s).
1988 static __isl_give isl_map *construct_power(__isl_keep isl_map *map,
1989 isl_bool *exact, int project)
1991 struct isl_map *app = NULL;
1992 isl_space *space = NULL;
1994 if (!map)
1995 return NULL;
1997 space = isl_map_get_space(map);
1999 space = isl_space_add_dims(space, isl_dim_in, 1);
2000 space = isl_space_add_dims(space, isl_dim_out, 1);
2002 app = construct_power_components(isl_space_copy(space), map,
2003 exact, project);
2005 isl_space_free(space);
2007 return app;
2010 /* Compute the positive powers of "map", or an overapproximation.
2011 * If the result is exact, then *exact is set to 1.
2013 * If project is set, then we are actually interested in the transitive
2014 * closure, so we can use a more relaxed exactness check.
2015 * The lengths of the paths are also projected out instead of being
2016 * encoded as the difference between an extra pair of final coordinates.
2018 static __isl_give isl_map *map_power(__isl_take isl_map *map,
2019 isl_bool *exact, int project)
2021 struct isl_map *app = NULL;
2023 if (exact)
2024 *exact = isl_bool_true;
2026 if (isl_map_check_transformation(map) < 0)
2027 return isl_map_free(map);
2029 app = construct_power(map, exact, project);
2031 isl_map_free(map);
2032 return app;
2035 /* Compute the positive powers of "map", or an overapproximation.
2036 * The result maps the exponent to a nested copy of the corresponding power.
2037 * If the result is exact, then *exact is set to 1.
2038 * map_power constructs an extended relation with the path lengths
2039 * encoded as the difference between the final coordinates.
2040 * In the final step, this difference is equated to an extra parameter
2041 * and made positive. The extra coordinates are subsequently projected out
2042 * and the parameter is turned into the domain of the result.
2044 __isl_give isl_map *isl_map_power(__isl_take isl_map *map, isl_bool *exact)
2046 isl_space *target_space;
2047 isl_space *space;
2048 isl_map *diff;
2049 isl_size d;
2050 isl_size param;
2052 d = isl_map_dim(map, isl_dim_in);
2053 param = isl_map_dim(map, isl_dim_param);
2054 if (d < 0 || param < 0)
2055 return isl_map_free(map);
2057 map = isl_map_compute_divs(map);
2058 map = isl_map_coalesce(map);
2060 if (isl_map_plain_is_empty(map)) {
2061 map = isl_map_from_range(isl_map_wrap(map));
2062 map = isl_map_add_dims(map, isl_dim_in, 1);
2063 map = isl_map_set_dim_name(map, isl_dim_in, 0, "k");
2064 return map;
2067 target_space = isl_map_get_space(map);
2068 target_space = isl_space_from_range(isl_space_wrap(target_space));
2069 target_space = isl_space_add_dims(target_space, isl_dim_in, 1);
2070 target_space = isl_space_set_dim_name(target_space, isl_dim_in, 0, "k");
2072 map = map_power(map, exact, 0);
2074 map = isl_map_add_dims(map, isl_dim_param, 1);
2075 space = isl_map_get_space(map);
2076 diff = equate_parameter_to_length(space, param);
2077 map = isl_map_intersect(map, diff);
2078 map = isl_map_project_out(map, isl_dim_in, d, 1);
2079 map = isl_map_project_out(map, isl_dim_out, d, 1);
2080 map = isl_map_from_range(isl_map_wrap(map));
2081 map = isl_map_move_dims(map, isl_dim_in, 0, isl_dim_param, param, 1);
2083 map = isl_map_reset_space(map, target_space);
2085 return map;
2088 /* Compute a relation that maps each element in the range of the input
2089 * relation to the lengths of all paths composed of edges in the input
2090 * relation that end up in the given range element.
2091 * The result may be an overapproximation, in which case *exact is set to 0.
2092 * The resulting relation is very similar to the power relation.
2093 * The difference are that the domain has been projected out, the
2094 * range has become the domain and the exponent is the range instead
2095 * of a parameter.
2097 __isl_give isl_map *isl_map_reaching_path_lengths(__isl_take isl_map *map,
2098 isl_bool *exact)
2100 isl_space *space;
2101 isl_map *diff;
2102 isl_size d;
2103 isl_size param;
2105 d = isl_map_dim(map, isl_dim_in);
2106 param = isl_map_dim(map, isl_dim_param);
2107 if (d < 0 || param < 0)
2108 return isl_map_free(map);
2110 map = isl_map_compute_divs(map);
2111 map = isl_map_coalesce(map);
2113 if (isl_map_plain_is_empty(map)) {
2114 if (exact)
2115 *exact = isl_bool_true;
2116 map = isl_map_project_out(map, isl_dim_out, 0, d);
2117 map = isl_map_add_dims(map, isl_dim_out, 1);
2118 return map;
2121 map = map_power(map, exact, 0);
2123 map = isl_map_add_dims(map, isl_dim_param, 1);
2124 space = isl_map_get_space(map);
2125 diff = equate_parameter_to_length(space, param);
2126 map = isl_map_intersect(map, diff);
2127 map = isl_map_project_out(map, isl_dim_in, 0, d + 1);
2128 map = isl_map_project_out(map, isl_dim_out, d, 1);
2129 map = isl_map_reverse(map);
2130 map = isl_map_move_dims(map, isl_dim_out, 0, isl_dim_param, param, 1);
2132 return map;
2135 /* Given a map, compute the smallest superset of this map that is of the form
2137 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2139 * (where p ranges over the (non-parametric) dimensions),
2140 * compute the transitive closure of this map, i.e.,
2142 * { i -> j : exists k > 0:
2143 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2145 * and intersect domain and range of this transitive closure with
2146 * the given domain and range.
2148 * If with_id is set, then try to include as much of the identity mapping
2149 * as possible, by computing
2151 * { i -> j : exists k >= 0:
2152 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2154 * instead (i.e., allow k = 0).
2156 * In practice, we compute the difference set
2158 * delta = { j - i | i -> j in map },
2160 * look for stride constraint on the individual dimensions and compute
2161 * (constant) lower and upper bounds for each individual dimension,
2162 * adding a constraint for each bound not equal to infinity.
2164 static __isl_give isl_map *box_closure_on_domain(__isl_take isl_map *map,
2165 __isl_take isl_set *dom, __isl_take isl_set *ran, int with_id)
2167 int i;
2168 int k;
2169 unsigned d;
2170 unsigned nparam;
2171 unsigned total;
2172 isl_space *space;
2173 isl_set *delta;
2174 isl_map *app = NULL;
2175 isl_basic_set *aff = NULL;
2176 isl_basic_map *bmap = NULL;
2177 isl_vec *obj = NULL;
2178 isl_int opt;
2180 isl_int_init(opt);
2182 delta = isl_map_deltas(isl_map_copy(map));
2184 aff = isl_set_affine_hull(isl_set_copy(delta));
2185 if (!aff)
2186 goto error;
2187 space = isl_map_get_space(map);
2188 d = isl_space_dim(space, isl_dim_in);
2189 nparam = isl_space_dim(space, isl_dim_param);
2190 total = isl_space_dim(space, isl_dim_all);
2191 bmap = isl_basic_map_alloc_space(space,
2192 aff->n_div + 1, aff->n_div, 2 * d + 1);
2193 for (i = 0; i < aff->n_div + 1; ++i) {
2194 k = isl_basic_map_alloc_div(bmap);
2195 if (k < 0)
2196 goto error;
2197 isl_int_set_si(bmap->div[k][0], 0);
2199 for (i = 0; i < aff->n_eq; ++i) {
2200 if (!isl_basic_set_eq_is_stride(aff, i))
2201 continue;
2202 k = isl_basic_map_alloc_equality(bmap);
2203 if (k < 0)
2204 goto error;
2205 isl_seq_clr(bmap->eq[k], 1 + nparam);
2206 isl_seq_cpy(bmap->eq[k] + 1 + nparam + d,
2207 aff->eq[i] + 1 + nparam, d);
2208 isl_seq_neg(bmap->eq[k] + 1 + nparam,
2209 aff->eq[i] + 1 + nparam, d);
2210 isl_seq_cpy(bmap->eq[k] + 1 + nparam + 2 * d,
2211 aff->eq[i] + 1 + nparam + d, aff->n_div);
2212 isl_int_set_si(bmap->eq[k][1 + total + aff->n_div], 0);
2214 obj = isl_vec_alloc(map->ctx, 1 + nparam + d);
2215 if (!obj)
2216 goto error;
2217 isl_seq_clr(obj->el, 1 + nparam + d);
2218 for (i = 0; i < d; ++ i) {
2219 enum isl_lp_result res;
2221 isl_int_set_si(obj->el[1 + nparam + i], 1);
2223 res = isl_set_solve_lp(delta, 0, obj->el, map->ctx->one, &opt,
2224 NULL, NULL);
2225 if (res == isl_lp_error)
2226 goto error;
2227 if (res == isl_lp_ok) {
2228 k = isl_basic_map_alloc_inequality(bmap);
2229 if (k < 0)
2230 goto error;
2231 isl_seq_clr(bmap->ineq[k],
2232 1 + nparam + 2 * d + bmap->n_div);
2233 isl_int_set_si(bmap->ineq[k][1 + nparam + i], -1);
2234 isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], 1);
2235 isl_int_neg(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
2238 res = isl_set_solve_lp(delta, 1, obj->el, map->ctx->one, &opt,
2239 NULL, NULL);
2240 if (res == isl_lp_error)
2241 goto error;
2242 if (res == isl_lp_ok) {
2243 k = isl_basic_map_alloc_inequality(bmap);
2244 if (k < 0)
2245 goto error;
2246 isl_seq_clr(bmap->ineq[k],
2247 1 + nparam + 2 * d + bmap->n_div);
2248 isl_int_set_si(bmap->ineq[k][1 + nparam + i], 1);
2249 isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], -1);
2250 isl_int_set(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
2253 isl_int_set_si(obj->el[1 + nparam + i], 0);
2255 k = isl_basic_map_alloc_inequality(bmap);
2256 if (k < 0)
2257 goto error;
2258 isl_seq_clr(bmap->ineq[k],
2259 1 + nparam + 2 * d + bmap->n_div);
2260 if (!with_id)
2261 isl_int_set_si(bmap->ineq[k][0], -1);
2262 isl_int_set_si(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], 1);
2264 app = isl_map_from_domain_and_range(dom, ran);
2266 isl_vec_free(obj);
2267 isl_basic_set_free(aff);
2268 isl_map_free(map);
2269 bmap = isl_basic_map_finalize(bmap);
2270 isl_set_free(delta);
2271 isl_int_clear(opt);
2273 map = isl_map_from_basic_map(bmap);
2274 map = isl_map_intersect(map, app);
2276 return map;
2277 error:
2278 isl_vec_free(obj);
2279 isl_basic_map_free(bmap);
2280 isl_basic_set_free(aff);
2281 isl_set_free(dom);
2282 isl_set_free(ran);
2283 isl_map_free(map);
2284 isl_set_free(delta);
2285 isl_int_clear(opt);
2286 return NULL;
2289 /* Given a map, compute the smallest superset of this map that is of the form
2291 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2293 * (where p ranges over the (non-parametric) dimensions),
2294 * compute the transitive closure of this map, i.e.,
2296 * { i -> j : exists k > 0:
2297 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2299 * and intersect domain and range of this transitive closure with
2300 * domain and range of the original map.
2302 static __isl_give isl_map *box_closure(__isl_take isl_map *map)
2304 isl_set *domain;
2305 isl_set *range;
2307 domain = isl_map_domain(isl_map_copy(map));
2308 domain = isl_set_coalesce(domain);
2309 range = isl_map_range(isl_map_copy(map));
2310 range = isl_set_coalesce(range);
2312 return box_closure_on_domain(map, domain, range, 0);
2315 /* Given a map, compute the smallest superset of this map that is of the form
2317 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2319 * (where p ranges over the (non-parametric) dimensions),
2320 * compute the transitive and partially reflexive closure of this map, i.e.,
2322 * { i -> j : exists k >= 0:
2323 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2325 * and intersect domain and range of this transitive closure with
2326 * the given domain.
2328 static __isl_give isl_map *box_closure_with_identity(__isl_take isl_map *map,
2329 __isl_take isl_set *dom)
2331 return box_closure_on_domain(map, dom, isl_set_copy(dom), 1);
2334 /* Check whether app is the transitive closure of map.
2335 * In particular, check that app is acyclic and, if so,
2336 * check that
2338 * app \subset (map \cup (map \circ app))
2340 static isl_bool check_exactness_omega(__isl_keep isl_map *map,
2341 __isl_keep isl_map *app)
2343 isl_set *delta;
2344 int i;
2345 isl_bool is_empty, is_exact;
2346 isl_size d;
2347 isl_map *test;
2349 delta = isl_map_deltas(isl_map_copy(app));
2350 d = isl_set_dim(delta, isl_dim_set);
2351 if (d < 0)
2352 delta = isl_set_free(delta);
2353 for (i = 0; i < d; ++i)
2354 delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
2355 is_empty = isl_set_is_empty(delta);
2356 isl_set_free(delta);
2357 if (is_empty < 0 || !is_empty)
2358 return is_empty;
2360 test = isl_map_apply_range(isl_map_copy(app), isl_map_copy(map));
2361 test = isl_map_union(test, isl_map_copy(map));
2362 is_exact = isl_map_is_subset(app, test);
2363 isl_map_free(test);
2365 return is_exact;
2368 /* Check if basic map M_i can be combined with all the other
2369 * basic maps such that
2371 * (\cup_j M_j)^+
2373 * can be computed as
2375 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2377 * In particular, check if we can compute a compact representation
2378 * of
2380 * M_i^* \circ M_j \circ M_i^*
2382 * for each j != i.
2383 * Let M_i^? be an extension of M_i^+ that allows paths
2384 * of length zero, i.e., the result of box_closure(., 1).
2385 * The criterion, as proposed by Kelly et al., is that
2386 * id = M_i^? - M_i^+ can be represented as a basic map
2387 * and that
2389 * id \circ M_j \circ id = M_j
2391 * for each j != i.
2393 * If this function returns 1, then tc and qc are set to
2394 * M_i^+ and M_i^?, respectively.
2396 static int can_be_split_off(__isl_keep isl_map *map, int i,
2397 __isl_give isl_map **tc, __isl_give isl_map **qc)
2399 isl_map *map_i, *id = NULL;
2400 int j = -1;
2401 isl_set *C;
2403 *tc = NULL;
2404 *qc = NULL;
2406 C = isl_set_union(isl_map_domain(isl_map_copy(map)),
2407 isl_map_range(isl_map_copy(map)));
2408 C = isl_set_from_basic_set(isl_set_simple_hull(C));
2409 if (!C)
2410 goto error;
2412 map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
2413 *tc = box_closure(isl_map_copy(map_i));
2414 *qc = box_closure_with_identity(map_i, C);
2415 id = isl_map_subtract(isl_map_copy(*qc), isl_map_copy(*tc));
2417 if (!id || !*qc)
2418 goto error;
2419 if (id->n != 1 || (*qc)->n != 1)
2420 goto done;
2422 for (j = 0; j < map->n; ++j) {
2423 isl_map *map_j, *test;
2424 int is_ok;
2426 if (i == j)
2427 continue;
2428 map_j = isl_map_from_basic_map(
2429 isl_basic_map_copy(map->p[j]));
2430 test = isl_map_apply_range(isl_map_copy(id),
2431 isl_map_copy(map_j));
2432 test = isl_map_apply_range(test, isl_map_copy(id));
2433 is_ok = isl_map_is_equal(test, map_j);
2434 isl_map_free(map_j);
2435 isl_map_free(test);
2436 if (is_ok < 0)
2437 goto error;
2438 if (!is_ok)
2439 break;
2442 done:
2443 isl_map_free(id);
2444 if (j == map->n)
2445 return 1;
2447 isl_map_free(*qc);
2448 isl_map_free(*tc);
2449 *qc = NULL;
2450 *tc = NULL;
2452 return 0;
2453 error:
2454 isl_map_free(id);
2455 isl_map_free(*qc);
2456 isl_map_free(*tc);
2457 *qc = NULL;
2458 *tc = NULL;
2459 return -1;
2462 static __isl_give isl_map *box_closure_with_check(__isl_take isl_map *map,
2463 isl_bool *exact)
2465 isl_map *app;
2467 app = box_closure(isl_map_copy(map));
2468 if (exact) {
2469 isl_bool is_exact = check_exactness_omega(map, app);
2471 if (is_exact < 0)
2472 app = isl_map_free(app);
2473 else
2474 *exact = is_exact;
2477 isl_map_free(map);
2478 return app;
2481 /* Compute an overapproximation of the transitive closure of "map"
2482 * using a variation of the algorithm from
2483 * "Transitive Closure of Infinite Graphs and its Applications"
2484 * by Kelly et al.
2486 * We first check whether we can can split of any basic map M_i and
2487 * compute
2489 * (\cup_j M_j)^+
2491 * as
2493 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2495 * using a recursive call on the remaining map.
2497 * If not, we simply call box_closure on the whole map.
2499 static __isl_give isl_map *transitive_closure_omega(__isl_take isl_map *map,
2500 isl_bool *exact)
2502 int i, j;
2503 isl_bool exact_i;
2504 isl_map *app;
2506 if (!map)
2507 return NULL;
2508 if (map->n == 1)
2509 return box_closure_with_check(map, exact);
2511 for (i = 0; i < map->n; ++i) {
2512 int ok;
2513 isl_map *qc, *tc;
2514 ok = can_be_split_off(map, i, &tc, &qc);
2515 if (ok < 0)
2516 goto error;
2517 if (!ok)
2518 continue;
2520 app = isl_map_alloc_space(isl_map_get_space(map), map->n - 1, 0);
2522 for (j = 0; j < map->n; ++j) {
2523 if (j == i)
2524 continue;
2525 app = isl_map_add_basic_map(app,
2526 isl_basic_map_copy(map->p[j]));
2529 app = isl_map_apply_range(isl_map_copy(qc), app);
2530 app = isl_map_apply_range(app, qc);
2532 app = isl_map_union(tc, transitive_closure_omega(app, NULL));
2533 exact_i = check_exactness_omega(map, app);
2534 if (exact_i == isl_bool_true) {
2535 if (exact)
2536 *exact = exact_i;
2537 isl_map_free(map);
2538 return app;
2540 isl_map_free(app);
2541 if (exact_i < 0)
2542 goto error;
2545 return box_closure_with_check(map, exact);
2546 error:
2547 isl_map_free(map);
2548 return NULL;
2551 /* Compute the transitive closure of "map", or an overapproximation.
2552 * If the result is exact, then *exact is set to 1.
2553 * Simply use map_power to compute the powers of map, but tell
2554 * it to project out the lengths of the paths instead of equating
2555 * the length to a parameter.
2557 __isl_give isl_map *isl_map_transitive_closure(__isl_take isl_map *map,
2558 isl_bool *exact)
2560 isl_space *target_dim;
2561 isl_bool closed;
2563 if (!map)
2564 goto error;
2566 if (map->ctx->opt->closure == ISL_CLOSURE_BOX)
2567 return transitive_closure_omega(map, exact);
2569 map = isl_map_compute_divs(map);
2570 map = isl_map_coalesce(map);
2571 closed = isl_map_is_transitively_closed(map);
2572 if (closed < 0)
2573 goto error;
2574 if (closed) {
2575 if (exact)
2576 *exact = isl_bool_true;
2577 return map;
2580 target_dim = isl_map_get_space(map);
2581 map = map_power(map, exact, 1);
2582 map = isl_map_reset_space(map, target_dim);
2584 return map;
2585 error:
2586 isl_map_free(map);
2587 return NULL;
2590 static isl_stat inc_count(__isl_take isl_map *map, void *user)
2592 int *n = user;
2594 *n += map->n;
2596 isl_map_free(map);
2598 return isl_stat_ok;
2601 static isl_stat collect_basic_map(__isl_take isl_map *map, void *user)
2603 int i;
2604 isl_basic_map ***next = user;
2606 for (i = 0; i < map->n; ++i) {
2607 **next = isl_basic_map_copy(map->p[i]);
2608 if (!**next)
2609 goto error;
2610 (*next)++;
2613 isl_map_free(map);
2614 return isl_stat_ok;
2615 error:
2616 isl_map_free(map);
2617 return isl_stat_error;
2620 /* Perform Floyd-Warshall on the given list of basic relations.
2621 * The basic relations may live in different dimensions,
2622 * but basic relations that get assigned to the diagonal of the
2623 * grid have domains and ranges of the same dimension and so
2624 * the standard algorithm can be used because the nested transitive
2625 * closures are only applied to diagonal elements and because all
2626 * compositions are performed on relations with compatible domains and ranges.
2628 static __isl_give isl_union_map *union_floyd_warshall_on_list(isl_ctx *ctx,
2629 __isl_keep isl_basic_map **list, int n, isl_bool *exact)
2631 int i, j, k;
2632 int n_group;
2633 int *group = NULL;
2634 isl_set **set = NULL;
2635 isl_map ***grid = NULL;
2636 isl_union_map *app;
2638 group = setup_groups(ctx, list, n, &set, &n_group);
2639 if (!group)
2640 goto error;
2642 grid = isl_calloc_array(ctx, isl_map **, n_group);
2643 if (!grid)
2644 goto error;
2645 for (i = 0; i < n_group; ++i) {
2646 grid[i] = isl_calloc_array(ctx, isl_map *, n_group);
2647 if (!grid[i])
2648 goto error;
2649 for (j = 0; j < n_group; ++j) {
2650 isl_space *space1, *space2, *space;
2651 space1 = isl_space_reverse(isl_set_get_space(set[i]));
2652 space2 = isl_set_get_space(set[j]);
2653 space = isl_space_join(space1, space2);
2654 grid[i][j] = isl_map_empty(space);
2658 for (k = 0; k < n; ++k) {
2659 i = group[2 * k];
2660 j = group[2 * k + 1];
2661 grid[i][j] = isl_map_union(grid[i][j],
2662 isl_map_from_basic_map(
2663 isl_basic_map_copy(list[k])));
2666 floyd_warshall_iterate(grid, n_group, exact);
2668 app = isl_union_map_empty(isl_map_get_space(grid[0][0]));
2670 for (i = 0; i < n_group; ++i) {
2671 for (j = 0; j < n_group; ++j)
2672 app = isl_union_map_add_map(app, grid[i][j]);
2673 free(grid[i]);
2675 free(grid);
2677 for (i = 0; i < 2 * n; ++i)
2678 isl_set_free(set[i]);
2679 free(set);
2681 free(group);
2682 return app;
2683 error:
2684 if (grid)
2685 for (i = 0; i < n_group; ++i) {
2686 if (!grid[i])
2687 continue;
2688 for (j = 0; j < n_group; ++j)
2689 isl_map_free(grid[i][j]);
2690 free(grid[i]);
2692 free(grid);
2693 if (set) {
2694 for (i = 0; i < 2 * n; ++i)
2695 isl_set_free(set[i]);
2696 free(set);
2698 free(group);
2699 return NULL;
2702 /* Perform Floyd-Warshall on the given union relation.
2703 * The implementation is very similar to that for non-unions.
2704 * The main difference is that it is applied unconditionally.
2705 * We first extract a list of basic maps from the union map
2706 * and then perform the algorithm on this list.
2708 static __isl_give isl_union_map *union_floyd_warshall(
2709 __isl_take isl_union_map *umap, isl_bool *exact)
2711 int i, n;
2712 isl_ctx *ctx;
2713 isl_basic_map **list = NULL;
2714 isl_basic_map **next;
2715 isl_union_map *res;
2717 n = 0;
2718 if (isl_union_map_foreach_map(umap, inc_count, &n) < 0)
2719 goto error;
2721 ctx = isl_union_map_get_ctx(umap);
2722 list = isl_calloc_array(ctx, isl_basic_map *, n);
2723 if (!list)
2724 goto error;
2726 next = list;
2727 if (isl_union_map_foreach_map(umap, collect_basic_map, &next) < 0)
2728 goto error;
2730 res = union_floyd_warshall_on_list(ctx, list, n, exact);
2732 if (list) {
2733 for (i = 0; i < n; ++i)
2734 isl_basic_map_free(list[i]);
2735 free(list);
2738 isl_union_map_free(umap);
2739 return res;
2740 error:
2741 if (list) {
2742 for (i = 0; i < n; ++i)
2743 isl_basic_map_free(list[i]);
2744 free(list);
2746 isl_union_map_free(umap);
2747 return NULL;
2750 /* Decompose the give union relation into strongly connected components.
2751 * The implementation is essentially the same as that of
2752 * construct_power_components with the major difference that all
2753 * operations are performed on union maps.
2755 static __isl_give isl_union_map *union_components(
2756 __isl_take isl_union_map *umap, isl_bool *exact)
2758 int i;
2759 int n;
2760 isl_ctx *ctx;
2761 isl_basic_map **list = NULL;
2762 isl_basic_map **next;
2763 isl_union_map *path = NULL;
2764 struct isl_tc_follows_data data;
2765 struct isl_tarjan_graph *g = NULL;
2766 int c, l;
2767 int recheck = 0;
2769 n = 0;
2770 if (isl_union_map_foreach_map(umap, inc_count, &n) < 0)
2771 goto error;
2773 if (n == 0)
2774 return umap;
2775 if (n <= 1)
2776 return union_floyd_warshall(umap, exact);
2778 ctx = isl_union_map_get_ctx(umap);
2779 list = isl_calloc_array(ctx, isl_basic_map *, n);
2780 if (!list)
2781 goto error;
2783 next = list;
2784 if (isl_union_map_foreach_map(umap, collect_basic_map, &next) < 0)
2785 goto error;
2787 data.list = list;
2788 data.check_closed = 0;
2789 g = isl_tarjan_graph_init(ctx, n, &basic_map_follows, &data);
2790 if (!g)
2791 goto error;
2793 c = 0;
2794 i = 0;
2795 l = n;
2796 path = isl_union_map_empty(isl_union_map_get_space(umap));
2797 while (l) {
2798 isl_union_map *comp;
2799 isl_union_map *path_comp, *path_comb;
2800 comp = isl_union_map_empty(isl_union_map_get_space(umap));
2801 while (g->order[i] != -1) {
2802 comp = isl_union_map_add_map(comp,
2803 isl_map_from_basic_map(
2804 isl_basic_map_copy(list[g->order[i]])));
2805 --l;
2806 ++i;
2808 path_comp = union_floyd_warshall(comp, exact);
2809 path_comb = isl_union_map_apply_range(isl_union_map_copy(path),
2810 isl_union_map_copy(path_comp));
2811 path = isl_union_map_union(path, path_comp);
2812 path = isl_union_map_union(path, path_comb);
2813 ++i;
2814 ++c;
2817 if (c > 1 && data.check_closed && !*exact) {
2818 isl_bool closed;
2820 closed = isl_union_map_is_transitively_closed(path);
2821 if (closed < 0)
2822 goto error;
2823 recheck = !closed;
2826 isl_tarjan_graph_free(g);
2828 for (i = 0; i < n; ++i)
2829 isl_basic_map_free(list[i]);
2830 free(list);
2832 if (recheck) {
2833 isl_union_map_free(path);
2834 return union_floyd_warshall(umap, exact);
2837 isl_union_map_free(umap);
2839 return path;
2840 error:
2841 isl_tarjan_graph_free(g);
2842 if (list) {
2843 for (i = 0; i < n; ++i)
2844 isl_basic_map_free(list[i]);
2845 free(list);
2847 isl_union_map_free(umap);
2848 isl_union_map_free(path);
2849 return NULL;
2852 /* Compute the transitive closure of "umap", or an overapproximation.
2853 * If the result is exact, then *exact is set to 1.
2855 __isl_give isl_union_map *isl_union_map_transitive_closure(
2856 __isl_take isl_union_map *umap, isl_bool *exact)
2858 isl_bool closed;
2860 if (!umap)
2861 return NULL;
2863 if (exact)
2864 *exact = isl_bool_true;
2866 umap = isl_union_map_compute_divs(umap);
2867 umap = isl_union_map_coalesce(umap);
2868 closed = isl_union_map_is_transitively_closed(umap);
2869 if (closed < 0)
2870 goto error;
2871 if (closed)
2872 return umap;
2873 umap = union_components(umap, exact);
2874 return umap;
2875 error:
2876 isl_union_map_free(umap);
2877 return NULL;
2880 struct isl_union_power {
2881 isl_union_map *pow;
2882 isl_bool *exact;
2885 static isl_stat power(__isl_take isl_map *map, void *user)
2887 struct isl_union_power *up = user;
2889 map = isl_map_power(map, up->exact);
2890 up->pow = isl_union_map_from_map(map);
2892 return isl_stat_error;
2895 /* Construct a map [[x]->[y]] -> [y-x], with parameters prescribed by "space".
2897 static __isl_give isl_union_map *deltas_map(__isl_take isl_space *space)
2899 isl_basic_map *bmap;
2901 space = isl_space_add_dims(space, isl_dim_in, 1);
2902 space = isl_space_add_dims(space, isl_dim_out, 1);
2903 bmap = isl_basic_map_universe(space);
2904 bmap = isl_basic_map_deltas_map(bmap);
2906 return isl_union_map_from_map(isl_map_from_basic_map(bmap));
2909 /* Compute the positive powers of "map", or an overapproximation.
2910 * The result maps the exponent to a nested copy of the corresponding power.
2911 * If the result is exact, then *exact is set to 1.
2913 __isl_give isl_union_map *isl_union_map_power(__isl_take isl_union_map *umap,
2914 isl_bool *exact)
2916 isl_size n;
2917 isl_union_map *inc;
2918 isl_union_map *dm;
2920 n = isl_union_map_n_map(umap);
2921 if (n < 0)
2922 return isl_union_map_free(umap);
2923 if (n == 0)
2924 return umap;
2925 if (n == 1) {
2926 struct isl_union_power up = { NULL, exact };
2927 isl_union_map_foreach_map(umap, &power, &up);
2928 isl_union_map_free(umap);
2929 return up.pow;
2931 inc = isl_union_map_from_map(increment(isl_union_map_get_space(umap)));
2932 umap = isl_union_map_product(inc, umap);
2933 umap = isl_union_map_transitive_closure(umap, exact);
2934 umap = isl_union_map_zip(umap);
2935 dm = deltas_map(isl_union_map_get_space(umap));
2936 umap = isl_union_map_apply_domain(umap, dm);
2938 return umap;
2941 #undef TYPE
2942 #define TYPE isl_map
2943 #include "isl_power_templ.c"
2945 #undef TYPE
2946 #define TYPE isl_union_map
2947 #include "isl_power_templ.c"