2 * Copyright 2010 INRIA Saclay
4 * Use of this software is governed by the GNU LGPLv2.1 license
6 * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
7 * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
12 #include "isl_map_private.h"
16 /* Given a map that represents a path with the length of the path
17 * encoded as the difference between the last output coordindate
18 * and the last input coordinate, set this length to either
19 * exactly "length" (if "exactly" is set) or at least "length"
20 * (if "exactly" is not set).
22 static __isl_give isl_map *set_path_length(__isl_take isl_map *map,
23 int exactly, int length)
26 struct isl_basic_map *bmap;
35 dim = isl_map_get_dim(map);
36 d = isl_dim_size(dim, isl_dim_in);
37 nparam = isl_dim_size(dim, isl_dim_param);
38 bmap = isl_basic_map_alloc_dim(dim, 0, 1, 1);
40 k = isl_basic_map_alloc_equality(bmap);
43 k = isl_basic_map_alloc_inequality(bmap);
48 isl_seq_clr(c, 1 + isl_basic_map_total_dim(bmap));
49 isl_int_set_si(c[0], -length);
50 isl_int_set_si(c[1 + nparam + d - 1], -1);
51 isl_int_set_si(c[1 + nparam + d + d - 1], 1);
53 bmap = isl_basic_map_finalize(bmap);
54 map = isl_map_intersect(map, isl_map_from_basic_map(bmap));
58 isl_basic_map_free(bmap);
63 /* Check whether the overapproximation of the power of "map" is exactly
64 * the power of "map". Let R be "map" and A_k the overapproximation.
65 * The approximation is exact if
68 * A_k = A_{k-1} \circ R k >= 2
70 * Since A_k is known to be an overapproximation, we only need to check
73 * A_k \subset A_{k-1} \circ R k >= 2
75 * In practice, "app" has an extra input and output coordinate
76 * to encode the length of the path. So, we first need to add
77 * this coordinate to "map" and set the length of the path to
80 static int check_power_exactness(__isl_take isl_map *map,
81 __isl_take isl_map *app)
87 map = isl_map_add(map, isl_dim_in, 1);
88 map = isl_map_add(map, isl_dim_out, 1);
89 map = set_path_length(map, 1, 1);
91 app_1 = set_path_length(isl_map_copy(app), 1, 1);
93 exact = isl_map_is_subset(app_1, map);
96 if (!exact || exact < 0) {
102 app_1 = set_path_length(isl_map_copy(app), 0, 1);
103 app_2 = set_path_length(app, 0, 2);
104 app_1 = isl_map_apply_range(map, app_1);
106 exact = isl_map_is_subset(app_2, app_1);
114 /* Check whether the overapproximation of the power of "map" is exactly
115 * the power of "map", possibly after projecting out the power (if "project"
118 * If "project" is set and if "steps" can only result in acyclic paths,
121 * A = R \cup (A \circ R)
123 * where A is the overapproximation with the power projected out, i.e.,
124 * an overapproximation of the transitive closure.
125 * More specifically, since A is known to be an overapproximation, we check
127 * A \subset R \cup (A \circ R)
129 * Otherwise, we check if the power is exact.
131 * Note that "app" has an extra input and output coordinate to encode
132 * the length of the part. If we are only interested in the transitive
133 * closure, then we can simply project out these coordinates first.
135 static int check_exactness(__isl_take isl_map *map, __isl_take isl_map *app,
143 return check_power_exactness(map, app);
145 d = isl_map_dim(map, isl_dim_in);
146 app = set_path_length(app, 0, 1);
147 app = isl_map_project_out(app, isl_dim_in, d, 1);
148 app = isl_map_project_out(app, isl_dim_out, d, 1);
150 test = isl_map_apply_range(isl_map_copy(map), isl_map_copy(app));
151 test = isl_map_union(test, isl_map_copy(map));
153 exact = isl_map_is_subset(app, test);
168 * The transitive closure implementation is based on the paper
169 * "Computing the Transitive Closure of a Union of Affine Integer
170 * Tuple Relations" by Anna Beletska, Denis Barthou, Wlodzimierz Bielecki and
174 /* Given a set of n offsets v_i (the rows of "steps"), construct a relation
175 * of the given dimension specification (Z^{n+1} -> Z^{n+1})
176 * that maps an element x to any element that can be reached
177 * by taking a non-negative number of steps along any of
178 * the extended offsets v'_i = [v_i 1].
181 * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v'_i }
183 * For any element in this relation, the number of steps taken
184 * is equal to the difference in the final coordinates.
186 static __isl_give isl_map *path_along_steps(__isl_take isl_dim *dim,
187 __isl_keep isl_mat *steps)
190 struct isl_basic_map *path = NULL;
198 d = isl_dim_size(dim, isl_dim_in);
200 nparam = isl_dim_size(dim, isl_dim_param);
202 path = isl_basic_map_alloc_dim(isl_dim_copy(dim), n, d, n);
204 for (i = 0; i < n; ++i) {
205 k = isl_basic_map_alloc_div(path);
208 isl_assert(steps->ctx, i == k, goto error);
209 isl_int_set_si(path->div[k][0], 0);
212 for (i = 0; i < d; ++i) {
213 k = isl_basic_map_alloc_equality(path);
216 isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
217 isl_int_set_si(path->eq[k][1 + nparam + i], 1);
218 isl_int_set_si(path->eq[k][1 + nparam + d + i], -1);
220 for (j = 0; j < n; ++j)
221 isl_int_set_si(path->eq[k][1 + nparam + 2 * d + j], 1);
223 for (j = 0; j < n; ++j)
224 isl_int_set(path->eq[k][1 + nparam + 2 * d + j],
228 for (i = 0; i < n; ++i) {
229 k = isl_basic_map_alloc_inequality(path);
232 isl_seq_clr(path->ineq[k], 1 + isl_basic_map_total_dim(path));
233 isl_int_set_si(path->ineq[k][1 + nparam + 2 * d + i], 1);
238 path = isl_basic_map_simplify(path);
239 path = isl_basic_map_finalize(path);
240 return isl_map_from_basic_map(path);
243 isl_basic_map_free(path);
252 /* Return PURE_PARAM if only the coefficients of the parameters are non-zero.
253 * Return PURE_VAR if only the coefficients of the set variables are non-zero.
254 * Return MIXED if only the coefficients of the parameters and the set
255 * variables are non-zero and if moreover the parametric constant
256 * can never attain positive values.
257 * Return IMPURE otherwise.
259 static int purity(__isl_keep isl_basic_set *bset, isl_int *c, int *div_purity,
270 n_div = isl_basic_set_dim(bset, isl_dim_div);
271 d = isl_basic_set_dim(bset, isl_dim_set);
272 nparam = isl_basic_set_dim(bset, isl_dim_param);
274 for (i = 0; i < n_div; ++i) {
275 if (isl_int_is_zero(c[1 + nparam + d + i]))
277 switch (div_purity[i]) {
278 case PURE_PARAM: p = 1; break;
279 case PURE_VAR: v = 1; break;
280 default: return IMPURE;
283 if (!p && isl_seq_first_non_zero(c + 1, nparam) == -1)
285 if (!v && isl_seq_first_non_zero(c + 1 + nparam, d) == -1)
290 bset = isl_basic_set_copy(bset);
291 bset = isl_basic_set_cow(bset);
292 bset = isl_basic_set_extend_constraints(bset, 0, 1);
293 k = isl_basic_set_alloc_inequality(bset);
296 isl_seq_clr(bset->ineq[k], 1 + isl_basic_set_total_dim(bset));
297 isl_seq_cpy(bset->ineq[k], c, 1 + nparam);
298 for (i = 0; i < n_div; ++i) {
299 if (div_purity[i] != PURE_PARAM)
301 isl_int_set(bset->ineq[k][1 + nparam + d + i],
302 c[1 + nparam + d + i]);
304 isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);
305 empty = isl_basic_set_is_empty(bset);
306 isl_basic_set_free(bset);
308 return empty < 0 ? -1 : empty ? MIXED : IMPURE;
310 isl_basic_set_free(bset);
314 /* Return an array of integers indicating the type of each div in bset.
315 * If the div is (recursively) defined in terms of only the parameters,
316 * then the type is PURE_PARAM.
317 * If the div is (recursively) defined in terms of only the set variables,
318 * then the type is PURE_VAR.
319 * Otherwise, the type is IMPURE.
321 static __isl_give int *get_div_purity(__isl_keep isl_basic_set *bset)
332 n_div = isl_basic_set_dim(bset, isl_dim_div);
333 d = isl_basic_set_dim(bset, isl_dim_set);
334 nparam = isl_basic_set_dim(bset, isl_dim_param);
336 div_purity = isl_alloc_array(bset->ctx, int, n_div);
340 for (i = 0; i < bset->n_div; ++i) {
342 if (isl_int_is_zero(bset->div[i][0])) {
343 div_purity[i] = IMPURE;
346 if (isl_seq_first_non_zero(bset->div[i] + 2, nparam) != -1)
348 if (isl_seq_first_non_zero(bset->div[i] + 2 + nparam, d) != -1)
350 for (j = 0; j < i; ++j) {
351 if (isl_int_is_zero(bset->div[i][2 + nparam + d + j]))
353 switch (div_purity[j]) {
354 case PURE_PARAM: p = 1; break;
355 case PURE_VAR: v = 1; break;
356 default: p = v = 1; break;
359 div_purity[i] = v ? p ? IMPURE : PURE_VAR : PURE_PARAM;
365 /* Given a path with the as yet unconstrained length at position "pos",
366 * check if setting the length to zero results in only the identity
369 int empty_path_is_identity(__isl_keep isl_basic_map *path, unsigned pos)
371 isl_basic_map *test = NULL;
372 isl_basic_map *id = NULL;
376 test = isl_basic_map_copy(path);
377 test = isl_basic_map_extend_constraints(test, 1, 0);
378 k = isl_basic_map_alloc_equality(test);
381 isl_seq_clr(test->eq[k], 1 + isl_basic_map_total_dim(test));
382 isl_int_set_si(test->eq[k][pos], 1);
383 id = isl_basic_map_identity(isl_dim_domain(isl_basic_map_get_dim(path)));
384 is_id = isl_basic_map_is_equal(test, id);
385 isl_basic_map_free(test);
386 isl_basic_map_free(id);
389 isl_basic_map_free(test);
393 __isl_give isl_basic_map *add_delta_constraints(__isl_take isl_basic_map *path,
394 __isl_keep isl_basic_set *delta, unsigned off, unsigned nparam,
395 unsigned d, int *div_purity, int eq)
398 int n = eq ? delta->n_eq : delta->n_ineq;
399 isl_int **delta_c = eq ? delta->eq : delta->ineq;
400 isl_int **path_c = eq ? path->eq : path->ineq;
403 n_div = isl_basic_set_dim(delta, isl_dim_div);
405 for (i = 0; i < n; ++i) {
406 int p = purity(delta, delta_c[i], div_purity, eq);
412 k = isl_basic_map_alloc_equality(path);
414 k = isl_basic_map_alloc_inequality(path);
417 isl_seq_clr(path_c[k], 1 + isl_basic_map_total_dim(path));
419 isl_seq_cpy(path_c[k] + off,
420 delta_c[i] + 1 + nparam, d);
421 isl_int_set(path_c[k][off + d], delta_c[i][0]);
422 } else if (p == PURE_PARAM) {
423 isl_seq_cpy(path_c[k], delta_c[i], 1 + nparam);
425 isl_seq_cpy(path_c[k] + off,
426 delta_c[i] + 1 + nparam, d);
427 isl_seq_cpy(path_c[k], delta_c[i], 1 + nparam);
429 isl_seq_cpy(path_c[k] + off - n_div,
430 delta_c[i] + 1 + nparam + d, n_div);
435 isl_basic_map_free(path);
439 /* Given a set of offsets "delta", construct a relation of the
440 * given dimension specification (Z^{n+1} -> Z^{n+1}) that
441 * is an overapproximation of the relations that
442 * maps an element x to any element that can be reached
443 * by taking a non-negative number of steps along any of
444 * the elements in "delta".
445 * That is, construct an approximation of
447 * { [x] -> [y] : exists f \in \delta, k \in Z :
448 * y = x + k [f, 1] and k >= 0 }
450 * For any element in this relation, the number of steps taken
451 * is equal to the difference in the final coordinates.
453 * In particular, let delta be defined as
455 * \delta = [p] -> { [x] : A x + a >= and B p + b >= 0 and
456 * C x + C'p + c >= 0 and
457 * D x + D'p + d >= 0 }
459 * where the constraints C x + C'p + c >= 0 are such that the parametric
460 * constant term of each constraint j, "C_j x + C'_j p + c_j",
461 * can never attain positive values, then the relation is constructed as
463 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
464 * A f + k a >= 0 and B p + b >= 0 and
465 * C f + C'p + c >= 0 and k >= 1 }
466 * union { [x] -> [x] }
468 * If the zero-length paths happen to correspond exactly to the identity
469 * mapping, then we return
471 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
472 * A f + k a >= 0 and B p + b >= 0 and
473 * C f + C'p + c >= 0 and k >= 0 }
477 * Existentially quantified variables in \delta are handled by
478 * classifying them as independent of the parameters, purely
479 * parameter dependent and others. Constraints containing
480 * any of the other existentially quantified variables are removed.
481 * This is safe, but leads to an additional overapproximation.
483 static __isl_give isl_map *path_along_delta(__isl_take isl_dim *dim,
484 __isl_take isl_basic_set *delta)
486 isl_basic_map *path = NULL;
493 int *div_purity = NULL;
497 n_div = isl_basic_set_dim(delta, isl_dim_div);
498 d = isl_basic_set_dim(delta, isl_dim_set);
499 nparam = isl_basic_set_dim(delta, isl_dim_param);
500 path = isl_basic_map_alloc_dim(isl_dim_copy(dim), n_div + d + 1,
501 d + 1 + delta->n_eq, delta->n_ineq + 1);
502 off = 1 + nparam + 2 * (d + 1) + n_div;
504 for (i = 0; i < n_div + d + 1; ++i) {
505 k = isl_basic_map_alloc_div(path);
508 isl_int_set_si(path->div[k][0], 0);
511 for (i = 0; i < d + 1; ++i) {
512 k = isl_basic_map_alloc_equality(path);
515 isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
516 isl_int_set_si(path->eq[k][1 + nparam + i], 1);
517 isl_int_set_si(path->eq[k][1 + nparam + d + 1 + i], -1);
518 isl_int_set_si(path->eq[k][off + i], 1);
521 div_purity = get_div_purity(delta);
525 path = add_delta_constraints(path, delta, off, nparam, d, div_purity, 1);
526 path = add_delta_constraints(path, delta, off, nparam, d, div_purity, 0);
528 is_id = empty_path_is_identity(path, off + d);
532 k = isl_basic_map_alloc_inequality(path);
535 isl_seq_clr(path->ineq[k], 1 + isl_basic_map_total_dim(path));
537 isl_int_set_si(path->ineq[k][0], -1);
538 isl_int_set_si(path->ineq[k][off + d], 1);
541 isl_basic_set_free(delta);
542 path = isl_basic_map_finalize(path);
545 return isl_map_from_basic_map(path);
547 return isl_basic_map_union(path,
548 isl_basic_map_identity(isl_dim_domain(dim)));
552 isl_basic_set_free(delta);
553 isl_basic_map_free(path);
557 /* Given a dimenion specification Z^{n+1} -> Z^{n+1} and a parameter "param",
558 * construct a map that equates the parameter to the difference
559 * in the final coordinates and imposes that this difference is positive.
562 * { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
564 static __isl_give isl_map *equate_parameter_to_length(__isl_take isl_dim *dim,
567 struct isl_basic_map *bmap;
572 d = isl_dim_size(dim, isl_dim_in);
573 nparam = isl_dim_size(dim, isl_dim_param);
574 bmap = isl_basic_map_alloc_dim(dim, 0, 1, 1);
575 k = isl_basic_map_alloc_equality(bmap);
578 isl_seq_clr(bmap->eq[k], 1 + isl_basic_map_total_dim(bmap));
579 isl_int_set_si(bmap->eq[k][1 + param], -1);
580 isl_int_set_si(bmap->eq[k][1 + nparam + d - 1], -1);
581 isl_int_set_si(bmap->eq[k][1 + nparam + d + d - 1], 1);
583 k = isl_basic_map_alloc_inequality(bmap);
586 isl_seq_clr(bmap->ineq[k], 1 + isl_basic_map_total_dim(bmap));
587 isl_int_set_si(bmap->ineq[k][1 + param], 1);
588 isl_int_set_si(bmap->ineq[k][0], -1);
590 bmap = isl_basic_map_finalize(bmap);
591 return isl_map_from_basic_map(bmap);
593 isl_basic_map_free(bmap);
597 /* Check whether "path" is acyclic, where the last coordinates of domain
598 * and range of path encode the number of steps taken.
599 * That is, check whether
601 * { d | d = y - x and (x,y) in path }
603 * does not contain any element with positive last coordinate (positive length)
604 * and zero remaining coordinates (cycle).
606 static int is_acyclic(__isl_take isl_map *path)
611 struct isl_set *delta;
613 delta = isl_map_deltas(path);
614 dim = isl_set_dim(delta, isl_dim_set);
615 for (i = 0; i < dim; ++i) {
617 delta = isl_set_lower_bound_si(delta, isl_dim_set, i, 1);
619 delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
622 acyclic = isl_set_is_empty(delta);
628 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
629 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
630 * construct a map that is an overapproximation of the map
631 * that takes an element from the space D \times Z to another
632 * element from the same space, such that the first n coordinates of the
633 * difference between them is a sum of differences between images
634 * and pre-images in one of the R_i and such that the last coordinate
635 * is equal to the number of steps taken.
638 * \Delta_i = { y - x | (x, y) in R_i }
640 * then the constructed map is an overapproximation of
642 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
643 * d = (\sum_i k_i \delta_i, \sum_i k_i) }
645 * The elements of the singleton \Delta_i's are collected as the
646 * rows of the steps matrix. For all these \Delta_i's together,
647 * a single path is constructed.
648 * For each of the other \Delta_i's, we compute an overapproximation
649 * of the paths along elements of \Delta_i.
650 * Since each of these paths performs an addition, composition is
651 * symmetric and we can simply compose all resulting paths in any order.
653 static __isl_give isl_map *construct_extended_path(__isl_take isl_dim *dim,
654 __isl_keep isl_map *map, int *project)
656 struct isl_mat *steps = NULL;
657 struct isl_map *path = NULL;
661 d = isl_map_dim(map, isl_dim_in);
663 path = isl_map_identity(isl_dim_domain(isl_dim_copy(dim)));
665 steps = isl_mat_alloc(map->ctx, map->n, d);
670 for (i = 0; i < map->n; ++i) {
671 struct isl_basic_set *delta;
673 delta = isl_basic_map_deltas(isl_basic_map_copy(map->p[i]));
675 for (j = 0; j < d; ++j) {
678 fixed = isl_basic_set_fast_dim_is_fixed(delta, j,
681 isl_basic_set_free(delta);
690 path = isl_map_apply_range(path,
691 path_along_delta(isl_dim_copy(dim), delta));
692 path = isl_map_coalesce(path);
694 isl_basic_set_free(delta);
701 path = isl_map_apply_range(path,
702 path_along_steps(isl_dim_copy(dim), steps));
705 if (project && *project) {
706 *project = is_acyclic(isl_map_copy(path));
721 static int isl_set_overlaps(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
726 i = isl_set_intersect(isl_set_copy(set1), isl_set_copy(set2));
727 no_overlap = isl_set_is_empty(i);
730 return no_overlap < 0 ? -1 : !no_overlap;
733 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
734 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
735 * construct a map that is an overapproximation of the map
736 * that takes an element from the dom R \times Z to an
737 * element from ran R \times Z, such that the first n coordinates of the
738 * difference between them is a sum of differences between images
739 * and pre-images in one of the R_i and such that the last coordinate
740 * is equal to the number of steps taken.
743 * \Delta_i = { y - x | (x, y) in R_i }
745 * then the constructed map is an overapproximation of
747 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
748 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
749 * x in dom R and x + d in ran R and
752 static __isl_give isl_map *construct_component(__isl_take isl_dim *dim,
753 __isl_keep isl_map *map, int *exact, int project)
755 struct isl_set *domain = NULL;
756 struct isl_set *range = NULL;
757 struct isl_map *app = NULL;
758 struct isl_map *path = NULL;
760 domain = isl_map_domain(isl_map_copy(map));
761 domain = isl_set_coalesce(domain);
762 range = isl_map_range(isl_map_copy(map));
763 range = isl_set_coalesce(range);
764 if (!isl_set_overlaps(domain, range)) {
765 isl_set_free(domain);
769 map = isl_map_copy(map);
770 map = isl_map_add(map, isl_dim_in, 1);
771 map = isl_map_add(map, isl_dim_out, 1);
772 map = set_path_length(map, 1, 1);
775 app = isl_map_from_domain_and_range(domain, range);
776 app = isl_map_add(app, isl_dim_in, 1);
777 app = isl_map_add(app, isl_dim_out, 1);
779 path = construct_extended_path(isl_dim_copy(dim), map,
780 exact && *exact ? &project : NULL);
781 app = isl_map_intersect(app, path);
783 if (exact && *exact &&
784 (*exact = check_exactness(isl_map_copy(map), isl_map_copy(app),
789 app = set_path_length(app, 0, 1);
797 /* Call construct_component and, if "project" is set, project out
798 * the final coordinates.
800 static __isl_give isl_map *construct_projected_component(
801 __isl_take isl_dim *dim,
802 __isl_keep isl_map *map, int *exact, int project)
809 d = isl_dim_size(dim, isl_dim_in);
811 app = construct_component(dim, map, exact, project);
813 app = isl_map_project_out(app, isl_dim_in, d - 1, 1);
814 app = isl_map_project_out(app, isl_dim_out, d - 1, 1);
819 /* Compute an extended version, i.e., with path lengths, of
820 * an overapproximation of the transitive closure of "bmap"
821 * with path lengths greater than or equal to zero and with
822 * domain and range equal to "dom".
824 static __isl_give isl_map *q_closure(__isl_take isl_dim *dim,
825 __isl_take isl_set *dom, __isl_keep isl_basic_map *bmap, int *exact)
832 dom = isl_set_add(dom, isl_dim_set, 1);
833 app = isl_map_from_domain_and_range(dom, isl_set_copy(dom));
834 map = isl_map_from_basic_map(isl_basic_map_copy(bmap));
835 path = construct_extended_path(dim, map, &project);
836 app = isl_map_intersect(app, path);
838 if ((*exact = check_exactness(map, isl_map_copy(app), project)) < 0)
847 /* Check whether qc has any elements of length at least one
848 * with domain and/or range outside of dom and ran.
850 static int has_spurious_elements(__isl_keep isl_map *qc,
851 __isl_keep isl_set *dom, __isl_keep isl_set *ran)
857 if (!qc || !dom || !ran)
860 d = isl_map_dim(qc, isl_dim_in);
862 qc = isl_map_copy(qc);
863 qc = set_path_length(qc, 0, 1);
864 qc = isl_map_project_out(qc, isl_dim_in, d - 1, 1);
865 qc = isl_map_project_out(qc, isl_dim_out, d - 1, 1);
867 s = isl_map_domain(isl_map_copy(qc));
868 subset = isl_set_is_subset(s, dom);
877 s = isl_map_range(qc);
878 subset = isl_set_is_subset(s, ran);
881 return subset < 0 ? -1 : !subset;
890 /* For each basic map in "map", except i, check whether it combines
891 * with the transitive closure that is reflexive on C combines
892 * to the left and to the right.
896 * dom map_j \subseteq C
898 * then right[j] is set to 1. Otherwise, if
900 * ran map_i \cap dom map_j = \emptyset
902 * then right[j] is set to 0. Otherwise, composing to the right
905 * Similar, for composing to the left, we have if
907 * ran map_j \subseteq C
909 * then left[j] is set to 1. Otherwise, if
911 * dom map_i \cap ran map_j = \emptyset
913 * then left[j] is set to 0. Otherwise, composing to the left
916 * The return value is or'd with LEFT if composing to the left
917 * is possible and with RIGHT if composing to the right is possible.
919 static int composability(__isl_keep isl_set *C, int i,
920 isl_set **dom, isl_set **ran, int *left, int *right,
921 __isl_keep isl_map *map)
927 for (j = 0; j < map->n && ok; ++j) {
928 int overlaps, subset;
934 dom[j] = isl_set_from_basic_set(
935 isl_basic_map_domain(
936 isl_basic_map_copy(map->p[j])));
939 overlaps = isl_set_overlaps(ran[i], dom[j]);
945 subset = isl_set_is_subset(dom[j], C);
957 ran[j] = isl_set_from_basic_set(
959 isl_basic_map_copy(map->p[j])));
962 overlaps = isl_set_overlaps(dom[i], ran[j]);
968 subset = isl_set_is_subset(ran[j], C);
982 /* Return a map that is a union of the basic maps in "map", except i,
983 * composed to left and right with qc based on the entries of "left"
986 static __isl_give isl_map *compose(__isl_keep isl_map *map, int i,
987 __isl_take isl_map *qc, int *left, int *right)
992 comp = isl_map_empty(isl_map_get_dim(map));
993 for (j = 0; j < map->n; ++j) {
999 map_j = isl_map_from_basic_map(isl_basic_map_copy(map->p[j]));
1000 if (left && left[j])
1001 map_j = isl_map_apply_range(map_j, isl_map_copy(qc));
1002 if (right && right[j])
1003 map_j = isl_map_apply_range(isl_map_copy(qc), map_j);
1004 comp = isl_map_union(comp, map_j);
1007 comp = isl_map_compute_divs(comp);
1008 comp = isl_map_coalesce(comp);
1015 /* Compute the transitive closure of "map" incrementally by
1022 * map_i^+ \cup ((id \cup map_i^) \circ qc^+)
1026 * map_i^+ \cup (qc^+ \circ (id \cup map_i^))
1028 * depending on whether left or right are NULL.
1030 static __isl_give isl_map *compute_incremental(
1031 __isl_take isl_dim *dim, __isl_keep isl_map *map,
1032 int i, __isl_take isl_map *qc, int *left, int *right, int *exact)
1036 isl_map *rtc = NULL;
1040 isl_assert(map->ctx, left || right, goto error);
1042 map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
1043 tc = construct_projected_component(isl_dim_copy(dim), map_i,
1045 isl_map_free(map_i);
1048 qc = isl_map_transitive_closure(qc, exact);
1054 return isl_map_universe(isl_map_get_dim(map));
1057 if (!left || !right)
1058 rtc = isl_map_union(isl_map_copy(tc),
1059 isl_map_identity(isl_dim_domain(isl_map_get_dim(tc))));
1061 qc = isl_map_apply_range(rtc, qc);
1063 qc = isl_map_apply_range(qc, rtc);
1064 qc = isl_map_union(tc, qc);
1075 /* Given a map "map", try to find a basic map such that
1076 * map^+ can be computed as
1078 * map^+ = map_i^+ \cup
1079 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1081 * with C the simple hull of the domain and range of the input map.
1082 * map_i^ \cup Id_C is computed by allowing the path lengths to be zero
1083 * and by intersecting domain and range with C.
1084 * Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
1085 * Also, we only use the incremental computation if all the transitive
1086 * closures are exact and if the number of basic maps in the union,
1087 * after computing the integer divisions, is smaller than the number
1088 * of basic maps in the input map.
1090 static int incemental_on_entire_domain(__isl_keep isl_dim *dim,
1091 __isl_keep isl_map *map,
1092 isl_set **dom, isl_set **ran, int *left, int *right,
1093 __isl_give isl_map **res)
1101 C = isl_set_union(isl_map_domain(isl_map_copy(map)),
1102 isl_map_range(isl_map_copy(map)));
1103 C = isl_set_from_basic_set(isl_set_simple_hull(C));
1111 d = isl_map_dim(map, isl_dim_in);
1113 for (i = 0; i < map->n; ++i) {
1115 int exact_i, spurious;
1117 dom[i] = isl_set_from_basic_set(isl_basic_map_domain(
1118 isl_basic_map_copy(map->p[i])));
1119 ran[i] = isl_set_from_basic_set(isl_basic_map_range(
1120 isl_basic_map_copy(map->p[i])));
1121 qc = q_closure(isl_dim_copy(dim), isl_set_copy(C),
1122 map->p[i], &exact_i);
1129 spurious = has_spurious_elements(qc, dom[i], ran[i]);
1136 qc = isl_map_project_out(qc, isl_dim_in, d, 1);
1137 qc = isl_map_project_out(qc, isl_dim_out, d, 1);
1138 qc = isl_map_compute_divs(qc);
1139 for (j = 0; j < map->n; ++j)
1140 left[j] = right[j] = 1;
1141 qc = compose(map, i, qc, left, right);
1144 if (qc->n >= map->n) {
1148 *res = compute_incremental(isl_dim_copy(dim), map, i, qc,
1149 left, right, &exact_i);
1160 return *res != NULL;
1166 /* Try and compute the transitive closure of "map" as
1168 * map^+ = map_i^+ \cup
1169 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1171 * with C either the simple hull of the domain and range of the entire
1172 * map or the simple hull of domain and range of map_i.
1174 static __isl_give isl_map *incremental_closure(__isl_take isl_dim *dim,
1175 __isl_keep isl_map *map, int *exact, int project)
1178 isl_set **dom = NULL;
1179 isl_set **ran = NULL;
1184 isl_map *res = NULL;
1187 return construct_projected_component(dim, map, exact, project);
1192 return construct_projected_component(dim, map, exact, project);
1194 d = isl_map_dim(map, isl_dim_in);
1196 dom = isl_calloc_array(map->ctx, isl_set *, map->n);
1197 ran = isl_calloc_array(map->ctx, isl_set *, map->n);
1198 left = isl_calloc_array(map->ctx, int, map->n);
1199 right = isl_calloc_array(map->ctx, int, map->n);
1200 if (!ran || !dom || !left || !right)
1203 if (incemental_on_entire_domain(dim, map, dom, ran, left, right, &res) < 0)
1206 for (i = 0; !res && i < map->n; ++i) {
1208 int exact_i, spurious, comp;
1210 dom[i] = isl_set_from_basic_set(
1211 isl_basic_map_domain(
1212 isl_basic_map_copy(map->p[i])));
1216 ran[i] = isl_set_from_basic_set(
1217 isl_basic_map_range(
1218 isl_basic_map_copy(map->p[i])));
1221 C = isl_set_union(isl_set_copy(dom[i]),
1222 isl_set_copy(ran[i]));
1223 C = isl_set_from_basic_set(isl_set_simple_hull(C));
1230 comp = composability(C, i, dom, ran, left, right, map);
1231 if (!comp || comp < 0) {
1237 qc = q_closure(isl_dim_copy(dim), C, map->p[i], &exact_i);
1244 spurious = has_spurious_elements(qc, dom[i], ran[i]);
1251 qc = isl_map_project_out(qc, isl_dim_in, d, 1);
1252 qc = isl_map_project_out(qc, isl_dim_out, d, 1);
1253 qc = isl_map_compute_divs(qc);
1254 qc = compose(map, i, qc, (comp & LEFT) ? left : NULL,
1255 (comp & RIGHT) ? right : NULL);
1258 if (qc->n >= map->n) {
1262 res = compute_incremental(isl_dim_copy(dim), map, i, qc,
1263 (comp & LEFT) ? left : NULL,
1264 (comp & RIGHT) ? right : NULL, &exact_i);
1273 for (i = 0; i < map->n; ++i) {
1274 isl_set_free(dom[i]);
1275 isl_set_free(ran[i]);
1287 return construct_projected_component(dim, map, exact, project);
1290 for (i = 0; i < map->n; ++i)
1291 isl_set_free(dom[i]);
1294 for (i = 0; i < map->n; ++i)
1295 isl_set_free(ran[i]);
1303 /* Given an array of sets "set", add "dom" at position "pos"
1304 * and search for elements at earlier positions that overlap with "dom".
1305 * If any can be found, then merge all of them, together with "dom", into
1306 * a single set and assign the union to the first in the array,
1307 * which becomes the new group leader for all groups involved in the merge.
1308 * During the search, we only consider group leaders, i.e., those with
1309 * group[i] = i, as the other sets have already been combined
1310 * with one of the group leaders.
1312 static int merge(isl_set **set, int *group, __isl_take isl_set *dom, int pos)
1317 set[pos] = isl_set_copy(dom);
1319 for (i = pos - 1; i >= 0; --i) {
1325 o = isl_set_overlaps(set[i], dom);
1331 set[i] = isl_set_union(set[i], set[group[pos]]);
1334 set[group[pos]] = NULL;
1335 group[group[pos]] = i;
1346 /* Given a partition of the domains and ranges of the basic maps in "map",
1347 * apply the Floyd-Warshall algorithm with the elements in the partition
1350 * In particular, there are "n" elements in the partition and "group" is
1351 * an array of length 2 * map->n with entries in [0,n-1].
1353 * We first construct a matrix of relations based on the partition information,
1354 * apply Floyd-Warshall on this matrix of relations and then take the
1355 * union of all entries in the matrix as the final result.
1357 * The algorithm iterates over all vertices. In each step, the whole
1358 * matrix is updated to include all paths that go to the current vertex,
1359 * possibly stay there a while (including passing through earlier vertices)
1360 * and then come back. At the start of each iteration, the diagonal
1361 * element corresponding to the current vertex is replaced by its
1362 * transitive closure to account for all indirect paths that stay
1363 * in the current vertex.
1365 static __isl_give isl_map *floyd_warshall_with_groups(__isl_take isl_dim *dim,
1366 __isl_keep isl_map *map, int *exact, int project, int *group, int n)
1370 isl_map ***grid = NULL;
1378 return incremental_closure(dim, map, exact, project);
1381 grid = isl_calloc_array(map->ctx, isl_map **, n);
1384 for (i = 0; i < n; ++i) {
1385 grid[i] = isl_calloc_array(map->ctx, isl_map *, n);
1388 for (j = 0; j < n; ++j)
1389 grid[i][j] = isl_map_empty(isl_map_get_dim(map));
1392 for (k = 0; k < map->n; ++k) {
1394 j = group[2 * k + 1];
1395 grid[i][j] = isl_map_union(grid[i][j],
1396 isl_map_from_basic_map(
1397 isl_basic_map_copy(map->p[k])));
1400 for (r = 0; r < n; ++r) {
1402 grid[r][r] = isl_map_transitive_closure(grid[r][r],
1403 (exact && *exact) ? &r_exact : NULL);
1404 if (exact && *exact && !r_exact)
1407 for (p = 0; p < n; ++p)
1408 for (q = 0; q < n; ++q) {
1410 if (p == r && q == r)
1412 loop = isl_map_apply_range(
1413 isl_map_copy(grid[p][r]),
1414 isl_map_copy(grid[r][q]));
1415 grid[p][q] = isl_map_union(grid[p][q], loop);
1416 loop = isl_map_apply_range(
1417 isl_map_copy(grid[p][r]),
1418 isl_map_apply_range(
1419 isl_map_copy(grid[r][r]),
1420 isl_map_copy(grid[r][q])));
1421 grid[p][q] = isl_map_union(grid[p][q], loop);
1422 grid[p][q] = isl_map_coalesce(grid[p][q]);
1426 app = isl_map_empty(isl_map_get_dim(map));
1428 for (i = 0; i < n; ++i) {
1429 for (j = 0; j < n; ++j)
1430 app = isl_map_union(app, grid[i][j]);
1441 for (i = 0; i < n; ++i) {
1444 for (j = 0; j < n; ++j)
1445 isl_map_free(grid[i][j]);
1454 /* Check if the domains and ranges of the basic maps in "map" can
1455 * be partitioned, and if so, apply Floyd-Warshall on the elements
1456 * of the partition. Note that we can only apply this algorithm
1457 * if we want to compute the transitive closure, i.e., when "project"
1458 * is set. If we want to compute the power, we need to keep track
1459 * of the lengths and the recursive calls inside the Floyd-Warshall
1460 * would result in non-linear lengths.
1462 * To find the partition, we simply consider all of the domains
1463 * and ranges in turn and combine those that overlap.
1464 * "set" contains the partition elements and "group" indicates
1465 * to which partition element a given domain or range belongs.
1466 * The domain of basic map i corresponds to element 2 * i in these arrays,
1467 * while the domain corresponds to element 2 * i + 1.
1468 * During the construction group[k] is either equal to k,
1469 * in which case set[k] contains the union of all the domains and
1470 * ranges in the corresponding group, or is equal to some l < k,
1471 * with l another domain or range in the same group.
1473 static __isl_give isl_map *floyd_warshall(__isl_take isl_dim *dim,
1474 __isl_keep isl_map *map, int *exact, int project)
1477 isl_set **set = NULL;
1483 if (!project || map->n <= 1)
1484 return incremental_closure(dim, map, exact, project);
1486 set = isl_calloc_array(map->ctx, isl_set *, 2 * map->n);
1487 group = isl_alloc_array(map->ctx, int, 2 * map->n);
1492 for (i = 0; i < map->n; ++i) {
1494 dom = isl_set_from_basic_set(isl_basic_map_domain(
1495 isl_basic_map_copy(map->p[i])));
1496 if (merge(set, group, dom, 2 * i) < 0)
1498 dom = isl_set_from_basic_set(isl_basic_map_range(
1499 isl_basic_map_copy(map->p[i])));
1500 if (merge(set, group, dom, 2 * i + 1) < 0)
1505 for (i = 0; i < 2 * map->n; ++i)
1509 group[i] = group[group[i]];
1511 for (i = 0; i < 2 * map->n; ++i)
1512 isl_set_free(set[i]);
1516 return floyd_warshall_with_groups(dim, map, exact, project, group, n);
1518 for (i = 0; i < 2 * map->n; ++i)
1519 isl_set_free(set[i]);
1526 /* Structure for representing the nodes in the graph being traversed
1527 * using Tarjan's algorithm.
1528 * index represents the order in which nodes are visited.
1529 * min_index is the index of the root of a (sub)component.
1530 * on_stack indicates whether the node is currently on the stack.
1532 struct basic_map_sort_node {
1537 /* Structure for representing the graph being traversed
1538 * using Tarjan's algorithm.
1539 * len is the number of nodes
1540 * node is an array of nodes
1541 * stack contains the nodes on the path from the root to the current node
1542 * sp is the stack pointer
1543 * index is the index of the last node visited
1544 * order contains the elements of the components separated by -1
1545 * op represents the current position in order
1547 struct basic_map_sort {
1549 struct basic_map_sort_node *node;
1557 static void basic_map_sort_free(struct basic_map_sort *s)
1567 static struct basic_map_sort *basic_map_sort_alloc(struct isl_ctx *ctx, int len)
1569 struct basic_map_sort *s;
1572 s = isl_calloc_type(ctx, struct basic_map_sort);
1576 s->node = isl_alloc_array(ctx, struct basic_map_sort_node, len);
1579 for (i = 0; i < len; ++i)
1580 s->node[i].index = -1;
1581 s->stack = isl_alloc_array(ctx, int, len);
1584 s->order = isl_alloc_array(ctx, int, 2 * len);
1594 basic_map_sort_free(s);
1598 /* Check whether in the computation of the transitive closure
1599 * "bmap1" (R_1) should follow (or be part of the same component as)
1602 * That is check whether
1610 * If so, then there is no reason for R_1 to immediately follow R_2
1613 static int basic_map_follows(__isl_keep isl_basic_map *bmap1,
1614 __isl_keep isl_basic_map *bmap2)
1616 struct isl_map *map12 = NULL;
1617 struct isl_map *map21 = NULL;
1620 map21 = isl_map_from_basic_map(
1621 isl_basic_map_apply_range(
1622 isl_basic_map_copy(bmap2),
1623 isl_basic_map_copy(bmap1)));
1624 subset = isl_map_is_empty(map21);
1628 isl_map_free(map21);
1632 map12 = isl_map_from_basic_map(
1633 isl_basic_map_apply_range(
1634 isl_basic_map_copy(bmap1),
1635 isl_basic_map_copy(bmap2)));
1637 subset = isl_map_is_subset(map21, map12);
1639 isl_map_free(map12);
1640 isl_map_free(map21);
1642 return subset < 0 ? -1 : !subset;
1644 isl_map_free(map21);
1648 /* Perform Tarjan's algorithm for computing the strongly connected components
1649 * in the graph with the disjuncts of "map" as vertices and with an
1650 * edge between any pair of disjuncts such that the first has
1651 * to be applied after the second.
1653 static int power_components_tarjan(struct basic_map_sort *s,
1654 __isl_keep isl_map *map, int i)
1658 s->node[i].index = s->index;
1659 s->node[i].min_index = s->index;
1660 s->node[i].on_stack = 1;
1662 s->stack[s->sp++] = i;
1664 for (j = s->len - 1; j >= 0; --j) {
1669 if (s->node[j].index >= 0 &&
1670 (!s->node[j].on_stack ||
1671 s->node[j].index > s->node[i].min_index))
1674 f = basic_map_follows(map->p[i], map->p[j]);
1680 if (s->node[j].index < 0) {
1681 power_components_tarjan(s, map, j);
1682 if (s->node[j].min_index < s->node[i].min_index)
1683 s->node[i].min_index = s->node[j].min_index;
1684 } else if (s->node[j].index < s->node[i].min_index)
1685 s->node[i].min_index = s->node[j].index;
1688 if (s->node[i].index != s->node[i].min_index)
1692 j = s->stack[--s->sp];
1693 s->node[j].on_stack = 0;
1694 s->order[s->op++] = j;
1696 s->order[s->op++] = -1;
1701 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
1702 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
1703 * construct a map that is an overapproximation of the map
1704 * that takes an element from the dom R \times Z to an
1705 * element from ran R \times Z, such that the first n coordinates of the
1706 * difference between them is a sum of differences between images
1707 * and pre-images in one of the R_i and such that the last coordinate
1708 * is equal to the number of steps taken.
1709 * If "project" is set, then these final coordinates are not included,
1710 * i.e., a relation of type Z^n -> Z^n is returned.
1713 * \Delta_i = { y - x | (x, y) in R_i }
1715 * then the constructed map is an overapproximation of
1717 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1718 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
1719 * x in dom R and x + d in ran R }
1723 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1724 * d = (\sum_i k_i \delta_i) and
1725 * x in dom R and x + d in ran R }
1727 * if "project" is set.
1729 * We first split the map into strongly connected components, perform
1730 * the above on each component and then join the results in the correct
1731 * order, at each join also taking in the union of both arguments
1732 * to allow for paths that do not go through one of the two arguments.
1734 static __isl_give isl_map *construct_power_components(__isl_take isl_dim *dim,
1735 __isl_keep isl_map *map, int *exact, int project)
1738 struct isl_map *path = NULL;
1739 struct basic_map_sort *s = NULL;
1744 return floyd_warshall(dim, map, exact, project);
1746 s = basic_map_sort_alloc(map->ctx, map->n);
1749 for (i = map->n - 1; i >= 0; --i) {
1750 if (s->node[i].index >= 0)
1752 if (power_components_tarjan(s, map, i) < 0)
1759 path = isl_map_empty(isl_map_get_dim(map));
1761 path = isl_map_empty(isl_dim_copy(dim));
1763 struct isl_map *comp;
1764 isl_map *path_comp, *path_comb;
1765 comp = isl_map_alloc_dim(isl_map_get_dim(map), n, 0);
1766 while (s->order[i] != -1) {
1767 comp = isl_map_add_basic_map(comp,
1768 isl_basic_map_copy(map->p[s->order[i]]));
1772 path_comp = floyd_warshall(isl_dim_copy(dim),
1773 comp, exact, project);
1774 path_comb = isl_map_apply_range(isl_map_copy(path),
1775 isl_map_copy(path_comp));
1776 path = isl_map_union(path, path_comp);
1777 path = isl_map_union(path, path_comb);
1782 basic_map_sort_free(s);
1787 basic_map_sort_free(s);
1792 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
1793 * construct a map that is an overapproximation of the map
1794 * that takes an element from the space D to another
1795 * element from the same space, such that the difference between
1796 * them is a strictly positive sum of differences between images
1797 * and pre-images in one of the R_i.
1798 * The number of differences in the sum is equated to parameter "param".
1801 * \Delta_i = { y - x | (x, y) in R_i }
1803 * then the constructed map is an overapproximation of
1805 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1806 * d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
1809 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1810 * d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
1812 * if "project" is set.
1814 * If "project" is not set, then
1815 * we first construct an extended mapping with an extra coordinate
1816 * that indicates the number of steps taken. In particular,
1817 * the difference in the last coordinate is equal to the number
1818 * of steps taken to move from a domain element to the corresponding
1820 * In the final step, this difference is equated to the parameter "param"
1821 * and made positive. The extra coordinates are subsequently projected out.
1823 static __isl_give isl_map *construct_power(__isl_keep isl_map *map,
1824 unsigned param, int *exact, int project)
1826 struct isl_map *app = NULL;
1827 struct isl_map *diff;
1828 struct isl_dim *dim = NULL;
1834 dim = isl_map_get_dim(map);
1836 d = isl_dim_size(dim, isl_dim_in);
1837 dim = isl_dim_add(dim, isl_dim_in, 1);
1838 dim = isl_dim_add(dim, isl_dim_out, 1);
1840 app = construct_power_components(isl_dim_copy(dim), map,
1846 diff = equate_parameter_to_length(dim, param);
1847 app = isl_map_intersect(app, diff);
1848 app = isl_map_project_out(app, isl_dim_in, d, 1);
1849 app = isl_map_project_out(app, isl_dim_out, d, 1);
1855 /* Compute the positive powers of "map", or an overapproximation.
1856 * The power is given by parameter "param". If the result is exact,
1857 * then *exact is set to 1.
1859 * If project is set, then we are actually interested in the transitive
1860 * closure, so we can use a more relaxed exactness check.
1861 * The lengths of the paths are also projected out instead of being
1862 * equated to "param" (which is then ignored in this case).
1864 static __isl_give isl_map *map_power(__isl_take isl_map *map, unsigned param,
1865 int *exact, int project)
1867 struct isl_map *app = NULL;
1872 map = isl_map_compute_divs(map);
1873 map = isl_map_coalesce(map);
1877 if (isl_map_fast_is_empty(map))
1880 isl_assert(map->ctx, project || param < isl_map_dim(map, isl_dim_param),
1882 isl_assert(map->ctx,
1883 isl_map_dim(map, isl_dim_in) == isl_map_dim(map, isl_dim_out),
1886 app = construct_power(map, param, exact, project);
1896 /* Compute the positive powers of "map", or an overapproximation.
1897 * The power is given by parameter "param". If the result is exact,
1898 * then *exact is set to 1.
1900 __isl_give isl_map *isl_map_power(__isl_take isl_map *map, unsigned param,
1903 return map_power(map, param, exact, 0);
1906 /* Check whether equality i of bset is a pure stride constraint
1907 * on a single dimensions, i.e., of the form
1911 * with k a constant and e an existentially quantified variable.
1913 static int is_eq_stride(__isl_keep isl_basic_set *bset, int i)
1925 if (!isl_int_is_zero(bset->eq[i][0]))
1928 nparam = isl_basic_set_dim(bset, isl_dim_param);
1929 d = isl_basic_set_dim(bset, isl_dim_set);
1930 n_div = isl_basic_set_dim(bset, isl_dim_div);
1932 if (isl_seq_first_non_zero(bset->eq[i] + 1, nparam) != -1)
1934 pos1 = isl_seq_first_non_zero(bset->eq[i] + 1 + nparam, d);
1937 if (isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + pos1 + 1,
1938 d - pos1 - 1) != -1)
1941 pos2 = isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + d, n_div);
1944 if (isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + d + pos2 + 1,
1945 n_div - pos2 - 1) != -1)
1947 if (!isl_int_is_one(bset->eq[i][1 + nparam + pos1]) &&
1948 !isl_int_is_negone(bset->eq[i][1 + nparam + pos1]))
1954 /* Given a map, compute the smallest superset of this map that is of the form
1956 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
1958 * (where p ranges over the (non-parametric) dimensions),
1959 * compute the transitive closure of this map, i.e.,
1961 * { i -> j : exists k > 0:
1962 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
1964 * and intersect domain and range of this transitive closure with
1965 * the given domain and range.
1967 * If with_id is set, then try to include as much of the identity mapping
1968 * as possible, by computing
1970 * { i -> j : exists k >= 0:
1971 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
1973 * instead (i.e., allow k = 0).
1975 * In practice, we compute the difference set
1977 * delta = { j - i | i -> j in map },
1979 * look for stride constraint on the individual dimensions and compute
1980 * (constant) lower and upper bounds for each individual dimension,
1981 * adding a constraint for each bound not equal to infinity.
1983 static __isl_give isl_map *box_closure_on_domain(__isl_take isl_map *map,
1984 __isl_take isl_set *dom, __isl_take isl_set *ran, int with_id)
1993 isl_map *app = NULL;
1994 isl_basic_set *aff = NULL;
1995 isl_basic_map *bmap = NULL;
1996 isl_vec *obj = NULL;
2001 delta = isl_map_deltas(isl_map_copy(map));
2003 aff = isl_set_affine_hull(isl_set_copy(delta));
2006 dim = isl_map_get_dim(map);
2007 d = isl_dim_size(dim, isl_dim_in);
2008 nparam = isl_dim_size(dim, isl_dim_param);
2009 total = isl_dim_total(dim);
2010 bmap = isl_basic_map_alloc_dim(dim,
2011 aff->n_div + 1, aff->n_div, 2 * d + 1);
2012 for (i = 0; i < aff->n_div + 1; ++i) {
2013 k = isl_basic_map_alloc_div(bmap);
2016 isl_int_set_si(bmap->div[k][0], 0);
2018 for (i = 0; i < aff->n_eq; ++i) {
2019 if (!is_eq_stride(aff, i))
2021 k = isl_basic_map_alloc_equality(bmap);
2024 isl_seq_clr(bmap->eq[k], 1 + nparam);
2025 isl_seq_cpy(bmap->eq[k] + 1 + nparam + d,
2026 aff->eq[i] + 1 + nparam, d);
2027 isl_seq_neg(bmap->eq[k] + 1 + nparam,
2028 aff->eq[i] + 1 + nparam, d);
2029 isl_seq_cpy(bmap->eq[k] + 1 + nparam + 2 * d,
2030 aff->eq[i] + 1 + nparam + d, aff->n_div);
2031 isl_int_set_si(bmap->eq[k][1 + total + aff->n_div], 0);
2033 obj = isl_vec_alloc(map->ctx, 1 + nparam + d);
2036 isl_seq_clr(obj->el, 1 + nparam + d);
2037 for (i = 0; i < d; ++ i) {
2038 enum isl_lp_result res;
2040 isl_int_set_si(obj->el[1 + nparam + i], 1);
2042 res = isl_set_solve_lp(delta, 0, obj->el, map->ctx->one, &opt,
2044 if (res == isl_lp_error)
2046 if (res == isl_lp_ok) {
2047 k = isl_basic_map_alloc_inequality(bmap);
2050 isl_seq_clr(bmap->ineq[k],
2051 1 + nparam + 2 * d + bmap->n_div);
2052 isl_int_set_si(bmap->ineq[k][1 + nparam + i], -1);
2053 isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], 1);
2054 isl_int_neg(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
2057 res = isl_set_solve_lp(delta, 1, obj->el, map->ctx->one, &opt,
2059 if (res == isl_lp_error)
2061 if (res == isl_lp_ok) {
2062 k = isl_basic_map_alloc_inequality(bmap);
2065 isl_seq_clr(bmap->ineq[k],
2066 1 + nparam + 2 * d + bmap->n_div);
2067 isl_int_set_si(bmap->ineq[k][1 + nparam + i], 1);
2068 isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], -1);
2069 isl_int_set(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
2072 isl_int_set_si(obj->el[1 + nparam + i], 0);
2074 k = isl_basic_map_alloc_inequality(bmap);
2077 isl_seq_clr(bmap->ineq[k],
2078 1 + nparam + 2 * d + bmap->n_div);
2080 isl_int_set_si(bmap->ineq[k][0], -1);
2081 isl_int_set_si(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], 1);
2083 app = isl_map_from_domain_and_range(dom, ran);
2086 isl_basic_set_free(aff);
2088 bmap = isl_basic_map_finalize(bmap);
2089 isl_set_free(delta);
2092 map = isl_map_from_basic_map(bmap);
2093 map = isl_map_intersect(map, app);
2098 isl_basic_map_free(bmap);
2099 isl_basic_set_free(aff);
2103 isl_set_free(delta);
2108 /* Given a map, compute the smallest superset of this map that is of the form
2110 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2112 * (where p ranges over the (non-parametric) dimensions),
2113 * compute the transitive closure of this map, i.e.,
2115 * { i -> j : exists k > 0:
2116 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2118 * and intersect domain and range of this transitive closure with
2119 * domain and range of the original map.
2121 static __isl_give isl_map *box_closure(__isl_take isl_map *map)
2126 domain = isl_map_domain(isl_map_copy(map));
2127 domain = isl_set_coalesce(domain);
2128 range = isl_map_range(isl_map_copy(map));
2129 range = isl_set_coalesce(range);
2131 return box_closure_on_domain(map, domain, range, 0);
2134 /* Given a map, compute the smallest superset of this map that is of the form
2136 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2138 * (where p ranges over the (non-parametric) dimensions),
2139 * compute the transitive and partially reflexive closure of this map, i.e.,
2141 * { i -> j : exists k >= 0:
2142 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2144 * and intersect domain and range of this transitive closure with
2147 static __isl_give isl_map *box_closure_with_identity(__isl_take isl_map *map,
2148 __isl_take isl_set *dom)
2150 return box_closure_on_domain(map, dom, isl_set_copy(dom), 1);
2153 /* Check whether app is the transitive closure of map.
2154 * In particular, check that app is acyclic and, if so,
2157 * app \subset (map \cup (map \circ app))
2159 static int check_exactness_omega(__isl_keep isl_map *map,
2160 __isl_keep isl_map *app)
2164 int is_empty, is_exact;
2168 delta = isl_map_deltas(isl_map_copy(app));
2169 d = isl_set_dim(delta, isl_dim_set);
2170 for (i = 0; i < d; ++i)
2171 delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
2172 is_empty = isl_set_is_empty(delta);
2173 isl_set_free(delta);
2179 test = isl_map_apply_range(isl_map_copy(app), isl_map_copy(map));
2180 test = isl_map_union(test, isl_map_copy(map));
2181 is_exact = isl_map_is_subset(app, test);
2187 /* Check if basic map M_i can be combined with all the other
2188 * basic maps such that
2192 * can be computed as
2194 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2196 * In particular, check if we can compute a compact representation
2199 * M_i^* \circ M_j \circ M_i^*
2202 * Let M_i^? be an extension of M_i^+ that allows paths
2203 * of length zero, i.e., the result of box_closure(., 1).
2204 * The criterion, as proposed by Kelly et al., is that
2205 * id = M_i^? - M_i^+ can be represented as a basic map
2208 * id \circ M_j \circ id = M_j
2212 * If this function returns 1, then tc and qc are set to
2213 * M_i^+ and M_i^?, respectively.
2215 static int can_be_split_off(__isl_keep isl_map *map, int i,
2216 __isl_give isl_map **tc, __isl_give isl_map **qc)
2218 isl_map *map_i, *id = NULL;
2225 C = isl_set_union(isl_map_domain(isl_map_copy(map)),
2226 isl_map_range(isl_map_copy(map)));
2227 C = isl_set_from_basic_set(isl_set_simple_hull(C));
2231 map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
2232 *tc = box_closure(isl_map_copy(map_i));
2233 *qc = box_closure_with_identity(map_i, C);
2234 id = isl_map_subtract(isl_map_copy(*qc), isl_map_copy(*tc));
2238 if (id->n != 1 || (*qc)->n != 1)
2241 for (j = 0; j < map->n; ++j) {
2242 isl_map *map_j, *test;
2247 map_j = isl_map_from_basic_map(
2248 isl_basic_map_copy(map->p[j]));
2249 test = isl_map_apply_range(isl_map_copy(id),
2250 isl_map_copy(map_j));
2251 test = isl_map_apply_range(test, isl_map_copy(id));
2252 is_ok = isl_map_is_equal(test, map_j);
2253 isl_map_free(map_j);
2281 /* Compute an overapproximation of the transitive closure of "map"
2282 * using a variation of the algorithm from
2283 * "Transitive Closure of Infinite Graphs and its Applications"
2286 * We first check whether we can can split of any basic map M_i and
2293 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2295 * using a recursive call on the remaining map.
2297 * If not, we simply call box_closure on the whole map.
2299 static __isl_give isl_map *compute_closure_omega(__isl_take isl_map *map)
2306 return box_closure(map);
2308 map = isl_map_cow(map);
2312 for (i = 0; i < map->n; ++i) {
2315 ok = can_be_split_off(map, i, &tc, &qc);
2321 isl_basic_map_free(map->p[i]);
2322 if (i != map->n - 1)
2323 map->p[i] = map->p[map->n - 1];
2326 map = isl_map_apply_range(isl_map_copy(qc), map);
2327 map = isl_map_apply_range(map, qc);
2329 return isl_map_union(tc, compute_closure_omega(map));
2332 return box_closure(map);
2338 /* Compute an overapproximation of the transitive closure of "map"
2339 * using a variation of the algorithm from
2340 * "Transitive Closure of Infinite Graphs and its Applications"
2341 * by Kelly et al. and check whether the result is definitely exact.
2343 static __isl_give isl_map *transitive_closure_omega(__isl_take isl_map *map,
2348 app = compute_closure_omega(isl_map_copy(map));
2351 *exact = check_exactness_omega(map, app);
2357 /* Compute the transitive closure of "map", or an overapproximation.
2358 * If the result is exact, then *exact is set to 1.
2359 * Simply use map_power to compute the powers of map, but tell
2360 * it to project out the lengths of the paths instead of equating
2361 * the length to a parameter.
2363 __isl_give isl_map *isl_map_transitive_closure(__isl_take isl_map *map,
2371 if (map->ctx->opt->closure == ISL_CLOSURE_OMEGA)
2372 return transitive_closure_omega(map, exact);
2374 param = isl_map_dim(map, isl_dim_param);
2375 map = map_power(map, param, exact, 1);