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 int isl_map_is_transitively_closed(__isl_keep isl_map *map)
21 map2 = isl_map_apply_range(isl_map_copy(map), isl_map_copy(map));
22 closed = isl_map_is_subset(map2, map);
28 /* Given a map that represents a path with the length of the path
29 * encoded as the difference between the last output coordindate
30 * and the last input coordinate, set this length to either
31 * exactly "length" (if "exactly" is set) or at least "length"
32 * (if "exactly" is not set).
34 static __isl_give isl_map *set_path_length(__isl_take isl_map *map,
35 int exactly, int length)
38 struct isl_basic_map *bmap;
47 dim = isl_map_get_dim(map);
48 d = isl_dim_size(dim, isl_dim_in);
49 nparam = isl_dim_size(dim, isl_dim_param);
50 bmap = isl_basic_map_alloc_dim(dim, 0, 1, 1);
52 k = isl_basic_map_alloc_equality(bmap);
55 k = isl_basic_map_alloc_inequality(bmap);
60 isl_seq_clr(c, 1 + isl_basic_map_total_dim(bmap));
61 isl_int_set_si(c[0], -length);
62 isl_int_set_si(c[1 + nparam + d - 1], -1);
63 isl_int_set_si(c[1 + nparam + d + d - 1], 1);
65 bmap = isl_basic_map_finalize(bmap);
66 map = isl_map_intersect(map, isl_map_from_basic_map(bmap));
70 isl_basic_map_free(bmap);
75 /* Check whether the overapproximation of the power of "map" is exactly
76 * the power of "map". Let R be "map" and A_k the overapproximation.
77 * The approximation is exact if
80 * A_k = A_{k-1} \circ R k >= 2
82 * Since A_k is known to be an overapproximation, we only need to check
85 * A_k \subset A_{k-1} \circ R k >= 2
87 * In practice, "app" has an extra input and output coordinate
88 * to encode the length of the path. So, we first need to add
89 * this coordinate to "map" and set the length of the path to
92 static int check_power_exactness(__isl_take isl_map *map,
93 __isl_take isl_map *app)
99 map = isl_map_add(map, isl_dim_in, 1);
100 map = isl_map_add(map, isl_dim_out, 1);
101 map = set_path_length(map, 1, 1);
103 app_1 = set_path_length(isl_map_copy(app), 1, 1);
105 exact = isl_map_is_subset(app_1, map);
108 if (!exact || exact < 0) {
114 app_1 = set_path_length(isl_map_copy(app), 0, 1);
115 app_2 = set_path_length(app, 0, 2);
116 app_1 = isl_map_apply_range(map, app_1);
118 exact = isl_map_is_subset(app_2, app_1);
126 /* Check whether the overapproximation of the power of "map" is exactly
127 * the power of "map", possibly after projecting out the power (if "project"
130 * If "project" is set and if "steps" can only result in acyclic paths,
133 * A = R \cup (A \circ R)
135 * where A is the overapproximation with the power projected out, i.e.,
136 * an overapproximation of the transitive closure.
137 * More specifically, since A is known to be an overapproximation, we check
139 * A \subset R \cup (A \circ R)
141 * Otherwise, we check if the power is exact.
143 * Note that "app" has an extra input and output coordinate to encode
144 * the length of the part. If we are only interested in the transitive
145 * closure, then we can simply project out these coordinates first.
147 static int check_exactness(__isl_take isl_map *map, __isl_take isl_map *app,
155 return check_power_exactness(map, app);
157 d = isl_map_dim(map, isl_dim_in);
158 app = set_path_length(app, 0, 1);
159 app = isl_map_project_out(app, isl_dim_in, d, 1);
160 app = isl_map_project_out(app, isl_dim_out, d, 1);
162 test = isl_map_apply_range(isl_map_copy(map), isl_map_copy(app));
163 test = isl_map_union(test, isl_map_copy(map));
165 exact = isl_map_is_subset(app, test);
176 * The transitive closure implementation is based on the paper
177 * "Computing the Transitive Closure of a Union of Affine Integer
178 * Tuple Relations" by Anna Beletska, Denis Barthou, Wlodzimierz Bielecki and
182 /* Given a set of n offsets v_i (the rows of "steps"), construct a relation
183 * of the given dimension specification (Z^{n+1} -> Z^{n+1})
184 * that maps an element x to any element that can be reached
185 * by taking a non-negative number of steps along any of
186 * the extended offsets v'_i = [v_i 1].
189 * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v'_i }
191 * For any element in this relation, the number of steps taken
192 * is equal to the difference in the final coordinates.
194 static __isl_give isl_map *path_along_steps(__isl_take isl_dim *dim,
195 __isl_keep isl_mat *steps)
198 struct isl_basic_map *path = NULL;
206 d = isl_dim_size(dim, isl_dim_in);
208 nparam = isl_dim_size(dim, isl_dim_param);
210 path = isl_basic_map_alloc_dim(isl_dim_copy(dim), n, d, n);
212 for (i = 0; i < n; ++i) {
213 k = isl_basic_map_alloc_div(path);
216 isl_assert(steps->ctx, i == k, goto error);
217 isl_int_set_si(path->div[k][0], 0);
220 for (i = 0; i < d; ++i) {
221 k = isl_basic_map_alloc_equality(path);
224 isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
225 isl_int_set_si(path->eq[k][1 + nparam + i], 1);
226 isl_int_set_si(path->eq[k][1 + nparam + d + i], -1);
228 for (j = 0; j < n; ++j)
229 isl_int_set_si(path->eq[k][1 + nparam + 2 * d + j], 1);
231 for (j = 0; j < n; ++j)
232 isl_int_set(path->eq[k][1 + nparam + 2 * d + j],
236 for (i = 0; i < n; ++i) {
237 k = isl_basic_map_alloc_inequality(path);
240 isl_seq_clr(path->ineq[k], 1 + isl_basic_map_total_dim(path));
241 isl_int_set_si(path->ineq[k][1 + nparam + 2 * d + i], 1);
246 path = isl_basic_map_simplify(path);
247 path = isl_basic_map_finalize(path);
248 return isl_map_from_basic_map(path);
251 isl_basic_map_free(path);
260 /* Check whether the parametric constant term of constraint c is never
261 * positive in "bset".
263 static int parametric_constant_never_positive(__isl_keep isl_basic_set *bset,
264 isl_int *c, int *div_purity)
273 n_div = isl_basic_set_dim(bset, isl_dim_div);
274 d = isl_basic_set_dim(bset, isl_dim_set);
275 nparam = isl_basic_set_dim(bset, isl_dim_param);
277 bset = isl_basic_set_copy(bset);
278 bset = isl_basic_set_cow(bset);
279 bset = isl_basic_set_extend_constraints(bset, 0, 1);
280 k = isl_basic_set_alloc_inequality(bset);
283 isl_seq_clr(bset->ineq[k], 1 + isl_basic_set_total_dim(bset));
284 isl_seq_cpy(bset->ineq[k], c, 1 + nparam);
285 for (i = 0; i < n_div; ++i) {
286 if (div_purity[i] != PURE_PARAM)
288 isl_int_set(bset->ineq[k][1 + nparam + d + i],
289 c[1 + nparam + d + i]);
291 isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);
292 empty = isl_basic_set_is_empty(bset);
293 isl_basic_set_free(bset);
297 isl_basic_set_free(bset);
301 /* Return PURE_PARAM if only the coefficients of the parameters are non-zero.
302 * Return PURE_VAR if only the coefficients of the set variables are non-zero.
303 * Return MIXED if only the coefficients of the parameters and the set
304 * variables are non-zero and if moreover the parametric constant
305 * can never attain positive values.
306 * Return IMPURE otherwise.
308 static int purity(__isl_keep isl_basic_set *bset, isl_int *c, int *div_purity,
318 n_div = isl_basic_set_dim(bset, isl_dim_div);
319 d = isl_basic_set_dim(bset, isl_dim_set);
320 nparam = isl_basic_set_dim(bset, isl_dim_param);
322 for (i = 0; i < n_div; ++i) {
323 if (isl_int_is_zero(c[1 + nparam + d + i]))
325 switch (div_purity[i]) {
326 case PURE_PARAM: p = 1; break;
327 case PURE_VAR: v = 1; break;
328 default: return IMPURE;
331 if (!p && isl_seq_first_non_zero(c + 1, nparam) == -1)
333 if (!v && isl_seq_first_non_zero(c + 1 + nparam, d) == -1)
336 empty = parametric_constant_never_positive(bset, c, div_purity);
337 if (eq && empty >= 0 && !empty) {
338 isl_seq_neg(c, c, 1 + nparam + d + n_div);
339 empty = parametric_constant_never_positive(bset, c, div_purity);
342 return empty < 0 ? -1 : empty ? MIXED : IMPURE;
345 /* Return an array of integers indicating the type of each div in bset.
346 * If the div is (recursively) defined in terms of only the parameters,
347 * then the type is PURE_PARAM.
348 * If the div is (recursively) defined in terms of only the set variables,
349 * then the type is PURE_VAR.
350 * Otherwise, the type is IMPURE.
352 static __isl_give int *get_div_purity(__isl_keep isl_basic_set *bset)
363 n_div = isl_basic_set_dim(bset, isl_dim_div);
364 d = isl_basic_set_dim(bset, isl_dim_set);
365 nparam = isl_basic_set_dim(bset, isl_dim_param);
367 div_purity = isl_alloc_array(bset->ctx, int, n_div);
371 for (i = 0; i < bset->n_div; ++i) {
373 if (isl_int_is_zero(bset->div[i][0])) {
374 div_purity[i] = IMPURE;
377 if (isl_seq_first_non_zero(bset->div[i] + 2, nparam) != -1)
379 if (isl_seq_first_non_zero(bset->div[i] + 2 + nparam, d) != -1)
381 for (j = 0; j < i; ++j) {
382 if (isl_int_is_zero(bset->div[i][2 + nparam + d + j]))
384 switch (div_purity[j]) {
385 case PURE_PARAM: p = 1; break;
386 case PURE_VAR: v = 1; break;
387 default: p = v = 1; break;
390 div_purity[i] = v ? p ? IMPURE : PURE_VAR : PURE_PARAM;
396 /* Given a path with the as yet unconstrained length at position "pos",
397 * check if setting the length to zero results in only the identity
400 int empty_path_is_identity(__isl_keep isl_basic_map *path, unsigned pos)
402 isl_basic_map *test = NULL;
403 isl_basic_map *id = NULL;
407 test = isl_basic_map_copy(path);
408 test = isl_basic_map_extend_constraints(test, 1, 0);
409 k = isl_basic_map_alloc_equality(test);
412 isl_seq_clr(test->eq[k], 1 + isl_basic_map_total_dim(test));
413 isl_int_set_si(test->eq[k][pos], 1);
414 id = isl_basic_map_identity(isl_dim_domain(isl_basic_map_get_dim(path)));
415 is_id = isl_basic_map_is_equal(test, id);
416 isl_basic_map_free(test);
417 isl_basic_map_free(id);
420 isl_basic_map_free(test);
424 __isl_give isl_basic_map *add_delta_constraints(__isl_take isl_basic_map *path,
425 __isl_keep isl_basic_set *delta, unsigned off, unsigned nparam,
426 unsigned d, int *div_purity, int eq)
429 int n = eq ? delta->n_eq : delta->n_ineq;
430 isl_int **delta_c = eq ? delta->eq : delta->ineq;
433 n_div = isl_basic_set_dim(delta, isl_dim_div);
435 for (i = 0; i < n; ++i) {
437 int p = purity(delta, delta_c[i], div_purity, eq);
442 if (eq && p != MIXED) {
443 k = isl_basic_map_alloc_equality(path);
444 path_c = path->eq[k];
446 k = isl_basic_map_alloc_inequality(path);
447 path_c = path->ineq[k];
451 isl_seq_clr(path_c, 1 + isl_basic_map_total_dim(path));
453 isl_seq_cpy(path_c + off,
454 delta_c[i] + 1 + nparam, d);
455 isl_int_set(path_c[off + d], delta_c[i][0]);
456 } else if (p == PURE_PARAM) {
457 isl_seq_cpy(path_c, delta_c[i], 1 + nparam);
459 isl_seq_cpy(path_c + off,
460 delta_c[i] + 1 + nparam, d);
461 isl_seq_cpy(path_c, delta_c[i], 1 + nparam);
463 isl_seq_cpy(path_c + off - n_div,
464 delta_c[i] + 1 + nparam + d, n_div);
469 isl_basic_map_free(path);
473 /* Given a set of offsets "delta", construct a relation of the
474 * given dimension specification (Z^{n+1} -> Z^{n+1}) that
475 * is an overapproximation of the relations that
476 * maps an element x to any element that can be reached
477 * by taking a non-negative number of steps along any of
478 * the elements in "delta".
479 * That is, construct an approximation of
481 * { [x] -> [y] : exists f \in \delta, k \in Z :
482 * y = x + k [f, 1] and k >= 0 }
484 * For any element in this relation, the number of steps taken
485 * is equal to the difference in the final coordinates.
487 * In particular, let delta be defined as
489 * \delta = [p] -> { [x] : A x + a >= and B p + b >= 0 and
490 * C x + C'p + c >= 0 and
491 * D x + D'p + d >= 0 }
493 * where the constraints C x + C'p + c >= 0 are such that the parametric
494 * constant term of each constraint j, "C_j x + C'_j p + c_j",
495 * can never attain positive values, then the relation is constructed as
497 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
498 * A f + k a >= 0 and B p + b >= 0 and
499 * C f + C'p + c >= 0 and k >= 1 }
500 * union { [x] -> [x] }
502 * If the zero-length paths happen to correspond exactly to the identity
503 * mapping, then we return
505 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
506 * A f + k a >= 0 and B p + b >= 0 and
507 * C f + C'p + c >= 0 and k >= 0 }
511 * Existentially quantified variables in \delta are handled by
512 * classifying them as independent of the parameters, purely
513 * parameter dependent and others. Constraints containing
514 * any of the other existentially quantified variables are removed.
515 * This is safe, but leads to an additional overapproximation.
517 static __isl_give isl_map *path_along_delta(__isl_take isl_dim *dim,
518 __isl_take isl_basic_set *delta)
520 isl_basic_map *path = NULL;
527 int *div_purity = NULL;
531 n_div = isl_basic_set_dim(delta, isl_dim_div);
532 d = isl_basic_set_dim(delta, isl_dim_set);
533 nparam = isl_basic_set_dim(delta, isl_dim_param);
534 path = isl_basic_map_alloc_dim(isl_dim_copy(dim), n_div + d + 1,
535 d + 1 + delta->n_eq, delta->n_eq + delta->n_ineq + 1);
536 off = 1 + nparam + 2 * (d + 1) + n_div;
538 for (i = 0; i < n_div + d + 1; ++i) {
539 k = isl_basic_map_alloc_div(path);
542 isl_int_set_si(path->div[k][0], 0);
545 for (i = 0; i < d + 1; ++i) {
546 k = isl_basic_map_alloc_equality(path);
549 isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
550 isl_int_set_si(path->eq[k][1 + nparam + i], 1);
551 isl_int_set_si(path->eq[k][1 + nparam + d + 1 + i], -1);
552 isl_int_set_si(path->eq[k][off + i], 1);
555 div_purity = get_div_purity(delta);
559 path = add_delta_constraints(path, delta, off, nparam, d, div_purity, 1);
560 path = add_delta_constraints(path, delta, off, nparam, d, div_purity, 0);
562 is_id = empty_path_is_identity(path, off + d);
566 k = isl_basic_map_alloc_inequality(path);
569 isl_seq_clr(path->ineq[k], 1 + isl_basic_map_total_dim(path));
571 isl_int_set_si(path->ineq[k][0], -1);
572 isl_int_set_si(path->ineq[k][off + d], 1);
575 isl_basic_set_free(delta);
576 path = isl_basic_map_finalize(path);
579 return isl_map_from_basic_map(path);
581 return isl_basic_map_union(path,
582 isl_basic_map_identity(isl_dim_domain(dim)));
586 isl_basic_set_free(delta);
587 isl_basic_map_free(path);
591 /* Given a dimension specification Z^{n+1} -> Z^{n+1} and a parameter "param",
592 * construct a map that equates the parameter to the difference
593 * in the final coordinates and imposes that this difference is positive.
596 * { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
598 static __isl_give isl_map *equate_parameter_to_length(__isl_take isl_dim *dim,
601 struct isl_basic_map *bmap;
606 d = isl_dim_size(dim, isl_dim_in);
607 nparam = isl_dim_size(dim, isl_dim_param);
608 bmap = isl_basic_map_alloc_dim(dim, 0, 1, 1);
609 k = isl_basic_map_alloc_equality(bmap);
612 isl_seq_clr(bmap->eq[k], 1 + isl_basic_map_total_dim(bmap));
613 isl_int_set_si(bmap->eq[k][1 + param], -1);
614 isl_int_set_si(bmap->eq[k][1 + nparam + d - 1], -1);
615 isl_int_set_si(bmap->eq[k][1 + nparam + d + d - 1], 1);
617 k = isl_basic_map_alloc_inequality(bmap);
620 isl_seq_clr(bmap->ineq[k], 1 + isl_basic_map_total_dim(bmap));
621 isl_int_set_si(bmap->ineq[k][1 + param], 1);
622 isl_int_set_si(bmap->ineq[k][0], -1);
624 bmap = isl_basic_map_finalize(bmap);
625 return isl_map_from_basic_map(bmap);
627 isl_basic_map_free(bmap);
631 /* Check whether "path" is acyclic, where the last coordinates of domain
632 * and range of path encode the number of steps taken.
633 * That is, check whether
635 * { d | d = y - x and (x,y) in path }
637 * does not contain any element with positive last coordinate (positive length)
638 * and zero remaining coordinates (cycle).
640 static int is_acyclic(__isl_take isl_map *path)
645 struct isl_set *delta;
647 delta = isl_map_deltas(path);
648 dim = isl_set_dim(delta, isl_dim_set);
649 for (i = 0; i < dim; ++i) {
651 delta = isl_set_lower_bound_si(delta, isl_dim_set, i, 1);
653 delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
656 acyclic = isl_set_is_empty(delta);
662 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
663 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
664 * construct a map that is an overapproximation of the map
665 * that takes an element from the space D \times Z to another
666 * element from the same space, such that the first n coordinates of the
667 * difference between them is a sum of differences between images
668 * and pre-images in one of the R_i and such that the last coordinate
669 * is equal to the number of steps taken.
672 * \Delta_i = { y - x | (x, y) in R_i }
674 * then the constructed map is an overapproximation of
676 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
677 * d = (\sum_i k_i \delta_i, \sum_i k_i) }
679 * The elements of the singleton \Delta_i's are collected as the
680 * rows of the steps matrix. For all these \Delta_i's together,
681 * a single path is constructed.
682 * For each of the other \Delta_i's, we compute an overapproximation
683 * of the paths along elements of \Delta_i.
684 * Since each of these paths performs an addition, composition is
685 * symmetric and we can simply compose all resulting paths in any order.
687 static __isl_give isl_map *construct_extended_path(__isl_take isl_dim *dim,
688 __isl_keep isl_map *map, int *project)
690 struct isl_mat *steps = NULL;
691 struct isl_map *path = NULL;
695 d = isl_map_dim(map, isl_dim_in);
697 path = isl_map_identity(isl_dim_domain(isl_dim_copy(dim)));
699 steps = isl_mat_alloc(map->ctx, map->n, d);
704 for (i = 0; i < map->n; ++i) {
705 struct isl_basic_set *delta;
707 delta = isl_basic_map_deltas(isl_basic_map_copy(map->p[i]));
709 for (j = 0; j < d; ++j) {
712 fixed = isl_basic_set_fast_dim_is_fixed(delta, j,
715 isl_basic_set_free(delta);
724 path = isl_map_apply_range(path,
725 path_along_delta(isl_dim_copy(dim), delta));
726 path = isl_map_coalesce(path);
728 isl_basic_set_free(delta);
735 path = isl_map_apply_range(path,
736 path_along_steps(isl_dim_copy(dim), steps));
739 if (project && *project) {
740 *project = is_acyclic(isl_map_copy(path));
755 static int isl_set_overlaps(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
760 i = isl_set_intersect(isl_set_copy(set1), isl_set_copy(set2));
761 no_overlap = isl_set_is_empty(i);
764 return no_overlap < 0 ? -1 : !no_overlap;
767 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
768 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
769 * construct a map that is an overapproximation of the map
770 * that takes an element from the dom R \times Z to an
771 * element from ran R \times Z, such that the first n coordinates of the
772 * difference between them is a sum of differences between images
773 * and pre-images in one of the R_i and such that the last coordinate
774 * is equal to the number of steps taken.
777 * \Delta_i = { y - x | (x, y) in R_i }
779 * then the constructed map is an overapproximation of
781 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
782 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
783 * x in dom R and x + d in ran R and
786 static __isl_give isl_map *construct_component(__isl_take isl_dim *dim,
787 __isl_keep isl_map *map, int *exact, int project)
789 struct isl_set *domain = NULL;
790 struct isl_set *range = NULL;
791 struct isl_map *app = NULL;
792 struct isl_map *path = NULL;
794 domain = isl_map_domain(isl_map_copy(map));
795 domain = isl_set_coalesce(domain);
796 range = isl_map_range(isl_map_copy(map));
797 range = isl_set_coalesce(range);
798 if (!isl_set_overlaps(domain, range)) {
799 isl_set_free(domain);
803 map = isl_map_copy(map);
804 map = isl_map_add(map, isl_dim_in, 1);
805 map = isl_map_add(map, isl_dim_out, 1);
806 map = set_path_length(map, 1, 1);
809 app = isl_map_from_domain_and_range(domain, range);
810 app = isl_map_add(app, isl_dim_in, 1);
811 app = isl_map_add(app, isl_dim_out, 1);
813 path = construct_extended_path(isl_dim_copy(dim), map,
814 exact && *exact ? &project : NULL);
815 app = isl_map_intersect(app, path);
817 if (exact && *exact &&
818 (*exact = check_exactness(isl_map_copy(map), isl_map_copy(app),
823 app = set_path_length(app, 0, 1);
831 /* Call construct_component and, if "project" is set, project out
832 * the final coordinates.
834 static __isl_give isl_map *construct_projected_component(
835 __isl_take isl_dim *dim,
836 __isl_keep isl_map *map, int *exact, int project)
843 d = isl_dim_size(dim, isl_dim_in);
845 app = construct_component(dim, map, exact, project);
847 app = isl_map_project_out(app, isl_dim_in, d - 1, 1);
848 app = isl_map_project_out(app, isl_dim_out, d - 1, 1);
853 /* Compute an extended version, i.e., with path lengths, of
854 * an overapproximation of the transitive closure of "bmap"
855 * with path lengths greater than or equal to zero and with
856 * domain and range equal to "dom".
858 static __isl_give isl_map *q_closure(__isl_take isl_dim *dim,
859 __isl_take isl_set *dom, __isl_keep isl_basic_map *bmap, int *exact)
866 dom = isl_set_add(dom, isl_dim_set, 1);
867 app = isl_map_from_domain_and_range(dom, isl_set_copy(dom));
868 map = isl_map_from_basic_map(isl_basic_map_copy(bmap));
869 path = construct_extended_path(dim, map, &project);
870 app = isl_map_intersect(app, path);
872 if ((*exact = check_exactness(map, isl_map_copy(app), project)) < 0)
881 /* Check whether qc has any elements of length at least one
882 * with domain and/or range outside of dom and ran.
884 static int has_spurious_elements(__isl_keep isl_map *qc,
885 __isl_keep isl_set *dom, __isl_keep isl_set *ran)
891 if (!qc || !dom || !ran)
894 d = isl_map_dim(qc, isl_dim_in);
896 qc = isl_map_copy(qc);
897 qc = set_path_length(qc, 0, 1);
898 qc = isl_map_project_out(qc, isl_dim_in, d - 1, 1);
899 qc = isl_map_project_out(qc, isl_dim_out, d - 1, 1);
901 s = isl_map_domain(isl_map_copy(qc));
902 subset = isl_set_is_subset(s, dom);
911 s = isl_map_range(qc);
912 subset = isl_set_is_subset(s, ran);
915 return subset < 0 ? -1 : !subset;
924 /* For each basic map in "map", except i, check whether it combines
925 * with the transitive closure that is reflexive on C combines
926 * to the left and to the right.
930 * dom map_j \subseteq C
932 * then right[j] is set to 1. Otherwise, if
934 * ran map_i \cap dom map_j = \emptyset
936 * then right[j] is set to 0. Otherwise, composing to the right
939 * Similar, for composing to the left, we have if
941 * ran map_j \subseteq C
943 * then left[j] is set to 1. Otherwise, if
945 * dom map_i \cap ran map_j = \emptyset
947 * then left[j] is set to 0. Otherwise, composing to the left
950 * The return value is or'd with LEFT if composing to the left
951 * is possible and with RIGHT if composing to the right is possible.
953 static int composability(__isl_keep isl_set *C, int i,
954 isl_set **dom, isl_set **ran, int *left, int *right,
955 __isl_keep isl_map *map)
961 for (j = 0; j < map->n && ok; ++j) {
962 int overlaps, subset;
968 dom[j] = isl_set_from_basic_set(
969 isl_basic_map_domain(
970 isl_basic_map_copy(map->p[j])));
973 overlaps = isl_set_overlaps(ran[i], dom[j]);
979 subset = isl_set_is_subset(dom[j], C);
991 ran[j] = isl_set_from_basic_set(
993 isl_basic_map_copy(map->p[j])));
996 overlaps = isl_set_overlaps(dom[i], ran[j]);
1002 subset = isl_set_is_subset(ran[j], C);
1016 /* Return a map that is a union of the basic maps in "map", except i,
1017 * composed to left and right with qc based on the entries of "left"
1020 static __isl_give isl_map *compose(__isl_keep isl_map *map, int i,
1021 __isl_take isl_map *qc, int *left, int *right)
1026 comp = isl_map_empty(isl_map_get_dim(map));
1027 for (j = 0; j < map->n; ++j) {
1033 map_j = isl_map_from_basic_map(isl_basic_map_copy(map->p[j]));
1034 if (left && left[j])
1035 map_j = isl_map_apply_range(map_j, isl_map_copy(qc));
1036 if (right && right[j])
1037 map_j = isl_map_apply_range(isl_map_copy(qc), map_j);
1038 comp = isl_map_union(comp, map_j);
1041 comp = isl_map_compute_divs(comp);
1042 comp = isl_map_coalesce(comp);
1049 /* Compute the transitive closure of "map" incrementally by
1056 * map_i^+ \cup ((id \cup map_i^) \circ qc^+)
1060 * map_i^+ \cup (qc^+ \circ (id \cup map_i^))
1062 * depending on whether left or right are NULL.
1064 static __isl_give isl_map *compute_incremental(
1065 __isl_take isl_dim *dim, __isl_keep isl_map *map,
1066 int i, __isl_take isl_map *qc, int *left, int *right, int *exact)
1070 isl_map *rtc = NULL;
1074 isl_assert(map->ctx, left || right, goto error);
1076 map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
1077 tc = construct_projected_component(isl_dim_copy(dim), map_i,
1079 isl_map_free(map_i);
1082 qc = isl_map_transitive_closure(qc, exact);
1088 return isl_map_universe(isl_map_get_dim(map));
1091 if (!left || !right)
1092 rtc = isl_map_union(isl_map_copy(tc),
1093 isl_map_identity(isl_dim_domain(isl_map_get_dim(tc))));
1095 qc = isl_map_apply_range(rtc, qc);
1097 qc = isl_map_apply_range(qc, rtc);
1098 qc = isl_map_union(tc, qc);
1109 /* Given a map "map", try to find a basic map such that
1110 * map^+ can be computed as
1112 * map^+ = map_i^+ \cup
1113 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1115 * with C the simple hull of the domain and range of the input map.
1116 * map_i^ \cup Id_C is computed by allowing the path lengths to be zero
1117 * and by intersecting domain and range with C.
1118 * Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
1119 * Also, we only use the incremental computation if all the transitive
1120 * closures are exact and if the number of basic maps in the union,
1121 * after computing the integer divisions, is smaller than the number
1122 * of basic maps in the input map.
1124 static int incemental_on_entire_domain(__isl_keep isl_dim *dim,
1125 __isl_keep isl_map *map,
1126 isl_set **dom, isl_set **ran, int *left, int *right,
1127 __isl_give isl_map **res)
1135 C = isl_set_union(isl_map_domain(isl_map_copy(map)),
1136 isl_map_range(isl_map_copy(map)));
1137 C = isl_set_from_basic_set(isl_set_simple_hull(C));
1145 d = isl_map_dim(map, isl_dim_in);
1147 for (i = 0; i < map->n; ++i) {
1149 int exact_i, spurious;
1151 dom[i] = isl_set_from_basic_set(isl_basic_map_domain(
1152 isl_basic_map_copy(map->p[i])));
1153 ran[i] = isl_set_from_basic_set(isl_basic_map_range(
1154 isl_basic_map_copy(map->p[i])));
1155 qc = q_closure(isl_dim_copy(dim), isl_set_copy(C),
1156 map->p[i], &exact_i);
1163 spurious = has_spurious_elements(qc, dom[i], ran[i]);
1170 qc = isl_map_project_out(qc, isl_dim_in, d, 1);
1171 qc = isl_map_project_out(qc, isl_dim_out, d, 1);
1172 qc = isl_map_compute_divs(qc);
1173 for (j = 0; j < map->n; ++j)
1174 left[j] = right[j] = 1;
1175 qc = compose(map, i, qc, left, right);
1178 if (qc->n >= map->n) {
1182 *res = compute_incremental(isl_dim_copy(dim), map, i, qc,
1183 left, right, &exact_i);
1194 return *res != NULL;
1200 /* Try and compute the transitive closure of "map" as
1202 * map^+ = map_i^+ \cup
1203 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1205 * with C either the simple hull of the domain and range of the entire
1206 * map or the simple hull of domain and range of map_i.
1208 static __isl_give isl_map *incremental_closure(__isl_take isl_dim *dim,
1209 __isl_keep isl_map *map, int *exact, int project)
1212 isl_set **dom = NULL;
1213 isl_set **ran = NULL;
1218 isl_map *res = NULL;
1221 return construct_projected_component(dim, map, exact, project);
1226 return construct_projected_component(dim, map, exact, project);
1228 d = isl_map_dim(map, isl_dim_in);
1230 dom = isl_calloc_array(map->ctx, isl_set *, map->n);
1231 ran = isl_calloc_array(map->ctx, isl_set *, map->n);
1232 left = isl_calloc_array(map->ctx, int, map->n);
1233 right = isl_calloc_array(map->ctx, int, map->n);
1234 if (!ran || !dom || !left || !right)
1237 if (incemental_on_entire_domain(dim, map, dom, ran, left, right, &res) < 0)
1240 for (i = 0; !res && i < map->n; ++i) {
1242 int exact_i, spurious, comp;
1244 dom[i] = isl_set_from_basic_set(
1245 isl_basic_map_domain(
1246 isl_basic_map_copy(map->p[i])));
1250 ran[i] = isl_set_from_basic_set(
1251 isl_basic_map_range(
1252 isl_basic_map_copy(map->p[i])));
1255 C = isl_set_union(isl_set_copy(dom[i]),
1256 isl_set_copy(ran[i]));
1257 C = isl_set_from_basic_set(isl_set_simple_hull(C));
1264 comp = composability(C, i, dom, ran, left, right, map);
1265 if (!comp || comp < 0) {
1271 qc = q_closure(isl_dim_copy(dim), C, map->p[i], &exact_i);
1278 spurious = has_spurious_elements(qc, dom[i], ran[i]);
1285 qc = isl_map_project_out(qc, isl_dim_in, d, 1);
1286 qc = isl_map_project_out(qc, isl_dim_out, d, 1);
1287 qc = isl_map_compute_divs(qc);
1288 qc = compose(map, i, qc, (comp & LEFT) ? left : NULL,
1289 (comp & RIGHT) ? right : NULL);
1292 if (qc->n >= map->n) {
1296 res = compute_incremental(isl_dim_copy(dim), map, i, qc,
1297 (comp & LEFT) ? left : NULL,
1298 (comp & RIGHT) ? right : NULL, &exact_i);
1307 for (i = 0; i < map->n; ++i) {
1308 isl_set_free(dom[i]);
1309 isl_set_free(ran[i]);
1321 return construct_projected_component(dim, map, exact, project);
1324 for (i = 0; i < map->n; ++i)
1325 isl_set_free(dom[i]);
1328 for (i = 0; i < map->n; ++i)
1329 isl_set_free(ran[i]);
1337 /* Given an array of sets "set", add "dom" at position "pos"
1338 * and search for elements at earlier positions that overlap with "dom".
1339 * If any can be found, then merge all of them, together with "dom", into
1340 * a single set and assign the union to the first in the array,
1341 * which becomes the new group leader for all groups involved in the merge.
1342 * During the search, we only consider group leaders, i.e., those with
1343 * group[i] = i, as the other sets have already been combined
1344 * with one of the group leaders.
1346 static int merge(isl_set **set, int *group, __isl_take isl_set *dom, int pos)
1351 set[pos] = isl_set_copy(dom);
1353 for (i = pos - 1; i >= 0; --i) {
1359 o = isl_set_overlaps(set[i], dom);
1365 set[i] = isl_set_union(set[i], set[group[pos]]);
1366 set[group[pos]] = NULL;
1369 group[group[pos]] = i;
1380 /* Replace each entry in the n by n grid of maps by the cross product
1381 * with the relation { [i] -> [i + 1] }.
1383 static int add_length(__isl_keep isl_map *map, isl_map ***grid, int n)
1387 isl_basic_map *bstep;
1394 dim = isl_map_get_dim(map);
1395 nparam = isl_dim_size(dim, isl_dim_param);
1396 dim = isl_dim_drop(dim, isl_dim_in, 0, isl_dim_size(dim, isl_dim_in));
1397 dim = isl_dim_drop(dim, isl_dim_out, 0, isl_dim_size(dim, isl_dim_out));
1398 dim = isl_dim_add(dim, isl_dim_in, 1);
1399 dim = isl_dim_add(dim, isl_dim_out, 1);
1400 bstep = isl_basic_map_alloc_dim(dim, 0, 1, 0);
1401 k = isl_basic_map_alloc_equality(bstep);
1403 isl_basic_map_free(bstep);
1406 isl_seq_clr(bstep->eq[k], 1 + isl_basic_map_total_dim(bstep));
1407 isl_int_set_si(bstep->eq[k][0], 1);
1408 isl_int_set_si(bstep->eq[k][1 + nparam], 1);
1409 isl_int_set_si(bstep->eq[k][1 + nparam + 1], -1);
1410 bstep = isl_basic_map_finalize(bstep);
1411 step = isl_map_from_basic_map(bstep);
1413 for (i = 0; i < n; ++i)
1414 for (j = 0; j < n; ++j)
1415 grid[i][j] = isl_map_product(grid[i][j],
1416 isl_map_copy(step));
1423 /* Given a partition of the domains and ranges of the basic maps in "map",
1424 * apply the Floyd-Warshall algorithm with the elements in the partition
1427 * In particular, there are "n" elements in the partition and "group" is
1428 * an array of length 2 * map->n with entries in [0,n-1].
1430 * We first construct a matrix of relations based on the partition information,
1431 * apply Floyd-Warshall on this matrix of relations and then take the
1432 * union of all entries in the matrix as the final result.
1434 * The algorithm iterates over all vertices. In each step, the whole
1435 * matrix is updated to include all paths that go to the current vertex,
1436 * possibly stay there a while (including passing through earlier vertices)
1437 * and then come back. At the start of each iteration, the diagonal
1438 * element corresponding to the current vertex is replaced by its
1439 * transitive closure to account for all indirect paths that stay
1440 * in the current vertex.
1442 * If we are actually computing the power instead of the transitive closure,
1443 * i.e., when "project" is not set, then the result should have the
1444 * path lengths encoded as the difference between an extra pair of
1445 * coordinates. We therefore apply the nested transitive closures
1446 * to relations that include these lengths. In particular, we replace
1447 * the input relation by the cross product with the unit length relation
1448 * { [i] -> [i + 1] }.
1450 static __isl_give isl_map *floyd_warshall_with_groups(__isl_take isl_dim *dim,
1451 __isl_keep isl_map *map, int *exact, int project, int *group, int n)
1455 isl_map ***grid = NULL;
1463 return incremental_closure(dim, map, exact, project);
1466 grid = isl_calloc_array(map->ctx, isl_map **, n);
1469 for (i = 0; i < n; ++i) {
1470 grid[i] = isl_calloc_array(map->ctx, isl_map *, n);
1473 for (j = 0; j < n; ++j)
1474 grid[i][j] = isl_map_empty(isl_map_get_dim(map));
1477 for (k = 0; k < map->n; ++k) {
1479 j = group[2 * k + 1];
1480 grid[i][j] = isl_map_union(grid[i][j],
1481 isl_map_from_basic_map(
1482 isl_basic_map_copy(map->p[k])));
1485 if (!project && add_length(map, grid, n) < 0)
1488 for (r = 0; r < n; ++r) {
1490 grid[r][r] = isl_map_transitive_closure(grid[r][r],
1491 (exact && *exact) ? &r_exact : NULL);
1492 if (exact && *exact && !r_exact)
1495 for (p = 0; p < n; ++p)
1496 for (q = 0; q < n; ++q) {
1498 if (p == r && q == r)
1500 loop = isl_map_apply_range(
1501 isl_map_copy(grid[p][r]),
1502 isl_map_copy(grid[r][q]));
1503 grid[p][q] = isl_map_union(grid[p][q], loop);
1504 loop = isl_map_apply_range(
1505 isl_map_copy(grid[p][r]),
1506 isl_map_apply_range(
1507 isl_map_copy(grid[r][r]),
1508 isl_map_copy(grid[r][q])));
1509 grid[p][q] = isl_map_union(grid[p][q], loop);
1510 grid[p][q] = isl_map_coalesce(grid[p][q]);
1514 app = isl_map_empty(isl_map_get_dim(map));
1516 for (i = 0; i < n; ++i) {
1517 for (j = 0; j < n; ++j)
1518 app = isl_map_union(app, grid[i][j]);
1529 for (i = 0; i < n; ++i) {
1532 for (j = 0; j < n; ++j)
1533 isl_map_free(grid[i][j]);
1542 /* Check if the domains and ranges of the basic maps in "map" can
1543 * be partitioned, and if so, apply Floyd-Warshall on the elements
1544 * of the partition. Note that we also apply this algorithm
1545 * if we want to compute the power, i.e., when "project" is not set.
1546 * However, the results are unlikely to be exact since the recursive
1547 * calls inside the Floyd-Warshall algorithm typically result in
1548 * non-linear path lengths quite quickly.
1550 * To find the partition, we simply consider all of the domains
1551 * and ranges in turn and combine those that overlap.
1552 * "set" contains the partition elements and "group" indicates
1553 * to which partition element a given domain or range belongs.
1554 * The domain of basic map i corresponds to element 2 * i in these arrays,
1555 * while the domain corresponds to element 2 * i + 1.
1556 * During the construction group[k] is either equal to k,
1557 * in which case set[k] contains the union of all the domains and
1558 * ranges in the corresponding group, or is equal to some l < k,
1559 * with l another domain or range in the same group.
1561 static __isl_give isl_map *floyd_warshall(__isl_take isl_dim *dim,
1562 __isl_keep isl_map *map, int *exact, int project)
1565 isl_set **set = NULL;
1572 return incremental_closure(dim, map, exact, project);
1574 set = isl_calloc_array(map->ctx, isl_set *, 2 * map->n);
1575 group = isl_alloc_array(map->ctx, int, 2 * map->n);
1580 for (i = 0; i < map->n; ++i) {
1582 dom = isl_set_from_basic_set(isl_basic_map_domain(
1583 isl_basic_map_copy(map->p[i])));
1584 if (merge(set, group, dom, 2 * i) < 0)
1586 dom = isl_set_from_basic_set(isl_basic_map_range(
1587 isl_basic_map_copy(map->p[i])));
1588 if (merge(set, group, dom, 2 * i + 1) < 0)
1593 for (i = 0; i < 2 * map->n; ++i)
1597 group[i] = group[group[i]];
1599 for (i = 0; i < 2 * map->n; ++i)
1600 isl_set_free(set[i]);
1604 return floyd_warshall_with_groups(dim, map, exact, project, group, n);
1606 for (i = 0; i < 2 * map->n; ++i)
1607 isl_set_free(set[i]);
1614 /* Structure for representing the nodes in the graph being traversed
1615 * using Tarjan's algorithm.
1616 * index represents the order in which nodes are visited.
1617 * min_index is the index of the root of a (sub)component.
1618 * on_stack indicates whether the node is currently on the stack.
1620 struct basic_map_sort_node {
1625 /* Structure for representing the graph being traversed
1626 * using Tarjan's algorithm.
1627 * len is the number of nodes
1628 * node is an array of nodes
1629 * stack contains the nodes on the path from the root to the current node
1630 * sp is the stack pointer
1631 * index is the index of the last node visited
1632 * order contains the elements of the components separated by -1
1633 * op represents the current position in order
1635 * check_closed is set if we may have used the fact that
1636 * a pair of basic maps can be interchanged
1638 struct basic_map_sort {
1640 struct basic_map_sort_node *node;
1649 static void basic_map_sort_free(struct basic_map_sort *s)
1659 static struct basic_map_sort *basic_map_sort_alloc(struct isl_ctx *ctx, int len)
1661 struct basic_map_sort *s;
1664 s = isl_calloc_type(ctx, struct basic_map_sort);
1668 s->node = isl_alloc_array(ctx, struct basic_map_sort_node, len);
1671 for (i = 0; i < len; ++i)
1672 s->node[i].index = -1;
1673 s->stack = isl_alloc_array(ctx, int, len);
1676 s->order = isl_alloc_array(ctx, int, 2 * len);
1684 s->check_closed = 0;
1688 basic_map_sort_free(s);
1692 /* Check whether in the computation of the transitive closure
1693 * "bmap1" (R_1) should follow (or be part of the same component as)
1696 * That is check whether
1704 * If so, then there is no reason for R_1 to immediately follow R_2
1707 * *check_closed is set if the subset relation holds while
1708 * R_1 \circ R_2 is not empty.
1710 static int basic_map_follows(__isl_keep isl_basic_map *bmap1,
1711 __isl_keep isl_basic_map *bmap2, int *check_closed)
1713 struct isl_map *map12 = NULL;
1714 struct isl_map *map21 = NULL;
1717 map21 = isl_map_from_basic_map(
1718 isl_basic_map_apply_range(
1719 isl_basic_map_copy(bmap2),
1720 isl_basic_map_copy(bmap1)));
1721 subset = isl_map_is_empty(map21);
1725 isl_map_free(map21);
1729 map12 = isl_map_from_basic_map(
1730 isl_basic_map_apply_range(
1731 isl_basic_map_copy(bmap1),
1732 isl_basic_map_copy(bmap2)));
1734 subset = isl_map_is_subset(map21, map12);
1736 isl_map_free(map12);
1737 isl_map_free(map21);
1742 return subset < 0 ? -1 : !subset;
1744 isl_map_free(map21);
1748 /* Perform Tarjan's algorithm for computing the strongly connected components
1749 * in the graph with the disjuncts of "map" as vertices and with an
1750 * edge between any pair of disjuncts such that the first has
1751 * to be applied after the second.
1753 static int power_components_tarjan(struct basic_map_sort *s,
1754 __isl_keep isl_map *map, int i)
1758 s->node[i].index = s->index;
1759 s->node[i].min_index = s->index;
1760 s->node[i].on_stack = 1;
1762 s->stack[s->sp++] = i;
1764 for (j = s->len - 1; j >= 0; --j) {
1769 if (s->node[j].index >= 0 &&
1770 (!s->node[j].on_stack ||
1771 s->node[j].index > s->node[i].min_index))
1774 f = basic_map_follows(map->p[i], map->p[j], &s->check_closed);
1780 if (s->node[j].index < 0) {
1781 power_components_tarjan(s, map, j);
1782 if (s->node[j].min_index < s->node[i].min_index)
1783 s->node[i].min_index = s->node[j].min_index;
1784 } else if (s->node[j].index < s->node[i].min_index)
1785 s->node[i].min_index = s->node[j].index;
1788 if (s->node[i].index != s->node[i].min_index)
1792 j = s->stack[--s->sp];
1793 s->node[j].on_stack = 0;
1794 s->order[s->op++] = j;
1796 s->order[s->op++] = -1;
1801 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
1802 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
1803 * construct a map that is an overapproximation of the map
1804 * that takes an element from the dom R \times Z to an
1805 * element from ran R \times Z, such that the first n coordinates of the
1806 * difference between them is a sum of differences between images
1807 * and pre-images in one of the R_i and such that the last coordinate
1808 * is equal to the number of steps taken.
1809 * If "project" is set, then these final coordinates are not included,
1810 * i.e., a relation of type Z^n -> Z^n is returned.
1813 * \Delta_i = { y - x | (x, y) in R_i }
1815 * then the constructed map is an overapproximation of
1817 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1818 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
1819 * x in dom R and x + d in ran R }
1823 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1824 * d = (\sum_i k_i \delta_i) and
1825 * x in dom R and x + d in ran R }
1827 * if "project" is set.
1829 * We first split the map into strongly connected components, perform
1830 * the above on each component and then join the results in the correct
1831 * order, at each join also taking in the union of both arguments
1832 * to allow for paths that do not go through one of the two arguments.
1834 static __isl_give isl_map *construct_power_components(__isl_take isl_dim *dim,
1835 __isl_keep isl_map *map, int *exact, int project)
1838 struct isl_map *path = NULL;
1839 struct basic_map_sort *s = NULL;
1846 return floyd_warshall(dim, map, exact, project);
1848 s = basic_map_sort_alloc(map->ctx, map->n);
1851 for (i = map->n - 1; i >= 0; --i) {
1852 if (s->node[i].index >= 0)
1854 if (power_components_tarjan(s, map, i) < 0)
1859 if (s->check_closed && !exact)
1860 exact = &local_exact;
1866 path = isl_map_empty(isl_map_get_dim(map));
1868 path = isl_map_empty(isl_dim_copy(dim));
1870 struct isl_map *comp;
1871 isl_map *path_comp, *path_comb;
1872 comp = isl_map_alloc_dim(isl_map_get_dim(map), n, 0);
1873 while (s->order[i] != -1) {
1874 comp = isl_map_add_basic_map(comp,
1875 isl_basic_map_copy(map->p[s->order[i]]));
1879 path_comp = floyd_warshall(isl_dim_copy(dim),
1880 comp, exact, project);
1881 path_comb = isl_map_apply_range(isl_map_copy(path),
1882 isl_map_copy(path_comp));
1883 path = isl_map_union(path, path_comp);
1884 path = isl_map_union(path, path_comb);
1890 if (c > 1 && s->check_closed && !*exact) {
1893 closed = isl_map_is_transitively_closed(path);
1897 basic_map_sort_free(s);
1899 return floyd_warshall(dim, map, orig_exact, project);
1903 basic_map_sort_free(s);
1908 basic_map_sort_free(s);
1914 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
1915 * construct a map that is an overapproximation of the map
1916 * that takes an element from the space D to another
1917 * element from the same space, such that the difference between
1918 * them is a strictly positive sum of differences between images
1919 * and pre-images in one of the R_i.
1920 * The number of differences in the sum is equated to parameter "param".
1923 * \Delta_i = { y - x | (x, y) in R_i }
1925 * then the constructed map is an overapproximation of
1927 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1928 * d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
1931 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1932 * d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
1934 * if "project" is set.
1936 * If "project" is not set, then
1937 * we construct an extended mapping with an extra coordinate
1938 * that indicates the number of steps taken. In particular,
1939 * the difference in the last coordinate is equal to the number
1940 * of steps taken to move from a domain element to the corresponding
1943 static __isl_give isl_map *construct_power(__isl_keep isl_map *map,
1944 int *exact, int project)
1946 struct isl_map *app = NULL;
1947 struct isl_dim *dim = NULL;
1953 dim = isl_map_get_dim(map);
1955 d = isl_dim_size(dim, isl_dim_in);
1956 dim = isl_dim_add(dim, isl_dim_in, 1);
1957 dim = isl_dim_add(dim, isl_dim_out, 1);
1959 app = construct_power_components(isl_dim_copy(dim), map,
1967 /* Compute the positive powers of "map", or an overapproximation.
1968 * If the result is exact, then *exact is set to 1.
1970 * If project is set, then we are actually interested in the transitive
1971 * closure, so we can use a more relaxed exactness check.
1972 * The lengths of the paths are also projected out instead of being
1973 * encoded as the difference between an extra pair of final coordinates.
1975 static __isl_give isl_map *map_power(__isl_take isl_map *map,
1976 int *exact, int project)
1978 struct isl_map *app = NULL;
1986 isl_assert(map->ctx,
1987 isl_map_dim(map, isl_dim_in) == isl_map_dim(map, isl_dim_out),
1990 app = construct_power(map, exact, project);
2000 /* Compute the positive powers of "map", or an overapproximation.
2001 * The power is given by parameter "param". If the result is exact,
2002 * then *exact is set to 1.
2003 * map_power constructs an extended relation with the path lengths
2004 * encoded as the difference between the final coordinates.
2005 * In the final step, this difference is equated to the parameter "param"
2006 * and made positive. The extra coordinates are subsequently projected out.
2008 __isl_give isl_map *isl_map_power(__isl_take isl_map *map, unsigned param,
2018 isl_assert(map->ctx, param < isl_map_dim(map, isl_dim_param),
2021 d = isl_map_dim(map, isl_dim_in);
2023 map = isl_map_compute_divs(map);
2024 map = isl_map_coalesce(map);
2026 if (isl_map_fast_is_empty(map))
2029 map = map_power(map, exact, 0);
2031 dim = isl_map_get_dim(map);
2032 diff = equate_parameter_to_length(dim, param);
2033 map = isl_map_intersect(map, diff);
2034 map = isl_map_project_out(map, isl_dim_in, d, 1);
2035 map = isl_map_project_out(map, isl_dim_out, d, 1);
2043 /* Compute a relation that maps each element in the range of the input
2044 * relation to the lengths of all paths composed of edges in the input
2045 * relation that end up in the given range element.
2046 * The result may be an overapproximation, in which case *exact is set to 0.
2047 * The resulting relation is very similar to the power relation.
2048 * The difference are that the domain has been projected out, the
2049 * range has become the domain and the exponent is the range instead
2052 __isl_give isl_map *isl_map_reaching_path_lengths(__isl_take isl_map *map,
2063 d = isl_map_dim(map, isl_dim_in);
2064 param = isl_map_dim(map, isl_dim_param);
2066 map = isl_map_compute_divs(map);
2067 map = isl_map_coalesce(map);
2069 if (isl_map_fast_is_empty(map)) {
2072 map = isl_map_project_out(map, isl_dim_out, 0, d);
2073 map = isl_map_add(map, isl_dim_out, 1);
2077 map = map_power(map, exact, 0);
2079 map = isl_map_add(map, isl_dim_param, 1);
2080 dim = isl_map_get_dim(map);
2081 diff = equate_parameter_to_length(dim, param);
2082 map = isl_map_intersect(map, diff);
2083 map = isl_map_project_out(map, isl_dim_in, 0, d + 1);
2084 map = isl_map_project_out(map, isl_dim_out, d, 1);
2085 map = isl_map_reverse(map);
2086 map = isl_map_move_dims(map, isl_dim_out, 0, isl_dim_param, param, 1);
2091 /* Check whether equality i of bset is a pure stride constraint
2092 * on a single dimensions, i.e., of the form
2096 * with k a constant and e an existentially quantified variable.
2098 static int is_eq_stride(__isl_keep isl_basic_set *bset, int i)
2110 if (!isl_int_is_zero(bset->eq[i][0]))
2113 nparam = isl_basic_set_dim(bset, isl_dim_param);
2114 d = isl_basic_set_dim(bset, isl_dim_set);
2115 n_div = isl_basic_set_dim(bset, isl_dim_div);
2117 if (isl_seq_first_non_zero(bset->eq[i] + 1, nparam) != -1)
2119 pos1 = isl_seq_first_non_zero(bset->eq[i] + 1 + nparam, d);
2122 if (isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + pos1 + 1,
2123 d - pos1 - 1) != -1)
2126 pos2 = isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + d, n_div);
2129 if (isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + d + pos2 + 1,
2130 n_div - pos2 - 1) != -1)
2132 if (!isl_int_is_one(bset->eq[i][1 + nparam + pos1]) &&
2133 !isl_int_is_negone(bset->eq[i][1 + nparam + pos1]))
2139 /* Given a map, compute the smallest superset of this map that is of the form
2141 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2143 * (where p ranges over the (non-parametric) dimensions),
2144 * compute the transitive closure of this map, i.e.,
2146 * { i -> j : exists k > 0:
2147 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2149 * and intersect domain and range of this transitive closure with
2150 * the given domain and range.
2152 * If with_id is set, then try to include as much of the identity mapping
2153 * as possible, by computing
2155 * { i -> j : exists k >= 0:
2156 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2158 * instead (i.e., allow k = 0).
2160 * In practice, we compute the difference set
2162 * delta = { j - i | i -> j in map },
2164 * look for stride constraint on the individual dimensions and compute
2165 * (constant) lower and upper bounds for each individual dimension,
2166 * adding a constraint for each bound not equal to infinity.
2168 static __isl_give isl_map *box_closure_on_domain(__isl_take isl_map *map,
2169 __isl_take isl_set *dom, __isl_take isl_set *ran, int with_id)
2178 isl_map *app = NULL;
2179 isl_basic_set *aff = NULL;
2180 isl_basic_map *bmap = NULL;
2181 isl_vec *obj = NULL;
2186 delta = isl_map_deltas(isl_map_copy(map));
2188 aff = isl_set_affine_hull(isl_set_copy(delta));
2191 dim = isl_map_get_dim(map);
2192 d = isl_dim_size(dim, isl_dim_in);
2193 nparam = isl_dim_size(dim, isl_dim_param);
2194 total = isl_dim_total(dim);
2195 bmap = isl_basic_map_alloc_dim(dim,
2196 aff->n_div + 1, aff->n_div, 2 * d + 1);
2197 for (i = 0; i < aff->n_div + 1; ++i) {
2198 k = isl_basic_map_alloc_div(bmap);
2201 isl_int_set_si(bmap->div[k][0], 0);
2203 for (i = 0; i < aff->n_eq; ++i) {
2204 if (!is_eq_stride(aff, i))
2206 k = isl_basic_map_alloc_equality(bmap);
2209 isl_seq_clr(bmap->eq[k], 1 + nparam);
2210 isl_seq_cpy(bmap->eq[k] + 1 + nparam + d,
2211 aff->eq[i] + 1 + nparam, d);
2212 isl_seq_neg(bmap->eq[k] + 1 + nparam,
2213 aff->eq[i] + 1 + nparam, d);
2214 isl_seq_cpy(bmap->eq[k] + 1 + nparam + 2 * d,
2215 aff->eq[i] + 1 + nparam + d, aff->n_div);
2216 isl_int_set_si(bmap->eq[k][1 + total + aff->n_div], 0);
2218 obj = isl_vec_alloc(map->ctx, 1 + nparam + d);
2221 isl_seq_clr(obj->el, 1 + nparam + d);
2222 for (i = 0; i < d; ++ i) {
2223 enum isl_lp_result res;
2225 isl_int_set_si(obj->el[1 + nparam + i], 1);
2227 res = isl_set_solve_lp(delta, 0, obj->el, map->ctx->one, &opt,
2229 if (res == isl_lp_error)
2231 if (res == isl_lp_ok) {
2232 k = isl_basic_map_alloc_inequality(bmap);
2235 isl_seq_clr(bmap->ineq[k],
2236 1 + nparam + 2 * d + bmap->n_div);
2237 isl_int_set_si(bmap->ineq[k][1 + nparam + i], -1);
2238 isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], 1);
2239 isl_int_neg(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
2242 res = isl_set_solve_lp(delta, 1, obj->el, map->ctx->one, &opt,
2244 if (res == isl_lp_error)
2246 if (res == isl_lp_ok) {
2247 k = isl_basic_map_alloc_inequality(bmap);
2250 isl_seq_clr(bmap->ineq[k],
2251 1 + nparam + 2 * d + bmap->n_div);
2252 isl_int_set_si(bmap->ineq[k][1 + nparam + i], 1);
2253 isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], -1);
2254 isl_int_set(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
2257 isl_int_set_si(obj->el[1 + nparam + i], 0);
2259 k = isl_basic_map_alloc_inequality(bmap);
2262 isl_seq_clr(bmap->ineq[k],
2263 1 + nparam + 2 * d + bmap->n_div);
2265 isl_int_set_si(bmap->ineq[k][0], -1);
2266 isl_int_set_si(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], 1);
2268 app = isl_map_from_domain_and_range(dom, ran);
2271 isl_basic_set_free(aff);
2273 bmap = isl_basic_map_finalize(bmap);
2274 isl_set_free(delta);
2277 map = isl_map_from_basic_map(bmap);
2278 map = isl_map_intersect(map, app);
2283 isl_basic_map_free(bmap);
2284 isl_basic_set_free(aff);
2288 isl_set_free(delta);
2293 /* Given a map, compute the smallest superset of this map that is of the form
2295 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2297 * (where p ranges over the (non-parametric) dimensions),
2298 * compute the transitive closure of this map, i.e.,
2300 * { i -> j : exists k > 0:
2301 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2303 * and intersect domain and range of this transitive closure with
2304 * domain and range of the original map.
2306 static __isl_give isl_map *box_closure(__isl_take isl_map *map)
2311 domain = isl_map_domain(isl_map_copy(map));
2312 domain = isl_set_coalesce(domain);
2313 range = isl_map_range(isl_map_copy(map));
2314 range = isl_set_coalesce(range);
2316 return box_closure_on_domain(map, domain, range, 0);
2319 /* Given a map, compute the smallest superset of this map that is of the form
2321 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2323 * (where p ranges over the (non-parametric) dimensions),
2324 * compute the transitive and partially reflexive closure of this map, i.e.,
2326 * { i -> j : exists k >= 0:
2327 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2329 * and intersect domain and range of this transitive closure with
2332 static __isl_give isl_map *box_closure_with_identity(__isl_take isl_map *map,
2333 __isl_take isl_set *dom)
2335 return box_closure_on_domain(map, dom, isl_set_copy(dom), 1);
2338 /* Check whether app is the transitive closure of map.
2339 * In particular, check that app is acyclic and, if so,
2342 * app \subset (map \cup (map \circ app))
2344 static int check_exactness_omega(__isl_keep isl_map *map,
2345 __isl_keep isl_map *app)
2349 int is_empty, is_exact;
2353 delta = isl_map_deltas(isl_map_copy(app));
2354 d = isl_set_dim(delta, isl_dim_set);
2355 for (i = 0; i < d; ++i)
2356 delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
2357 is_empty = isl_set_is_empty(delta);
2358 isl_set_free(delta);
2364 test = isl_map_apply_range(isl_map_copy(app), isl_map_copy(map));
2365 test = isl_map_union(test, isl_map_copy(map));
2366 is_exact = isl_map_is_subset(app, test);
2372 /* Check if basic map M_i can be combined with all the other
2373 * basic maps such that
2377 * can be computed as
2379 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2381 * In particular, check if we can compute a compact representation
2384 * M_i^* \circ M_j \circ M_i^*
2387 * Let M_i^? be an extension of M_i^+ that allows paths
2388 * of length zero, i.e., the result of box_closure(., 1).
2389 * The criterion, as proposed by Kelly et al., is that
2390 * id = M_i^? - M_i^+ can be represented as a basic map
2393 * id \circ M_j \circ id = M_j
2397 * If this function returns 1, then tc and qc are set to
2398 * M_i^+ and M_i^?, respectively.
2400 static int can_be_split_off(__isl_keep isl_map *map, int i,
2401 __isl_give isl_map **tc, __isl_give isl_map **qc)
2403 isl_map *map_i, *id = NULL;
2410 C = isl_set_union(isl_map_domain(isl_map_copy(map)),
2411 isl_map_range(isl_map_copy(map)));
2412 C = isl_set_from_basic_set(isl_set_simple_hull(C));
2416 map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
2417 *tc = box_closure(isl_map_copy(map_i));
2418 *qc = box_closure_with_identity(map_i, C);
2419 id = isl_map_subtract(isl_map_copy(*qc), isl_map_copy(*tc));
2423 if (id->n != 1 || (*qc)->n != 1)
2426 for (j = 0; j < map->n; ++j) {
2427 isl_map *map_j, *test;
2432 map_j = isl_map_from_basic_map(
2433 isl_basic_map_copy(map->p[j]));
2434 test = isl_map_apply_range(isl_map_copy(id),
2435 isl_map_copy(map_j));
2436 test = isl_map_apply_range(test, isl_map_copy(id));
2437 is_ok = isl_map_is_equal(test, map_j);
2438 isl_map_free(map_j);
2466 static __isl_give isl_map *box_closure_with_check(__isl_take isl_map *map,
2471 app = box_closure(isl_map_copy(map));
2473 *exact = check_exactness_omega(map, app);
2479 /* Compute an overapproximation of the transitive closure of "map"
2480 * using a variation of the algorithm from
2481 * "Transitive Closure of Infinite Graphs and its Applications"
2484 * We first check whether we can can split of any basic map M_i and
2491 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2493 * using a recursive call on the remaining map.
2495 * If not, we simply call box_closure on the whole map.
2497 static __isl_give isl_map *transitive_closure_omega(__isl_take isl_map *map,
2507 return box_closure_with_check(map, exact);
2509 for (i = 0; i < map->n; ++i) {
2512 ok = can_be_split_off(map, i, &tc, &qc);
2518 app = isl_map_alloc_dim(isl_map_get_dim(map), map->n - 1, 0);
2520 for (j = 0; j < map->n; ++j) {
2523 app = isl_map_add_basic_map(app,
2524 isl_basic_map_copy(map->p[j]));
2527 app = isl_map_apply_range(isl_map_copy(qc), app);
2528 app = isl_map_apply_range(app, qc);
2530 app = isl_map_union(tc, transitive_closure_omega(app, NULL));
2531 exact_i = check_exactness_omega(map, app);
2543 return box_closure_with_check(map, exact);
2549 /* Compute the transitive closure of "map", or an overapproximation.
2550 * If the result is exact, then *exact is set to 1.
2551 * Simply use map_power to compute the powers of map, but tell
2552 * it to project out the lengths of the paths instead of equating
2553 * the length to a parameter.
2555 __isl_give isl_map *isl_map_transitive_closure(__isl_take isl_map *map,
2563 if (map->ctx->opt->closure == ISL_CLOSURE_OMEGA)
2564 return transitive_closure_omega(map, exact);
2566 map = isl_map_compute_divs(map);
2567 map = isl_map_coalesce(map);
2568 closed = isl_map_is_transitively_closed(map);
2577 map = map_power(map, exact, 1);