2 * Copyright 2010 INRIA Saclay
4 * Use of this software is governed by the MIT license
6 * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
7 * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
11 #include <isl_ctx_private.h>
12 #include <isl_map_private.h>
15 #include <isl_space_private.h>
16 #include <isl_lp_private.h>
17 #include <isl/union_map.h>
18 #include <isl_mat_private.h>
19 #include <isl_vec_private.h>
20 #include <isl_options_private.h>
21 #include <isl_tarjan.h>
23 int isl_map_is_transitively_closed(__isl_keep isl_map
*map
)
28 map2
= isl_map_apply_range(isl_map_copy(map
), isl_map_copy(map
));
29 closed
= isl_map_is_subset(map2
, map
);
35 int isl_union_map_is_transitively_closed(__isl_keep isl_union_map
*umap
)
40 umap2
= isl_union_map_apply_range(isl_union_map_copy(umap
),
41 isl_union_map_copy(umap
));
42 closed
= isl_union_map_is_subset(umap2
, umap
);
43 isl_union_map_free(umap2
);
48 /* Given a map that represents a path with the length of the path
49 * encoded as the difference between the last output coordindate
50 * and the last input coordinate, set this length to either
51 * exactly "length" (if "exactly" is set) or at least "length"
52 * (if "exactly" is not set).
54 static __isl_give isl_map
*set_path_length(__isl_take isl_map
*map
,
55 int exactly
, int length
)
58 struct isl_basic_map
*bmap
;
67 space
= isl_map_get_space(map
);
68 d
= isl_space_dim(space
, isl_dim_in
);
69 nparam
= isl_space_dim(space
, isl_dim_param
);
70 bmap
= isl_basic_map_alloc_space(space
, 0, 1, 1);
72 k
= isl_basic_map_alloc_equality(bmap
);
77 k
= isl_basic_map_alloc_inequality(bmap
);
82 isl_seq_clr(c
, 1 + isl_basic_map_total_dim(bmap
));
83 isl_int_set_si(c
[0], -length
);
84 isl_int_set_si(c
[1 + nparam
+ d
- 1], -1);
85 isl_int_set_si(c
[1 + nparam
+ d
+ d
- 1], 1);
87 bmap
= isl_basic_map_finalize(bmap
);
88 map
= isl_map_intersect(map
, isl_map_from_basic_map(bmap
));
92 isl_basic_map_free(bmap
);
97 /* Check whether the overapproximation of the power of "map" is exactly
98 * the power of "map". Let R be "map" and A_k the overapproximation.
99 * The approximation is exact if
102 * A_k = A_{k-1} \circ R k >= 2
104 * Since A_k is known to be an overapproximation, we only need to check
107 * A_k \subset A_{k-1} \circ R k >= 2
109 * In practice, "app" has an extra input and output coordinate
110 * to encode the length of the path. So, we first need to add
111 * this coordinate to "map" and set the length of the path to
114 static int check_power_exactness(__isl_take isl_map
*map
,
115 __isl_take isl_map
*app
)
121 map
= isl_map_add_dims(map
, isl_dim_in
, 1);
122 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
123 map
= set_path_length(map
, 1, 1);
125 app_1
= set_path_length(isl_map_copy(app
), 1, 1);
127 exact
= isl_map_is_subset(app_1
, map
);
130 if (!exact
|| exact
< 0) {
136 app_1
= set_path_length(isl_map_copy(app
), 0, 1);
137 app_2
= set_path_length(app
, 0, 2);
138 app_1
= isl_map_apply_range(map
, app_1
);
140 exact
= isl_map_is_subset(app_2
, app_1
);
148 /* Check whether the overapproximation of the power of "map" is exactly
149 * the power of "map", possibly after projecting out the power (if "project"
152 * If "project" is set and if "steps" can only result in acyclic paths,
155 * A = R \cup (A \circ R)
157 * where A is the overapproximation with the power projected out, i.e.,
158 * an overapproximation of the transitive closure.
159 * More specifically, since A is known to be an overapproximation, we check
161 * A \subset R \cup (A \circ R)
163 * Otherwise, we check if the power is exact.
165 * Note that "app" has an extra input and output coordinate to encode
166 * the length of the part. If we are only interested in the transitive
167 * closure, then we can simply project out these coordinates first.
169 static int check_exactness(__isl_take isl_map
*map
, __isl_take isl_map
*app
,
177 return check_power_exactness(map
, app
);
179 d
= isl_map_dim(map
, isl_dim_in
);
180 app
= set_path_length(app
, 0, 1);
181 app
= isl_map_project_out(app
, isl_dim_in
, d
, 1);
182 app
= isl_map_project_out(app
, isl_dim_out
, d
, 1);
184 app
= isl_map_reset_space(app
, isl_map_get_space(map
));
186 test
= isl_map_apply_range(isl_map_copy(map
), isl_map_copy(app
));
187 test
= isl_map_union(test
, isl_map_copy(map
));
189 exact
= isl_map_is_subset(app
, test
);
200 * The transitive closure implementation is based on the paper
201 * "Computing the Transitive Closure of a Union of Affine Integer
202 * Tuple Relations" by Anna Beletska, Denis Barthou, Wlodzimierz Bielecki and
206 /* Given a set of n offsets v_i (the rows of "steps"), construct a relation
207 * of the given dimension specification (Z^{n+1} -> Z^{n+1})
208 * that maps an element x to any element that can be reached
209 * by taking a non-negative number of steps along any of
210 * the extended offsets v'_i = [v_i 1].
213 * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v'_i }
215 * For any element in this relation, the number of steps taken
216 * is equal to the difference in the final coordinates.
218 static __isl_give isl_map
*path_along_steps(__isl_take isl_space
*space
,
219 __isl_keep isl_mat
*steps
)
222 struct isl_basic_map
*path
= NULL
;
227 if (!space
|| !steps
)
230 d
= isl_space_dim(space
, isl_dim_in
);
232 nparam
= isl_space_dim(space
, isl_dim_param
);
234 path
= isl_basic_map_alloc_space(isl_space_copy(space
), n
, d
, n
);
236 for (i
= 0; i
< n
; ++i
) {
237 k
= isl_basic_map_alloc_div(path
);
240 isl_assert(steps
->ctx
, i
== k
, goto error
);
241 isl_int_set_si(path
->div
[k
][0], 0);
244 for (i
= 0; i
< d
; ++i
) {
245 k
= isl_basic_map_alloc_equality(path
);
248 isl_seq_clr(path
->eq
[k
], 1 + isl_basic_map_total_dim(path
));
249 isl_int_set_si(path
->eq
[k
][1 + nparam
+ i
], 1);
250 isl_int_set_si(path
->eq
[k
][1 + nparam
+ d
+ i
], -1);
252 for (j
= 0; j
< n
; ++j
)
253 isl_int_set_si(path
->eq
[k
][1 + nparam
+ 2 * d
+ j
], 1);
255 for (j
= 0; j
< n
; ++j
)
256 isl_int_set(path
->eq
[k
][1 + nparam
+ 2 * d
+ j
],
260 for (i
= 0; i
< n
; ++i
) {
261 k
= isl_basic_map_alloc_inequality(path
);
264 isl_seq_clr(path
->ineq
[k
], 1 + isl_basic_map_total_dim(path
));
265 isl_int_set_si(path
->ineq
[k
][1 + nparam
+ 2 * d
+ i
], 1);
268 isl_space_free(space
);
270 path
= isl_basic_map_simplify(path
);
271 path
= isl_basic_map_finalize(path
);
272 return isl_map_from_basic_map(path
);
274 isl_space_free(space
);
275 isl_basic_map_free(path
);
284 /* Check whether the parametric constant term of constraint c is never
285 * positive in "bset".
287 static isl_bool
parametric_constant_never_positive(
288 __isl_keep isl_basic_set
*bset
, isl_int
*c
, int *div_purity
)
297 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
298 d
= isl_basic_set_dim(bset
, isl_dim_set
);
299 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
301 bset
= isl_basic_set_copy(bset
);
302 bset
= isl_basic_set_cow(bset
);
303 bset
= isl_basic_set_extend_constraints(bset
, 0, 1);
304 k
= isl_basic_set_alloc_inequality(bset
);
307 isl_seq_clr(bset
->ineq
[k
], 1 + isl_basic_set_total_dim(bset
));
308 isl_seq_cpy(bset
->ineq
[k
], c
, 1 + nparam
);
309 for (i
= 0; i
< n_div
; ++i
) {
310 if (div_purity
[i
] != PURE_PARAM
)
312 isl_int_set(bset
->ineq
[k
][1 + nparam
+ d
+ i
],
313 c
[1 + nparam
+ d
+ i
]);
315 isl_int_sub_ui(bset
->ineq
[k
][0], bset
->ineq
[k
][0], 1);
316 empty
= isl_basic_set_is_empty(bset
);
317 isl_basic_set_free(bset
);
321 isl_basic_set_free(bset
);
322 return isl_bool_error
;
325 /* Return PURE_PARAM if only the coefficients of the parameters are non-zero.
326 * Return PURE_VAR if only the coefficients of the set variables are non-zero.
327 * Return MIXED if only the coefficients of the parameters and the set
328 * variables are non-zero and if moreover the parametric constant
329 * can never attain positive values.
330 * Return IMPURE otherwise.
332 static int purity(__isl_keep isl_basic_set
*bset
, isl_int
*c
, int *div_purity
,
342 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
343 d
= isl_basic_set_dim(bset
, isl_dim_set
);
344 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
346 for (i
= 0; i
< n_div
; ++i
) {
347 if (isl_int_is_zero(c
[1 + nparam
+ d
+ i
]))
349 switch (div_purity
[i
]) {
350 case PURE_PARAM
: p
= 1; break;
351 case PURE_VAR
: v
= 1; break;
352 default: return IMPURE
;
355 if (!p
&& isl_seq_first_non_zero(c
+ 1, nparam
) == -1)
357 if (!v
&& isl_seq_first_non_zero(c
+ 1 + nparam
, d
) == -1)
360 empty
= parametric_constant_never_positive(bset
, c
, div_purity
);
361 if (eq
&& empty
>= 0 && !empty
) {
362 isl_seq_neg(c
, c
, 1 + nparam
+ d
+ n_div
);
363 empty
= parametric_constant_never_positive(bset
, c
, div_purity
);
366 return empty
< 0 ? -1 : empty
? MIXED
: IMPURE
;
369 /* Return an array of integers indicating the type of each div in bset.
370 * If the div is (recursively) defined in terms of only the parameters,
371 * then the type is PURE_PARAM.
372 * If the div is (recursively) defined in terms of only the set variables,
373 * then the type is PURE_VAR.
374 * Otherwise, the type is IMPURE.
376 static __isl_give
int *get_div_purity(__isl_keep isl_basic_set
*bset
)
387 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
388 d
= isl_basic_set_dim(bset
, isl_dim_set
);
389 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
391 div_purity
= isl_alloc_array(bset
->ctx
, int, n_div
);
392 if (n_div
&& !div_purity
)
395 for (i
= 0; i
< bset
->n_div
; ++i
) {
397 if (isl_int_is_zero(bset
->div
[i
][0])) {
398 div_purity
[i
] = IMPURE
;
401 if (isl_seq_first_non_zero(bset
->div
[i
] + 2, nparam
) != -1)
403 if (isl_seq_first_non_zero(bset
->div
[i
] + 2 + nparam
, d
) != -1)
405 for (j
= 0; j
< i
; ++j
) {
406 if (isl_int_is_zero(bset
->div
[i
][2 + nparam
+ d
+ j
]))
408 switch (div_purity
[j
]) {
409 case PURE_PARAM
: p
= 1; break;
410 case PURE_VAR
: v
= 1; break;
411 default: p
= v
= 1; break;
414 div_purity
[i
] = v
? p
? IMPURE
: PURE_VAR
: PURE_PARAM
;
420 /* Given a path with the as yet unconstrained length at position "pos",
421 * check if setting the length to zero results in only the identity
424 static isl_bool
empty_path_is_identity(__isl_keep isl_basic_map
*path
,
427 isl_basic_map
*test
= NULL
;
428 isl_basic_map
*id
= NULL
;
432 test
= isl_basic_map_copy(path
);
433 test
= isl_basic_map_extend_constraints(test
, 1, 0);
434 k
= isl_basic_map_alloc_equality(test
);
437 isl_seq_clr(test
->eq
[k
], 1 + isl_basic_map_total_dim(test
));
438 isl_int_set_si(test
->eq
[k
][pos
], 1);
439 test
= isl_basic_map_gauss(test
, NULL
);
440 id
= isl_basic_map_identity(isl_basic_map_get_space(path
));
441 is_id
= isl_basic_map_is_equal(test
, id
);
442 isl_basic_map_free(test
);
443 isl_basic_map_free(id
);
446 isl_basic_map_free(test
);
447 return isl_bool_error
;
450 /* If any of the constraints is found to be impure then this function
451 * sets *impurity to 1.
453 * If impurity is NULL then we are dealing with a non-parametric set
454 * and so the constraints are obviously PURE_VAR.
456 static __isl_give isl_basic_map
*add_delta_constraints(
457 __isl_take isl_basic_map
*path
,
458 __isl_keep isl_basic_set
*delta
, unsigned off
, unsigned nparam
,
459 unsigned d
, int *div_purity
, int eq
, int *impurity
)
462 int n
= eq
? delta
->n_eq
: delta
->n_ineq
;
463 isl_int
**delta_c
= eq
? delta
->eq
: delta
->ineq
;
466 n_div
= isl_basic_set_dim(delta
, isl_dim_div
);
468 for (i
= 0; i
< n
; ++i
) {
472 p
= purity(delta
, delta_c
[i
], div_purity
, eq
);
475 if (p
!= PURE_VAR
&& p
!= PURE_PARAM
&& !*impurity
)
479 if (eq
&& p
!= MIXED
) {
480 k
= isl_basic_map_alloc_equality(path
);
483 path_c
= path
->eq
[k
];
485 k
= isl_basic_map_alloc_inequality(path
);
488 path_c
= path
->ineq
[k
];
490 isl_seq_clr(path_c
, 1 + isl_basic_map_total_dim(path
));
492 isl_seq_cpy(path_c
+ off
,
493 delta_c
[i
] + 1 + nparam
, d
);
494 isl_int_set(path_c
[off
+ d
], delta_c
[i
][0]);
495 } else if (p
== PURE_PARAM
) {
496 isl_seq_cpy(path_c
, delta_c
[i
], 1 + nparam
);
498 isl_seq_cpy(path_c
+ off
,
499 delta_c
[i
] + 1 + nparam
, d
);
500 isl_seq_cpy(path_c
, delta_c
[i
], 1 + nparam
);
502 isl_seq_cpy(path_c
+ off
- n_div
,
503 delta_c
[i
] + 1 + nparam
+ d
, n_div
);
508 isl_basic_map_free(path
);
512 /* Given a set of offsets "delta", construct a relation of the
513 * given dimension specification (Z^{n+1} -> Z^{n+1}) that
514 * is an overapproximation of the relations that
515 * maps an element x to any element that can be reached
516 * by taking a non-negative number of steps along any of
517 * the elements in "delta".
518 * That is, construct an approximation of
520 * { [x] -> [y] : exists f \in \delta, k \in Z :
521 * y = x + k [f, 1] and k >= 0 }
523 * For any element in this relation, the number of steps taken
524 * is equal to the difference in the final coordinates.
526 * In particular, let delta be defined as
528 * \delta = [p] -> { [x] : A x + a >= 0 and B p + b >= 0 and
529 * C x + C'p + c >= 0 and
530 * D x + D'p + d >= 0 }
532 * where the constraints C x + C'p + c >= 0 are such that the parametric
533 * constant term of each constraint j, "C_j x + C'_j p + c_j",
534 * can never attain positive values, then the relation is constructed as
536 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
537 * A f + k a >= 0 and B p + b >= 0 and
538 * C f + C'p + c >= 0 and k >= 1 }
539 * union { [x] -> [x] }
541 * If the zero-length paths happen to correspond exactly to the identity
542 * mapping, then we return
544 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
545 * A f + k a >= 0 and B p + b >= 0 and
546 * C f + C'p + c >= 0 and k >= 0 }
550 * Existentially quantified variables in \delta are handled by
551 * classifying them as independent of the parameters, purely
552 * parameter dependent and others. Constraints containing
553 * any of the other existentially quantified variables are removed.
554 * This is safe, but leads to an additional overapproximation.
556 * If there are any impure constraints, then we also eliminate
557 * the parameters from \delta, resulting in a set
559 * \delta' = { [x] : E x + e >= 0 }
561 * and add the constraints
565 * to the constructed relation.
567 static __isl_give isl_map
*path_along_delta(__isl_take isl_space
*space
,
568 __isl_take isl_basic_set
*delta
)
570 isl_basic_map
*path
= NULL
;
577 int *div_purity
= NULL
;
582 n_div
= isl_basic_set_dim(delta
, isl_dim_div
);
583 d
= isl_basic_set_dim(delta
, isl_dim_set
);
584 nparam
= isl_basic_set_dim(delta
, isl_dim_param
);
585 path
= isl_basic_map_alloc_space(isl_space_copy(space
), n_div
+ d
+ 1,
586 d
+ 1 + delta
->n_eq
, delta
->n_eq
+ delta
->n_ineq
+ 1);
587 off
= 1 + nparam
+ 2 * (d
+ 1) + n_div
;
589 for (i
= 0; i
< n_div
+ d
+ 1; ++i
) {
590 k
= isl_basic_map_alloc_div(path
);
593 isl_int_set_si(path
->div
[k
][0], 0);
596 for (i
= 0; i
< d
+ 1; ++i
) {
597 k
= isl_basic_map_alloc_equality(path
);
600 isl_seq_clr(path
->eq
[k
], 1 + isl_basic_map_total_dim(path
));
601 isl_int_set_si(path
->eq
[k
][1 + nparam
+ i
], 1);
602 isl_int_set_si(path
->eq
[k
][1 + nparam
+ d
+ 1 + i
], -1);
603 isl_int_set_si(path
->eq
[k
][off
+ i
], 1);
606 div_purity
= get_div_purity(delta
);
607 if (n_div
&& !div_purity
)
610 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
611 div_purity
, 1, &impurity
);
612 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
613 div_purity
, 0, &impurity
);
615 isl_space
*space
= isl_basic_set_get_space(delta
);
616 delta
= isl_basic_set_project_out(delta
,
617 isl_dim_param
, 0, nparam
);
618 delta
= isl_basic_set_add_dims(delta
, isl_dim_param
, nparam
);
619 delta
= isl_basic_set_reset_space(delta
, space
);
622 path
= isl_basic_map_extend_constraints(path
, delta
->n_eq
,
624 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
626 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
628 path
= isl_basic_map_gauss(path
, NULL
);
631 is_id
= empty_path_is_identity(path
, off
+ d
);
635 k
= isl_basic_map_alloc_inequality(path
);
638 isl_seq_clr(path
->ineq
[k
], 1 + isl_basic_map_total_dim(path
));
640 isl_int_set_si(path
->ineq
[k
][0], -1);
641 isl_int_set_si(path
->ineq
[k
][off
+ d
], 1);
644 isl_basic_set_free(delta
);
645 path
= isl_basic_map_finalize(path
);
647 isl_space_free(space
);
648 return isl_map_from_basic_map(path
);
650 return isl_basic_map_union(path
, isl_basic_map_identity(space
));
653 isl_space_free(space
);
654 isl_basic_set_free(delta
);
655 isl_basic_map_free(path
);
659 /* Given a dimension specification Z^{n+1} -> Z^{n+1} and a parameter "param",
660 * construct a map that equates the parameter to the difference
661 * in the final coordinates and imposes that this difference is positive.
664 * { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
666 static __isl_give isl_map
*equate_parameter_to_length(
667 __isl_take isl_space
*space
, unsigned param
)
669 struct isl_basic_map
*bmap
;
674 d
= isl_space_dim(space
, isl_dim_in
);
675 nparam
= isl_space_dim(space
, isl_dim_param
);
676 bmap
= isl_basic_map_alloc_space(space
, 0, 1, 1);
677 k
= isl_basic_map_alloc_equality(bmap
);
680 isl_seq_clr(bmap
->eq
[k
], 1 + isl_basic_map_total_dim(bmap
));
681 isl_int_set_si(bmap
->eq
[k
][1 + param
], -1);
682 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
- 1], -1);
683 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
+ d
- 1], 1);
685 k
= isl_basic_map_alloc_inequality(bmap
);
688 isl_seq_clr(bmap
->ineq
[k
], 1 + isl_basic_map_total_dim(bmap
));
689 isl_int_set_si(bmap
->ineq
[k
][1 + param
], 1);
690 isl_int_set_si(bmap
->ineq
[k
][0], -1);
692 bmap
= isl_basic_map_finalize(bmap
);
693 return isl_map_from_basic_map(bmap
);
695 isl_basic_map_free(bmap
);
699 /* Check whether "path" is acyclic, where the last coordinates of domain
700 * and range of path encode the number of steps taken.
701 * That is, check whether
703 * { d | d = y - x and (x,y) in path }
705 * does not contain any element with positive last coordinate (positive length)
706 * and zero remaining coordinates (cycle).
708 static int is_acyclic(__isl_take isl_map
*path
)
713 struct isl_set
*delta
;
715 delta
= isl_map_deltas(path
);
716 dim
= isl_set_dim(delta
, isl_dim_set
);
717 for (i
= 0; i
< dim
; ++i
) {
719 delta
= isl_set_lower_bound_si(delta
, isl_dim_set
, i
, 1);
721 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
724 acyclic
= isl_set_is_empty(delta
);
730 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
731 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
732 * construct a map that is an overapproximation of the map
733 * that takes an element from the space D \times Z to another
734 * element from the same space, such that the first n coordinates of the
735 * difference between them is a sum of differences between images
736 * and pre-images in one of the R_i and such that the last coordinate
737 * is equal to the number of steps taken.
740 * \Delta_i = { y - x | (x, y) in R_i }
742 * then the constructed map is an overapproximation of
744 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
745 * d = (\sum_i k_i \delta_i, \sum_i k_i) }
747 * The elements of the singleton \Delta_i's are collected as the
748 * rows of the steps matrix. For all these \Delta_i's together,
749 * a single path is constructed.
750 * For each of the other \Delta_i's, we compute an overapproximation
751 * of the paths along elements of \Delta_i.
752 * Since each of these paths performs an addition, composition is
753 * symmetric and we can simply compose all resulting paths in any order.
755 static __isl_give isl_map
*construct_extended_path(__isl_take isl_space
*space
,
756 __isl_keep isl_map
*map
, int *project
)
758 struct isl_mat
*steps
= NULL
;
759 struct isl_map
*path
= NULL
;
766 d
= isl_map_dim(map
, isl_dim_in
);
768 path
= isl_map_identity(isl_space_copy(space
));
770 steps
= isl_mat_alloc(map
->ctx
, map
->n
, d
);
775 for (i
= 0; i
< map
->n
; ++i
) {
776 struct isl_basic_set
*delta
;
778 delta
= isl_basic_map_deltas(isl_basic_map_copy(map
->p
[i
]));
780 for (j
= 0; j
< d
; ++j
) {
783 fixed
= isl_basic_set_plain_dim_is_fixed(delta
, j
,
786 isl_basic_set_free(delta
);
795 path
= isl_map_apply_range(path
,
796 path_along_delta(isl_space_copy(space
), delta
));
797 path
= isl_map_coalesce(path
);
799 isl_basic_set_free(delta
);
806 path
= isl_map_apply_range(path
,
807 path_along_steps(isl_space_copy(space
), steps
));
810 if (project
&& *project
) {
811 *project
= is_acyclic(isl_map_copy(path
));
816 isl_space_free(space
);
820 isl_space_free(space
);
826 static isl_bool
isl_set_overlaps(__isl_keep isl_set
*set1
,
827 __isl_keep isl_set
*set2
)
833 return isl_bool_error
;
835 if (!isl_space_tuple_is_equal(set1
->dim
, isl_dim_set
,
836 set2
->dim
, isl_dim_set
))
837 return isl_bool_false
;
839 i
= isl_set_intersect(isl_set_copy(set1
), isl_set_copy(set2
));
840 no_overlap
= isl_set_is_empty(i
);
843 return isl_bool_not(no_overlap
);
846 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
847 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
848 * construct a map that is an overapproximation of the map
849 * that takes an element from the dom R \times Z to an
850 * element from ran R \times Z, such that the first n coordinates of the
851 * difference between them is a sum of differences between images
852 * and pre-images in one of the R_i and such that the last coordinate
853 * is equal to the number of steps taken.
856 * \Delta_i = { y - x | (x, y) in R_i }
858 * then the constructed map is an overapproximation of
860 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
861 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
862 * x in dom R and x + d in ran R and
865 static __isl_give isl_map
*construct_component(__isl_take isl_space
*dim
,
866 __isl_keep isl_map
*map
, int *exact
, int project
)
868 struct isl_set
*domain
= NULL
;
869 struct isl_set
*range
= NULL
;
870 struct isl_map
*app
= NULL
;
871 struct isl_map
*path
= NULL
;
874 domain
= isl_map_domain(isl_map_copy(map
));
875 domain
= isl_set_coalesce(domain
);
876 range
= isl_map_range(isl_map_copy(map
));
877 range
= isl_set_coalesce(range
);
878 overlaps
= isl_set_overlaps(domain
, range
);
879 if (overlaps
< 0 || !overlaps
) {
880 isl_set_free(domain
);
886 map
= isl_map_copy(map
);
887 map
= isl_map_add_dims(map
, isl_dim_in
, 1);
888 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
889 map
= set_path_length(map
, 1, 1);
892 app
= isl_map_from_domain_and_range(domain
, range
);
893 app
= isl_map_add_dims(app
, isl_dim_in
, 1);
894 app
= isl_map_add_dims(app
, isl_dim_out
, 1);
896 path
= construct_extended_path(isl_space_copy(dim
), map
,
897 exact
&& *exact
? &project
: NULL
);
898 app
= isl_map_intersect(app
, path
);
900 if (exact
&& *exact
&&
901 (*exact
= check_exactness(isl_map_copy(map
), isl_map_copy(app
),
906 app
= set_path_length(app
, 0, 1);
914 /* Call construct_component and, if "project" is set, project out
915 * the final coordinates.
917 static __isl_give isl_map
*construct_projected_component(
918 __isl_take isl_space
*space
,
919 __isl_keep isl_map
*map
, int *exact
, int project
)
926 d
= isl_space_dim(space
, isl_dim_in
);
928 app
= construct_component(space
, map
, exact
, project
);
930 app
= isl_map_project_out(app
, isl_dim_in
, d
- 1, 1);
931 app
= isl_map_project_out(app
, isl_dim_out
, d
- 1, 1);
936 /* Compute an extended version, i.e., with path lengths, of
937 * an overapproximation of the transitive closure of "bmap"
938 * with path lengths greater than or equal to zero and with
939 * domain and range equal to "dom".
941 static __isl_give isl_map
*q_closure(__isl_take isl_space
*dim
,
942 __isl_take isl_set
*dom
, __isl_keep isl_basic_map
*bmap
, int *exact
)
949 dom
= isl_set_add_dims(dom
, isl_dim_set
, 1);
950 app
= isl_map_from_domain_and_range(dom
, isl_set_copy(dom
));
951 map
= isl_map_from_basic_map(isl_basic_map_copy(bmap
));
952 path
= construct_extended_path(dim
, map
, &project
);
953 app
= isl_map_intersect(app
, path
);
955 if ((*exact
= check_exactness(map
, isl_map_copy(app
), project
)) < 0)
964 /* Check whether qc has any elements of length at least one
965 * with domain and/or range outside of dom and ran.
967 static int has_spurious_elements(__isl_keep isl_map
*qc
,
968 __isl_keep isl_set
*dom
, __isl_keep isl_set
*ran
)
974 if (!qc
|| !dom
|| !ran
)
977 d
= isl_map_dim(qc
, isl_dim_in
);
979 qc
= isl_map_copy(qc
);
980 qc
= set_path_length(qc
, 0, 1);
981 qc
= isl_map_project_out(qc
, isl_dim_in
, d
- 1, 1);
982 qc
= isl_map_project_out(qc
, isl_dim_out
, d
- 1, 1);
984 s
= isl_map_domain(isl_map_copy(qc
));
985 subset
= isl_set_is_subset(s
, dom
);
994 s
= isl_map_range(qc
);
995 subset
= isl_set_is_subset(s
, ran
);
998 return subset
< 0 ? -1 : !subset
;
1007 /* For each basic map in "map", except i, check whether it combines
1008 * with the transitive closure that is reflexive on C combines
1009 * to the left and to the right.
1013 * dom map_j \subseteq C
1015 * then right[j] is set to 1. Otherwise, if
1017 * ran map_i \cap dom map_j = \emptyset
1019 * then right[j] is set to 0. Otherwise, composing to the right
1022 * Similar, for composing to the left, we have if
1024 * ran map_j \subseteq C
1026 * then left[j] is set to 1. Otherwise, if
1028 * dom map_i \cap ran map_j = \emptyset
1030 * then left[j] is set to 0. Otherwise, composing to the left
1033 * The return value is or'd with LEFT if composing to the left
1034 * is possible and with RIGHT if composing to the right is possible.
1036 static int composability(__isl_keep isl_set
*C
, int i
,
1037 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
1038 __isl_keep isl_map
*map
)
1044 for (j
= 0; j
< map
->n
&& ok
; ++j
) {
1045 isl_bool overlaps
, subset
;
1051 dom
[j
] = isl_set_from_basic_set(
1052 isl_basic_map_domain(
1053 isl_basic_map_copy(map
->p
[j
])));
1056 overlaps
= isl_set_overlaps(ran
[i
], dom
[j
]);
1062 subset
= isl_set_is_subset(dom
[j
], C
);
1074 ran
[j
] = isl_set_from_basic_set(
1075 isl_basic_map_range(
1076 isl_basic_map_copy(map
->p
[j
])));
1079 overlaps
= isl_set_overlaps(dom
[i
], ran
[j
]);
1085 subset
= isl_set_is_subset(ran
[j
], C
);
1099 static __isl_give isl_map
*anonymize(__isl_take isl_map
*map
)
1101 map
= isl_map_reset(map
, isl_dim_in
);
1102 map
= isl_map_reset(map
, isl_dim_out
);
1106 /* Return a map that is a union of the basic maps in "map", except i,
1107 * composed to left and right with qc based on the entries of "left"
1110 static __isl_give isl_map
*compose(__isl_keep isl_map
*map
, int i
,
1111 __isl_take isl_map
*qc
, int *left
, int *right
)
1116 comp
= isl_map_empty(isl_map_get_space(map
));
1117 for (j
= 0; j
< map
->n
; ++j
) {
1123 map_j
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[j
]));
1124 map_j
= anonymize(map_j
);
1125 if (left
&& left
[j
])
1126 map_j
= isl_map_apply_range(map_j
, isl_map_copy(qc
));
1127 if (right
&& right
[j
])
1128 map_j
= isl_map_apply_range(isl_map_copy(qc
), map_j
);
1129 comp
= isl_map_union(comp
, map_j
);
1132 comp
= isl_map_compute_divs(comp
);
1133 comp
= isl_map_coalesce(comp
);
1140 /* Compute the transitive closure of "map" incrementally by
1147 * map_i^+ \cup ((id \cup map_i^) \circ qc^+)
1151 * map_i^+ \cup (qc^+ \circ (id \cup map_i^))
1153 * depending on whether left or right are NULL.
1155 static __isl_give isl_map
*compute_incremental(
1156 __isl_take isl_space
*space
, __isl_keep isl_map
*map
,
1157 int i
, __isl_take isl_map
*qc
, int *left
, int *right
, int *exact
)
1161 isl_map
*rtc
= NULL
;
1165 isl_assert(map
->ctx
, left
|| right
, goto error
);
1167 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
1168 tc
= construct_projected_component(isl_space_copy(space
), map_i
,
1170 isl_map_free(map_i
);
1173 qc
= isl_map_transitive_closure(qc
, exact
);
1176 isl_space_free(space
);
1179 return isl_map_universe(isl_map_get_space(map
));
1182 if (!left
|| !right
)
1183 rtc
= isl_map_union(isl_map_copy(tc
),
1184 isl_map_identity(isl_map_get_space(tc
)));
1186 qc
= isl_map_apply_range(rtc
, qc
);
1188 qc
= isl_map_apply_range(qc
, rtc
);
1189 qc
= isl_map_union(tc
, qc
);
1191 isl_space_free(space
);
1195 isl_space_free(space
);
1200 /* Given a map "map", try to find a basic map such that
1201 * map^+ can be computed as
1203 * map^+ = map_i^+ \cup
1204 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1206 * with C the simple hull of the domain and range of the input map.
1207 * map_i^ \cup Id_C is computed by allowing the path lengths to be zero
1208 * and by intersecting domain and range with C.
1209 * Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
1210 * Also, we only use the incremental computation if all the transitive
1211 * closures are exact and if the number of basic maps in the union,
1212 * after computing the integer divisions, is smaller than the number
1213 * of basic maps in the input map.
1215 static int incremental_on_entire_domain(__isl_keep isl_space
*space
,
1216 __isl_keep isl_map
*map
,
1217 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
1218 __isl_give isl_map
**res
)
1226 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
1227 isl_map_range(isl_map_copy(map
)));
1228 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1236 d
= isl_map_dim(map
, isl_dim_in
);
1238 for (i
= 0; i
< map
->n
; ++i
) {
1240 int exact_i
, spurious
;
1242 dom
[i
] = isl_set_from_basic_set(isl_basic_map_domain(
1243 isl_basic_map_copy(map
->p
[i
])));
1244 ran
[i
] = isl_set_from_basic_set(isl_basic_map_range(
1245 isl_basic_map_copy(map
->p
[i
])));
1246 qc
= q_closure(isl_space_copy(space
), isl_set_copy(C
),
1247 map
->p
[i
], &exact_i
);
1254 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1261 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1262 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1263 qc
= isl_map_compute_divs(qc
);
1264 for (j
= 0; j
< map
->n
; ++j
)
1265 left
[j
] = right
[j
] = 1;
1266 qc
= compose(map
, i
, qc
, left
, right
);
1269 if (qc
->n
>= map
->n
) {
1273 *res
= compute_incremental(isl_space_copy(space
), map
, i
, qc
,
1274 left
, right
, &exact_i
);
1285 return *res
!= NULL
;
1291 /* Try and compute the transitive closure of "map" as
1293 * map^+ = map_i^+ \cup
1294 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1296 * with C either the simple hull of the domain and range of the entire
1297 * map or the simple hull of domain and range of map_i.
1299 static __isl_give isl_map
*incremental_closure(__isl_take isl_space
*space
,
1300 __isl_keep isl_map
*map
, int *exact
, int project
)
1303 isl_set
**dom
= NULL
;
1304 isl_set
**ran
= NULL
;
1309 isl_map
*res
= NULL
;
1312 return construct_projected_component(space
, map
, exact
,
1318 return construct_projected_component(space
, map
, exact
,
1321 d
= isl_map_dim(map
, isl_dim_in
);
1323 dom
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1324 ran
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1325 left
= isl_calloc_array(map
->ctx
, int, map
->n
);
1326 right
= isl_calloc_array(map
->ctx
, int, map
->n
);
1327 if (!ran
|| !dom
|| !left
|| !right
)
1330 if (incremental_on_entire_domain(space
, map
, dom
, ran
, left
, right
,
1334 for (i
= 0; !res
&& i
< map
->n
; ++i
) {
1336 int exact_i
, spurious
, comp
;
1338 dom
[i
] = isl_set_from_basic_set(
1339 isl_basic_map_domain(
1340 isl_basic_map_copy(map
->p
[i
])));
1344 ran
[i
] = isl_set_from_basic_set(
1345 isl_basic_map_range(
1346 isl_basic_map_copy(map
->p
[i
])));
1349 C
= isl_set_union(isl_set_copy(dom
[i
]),
1350 isl_set_copy(ran
[i
]));
1351 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1358 comp
= composability(C
, i
, dom
, ran
, left
, right
, map
);
1359 if (!comp
|| comp
< 0) {
1365 qc
= q_closure(isl_space_copy(space
), C
, map
->p
[i
], &exact_i
);
1372 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1379 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1380 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1381 qc
= isl_map_compute_divs(qc
);
1382 qc
= compose(map
, i
, qc
, (comp
& LEFT
) ? left
: NULL
,
1383 (comp
& RIGHT
) ? right
: NULL
);
1386 if (qc
->n
>= map
->n
) {
1390 res
= compute_incremental(isl_space_copy(space
), map
, i
, qc
,
1391 (comp
& LEFT
) ? left
: NULL
,
1392 (comp
& RIGHT
) ? right
: NULL
, &exact_i
);
1401 for (i
= 0; i
< map
->n
; ++i
) {
1402 isl_set_free(dom
[i
]);
1403 isl_set_free(ran
[i
]);
1411 isl_space_free(space
);
1415 return construct_projected_component(space
, map
, exact
, project
);
1418 for (i
= 0; i
< map
->n
; ++i
)
1419 isl_set_free(dom
[i
]);
1422 for (i
= 0; i
< map
->n
; ++i
)
1423 isl_set_free(ran
[i
]);
1427 isl_space_free(space
);
1431 /* Given an array of sets "set", add "dom" at position "pos"
1432 * and search for elements at earlier positions that overlap with "dom".
1433 * If any can be found, then merge all of them, together with "dom", into
1434 * a single set and assign the union to the first in the array,
1435 * which becomes the new group leader for all groups involved in the merge.
1436 * During the search, we only consider group leaders, i.e., those with
1437 * group[i] = i, as the other sets have already been combined
1438 * with one of the group leaders.
1440 static int merge(isl_set
**set
, int *group
, __isl_take isl_set
*dom
, int pos
)
1445 set
[pos
] = isl_set_copy(dom
);
1447 for (i
= pos
- 1; i
>= 0; --i
) {
1453 o
= isl_set_overlaps(set
[i
], dom
);
1459 set
[i
] = isl_set_union(set
[i
], set
[group
[pos
]]);
1460 set
[group
[pos
]] = NULL
;
1463 group
[group
[pos
]] = i
;
1474 /* Replace each entry in the n by n grid of maps by the cross product
1475 * with the relation { [i] -> [i + 1] }.
1477 static int add_length(__isl_keep isl_map
*map
, isl_map
***grid
, int n
)
1481 isl_basic_map
*bstep
;
1488 space
= isl_map_get_space(map
);
1489 nparam
= isl_space_dim(space
, isl_dim_param
);
1490 space
= isl_space_drop_dims(space
, isl_dim_in
, 0,
1491 isl_space_dim(space
, isl_dim_in
));
1492 space
= isl_space_drop_dims(space
, isl_dim_out
, 0,
1493 isl_space_dim(space
, isl_dim_out
));
1494 space
= isl_space_add_dims(space
, isl_dim_in
, 1);
1495 space
= isl_space_add_dims(space
, isl_dim_out
, 1);
1496 bstep
= isl_basic_map_alloc_space(space
, 0, 1, 0);
1497 k
= isl_basic_map_alloc_equality(bstep
);
1499 isl_basic_map_free(bstep
);
1502 isl_seq_clr(bstep
->eq
[k
], 1 + isl_basic_map_total_dim(bstep
));
1503 isl_int_set_si(bstep
->eq
[k
][0], 1);
1504 isl_int_set_si(bstep
->eq
[k
][1 + nparam
], 1);
1505 isl_int_set_si(bstep
->eq
[k
][1 + nparam
+ 1], -1);
1506 bstep
= isl_basic_map_finalize(bstep
);
1507 step
= isl_map_from_basic_map(bstep
);
1509 for (i
= 0; i
< n
; ++i
)
1510 for (j
= 0; j
< n
; ++j
)
1511 grid
[i
][j
] = isl_map_product(grid
[i
][j
],
1512 isl_map_copy(step
));
1519 /* The core of the Floyd-Warshall algorithm.
1520 * Updates the given n x x matrix of relations in place.
1522 * The algorithm iterates over all vertices. In each step, the whole
1523 * matrix is updated to include all paths that go to the current vertex,
1524 * possibly stay there a while (including passing through earlier vertices)
1525 * and then come back. At the start of each iteration, the diagonal
1526 * element corresponding to the current vertex is replaced by its
1527 * transitive closure to account for all indirect paths that stay
1528 * in the current vertex.
1530 static void floyd_warshall_iterate(isl_map
***grid
, int n
, int *exact
)
1534 for (r
= 0; r
< n
; ++r
) {
1536 grid
[r
][r
] = isl_map_transitive_closure(grid
[r
][r
],
1537 (exact
&& *exact
) ? &r_exact
: NULL
);
1538 if (exact
&& *exact
&& !r_exact
)
1541 for (p
= 0; p
< n
; ++p
)
1542 for (q
= 0; q
< n
; ++q
) {
1544 if (p
== r
&& q
== r
)
1546 loop
= isl_map_apply_range(
1547 isl_map_copy(grid
[p
][r
]),
1548 isl_map_copy(grid
[r
][q
]));
1549 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1550 loop
= isl_map_apply_range(
1551 isl_map_copy(grid
[p
][r
]),
1552 isl_map_apply_range(
1553 isl_map_copy(grid
[r
][r
]),
1554 isl_map_copy(grid
[r
][q
])));
1555 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1556 grid
[p
][q
] = isl_map_coalesce(grid
[p
][q
]);
1561 /* Given a partition of the domains and ranges of the basic maps in "map",
1562 * apply the Floyd-Warshall algorithm with the elements in the partition
1565 * In particular, there are "n" elements in the partition and "group" is
1566 * an array of length 2 * map->n with entries in [0,n-1].
1568 * We first construct a matrix of relations based on the partition information,
1569 * apply Floyd-Warshall on this matrix of relations and then take the
1570 * union of all entries in the matrix as the final result.
1572 * If we are actually computing the power instead of the transitive closure,
1573 * i.e., when "project" is not set, then the result should have the
1574 * path lengths encoded as the difference between an extra pair of
1575 * coordinates. We therefore apply the nested transitive closures
1576 * to relations that include these lengths. In particular, we replace
1577 * the input relation by the cross product with the unit length relation
1578 * { [i] -> [i + 1] }.
1580 static __isl_give isl_map
*floyd_warshall_with_groups(
1581 __isl_take isl_space
*space
, __isl_keep isl_map
*map
,
1582 int *exact
, int project
, int *group
, int n
)
1585 isl_map
***grid
= NULL
;
1593 return incremental_closure(space
, map
, exact
, project
);
1596 grid
= isl_calloc_array(map
->ctx
, isl_map
**, n
);
1599 for (i
= 0; i
< n
; ++i
) {
1600 grid
[i
] = isl_calloc_array(map
->ctx
, isl_map
*, n
);
1603 for (j
= 0; j
< n
; ++j
)
1604 grid
[i
][j
] = isl_map_empty(isl_map_get_space(map
));
1607 for (k
= 0; k
< map
->n
; ++k
) {
1609 j
= group
[2 * k
+ 1];
1610 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
1611 isl_map_from_basic_map(
1612 isl_basic_map_copy(map
->p
[k
])));
1615 if (!project
&& add_length(map
, grid
, n
) < 0)
1618 floyd_warshall_iterate(grid
, n
, exact
);
1620 app
= isl_map_empty(isl_map_get_space(grid
[0][0]));
1622 for (i
= 0; i
< n
; ++i
) {
1623 for (j
= 0; j
< n
; ++j
)
1624 app
= isl_map_union(app
, grid
[i
][j
]);
1630 isl_space_free(space
);
1635 for (i
= 0; i
< n
; ++i
) {
1638 for (j
= 0; j
< n
; ++j
)
1639 isl_map_free(grid
[i
][j
]);
1644 isl_space_free(space
);
1648 /* Partition the domains and ranges of the n basic relations in list
1649 * into disjoint cells.
1651 * To find the partition, we simply consider all of the domains
1652 * and ranges in turn and combine those that overlap.
1653 * "set" contains the partition elements and "group" indicates
1654 * to which partition element a given domain or range belongs.
1655 * The domain of basic map i corresponds to element 2 * i in these arrays,
1656 * while the domain corresponds to element 2 * i + 1.
1657 * During the construction group[k] is either equal to k,
1658 * in which case set[k] contains the union of all the domains and
1659 * ranges in the corresponding group, or is equal to some l < k,
1660 * with l another domain or range in the same group.
1662 static int *setup_groups(isl_ctx
*ctx
, __isl_keep isl_basic_map
**list
, int n
,
1663 isl_set
***set
, int *n_group
)
1669 *set
= isl_calloc_array(ctx
, isl_set
*, 2 * n
);
1670 group
= isl_alloc_array(ctx
, int, 2 * n
);
1672 if (!*set
|| !group
)
1675 for (i
= 0; i
< n
; ++i
) {
1677 dom
= isl_set_from_basic_set(isl_basic_map_domain(
1678 isl_basic_map_copy(list
[i
])));
1679 if (merge(*set
, group
, dom
, 2 * i
) < 0)
1681 dom
= isl_set_from_basic_set(isl_basic_map_range(
1682 isl_basic_map_copy(list
[i
])));
1683 if (merge(*set
, group
, dom
, 2 * i
+ 1) < 0)
1688 for (i
= 0; i
< 2 * n
; ++i
)
1689 if (group
[i
] == i
) {
1691 (*set
)[g
] = (*set
)[i
];
1696 group
[i
] = group
[group
[i
]];
1703 for (i
= 0; i
< 2 * n
; ++i
)
1704 isl_set_free((*set
)[i
]);
1712 /* Check if the domains and ranges of the basic maps in "map" can
1713 * be partitioned, and if so, apply Floyd-Warshall on the elements
1714 * of the partition. Note that we also apply this algorithm
1715 * if we want to compute the power, i.e., when "project" is not set.
1716 * However, the results are unlikely to be exact since the recursive
1717 * calls inside the Floyd-Warshall algorithm typically result in
1718 * non-linear path lengths quite quickly.
1720 static __isl_give isl_map
*floyd_warshall(__isl_take isl_space
*space
,
1721 __isl_keep isl_map
*map
, int *exact
, int project
)
1724 isl_set
**set
= NULL
;
1731 return incremental_closure(space
, map
, exact
, project
);
1733 group
= setup_groups(map
->ctx
, map
->p
, map
->n
, &set
, &n
);
1737 for (i
= 0; i
< 2 * map
->n
; ++i
)
1738 isl_set_free(set
[i
]);
1742 return floyd_warshall_with_groups(space
, map
, exact
, project
, group
, n
);
1744 isl_space_free(space
);
1748 /* Structure for representing the nodes of the graph of which
1749 * strongly connected components are being computed.
1751 * list contains the actual nodes
1752 * check_closed is set if we may have used the fact that
1753 * a pair of basic maps can be interchanged
1755 struct isl_tc_follows_data
{
1756 isl_basic_map
**list
;
1760 /* Check whether in the computation of the transitive closure
1761 * "list[i]" (R_1) should follow (or be part of the same component as)
1764 * That is check whether
1772 * If so, then there is no reason for R_1 to immediately follow R_2
1775 * *check_closed is set if the subset relation holds while
1776 * R_1 \circ R_2 is not empty.
1778 static isl_bool
basic_map_follows(int i
, int j
, void *user
)
1780 struct isl_tc_follows_data
*data
= user
;
1781 struct isl_map
*map12
= NULL
;
1782 struct isl_map
*map21
= NULL
;
1785 if (!isl_space_tuple_is_equal(data
->list
[i
]->dim
, isl_dim_in
,
1786 data
->list
[j
]->dim
, isl_dim_out
))
1787 return isl_bool_false
;
1789 map21
= isl_map_from_basic_map(
1790 isl_basic_map_apply_range(
1791 isl_basic_map_copy(data
->list
[j
]),
1792 isl_basic_map_copy(data
->list
[i
])));
1793 subset
= isl_map_is_empty(map21
);
1797 isl_map_free(map21
);
1798 return isl_bool_false
;
1801 if (!isl_space_tuple_is_equal(data
->list
[i
]->dim
, isl_dim_in
,
1802 data
->list
[i
]->dim
, isl_dim_out
) ||
1803 !isl_space_tuple_is_equal(data
->list
[j
]->dim
, isl_dim_in
,
1804 data
->list
[j
]->dim
, isl_dim_out
)) {
1805 isl_map_free(map21
);
1806 return isl_bool_true
;
1809 map12
= isl_map_from_basic_map(
1810 isl_basic_map_apply_range(
1811 isl_basic_map_copy(data
->list
[i
]),
1812 isl_basic_map_copy(data
->list
[j
])));
1814 subset
= isl_map_is_subset(map21
, map12
);
1816 isl_map_free(map12
);
1817 isl_map_free(map21
);
1820 data
->check_closed
= 1;
1822 return isl_bool_not(subset
);
1824 isl_map_free(map21
);
1825 return isl_bool_error
;
1828 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
1829 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
1830 * construct a map that is an overapproximation of the map
1831 * that takes an element from the dom R \times Z to an
1832 * element from ran R \times Z, such that the first n coordinates of the
1833 * difference between them is a sum of differences between images
1834 * and pre-images in one of the R_i and such that the last coordinate
1835 * is equal to the number of steps taken.
1836 * If "project" is set, then these final coordinates are not included,
1837 * i.e., a relation of type Z^n -> Z^n is returned.
1840 * \Delta_i = { y - x | (x, y) in R_i }
1842 * then the constructed map is an overapproximation of
1844 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1845 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
1846 * x in dom R and x + d in ran R }
1850 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1851 * d = (\sum_i k_i \delta_i) and
1852 * x in dom R and x + d in ran R }
1854 * if "project" is set.
1856 * We first split the map into strongly connected components, perform
1857 * the above on each component and then join the results in the correct
1858 * order, at each join also taking in the union of both arguments
1859 * to allow for paths that do not go through one of the two arguments.
1861 static __isl_give isl_map
*construct_power_components(
1862 __isl_take isl_space
*space
, __isl_keep isl_map
*map
, int *exact
,
1866 struct isl_map
*path
= NULL
;
1867 struct isl_tc_follows_data data
;
1868 struct isl_tarjan_graph
*g
= NULL
;
1875 return floyd_warshall(space
, map
, exact
, project
);
1878 data
.check_closed
= 0;
1879 g
= isl_tarjan_graph_init(map
->ctx
, map
->n
, &basic_map_follows
, &data
);
1884 if (data
.check_closed
&& !exact
)
1885 exact
= &local_exact
;
1891 path
= isl_map_empty(isl_map_get_space(map
));
1893 path
= isl_map_empty(isl_space_copy(space
));
1894 path
= anonymize(path
);
1896 struct isl_map
*comp
;
1897 isl_map
*path_comp
, *path_comb
;
1898 comp
= isl_map_alloc_space(isl_map_get_space(map
), n
, 0);
1899 while (g
->order
[i
] != -1) {
1900 comp
= isl_map_add_basic_map(comp
,
1901 isl_basic_map_copy(map
->p
[g
->order
[i
]]));
1905 path_comp
= floyd_warshall(isl_space_copy(space
),
1906 comp
, exact
, project
);
1907 path_comp
= anonymize(path_comp
);
1908 path_comb
= isl_map_apply_range(isl_map_copy(path
),
1909 isl_map_copy(path_comp
));
1910 path
= isl_map_union(path
, path_comp
);
1911 path
= isl_map_union(path
, path_comb
);
1917 if (c
> 1 && data
.check_closed
&& !*exact
) {
1920 closed
= isl_map_is_transitively_closed(path
);
1924 isl_tarjan_graph_free(g
);
1926 return floyd_warshall(space
, map
, orig_exact
, project
);
1930 isl_tarjan_graph_free(g
);
1931 isl_space_free(space
);
1935 isl_tarjan_graph_free(g
);
1936 isl_space_free(space
);
1941 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
1942 * construct a map that is an overapproximation of the map
1943 * that takes an element from the space D to another
1944 * element from the same space, such that the difference between
1945 * them is a strictly positive sum of differences between images
1946 * and pre-images in one of the R_i.
1947 * The number of differences in the sum is equated to parameter "param".
1950 * \Delta_i = { y - x | (x, y) in R_i }
1952 * then the constructed map is an overapproximation of
1954 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1955 * d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
1958 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1959 * d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
1961 * if "project" is set.
1963 * If "project" is not set, then
1964 * we construct an extended mapping with an extra coordinate
1965 * that indicates the number of steps taken. In particular,
1966 * the difference in the last coordinate is equal to the number
1967 * of steps taken to move from a domain element to the corresponding
1970 static __isl_give isl_map
*construct_power(__isl_keep isl_map
*map
,
1971 int *exact
, int project
)
1973 struct isl_map
*app
= NULL
;
1974 isl_space
*space
= NULL
;
1979 space
= isl_map_get_space(map
);
1981 space
= isl_space_add_dims(space
, isl_dim_in
, 1);
1982 space
= isl_space_add_dims(space
, isl_dim_out
, 1);
1984 app
= construct_power_components(isl_space_copy(space
), map
,
1987 isl_space_free(space
);
1992 /* Compute the positive powers of "map", or an overapproximation.
1993 * If the result is exact, then *exact is set to 1.
1995 * If project is set, then we are actually interested in the transitive
1996 * closure, so we can use a more relaxed exactness check.
1997 * The lengths of the paths are also projected out instead of being
1998 * encoded as the difference between an extra pair of final coordinates.
2000 static __isl_give isl_map
*map_power(__isl_take isl_map
*map
,
2001 int *exact
, int project
)
2003 struct isl_map
*app
= NULL
;
2011 isl_assert(map
->ctx
,
2012 isl_map_dim(map
, isl_dim_in
) == isl_map_dim(map
, isl_dim_out
),
2015 app
= construct_power(map
, exact
, project
);
2025 /* Compute the positive powers of "map", or an overapproximation.
2026 * The result maps the exponent to a nested copy of the corresponding power.
2027 * If the result is exact, then *exact is set to 1.
2028 * map_power constructs an extended relation with the path lengths
2029 * encoded as the difference between the final coordinates.
2030 * In the final step, this difference is equated to an extra parameter
2031 * and made positive. The extra coordinates are subsequently projected out
2032 * and the parameter is turned into the domain of the result.
2034 __isl_give isl_map
*isl_map_power(__isl_take isl_map
*map
, int *exact
)
2036 isl_space
*target_space
;
2045 d
= isl_map_dim(map
, isl_dim_in
);
2046 param
= isl_map_dim(map
, isl_dim_param
);
2048 map
= isl_map_compute_divs(map
);
2049 map
= isl_map_coalesce(map
);
2051 if (isl_map_plain_is_empty(map
)) {
2052 map
= isl_map_from_range(isl_map_wrap(map
));
2053 map
= isl_map_add_dims(map
, isl_dim_in
, 1);
2054 map
= isl_map_set_dim_name(map
, isl_dim_in
, 0, "k");
2058 target_space
= isl_map_get_space(map
);
2059 target_space
= isl_space_from_range(isl_space_wrap(target_space
));
2060 target_space
= isl_space_add_dims(target_space
, isl_dim_in
, 1);
2061 target_space
= isl_space_set_dim_name(target_space
, isl_dim_in
, 0, "k");
2063 map
= map_power(map
, exact
, 0);
2065 map
= isl_map_add_dims(map
, isl_dim_param
, 1);
2066 space
= isl_map_get_space(map
);
2067 diff
= equate_parameter_to_length(space
, param
);
2068 map
= isl_map_intersect(map
, diff
);
2069 map
= isl_map_project_out(map
, isl_dim_in
, d
, 1);
2070 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2071 map
= isl_map_from_range(isl_map_wrap(map
));
2072 map
= isl_map_move_dims(map
, isl_dim_in
, 0, isl_dim_param
, param
, 1);
2074 map
= isl_map_reset_space(map
, target_space
);
2079 /* Compute a relation that maps each element in the range of the input
2080 * relation to the lengths of all paths composed of edges in the input
2081 * relation that end up in the given range element.
2082 * The result may be an overapproximation, in which case *exact is set to 0.
2083 * The resulting relation is very similar to the power relation.
2084 * The difference are that the domain has been projected out, the
2085 * range has become the domain and the exponent is the range instead
2088 __isl_give isl_map
*isl_map_reaching_path_lengths(__isl_take isl_map
*map
,
2099 d
= isl_map_dim(map
, isl_dim_in
);
2100 param
= isl_map_dim(map
, isl_dim_param
);
2102 map
= isl_map_compute_divs(map
);
2103 map
= isl_map_coalesce(map
);
2105 if (isl_map_plain_is_empty(map
)) {
2108 map
= isl_map_project_out(map
, isl_dim_out
, 0, d
);
2109 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
2113 map
= map_power(map
, exact
, 0);
2115 map
= isl_map_add_dims(map
, isl_dim_param
, 1);
2116 space
= isl_map_get_space(map
);
2117 diff
= equate_parameter_to_length(space
, param
);
2118 map
= isl_map_intersect(map
, diff
);
2119 map
= isl_map_project_out(map
, isl_dim_in
, 0, d
+ 1);
2120 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2121 map
= isl_map_reverse(map
);
2122 map
= isl_map_move_dims(map
, isl_dim_out
, 0, isl_dim_param
, param
, 1);
2127 /* Given a map, compute the smallest superset of this map that is of the form
2129 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2131 * (where p ranges over the (non-parametric) dimensions),
2132 * compute the transitive closure of this map, i.e.,
2134 * { i -> j : exists k > 0:
2135 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2137 * and intersect domain and range of this transitive closure with
2138 * the given domain and range.
2140 * If with_id is set, then try to include as much of the identity mapping
2141 * as possible, by computing
2143 * { i -> j : exists k >= 0:
2144 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2146 * instead (i.e., allow k = 0).
2148 * In practice, we compute the difference set
2150 * delta = { j - i | i -> j in map },
2152 * look for stride constraint on the individual dimensions and compute
2153 * (constant) lower and upper bounds for each individual dimension,
2154 * adding a constraint for each bound not equal to infinity.
2156 static __isl_give isl_map
*box_closure_on_domain(__isl_take isl_map
*map
,
2157 __isl_take isl_set
*dom
, __isl_take isl_set
*ran
, int with_id
)
2166 isl_map
*app
= NULL
;
2167 isl_basic_set
*aff
= NULL
;
2168 isl_basic_map
*bmap
= NULL
;
2169 isl_vec
*obj
= NULL
;
2174 delta
= isl_map_deltas(isl_map_copy(map
));
2176 aff
= isl_set_affine_hull(isl_set_copy(delta
));
2179 dim
= isl_map_get_space(map
);
2180 d
= isl_space_dim(dim
, isl_dim_in
);
2181 nparam
= isl_space_dim(dim
, isl_dim_param
);
2182 total
= isl_space_dim(dim
, isl_dim_all
);
2183 bmap
= isl_basic_map_alloc_space(dim
,
2184 aff
->n_div
+ 1, aff
->n_div
, 2 * d
+ 1);
2185 for (i
= 0; i
< aff
->n_div
+ 1; ++i
) {
2186 k
= isl_basic_map_alloc_div(bmap
);
2189 isl_int_set_si(bmap
->div
[k
][0], 0);
2191 for (i
= 0; i
< aff
->n_eq
; ++i
) {
2192 if (!isl_basic_set_eq_is_stride(aff
, i
))
2194 k
= isl_basic_map_alloc_equality(bmap
);
2197 isl_seq_clr(bmap
->eq
[k
], 1 + nparam
);
2198 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ d
,
2199 aff
->eq
[i
] + 1 + nparam
, d
);
2200 isl_seq_neg(bmap
->eq
[k
] + 1 + nparam
,
2201 aff
->eq
[i
] + 1 + nparam
, d
);
2202 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ 2 * d
,
2203 aff
->eq
[i
] + 1 + nparam
+ d
, aff
->n_div
);
2204 isl_int_set_si(bmap
->eq
[k
][1 + total
+ aff
->n_div
], 0);
2206 obj
= isl_vec_alloc(map
->ctx
, 1 + nparam
+ d
);
2209 isl_seq_clr(obj
->el
, 1 + nparam
+ d
);
2210 for (i
= 0; i
< d
; ++ i
) {
2211 enum isl_lp_result res
;
2213 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 1);
2215 res
= isl_set_solve_lp(delta
, 0, obj
->el
, map
->ctx
->one
, &opt
,
2217 if (res
== isl_lp_error
)
2219 if (res
== isl_lp_ok
) {
2220 k
= isl_basic_map_alloc_inequality(bmap
);
2223 isl_seq_clr(bmap
->ineq
[k
],
2224 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2225 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], -1);
2226 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], 1);
2227 isl_int_neg(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2230 res
= isl_set_solve_lp(delta
, 1, obj
->el
, map
->ctx
->one
, &opt
,
2232 if (res
== isl_lp_error
)
2234 if (res
== isl_lp_ok
) {
2235 k
= isl_basic_map_alloc_inequality(bmap
);
2238 isl_seq_clr(bmap
->ineq
[k
],
2239 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2240 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], 1);
2241 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], -1);
2242 isl_int_set(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2245 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 0);
2247 k
= isl_basic_map_alloc_inequality(bmap
);
2250 isl_seq_clr(bmap
->ineq
[k
],
2251 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2253 isl_int_set_si(bmap
->ineq
[k
][0], -1);
2254 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], 1);
2256 app
= isl_map_from_domain_and_range(dom
, ran
);
2259 isl_basic_set_free(aff
);
2261 bmap
= isl_basic_map_finalize(bmap
);
2262 isl_set_free(delta
);
2265 map
= isl_map_from_basic_map(bmap
);
2266 map
= isl_map_intersect(map
, app
);
2271 isl_basic_map_free(bmap
);
2272 isl_basic_set_free(aff
);
2276 isl_set_free(delta
);
2281 /* Given a map, compute the smallest superset of this map that is of the form
2283 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2285 * (where p ranges over the (non-parametric) dimensions),
2286 * compute the transitive closure of this map, i.e.,
2288 * { i -> j : exists k > 0:
2289 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2291 * and intersect domain and range of this transitive closure with
2292 * domain and range of the original map.
2294 static __isl_give isl_map
*box_closure(__isl_take isl_map
*map
)
2299 domain
= isl_map_domain(isl_map_copy(map
));
2300 domain
= isl_set_coalesce(domain
);
2301 range
= isl_map_range(isl_map_copy(map
));
2302 range
= isl_set_coalesce(range
);
2304 return box_closure_on_domain(map
, domain
, range
, 0);
2307 /* Given a map, compute the smallest superset of this map that is of the form
2309 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2311 * (where p ranges over the (non-parametric) dimensions),
2312 * compute the transitive and partially reflexive closure of this map, i.e.,
2314 * { i -> j : exists k >= 0:
2315 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2317 * and intersect domain and range of this transitive closure with
2320 static __isl_give isl_map
*box_closure_with_identity(__isl_take isl_map
*map
,
2321 __isl_take isl_set
*dom
)
2323 return box_closure_on_domain(map
, dom
, isl_set_copy(dom
), 1);
2326 /* Check whether app is the transitive closure of map.
2327 * In particular, check that app is acyclic and, if so,
2330 * app \subset (map \cup (map \circ app))
2332 static int check_exactness_omega(__isl_keep isl_map
*map
,
2333 __isl_keep isl_map
*app
)
2337 isl_bool is_empty
, is_exact
;
2341 delta
= isl_map_deltas(isl_map_copy(app
));
2342 d
= isl_set_dim(delta
, isl_dim_set
);
2343 for (i
= 0; i
< d
; ++i
)
2344 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
2345 is_empty
= isl_set_is_empty(delta
);
2346 isl_set_free(delta
);
2352 test
= isl_map_apply_range(isl_map_copy(app
), isl_map_copy(map
));
2353 test
= isl_map_union(test
, isl_map_copy(map
));
2354 is_exact
= isl_map_is_subset(app
, test
);
2360 /* Check if basic map M_i can be combined with all the other
2361 * basic maps such that
2365 * can be computed as
2367 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2369 * In particular, check if we can compute a compact representation
2372 * M_i^* \circ M_j \circ M_i^*
2375 * Let M_i^? be an extension of M_i^+ that allows paths
2376 * of length zero, i.e., the result of box_closure(., 1).
2377 * The criterion, as proposed by Kelly et al., is that
2378 * id = M_i^? - M_i^+ can be represented as a basic map
2381 * id \circ M_j \circ id = M_j
2385 * If this function returns 1, then tc and qc are set to
2386 * M_i^+ and M_i^?, respectively.
2388 static int can_be_split_off(__isl_keep isl_map
*map
, int i
,
2389 __isl_give isl_map
**tc
, __isl_give isl_map
**qc
)
2391 isl_map
*map_i
, *id
= NULL
;
2398 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
2399 isl_map_range(isl_map_copy(map
)));
2400 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
2404 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
2405 *tc
= box_closure(isl_map_copy(map_i
));
2406 *qc
= box_closure_with_identity(map_i
, C
);
2407 id
= isl_map_subtract(isl_map_copy(*qc
), isl_map_copy(*tc
));
2411 if (id
->n
!= 1 || (*qc
)->n
!= 1)
2414 for (j
= 0; j
< map
->n
; ++j
) {
2415 isl_map
*map_j
, *test
;
2420 map_j
= isl_map_from_basic_map(
2421 isl_basic_map_copy(map
->p
[j
]));
2422 test
= isl_map_apply_range(isl_map_copy(id
),
2423 isl_map_copy(map_j
));
2424 test
= isl_map_apply_range(test
, isl_map_copy(id
));
2425 is_ok
= isl_map_is_equal(test
, map_j
);
2426 isl_map_free(map_j
);
2454 static __isl_give isl_map
*box_closure_with_check(__isl_take isl_map
*map
,
2459 app
= box_closure(isl_map_copy(map
));
2461 *exact
= check_exactness_omega(map
, app
);
2467 /* Compute an overapproximation of the transitive closure of "map"
2468 * using a variation of the algorithm from
2469 * "Transitive Closure of Infinite Graphs and its Applications"
2472 * We first check whether we can can split of any basic map M_i and
2479 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2481 * using a recursive call on the remaining map.
2483 * If not, we simply call box_closure on the whole map.
2485 static __isl_give isl_map
*transitive_closure_omega(__isl_take isl_map
*map
,
2495 return box_closure_with_check(map
, exact
);
2497 for (i
= 0; i
< map
->n
; ++i
) {
2500 ok
= can_be_split_off(map
, i
, &tc
, &qc
);
2506 app
= isl_map_alloc_space(isl_map_get_space(map
), map
->n
- 1, 0);
2508 for (j
= 0; j
< map
->n
; ++j
) {
2511 app
= isl_map_add_basic_map(app
,
2512 isl_basic_map_copy(map
->p
[j
]));
2515 app
= isl_map_apply_range(isl_map_copy(qc
), app
);
2516 app
= isl_map_apply_range(app
, qc
);
2518 app
= isl_map_union(tc
, transitive_closure_omega(app
, NULL
));
2519 exact_i
= check_exactness_omega(map
, app
);
2531 return box_closure_with_check(map
, exact
);
2537 /* Compute the transitive closure of "map", or an overapproximation.
2538 * If the result is exact, then *exact is set to 1.
2539 * Simply use map_power to compute the powers of map, but tell
2540 * it to project out the lengths of the paths instead of equating
2541 * the length to a parameter.
2543 __isl_give isl_map
*isl_map_transitive_closure(__isl_take isl_map
*map
,
2546 isl_space
*target_dim
;
2552 if (map
->ctx
->opt
->closure
== ISL_CLOSURE_BOX
)
2553 return transitive_closure_omega(map
, exact
);
2555 map
= isl_map_compute_divs(map
);
2556 map
= isl_map_coalesce(map
);
2557 closed
= isl_map_is_transitively_closed(map
);
2566 target_dim
= isl_map_get_space(map
);
2567 map
= map_power(map
, exact
, 1);
2568 map
= isl_map_reset_space(map
, target_dim
);
2576 static isl_stat
inc_count(__isl_take isl_map
*map
, void *user
)
2587 static isl_stat
collect_basic_map(__isl_take isl_map
*map
, void *user
)
2590 isl_basic_map
***next
= user
;
2592 for (i
= 0; i
< map
->n
; ++i
) {
2593 **next
= isl_basic_map_copy(map
->p
[i
]);
2603 return isl_stat_error
;
2606 /* Perform Floyd-Warshall on the given list of basic relations.
2607 * The basic relations may live in different dimensions,
2608 * but basic relations that get assigned to the diagonal of the
2609 * grid have domains and ranges of the same dimension and so
2610 * the standard algorithm can be used because the nested transitive
2611 * closures are only applied to diagonal elements and because all
2612 * compositions are peformed on relations with compatible domains and ranges.
2614 static __isl_give isl_union_map
*union_floyd_warshall_on_list(isl_ctx
*ctx
,
2615 __isl_keep isl_basic_map
**list
, int n
, int *exact
)
2620 isl_set
**set
= NULL
;
2621 isl_map
***grid
= NULL
;
2624 group
= setup_groups(ctx
, list
, n
, &set
, &n_group
);
2628 grid
= isl_calloc_array(ctx
, isl_map
**, n_group
);
2631 for (i
= 0; i
< n_group
; ++i
) {
2632 grid
[i
] = isl_calloc_array(ctx
, isl_map
*, n_group
);
2635 for (j
= 0; j
< n_group
; ++j
) {
2636 isl_space
*space1
, *space2
, *space
;
2637 space1
= isl_space_reverse(isl_set_get_space(set
[i
]));
2638 space2
= isl_set_get_space(set
[j
]);
2639 space
= isl_space_join(space1
, space2
);
2640 grid
[i
][j
] = isl_map_empty(space
);
2644 for (k
= 0; k
< n
; ++k
) {
2646 j
= group
[2 * k
+ 1];
2647 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
2648 isl_map_from_basic_map(
2649 isl_basic_map_copy(list
[k
])));
2652 floyd_warshall_iterate(grid
, n_group
, exact
);
2654 app
= isl_union_map_empty(isl_map_get_space(grid
[0][0]));
2656 for (i
= 0; i
< n_group
; ++i
) {
2657 for (j
= 0; j
< n_group
; ++j
)
2658 app
= isl_union_map_add_map(app
, grid
[i
][j
]);
2663 for (i
= 0; i
< 2 * n
; ++i
)
2664 isl_set_free(set
[i
]);
2671 for (i
= 0; i
< n_group
; ++i
) {
2674 for (j
= 0; j
< n_group
; ++j
)
2675 isl_map_free(grid
[i
][j
]);
2680 for (i
= 0; i
< 2 * n
; ++i
)
2681 isl_set_free(set
[i
]);
2688 /* Perform Floyd-Warshall on the given union relation.
2689 * The implementation is very similar to that for non-unions.
2690 * The main difference is that it is applied unconditionally.
2691 * We first extract a list of basic maps from the union map
2692 * and then perform the algorithm on this list.
2694 static __isl_give isl_union_map
*union_floyd_warshall(
2695 __isl_take isl_union_map
*umap
, int *exact
)
2699 isl_basic_map
**list
= NULL
;
2700 isl_basic_map
**next
;
2704 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2707 ctx
= isl_union_map_get_ctx(umap
);
2708 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2713 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2716 res
= union_floyd_warshall_on_list(ctx
, list
, n
, exact
);
2719 for (i
= 0; i
< n
; ++i
)
2720 isl_basic_map_free(list
[i
]);
2724 isl_union_map_free(umap
);
2728 for (i
= 0; i
< n
; ++i
)
2729 isl_basic_map_free(list
[i
]);
2732 isl_union_map_free(umap
);
2736 /* Decompose the give union relation into strongly connected components.
2737 * The implementation is essentially the same as that of
2738 * construct_power_components with the major difference that all
2739 * operations are performed on union maps.
2741 static __isl_give isl_union_map
*union_components(
2742 __isl_take isl_union_map
*umap
, int *exact
)
2747 isl_basic_map
**list
= NULL
;
2748 isl_basic_map
**next
;
2749 isl_union_map
*path
= NULL
;
2750 struct isl_tc_follows_data data
;
2751 struct isl_tarjan_graph
*g
= NULL
;
2756 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2762 return union_floyd_warshall(umap
, exact
);
2764 ctx
= isl_union_map_get_ctx(umap
);
2765 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2770 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2774 data
.check_closed
= 0;
2775 g
= isl_tarjan_graph_init(ctx
, n
, &basic_map_follows
, &data
);
2782 path
= isl_union_map_empty(isl_union_map_get_space(umap
));
2784 isl_union_map
*comp
;
2785 isl_union_map
*path_comp
, *path_comb
;
2786 comp
= isl_union_map_empty(isl_union_map_get_space(umap
));
2787 while (g
->order
[i
] != -1) {
2788 comp
= isl_union_map_add_map(comp
,
2789 isl_map_from_basic_map(
2790 isl_basic_map_copy(list
[g
->order
[i
]])));
2794 path_comp
= union_floyd_warshall(comp
, exact
);
2795 path_comb
= isl_union_map_apply_range(isl_union_map_copy(path
),
2796 isl_union_map_copy(path_comp
));
2797 path
= isl_union_map_union(path
, path_comp
);
2798 path
= isl_union_map_union(path
, path_comb
);
2803 if (c
> 1 && data
.check_closed
&& !*exact
) {
2806 closed
= isl_union_map_is_transitively_closed(path
);
2812 isl_tarjan_graph_free(g
);
2814 for (i
= 0; i
< n
; ++i
)
2815 isl_basic_map_free(list
[i
]);
2819 isl_union_map_free(path
);
2820 return union_floyd_warshall(umap
, exact
);
2823 isl_union_map_free(umap
);
2827 isl_tarjan_graph_free(g
);
2829 for (i
= 0; i
< n
; ++i
)
2830 isl_basic_map_free(list
[i
]);
2833 isl_union_map_free(umap
);
2834 isl_union_map_free(path
);
2838 /* Compute the transitive closure of "umap", or an overapproximation.
2839 * If the result is exact, then *exact is set to 1.
2841 __isl_give isl_union_map
*isl_union_map_transitive_closure(
2842 __isl_take isl_union_map
*umap
, int *exact
)
2852 umap
= isl_union_map_compute_divs(umap
);
2853 umap
= isl_union_map_coalesce(umap
);
2854 closed
= isl_union_map_is_transitively_closed(umap
);
2859 umap
= union_components(umap
, exact
);
2862 isl_union_map_free(umap
);
2866 struct isl_union_power
{
2871 static isl_stat
power(__isl_take isl_map
*map
, void *user
)
2873 struct isl_union_power
*up
= user
;
2875 map
= isl_map_power(map
, up
->exact
);
2876 up
->pow
= isl_union_map_from_map(map
);
2878 return isl_stat_error
;
2881 /* Construct a map [x] -> [x+1], with parameters prescribed by "space".
2883 static __isl_give isl_union_map
*increment(__isl_take isl_space
*space
)
2886 isl_basic_map
*bmap
;
2888 space
= isl_space_add_dims(space
, isl_dim_in
, 1);
2889 space
= isl_space_add_dims(space
, isl_dim_out
, 1);
2890 bmap
= isl_basic_map_alloc_space(space
, 0, 1, 0);
2891 k
= isl_basic_map_alloc_equality(bmap
);
2894 isl_seq_clr(bmap
->eq
[k
], 1 + isl_basic_map_total_dim(bmap
));
2895 isl_int_set_si(bmap
->eq
[k
][0], 1);
2896 isl_int_set_si(bmap
->eq
[k
][isl_basic_map_offset(bmap
, isl_dim_in
)], 1);
2897 isl_int_set_si(bmap
->eq
[k
][isl_basic_map_offset(bmap
, isl_dim_out
)], -1);
2898 return isl_union_map_from_map(isl_map_from_basic_map(bmap
));
2900 isl_basic_map_free(bmap
);
2904 /* Construct a map [[x]->[y]] -> [y-x], with parameters prescribed by "dim".
2906 static __isl_give isl_union_map
*deltas_map(__isl_take isl_space
*dim
)
2908 isl_basic_map
*bmap
;
2910 dim
= isl_space_add_dims(dim
, isl_dim_in
, 1);
2911 dim
= isl_space_add_dims(dim
, isl_dim_out
, 1);
2912 bmap
= isl_basic_map_universe(dim
);
2913 bmap
= isl_basic_map_deltas_map(bmap
);
2915 return isl_union_map_from_map(isl_map_from_basic_map(bmap
));
2918 /* Compute the positive powers of "map", or an overapproximation.
2919 * The result maps the exponent to a nested copy of the corresponding power.
2920 * If the result is exact, then *exact is set to 1.
2922 __isl_give isl_union_map
*isl_union_map_power(__isl_take isl_union_map
*umap
,
2931 n
= isl_union_map_n_map(umap
);
2935 struct isl_union_power up
= { NULL
, exact
};
2936 isl_union_map_foreach_map(umap
, &power
, &up
);
2937 isl_union_map_free(umap
);
2940 inc
= increment(isl_union_map_get_space(umap
));
2941 umap
= isl_union_map_product(inc
, umap
);
2942 umap
= isl_union_map_transitive_closure(umap
, exact
);
2943 umap
= isl_union_map_zip(umap
);
2944 dm
= deltas_map(isl_union_map_get_space(umap
));
2945 umap
= isl_union_map_apply_domain(umap
, dm
);
2951 #define TYPE isl_map
2952 #include "isl_power_templ.c"
2955 #define TYPE isl_union_map
2956 #include "isl_power_templ.c"