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>
17 #include <isl_mat_private.h>
19 int isl_map_is_transitively_closed(__isl_keep isl_map
*map
)
24 map2
= isl_map_apply_range(isl_map_copy(map
), isl_map_copy(map
));
25 closed
= isl_map_is_subset(map2
, map
);
31 int isl_union_map_is_transitively_closed(__isl_keep isl_union_map
*umap
)
36 umap2
= isl_union_map_apply_range(isl_union_map_copy(umap
),
37 isl_union_map_copy(umap
));
38 closed
= isl_union_map_is_subset(umap2
, umap
);
39 isl_union_map_free(umap2
);
44 /* Given a map that represents a path with the length of the path
45 * encoded as the difference between the last output coordindate
46 * and the last input coordinate, set this length to either
47 * exactly "length" (if "exactly" is set) or at least "length"
48 * (if "exactly" is not set).
50 static __isl_give isl_map
*set_path_length(__isl_take isl_map
*map
,
51 int exactly
, int length
)
54 struct isl_basic_map
*bmap
;
63 dim
= isl_map_get_dim(map
);
64 d
= isl_dim_size(dim
, isl_dim_in
);
65 nparam
= isl_dim_size(dim
, isl_dim_param
);
66 bmap
= isl_basic_map_alloc_dim(dim
, 0, 1, 1);
68 k
= isl_basic_map_alloc_equality(bmap
);
71 k
= isl_basic_map_alloc_inequality(bmap
);
76 isl_seq_clr(c
, 1 + isl_basic_map_total_dim(bmap
));
77 isl_int_set_si(c
[0], -length
);
78 isl_int_set_si(c
[1 + nparam
+ d
- 1], -1);
79 isl_int_set_si(c
[1 + nparam
+ d
+ d
- 1], 1);
81 bmap
= isl_basic_map_finalize(bmap
);
82 map
= isl_map_intersect(map
, isl_map_from_basic_map(bmap
));
86 isl_basic_map_free(bmap
);
91 /* Check whether the overapproximation of the power of "map" is exactly
92 * the power of "map". Let R be "map" and A_k the overapproximation.
93 * The approximation is exact if
96 * A_k = A_{k-1} \circ R k >= 2
98 * Since A_k is known to be an overapproximation, we only need to check
101 * A_k \subset A_{k-1} \circ R k >= 2
103 * In practice, "app" has an extra input and output coordinate
104 * to encode the length of the path. So, we first need to add
105 * this coordinate to "map" and set the length of the path to
108 static int check_power_exactness(__isl_take isl_map
*map
,
109 __isl_take isl_map
*app
)
115 map
= isl_map_add_dims(map
, isl_dim_in
, 1);
116 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
117 map
= set_path_length(map
, 1, 1);
119 app_1
= set_path_length(isl_map_copy(app
), 1, 1);
121 exact
= isl_map_is_subset(app_1
, map
);
124 if (!exact
|| exact
< 0) {
130 app_1
= set_path_length(isl_map_copy(app
), 0, 1);
131 app_2
= set_path_length(app
, 0, 2);
132 app_1
= isl_map_apply_range(map
, app_1
);
134 exact
= isl_map_is_subset(app_2
, app_1
);
142 /* Check whether the overapproximation of the power of "map" is exactly
143 * the power of "map", possibly after projecting out the power (if "project"
146 * If "project" is set and if "steps" can only result in acyclic paths,
149 * A = R \cup (A \circ R)
151 * where A is the overapproximation with the power projected out, i.e.,
152 * an overapproximation of the transitive closure.
153 * More specifically, since A is known to be an overapproximation, we check
155 * A \subset R \cup (A \circ R)
157 * Otherwise, we check if the power is exact.
159 * Note that "app" has an extra input and output coordinate to encode
160 * the length of the part. If we are only interested in the transitive
161 * closure, then we can simply project out these coordinates first.
163 static int check_exactness(__isl_take isl_map
*map
, __isl_take isl_map
*app
,
171 return check_power_exactness(map
, app
);
173 d
= isl_map_dim(map
, isl_dim_in
);
174 app
= set_path_length(app
, 0, 1);
175 app
= isl_map_project_out(app
, isl_dim_in
, d
, 1);
176 app
= isl_map_project_out(app
, isl_dim_out
, d
, 1);
178 app
= isl_map_reset_dim(app
, isl_map_get_dim(map
));
180 test
= isl_map_apply_range(isl_map_copy(map
), isl_map_copy(app
));
181 test
= isl_map_union(test
, isl_map_copy(map
));
183 exact
= isl_map_is_subset(app
, test
);
194 * The transitive closure implementation is based on the paper
195 * "Computing the Transitive Closure of a Union of Affine Integer
196 * Tuple Relations" by Anna Beletska, Denis Barthou, Wlodzimierz Bielecki and
200 /* Given a set of n offsets v_i (the rows of "steps"), construct a relation
201 * of the given dimension specification (Z^{n+1} -> Z^{n+1})
202 * that maps an element x to any element that can be reached
203 * by taking a non-negative number of steps along any of
204 * the extended offsets v'_i = [v_i 1].
207 * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v'_i }
209 * For any element in this relation, the number of steps taken
210 * is equal to the difference in the final coordinates.
212 static __isl_give isl_map
*path_along_steps(__isl_take isl_dim
*dim
,
213 __isl_keep isl_mat
*steps
)
216 struct isl_basic_map
*path
= NULL
;
224 d
= isl_dim_size(dim
, isl_dim_in
);
226 nparam
= isl_dim_size(dim
, isl_dim_param
);
228 path
= isl_basic_map_alloc_dim(isl_dim_copy(dim
), n
, d
, n
);
230 for (i
= 0; i
< n
; ++i
) {
231 k
= isl_basic_map_alloc_div(path
);
234 isl_assert(steps
->ctx
, i
== k
, goto error
);
235 isl_int_set_si(path
->div
[k
][0], 0);
238 for (i
= 0; i
< d
; ++i
) {
239 k
= isl_basic_map_alloc_equality(path
);
242 isl_seq_clr(path
->eq
[k
], 1 + isl_basic_map_total_dim(path
));
243 isl_int_set_si(path
->eq
[k
][1 + nparam
+ i
], 1);
244 isl_int_set_si(path
->eq
[k
][1 + nparam
+ d
+ i
], -1);
246 for (j
= 0; j
< n
; ++j
)
247 isl_int_set_si(path
->eq
[k
][1 + nparam
+ 2 * d
+ j
], 1);
249 for (j
= 0; j
< n
; ++j
)
250 isl_int_set(path
->eq
[k
][1 + nparam
+ 2 * d
+ j
],
254 for (i
= 0; i
< n
; ++i
) {
255 k
= isl_basic_map_alloc_inequality(path
);
258 isl_seq_clr(path
->ineq
[k
], 1 + isl_basic_map_total_dim(path
));
259 isl_int_set_si(path
->ineq
[k
][1 + nparam
+ 2 * d
+ i
], 1);
264 path
= isl_basic_map_simplify(path
);
265 path
= isl_basic_map_finalize(path
);
266 return isl_map_from_basic_map(path
);
269 isl_basic_map_free(path
);
278 /* Check whether the parametric constant term of constraint c is never
279 * positive in "bset".
281 static int parametric_constant_never_positive(__isl_keep isl_basic_set
*bset
,
282 isl_int
*c
, int *div_purity
)
291 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
292 d
= isl_basic_set_dim(bset
, isl_dim_set
);
293 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
295 bset
= isl_basic_set_copy(bset
);
296 bset
= isl_basic_set_cow(bset
);
297 bset
= isl_basic_set_extend_constraints(bset
, 0, 1);
298 k
= isl_basic_set_alloc_inequality(bset
);
301 isl_seq_clr(bset
->ineq
[k
], 1 + isl_basic_set_total_dim(bset
));
302 isl_seq_cpy(bset
->ineq
[k
], c
, 1 + nparam
);
303 for (i
= 0; i
< n_div
; ++i
) {
304 if (div_purity
[i
] != PURE_PARAM
)
306 isl_int_set(bset
->ineq
[k
][1 + nparam
+ d
+ i
],
307 c
[1 + nparam
+ d
+ i
]);
309 isl_int_sub_ui(bset
->ineq
[k
][0], bset
->ineq
[k
][0], 1);
310 empty
= isl_basic_set_is_empty(bset
);
311 isl_basic_set_free(bset
);
315 isl_basic_set_free(bset
);
319 /* Return PURE_PARAM if only the coefficients of the parameters are non-zero.
320 * Return PURE_VAR if only the coefficients of the set variables are non-zero.
321 * Return MIXED if only the coefficients of the parameters and the set
322 * variables are non-zero and if moreover the parametric constant
323 * can never attain positive values.
324 * Return IMPURE otherwise.
326 static int purity(__isl_keep isl_basic_set
*bset
, isl_int
*c
, int *div_purity
,
336 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
337 d
= isl_basic_set_dim(bset
, isl_dim_set
);
338 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
340 for (i
= 0; i
< n_div
; ++i
) {
341 if (isl_int_is_zero(c
[1 + nparam
+ d
+ i
]))
343 switch (div_purity
[i
]) {
344 case PURE_PARAM
: p
= 1; break;
345 case PURE_VAR
: v
= 1; break;
346 default: return IMPURE
;
349 if (!p
&& isl_seq_first_non_zero(c
+ 1, nparam
) == -1)
351 if (!v
&& isl_seq_first_non_zero(c
+ 1 + nparam
, d
) == -1)
354 empty
= parametric_constant_never_positive(bset
, c
, div_purity
);
355 if (eq
&& empty
>= 0 && !empty
) {
356 isl_seq_neg(c
, c
, 1 + nparam
+ d
+ n_div
);
357 empty
= parametric_constant_never_positive(bset
, c
, div_purity
);
360 return empty
< 0 ? -1 : empty
? MIXED
: IMPURE
;
363 /* Return an array of integers indicating the type of each div in bset.
364 * If the div is (recursively) defined in terms of only the parameters,
365 * then the type is PURE_PARAM.
366 * If the div is (recursively) defined in terms of only the set variables,
367 * then the type is PURE_VAR.
368 * Otherwise, the type is IMPURE.
370 static __isl_give
int *get_div_purity(__isl_keep isl_basic_set
*bset
)
381 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
382 d
= isl_basic_set_dim(bset
, isl_dim_set
);
383 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
385 div_purity
= isl_alloc_array(bset
->ctx
, int, n_div
);
389 for (i
= 0; i
< bset
->n_div
; ++i
) {
391 if (isl_int_is_zero(bset
->div
[i
][0])) {
392 div_purity
[i
] = IMPURE
;
395 if (isl_seq_first_non_zero(bset
->div
[i
] + 2, nparam
) != -1)
397 if (isl_seq_first_non_zero(bset
->div
[i
] + 2 + nparam
, d
) != -1)
399 for (j
= 0; j
< i
; ++j
) {
400 if (isl_int_is_zero(bset
->div
[i
][2 + nparam
+ d
+ j
]))
402 switch (div_purity
[j
]) {
403 case PURE_PARAM
: p
= 1; break;
404 case PURE_VAR
: v
= 1; break;
405 default: p
= v
= 1; break;
408 div_purity
[i
] = v
? p
? IMPURE
: PURE_VAR
: PURE_PARAM
;
414 /* Given a path with the as yet unconstrained length at position "pos",
415 * check if setting the length to zero results in only the identity
418 int empty_path_is_identity(__isl_keep isl_basic_map
*path
, unsigned pos
)
420 isl_basic_map
*test
= NULL
;
421 isl_basic_map
*id
= NULL
;
425 test
= isl_basic_map_copy(path
);
426 test
= isl_basic_map_extend_constraints(test
, 1, 0);
427 k
= isl_basic_map_alloc_equality(test
);
430 isl_seq_clr(test
->eq
[k
], 1 + isl_basic_map_total_dim(test
));
431 isl_int_set_si(test
->eq
[k
][pos
], 1);
432 id
= isl_basic_map_identity(isl_dim_domain(isl_basic_map_get_dim(path
)));
433 is_id
= isl_basic_map_is_equal(test
, id
);
434 isl_basic_map_free(test
);
435 isl_basic_map_free(id
);
438 isl_basic_map_free(test
);
442 __isl_give isl_basic_map
*add_delta_constraints(__isl_take isl_basic_map
*path
,
443 __isl_keep isl_basic_set
*delta
, unsigned off
, unsigned nparam
,
444 unsigned d
, int *div_purity
, int eq
)
447 int n
= eq
? delta
->n_eq
: delta
->n_ineq
;
448 isl_int
**delta_c
= eq
? delta
->eq
: delta
->ineq
;
451 n_div
= isl_basic_set_dim(delta
, isl_dim_div
);
453 for (i
= 0; i
< n
; ++i
) {
455 int p
= purity(delta
, delta_c
[i
], div_purity
, eq
);
460 if (eq
&& p
!= MIXED
) {
461 k
= isl_basic_map_alloc_equality(path
);
462 path_c
= path
->eq
[k
];
464 k
= isl_basic_map_alloc_inequality(path
);
465 path_c
= path
->ineq
[k
];
469 isl_seq_clr(path_c
, 1 + isl_basic_map_total_dim(path
));
471 isl_seq_cpy(path_c
+ off
,
472 delta_c
[i
] + 1 + nparam
, d
);
473 isl_int_set(path_c
[off
+ d
], delta_c
[i
][0]);
474 } else if (p
== PURE_PARAM
) {
475 isl_seq_cpy(path_c
, delta_c
[i
], 1 + nparam
);
477 isl_seq_cpy(path_c
+ off
,
478 delta_c
[i
] + 1 + nparam
, d
);
479 isl_seq_cpy(path_c
, delta_c
[i
], 1 + nparam
);
481 isl_seq_cpy(path_c
+ off
- n_div
,
482 delta_c
[i
] + 1 + nparam
+ d
, n_div
);
487 isl_basic_map_free(path
);
491 /* Given a set of offsets "delta", construct a relation of the
492 * given dimension specification (Z^{n+1} -> Z^{n+1}) that
493 * is an overapproximation of the relations that
494 * maps an element x to any element that can be reached
495 * by taking a non-negative number of steps along any of
496 * the elements in "delta".
497 * That is, construct an approximation of
499 * { [x] -> [y] : exists f \in \delta, k \in Z :
500 * y = x + k [f, 1] and k >= 0 }
502 * For any element in this relation, the number of steps taken
503 * is equal to the difference in the final coordinates.
505 * In particular, let delta be defined as
507 * \delta = [p] -> { [x] : A x + a >= and B p + b >= 0 and
508 * C x + C'p + c >= 0 and
509 * D x + D'p + d >= 0 }
511 * where the constraints C x + C'p + c >= 0 are such that the parametric
512 * constant term of each constraint j, "C_j x + C'_j p + c_j",
513 * can never attain positive values, then the relation is constructed as
515 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
516 * A f + k a >= 0 and B p + b >= 0 and
517 * C f + C'p + c >= 0 and k >= 1 }
518 * union { [x] -> [x] }
520 * If the zero-length paths happen to correspond exactly to the identity
521 * mapping, then we return
523 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
524 * A f + k a >= 0 and B p + b >= 0 and
525 * C f + C'p + c >= 0 and k >= 0 }
529 * Existentially quantified variables in \delta are handled by
530 * classifying them as independent of the parameters, purely
531 * parameter dependent and others. Constraints containing
532 * any of the other existentially quantified variables are removed.
533 * This is safe, but leads to an additional overapproximation.
535 static __isl_give isl_map
*path_along_delta(__isl_take isl_dim
*dim
,
536 __isl_take isl_basic_set
*delta
)
538 isl_basic_map
*path
= NULL
;
545 int *div_purity
= NULL
;
549 n_div
= isl_basic_set_dim(delta
, isl_dim_div
);
550 d
= isl_basic_set_dim(delta
, isl_dim_set
);
551 nparam
= isl_basic_set_dim(delta
, isl_dim_param
);
552 path
= isl_basic_map_alloc_dim(isl_dim_copy(dim
), n_div
+ d
+ 1,
553 d
+ 1 + delta
->n_eq
, delta
->n_eq
+ delta
->n_ineq
+ 1);
554 off
= 1 + nparam
+ 2 * (d
+ 1) + n_div
;
556 for (i
= 0; i
< n_div
+ d
+ 1; ++i
) {
557 k
= isl_basic_map_alloc_div(path
);
560 isl_int_set_si(path
->div
[k
][0], 0);
563 for (i
= 0; i
< d
+ 1; ++i
) {
564 k
= isl_basic_map_alloc_equality(path
);
567 isl_seq_clr(path
->eq
[k
], 1 + isl_basic_map_total_dim(path
));
568 isl_int_set_si(path
->eq
[k
][1 + nparam
+ i
], 1);
569 isl_int_set_si(path
->eq
[k
][1 + nparam
+ d
+ 1 + i
], -1);
570 isl_int_set_si(path
->eq
[k
][off
+ i
], 1);
573 div_purity
= get_div_purity(delta
);
577 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
, div_purity
, 1);
578 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
, div_purity
, 0);
580 is_id
= empty_path_is_identity(path
, off
+ d
);
584 k
= isl_basic_map_alloc_inequality(path
);
587 isl_seq_clr(path
->ineq
[k
], 1 + isl_basic_map_total_dim(path
));
589 isl_int_set_si(path
->ineq
[k
][0], -1);
590 isl_int_set_si(path
->ineq
[k
][off
+ d
], 1);
593 isl_basic_set_free(delta
);
594 path
= isl_basic_map_finalize(path
);
597 return isl_map_from_basic_map(path
);
599 return isl_basic_map_union(path
,
600 isl_basic_map_identity(isl_dim_domain(dim
)));
604 isl_basic_set_free(delta
);
605 isl_basic_map_free(path
);
609 /* Given a dimension specification Z^{n+1} -> Z^{n+1} and a parameter "param",
610 * construct a map that equates the parameter to the difference
611 * in the final coordinates and imposes that this difference is positive.
614 * { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
616 static __isl_give isl_map
*equate_parameter_to_length(__isl_take isl_dim
*dim
,
619 struct isl_basic_map
*bmap
;
624 d
= isl_dim_size(dim
, isl_dim_in
);
625 nparam
= isl_dim_size(dim
, isl_dim_param
);
626 bmap
= isl_basic_map_alloc_dim(dim
, 0, 1, 1);
627 k
= isl_basic_map_alloc_equality(bmap
);
630 isl_seq_clr(bmap
->eq
[k
], 1 + isl_basic_map_total_dim(bmap
));
631 isl_int_set_si(bmap
->eq
[k
][1 + param
], -1);
632 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
- 1], -1);
633 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
+ d
- 1], 1);
635 k
= isl_basic_map_alloc_inequality(bmap
);
638 isl_seq_clr(bmap
->ineq
[k
], 1 + isl_basic_map_total_dim(bmap
));
639 isl_int_set_si(bmap
->ineq
[k
][1 + param
], 1);
640 isl_int_set_si(bmap
->ineq
[k
][0], -1);
642 bmap
= isl_basic_map_finalize(bmap
);
643 return isl_map_from_basic_map(bmap
);
645 isl_basic_map_free(bmap
);
649 /* Check whether "path" is acyclic, where the last coordinates of domain
650 * and range of path encode the number of steps taken.
651 * That is, check whether
653 * { d | d = y - x and (x,y) in path }
655 * does not contain any element with positive last coordinate (positive length)
656 * and zero remaining coordinates (cycle).
658 static int is_acyclic(__isl_take isl_map
*path
)
663 struct isl_set
*delta
;
665 delta
= isl_map_deltas(path
);
666 dim
= isl_set_dim(delta
, isl_dim_set
);
667 for (i
= 0; i
< dim
; ++i
) {
669 delta
= isl_set_lower_bound_si(delta
, isl_dim_set
, i
, 1);
671 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
674 acyclic
= isl_set_is_empty(delta
);
680 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
681 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
682 * construct a map that is an overapproximation of the map
683 * that takes an element from the space D \times Z to another
684 * element from the same space, such that the first n coordinates of the
685 * difference between them is a sum of differences between images
686 * and pre-images in one of the R_i and such that the last coordinate
687 * is equal to the number of steps taken.
690 * \Delta_i = { y - x | (x, y) in R_i }
692 * then the constructed map is an overapproximation of
694 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
695 * d = (\sum_i k_i \delta_i, \sum_i k_i) }
697 * The elements of the singleton \Delta_i's are collected as the
698 * rows of the steps matrix. For all these \Delta_i's together,
699 * a single path is constructed.
700 * For each of the other \Delta_i's, we compute an overapproximation
701 * of the paths along elements of \Delta_i.
702 * Since each of these paths performs an addition, composition is
703 * symmetric and we can simply compose all resulting paths in any order.
705 static __isl_give isl_map
*construct_extended_path(__isl_take isl_dim
*dim
,
706 __isl_keep isl_map
*map
, int *project
)
708 struct isl_mat
*steps
= NULL
;
709 struct isl_map
*path
= NULL
;
713 d
= isl_map_dim(map
, isl_dim_in
);
715 path
= isl_map_identity(isl_dim_domain(isl_dim_copy(dim
)));
717 steps
= isl_mat_alloc(map
->ctx
, map
->n
, d
);
722 for (i
= 0; i
< map
->n
; ++i
) {
723 struct isl_basic_set
*delta
;
725 delta
= isl_basic_map_deltas(isl_basic_map_copy(map
->p
[i
]));
727 for (j
= 0; j
< d
; ++j
) {
730 fixed
= isl_basic_set_fast_dim_is_fixed(delta
, j
,
733 isl_basic_set_free(delta
);
742 path
= isl_map_apply_range(path
,
743 path_along_delta(isl_dim_copy(dim
), delta
));
744 path
= isl_map_coalesce(path
);
746 isl_basic_set_free(delta
);
753 path
= isl_map_apply_range(path
,
754 path_along_steps(isl_dim_copy(dim
), steps
));
757 if (project
&& *project
) {
758 *project
= is_acyclic(isl_map_copy(path
));
773 static int isl_set_overlaps(__isl_keep isl_set
*set1
, __isl_keep isl_set
*set2
)
778 if (!isl_dim_tuple_match(set1
->dim
, isl_dim_set
, set2
->dim
, isl_dim_set
))
781 i
= isl_set_intersect(isl_set_copy(set1
), isl_set_copy(set2
));
782 no_overlap
= isl_set_is_empty(i
);
785 return no_overlap
< 0 ? -1 : !no_overlap
;
788 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
789 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
790 * construct a map that is an overapproximation of the map
791 * that takes an element from the dom R \times Z to an
792 * element from ran R \times Z, such that the first n coordinates of the
793 * difference between them is a sum of differences between images
794 * and pre-images in one of the R_i and such that the last coordinate
795 * is equal to the number of steps taken.
798 * \Delta_i = { y - x | (x, y) in R_i }
800 * then the constructed map is an overapproximation of
802 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
803 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
804 * x in dom R and x + d in ran R and
807 static __isl_give isl_map
*construct_component(__isl_take isl_dim
*dim
,
808 __isl_keep isl_map
*map
, int *exact
, int project
)
810 struct isl_set
*domain
= NULL
;
811 struct isl_set
*range
= NULL
;
812 struct isl_map
*app
= NULL
;
813 struct isl_map
*path
= NULL
;
815 domain
= isl_map_domain(isl_map_copy(map
));
816 domain
= isl_set_coalesce(domain
);
817 range
= isl_map_range(isl_map_copy(map
));
818 range
= isl_set_coalesce(range
);
819 if (!isl_set_overlaps(domain
, range
)) {
820 isl_set_free(domain
);
824 map
= isl_map_copy(map
);
825 map
= isl_map_add_dims(map
, isl_dim_in
, 1);
826 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
827 map
= set_path_length(map
, 1, 1);
830 app
= isl_map_from_domain_and_range(domain
, range
);
831 app
= isl_map_add_dims(app
, isl_dim_in
, 1);
832 app
= isl_map_add_dims(app
, isl_dim_out
, 1);
834 path
= construct_extended_path(isl_dim_copy(dim
), map
,
835 exact
&& *exact
? &project
: NULL
);
836 app
= isl_map_intersect(app
, path
);
838 if (exact
&& *exact
&&
839 (*exact
= check_exactness(isl_map_copy(map
), isl_map_copy(app
),
844 app
= set_path_length(app
, 0, 1);
852 /* Call construct_component and, if "project" is set, project out
853 * the final coordinates.
855 static __isl_give isl_map
*construct_projected_component(
856 __isl_take isl_dim
*dim
,
857 __isl_keep isl_map
*map
, int *exact
, int project
)
864 d
= isl_dim_size(dim
, isl_dim_in
);
866 app
= construct_component(dim
, map
, exact
, project
);
868 app
= isl_map_project_out(app
, isl_dim_in
, d
- 1, 1);
869 app
= isl_map_project_out(app
, isl_dim_out
, d
- 1, 1);
874 /* Compute an extended version, i.e., with path lengths, of
875 * an overapproximation of the transitive closure of "bmap"
876 * with path lengths greater than or equal to zero and with
877 * domain and range equal to "dom".
879 static __isl_give isl_map
*q_closure(__isl_take isl_dim
*dim
,
880 __isl_take isl_set
*dom
, __isl_keep isl_basic_map
*bmap
, int *exact
)
887 dom
= isl_set_add_dims(dom
, isl_dim_set
, 1);
888 app
= isl_map_from_domain_and_range(dom
, isl_set_copy(dom
));
889 map
= isl_map_from_basic_map(isl_basic_map_copy(bmap
));
890 path
= construct_extended_path(dim
, map
, &project
);
891 app
= isl_map_intersect(app
, path
);
893 if ((*exact
= check_exactness(map
, isl_map_copy(app
), project
)) < 0)
902 /* Check whether qc has any elements of length at least one
903 * with domain and/or range outside of dom and ran.
905 static int has_spurious_elements(__isl_keep isl_map
*qc
,
906 __isl_keep isl_set
*dom
, __isl_keep isl_set
*ran
)
912 if (!qc
|| !dom
|| !ran
)
915 d
= isl_map_dim(qc
, isl_dim_in
);
917 qc
= isl_map_copy(qc
);
918 qc
= set_path_length(qc
, 0, 1);
919 qc
= isl_map_project_out(qc
, isl_dim_in
, d
- 1, 1);
920 qc
= isl_map_project_out(qc
, isl_dim_out
, d
- 1, 1);
922 s
= isl_map_domain(isl_map_copy(qc
));
923 subset
= isl_set_is_subset(s
, dom
);
932 s
= isl_map_range(qc
);
933 subset
= isl_set_is_subset(s
, ran
);
936 return subset
< 0 ? -1 : !subset
;
945 /* For each basic map in "map", except i, check whether it combines
946 * with the transitive closure that is reflexive on C combines
947 * to the left and to the right.
951 * dom map_j \subseteq C
953 * then right[j] is set to 1. Otherwise, if
955 * ran map_i \cap dom map_j = \emptyset
957 * then right[j] is set to 0. Otherwise, composing to the right
960 * Similar, for composing to the left, we have if
962 * ran map_j \subseteq C
964 * then left[j] is set to 1. Otherwise, if
966 * dom map_i \cap ran map_j = \emptyset
968 * then left[j] is set to 0. Otherwise, composing to the left
971 * The return value is or'd with LEFT if composing to the left
972 * is possible and with RIGHT if composing to the right is possible.
974 static int composability(__isl_keep isl_set
*C
, int i
,
975 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
976 __isl_keep isl_map
*map
)
982 for (j
= 0; j
< map
->n
&& ok
; ++j
) {
983 int overlaps
, subset
;
989 dom
[j
] = isl_set_from_basic_set(
990 isl_basic_map_domain(
991 isl_basic_map_copy(map
->p
[j
])));
994 overlaps
= isl_set_overlaps(ran
[i
], dom
[j
]);
1000 subset
= isl_set_is_subset(dom
[j
], C
);
1012 ran
[j
] = isl_set_from_basic_set(
1013 isl_basic_map_range(
1014 isl_basic_map_copy(map
->p
[j
])));
1017 overlaps
= isl_set_overlaps(dom
[i
], ran
[j
]);
1023 subset
= isl_set_is_subset(ran
[j
], C
);
1037 static __isl_give isl_map
*anonymize(__isl_take isl_map
*map
)
1039 map
= isl_map_reset(map
, isl_dim_in
);
1040 map
= isl_map_reset(map
, isl_dim_out
);
1044 /* Return a map that is a union of the basic maps in "map", except i,
1045 * composed to left and right with qc based on the entries of "left"
1048 static __isl_give isl_map
*compose(__isl_keep isl_map
*map
, int i
,
1049 __isl_take isl_map
*qc
, int *left
, int *right
)
1054 comp
= isl_map_empty(isl_map_get_dim(map
));
1055 for (j
= 0; j
< map
->n
; ++j
) {
1061 map_j
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[j
]));
1062 map_j
= anonymize(map_j
);
1063 if (left
&& left
[j
])
1064 map_j
= isl_map_apply_range(map_j
, isl_map_copy(qc
));
1065 if (right
&& right
[j
])
1066 map_j
= isl_map_apply_range(isl_map_copy(qc
), map_j
);
1067 comp
= isl_map_union(comp
, map_j
);
1070 comp
= isl_map_compute_divs(comp
);
1071 comp
= isl_map_coalesce(comp
);
1078 /* Compute the transitive closure of "map" incrementally by
1085 * map_i^+ \cup ((id \cup map_i^) \circ qc^+)
1089 * map_i^+ \cup (qc^+ \circ (id \cup map_i^))
1091 * depending on whether left or right are NULL.
1093 static __isl_give isl_map
*compute_incremental(
1094 __isl_take isl_dim
*dim
, __isl_keep isl_map
*map
,
1095 int i
, __isl_take isl_map
*qc
, int *left
, int *right
, int *exact
)
1099 isl_map
*rtc
= NULL
;
1103 isl_assert(map
->ctx
, left
|| right
, goto error
);
1105 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
1106 tc
= construct_projected_component(isl_dim_copy(dim
), map_i
,
1108 isl_map_free(map_i
);
1111 qc
= isl_map_transitive_closure(qc
, exact
);
1117 return isl_map_universe(isl_map_get_dim(map
));
1120 if (!left
|| !right
)
1121 rtc
= isl_map_union(isl_map_copy(tc
),
1122 isl_map_identity(isl_dim_domain(isl_map_get_dim(tc
))));
1124 qc
= isl_map_apply_range(rtc
, qc
);
1126 qc
= isl_map_apply_range(qc
, rtc
);
1127 qc
= isl_map_union(tc
, qc
);
1138 /* Given a map "map", try to find a basic map such that
1139 * map^+ can be computed as
1141 * map^+ = map_i^+ \cup
1142 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1144 * with C the simple hull of the domain and range of the input map.
1145 * map_i^ \cup Id_C is computed by allowing the path lengths to be zero
1146 * and by intersecting domain and range with C.
1147 * Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
1148 * Also, we only use the incremental computation if all the transitive
1149 * closures are exact and if the number of basic maps in the union,
1150 * after computing the integer divisions, is smaller than the number
1151 * of basic maps in the input map.
1153 static int incemental_on_entire_domain(__isl_keep isl_dim
*dim
,
1154 __isl_keep isl_map
*map
,
1155 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
1156 __isl_give isl_map
**res
)
1164 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
1165 isl_map_range(isl_map_copy(map
)));
1166 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1174 d
= isl_map_dim(map
, isl_dim_in
);
1176 for (i
= 0; i
< map
->n
; ++i
) {
1178 int exact_i
, spurious
;
1180 dom
[i
] = isl_set_from_basic_set(isl_basic_map_domain(
1181 isl_basic_map_copy(map
->p
[i
])));
1182 ran
[i
] = isl_set_from_basic_set(isl_basic_map_range(
1183 isl_basic_map_copy(map
->p
[i
])));
1184 qc
= q_closure(isl_dim_copy(dim
), isl_set_copy(C
),
1185 map
->p
[i
], &exact_i
);
1192 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1199 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1200 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1201 qc
= isl_map_compute_divs(qc
);
1202 for (j
= 0; j
< map
->n
; ++j
)
1203 left
[j
] = right
[j
] = 1;
1204 qc
= compose(map
, i
, qc
, left
, right
);
1207 if (qc
->n
>= map
->n
) {
1211 *res
= compute_incremental(isl_dim_copy(dim
), map
, i
, qc
,
1212 left
, right
, &exact_i
);
1223 return *res
!= NULL
;
1229 /* Try and compute the transitive closure of "map" as
1231 * map^+ = map_i^+ \cup
1232 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1234 * with C either the simple hull of the domain and range of the entire
1235 * map or the simple hull of domain and range of map_i.
1237 static __isl_give isl_map
*incremental_closure(__isl_take isl_dim
*dim
,
1238 __isl_keep isl_map
*map
, int *exact
, int project
)
1241 isl_set
**dom
= NULL
;
1242 isl_set
**ran
= NULL
;
1247 isl_map
*res
= NULL
;
1250 return construct_projected_component(dim
, map
, exact
, project
);
1255 return construct_projected_component(dim
, map
, exact
, project
);
1257 d
= isl_map_dim(map
, isl_dim_in
);
1259 dom
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1260 ran
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1261 left
= isl_calloc_array(map
->ctx
, int, map
->n
);
1262 right
= isl_calloc_array(map
->ctx
, int, map
->n
);
1263 if (!ran
|| !dom
|| !left
|| !right
)
1266 if (incemental_on_entire_domain(dim
, map
, dom
, ran
, left
, right
, &res
) < 0)
1269 for (i
= 0; !res
&& i
< map
->n
; ++i
) {
1271 int exact_i
, spurious
, comp
;
1273 dom
[i
] = isl_set_from_basic_set(
1274 isl_basic_map_domain(
1275 isl_basic_map_copy(map
->p
[i
])));
1279 ran
[i
] = isl_set_from_basic_set(
1280 isl_basic_map_range(
1281 isl_basic_map_copy(map
->p
[i
])));
1284 C
= isl_set_union(isl_set_copy(dom
[i
]),
1285 isl_set_copy(ran
[i
]));
1286 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1293 comp
= composability(C
, i
, dom
, ran
, left
, right
, map
);
1294 if (!comp
|| comp
< 0) {
1300 qc
= q_closure(isl_dim_copy(dim
), C
, map
->p
[i
], &exact_i
);
1307 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1314 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1315 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1316 qc
= isl_map_compute_divs(qc
);
1317 qc
= compose(map
, i
, qc
, (comp
& LEFT
) ? left
: NULL
,
1318 (comp
& RIGHT
) ? right
: NULL
);
1321 if (qc
->n
>= map
->n
) {
1325 res
= compute_incremental(isl_dim_copy(dim
), map
, i
, qc
,
1326 (comp
& LEFT
) ? left
: NULL
,
1327 (comp
& RIGHT
) ? right
: NULL
, &exact_i
);
1336 for (i
= 0; i
< map
->n
; ++i
) {
1337 isl_set_free(dom
[i
]);
1338 isl_set_free(ran
[i
]);
1350 return construct_projected_component(dim
, map
, exact
, project
);
1353 for (i
= 0; i
< map
->n
; ++i
)
1354 isl_set_free(dom
[i
]);
1357 for (i
= 0; i
< map
->n
; ++i
)
1358 isl_set_free(ran
[i
]);
1366 /* Given an array of sets "set", add "dom" at position "pos"
1367 * and search for elements at earlier positions that overlap with "dom".
1368 * If any can be found, then merge all of them, together with "dom", into
1369 * a single set and assign the union to the first in the array,
1370 * which becomes the new group leader for all groups involved in the merge.
1371 * During the search, we only consider group leaders, i.e., those with
1372 * group[i] = i, as the other sets have already been combined
1373 * with one of the group leaders.
1375 static int merge(isl_set
**set
, int *group
, __isl_take isl_set
*dom
, int pos
)
1380 set
[pos
] = isl_set_copy(dom
);
1382 for (i
= pos
- 1; i
>= 0; --i
) {
1388 o
= isl_set_overlaps(set
[i
], dom
);
1394 set
[i
] = isl_set_union(set
[i
], set
[group
[pos
]]);
1395 set
[group
[pos
]] = NULL
;
1398 group
[group
[pos
]] = i
;
1409 /* Replace each entry in the n by n grid of maps by the cross product
1410 * with the relation { [i] -> [i + 1] }.
1412 static int add_length(__isl_keep isl_map
*map
, isl_map
***grid
, int n
)
1416 isl_basic_map
*bstep
;
1423 dim
= isl_map_get_dim(map
);
1424 nparam
= isl_dim_size(dim
, isl_dim_param
);
1425 dim
= isl_dim_drop(dim
, isl_dim_in
, 0, isl_dim_size(dim
, isl_dim_in
));
1426 dim
= isl_dim_drop(dim
, isl_dim_out
, 0, isl_dim_size(dim
, isl_dim_out
));
1427 dim
= isl_dim_add(dim
, isl_dim_in
, 1);
1428 dim
= isl_dim_add(dim
, isl_dim_out
, 1);
1429 bstep
= isl_basic_map_alloc_dim(dim
, 0, 1, 0);
1430 k
= isl_basic_map_alloc_equality(bstep
);
1432 isl_basic_map_free(bstep
);
1435 isl_seq_clr(bstep
->eq
[k
], 1 + isl_basic_map_total_dim(bstep
));
1436 isl_int_set_si(bstep
->eq
[k
][0], 1);
1437 isl_int_set_si(bstep
->eq
[k
][1 + nparam
], 1);
1438 isl_int_set_si(bstep
->eq
[k
][1 + nparam
+ 1], -1);
1439 bstep
= isl_basic_map_finalize(bstep
);
1440 step
= isl_map_from_basic_map(bstep
);
1442 for (i
= 0; i
< n
; ++i
)
1443 for (j
= 0; j
< n
; ++j
)
1444 grid
[i
][j
] = isl_map_product(grid
[i
][j
],
1445 isl_map_copy(step
));
1452 /* The core of the Floyd-Warshall algorithm.
1453 * Updates the given n x x matrix of relations in place.
1455 * The algorithm iterates over all vertices. In each step, the whole
1456 * matrix is updated to include all paths that go to the current vertex,
1457 * possibly stay there a while (including passing through earlier vertices)
1458 * and then come back. At the start of each iteration, the diagonal
1459 * element corresponding to the current vertex is replaced by its
1460 * transitive closure to account for all indirect paths that stay
1461 * in the current vertex.
1463 static void floyd_warshall_iterate(isl_map
***grid
, int n
, int *exact
)
1467 for (r
= 0; r
< n
; ++r
) {
1469 grid
[r
][r
] = isl_map_transitive_closure(grid
[r
][r
],
1470 (exact
&& *exact
) ? &r_exact
: NULL
);
1471 if (exact
&& *exact
&& !r_exact
)
1474 for (p
= 0; p
< n
; ++p
)
1475 for (q
= 0; q
< n
; ++q
) {
1477 if (p
== r
&& q
== r
)
1479 loop
= isl_map_apply_range(
1480 isl_map_copy(grid
[p
][r
]),
1481 isl_map_copy(grid
[r
][q
]));
1482 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1483 loop
= isl_map_apply_range(
1484 isl_map_copy(grid
[p
][r
]),
1485 isl_map_apply_range(
1486 isl_map_copy(grid
[r
][r
]),
1487 isl_map_copy(grid
[r
][q
])));
1488 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1489 grid
[p
][q
] = isl_map_coalesce(grid
[p
][q
]);
1494 /* Given a partition of the domains and ranges of the basic maps in "map",
1495 * apply the Floyd-Warshall algorithm with the elements in the partition
1498 * In particular, there are "n" elements in the partition and "group" is
1499 * an array of length 2 * map->n with entries in [0,n-1].
1501 * We first construct a matrix of relations based on the partition information,
1502 * apply Floyd-Warshall on this matrix of relations and then take the
1503 * union of all entries in the matrix as the final result.
1505 * If we are actually computing the power instead of the transitive closure,
1506 * i.e., when "project" is not set, then the result should have the
1507 * path lengths encoded as the difference between an extra pair of
1508 * coordinates. We therefore apply the nested transitive closures
1509 * to relations that include these lengths. In particular, we replace
1510 * the input relation by the cross product with the unit length relation
1511 * { [i] -> [i + 1] }.
1513 static __isl_give isl_map
*floyd_warshall_with_groups(__isl_take isl_dim
*dim
,
1514 __isl_keep isl_map
*map
, int *exact
, int project
, int *group
, int n
)
1517 isl_map
***grid
= NULL
;
1525 return incremental_closure(dim
, map
, exact
, project
);
1528 grid
= isl_calloc_array(map
->ctx
, isl_map
**, n
);
1531 for (i
= 0; i
< n
; ++i
) {
1532 grid
[i
] = isl_calloc_array(map
->ctx
, isl_map
*, n
);
1535 for (j
= 0; j
< n
; ++j
)
1536 grid
[i
][j
] = isl_map_empty(isl_map_get_dim(map
));
1539 for (k
= 0; k
< map
->n
; ++k
) {
1541 j
= group
[2 * k
+ 1];
1542 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
1543 isl_map_from_basic_map(
1544 isl_basic_map_copy(map
->p
[k
])));
1547 if (!project
&& add_length(map
, grid
, n
) < 0)
1550 floyd_warshall_iterate(grid
, n
, exact
);
1552 app
= isl_map_empty(isl_map_get_dim(map
));
1554 for (i
= 0; i
< n
; ++i
) {
1555 for (j
= 0; j
< n
; ++j
)
1556 app
= isl_map_union(app
, grid
[i
][j
]);
1567 for (i
= 0; i
< n
; ++i
) {
1570 for (j
= 0; j
< n
; ++j
)
1571 isl_map_free(grid
[i
][j
]);
1580 /* Partition the domains and ranges of the n basic relations in list
1581 * into disjoint cells.
1583 * To find the partition, we simply consider all of the domains
1584 * and ranges in turn and combine those that overlap.
1585 * "set" contains the partition elements and "group" indicates
1586 * to which partition element a given domain or range belongs.
1587 * The domain of basic map i corresponds to element 2 * i in these arrays,
1588 * while the domain corresponds to element 2 * i + 1.
1589 * During the construction group[k] is either equal to k,
1590 * in which case set[k] contains the union of all the domains and
1591 * ranges in the corresponding group, or is equal to some l < k,
1592 * with l another domain or range in the same group.
1594 static int *setup_groups(isl_ctx
*ctx
, __isl_keep isl_basic_map
**list
, int n
,
1595 isl_set
***set
, int *n_group
)
1601 *set
= isl_calloc_array(ctx
, isl_set
*, 2 * n
);
1602 group
= isl_alloc_array(ctx
, int, 2 * n
);
1604 if (!*set
|| !group
)
1607 for (i
= 0; i
< n
; ++i
) {
1609 dom
= isl_set_from_basic_set(isl_basic_map_domain(
1610 isl_basic_map_copy(list
[i
])));
1611 if (merge(*set
, group
, dom
, 2 * i
) < 0)
1613 dom
= isl_set_from_basic_set(isl_basic_map_range(
1614 isl_basic_map_copy(list
[i
])));
1615 if (merge(*set
, group
, dom
, 2 * i
+ 1) < 0)
1620 for (i
= 0; i
< 2 * n
; ++i
)
1621 if (group
[i
] == i
) {
1623 (*set
)[g
] = (*set
)[i
];
1628 group
[i
] = group
[group
[i
]];
1635 for (i
= 0; i
< 2 * n
; ++i
)
1636 isl_set_free((*set
)[i
]);
1644 /* Check if the domains and ranges of the basic maps in "map" can
1645 * be partitioned, and if so, apply Floyd-Warshall on the elements
1646 * of the partition. Note that we also apply this algorithm
1647 * if we want to compute the power, i.e., when "project" is not set.
1648 * However, the results are unlikely to be exact since the recursive
1649 * calls inside the Floyd-Warshall algorithm typically result in
1650 * non-linear path lengths quite quickly.
1652 static __isl_give isl_map
*floyd_warshall(__isl_take isl_dim
*dim
,
1653 __isl_keep isl_map
*map
, int *exact
, int project
)
1656 isl_set
**set
= NULL
;
1663 return incremental_closure(dim
, map
, exact
, project
);
1665 group
= setup_groups(map
->ctx
, map
->p
, map
->n
, &set
, &n
);
1669 for (i
= 0; i
< 2 * map
->n
; ++i
)
1670 isl_set_free(set
[i
]);
1674 return floyd_warshall_with_groups(dim
, map
, exact
, project
, group
, n
);
1680 /* Structure for representing the nodes in the graph being traversed
1681 * using Tarjan's algorithm.
1682 * index represents the order in which nodes are visited.
1683 * min_index is the index of the root of a (sub)component.
1684 * on_stack indicates whether the node is currently on the stack.
1686 struct basic_map_sort_node
{
1691 /* Structure for representing the graph being traversed
1692 * using Tarjan's algorithm.
1693 * len is the number of nodes
1694 * node is an array of nodes
1695 * stack contains the nodes on the path from the root to the current node
1696 * sp is the stack pointer
1697 * index is the index of the last node visited
1698 * order contains the elements of the components separated by -1
1699 * op represents the current position in order
1701 * check_closed is set if we may have used the fact that
1702 * a pair of basic maps can be interchanged
1704 struct basic_map_sort
{
1706 struct basic_map_sort_node
*node
;
1715 static void basic_map_sort_free(struct basic_map_sort
*s
)
1725 static struct basic_map_sort
*basic_map_sort_alloc(struct isl_ctx
*ctx
, int len
)
1727 struct basic_map_sort
*s
;
1730 s
= isl_calloc_type(ctx
, struct basic_map_sort
);
1734 s
->node
= isl_alloc_array(ctx
, struct basic_map_sort_node
, len
);
1737 for (i
= 0; i
< len
; ++i
)
1738 s
->node
[i
].index
= -1;
1739 s
->stack
= isl_alloc_array(ctx
, int, len
);
1742 s
->order
= isl_alloc_array(ctx
, int, 2 * len
);
1750 s
->check_closed
= 0;
1754 basic_map_sort_free(s
);
1758 /* Check whether in the computation of the transitive closure
1759 * "bmap1" (R_1) should follow (or be part of the same component as)
1762 * That is check whether
1770 * If so, then there is no reason for R_1 to immediately follow R_2
1773 * *check_closed is set if the subset relation holds while
1774 * R_1 \circ R_2 is not empty.
1776 static int basic_map_follows(__isl_keep isl_basic_map
*bmap1
,
1777 __isl_keep isl_basic_map
*bmap2
, int *check_closed
)
1779 struct isl_map
*map12
= NULL
;
1780 struct isl_map
*map21
= NULL
;
1783 if (!isl_dim_tuple_match(bmap1
->dim
, isl_dim_in
, bmap2
->dim
, isl_dim_out
))
1786 map21
= isl_map_from_basic_map(
1787 isl_basic_map_apply_range(
1788 isl_basic_map_copy(bmap2
),
1789 isl_basic_map_copy(bmap1
)));
1790 subset
= isl_map_is_empty(map21
);
1794 isl_map_free(map21
);
1798 if (!isl_dim_tuple_match(bmap1
->dim
, isl_dim_in
, bmap1
->dim
, isl_dim_out
) ||
1799 !isl_dim_tuple_match(bmap2
->dim
, isl_dim_in
, bmap2
->dim
, isl_dim_out
)) {
1800 isl_map_free(map21
);
1804 map12
= isl_map_from_basic_map(
1805 isl_basic_map_apply_range(
1806 isl_basic_map_copy(bmap1
),
1807 isl_basic_map_copy(bmap2
)));
1809 subset
= isl_map_is_subset(map21
, map12
);
1811 isl_map_free(map12
);
1812 isl_map_free(map21
);
1817 return subset
< 0 ? -1 : !subset
;
1819 isl_map_free(map21
);
1823 /* Perform Tarjan's algorithm for computing the strongly connected components
1824 * in the graph with the disjuncts of "map" as vertices and with an
1825 * edge between any pair of disjuncts such that the first has
1826 * to be applied after the second.
1828 static int power_components_tarjan(struct basic_map_sort
*s
,
1829 __isl_keep isl_basic_map
**list
, int i
)
1833 s
->node
[i
].index
= s
->index
;
1834 s
->node
[i
].min_index
= s
->index
;
1835 s
->node
[i
].on_stack
= 1;
1837 s
->stack
[s
->sp
++] = i
;
1839 for (j
= s
->len
- 1; j
>= 0; --j
) {
1844 if (s
->node
[j
].index
>= 0 &&
1845 (!s
->node
[j
].on_stack
||
1846 s
->node
[j
].index
> s
->node
[i
].min_index
))
1849 f
= basic_map_follows(list
[i
], list
[j
], &s
->check_closed
);
1855 if (s
->node
[j
].index
< 0) {
1856 power_components_tarjan(s
, list
, j
);
1857 if (s
->node
[j
].min_index
< s
->node
[i
].min_index
)
1858 s
->node
[i
].min_index
= s
->node
[j
].min_index
;
1859 } else if (s
->node
[j
].index
< s
->node
[i
].min_index
)
1860 s
->node
[i
].min_index
= s
->node
[j
].index
;
1863 if (s
->node
[i
].index
!= s
->node
[i
].min_index
)
1867 j
= s
->stack
[--s
->sp
];
1868 s
->node
[j
].on_stack
= 0;
1869 s
->order
[s
->op
++] = j
;
1871 s
->order
[s
->op
++] = -1;
1876 /* Decompose the "len" basic relations in "list" into strongly connected
1879 static struct basic_map_sort
*basic_map_sort_init(isl_ctx
*ctx
, int len
,
1880 __isl_keep isl_basic_map
**list
)
1883 struct basic_map_sort
*s
= NULL
;
1885 s
= basic_map_sort_alloc(ctx
, len
);
1888 for (i
= len
- 1; i
>= 0; --i
) {
1889 if (s
->node
[i
].index
>= 0)
1891 if (power_components_tarjan(s
, list
, i
) < 0)
1897 basic_map_sort_free(s
);
1901 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
1902 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
1903 * construct a map that is an overapproximation of the map
1904 * that takes an element from the dom R \times Z to an
1905 * element from ran R \times Z, such that the first n coordinates of the
1906 * difference between them is a sum of differences between images
1907 * and pre-images in one of the R_i and such that the last coordinate
1908 * is equal to the number of steps taken.
1909 * If "project" is set, then these final coordinates are not included,
1910 * i.e., a relation of type Z^n -> Z^n is returned.
1913 * \Delta_i = { y - x | (x, y) in R_i }
1915 * then the constructed map is an overapproximation of
1917 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1918 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
1919 * x in dom R and x + d in ran R }
1923 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1924 * d = (\sum_i k_i \delta_i) and
1925 * x in dom R and x + d in ran R }
1927 * if "project" is set.
1929 * We first split the map into strongly connected components, perform
1930 * the above on each component and then join the results in the correct
1931 * order, at each join also taking in the union of both arguments
1932 * to allow for paths that do not go through one of the two arguments.
1934 static __isl_give isl_map
*construct_power_components(__isl_take isl_dim
*dim
,
1935 __isl_keep isl_map
*map
, int *exact
, int project
)
1938 struct isl_map
*path
= NULL
;
1939 struct basic_map_sort
*s
= NULL
;
1946 return floyd_warshall(dim
, map
, exact
, project
);
1948 s
= basic_map_sort_init(map
->ctx
, map
->n
, map
->p
);
1953 if (s
->check_closed
&& !exact
)
1954 exact
= &local_exact
;
1960 path
= isl_map_empty(isl_map_get_dim(map
));
1962 path
= isl_map_empty(isl_dim_copy(dim
));
1963 path
= anonymize(path
);
1965 struct isl_map
*comp
;
1966 isl_map
*path_comp
, *path_comb
;
1967 comp
= isl_map_alloc_dim(isl_map_get_dim(map
), n
, 0);
1968 while (s
->order
[i
] != -1) {
1969 comp
= isl_map_add_basic_map(comp
,
1970 isl_basic_map_copy(map
->p
[s
->order
[i
]]));
1974 path_comp
= floyd_warshall(isl_dim_copy(dim
),
1975 comp
, exact
, project
);
1976 path_comb
= isl_map_apply_range(isl_map_copy(path
),
1977 isl_map_copy(path_comp
));
1978 path
= isl_map_union(path
, path_comp
);
1979 path
= isl_map_union(path
, path_comb
);
1985 if (c
> 1 && s
->check_closed
&& !*exact
) {
1988 closed
= isl_map_is_transitively_closed(path
);
1992 basic_map_sort_free(s
);
1994 return floyd_warshall(dim
, map
, orig_exact
, project
);
1998 basic_map_sort_free(s
);
2003 basic_map_sort_free(s
);
2009 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
2010 * construct a map that is an overapproximation of the map
2011 * that takes an element from the space D to another
2012 * element from the same space, such that the difference between
2013 * them is a strictly positive sum of differences between images
2014 * and pre-images in one of the R_i.
2015 * The number of differences in the sum is equated to parameter "param".
2018 * \Delta_i = { y - x | (x, y) in R_i }
2020 * then the constructed map is an overapproximation of
2022 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
2023 * d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
2026 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
2027 * d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
2029 * if "project" is set.
2031 * If "project" is not set, then
2032 * we construct an extended mapping with an extra coordinate
2033 * that indicates the number of steps taken. In particular,
2034 * the difference in the last coordinate is equal to the number
2035 * of steps taken to move from a domain element to the corresponding
2038 static __isl_give isl_map
*construct_power(__isl_keep isl_map
*map
,
2039 int *exact
, int project
)
2041 struct isl_map
*app
= NULL
;
2042 struct isl_dim
*dim
= NULL
;
2048 dim
= isl_map_get_dim(map
);
2050 d
= isl_dim_size(dim
, isl_dim_in
);
2051 dim
= isl_dim_add(dim
, isl_dim_in
, 1);
2052 dim
= isl_dim_add(dim
, isl_dim_out
, 1);
2054 app
= construct_power_components(isl_dim_copy(dim
), map
,
2062 /* Compute the positive powers of "map", or an overapproximation.
2063 * If the result is exact, then *exact is set to 1.
2065 * If project is set, then we are actually interested in the transitive
2066 * closure, so we can use a more relaxed exactness check.
2067 * The lengths of the paths are also projected out instead of being
2068 * encoded as the difference between an extra pair of final coordinates.
2070 static __isl_give isl_map
*map_power(__isl_take isl_map
*map
,
2071 int *exact
, int project
)
2073 struct isl_map
*app
= NULL
;
2081 isl_assert(map
->ctx
,
2082 isl_map_dim(map
, isl_dim_in
) == isl_map_dim(map
, isl_dim_out
),
2085 app
= construct_power(map
, exact
, project
);
2095 /* Compute the positive powers of "map", or an overapproximation.
2096 * The power is given by parameter "param". If the result is exact,
2097 * then *exact is set to 1.
2098 * map_power constructs an extended relation with the path lengths
2099 * encoded as the difference between the final coordinates.
2100 * In the final step, this difference is equated to the parameter "param"
2101 * and made positive. The extra coordinates are subsequently projected out.
2103 __isl_give isl_map
*isl_map_power(__isl_take isl_map
*map
, unsigned param
,
2106 isl_dim
*target_dim
;
2114 isl_assert(map
->ctx
, param
< isl_map_dim(map
, isl_dim_param
),
2117 d
= isl_map_dim(map
, isl_dim_in
);
2119 map
= isl_map_compute_divs(map
);
2120 map
= isl_map_coalesce(map
);
2122 if (isl_map_fast_is_empty(map
))
2125 target_dim
= isl_map_get_dim(map
);
2126 map
= map_power(map
, exact
, 0);
2128 dim
= isl_map_get_dim(map
);
2129 diff
= equate_parameter_to_length(dim
, param
);
2130 map
= isl_map_intersect(map
, diff
);
2131 map
= isl_map_project_out(map
, isl_dim_in
, d
, 1);
2132 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2134 map
= isl_map_reset_dim(map
, target_dim
);
2142 /* Compute a relation that maps each element in the range of the input
2143 * relation to the lengths of all paths composed of edges in the input
2144 * relation that end up in the given range element.
2145 * The result may be an overapproximation, in which case *exact is set to 0.
2146 * The resulting relation is very similar to the power relation.
2147 * The difference are that the domain has been projected out, the
2148 * range has become the domain and the exponent is the range instead
2151 __isl_give isl_map
*isl_map_reaching_path_lengths(__isl_take isl_map
*map
,
2162 d
= isl_map_dim(map
, isl_dim_in
);
2163 param
= isl_map_dim(map
, isl_dim_param
);
2165 map
= isl_map_compute_divs(map
);
2166 map
= isl_map_coalesce(map
);
2168 if (isl_map_fast_is_empty(map
)) {
2171 map
= isl_map_project_out(map
, isl_dim_out
, 0, d
);
2172 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
2176 map
= map_power(map
, exact
, 0);
2178 map
= isl_map_add_dims(map
, isl_dim_param
, 1);
2179 dim
= isl_map_get_dim(map
);
2180 diff
= equate_parameter_to_length(dim
, param
);
2181 map
= isl_map_intersect(map
, diff
);
2182 map
= isl_map_project_out(map
, isl_dim_in
, 0, d
+ 1);
2183 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2184 map
= isl_map_reverse(map
);
2185 map
= isl_map_move_dims(map
, isl_dim_out
, 0, isl_dim_param
, param
, 1);
2190 /* Check whether equality i of bset is a pure stride constraint
2191 * on a single dimensions, i.e., of the form
2195 * with k a constant and e an existentially quantified variable.
2197 static int is_eq_stride(__isl_keep isl_basic_set
*bset
, int i
)
2209 if (!isl_int_is_zero(bset
->eq
[i
][0]))
2212 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
2213 d
= isl_basic_set_dim(bset
, isl_dim_set
);
2214 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
2216 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1, nparam
) != -1)
2218 pos1
= isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
, d
);
2221 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ pos1
+ 1,
2222 d
- pos1
- 1) != -1)
2225 pos2
= isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ d
, n_div
);
2228 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ d
+ pos2
+ 1,
2229 n_div
- pos2
- 1) != -1)
2231 if (!isl_int_is_one(bset
->eq
[i
][1 + nparam
+ pos1
]) &&
2232 !isl_int_is_negone(bset
->eq
[i
][1 + nparam
+ pos1
]))
2238 /* Given a map, compute the smallest superset of this map that is of the form
2240 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2242 * (where p ranges over the (non-parametric) dimensions),
2243 * compute the transitive closure of this map, i.e.,
2245 * { i -> j : exists k > 0:
2246 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2248 * and intersect domain and range of this transitive closure with
2249 * the given domain and range.
2251 * If with_id is set, then try to include as much of the identity mapping
2252 * as possible, by computing
2254 * { i -> j : exists k >= 0:
2255 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2257 * instead (i.e., allow k = 0).
2259 * In practice, we compute the difference set
2261 * delta = { j - i | i -> j in map },
2263 * look for stride constraint on the individual dimensions and compute
2264 * (constant) lower and upper bounds for each individual dimension,
2265 * adding a constraint for each bound not equal to infinity.
2267 static __isl_give isl_map
*box_closure_on_domain(__isl_take isl_map
*map
,
2268 __isl_take isl_set
*dom
, __isl_take isl_set
*ran
, int with_id
)
2277 isl_map
*app
= NULL
;
2278 isl_basic_set
*aff
= NULL
;
2279 isl_basic_map
*bmap
= NULL
;
2280 isl_vec
*obj
= NULL
;
2285 delta
= isl_map_deltas(isl_map_copy(map
));
2287 aff
= isl_set_affine_hull(isl_set_copy(delta
));
2290 dim
= isl_map_get_dim(map
);
2291 d
= isl_dim_size(dim
, isl_dim_in
);
2292 nparam
= isl_dim_size(dim
, isl_dim_param
);
2293 total
= isl_dim_total(dim
);
2294 bmap
= isl_basic_map_alloc_dim(dim
,
2295 aff
->n_div
+ 1, aff
->n_div
, 2 * d
+ 1);
2296 for (i
= 0; i
< aff
->n_div
+ 1; ++i
) {
2297 k
= isl_basic_map_alloc_div(bmap
);
2300 isl_int_set_si(bmap
->div
[k
][0], 0);
2302 for (i
= 0; i
< aff
->n_eq
; ++i
) {
2303 if (!is_eq_stride(aff
, i
))
2305 k
= isl_basic_map_alloc_equality(bmap
);
2308 isl_seq_clr(bmap
->eq
[k
], 1 + nparam
);
2309 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ d
,
2310 aff
->eq
[i
] + 1 + nparam
, d
);
2311 isl_seq_neg(bmap
->eq
[k
] + 1 + nparam
,
2312 aff
->eq
[i
] + 1 + nparam
, d
);
2313 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ 2 * d
,
2314 aff
->eq
[i
] + 1 + nparam
+ d
, aff
->n_div
);
2315 isl_int_set_si(bmap
->eq
[k
][1 + total
+ aff
->n_div
], 0);
2317 obj
= isl_vec_alloc(map
->ctx
, 1 + nparam
+ d
);
2320 isl_seq_clr(obj
->el
, 1 + nparam
+ d
);
2321 for (i
= 0; i
< d
; ++ i
) {
2322 enum isl_lp_result res
;
2324 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 1);
2326 res
= isl_set_solve_lp(delta
, 0, obj
->el
, map
->ctx
->one
, &opt
,
2328 if (res
== isl_lp_error
)
2330 if (res
== isl_lp_ok
) {
2331 k
= isl_basic_map_alloc_inequality(bmap
);
2334 isl_seq_clr(bmap
->ineq
[k
],
2335 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2336 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], -1);
2337 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], 1);
2338 isl_int_neg(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2341 res
= isl_set_solve_lp(delta
, 1, obj
->el
, map
->ctx
->one
, &opt
,
2343 if (res
== isl_lp_error
)
2345 if (res
== isl_lp_ok
) {
2346 k
= isl_basic_map_alloc_inequality(bmap
);
2349 isl_seq_clr(bmap
->ineq
[k
],
2350 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2351 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], 1);
2352 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], -1);
2353 isl_int_set(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2356 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 0);
2358 k
= isl_basic_map_alloc_inequality(bmap
);
2361 isl_seq_clr(bmap
->ineq
[k
],
2362 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2364 isl_int_set_si(bmap
->ineq
[k
][0], -1);
2365 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], 1);
2367 app
= isl_map_from_domain_and_range(dom
, ran
);
2370 isl_basic_set_free(aff
);
2372 bmap
= isl_basic_map_finalize(bmap
);
2373 isl_set_free(delta
);
2376 map
= isl_map_from_basic_map(bmap
);
2377 map
= isl_map_intersect(map
, app
);
2382 isl_basic_map_free(bmap
);
2383 isl_basic_set_free(aff
);
2387 isl_set_free(delta
);
2392 /* Given a map, compute the smallest superset of this map that is of the form
2394 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2396 * (where p ranges over the (non-parametric) dimensions),
2397 * compute the transitive closure of this map, i.e.,
2399 * { i -> j : exists k > 0:
2400 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2402 * and intersect domain and range of this transitive closure with
2403 * domain and range of the original map.
2405 static __isl_give isl_map
*box_closure(__isl_take isl_map
*map
)
2410 domain
= isl_map_domain(isl_map_copy(map
));
2411 domain
= isl_set_coalesce(domain
);
2412 range
= isl_map_range(isl_map_copy(map
));
2413 range
= isl_set_coalesce(range
);
2415 return box_closure_on_domain(map
, domain
, range
, 0);
2418 /* Given a map, compute the smallest superset of this map that is of the form
2420 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2422 * (where p ranges over the (non-parametric) dimensions),
2423 * compute the transitive and partially reflexive closure of this map, i.e.,
2425 * { i -> j : exists k >= 0:
2426 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2428 * and intersect domain and range of this transitive closure with
2431 static __isl_give isl_map
*box_closure_with_identity(__isl_take isl_map
*map
,
2432 __isl_take isl_set
*dom
)
2434 return box_closure_on_domain(map
, dom
, isl_set_copy(dom
), 1);
2437 /* Check whether app is the transitive closure of map.
2438 * In particular, check that app is acyclic and, if so,
2441 * app \subset (map \cup (map \circ app))
2443 static int check_exactness_omega(__isl_keep isl_map
*map
,
2444 __isl_keep isl_map
*app
)
2448 int is_empty
, is_exact
;
2452 delta
= isl_map_deltas(isl_map_copy(app
));
2453 d
= isl_set_dim(delta
, isl_dim_set
);
2454 for (i
= 0; i
< d
; ++i
)
2455 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
2456 is_empty
= isl_set_is_empty(delta
);
2457 isl_set_free(delta
);
2463 test
= isl_map_apply_range(isl_map_copy(app
), isl_map_copy(map
));
2464 test
= isl_map_union(test
, isl_map_copy(map
));
2465 is_exact
= isl_map_is_subset(app
, test
);
2471 /* Check if basic map M_i can be combined with all the other
2472 * basic maps such that
2476 * can be computed as
2478 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2480 * In particular, check if we can compute a compact representation
2483 * M_i^* \circ M_j \circ M_i^*
2486 * Let M_i^? be an extension of M_i^+ that allows paths
2487 * of length zero, i.e., the result of box_closure(., 1).
2488 * The criterion, as proposed by Kelly et al., is that
2489 * id = M_i^? - M_i^+ can be represented as a basic map
2492 * id \circ M_j \circ id = M_j
2496 * If this function returns 1, then tc and qc are set to
2497 * M_i^+ and M_i^?, respectively.
2499 static int can_be_split_off(__isl_keep isl_map
*map
, int i
,
2500 __isl_give isl_map
**tc
, __isl_give isl_map
**qc
)
2502 isl_map
*map_i
, *id
= NULL
;
2509 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
2510 isl_map_range(isl_map_copy(map
)));
2511 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
2515 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
2516 *tc
= box_closure(isl_map_copy(map_i
));
2517 *qc
= box_closure_with_identity(map_i
, C
);
2518 id
= isl_map_subtract(isl_map_copy(*qc
), isl_map_copy(*tc
));
2522 if (id
->n
!= 1 || (*qc
)->n
!= 1)
2525 for (j
= 0; j
< map
->n
; ++j
) {
2526 isl_map
*map_j
, *test
;
2531 map_j
= isl_map_from_basic_map(
2532 isl_basic_map_copy(map
->p
[j
]));
2533 test
= isl_map_apply_range(isl_map_copy(id
),
2534 isl_map_copy(map_j
));
2535 test
= isl_map_apply_range(test
, isl_map_copy(id
));
2536 is_ok
= isl_map_is_equal(test
, map_j
);
2537 isl_map_free(map_j
);
2565 static __isl_give isl_map
*box_closure_with_check(__isl_take isl_map
*map
,
2570 app
= box_closure(isl_map_copy(map
));
2572 *exact
= check_exactness_omega(map
, app
);
2578 /* Compute an overapproximation of the transitive closure of "map"
2579 * using a variation of the algorithm from
2580 * "Transitive Closure of Infinite Graphs and its Applications"
2583 * We first check whether we can can split of any basic map M_i and
2590 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2592 * using a recursive call on the remaining map.
2594 * If not, we simply call box_closure on the whole map.
2596 static __isl_give isl_map
*transitive_closure_omega(__isl_take isl_map
*map
,
2606 return box_closure_with_check(map
, exact
);
2608 for (i
= 0; i
< map
->n
; ++i
) {
2611 ok
= can_be_split_off(map
, i
, &tc
, &qc
);
2617 app
= isl_map_alloc_dim(isl_map_get_dim(map
), map
->n
- 1, 0);
2619 for (j
= 0; j
< map
->n
; ++j
) {
2622 app
= isl_map_add_basic_map(app
,
2623 isl_basic_map_copy(map
->p
[j
]));
2626 app
= isl_map_apply_range(isl_map_copy(qc
), app
);
2627 app
= isl_map_apply_range(app
, qc
);
2629 app
= isl_map_union(tc
, transitive_closure_omega(app
, NULL
));
2630 exact_i
= check_exactness_omega(map
, app
);
2642 return box_closure_with_check(map
, exact
);
2648 /* Compute the transitive closure of "map", or an overapproximation.
2649 * If the result is exact, then *exact is set to 1.
2650 * Simply use map_power to compute the powers of map, but tell
2651 * it to project out the lengths of the paths instead of equating
2652 * the length to a parameter.
2654 __isl_give isl_map
*isl_map_transitive_closure(__isl_take isl_map
*map
,
2657 isl_dim
*target_dim
;
2663 if (map
->ctx
->opt
->closure
== ISL_CLOSURE_OMEGA
)
2664 return transitive_closure_omega(map
, exact
);
2666 map
= isl_map_compute_divs(map
);
2667 map
= isl_map_coalesce(map
);
2668 closed
= isl_map_is_transitively_closed(map
);
2677 target_dim
= isl_map_get_dim(map
);
2678 map
= map_power(map
, exact
, 1);
2679 map
= isl_map_reset_dim(map
, target_dim
);
2687 static int inc_count(__isl_take isl_map
*map
, void *user
)
2698 static int collect_basic_map(__isl_take isl_map
*map
, void *user
)
2701 isl_basic_map
***next
= user
;
2703 for (i
= 0; i
< map
->n
; ++i
) {
2704 **next
= isl_basic_map_copy(map
->p
[i
]);
2717 /* Perform Floyd-Warshall on the given list of basic relations.
2718 * The basic relations may live in different dimensions,
2719 * but basic relations that get assigned to the diagonal of the
2720 * grid have domains and ranges of the same dimension and so
2721 * the standard algorithm can be used because the nested transitive
2722 * closures are only applied to diagonal elements and because all
2723 * compositions are peformed on relations with compatible domains and ranges.
2725 static __isl_give isl_union_map
*union_floyd_warshall_on_list(isl_ctx
*ctx
,
2726 __isl_keep isl_basic_map
**list
, int n
, int *exact
)
2731 isl_set
**set
= NULL
;
2732 isl_map
***grid
= NULL
;
2735 group
= setup_groups(ctx
, list
, n
, &set
, &n_group
);
2739 grid
= isl_calloc_array(ctx
, isl_map
**, n_group
);
2742 for (i
= 0; i
< n_group
; ++i
) {
2743 grid
[i
] = isl_calloc_array(map
->ctx
, isl_map
*, n_group
);
2746 for (j
= 0; j
< n_group
; ++j
) {
2747 isl_dim
*dim1
, *dim2
, *dim
;
2748 dim1
= isl_dim_reverse(isl_set_get_dim(set
[i
]));
2749 dim2
= isl_set_get_dim(set
[j
]);
2750 dim
= isl_dim_join(dim1
, dim2
);
2751 grid
[i
][j
] = isl_map_empty(dim
);
2755 for (k
= 0; k
< n
; ++k
) {
2757 j
= group
[2 * k
+ 1];
2758 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
2759 isl_map_from_basic_map(
2760 isl_basic_map_copy(list
[k
])));
2763 floyd_warshall_iterate(grid
, n_group
, exact
);
2765 app
= isl_union_map_empty(isl_map_get_dim(grid
[0][0]));
2767 for (i
= 0; i
< n_group
; ++i
) {
2768 for (j
= 0; j
< n_group
; ++j
)
2769 app
= isl_union_map_add_map(app
, grid
[i
][j
]);
2774 for (i
= 0; i
< 2 * n
; ++i
)
2775 isl_set_free(set
[i
]);
2782 for (i
= 0; i
< n_group
; ++i
) {
2785 for (j
= 0; j
< n_group
; ++j
)
2786 isl_map_free(grid
[i
][j
]);
2791 for (i
= 0; i
< 2 * n
; ++i
)
2792 isl_set_free(set
[i
]);
2799 /* Perform Floyd-Warshall on the given union relation.
2800 * The implementation is very similar to that for non-unions.
2801 * The main difference is that it is applied unconditionally.
2802 * We first extract a list of basic maps from the union map
2803 * and then perform the algorithm on this list.
2805 static __isl_give isl_union_map
*union_floyd_warshall(
2806 __isl_take isl_union_map
*umap
, int *exact
)
2810 isl_basic_map
**list
;
2811 isl_basic_map
**next
;
2815 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2818 ctx
= isl_union_map_get_ctx(umap
);
2819 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2824 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2827 res
= union_floyd_warshall_on_list(ctx
, list
, n
, exact
);
2830 for (i
= 0; i
< n
; ++i
)
2831 isl_basic_map_free(list
[i
]);
2835 isl_union_map_free(umap
);
2839 for (i
= 0; i
< n
; ++i
)
2840 isl_basic_map_free(list
[i
]);
2843 isl_union_map_free(umap
);
2847 /* Decompose the give union relation into strongly connected components.
2848 * The implementation is essentially the same as that of
2849 * construct_power_components with the major difference that all
2850 * operations are performed on union maps.
2852 static __isl_give isl_union_map
*union_components(
2853 __isl_take isl_union_map
*umap
, int *exact
)
2858 isl_basic_map
**list
;
2859 isl_basic_map
**next
;
2860 isl_union_map
*path
= NULL
;
2861 struct basic_map_sort
*s
= NULL
;
2866 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2870 return union_floyd_warshall(umap
, exact
);
2872 ctx
= isl_union_map_get_ctx(umap
);
2873 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2878 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2881 s
= basic_map_sort_init(ctx
, n
, list
);
2888 path
= isl_union_map_empty(isl_union_map_get_dim(umap
));
2890 isl_union_map
*comp
;
2891 isl_union_map
*path_comp
, *path_comb
;
2892 comp
= isl_union_map_empty(isl_union_map_get_dim(umap
));
2893 while (s
->order
[i
] != -1) {
2894 comp
= isl_union_map_add_map(comp
,
2895 isl_map_from_basic_map(
2896 isl_basic_map_copy(list
[s
->order
[i
]])));
2900 path_comp
= union_floyd_warshall(comp
, exact
);
2901 path_comb
= isl_union_map_apply_range(isl_union_map_copy(path
),
2902 isl_union_map_copy(path_comp
));
2903 path
= isl_union_map_union(path
, path_comp
);
2904 path
= isl_union_map_union(path
, path_comb
);
2909 if (c
> 1 && s
->check_closed
&& !*exact
) {
2912 closed
= isl_union_map_is_transitively_closed(path
);
2918 basic_map_sort_free(s
);
2920 for (i
= 0; i
< n
; ++i
)
2921 isl_basic_map_free(list
[i
]);
2925 isl_union_map_free(path
);
2926 return union_floyd_warshall(umap
, exact
);
2929 isl_union_map_free(umap
);
2933 basic_map_sort_free(s
);
2935 for (i
= 0; i
< n
; ++i
)
2936 isl_basic_map_free(list
[i
]);
2939 isl_union_map_free(umap
);
2940 isl_union_map_free(path
);
2944 /* Compute the transitive closure of "umap", or an overapproximation.
2945 * If the result is exact, then *exact is set to 1.
2947 __isl_give isl_union_map
*isl_union_map_transitive_closure(
2948 __isl_take isl_union_map
*umap
, int *exact
)
2958 umap
= isl_union_map_compute_divs(umap
);
2959 umap
= isl_union_map_coalesce(umap
);
2960 closed
= isl_union_map_is_transitively_closed(umap
);
2965 umap
= union_components(umap
, exact
);
2968 isl_union_map_free(umap
);