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"
14 #include <isl_dim_private.h>
16 #include <isl_union_map.h>
18 int isl_map_is_transitively_closed(__isl_keep isl_map
*map
)
23 map2
= isl_map_apply_range(isl_map_copy(map
), isl_map_copy(map
));
24 closed
= isl_map_is_subset(map2
, map
);
30 int isl_union_map_is_transitively_closed(__isl_keep isl_union_map
*umap
)
35 umap2
= isl_union_map_apply_range(isl_union_map_copy(umap
),
36 isl_union_map_copy(umap
));
37 closed
= isl_union_map_is_subset(umap2
, umap
);
38 isl_union_map_free(umap2
);
43 /* Given a map that represents a path with the length of the path
44 * encoded as the difference between the last output coordindate
45 * and the last input coordinate, set this length to either
46 * exactly "length" (if "exactly" is set) or at least "length"
47 * (if "exactly" is not set).
49 static __isl_give isl_map
*set_path_length(__isl_take isl_map
*map
,
50 int exactly
, int length
)
53 struct isl_basic_map
*bmap
;
62 dim
= isl_map_get_dim(map
);
63 d
= isl_dim_size(dim
, isl_dim_in
);
64 nparam
= isl_dim_size(dim
, isl_dim_param
);
65 bmap
= isl_basic_map_alloc_dim(dim
, 0, 1, 1);
67 k
= isl_basic_map_alloc_equality(bmap
);
70 k
= isl_basic_map_alloc_inequality(bmap
);
75 isl_seq_clr(c
, 1 + isl_basic_map_total_dim(bmap
));
76 isl_int_set_si(c
[0], -length
);
77 isl_int_set_si(c
[1 + nparam
+ d
- 1], -1);
78 isl_int_set_si(c
[1 + nparam
+ d
+ d
- 1], 1);
80 bmap
= isl_basic_map_finalize(bmap
);
81 map
= isl_map_intersect(map
, isl_map_from_basic_map(bmap
));
85 isl_basic_map_free(bmap
);
90 /* Check whether the overapproximation of the power of "map" is exactly
91 * the power of "map". Let R be "map" and A_k the overapproximation.
92 * The approximation is exact if
95 * A_k = A_{k-1} \circ R k >= 2
97 * Since A_k is known to be an overapproximation, we only need to check
100 * A_k \subset A_{k-1} \circ R k >= 2
102 * In practice, "app" has an extra input and output coordinate
103 * to encode the length of the path. So, we first need to add
104 * this coordinate to "map" and set the length of the path to
107 static int check_power_exactness(__isl_take isl_map
*map
,
108 __isl_take isl_map
*app
)
114 map
= isl_map_add(map
, isl_dim_in
, 1);
115 map
= isl_map_add(map
, isl_dim_out
, 1);
116 map
= set_path_length(map
, 1, 1);
118 app_1
= set_path_length(isl_map_copy(app
), 1, 1);
120 exact
= isl_map_is_subset(app_1
, map
);
123 if (!exact
|| exact
< 0) {
129 app_1
= set_path_length(isl_map_copy(app
), 0, 1);
130 app_2
= set_path_length(app
, 0, 2);
131 app_1
= isl_map_apply_range(map
, app_1
);
133 exact
= isl_map_is_subset(app_2
, app_1
);
141 /* Check whether the overapproximation of the power of "map" is exactly
142 * the power of "map", possibly after projecting out the power (if "project"
145 * If "project" is set and if "steps" can only result in acyclic paths,
148 * A = R \cup (A \circ R)
150 * where A is the overapproximation with the power projected out, i.e.,
151 * an overapproximation of the transitive closure.
152 * More specifically, since A is known to be an overapproximation, we check
154 * A \subset R \cup (A \circ R)
156 * Otherwise, we check if the power is exact.
158 * Note that "app" has an extra input and output coordinate to encode
159 * the length of the part. If we are only interested in the transitive
160 * closure, then we can simply project out these coordinates first.
162 static int check_exactness(__isl_take isl_map
*map
, __isl_take isl_map
*app
,
170 return check_power_exactness(map
, app
);
172 d
= isl_map_dim(map
, isl_dim_in
);
173 app
= set_path_length(app
, 0, 1);
174 app
= isl_map_project_out(app
, isl_dim_in
, d
, 1);
175 app
= isl_map_project_out(app
, isl_dim_out
, d
, 1);
177 app
= isl_map_reset_dim(app
, isl_map_get_dim(map
));
179 test
= isl_map_apply_range(isl_map_copy(map
), isl_map_copy(app
));
180 test
= isl_map_union(test
, isl_map_copy(map
));
182 exact
= isl_map_is_subset(app
, test
);
193 * The transitive closure implementation is based on the paper
194 * "Computing the Transitive Closure of a Union of Affine Integer
195 * Tuple Relations" by Anna Beletska, Denis Barthou, Wlodzimierz Bielecki and
199 /* Given a set of n offsets v_i (the rows of "steps"), construct a relation
200 * of the given dimension specification (Z^{n+1} -> Z^{n+1})
201 * that maps an element x to any element that can be reached
202 * by taking a non-negative number of steps along any of
203 * the extended offsets v'_i = [v_i 1].
206 * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v'_i }
208 * For any element in this relation, the number of steps taken
209 * is equal to the difference in the final coordinates.
211 static __isl_give isl_map
*path_along_steps(__isl_take isl_dim
*dim
,
212 __isl_keep isl_mat
*steps
)
215 struct isl_basic_map
*path
= NULL
;
223 d
= isl_dim_size(dim
, isl_dim_in
);
225 nparam
= isl_dim_size(dim
, isl_dim_param
);
227 path
= isl_basic_map_alloc_dim(isl_dim_copy(dim
), n
, d
, n
);
229 for (i
= 0; i
< n
; ++i
) {
230 k
= isl_basic_map_alloc_div(path
);
233 isl_assert(steps
->ctx
, i
== k
, goto error
);
234 isl_int_set_si(path
->div
[k
][0], 0);
237 for (i
= 0; i
< d
; ++i
) {
238 k
= isl_basic_map_alloc_equality(path
);
241 isl_seq_clr(path
->eq
[k
], 1 + isl_basic_map_total_dim(path
));
242 isl_int_set_si(path
->eq
[k
][1 + nparam
+ i
], 1);
243 isl_int_set_si(path
->eq
[k
][1 + nparam
+ d
+ i
], -1);
245 for (j
= 0; j
< n
; ++j
)
246 isl_int_set_si(path
->eq
[k
][1 + nparam
+ 2 * d
+ j
], 1);
248 for (j
= 0; j
< n
; ++j
)
249 isl_int_set(path
->eq
[k
][1 + nparam
+ 2 * d
+ j
],
253 for (i
= 0; i
< n
; ++i
) {
254 k
= isl_basic_map_alloc_inequality(path
);
257 isl_seq_clr(path
->ineq
[k
], 1 + isl_basic_map_total_dim(path
));
258 isl_int_set_si(path
->ineq
[k
][1 + nparam
+ 2 * d
+ i
], 1);
263 path
= isl_basic_map_simplify(path
);
264 path
= isl_basic_map_finalize(path
);
265 return isl_map_from_basic_map(path
);
268 isl_basic_map_free(path
);
277 /* Check whether the parametric constant term of constraint c is never
278 * positive in "bset".
280 static int parametric_constant_never_positive(__isl_keep isl_basic_set
*bset
,
281 isl_int
*c
, int *div_purity
)
290 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
291 d
= isl_basic_set_dim(bset
, isl_dim_set
);
292 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
294 bset
= isl_basic_set_copy(bset
);
295 bset
= isl_basic_set_cow(bset
);
296 bset
= isl_basic_set_extend_constraints(bset
, 0, 1);
297 k
= isl_basic_set_alloc_inequality(bset
);
300 isl_seq_clr(bset
->ineq
[k
], 1 + isl_basic_set_total_dim(bset
));
301 isl_seq_cpy(bset
->ineq
[k
], c
, 1 + nparam
);
302 for (i
= 0; i
< n_div
; ++i
) {
303 if (div_purity
[i
] != PURE_PARAM
)
305 isl_int_set(bset
->ineq
[k
][1 + nparam
+ d
+ i
],
306 c
[1 + nparam
+ d
+ i
]);
308 isl_int_sub_ui(bset
->ineq
[k
][0], bset
->ineq
[k
][0], 1);
309 empty
= isl_basic_set_is_empty(bset
);
310 isl_basic_set_free(bset
);
314 isl_basic_set_free(bset
);
318 /* Return PURE_PARAM if only the coefficients of the parameters are non-zero.
319 * Return PURE_VAR if only the coefficients of the set variables are non-zero.
320 * Return MIXED if only the coefficients of the parameters and the set
321 * variables are non-zero and if moreover the parametric constant
322 * can never attain positive values.
323 * Return IMPURE otherwise.
325 static int purity(__isl_keep isl_basic_set
*bset
, isl_int
*c
, int *div_purity
,
335 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
336 d
= isl_basic_set_dim(bset
, isl_dim_set
);
337 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
339 for (i
= 0; i
< n_div
; ++i
) {
340 if (isl_int_is_zero(c
[1 + nparam
+ d
+ i
]))
342 switch (div_purity
[i
]) {
343 case PURE_PARAM
: p
= 1; break;
344 case PURE_VAR
: v
= 1; break;
345 default: return IMPURE
;
348 if (!p
&& isl_seq_first_non_zero(c
+ 1, nparam
) == -1)
350 if (!v
&& isl_seq_first_non_zero(c
+ 1 + nparam
, d
) == -1)
353 empty
= parametric_constant_never_positive(bset
, c
, div_purity
);
354 if (eq
&& empty
>= 0 && !empty
) {
355 isl_seq_neg(c
, c
, 1 + nparam
+ d
+ n_div
);
356 empty
= parametric_constant_never_positive(bset
, c
, div_purity
);
359 return empty
< 0 ? -1 : empty
? MIXED
: IMPURE
;
362 /* Return an array of integers indicating the type of each div in bset.
363 * If the div is (recursively) defined in terms of only the parameters,
364 * then the type is PURE_PARAM.
365 * If the div is (recursively) defined in terms of only the set variables,
366 * then the type is PURE_VAR.
367 * Otherwise, the type is IMPURE.
369 static __isl_give
int *get_div_purity(__isl_keep isl_basic_set
*bset
)
380 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
381 d
= isl_basic_set_dim(bset
, isl_dim_set
);
382 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
384 div_purity
= isl_alloc_array(bset
->ctx
, int, n_div
);
388 for (i
= 0; i
< bset
->n_div
; ++i
) {
390 if (isl_int_is_zero(bset
->div
[i
][0])) {
391 div_purity
[i
] = IMPURE
;
394 if (isl_seq_first_non_zero(bset
->div
[i
] + 2, nparam
) != -1)
396 if (isl_seq_first_non_zero(bset
->div
[i
] + 2 + nparam
, d
) != -1)
398 for (j
= 0; j
< i
; ++j
) {
399 if (isl_int_is_zero(bset
->div
[i
][2 + nparam
+ d
+ j
]))
401 switch (div_purity
[j
]) {
402 case PURE_PARAM
: p
= 1; break;
403 case PURE_VAR
: v
= 1; break;
404 default: p
= v
= 1; break;
407 div_purity
[i
] = v
? p
? IMPURE
: PURE_VAR
: PURE_PARAM
;
413 /* Given a path with the as yet unconstrained length at position "pos",
414 * check if setting the length to zero results in only the identity
417 int empty_path_is_identity(__isl_keep isl_basic_map
*path
, unsigned pos
)
419 isl_basic_map
*test
= NULL
;
420 isl_basic_map
*id
= NULL
;
424 test
= isl_basic_map_copy(path
);
425 test
= isl_basic_map_extend_constraints(test
, 1, 0);
426 k
= isl_basic_map_alloc_equality(test
);
429 isl_seq_clr(test
->eq
[k
], 1 + isl_basic_map_total_dim(test
));
430 isl_int_set_si(test
->eq
[k
][pos
], 1);
431 id
= isl_basic_map_identity(isl_dim_domain(isl_basic_map_get_dim(path
)));
432 is_id
= isl_basic_map_is_equal(test
, id
);
433 isl_basic_map_free(test
);
434 isl_basic_map_free(id
);
437 isl_basic_map_free(test
);
441 __isl_give isl_basic_map
*add_delta_constraints(__isl_take isl_basic_map
*path
,
442 __isl_keep isl_basic_set
*delta
, unsigned off
, unsigned nparam
,
443 unsigned d
, int *div_purity
, int eq
)
446 int n
= eq
? delta
->n_eq
: delta
->n_ineq
;
447 isl_int
**delta_c
= eq
? delta
->eq
: delta
->ineq
;
450 n_div
= isl_basic_set_dim(delta
, isl_dim_div
);
452 for (i
= 0; i
< n
; ++i
) {
454 int p
= purity(delta
, delta_c
[i
], div_purity
, eq
);
459 if (eq
&& p
!= MIXED
) {
460 k
= isl_basic_map_alloc_equality(path
);
461 path_c
= path
->eq
[k
];
463 k
= isl_basic_map_alloc_inequality(path
);
464 path_c
= path
->ineq
[k
];
468 isl_seq_clr(path_c
, 1 + isl_basic_map_total_dim(path
));
470 isl_seq_cpy(path_c
+ off
,
471 delta_c
[i
] + 1 + nparam
, d
);
472 isl_int_set(path_c
[off
+ d
], delta_c
[i
][0]);
473 } else if (p
== PURE_PARAM
) {
474 isl_seq_cpy(path_c
, delta_c
[i
], 1 + nparam
);
476 isl_seq_cpy(path_c
+ off
,
477 delta_c
[i
] + 1 + nparam
, d
);
478 isl_seq_cpy(path_c
, delta_c
[i
], 1 + nparam
);
480 isl_seq_cpy(path_c
+ off
- n_div
,
481 delta_c
[i
] + 1 + nparam
+ d
, n_div
);
486 isl_basic_map_free(path
);
490 /* Given a set of offsets "delta", construct a relation of the
491 * given dimension specification (Z^{n+1} -> Z^{n+1}) that
492 * is an overapproximation of the relations that
493 * maps an element x to any element that can be reached
494 * by taking a non-negative number of steps along any of
495 * the elements in "delta".
496 * That is, construct an approximation of
498 * { [x] -> [y] : exists f \in \delta, k \in Z :
499 * y = x + k [f, 1] and k >= 0 }
501 * For any element in this relation, the number of steps taken
502 * is equal to the difference in the final coordinates.
504 * In particular, let delta be defined as
506 * \delta = [p] -> { [x] : A x + a >= and B p + b >= 0 and
507 * C x + C'p + c >= 0 and
508 * D x + D'p + d >= 0 }
510 * where the constraints C x + C'p + c >= 0 are such that the parametric
511 * constant term of each constraint j, "C_j x + C'_j p + c_j",
512 * can never attain positive values, then the relation is constructed as
514 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
515 * A f + k a >= 0 and B p + b >= 0 and
516 * C f + C'p + c >= 0 and k >= 1 }
517 * union { [x] -> [x] }
519 * If the zero-length paths happen to correspond exactly to the identity
520 * mapping, then we return
522 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
523 * A f + k a >= 0 and B p + b >= 0 and
524 * C f + C'p + c >= 0 and k >= 0 }
528 * Existentially quantified variables in \delta are handled by
529 * classifying them as independent of the parameters, purely
530 * parameter dependent and others. Constraints containing
531 * any of the other existentially quantified variables are removed.
532 * This is safe, but leads to an additional overapproximation.
534 static __isl_give isl_map
*path_along_delta(__isl_take isl_dim
*dim
,
535 __isl_take isl_basic_set
*delta
)
537 isl_basic_map
*path
= NULL
;
544 int *div_purity
= NULL
;
548 n_div
= isl_basic_set_dim(delta
, isl_dim_div
);
549 d
= isl_basic_set_dim(delta
, isl_dim_set
);
550 nparam
= isl_basic_set_dim(delta
, isl_dim_param
);
551 path
= isl_basic_map_alloc_dim(isl_dim_copy(dim
), n_div
+ d
+ 1,
552 d
+ 1 + delta
->n_eq
, delta
->n_eq
+ delta
->n_ineq
+ 1);
553 off
= 1 + nparam
+ 2 * (d
+ 1) + n_div
;
555 for (i
= 0; i
< n_div
+ d
+ 1; ++i
) {
556 k
= isl_basic_map_alloc_div(path
);
559 isl_int_set_si(path
->div
[k
][0], 0);
562 for (i
= 0; i
< d
+ 1; ++i
) {
563 k
= isl_basic_map_alloc_equality(path
);
566 isl_seq_clr(path
->eq
[k
], 1 + isl_basic_map_total_dim(path
));
567 isl_int_set_si(path
->eq
[k
][1 + nparam
+ i
], 1);
568 isl_int_set_si(path
->eq
[k
][1 + nparam
+ d
+ 1 + i
], -1);
569 isl_int_set_si(path
->eq
[k
][off
+ i
], 1);
572 div_purity
= get_div_purity(delta
);
576 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
, div_purity
, 1);
577 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
, div_purity
, 0);
579 is_id
= empty_path_is_identity(path
, off
+ d
);
583 k
= isl_basic_map_alloc_inequality(path
);
586 isl_seq_clr(path
->ineq
[k
], 1 + isl_basic_map_total_dim(path
));
588 isl_int_set_si(path
->ineq
[k
][0], -1);
589 isl_int_set_si(path
->ineq
[k
][off
+ d
], 1);
592 isl_basic_set_free(delta
);
593 path
= isl_basic_map_finalize(path
);
596 return isl_map_from_basic_map(path
);
598 return isl_basic_map_union(path
,
599 isl_basic_map_identity(isl_dim_domain(dim
)));
603 isl_basic_set_free(delta
);
604 isl_basic_map_free(path
);
608 /* Given a dimension specification Z^{n+1} -> Z^{n+1} and a parameter "param",
609 * construct a map that equates the parameter to the difference
610 * in the final coordinates and imposes that this difference is positive.
613 * { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
615 static __isl_give isl_map
*equate_parameter_to_length(__isl_take isl_dim
*dim
,
618 struct isl_basic_map
*bmap
;
623 d
= isl_dim_size(dim
, isl_dim_in
);
624 nparam
= isl_dim_size(dim
, isl_dim_param
);
625 bmap
= isl_basic_map_alloc_dim(dim
, 0, 1, 1);
626 k
= isl_basic_map_alloc_equality(bmap
);
629 isl_seq_clr(bmap
->eq
[k
], 1 + isl_basic_map_total_dim(bmap
));
630 isl_int_set_si(bmap
->eq
[k
][1 + param
], -1);
631 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
- 1], -1);
632 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
+ d
- 1], 1);
634 k
= isl_basic_map_alloc_inequality(bmap
);
637 isl_seq_clr(bmap
->ineq
[k
], 1 + isl_basic_map_total_dim(bmap
));
638 isl_int_set_si(bmap
->ineq
[k
][1 + param
], 1);
639 isl_int_set_si(bmap
->ineq
[k
][0], -1);
641 bmap
= isl_basic_map_finalize(bmap
);
642 return isl_map_from_basic_map(bmap
);
644 isl_basic_map_free(bmap
);
648 /* Check whether "path" is acyclic, where the last coordinates of domain
649 * and range of path encode the number of steps taken.
650 * That is, check whether
652 * { d | d = y - x and (x,y) in path }
654 * does not contain any element with positive last coordinate (positive length)
655 * and zero remaining coordinates (cycle).
657 static int is_acyclic(__isl_take isl_map
*path
)
662 struct isl_set
*delta
;
664 delta
= isl_map_deltas(path
);
665 dim
= isl_set_dim(delta
, isl_dim_set
);
666 for (i
= 0; i
< dim
; ++i
) {
668 delta
= isl_set_lower_bound_si(delta
, isl_dim_set
, i
, 1);
670 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
673 acyclic
= isl_set_is_empty(delta
);
679 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
680 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
681 * construct a map that is an overapproximation of the map
682 * that takes an element from the space D \times Z to another
683 * element from the same space, such that the first n coordinates of the
684 * difference between them is a sum of differences between images
685 * and pre-images in one of the R_i and such that the last coordinate
686 * is equal to the number of steps taken.
689 * \Delta_i = { y - x | (x, y) in R_i }
691 * then the constructed map is an overapproximation of
693 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
694 * d = (\sum_i k_i \delta_i, \sum_i k_i) }
696 * The elements of the singleton \Delta_i's are collected as the
697 * rows of the steps matrix. For all these \Delta_i's together,
698 * a single path is constructed.
699 * For each of the other \Delta_i's, we compute an overapproximation
700 * of the paths along elements of \Delta_i.
701 * Since each of these paths performs an addition, composition is
702 * symmetric and we can simply compose all resulting paths in any order.
704 static __isl_give isl_map
*construct_extended_path(__isl_take isl_dim
*dim
,
705 __isl_keep isl_map
*map
, int *project
)
707 struct isl_mat
*steps
= NULL
;
708 struct isl_map
*path
= NULL
;
712 d
= isl_map_dim(map
, isl_dim_in
);
714 path
= isl_map_identity(isl_dim_domain(isl_dim_copy(dim
)));
716 steps
= isl_mat_alloc(map
->ctx
, map
->n
, d
);
721 for (i
= 0; i
< map
->n
; ++i
) {
722 struct isl_basic_set
*delta
;
724 delta
= isl_basic_map_deltas(isl_basic_map_copy(map
->p
[i
]));
726 for (j
= 0; j
< d
; ++j
) {
729 fixed
= isl_basic_set_fast_dim_is_fixed(delta
, j
,
732 isl_basic_set_free(delta
);
741 path
= isl_map_apply_range(path
,
742 path_along_delta(isl_dim_copy(dim
), delta
));
743 path
= isl_map_coalesce(path
);
745 isl_basic_set_free(delta
);
752 path
= isl_map_apply_range(path
,
753 path_along_steps(isl_dim_copy(dim
), steps
));
756 if (project
&& *project
) {
757 *project
= is_acyclic(isl_map_copy(path
));
772 static int isl_set_overlaps(__isl_keep isl_set
*set1
, __isl_keep isl_set
*set2
)
777 if (!isl_dim_tuple_match(set1
->dim
, isl_dim_set
, set2
->dim
, isl_dim_set
))
780 i
= isl_set_intersect(isl_set_copy(set1
), isl_set_copy(set2
));
781 no_overlap
= isl_set_is_empty(i
);
784 return no_overlap
< 0 ? -1 : !no_overlap
;
787 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
788 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
789 * construct a map that is an overapproximation of the map
790 * that takes an element from the dom R \times Z to an
791 * element from ran R \times Z, such that the first n coordinates of the
792 * difference between them is a sum of differences between images
793 * and pre-images in one of the R_i and such that the last coordinate
794 * is equal to the number of steps taken.
797 * \Delta_i = { y - x | (x, y) in R_i }
799 * then the constructed map is an overapproximation of
801 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
802 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
803 * x in dom R and x + d in ran R and
806 static __isl_give isl_map
*construct_component(__isl_take isl_dim
*dim
,
807 __isl_keep isl_map
*map
, int *exact
, int project
)
809 struct isl_set
*domain
= NULL
;
810 struct isl_set
*range
= NULL
;
811 struct isl_map
*app
= NULL
;
812 struct isl_map
*path
= NULL
;
814 domain
= isl_map_domain(isl_map_copy(map
));
815 domain
= isl_set_coalesce(domain
);
816 range
= isl_map_range(isl_map_copy(map
));
817 range
= isl_set_coalesce(range
);
818 if (!isl_set_overlaps(domain
, range
)) {
819 isl_set_free(domain
);
823 map
= isl_map_copy(map
);
824 map
= isl_map_add(map
, isl_dim_in
, 1);
825 map
= isl_map_add(map
, isl_dim_out
, 1);
826 map
= set_path_length(map
, 1, 1);
829 app
= isl_map_from_domain_and_range(domain
, range
);
830 app
= isl_map_add(app
, isl_dim_in
, 1);
831 app
= isl_map_add(app
, isl_dim_out
, 1);
833 path
= construct_extended_path(isl_dim_copy(dim
), map
,
834 exact
&& *exact
? &project
: NULL
);
835 app
= isl_map_intersect(app
, path
);
837 if (exact
&& *exact
&&
838 (*exact
= check_exactness(isl_map_copy(map
), isl_map_copy(app
),
843 app
= set_path_length(app
, 0, 1);
851 /* Call construct_component and, if "project" is set, project out
852 * the final coordinates.
854 static __isl_give isl_map
*construct_projected_component(
855 __isl_take isl_dim
*dim
,
856 __isl_keep isl_map
*map
, int *exact
, int project
)
863 d
= isl_dim_size(dim
, isl_dim_in
);
865 app
= construct_component(dim
, map
, exact
, project
);
867 app
= isl_map_project_out(app
, isl_dim_in
, d
- 1, 1);
868 app
= isl_map_project_out(app
, isl_dim_out
, d
- 1, 1);
873 /* Compute an extended version, i.e., with path lengths, of
874 * an overapproximation of the transitive closure of "bmap"
875 * with path lengths greater than or equal to zero and with
876 * domain and range equal to "dom".
878 static __isl_give isl_map
*q_closure(__isl_take isl_dim
*dim
,
879 __isl_take isl_set
*dom
, __isl_keep isl_basic_map
*bmap
, int *exact
)
886 dom
= isl_set_add(dom
, isl_dim_set
, 1);
887 app
= isl_map_from_domain_and_range(dom
, isl_set_copy(dom
));
888 map
= isl_map_from_basic_map(isl_basic_map_copy(bmap
));
889 path
= construct_extended_path(dim
, map
, &project
);
890 app
= isl_map_intersect(app
, path
);
892 if ((*exact
= check_exactness(map
, isl_map_copy(app
), project
)) < 0)
901 /* Check whether qc has any elements of length at least one
902 * with domain and/or range outside of dom and ran.
904 static int has_spurious_elements(__isl_keep isl_map
*qc
,
905 __isl_keep isl_set
*dom
, __isl_keep isl_set
*ran
)
911 if (!qc
|| !dom
|| !ran
)
914 d
= isl_map_dim(qc
, isl_dim_in
);
916 qc
= isl_map_copy(qc
);
917 qc
= set_path_length(qc
, 0, 1);
918 qc
= isl_map_project_out(qc
, isl_dim_in
, d
- 1, 1);
919 qc
= isl_map_project_out(qc
, isl_dim_out
, d
- 1, 1);
921 s
= isl_map_domain(isl_map_copy(qc
));
922 subset
= isl_set_is_subset(s
, dom
);
931 s
= isl_map_range(qc
);
932 subset
= isl_set_is_subset(s
, ran
);
935 return subset
< 0 ? -1 : !subset
;
944 /* For each basic map in "map", except i, check whether it combines
945 * with the transitive closure that is reflexive on C combines
946 * to the left and to the right.
950 * dom map_j \subseteq C
952 * then right[j] is set to 1. Otherwise, if
954 * ran map_i \cap dom map_j = \emptyset
956 * then right[j] is set to 0. Otherwise, composing to the right
959 * Similar, for composing to the left, we have if
961 * ran map_j \subseteq C
963 * then left[j] is set to 1. Otherwise, if
965 * dom map_i \cap ran map_j = \emptyset
967 * then left[j] is set to 0. Otherwise, composing to the left
970 * The return value is or'd with LEFT if composing to the left
971 * is possible and with RIGHT if composing to the right is possible.
973 static int composability(__isl_keep isl_set
*C
, int i
,
974 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
975 __isl_keep isl_map
*map
)
981 for (j
= 0; j
< map
->n
&& ok
; ++j
) {
982 int overlaps
, subset
;
988 dom
[j
] = isl_set_from_basic_set(
989 isl_basic_map_domain(
990 isl_basic_map_copy(map
->p
[j
])));
993 overlaps
= isl_set_overlaps(ran
[i
], dom
[j
]);
999 subset
= isl_set_is_subset(dom
[j
], C
);
1011 ran
[j
] = isl_set_from_basic_set(
1012 isl_basic_map_range(
1013 isl_basic_map_copy(map
->p
[j
])));
1016 overlaps
= isl_set_overlaps(dom
[i
], ran
[j
]);
1022 subset
= isl_set_is_subset(ran
[j
], C
);
1036 static __isl_give isl_map
*anonymize(__isl_take isl_map
*map
)
1038 map
= isl_map_reset(map
, isl_dim_in
);
1039 map
= isl_map_reset(map
, isl_dim_out
);
1043 /* Return a map that is a union of the basic maps in "map", except i,
1044 * composed to left and right with qc based on the entries of "left"
1047 static __isl_give isl_map
*compose(__isl_keep isl_map
*map
, int i
,
1048 __isl_take isl_map
*qc
, int *left
, int *right
)
1053 comp
= isl_map_empty(isl_map_get_dim(map
));
1054 for (j
= 0; j
< map
->n
; ++j
) {
1060 map_j
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[j
]));
1061 map_j
= anonymize(map_j
);
1062 if (left
&& left
[j
])
1063 map_j
= isl_map_apply_range(map_j
, isl_map_copy(qc
));
1064 if (right
&& right
[j
])
1065 map_j
= isl_map_apply_range(isl_map_copy(qc
), map_j
);
1066 comp
= isl_map_union(comp
, map_j
);
1069 comp
= isl_map_compute_divs(comp
);
1070 comp
= isl_map_coalesce(comp
);
1077 /* Compute the transitive closure of "map" incrementally by
1084 * map_i^+ \cup ((id \cup map_i^) \circ qc^+)
1088 * map_i^+ \cup (qc^+ \circ (id \cup map_i^))
1090 * depending on whether left or right are NULL.
1092 static __isl_give isl_map
*compute_incremental(
1093 __isl_take isl_dim
*dim
, __isl_keep isl_map
*map
,
1094 int i
, __isl_take isl_map
*qc
, int *left
, int *right
, int *exact
)
1098 isl_map
*rtc
= NULL
;
1102 isl_assert(map
->ctx
, left
|| right
, goto error
);
1104 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
1105 tc
= construct_projected_component(isl_dim_copy(dim
), map_i
,
1107 isl_map_free(map_i
);
1110 qc
= isl_map_transitive_closure(qc
, exact
);
1116 return isl_map_universe(isl_map_get_dim(map
));
1119 if (!left
|| !right
)
1120 rtc
= isl_map_union(isl_map_copy(tc
),
1121 isl_map_identity(isl_dim_domain(isl_map_get_dim(tc
))));
1123 qc
= isl_map_apply_range(rtc
, qc
);
1125 qc
= isl_map_apply_range(qc
, rtc
);
1126 qc
= isl_map_union(tc
, qc
);
1137 /* Given a map "map", try to find a basic map such that
1138 * map^+ can be computed as
1140 * map^+ = map_i^+ \cup
1141 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1143 * with C the simple hull of the domain and range of the input map.
1144 * map_i^ \cup Id_C is computed by allowing the path lengths to be zero
1145 * and by intersecting domain and range with C.
1146 * Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
1147 * Also, we only use the incremental computation if all the transitive
1148 * closures are exact and if the number of basic maps in the union,
1149 * after computing the integer divisions, is smaller than the number
1150 * of basic maps in the input map.
1152 static int incemental_on_entire_domain(__isl_keep isl_dim
*dim
,
1153 __isl_keep isl_map
*map
,
1154 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
1155 __isl_give isl_map
**res
)
1163 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
1164 isl_map_range(isl_map_copy(map
)));
1165 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1173 d
= isl_map_dim(map
, isl_dim_in
);
1175 for (i
= 0; i
< map
->n
; ++i
) {
1177 int exact_i
, spurious
;
1179 dom
[i
] = isl_set_from_basic_set(isl_basic_map_domain(
1180 isl_basic_map_copy(map
->p
[i
])));
1181 ran
[i
] = isl_set_from_basic_set(isl_basic_map_range(
1182 isl_basic_map_copy(map
->p
[i
])));
1183 qc
= q_closure(isl_dim_copy(dim
), isl_set_copy(C
),
1184 map
->p
[i
], &exact_i
);
1191 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1198 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1199 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1200 qc
= isl_map_compute_divs(qc
);
1201 for (j
= 0; j
< map
->n
; ++j
)
1202 left
[j
] = right
[j
] = 1;
1203 qc
= compose(map
, i
, qc
, left
, right
);
1206 if (qc
->n
>= map
->n
) {
1210 *res
= compute_incremental(isl_dim_copy(dim
), map
, i
, qc
,
1211 left
, right
, &exact_i
);
1222 return *res
!= NULL
;
1228 /* Try and compute the transitive closure of "map" as
1230 * map^+ = map_i^+ \cup
1231 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1233 * with C either the simple hull of the domain and range of the entire
1234 * map or the simple hull of domain and range of map_i.
1236 static __isl_give isl_map
*incremental_closure(__isl_take isl_dim
*dim
,
1237 __isl_keep isl_map
*map
, int *exact
, int project
)
1240 isl_set
**dom
= NULL
;
1241 isl_set
**ran
= NULL
;
1246 isl_map
*res
= NULL
;
1249 return construct_projected_component(dim
, map
, exact
, project
);
1254 return construct_projected_component(dim
, map
, exact
, project
);
1256 d
= isl_map_dim(map
, isl_dim_in
);
1258 dom
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1259 ran
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1260 left
= isl_calloc_array(map
->ctx
, int, map
->n
);
1261 right
= isl_calloc_array(map
->ctx
, int, map
->n
);
1262 if (!ran
|| !dom
|| !left
|| !right
)
1265 if (incemental_on_entire_domain(dim
, map
, dom
, ran
, left
, right
, &res
) < 0)
1268 for (i
= 0; !res
&& i
< map
->n
; ++i
) {
1270 int exact_i
, spurious
, comp
;
1272 dom
[i
] = isl_set_from_basic_set(
1273 isl_basic_map_domain(
1274 isl_basic_map_copy(map
->p
[i
])));
1278 ran
[i
] = isl_set_from_basic_set(
1279 isl_basic_map_range(
1280 isl_basic_map_copy(map
->p
[i
])));
1283 C
= isl_set_union(isl_set_copy(dom
[i
]),
1284 isl_set_copy(ran
[i
]));
1285 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1292 comp
= composability(C
, i
, dom
, ran
, left
, right
, map
);
1293 if (!comp
|| comp
< 0) {
1299 qc
= q_closure(isl_dim_copy(dim
), C
, map
->p
[i
], &exact_i
);
1306 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1313 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1314 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1315 qc
= isl_map_compute_divs(qc
);
1316 qc
= compose(map
, i
, qc
, (comp
& LEFT
) ? left
: NULL
,
1317 (comp
& RIGHT
) ? right
: NULL
);
1320 if (qc
->n
>= map
->n
) {
1324 res
= compute_incremental(isl_dim_copy(dim
), map
, i
, qc
,
1325 (comp
& LEFT
) ? left
: NULL
,
1326 (comp
& RIGHT
) ? right
: NULL
, &exact_i
);
1335 for (i
= 0; i
< map
->n
; ++i
) {
1336 isl_set_free(dom
[i
]);
1337 isl_set_free(ran
[i
]);
1349 return construct_projected_component(dim
, map
, exact
, project
);
1352 for (i
= 0; i
< map
->n
; ++i
)
1353 isl_set_free(dom
[i
]);
1356 for (i
= 0; i
< map
->n
; ++i
)
1357 isl_set_free(ran
[i
]);
1365 /* Given an array of sets "set", add "dom" at position "pos"
1366 * and search for elements at earlier positions that overlap with "dom".
1367 * If any can be found, then merge all of them, together with "dom", into
1368 * a single set and assign the union to the first in the array,
1369 * which becomes the new group leader for all groups involved in the merge.
1370 * During the search, we only consider group leaders, i.e., those with
1371 * group[i] = i, as the other sets have already been combined
1372 * with one of the group leaders.
1374 static int merge(isl_set
**set
, int *group
, __isl_take isl_set
*dom
, int pos
)
1379 set
[pos
] = isl_set_copy(dom
);
1381 for (i
= pos
- 1; i
>= 0; --i
) {
1387 o
= isl_set_overlaps(set
[i
], dom
);
1393 set
[i
] = isl_set_union(set
[i
], set
[group
[pos
]]);
1394 set
[group
[pos
]] = NULL
;
1397 group
[group
[pos
]] = i
;
1408 /* Replace each entry in the n by n grid of maps by the cross product
1409 * with the relation { [i] -> [i + 1] }.
1411 static int add_length(__isl_keep isl_map
*map
, isl_map
***grid
, int n
)
1415 isl_basic_map
*bstep
;
1422 dim
= isl_map_get_dim(map
);
1423 nparam
= isl_dim_size(dim
, isl_dim_param
);
1424 dim
= isl_dim_drop(dim
, isl_dim_in
, 0, isl_dim_size(dim
, isl_dim_in
));
1425 dim
= isl_dim_drop(dim
, isl_dim_out
, 0, isl_dim_size(dim
, isl_dim_out
));
1426 dim
= isl_dim_add(dim
, isl_dim_in
, 1);
1427 dim
= isl_dim_add(dim
, isl_dim_out
, 1);
1428 bstep
= isl_basic_map_alloc_dim(dim
, 0, 1, 0);
1429 k
= isl_basic_map_alloc_equality(bstep
);
1431 isl_basic_map_free(bstep
);
1434 isl_seq_clr(bstep
->eq
[k
], 1 + isl_basic_map_total_dim(bstep
));
1435 isl_int_set_si(bstep
->eq
[k
][0], 1);
1436 isl_int_set_si(bstep
->eq
[k
][1 + nparam
], 1);
1437 isl_int_set_si(bstep
->eq
[k
][1 + nparam
+ 1], -1);
1438 bstep
= isl_basic_map_finalize(bstep
);
1439 step
= isl_map_from_basic_map(bstep
);
1441 for (i
= 0; i
< n
; ++i
)
1442 for (j
= 0; j
< n
; ++j
)
1443 grid
[i
][j
] = isl_map_product(grid
[i
][j
],
1444 isl_map_copy(step
));
1451 /* The core of the Floyd-Warshall algorithm.
1452 * Updates the given n x x matrix of relations in place.
1454 * The algorithm iterates over all vertices. In each step, the whole
1455 * matrix is updated to include all paths that go to the current vertex,
1456 * possibly stay there a while (including passing through earlier vertices)
1457 * and then come back. At the start of each iteration, the diagonal
1458 * element corresponding to the current vertex is replaced by its
1459 * transitive closure to account for all indirect paths that stay
1460 * in the current vertex.
1462 static void floyd_warshall_iterate(isl_map
***grid
, int n
, int *exact
)
1466 for (r
= 0; r
< n
; ++r
) {
1468 grid
[r
][r
] = isl_map_transitive_closure(grid
[r
][r
],
1469 (exact
&& *exact
) ? &r_exact
: NULL
);
1470 if (exact
&& *exact
&& !r_exact
)
1473 for (p
= 0; p
< n
; ++p
)
1474 for (q
= 0; q
< n
; ++q
) {
1476 if (p
== r
&& q
== r
)
1478 loop
= isl_map_apply_range(
1479 isl_map_copy(grid
[p
][r
]),
1480 isl_map_copy(grid
[r
][q
]));
1481 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1482 loop
= isl_map_apply_range(
1483 isl_map_copy(grid
[p
][r
]),
1484 isl_map_apply_range(
1485 isl_map_copy(grid
[r
][r
]),
1486 isl_map_copy(grid
[r
][q
])));
1487 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1488 grid
[p
][q
] = isl_map_coalesce(grid
[p
][q
]);
1493 /* Given a partition of the domains and ranges of the basic maps in "map",
1494 * apply the Floyd-Warshall algorithm with the elements in the partition
1497 * In particular, there are "n" elements in the partition and "group" is
1498 * an array of length 2 * map->n with entries in [0,n-1].
1500 * We first construct a matrix of relations based on the partition information,
1501 * apply Floyd-Warshall on this matrix of relations and then take the
1502 * union of all entries in the matrix as the final result.
1504 * If we are actually computing the power instead of the transitive closure,
1505 * i.e., when "project" is not set, then the result should have the
1506 * path lengths encoded as the difference between an extra pair of
1507 * coordinates. We therefore apply the nested transitive closures
1508 * to relations that include these lengths. In particular, we replace
1509 * the input relation by the cross product with the unit length relation
1510 * { [i] -> [i + 1] }.
1512 static __isl_give isl_map
*floyd_warshall_with_groups(__isl_take isl_dim
*dim
,
1513 __isl_keep isl_map
*map
, int *exact
, int project
, int *group
, int n
)
1516 isl_map
***grid
= NULL
;
1524 return incremental_closure(dim
, map
, exact
, project
);
1527 grid
= isl_calloc_array(map
->ctx
, isl_map
**, n
);
1530 for (i
= 0; i
< n
; ++i
) {
1531 grid
[i
] = isl_calloc_array(map
->ctx
, isl_map
*, n
);
1534 for (j
= 0; j
< n
; ++j
)
1535 grid
[i
][j
] = isl_map_empty(isl_map_get_dim(map
));
1538 for (k
= 0; k
< map
->n
; ++k
) {
1540 j
= group
[2 * k
+ 1];
1541 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
1542 isl_map_from_basic_map(
1543 isl_basic_map_copy(map
->p
[k
])));
1546 if (!project
&& add_length(map
, grid
, n
) < 0)
1549 floyd_warshall_iterate(grid
, n
, exact
);
1551 app
= isl_map_empty(isl_map_get_dim(map
));
1553 for (i
= 0; i
< n
; ++i
) {
1554 for (j
= 0; j
< n
; ++j
)
1555 app
= isl_map_union(app
, grid
[i
][j
]);
1566 for (i
= 0; i
< n
; ++i
) {
1569 for (j
= 0; j
< n
; ++j
)
1570 isl_map_free(grid
[i
][j
]);
1579 /* Partition the domains and ranges of the n basic relations in list
1580 * into disjoint cells.
1582 * To find the partition, we simply consider all of the domains
1583 * and ranges in turn and combine those that overlap.
1584 * "set" contains the partition elements and "group" indicates
1585 * to which partition element a given domain or range belongs.
1586 * The domain of basic map i corresponds to element 2 * i in these arrays,
1587 * while the domain corresponds to element 2 * i + 1.
1588 * During the construction group[k] is either equal to k,
1589 * in which case set[k] contains the union of all the domains and
1590 * ranges in the corresponding group, or is equal to some l < k,
1591 * with l another domain or range in the same group.
1593 static int *setup_groups(isl_ctx
*ctx
, __isl_keep isl_basic_map
**list
, int n
,
1594 isl_set
***set
, int *n_group
)
1600 *set
= isl_calloc_array(ctx
, isl_set
*, 2 * n
);
1601 group
= isl_alloc_array(ctx
, int, 2 * n
);
1603 if (!*set
|| !group
)
1606 for (i
= 0; i
< n
; ++i
) {
1608 dom
= isl_set_from_basic_set(isl_basic_map_domain(
1609 isl_basic_map_copy(list
[i
])));
1610 if (merge(*set
, group
, dom
, 2 * i
) < 0)
1612 dom
= isl_set_from_basic_set(isl_basic_map_range(
1613 isl_basic_map_copy(list
[i
])));
1614 if (merge(*set
, group
, dom
, 2 * i
+ 1) < 0)
1619 for (i
= 0; i
< 2 * n
; ++i
)
1620 if (group
[i
] == i
) {
1622 (*set
)[g
] = (*set
)[i
];
1627 group
[i
] = group
[group
[i
]];
1634 for (i
= 0; i
< 2 * n
; ++i
)
1635 isl_set_free((*set
)[i
]);
1643 /* Check if the domains and ranges of the basic maps in "map" can
1644 * be partitioned, and if so, apply Floyd-Warshall on the elements
1645 * of the partition. Note that we also apply this algorithm
1646 * if we want to compute the power, i.e., when "project" is not set.
1647 * However, the results are unlikely to be exact since the recursive
1648 * calls inside the Floyd-Warshall algorithm typically result in
1649 * non-linear path lengths quite quickly.
1651 static __isl_give isl_map
*floyd_warshall(__isl_take isl_dim
*dim
,
1652 __isl_keep isl_map
*map
, int *exact
, int project
)
1655 isl_set
**set
= NULL
;
1662 return incremental_closure(dim
, map
, exact
, project
);
1664 group
= setup_groups(map
->ctx
, map
->p
, map
->n
, &set
, &n
);
1668 for (i
= 0; i
< 2 * map
->n
; ++i
)
1669 isl_set_free(set
[i
]);
1673 return floyd_warshall_with_groups(dim
, map
, exact
, project
, group
, n
);
1679 /* Structure for representing the nodes in the graph being traversed
1680 * using Tarjan's algorithm.
1681 * index represents the order in which nodes are visited.
1682 * min_index is the index of the root of a (sub)component.
1683 * on_stack indicates whether the node is currently on the stack.
1685 struct basic_map_sort_node
{
1690 /* Structure for representing the graph being traversed
1691 * using Tarjan's algorithm.
1692 * len is the number of nodes
1693 * node is an array of nodes
1694 * stack contains the nodes on the path from the root to the current node
1695 * sp is the stack pointer
1696 * index is the index of the last node visited
1697 * order contains the elements of the components separated by -1
1698 * op represents the current position in order
1700 * check_closed is set if we may have used the fact that
1701 * a pair of basic maps can be interchanged
1703 struct basic_map_sort
{
1705 struct basic_map_sort_node
*node
;
1714 static void basic_map_sort_free(struct basic_map_sort
*s
)
1724 static struct basic_map_sort
*basic_map_sort_alloc(struct isl_ctx
*ctx
, int len
)
1726 struct basic_map_sort
*s
;
1729 s
= isl_calloc_type(ctx
, struct basic_map_sort
);
1733 s
->node
= isl_alloc_array(ctx
, struct basic_map_sort_node
, len
);
1736 for (i
= 0; i
< len
; ++i
)
1737 s
->node
[i
].index
= -1;
1738 s
->stack
= isl_alloc_array(ctx
, int, len
);
1741 s
->order
= isl_alloc_array(ctx
, int, 2 * len
);
1749 s
->check_closed
= 0;
1753 basic_map_sort_free(s
);
1757 /* Check whether in the computation of the transitive closure
1758 * "bmap1" (R_1) should follow (or be part of the same component as)
1761 * That is check whether
1769 * If so, then there is no reason for R_1 to immediately follow R_2
1772 * *check_closed is set if the subset relation holds while
1773 * R_1 \circ R_2 is not empty.
1775 static int basic_map_follows(__isl_keep isl_basic_map
*bmap1
,
1776 __isl_keep isl_basic_map
*bmap2
, int *check_closed
)
1778 struct isl_map
*map12
= NULL
;
1779 struct isl_map
*map21
= NULL
;
1782 if (!isl_dim_tuple_match(bmap1
->dim
, isl_dim_in
, bmap2
->dim
, isl_dim_out
))
1785 map21
= isl_map_from_basic_map(
1786 isl_basic_map_apply_range(
1787 isl_basic_map_copy(bmap2
),
1788 isl_basic_map_copy(bmap1
)));
1789 subset
= isl_map_is_empty(map21
);
1793 isl_map_free(map21
);
1797 if (!isl_dim_tuple_match(bmap1
->dim
, isl_dim_in
, bmap1
->dim
, isl_dim_out
) ||
1798 !isl_dim_tuple_match(bmap2
->dim
, isl_dim_in
, bmap2
->dim
, isl_dim_out
)) {
1799 isl_map_free(map21
);
1803 map12
= isl_map_from_basic_map(
1804 isl_basic_map_apply_range(
1805 isl_basic_map_copy(bmap1
),
1806 isl_basic_map_copy(bmap2
)));
1808 subset
= isl_map_is_subset(map21
, map12
);
1810 isl_map_free(map12
);
1811 isl_map_free(map21
);
1816 return subset
< 0 ? -1 : !subset
;
1818 isl_map_free(map21
);
1822 /* Perform Tarjan's algorithm for computing the strongly connected components
1823 * in the graph with the disjuncts of "map" as vertices and with an
1824 * edge between any pair of disjuncts such that the first has
1825 * to be applied after the second.
1827 static int power_components_tarjan(struct basic_map_sort
*s
,
1828 __isl_keep isl_basic_map
**list
, int i
)
1832 s
->node
[i
].index
= s
->index
;
1833 s
->node
[i
].min_index
= s
->index
;
1834 s
->node
[i
].on_stack
= 1;
1836 s
->stack
[s
->sp
++] = i
;
1838 for (j
= s
->len
- 1; j
>= 0; --j
) {
1843 if (s
->node
[j
].index
>= 0 &&
1844 (!s
->node
[j
].on_stack
||
1845 s
->node
[j
].index
> s
->node
[i
].min_index
))
1848 f
= basic_map_follows(list
[i
], list
[j
], &s
->check_closed
);
1854 if (s
->node
[j
].index
< 0) {
1855 power_components_tarjan(s
, list
, j
);
1856 if (s
->node
[j
].min_index
< s
->node
[i
].min_index
)
1857 s
->node
[i
].min_index
= s
->node
[j
].min_index
;
1858 } else if (s
->node
[j
].index
< s
->node
[i
].min_index
)
1859 s
->node
[i
].min_index
= s
->node
[j
].index
;
1862 if (s
->node
[i
].index
!= s
->node
[i
].min_index
)
1866 j
= s
->stack
[--s
->sp
];
1867 s
->node
[j
].on_stack
= 0;
1868 s
->order
[s
->op
++] = j
;
1870 s
->order
[s
->op
++] = -1;
1875 /* Decompose the "len" basic relations in "list" into strongly connected
1878 static struct basic_map_sort
*basic_map_sort_init(isl_ctx
*ctx
, int len
,
1879 __isl_keep isl_basic_map
**list
)
1882 struct basic_map_sort
*s
= NULL
;
1884 s
= basic_map_sort_alloc(ctx
, len
);
1887 for (i
= len
- 1; i
>= 0; --i
) {
1888 if (s
->node
[i
].index
>= 0)
1890 if (power_components_tarjan(s
, list
, i
) < 0)
1896 basic_map_sort_free(s
);
1900 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
1901 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
1902 * construct a map that is an overapproximation of the map
1903 * that takes an element from the dom R \times Z to an
1904 * element from ran R \times Z, such that the first n coordinates of the
1905 * difference between them is a sum of differences between images
1906 * and pre-images in one of the R_i and such that the last coordinate
1907 * is equal to the number of steps taken.
1908 * If "project" is set, then these final coordinates are not included,
1909 * i.e., a relation of type Z^n -> Z^n is returned.
1912 * \Delta_i = { y - x | (x, y) in R_i }
1914 * then the constructed map is an overapproximation of
1916 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1917 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
1918 * x in dom R and x + d in ran R }
1922 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1923 * d = (\sum_i k_i \delta_i) and
1924 * x in dom R and x + d in ran R }
1926 * if "project" is set.
1928 * We first split the map into strongly connected components, perform
1929 * the above on each component and then join the results in the correct
1930 * order, at each join also taking in the union of both arguments
1931 * to allow for paths that do not go through one of the two arguments.
1933 static __isl_give isl_map
*construct_power_components(__isl_take isl_dim
*dim
,
1934 __isl_keep isl_map
*map
, int *exact
, int project
)
1937 struct isl_map
*path
= NULL
;
1938 struct basic_map_sort
*s
= NULL
;
1945 return floyd_warshall(dim
, map
, exact
, project
);
1947 s
= basic_map_sort_init(map
->ctx
, map
->n
, map
->p
);
1952 if (s
->check_closed
&& !exact
)
1953 exact
= &local_exact
;
1959 path
= isl_map_empty(isl_map_get_dim(map
));
1961 path
= isl_map_empty(isl_dim_copy(dim
));
1962 path
= anonymize(path
);
1964 struct isl_map
*comp
;
1965 isl_map
*path_comp
, *path_comb
;
1966 comp
= isl_map_alloc_dim(isl_map_get_dim(map
), n
, 0);
1967 while (s
->order
[i
] != -1) {
1968 comp
= isl_map_add_basic_map(comp
,
1969 isl_basic_map_copy(map
->p
[s
->order
[i
]]));
1973 path_comp
= floyd_warshall(isl_dim_copy(dim
),
1974 comp
, exact
, project
);
1975 path_comb
= isl_map_apply_range(isl_map_copy(path
),
1976 isl_map_copy(path_comp
));
1977 path
= isl_map_union(path
, path_comp
);
1978 path
= isl_map_union(path
, path_comb
);
1984 if (c
> 1 && s
->check_closed
&& !*exact
) {
1987 closed
= isl_map_is_transitively_closed(path
);
1991 basic_map_sort_free(s
);
1993 return floyd_warshall(dim
, map
, orig_exact
, project
);
1997 basic_map_sort_free(s
);
2002 basic_map_sort_free(s
);
2008 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
2009 * construct a map that is an overapproximation of the map
2010 * that takes an element from the space D to another
2011 * element from the same space, such that the difference between
2012 * them is a strictly positive sum of differences between images
2013 * and pre-images in one of the R_i.
2014 * The number of differences in the sum is equated to parameter "param".
2017 * \Delta_i = { y - x | (x, y) in R_i }
2019 * then the constructed map is an overapproximation of
2021 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
2022 * d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
2025 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
2026 * d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
2028 * if "project" is set.
2030 * If "project" is not set, then
2031 * we construct an extended mapping with an extra coordinate
2032 * that indicates the number of steps taken. In particular,
2033 * the difference in the last coordinate is equal to the number
2034 * of steps taken to move from a domain element to the corresponding
2037 static __isl_give isl_map
*construct_power(__isl_keep isl_map
*map
,
2038 int *exact
, int project
)
2040 struct isl_map
*app
= NULL
;
2041 struct isl_dim
*dim
= NULL
;
2047 dim
= isl_map_get_dim(map
);
2049 d
= isl_dim_size(dim
, isl_dim_in
);
2050 dim
= isl_dim_add(dim
, isl_dim_in
, 1);
2051 dim
= isl_dim_add(dim
, isl_dim_out
, 1);
2053 app
= construct_power_components(isl_dim_copy(dim
), map
,
2061 /* Compute the positive powers of "map", or an overapproximation.
2062 * If the result is exact, then *exact is set to 1.
2064 * If project is set, then we are actually interested in the transitive
2065 * closure, so we can use a more relaxed exactness check.
2066 * The lengths of the paths are also projected out instead of being
2067 * encoded as the difference between an extra pair of final coordinates.
2069 static __isl_give isl_map
*map_power(__isl_take isl_map
*map
,
2070 int *exact
, int project
)
2072 struct isl_map
*app
= NULL
;
2080 isl_assert(map
->ctx
,
2081 isl_map_dim(map
, isl_dim_in
) == isl_map_dim(map
, isl_dim_out
),
2084 app
= construct_power(map
, exact
, project
);
2094 /* Compute the positive powers of "map", or an overapproximation.
2095 * The power is given by parameter "param". If the result is exact,
2096 * then *exact is set to 1.
2097 * map_power constructs an extended relation with the path lengths
2098 * encoded as the difference between the final coordinates.
2099 * In the final step, this difference is equated to the parameter "param"
2100 * and made positive. The extra coordinates are subsequently projected out.
2102 __isl_give isl_map
*isl_map_power(__isl_take isl_map
*map
, unsigned param
,
2105 isl_dim
*target_dim
;
2113 isl_assert(map
->ctx
, param
< isl_map_dim(map
, isl_dim_param
),
2116 d
= isl_map_dim(map
, isl_dim_in
);
2118 map
= isl_map_compute_divs(map
);
2119 map
= isl_map_coalesce(map
);
2121 if (isl_map_fast_is_empty(map
))
2124 target_dim
= isl_map_get_dim(map
);
2125 map
= map_power(map
, exact
, 0);
2127 dim
= isl_map_get_dim(map
);
2128 diff
= equate_parameter_to_length(dim
, param
);
2129 map
= isl_map_intersect(map
, diff
);
2130 map
= isl_map_project_out(map
, isl_dim_in
, d
, 1);
2131 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2133 map
= isl_map_reset_dim(map
, target_dim
);
2141 /* Compute a relation that maps each element in the range of the input
2142 * relation to the lengths of all paths composed of edges in the input
2143 * relation that end up in the given range element.
2144 * The result may be an overapproximation, in which case *exact is set to 0.
2145 * The resulting relation is very similar to the power relation.
2146 * The difference are that the domain has been projected out, the
2147 * range has become the domain and the exponent is the range instead
2150 __isl_give isl_map
*isl_map_reaching_path_lengths(__isl_take isl_map
*map
,
2161 d
= isl_map_dim(map
, isl_dim_in
);
2162 param
= isl_map_dim(map
, isl_dim_param
);
2164 map
= isl_map_compute_divs(map
);
2165 map
= isl_map_coalesce(map
);
2167 if (isl_map_fast_is_empty(map
)) {
2170 map
= isl_map_project_out(map
, isl_dim_out
, 0, d
);
2171 map
= isl_map_add(map
, isl_dim_out
, 1);
2175 map
= map_power(map
, exact
, 0);
2177 map
= isl_map_add(map
, isl_dim_param
, 1);
2178 dim
= isl_map_get_dim(map
);
2179 diff
= equate_parameter_to_length(dim
, param
);
2180 map
= isl_map_intersect(map
, diff
);
2181 map
= isl_map_project_out(map
, isl_dim_in
, 0, d
+ 1);
2182 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2183 map
= isl_map_reverse(map
);
2184 map
= isl_map_move_dims(map
, isl_dim_out
, 0, isl_dim_param
, param
, 1);
2189 /* Check whether equality i of bset is a pure stride constraint
2190 * on a single dimensions, i.e., of the form
2194 * with k a constant and e an existentially quantified variable.
2196 static int is_eq_stride(__isl_keep isl_basic_set
*bset
, int i
)
2208 if (!isl_int_is_zero(bset
->eq
[i
][0]))
2211 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
2212 d
= isl_basic_set_dim(bset
, isl_dim_set
);
2213 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
2215 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1, nparam
) != -1)
2217 pos1
= isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
, d
);
2220 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ pos1
+ 1,
2221 d
- pos1
- 1) != -1)
2224 pos2
= isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ d
, n_div
);
2227 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ d
+ pos2
+ 1,
2228 n_div
- pos2
- 1) != -1)
2230 if (!isl_int_is_one(bset
->eq
[i
][1 + nparam
+ pos1
]) &&
2231 !isl_int_is_negone(bset
->eq
[i
][1 + nparam
+ pos1
]))
2237 /* Given a map, compute the smallest superset of this map that is of the form
2239 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2241 * (where p ranges over the (non-parametric) dimensions),
2242 * compute the transitive closure of this map, i.e.,
2244 * { i -> j : exists k > 0:
2245 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2247 * and intersect domain and range of this transitive closure with
2248 * the given domain and range.
2250 * If with_id is set, then try to include as much of the identity mapping
2251 * as possible, by computing
2253 * { i -> j : exists k >= 0:
2254 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2256 * instead (i.e., allow k = 0).
2258 * In practice, we compute the difference set
2260 * delta = { j - i | i -> j in map },
2262 * look for stride constraint on the individual dimensions and compute
2263 * (constant) lower and upper bounds for each individual dimension,
2264 * adding a constraint for each bound not equal to infinity.
2266 static __isl_give isl_map
*box_closure_on_domain(__isl_take isl_map
*map
,
2267 __isl_take isl_set
*dom
, __isl_take isl_set
*ran
, int with_id
)
2276 isl_map
*app
= NULL
;
2277 isl_basic_set
*aff
= NULL
;
2278 isl_basic_map
*bmap
= NULL
;
2279 isl_vec
*obj
= NULL
;
2284 delta
= isl_map_deltas(isl_map_copy(map
));
2286 aff
= isl_set_affine_hull(isl_set_copy(delta
));
2289 dim
= isl_map_get_dim(map
);
2290 d
= isl_dim_size(dim
, isl_dim_in
);
2291 nparam
= isl_dim_size(dim
, isl_dim_param
);
2292 total
= isl_dim_total(dim
);
2293 bmap
= isl_basic_map_alloc_dim(dim
,
2294 aff
->n_div
+ 1, aff
->n_div
, 2 * d
+ 1);
2295 for (i
= 0; i
< aff
->n_div
+ 1; ++i
) {
2296 k
= isl_basic_map_alloc_div(bmap
);
2299 isl_int_set_si(bmap
->div
[k
][0], 0);
2301 for (i
= 0; i
< aff
->n_eq
; ++i
) {
2302 if (!is_eq_stride(aff
, i
))
2304 k
= isl_basic_map_alloc_equality(bmap
);
2307 isl_seq_clr(bmap
->eq
[k
], 1 + nparam
);
2308 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ d
,
2309 aff
->eq
[i
] + 1 + nparam
, d
);
2310 isl_seq_neg(bmap
->eq
[k
] + 1 + nparam
,
2311 aff
->eq
[i
] + 1 + nparam
, d
);
2312 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ 2 * d
,
2313 aff
->eq
[i
] + 1 + nparam
+ d
, aff
->n_div
);
2314 isl_int_set_si(bmap
->eq
[k
][1 + total
+ aff
->n_div
], 0);
2316 obj
= isl_vec_alloc(map
->ctx
, 1 + nparam
+ d
);
2319 isl_seq_clr(obj
->el
, 1 + nparam
+ d
);
2320 for (i
= 0; i
< d
; ++ i
) {
2321 enum isl_lp_result res
;
2323 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 1);
2325 res
= isl_set_solve_lp(delta
, 0, obj
->el
, map
->ctx
->one
, &opt
,
2327 if (res
== isl_lp_error
)
2329 if (res
== isl_lp_ok
) {
2330 k
= isl_basic_map_alloc_inequality(bmap
);
2333 isl_seq_clr(bmap
->ineq
[k
],
2334 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2335 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], -1);
2336 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], 1);
2337 isl_int_neg(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2340 res
= isl_set_solve_lp(delta
, 1, obj
->el
, map
->ctx
->one
, &opt
,
2342 if (res
== isl_lp_error
)
2344 if (res
== isl_lp_ok
) {
2345 k
= isl_basic_map_alloc_inequality(bmap
);
2348 isl_seq_clr(bmap
->ineq
[k
],
2349 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2350 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], 1);
2351 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], -1);
2352 isl_int_set(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2355 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 0);
2357 k
= isl_basic_map_alloc_inequality(bmap
);
2360 isl_seq_clr(bmap
->ineq
[k
],
2361 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2363 isl_int_set_si(bmap
->ineq
[k
][0], -1);
2364 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], 1);
2366 app
= isl_map_from_domain_and_range(dom
, ran
);
2369 isl_basic_set_free(aff
);
2371 bmap
= isl_basic_map_finalize(bmap
);
2372 isl_set_free(delta
);
2375 map
= isl_map_from_basic_map(bmap
);
2376 map
= isl_map_intersect(map
, app
);
2381 isl_basic_map_free(bmap
);
2382 isl_basic_set_free(aff
);
2386 isl_set_free(delta
);
2391 /* Given a map, compute the smallest superset of this map that is of the form
2393 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2395 * (where p ranges over the (non-parametric) dimensions),
2396 * compute the transitive closure of this map, i.e.,
2398 * { i -> j : exists k > 0:
2399 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2401 * and intersect domain and range of this transitive closure with
2402 * domain and range of the original map.
2404 static __isl_give isl_map
*box_closure(__isl_take isl_map
*map
)
2409 domain
= isl_map_domain(isl_map_copy(map
));
2410 domain
= isl_set_coalesce(domain
);
2411 range
= isl_map_range(isl_map_copy(map
));
2412 range
= isl_set_coalesce(range
);
2414 return box_closure_on_domain(map
, domain
, range
, 0);
2417 /* Given a map, compute the smallest superset of this map that is of the form
2419 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2421 * (where p ranges over the (non-parametric) dimensions),
2422 * compute the transitive and partially reflexive closure of this map, i.e.,
2424 * { i -> j : exists k >= 0:
2425 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2427 * and intersect domain and range of this transitive closure with
2430 static __isl_give isl_map
*box_closure_with_identity(__isl_take isl_map
*map
,
2431 __isl_take isl_set
*dom
)
2433 return box_closure_on_domain(map
, dom
, isl_set_copy(dom
), 1);
2436 /* Check whether app is the transitive closure of map.
2437 * In particular, check that app is acyclic and, if so,
2440 * app \subset (map \cup (map \circ app))
2442 static int check_exactness_omega(__isl_keep isl_map
*map
,
2443 __isl_keep isl_map
*app
)
2447 int is_empty
, is_exact
;
2451 delta
= isl_map_deltas(isl_map_copy(app
));
2452 d
= isl_set_dim(delta
, isl_dim_set
);
2453 for (i
= 0; i
< d
; ++i
)
2454 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
2455 is_empty
= isl_set_is_empty(delta
);
2456 isl_set_free(delta
);
2462 test
= isl_map_apply_range(isl_map_copy(app
), isl_map_copy(map
));
2463 test
= isl_map_union(test
, isl_map_copy(map
));
2464 is_exact
= isl_map_is_subset(app
, test
);
2470 /* Check if basic map M_i can be combined with all the other
2471 * basic maps such that
2475 * can be computed as
2477 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2479 * In particular, check if we can compute a compact representation
2482 * M_i^* \circ M_j \circ M_i^*
2485 * Let M_i^? be an extension of M_i^+ that allows paths
2486 * of length zero, i.e., the result of box_closure(., 1).
2487 * The criterion, as proposed by Kelly et al., is that
2488 * id = M_i^? - M_i^+ can be represented as a basic map
2491 * id \circ M_j \circ id = M_j
2495 * If this function returns 1, then tc and qc are set to
2496 * M_i^+ and M_i^?, respectively.
2498 static int can_be_split_off(__isl_keep isl_map
*map
, int i
,
2499 __isl_give isl_map
**tc
, __isl_give isl_map
**qc
)
2501 isl_map
*map_i
, *id
= NULL
;
2508 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
2509 isl_map_range(isl_map_copy(map
)));
2510 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
2514 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
2515 *tc
= box_closure(isl_map_copy(map_i
));
2516 *qc
= box_closure_with_identity(map_i
, C
);
2517 id
= isl_map_subtract(isl_map_copy(*qc
), isl_map_copy(*tc
));
2521 if (id
->n
!= 1 || (*qc
)->n
!= 1)
2524 for (j
= 0; j
< map
->n
; ++j
) {
2525 isl_map
*map_j
, *test
;
2530 map_j
= isl_map_from_basic_map(
2531 isl_basic_map_copy(map
->p
[j
]));
2532 test
= isl_map_apply_range(isl_map_copy(id
),
2533 isl_map_copy(map_j
));
2534 test
= isl_map_apply_range(test
, isl_map_copy(id
));
2535 is_ok
= isl_map_is_equal(test
, map_j
);
2536 isl_map_free(map_j
);
2564 static __isl_give isl_map
*box_closure_with_check(__isl_take isl_map
*map
,
2569 app
= box_closure(isl_map_copy(map
));
2571 *exact
= check_exactness_omega(map
, app
);
2577 /* Compute an overapproximation of the transitive closure of "map"
2578 * using a variation of the algorithm from
2579 * "Transitive Closure of Infinite Graphs and its Applications"
2582 * We first check whether we can can split of any basic map M_i and
2589 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2591 * using a recursive call on the remaining map.
2593 * If not, we simply call box_closure on the whole map.
2595 static __isl_give isl_map
*transitive_closure_omega(__isl_take isl_map
*map
,
2605 return box_closure_with_check(map
, exact
);
2607 for (i
= 0; i
< map
->n
; ++i
) {
2610 ok
= can_be_split_off(map
, i
, &tc
, &qc
);
2616 app
= isl_map_alloc_dim(isl_map_get_dim(map
), map
->n
- 1, 0);
2618 for (j
= 0; j
< map
->n
; ++j
) {
2621 app
= isl_map_add_basic_map(app
,
2622 isl_basic_map_copy(map
->p
[j
]));
2625 app
= isl_map_apply_range(isl_map_copy(qc
), app
);
2626 app
= isl_map_apply_range(app
, qc
);
2628 app
= isl_map_union(tc
, transitive_closure_omega(app
, NULL
));
2629 exact_i
= check_exactness_omega(map
, app
);
2641 return box_closure_with_check(map
, exact
);
2647 /* Compute the transitive closure of "map", or an overapproximation.
2648 * If the result is exact, then *exact is set to 1.
2649 * Simply use map_power to compute the powers of map, but tell
2650 * it to project out the lengths of the paths instead of equating
2651 * the length to a parameter.
2653 __isl_give isl_map
*isl_map_transitive_closure(__isl_take isl_map
*map
,
2656 isl_dim
*target_dim
;
2662 if (map
->ctx
->opt
->closure
== ISL_CLOSURE_OMEGA
)
2663 return transitive_closure_omega(map
, exact
);
2665 map
= isl_map_compute_divs(map
);
2666 map
= isl_map_coalesce(map
);
2667 closed
= isl_map_is_transitively_closed(map
);
2676 target_dim
= isl_map_get_dim(map
);
2677 map
= map_power(map
, exact
, 1);
2678 map
= isl_map_reset_dim(map
, target_dim
);
2686 static int inc_count(__isl_take isl_map
*map
, void *user
)
2697 static int collect_basic_map(__isl_take isl_map
*map
, void *user
)
2700 isl_basic_map
***next
= user
;
2702 for (i
= 0; i
< map
->n
; ++i
) {
2703 **next
= isl_basic_map_copy(map
->p
[i
]);
2716 /* Perform Floyd-Warshall on the given list of basic relations.
2717 * The basic relations may live in different dimensions,
2718 * but basic relations that get assigned to the diagonal of the
2719 * grid have domains and ranges of the same dimension and so
2720 * the standard algorithm can be used because the nested transitive
2721 * closures are only applied to diagonal elements and because all
2722 * compositions are peformed on relations with compatible domains and ranges.
2724 static __isl_give isl_union_map
*union_floyd_warshall_on_list(isl_ctx
*ctx
,
2725 __isl_keep isl_basic_map
**list
, int n
, int *exact
)
2730 isl_set
**set
= NULL
;
2731 isl_map
***grid
= NULL
;
2734 group
= setup_groups(ctx
, list
, n
, &set
, &n_group
);
2738 grid
= isl_calloc_array(ctx
, isl_map
**, n_group
);
2741 for (i
= 0; i
< n_group
; ++i
) {
2742 grid
[i
] = isl_calloc_array(map
->ctx
, isl_map
*, n_group
);
2745 for (j
= 0; j
< n_group
; ++j
) {
2746 isl_dim
*dim1
, *dim2
, *dim
;
2747 dim1
= isl_dim_reverse(isl_set_get_dim(set
[i
]));
2748 dim2
= isl_set_get_dim(set
[j
]);
2749 dim
= isl_dim_join(dim1
, dim2
);
2750 grid
[i
][j
] = isl_map_empty(dim
);
2754 for (k
= 0; k
< n
; ++k
) {
2756 j
= group
[2 * k
+ 1];
2757 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
2758 isl_map_from_basic_map(
2759 isl_basic_map_copy(list
[k
])));
2762 floyd_warshall_iterate(grid
, n_group
, exact
);
2764 app
= isl_union_map_empty(isl_map_get_dim(grid
[0][0]));
2766 for (i
= 0; i
< n_group
; ++i
) {
2767 for (j
= 0; j
< n_group
; ++j
)
2768 app
= isl_union_map_add_map(app
, grid
[i
][j
]);
2773 for (i
= 0; i
< 2 * n
; ++i
)
2774 isl_set_free(set
[i
]);
2781 for (i
= 0; i
< n_group
; ++i
) {
2784 for (j
= 0; j
< n_group
; ++j
)
2785 isl_map_free(grid
[i
][j
]);
2790 for (i
= 0; i
< 2 * n
; ++i
)
2791 isl_set_free(set
[i
]);
2798 /* Perform Floyd-Warshall on the given union relation.
2799 * The implementation is very similar to that for non-unions.
2800 * The main difference is that it is applied unconditionally.
2801 * We first extract a list of basic maps from the union map
2802 * and then perform the algorithm on this list.
2804 static __isl_give isl_union_map
*union_floyd_warshall(
2805 __isl_take isl_union_map
*umap
, int *exact
)
2809 isl_basic_map
**list
;
2810 isl_basic_map
**next
;
2814 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2817 ctx
= isl_union_map_get_ctx(umap
);
2818 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2823 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2826 res
= union_floyd_warshall_on_list(ctx
, list
, n
, exact
);
2829 for (i
= 0; i
< n
; ++i
)
2830 isl_basic_map_free(list
[i
]);
2834 isl_union_map_free(umap
);
2838 for (i
= 0; i
< n
; ++i
)
2839 isl_basic_map_free(list
[i
]);
2842 isl_union_map_free(umap
);
2846 /* Decompose the give union relation into strongly connected components.
2847 * The implementation is essentially the same as that of
2848 * construct_power_components with the major difference that all
2849 * operations are performed on union maps.
2851 static __isl_give isl_union_map
*union_components(
2852 __isl_take isl_union_map
*umap
, int *exact
)
2857 isl_basic_map
**list
;
2858 isl_basic_map
**next
;
2859 isl_union_map
*path
= NULL
;
2860 struct basic_map_sort
*s
= NULL
;
2865 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2869 return union_floyd_warshall(umap
, exact
);
2871 ctx
= isl_union_map_get_ctx(umap
);
2872 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2877 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2880 s
= basic_map_sort_init(ctx
, n
, list
);
2887 path
= isl_union_map_empty(isl_union_map_get_dim(umap
));
2889 isl_union_map
*comp
;
2890 isl_union_map
*path_comp
, *path_comb
;
2891 comp
= isl_union_map_empty(isl_union_map_get_dim(umap
));
2892 while (s
->order
[i
] != -1) {
2893 comp
= isl_union_map_add_map(comp
,
2894 isl_map_from_basic_map(
2895 isl_basic_map_copy(list
[s
->order
[i
]])));
2899 path_comp
= union_floyd_warshall(comp
, exact
);
2900 path_comb
= isl_union_map_apply_range(isl_union_map_copy(path
),
2901 isl_union_map_copy(path_comp
));
2902 path
= isl_union_map_union(path
, path_comp
);
2903 path
= isl_union_map_union(path
, path_comb
);
2908 if (c
> 1 && s
->check_closed
&& !*exact
) {
2911 closed
= isl_union_map_is_transitively_closed(path
);
2917 basic_map_sort_free(s
);
2919 for (i
= 0; i
< n
; ++i
)
2920 isl_basic_map_free(list
[i
]);
2924 isl_union_map_free(path
);
2925 return union_floyd_warshall(umap
, exact
);
2928 isl_union_map_free(umap
);
2932 basic_map_sort_free(s
);
2934 for (i
= 0; i
< n
; ++i
)
2935 isl_basic_map_free(list
[i
]);
2938 isl_union_map_free(umap
);
2939 isl_union_map_free(path
);
2943 /* Compute the transitive closure of "umap", or an overapproximation.
2944 * If the result is exact, then *exact is set to 1.
2946 __isl_give isl_union_map
*isl_union_map_transitive_closure(
2947 __isl_take isl_union_map
*umap
, int *exact
)
2957 umap
= isl_union_map_compute_divs(umap
);
2958 umap
= isl_union_map_coalesce(umap
);
2959 closed
= isl_union_map_is_transitively_closed(umap
);
2964 umap
= union_components(umap
, exact
);
2967 isl_union_map_free(umap
);