2 * Copyright 2010 INRIA Saclay
4 * Use of this software is governed by the MIT license
6 * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
7 * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
11 #include <isl_ctx_private.h>
12 #include <isl_map_private.h>
15 #include <isl_space_private.h>
16 #include <isl_lp_private.h>
17 #include <isl/union_map.h>
18 #include <isl_mat_private.h>
19 #include <isl_vec_private.h>
20 #include <isl_options_private.h>
21 #include <isl_tarjan.h>
23 int isl_map_is_transitively_closed(__isl_keep isl_map
*map
)
28 map2
= isl_map_apply_range(isl_map_copy(map
), isl_map_copy(map
));
29 closed
= isl_map_is_subset(map2
, map
);
35 int isl_union_map_is_transitively_closed(__isl_keep isl_union_map
*umap
)
40 umap2
= isl_union_map_apply_range(isl_union_map_copy(umap
),
41 isl_union_map_copy(umap
));
42 closed
= isl_union_map_is_subset(umap2
, umap
);
43 isl_union_map_free(umap2
);
48 /* Given a map that represents a path with the length of the path
49 * encoded as the difference between the last output coordindate
50 * and the last input coordinate, set this length to either
51 * exactly "length" (if "exactly" is set) or at least "length"
52 * (if "exactly" is not set).
54 static __isl_give isl_map
*set_path_length(__isl_take isl_map
*map
,
55 int exactly
, int length
)
58 struct isl_basic_map
*bmap
;
67 dim
= isl_map_get_space(map
);
68 d
= isl_space_dim(dim
, isl_dim_in
);
69 nparam
= isl_space_dim(dim
, isl_dim_param
);
70 bmap
= isl_basic_map_alloc_space(dim
, 0, 1, 1);
72 k
= isl_basic_map_alloc_equality(bmap
);
77 k
= isl_basic_map_alloc_inequality(bmap
);
82 isl_seq_clr(c
, 1 + isl_basic_map_total_dim(bmap
));
83 isl_int_set_si(c
[0], -length
);
84 isl_int_set_si(c
[1 + nparam
+ d
- 1], -1);
85 isl_int_set_si(c
[1 + nparam
+ d
+ d
- 1], 1);
87 bmap
= isl_basic_map_finalize(bmap
);
88 map
= isl_map_intersect(map
, isl_map_from_basic_map(bmap
));
92 isl_basic_map_free(bmap
);
97 /* Check whether the overapproximation of the power of "map" is exactly
98 * the power of "map". Let R be "map" and A_k the overapproximation.
99 * The approximation is exact if
102 * A_k = A_{k-1} \circ R k >= 2
104 * Since A_k is known to be an overapproximation, we only need to check
107 * A_k \subset A_{k-1} \circ R k >= 2
109 * In practice, "app" has an extra input and output coordinate
110 * to encode the length of the path. So, we first need to add
111 * this coordinate to "map" and set the length of the path to
114 static int check_power_exactness(__isl_take isl_map
*map
,
115 __isl_take isl_map
*app
)
121 map
= isl_map_add_dims(map
, isl_dim_in
, 1);
122 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
123 map
= set_path_length(map
, 1, 1);
125 app_1
= set_path_length(isl_map_copy(app
), 1, 1);
127 exact
= isl_map_is_subset(app_1
, map
);
130 if (!exact
|| exact
< 0) {
136 app_1
= set_path_length(isl_map_copy(app
), 0, 1);
137 app_2
= set_path_length(app
, 0, 2);
138 app_1
= isl_map_apply_range(map
, app_1
);
140 exact
= isl_map_is_subset(app_2
, app_1
);
148 /* Check whether the overapproximation of the power of "map" is exactly
149 * the power of "map", possibly after projecting out the power (if "project"
152 * If "project" is set and if "steps" can only result in acyclic paths,
155 * A = R \cup (A \circ R)
157 * where A is the overapproximation with the power projected out, i.e.,
158 * an overapproximation of the transitive closure.
159 * More specifically, since A is known to be an overapproximation, we check
161 * A \subset R \cup (A \circ R)
163 * Otherwise, we check if the power is exact.
165 * Note that "app" has an extra input and output coordinate to encode
166 * the length of the part. If we are only interested in the transitive
167 * closure, then we can simply project out these coordinates first.
169 static int check_exactness(__isl_take isl_map
*map
, __isl_take isl_map
*app
,
177 return check_power_exactness(map
, app
);
179 d
= isl_map_dim(map
, isl_dim_in
);
180 app
= set_path_length(app
, 0, 1);
181 app
= isl_map_project_out(app
, isl_dim_in
, d
, 1);
182 app
= isl_map_project_out(app
, isl_dim_out
, d
, 1);
184 app
= isl_map_reset_space(app
, isl_map_get_space(map
));
186 test
= isl_map_apply_range(isl_map_copy(map
), isl_map_copy(app
));
187 test
= isl_map_union(test
, isl_map_copy(map
));
189 exact
= isl_map_is_subset(app
, test
);
200 * The transitive closure implementation is based on the paper
201 * "Computing the Transitive Closure of a Union of Affine Integer
202 * Tuple Relations" by Anna Beletska, Denis Barthou, Wlodzimierz Bielecki and
206 /* Given a set of n offsets v_i (the rows of "steps"), construct a relation
207 * of the given dimension specification (Z^{n+1} -> Z^{n+1})
208 * that maps an element x to any element that can be reached
209 * by taking a non-negative number of steps along any of
210 * the extended offsets v'_i = [v_i 1].
213 * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v'_i }
215 * For any element in this relation, the number of steps taken
216 * is equal to the difference in the final coordinates.
218 static __isl_give isl_map
*path_along_steps(__isl_take isl_space
*dim
,
219 __isl_keep isl_mat
*steps
)
222 struct isl_basic_map
*path
= NULL
;
230 d
= isl_space_dim(dim
, isl_dim_in
);
232 nparam
= isl_space_dim(dim
, isl_dim_param
);
234 path
= isl_basic_map_alloc_space(isl_space_copy(dim
), n
, d
, n
);
236 for (i
= 0; i
< n
; ++i
) {
237 k
= isl_basic_map_alloc_div(path
);
240 isl_assert(steps
->ctx
, i
== k
, goto error
);
241 isl_int_set_si(path
->div
[k
][0], 0);
244 for (i
= 0; i
< d
; ++i
) {
245 k
= isl_basic_map_alloc_equality(path
);
248 isl_seq_clr(path
->eq
[k
], 1 + isl_basic_map_total_dim(path
));
249 isl_int_set_si(path
->eq
[k
][1 + nparam
+ i
], 1);
250 isl_int_set_si(path
->eq
[k
][1 + nparam
+ d
+ i
], -1);
252 for (j
= 0; j
< n
; ++j
)
253 isl_int_set_si(path
->eq
[k
][1 + nparam
+ 2 * d
+ j
], 1);
255 for (j
= 0; j
< n
; ++j
)
256 isl_int_set(path
->eq
[k
][1 + nparam
+ 2 * d
+ j
],
260 for (i
= 0; i
< n
; ++i
) {
261 k
= isl_basic_map_alloc_inequality(path
);
264 isl_seq_clr(path
->ineq
[k
], 1 + isl_basic_map_total_dim(path
));
265 isl_int_set_si(path
->ineq
[k
][1 + nparam
+ 2 * d
+ i
], 1);
270 path
= isl_basic_map_simplify(path
);
271 path
= isl_basic_map_finalize(path
);
272 return isl_map_from_basic_map(path
);
275 isl_basic_map_free(path
);
284 /* Check whether the parametric constant term of constraint c is never
285 * positive in "bset".
287 static int parametric_constant_never_positive(__isl_keep isl_basic_set
*bset
,
288 isl_int
*c
, int *div_purity
)
297 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
298 d
= isl_basic_set_dim(bset
, isl_dim_set
);
299 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
301 bset
= isl_basic_set_copy(bset
);
302 bset
= isl_basic_set_cow(bset
);
303 bset
= isl_basic_set_extend_constraints(bset
, 0, 1);
304 k
= isl_basic_set_alloc_inequality(bset
);
307 isl_seq_clr(bset
->ineq
[k
], 1 + isl_basic_set_total_dim(bset
));
308 isl_seq_cpy(bset
->ineq
[k
], c
, 1 + nparam
);
309 for (i
= 0; i
< n_div
; ++i
) {
310 if (div_purity
[i
] != PURE_PARAM
)
312 isl_int_set(bset
->ineq
[k
][1 + nparam
+ d
+ i
],
313 c
[1 + nparam
+ d
+ i
]);
315 isl_int_sub_ui(bset
->ineq
[k
][0], bset
->ineq
[k
][0], 1);
316 empty
= isl_basic_set_is_empty(bset
);
317 isl_basic_set_free(bset
);
321 isl_basic_set_free(bset
);
325 /* Return PURE_PARAM if only the coefficients of the parameters are non-zero.
326 * Return PURE_VAR if only the coefficients of the set variables are non-zero.
327 * Return MIXED if only the coefficients of the parameters and the set
328 * variables are non-zero and if moreover the parametric constant
329 * can never attain positive values.
330 * Return IMPURE otherwise.
332 static int purity(__isl_keep isl_basic_set
*bset
, isl_int
*c
, int *div_purity
,
342 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
343 d
= isl_basic_set_dim(bset
, isl_dim_set
);
344 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
346 for (i
= 0; i
< n_div
; ++i
) {
347 if (isl_int_is_zero(c
[1 + nparam
+ d
+ i
]))
349 switch (div_purity
[i
]) {
350 case PURE_PARAM
: p
= 1; break;
351 case PURE_VAR
: v
= 1; break;
352 default: return IMPURE
;
355 if (!p
&& isl_seq_first_non_zero(c
+ 1, nparam
) == -1)
357 if (!v
&& isl_seq_first_non_zero(c
+ 1 + nparam
, d
) == -1)
360 empty
= parametric_constant_never_positive(bset
, c
, div_purity
);
361 if (eq
&& empty
>= 0 && !empty
) {
362 isl_seq_neg(c
, c
, 1 + nparam
+ d
+ n_div
);
363 empty
= parametric_constant_never_positive(bset
, c
, div_purity
);
366 return empty
< 0 ? -1 : empty
? MIXED
: IMPURE
;
369 /* Return an array of integers indicating the type of each div in bset.
370 * If the div is (recursively) defined in terms of only the parameters,
371 * then the type is PURE_PARAM.
372 * If the div is (recursively) defined in terms of only the set variables,
373 * then the type is PURE_VAR.
374 * Otherwise, the type is IMPURE.
376 static __isl_give
int *get_div_purity(__isl_keep isl_basic_set
*bset
)
387 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
388 d
= isl_basic_set_dim(bset
, isl_dim_set
);
389 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
391 div_purity
= isl_alloc_array(bset
->ctx
, int, n_div
);
392 if (n_div
&& !div_purity
)
395 for (i
= 0; i
< bset
->n_div
; ++i
) {
397 if (isl_int_is_zero(bset
->div
[i
][0])) {
398 div_purity
[i
] = IMPURE
;
401 if (isl_seq_first_non_zero(bset
->div
[i
] + 2, nparam
) != -1)
403 if (isl_seq_first_non_zero(bset
->div
[i
] + 2 + nparam
, d
) != -1)
405 for (j
= 0; j
< i
; ++j
) {
406 if (isl_int_is_zero(bset
->div
[i
][2 + nparam
+ d
+ j
]))
408 switch (div_purity
[j
]) {
409 case PURE_PARAM
: p
= 1; break;
410 case PURE_VAR
: v
= 1; break;
411 default: p
= v
= 1; break;
414 div_purity
[i
] = v
? p
? IMPURE
: PURE_VAR
: PURE_PARAM
;
420 /* Given a path with the as yet unconstrained length at position "pos",
421 * check if setting the length to zero results in only the identity
424 static int empty_path_is_identity(__isl_keep isl_basic_map
*path
, unsigned pos
)
426 isl_basic_map
*test
= NULL
;
427 isl_basic_map
*id
= NULL
;
431 test
= isl_basic_map_copy(path
);
432 test
= isl_basic_map_extend_constraints(test
, 1, 0);
433 k
= isl_basic_map_alloc_equality(test
);
436 isl_seq_clr(test
->eq
[k
], 1 + isl_basic_map_total_dim(test
));
437 isl_int_set_si(test
->eq
[k
][pos
], 1);
438 id
= isl_basic_map_identity(isl_basic_map_get_space(path
));
439 is_id
= isl_basic_map_is_equal(test
, id
);
440 isl_basic_map_free(test
);
441 isl_basic_map_free(id
);
444 isl_basic_map_free(test
);
448 /* If any of the constraints is found to be impure then this function
449 * sets *impurity to 1.
451 * If impurity is NULL then we are dealing with a non-parametric set
452 * and so the constraints are obviously PURE_VAR.
454 static __isl_give isl_basic_map
*add_delta_constraints(
455 __isl_take isl_basic_map
*path
,
456 __isl_keep isl_basic_set
*delta
, unsigned off
, unsigned nparam
,
457 unsigned d
, int *div_purity
, int eq
, int *impurity
)
460 int n
= eq
? delta
->n_eq
: delta
->n_ineq
;
461 isl_int
**delta_c
= eq
? delta
->eq
: delta
->ineq
;
464 n_div
= isl_basic_set_dim(delta
, isl_dim_div
);
466 for (i
= 0; i
< n
; ++i
) {
470 p
= purity(delta
, delta_c
[i
], div_purity
, eq
);
473 if (p
!= PURE_VAR
&& p
!= PURE_PARAM
&& !*impurity
)
477 if (eq
&& p
!= MIXED
) {
478 k
= isl_basic_map_alloc_equality(path
);
479 path_c
= path
->eq
[k
];
481 k
= isl_basic_map_alloc_inequality(path
);
482 path_c
= path
->ineq
[k
];
486 isl_seq_clr(path_c
, 1 + isl_basic_map_total_dim(path
));
488 isl_seq_cpy(path_c
+ off
,
489 delta_c
[i
] + 1 + nparam
, d
);
490 isl_int_set(path_c
[off
+ d
], delta_c
[i
][0]);
491 } else if (p
== PURE_PARAM
) {
492 isl_seq_cpy(path_c
, delta_c
[i
], 1 + nparam
);
494 isl_seq_cpy(path_c
+ off
,
495 delta_c
[i
] + 1 + nparam
, d
);
496 isl_seq_cpy(path_c
, delta_c
[i
], 1 + nparam
);
498 isl_seq_cpy(path_c
+ off
- n_div
,
499 delta_c
[i
] + 1 + nparam
+ d
, n_div
);
504 isl_basic_map_free(path
);
508 /* Given a set of offsets "delta", construct a relation of the
509 * given dimension specification (Z^{n+1} -> Z^{n+1}) that
510 * is an overapproximation of the relations that
511 * maps an element x to any element that can be reached
512 * by taking a non-negative number of steps along any of
513 * the elements in "delta".
514 * That is, construct an approximation of
516 * { [x] -> [y] : exists f \in \delta, k \in Z :
517 * y = x + k [f, 1] and k >= 0 }
519 * For any element in this relation, the number of steps taken
520 * is equal to the difference in the final coordinates.
522 * In particular, let delta be defined as
524 * \delta = [p] -> { [x] : A x + a >= 0 and B p + b >= 0 and
525 * C x + C'p + c >= 0 and
526 * D x + D'p + d >= 0 }
528 * where the constraints C x + C'p + c >= 0 are such that the parametric
529 * constant term of each constraint j, "C_j x + C'_j p + c_j",
530 * can never attain positive values, then the relation is constructed as
532 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
533 * A f + k a >= 0 and B p + b >= 0 and
534 * C f + C'p + c >= 0 and k >= 1 }
535 * union { [x] -> [x] }
537 * If the zero-length paths happen to correspond exactly to the identity
538 * mapping, then we return
540 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
541 * A f + k a >= 0 and B p + b >= 0 and
542 * C f + C'p + c >= 0 and k >= 0 }
546 * Existentially quantified variables in \delta are handled by
547 * classifying them as independent of the parameters, purely
548 * parameter dependent and others. Constraints containing
549 * any of the other existentially quantified variables are removed.
550 * This is safe, but leads to an additional overapproximation.
552 * If there are any impure constraints, then we also eliminate
553 * the parameters from \delta, resulting in a set
555 * \delta' = { [x] : E x + e >= 0 }
557 * and add the constraints
561 * to the constructed relation.
563 static __isl_give isl_map
*path_along_delta(__isl_take isl_space
*dim
,
564 __isl_take isl_basic_set
*delta
)
566 isl_basic_map
*path
= NULL
;
573 int *div_purity
= NULL
;
578 n_div
= isl_basic_set_dim(delta
, isl_dim_div
);
579 d
= isl_basic_set_dim(delta
, isl_dim_set
);
580 nparam
= isl_basic_set_dim(delta
, isl_dim_param
);
581 path
= isl_basic_map_alloc_space(isl_space_copy(dim
), n_div
+ d
+ 1,
582 d
+ 1 + delta
->n_eq
, delta
->n_eq
+ delta
->n_ineq
+ 1);
583 off
= 1 + nparam
+ 2 * (d
+ 1) + n_div
;
585 for (i
= 0; i
< n_div
+ d
+ 1; ++i
) {
586 k
= isl_basic_map_alloc_div(path
);
589 isl_int_set_si(path
->div
[k
][0], 0);
592 for (i
= 0; i
< d
+ 1; ++i
) {
593 k
= isl_basic_map_alloc_equality(path
);
596 isl_seq_clr(path
->eq
[k
], 1 + isl_basic_map_total_dim(path
));
597 isl_int_set_si(path
->eq
[k
][1 + nparam
+ i
], 1);
598 isl_int_set_si(path
->eq
[k
][1 + nparam
+ d
+ 1 + i
], -1);
599 isl_int_set_si(path
->eq
[k
][off
+ i
], 1);
602 div_purity
= get_div_purity(delta
);
603 if (n_div
&& !div_purity
)
606 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
607 div_purity
, 1, &impurity
);
608 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
609 div_purity
, 0, &impurity
);
611 isl_space
*dim
= isl_basic_set_get_space(delta
);
612 delta
= isl_basic_set_project_out(delta
,
613 isl_dim_param
, 0, nparam
);
614 delta
= isl_basic_set_add_dims(delta
, isl_dim_param
, nparam
);
615 delta
= isl_basic_set_reset_space(delta
, dim
);
618 path
= isl_basic_map_extend_constraints(path
, delta
->n_eq
,
620 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
622 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
624 path
= isl_basic_map_gauss(path
, NULL
);
627 is_id
= empty_path_is_identity(path
, off
+ d
);
631 k
= isl_basic_map_alloc_inequality(path
);
634 isl_seq_clr(path
->ineq
[k
], 1 + isl_basic_map_total_dim(path
));
636 isl_int_set_si(path
->ineq
[k
][0], -1);
637 isl_int_set_si(path
->ineq
[k
][off
+ d
], 1);
640 isl_basic_set_free(delta
);
641 path
= isl_basic_map_finalize(path
);
644 return isl_map_from_basic_map(path
);
646 return isl_basic_map_union(path
, isl_basic_map_identity(dim
));
650 isl_basic_set_free(delta
);
651 isl_basic_map_free(path
);
655 /* Given a dimension specification Z^{n+1} -> Z^{n+1} and a parameter "param",
656 * construct a map that equates the parameter to the difference
657 * in the final coordinates and imposes that this difference is positive.
660 * { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
662 static __isl_give isl_map
*equate_parameter_to_length(__isl_take isl_space
*dim
,
665 struct isl_basic_map
*bmap
;
670 d
= isl_space_dim(dim
, isl_dim_in
);
671 nparam
= isl_space_dim(dim
, isl_dim_param
);
672 bmap
= isl_basic_map_alloc_space(dim
, 0, 1, 1);
673 k
= isl_basic_map_alloc_equality(bmap
);
676 isl_seq_clr(bmap
->eq
[k
], 1 + isl_basic_map_total_dim(bmap
));
677 isl_int_set_si(bmap
->eq
[k
][1 + param
], -1);
678 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
- 1], -1);
679 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
+ d
- 1], 1);
681 k
= isl_basic_map_alloc_inequality(bmap
);
684 isl_seq_clr(bmap
->ineq
[k
], 1 + isl_basic_map_total_dim(bmap
));
685 isl_int_set_si(bmap
->ineq
[k
][1 + param
], 1);
686 isl_int_set_si(bmap
->ineq
[k
][0], -1);
688 bmap
= isl_basic_map_finalize(bmap
);
689 return isl_map_from_basic_map(bmap
);
691 isl_basic_map_free(bmap
);
695 /* Check whether "path" is acyclic, where the last coordinates of domain
696 * and range of path encode the number of steps taken.
697 * That is, check whether
699 * { d | d = y - x and (x,y) in path }
701 * does not contain any element with positive last coordinate (positive length)
702 * and zero remaining coordinates (cycle).
704 static int is_acyclic(__isl_take isl_map
*path
)
709 struct isl_set
*delta
;
711 delta
= isl_map_deltas(path
);
712 dim
= isl_set_dim(delta
, isl_dim_set
);
713 for (i
= 0; i
< dim
; ++i
) {
715 delta
= isl_set_lower_bound_si(delta
, isl_dim_set
, i
, 1);
717 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
720 acyclic
= isl_set_is_empty(delta
);
726 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
727 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
728 * construct a map that is an overapproximation of the map
729 * that takes an element from the space D \times Z to another
730 * element from the same space, such that the first n coordinates of the
731 * difference between them is a sum of differences between images
732 * and pre-images in one of the R_i and such that the last coordinate
733 * is equal to the number of steps taken.
736 * \Delta_i = { y - x | (x, y) in R_i }
738 * then the constructed map is an overapproximation of
740 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
741 * d = (\sum_i k_i \delta_i, \sum_i k_i) }
743 * The elements of the singleton \Delta_i's are collected as the
744 * rows of the steps matrix. For all these \Delta_i's together,
745 * a single path is constructed.
746 * For each of the other \Delta_i's, we compute an overapproximation
747 * of the paths along elements of \Delta_i.
748 * Since each of these paths performs an addition, composition is
749 * symmetric and we can simply compose all resulting paths in any order.
751 static __isl_give isl_map
*construct_extended_path(__isl_take isl_space
*dim
,
752 __isl_keep isl_map
*map
, int *project
)
754 struct isl_mat
*steps
= NULL
;
755 struct isl_map
*path
= NULL
;
759 d
= isl_map_dim(map
, isl_dim_in
);
761 path
= isl_map_identity(isl_space_copy(dim
));
763 steps
= isl_mat_alloc(map
->ctx
, map
->n
, d
);
768 for (i
= 0; i
< map
->n
; ++i
) {
769 struct isl_basic_set
*delta
;
771 delta
= isl_basic_map_deltas(isl_basic_map_copy(map
->p
[i
]));
773 for (j
= 0; j
< d
; ++j
) {
776 fixed
= isl_basic_set_plain_dim_is_fixed(delta
, j
,
779 isl_basic_set_free(delta
);
788 path
= isl_map_apply_range(path
,
789 path_along_delta(isl_space_copy(dim
), delta
));
790 path
= isl_map_coalesce(path
);
792 isl_basic_set_free(delta
);
799 path
= isl_map_apply_range(path
,
800 path_along_steps(isl_space_copy(dim
), steps
));
803 if (project
&& *project
) {
804 *project
= is_acyclic(isl_map_copy(path
));
819 static int isl_set_overlaps(__isl_keep isl_set
*set1
, __isl_keep isl_set
*set2
)
824 if (!isl_space_tuple_is_equal(set1
->dim
, isl_dim_set
,
825 set2
->dim
, isl_dim_set
))
828 i
= isl_set_intersect(isl_set_copy(set1
), isl_set_copy(set2
));
829 no_overlap
= isl_set_is_empty(i
);
832 return no_overlap
< 0 ? -1 : !no_overlap
;
835 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
836 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
837 * construct a map that is an overapproximation of the map
838 * that takes an element from the dom R \times Z to an
839 * element from ran R \times Z, such that the first n coordinates of the
840 * difference between them is a sum of differences between images
841 * and pre-images in one of the R_i and such that the last coordinate
842 * is equal to the number of steps taken.
845 * \Delta_i = { y - x | (x, y) in R_i }
847 * then the constructed map is an overapproximation of
849 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
850 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
851 * x in dom R and x + d in ran R and
854 static __isl_give isl_map
*construct_component(__isl_take isl_space
*dim
,
855 __isl_keep isl_map
*map
, int *exact
, int project
)
857 struct isl_set
*domain
= NULL
;
858 struct isl_set
*range
= NULL
;
859 struct isl_map
*app
= NULL
;
860 struct isl_map
*path
= NULL
;
862 domain
= isl_map_domain(isl_map_copy(map
));
863 domain
= isl_set_coalesce(domain
);
864 range
= isl_map_range(isl_map_copy(map
));
865 range
= isl_set_coalesce(range
);
866 if (!isl_set_overlaps(domain
, range
)) {
867 isl_set_free(domain
);
871 map
= isl_map_copy(map
);
872 map
= isl_map_add_dims(map
, isl_dim_in
, 1);
873 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
874 map
= set_path_length(map
, 1, 1);
877 app
= isl_map_from_domain_and_range(domain
, range
);
878 app
= isl_map_add_dims(app
, isl_dim_in
, 1);
879 app
= isl_map_add_dims(app
, isl_dim_out
, 1);
881 path
= construct_extended_path(isl_space_copy(dim
), map
,
882 exact
&& *exact
? &project
: NULL
);
883 app
= isl_map_intersect(app
, path
);
885 if (exact
&& *exact
&&
886 (*exact
= check_exactness(isl_map_copy(map
), isl_map_copy(app
),
891 app
= set_path_length(app
, 0, 1);
899 /* Call construct_component and, if "project" is set, project out
900 * the final coordinates.
902 static __isl_give isl_map
*construct_projected_component(
903 __isl_take isl_space
*dim
,
904 __isl_keep isl_map
*map
, int *exact
, int project
)
911 d
= isl_space_dim(dim
, isl_dim_in
);
913 app
= construct_component(dim
, map
, exact
, project
);
915 app
= isl_map_project_out(app
, isl_dim_in
, d
- 1, 1);
916 app
= isl_map_project_out(app
, isl_dim_out
, d
- 1, 1);
921 /* Compute an extended version, i.e., with path lengths, of
922 * an overapproximation of the transitive closure of "bmap"
923 * with path lengths greater than or equal to zero and with
924 * domain and range equal to "dom".
926 static __isl_give isl_map
*q_closure(__isl_take isl_space
*dim
,
927 __isl_take isl_set
*dom
, __isl_keep isl_basic_map
*bmap
, int *exact
)
934 dom
= isl_set_add_dims(dom
, isl_dim_set
, 1);
935 app
= isl_map_from_domain_and_range(dom
, isl_set_copy(dom
));
936 map
= isl_map_from_basic_map(isl_basic_map_copy(bmap
));
937 path
= construct_extended_path(dim
, map
, &project
);
938 app
= isl_map_intersect(app
, path
);
940 if ((*exact
= check_exactness(map
, isl_map_copy(app
), project
)) < 0)
949 /* Check whether qc has any elements of length at least one
950 * with domain and/or range outside of dom and ran.
952 static int has_spurious_elements(__isl_keep isl_map
*qc
,
953 __isl_keep isl_set
*dom
, __isl_keep isl_set
*ran
)
959 if (!qc
|| !dom
|| !ran
)
962 d
= isl_map_dim(qc
, isl_dim_in
);
964 qc
= isl_map_copy(qc
);
965 qc
= set_path_length(qc
, 0, 1);
966 qc
= isl_map_project_out(qc
, isl_dim_in
, d
- 1, 1);
967 qc
= isl_map_project_out(qc
, isl_dim_out
, d
- 1, 1);
969 s
= isl_map_domain(isl_map_copy(qc
));
970 subset
= isl_set_is_subset(s
, dom
);
979 s
= isl_map_range(qc
);
980 subset
= isl_set_is_subset(s
, ran
);
983 return subset
< 0 ? -1 : !subset
;
992 /* For each basic map in "map", except i, check whether it combines
993 * with the transitive closure that is reflexive on C combines
994 * to the left and to the right.
998 * dom map_j \subseteq C
1000 * then right[j] is set to 1. Otherwise, if
1002 * ran map_i \cap dom map_j = \emptyset
1004 * then right[j] is set to 0. Otherwise, composing to the right
1007 * Similar, for composing to the left, we have if
1009 * ran map_j \subseteq C
1011 * then left[j] is set to 1. Otherwise, if
1013 * dom map_i \cap ran map_j = \emptyset
1015 * then left[j] is set to 0. Otherwise, composing to the left
1018 * The return value is or'd with LEFT if composing to the left
1019 * is possible and with RIGHT if composing to the right is possible.
1021 static int composability(__isl_keep isl_set
*C
, int i
,
1022 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
1023 __isl_keep isl_map
*map
)
1029 for (j
= 0; j
< map
->n
&& ok
; ++j
) {
1030 int overlaps
, subset
;
1036 dom
[j
] = isl_set_from_basic_set(
1037 isl_basic_map_domain(
1038 isl_basic_map_copy(map
->p
[j
])));
1041 overlaps
= isl_set_overlaps(ran
[i
], dom
[j
]);
1047 subset
= isl_set_is_subset(dom
[j
], C
);
1059 ran
[j
] = isl_set_from_basic_set(
1060 isl_basic_map_range(
1061 isl_basic_map_copy(map
->p
[j
])));
1064 overlaps
= isl_set_overlaps(dom
[i
], ran
[j
]);
1070 subset
= isl_set_is_subset(ran
[j
], C
);
1084 static __isl_give isl_map
*anonymize(__isl_take isl_map
*map
)
1086 map
= isl_map_reset(map
, isl_dim_in
);
1087 map
= isl_map_reset(map
, isl_dim_out
);
1091 /* Return a map that is a union of the basic maps in "map", except i,
1092 * composed to left and right with qc based on the entries of "left"
1095 static __isl_give isl_map
*compose(__isl_keep isl_map
*map
, int i
,
1096 __isl_take isl_map
*qc
, int *left
, int *right
)
1101 comp
= isl_map_empty(isl_map_get_space(map
));
1102 for (j
= 0; j
< map
->n
; ++j
) {
1108 map_j
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[j
]));
1109 map_j
= anonymize(map_j
);
1110 if (left
&& left
[j
])
1111 map_j
= isl_map_apply_range(map_j
, isl_map_copy(qc
));
1112 if (right
&& right
[j
])
1113 map_j
= isl_map_apply_range(isl_map_copy(qc
), map_j
);
1114 comp
= isl_map_union(comp
, map_j
);
1117 comp
= isl_map_compute_divs(comp
);
1118 comp
= isl_map_coalesce(comp
);
1125 /* Compute the transitive closure of "map" incrementally by
1132 * map_i^+ \cup ((id \cup map_i^) \circ qc^+)
1136 * map_i^+ \cup (qc^+ \circ (id \cup map_i^))
1138 * depending on whether left or right are NULL.
1140 static __isl_give isl_map
*compute_incremental(
1141 __isl_take isl_space
*dim
, __isl_keep isl_map
*map
,
1142 int i
, __isl_take isl_map
*qc
, int *left
, int *right
, int *exact
)
1146 isl_map
*rtc
= NULL
;
1150 isl_assert(map
->ctx
, left
|| right
, goto error
);
1152 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
1153 tc
= construct_projected_component(isl_space_copy(dim
), map_i
,
1155 isl_map_free(map_i
);
1158 qc
= isl_map_transitive_closure(qc
, exact
);
1161 isl_space_free(dim
);
1164 return isl_map_universe(isl_map_get_space(map
));
1167 if (!left
|| !right
)
1168 rtc
= isl_map_union(isl_map_copy(tc
),
1169 isl_map_identity(isl_map_get_space(tc
)));
1171 qc
= isl_map_apply_range(rtc
, qc
);
1173 qc
= isl_map_apply_range(qc
, rtc
);
1174 qc
= isl_map_union(tc
, qc
);
1176 isl_space_free(dim
);
1180 isl_space_free(dim
);
1185 /* Given a map "map", try to find a basic map such that
1186 * map^+ can be computed as
1188 * map^+ = map_i^+ \cup
1189 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1191 * with C the simple hull of the domain and range of the input map.
1192 * map_i^ \cup Id_C is computed by allowing the path lengths to be zero
1193 * and by intersecting domain and range with C.
1194 * Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
1195 * Also, we only use the incremental computation if all the transitive
1196 * closures are exact and if the number of basic maps in the union,
1197 * after computing the integer divisions, is smaller than the number
1198 * of basic maps in the input map.
1200 static int incemental_on_entire_domain(__isl_keep isl_space
*dim
,
1201 __isl_keep isl_map
*map
,
1202 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
1203 __isl_give isl_map
**res
)
1211 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
1212 isl_map_range(isl_map_copy(map
)));
1213 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1221 d
= isl_map_dim(map
, isl_dim_in
);
1223 for (i
= 0; i
< map
->n
; ++i
) {
1225 int exact_i
, spurious
;
1227 dom
[i
] = isl_set_from_basic_set(isl_basic_map_domain(
1228 isl_basic_map_copy(map
->p
[i
])));
1229 ran
[i
] = isl_set_from_basic_set(isl_basic_map_range(
1230 isl_basic_map_copy(map
->p
[i
])));
1231 qc
= q_closure(isl_space_copy(dim
), isl_set_copy(C
),
1232 map
->p
[i
], &exact_i
);
1239 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1246 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1247 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1248 qc
= isl_map_compute_divs(qc
);
1249 for (j
= 0; j
< map
->n
; ++j
)
1250 left
[j
] = right
[j
] = 1;
1251 qc
= compose(map
, i
, qc
, left
, right
);
1254 if (qc
->n
>= map
->n
) {
1258 *res
= compute_incremental(isl_space_copy(dim
), map
, i
, qc
,
1259 left
, right
, &exact_i
);
1270 return *res
!= NULL
;
1276 /* Try and compute the transitive closure of "map" as
1278 * map^+ = map_i^+ \cup
1279 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1281 * with C either the simple hull of the domain and range of the entire
1282 * map or the simple hull of domain and range of map_i.
1284 static __isl_give isl_map
*incremental_closure(__isl_take isl_space
*dim
,
1285 __isl_keep isl_map
*map
, int *exact
, int project
)
1288 isl_set
**dom
= NULL
;
1289 isl_set
**ran
= NULL
;
1294 isl_map
*res
= NULL
;
1297 return construct_projected_component(dim
, map
, exact
, project
);
1302 return construct_projected_component(dim
, map
, exact
, project
);
1304 d
= isl_map_dim(map
, isl_dim_in
);
1306 dom
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1307 ran
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1308 left
= isl_calloc_array(map
->ctx
, int, map
->n
);
1309 right
= isl_calloc_array(map
->ctx
, int, map
->n
);
1310 if (!ran
|| !dom
|| !left
|| !right
)
1313 if (incemental_on_entire_domain(dim
, map
, dom
, ran
, left
, right
, &res
) < 0)
1316 for (i
= 0; !res
&& i
< map
->n
; ++i
) {
1318 int exact_i
, spurious
, comp
;
1320 dom
[i
] = isl_set_from_basic_set(
1321 isl_basic_map_domain(
1322 isl_basic_map_copy(map
->p
[i
])));
1326 ran
[i
] = isl_set_from_basic_set(
1327 isl_basic_map_range(
1328 isl_basic_map_copy(map
->p
[i
])));
1331 C
= isl_set_union(isl_set_copy(dom
[i
]),
1332 isl_set_copy(ran
[i
]));
1333 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1340 comp
= composability(C
, i
, dom
, ran
, left
, right
, map
);
1341 if (!comp
|| comp
< 0) {
1347 qc
= q_closure(isl_space_copy(dim
), C
, map
->p
[i
], &exact_i
);
1354 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1361 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1362 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1363 qc
= isl_map_compute_divs(qc
);
1364 qc
= compose(map
, i
, qc
, (comp
& LEFT
) ? left
: NULL
,
1365 (comp
& RIGHT
) ? right
: NULL
);
1368 if (qc
->n
>= map
->n
) {
1372 res
= compute_incremental(isl_space_copy(dim
), map
, i
, qc
,
1373 (comp
& LEFT
) ? left
: NULL
,
1374 (comp
& RIGHT
) ? right
: NULL
, &exact_i
);
1383 for (i
= 0; i
< map
->n
; ++i
) {
1384 isl_set_free(dom
[i
]);
1385 isl_set_free(ran
[i
]);
1393 isl_space_free(dim
);
1397 return construct_projected_component(dim
, map
, exact
, project
);
1400 for (i
= 0; i
< map
->n
; ++i
)
1401 isl_set_free(dom
[i
]);
1404 for (i
= 0; i
< map
->n
; ++i
)
1405 isl_set_free(ran
[i
]);
1409 isl_space_free(dim
);
1413 /* Given an array of sets "set", add "dom" at position "pos"
1414 * and search for elements at earlier positions that overlap with "dom".
1415 * If any can be found, then merge all of them, together with "dom", into
1416 * a single set and assign the union to the first in the array,
1417 * which becomes the new group leader for all groups involved in the merge.
1418 * During the search, we only consider group leaders, i.e., those with
1419 * group[i] = i, as the other sets have already been combined
1420 * with one of the group leaders.
1422 static int merge(isl_set
**set
, int *group
, __isl_take isl_set
*dom
, int pos
)
1427 set
[pos
] = isl_set_copy(dom
);
1429 for (i
= pos
- 1; i
>= 0; --i
) {
1435 o
= isl_set_overlaps(set
[i
], dom
);
1441 set
[i
] = isl_set_union(set
[i
], set
[group
[pos
]]);
1442 set
[group
[pos
]] = NULL
;
1445 group
[group
[pos
]] = i
;
1456 /* Replace each entry in the n by n grid of maps by the cross product
1457 * with the relation { [i] -> [i + 1] }.
1459 static int add_length(__isl_keep isl_map
*map
, isl_map
***grid
, int n
)
1463 isl_basic_map
*bstep
;
1470 dim
= isl_map_get_space(map
);
1471 nparam
= isl_space_dim(dim
, isl_dim_param
);
1472 dim
= isl_space_drop_dims(dim
, isl_dim_in
, 0, isl_space_dim(dim
, isl_dim_in
));
1473 dim
= isl_space_drop_dims(dim
, isl_dim_out
, 0, isl_space_dim(dim
, isl_dim_out
));
1474 dim
= isl_space_add_dims(dim
, isl_dim_in
, 1);
1475 dim
= isl_space_add_dims(dim
, isl_dim_out
, 1);
1476 bstep
= isl_basic_map_alloc_space(dim
, 0, 1, 0);
1477 k
= isl_basic_map_alloc_equality(bstep
);
1479 isl_basic_map_free(bstep
);
1482 isl_seq_clr(bstep
->eq
[k
], 1 + isl_basic_map_total_dim(bstep
));
1483 isl_int_set_si(bstep
->eq
[k
][0], 1);
1484 isl_int_set_si(bstep
->eq
[k
][1 + nparam
], 1);
1485 isl_int_set_si(bstep
->eq
[k
][1 + nparam
+ 1], -1);
1486 bstep
= isl_basic_map_finalize(bstep
);
1487 step
= isl_map_from_basic_map(bstep
);
1489 for (i
= 0; i
< n
; ++i
)
1490 for (j
= 0; j
< n
; ++j
)
1491 grid
[i
][j
] = isl_map_product(grid
[i
][j
],
1492 isl_map_copy(step
));
1499 /* The core of the Floyd-Warshall algorithm.
1500 * Updates the given n x x matrix of relations in place.
1502 * The algorithm iterates over all vertices. In each step, the whole
1503 * matrix is updated to include all paths that go to the current vertex,
1504 * possibly stay there a while (including passing through earlier vertices)
1505 * and then come back. At the start of each iteration, the diagonal
1506 * element corresponding to the current vertex is replaced by its
1507 * transitive closure to account for all indirect paths that stay
1508 * in the current vertex.
1510 static void floyd_warshall_iterate(isl_map
***grid
, int n
, int *exact
)
1514 for (r
= 0; r
< n
; ++r
) {
1516 grid
[r
][r
] = isl_map_transitive_closure(grid
[r
][r
],
1517 (exact
&& *exact
) ? &r_exact
: NULL
);
1518 if (exact
&& *exact
&& !r_exact
)
1521 for (p
= 0; p
< n
; ++p
)
1522 for (q
= 0; q
< n
; ++q
) {
1524 if (p
== r
&& q
== r
)
1526 loop
= isl_map_apply_range(
1527 isl_map_copy(grid
[p
][r
]),
1528 isl_map_copy(grid
[r
][q
]));
1529 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1530 loop
= isl_map_apply_range(
1531 isl_map_copy(grid
[p
][r
]),
1532 isl_map_apply_range(
1533 isl_map_copy(grid
[r
][r
]),
1534 isl_map_copy(grid
[r
][q
])));
1535 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1536 grid
[p
][q
] = isl_map_coalesce(grid
[p
][q
]);
1541 /* Given a partition of the domains and ranges of the basic maps in "map",
1542 * apply the Floyd-Warshall algorithm with the elements in the partition
1545 * In particular, there are "n" elements in the partition and "group" is
1546 * an array of length 2 * map->n with entries in [0,n-1].
1548 * We first construct a matrix of relations based on the partition information,
1549 * apply Floyd-Warshall on this matrix of relations and then take the
1550 * union of all entries in the matrix as the final result.
1552 * If we are actually computing the power instead of the transitive closure,
1553 * i.e., when "project" is not set, then the result should have the
1554 * path lengths encoded as the difference between an extra pair of
1555 * coordinates. We therefore apply the nested transitive closures
1556 * to relations that include these lengths. In particular, we replace
1557 * the input relation by the cross product with the unit length relation
1558 * { [i] -> [i + 1] }.
1560 static __isl_give isl_map
*floyd_warshall_with_groups(__isl_take isl_space
*dim
,
1561 __isl_keep isl_map
*map
, int *exact
, int project
, int *group
, int n
)
1564 isl_map
***grid
= NULL
;
1572 return incremental_closure(dim
, map
, exact
, project
);
1575 grid
= isl_calloc_array(map
->ctx
, isl_map
**, n
);
1578 for (i
= 0; i
< n
; ++i
) {
1579 grid
[i
] = isl_calloc_array(map
->ctx
, isl_map
*, n
);
1582 for (j
= 0; j
< n
; ++j
)
1583 grid
[i
][j
] = isl_map_empty(isl_map_get_space(map
));
1586 for (k
= 0; k
< map
->n
; ++k
) {
1588 j
= group
[2 * k
+ 1];
1589 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
1590 isl_map_from_basic_map(
1591 isl_basic_map_copy(map
->p
[k
])));
1594 if (!project
&& add_length(map
, grid
, n
) < 0)
1597 floyd_warshall_iterate(grid
, n
, exact
);
1599 app
= isl_map_empty(isl_map_get_space(map
));
1601 for (i
= 0; i
< n
; ++i
) {
1602 for (j
= 0; j
< n
; ++j
)
1603 app
= isl_map_union(app
, grid
[i
][j
]);
1609 isl_space_free(dim
);
1614 for (i
= 0; i
< n
; ++i
) {
1617 for (j
= 0; j
< n
; ++j
)
1618 isl_map_free(grid
[i
][j
]);
1623 isl_space_free(dim
);
1627 /* Partition the domains and ranges of the n basic relations in list
1628 * into disjoint cells.
1630 * To find the partition, we simply consider all of the domains
1631 * and ranges in turn and combine those that overlap.
1632 * "set" contains the partition elements and "group" indicates
1633 * to which partition element a given domain or range belongs.
1634 * The domain of basic map i corresponds to element 2 * i in these arrays,
1635 * while the domain corresponds to element 2 * i + 1.
1636 * During the construction group[k] is either equal to k,
1637 * in which case set[k] contains the union of all the domains and
1638 * ranges in the corresponding group, or is equal to some l < k,
1639 * with l another domain or range in the same group.
1641 static int *setup_groups(isl_ctx
*ctx
, __isl_keep isl_basic_map
**list
, int n
,
1642 isl_set
***set
, int *n_group
)
1648 *set
= isl_calloc_array(ctx
, isl_set
*, 2 * n
);
1649 group
= isl_alloc_array(ctx
, int, 2 * n
);
1651 if (!*set
|| !group
)
1654 for (i
= 0; i
< n
; ++i
) {
1656 dom
= isl_set_from_basic_set(isl_basic_map_domain(
1657 isl_basic_map_copy(list
[i
])));
1658 if (merge(*set
, group
, dom
, 2 * i
) < 0)
1660 dom
= isl_set_from_basic_set(isl_basic_map_range(
1661 isl_basic_map_copy(list
[i
])));
1662 if (merge(*set
, group
, dom
, 2 * i
+ 1) < 0)
1667 for (i
= 0; i
< 2 * n
; ++i
)
1668 if (group
[i
] == i
) {
1670 (*set
)[g
] = (*set
)[i
];
1675 group
[i
] = group
[group
[i
]];
1682 for (i
= 0; i
< 2 * n
; ++i
)
1683 isl_set_free((*set
)[i
]);
1691 /* Check if the domains and ranges of the basic maps in "map" can
1692 * be partitioned, and if so, apply Floyd-Warshall on the elements
1693 * of the partition. Note that we also apply this algorithm
1694 * if we want to compute the power, i.e., when "project" is not set.
1695 * However, the results are unlikely to be exact since the recursive
1696 * calls inside the Floyd-Warshall algorithm typically result in
1697 * non-linear path lengths quite quickly.
1699 static __isl_give isl_map
*floyd_warshall(__isl_take isl_space
*dim
,
1700 __isl_keep isl_map
*map
, int *exact
, int project
)
1703 isl_set
**set
= NULL
;
1710 return incremental_closure(dim
, map
, exact
, project
);
1712 group
= setup_groups(map
->ctx
, map
->p
, map
->n
, &set
, &n
);
1716 for (i
= 0; i
< 2 * map
->n
; ++i
)
1717 isl_set_free(set
[i
]);
1721 return floyd_warshall_with_groups(dim
, map
, exact
, project
, group
, n
);
1723 isl_space_free(dim
);
1727 /* Structure for representing the nodes of the graph of which
1728 * strongly connected components are being computed.
1730 * list contains the actual nodes
1731 * check_closed is set if we may have used the fact that
1732 * a pair of basic maps can be interchanged
1734 struct isl_tc_follows_data
{
1735 isl_basic_map
**list
;
1739 /* Check whether in the computation of the transitive closure
1740 * "list[i]" (R_1) should follow (or be part of the same component as)
1743 * That is check whether
1751 * If so, then there is no reason for R_1 to immediately follow R_2
1754 * *check_closed is set if the subset relation holds while
1755 * R_1 \circ R_2 is not empty.
1757 static int basic_map_follows(int i
, int j
, void *user
)
1759 struct isl_tc_follows_data
*data
= user
;
1760 struct isl_map
*map12
= NULL
;
1761 struct isl_map
*map21
= NULL
;
1764 if (!isl_space_tuple_is_equal(data
->list
[i
]->dim
, isl_dim_in
,
1765 data
->list
[j
]->dim
, isl_dim_out
))
1768 map21
= isl_map_from_basic_map(
1769 isl_basic_map_apply_range(
1770 isl_basic_map_copy(data
->list
[j
]),
1771 isl_basic_map_copy(data
->list
[i
])));
1772 subset
= isl_map_is_empty(map21
);
1776 isl_map_free(map21
);
1780 if (!isl_space_tuple_is_equal(data
->list
[i
]->dim
, isl_dim_in
,
1781 data
->list
[i
]->dim
, isl_dim_out
) ||
1782 !isl_space_tuple_is_equal(data
->list
[j
]->dim
, isl_dim_in
,
1783 data
->list
[j
]->dim
, isl_dim_out
)) {
1784 isl_map_free(map21
);
1788 map12
= isl_map_from_basic_map(
1789 isl_basic_map_apply_range(
1790 isl_basic_map_copy(data
->list
[i
]),
1791 isl_basic_map_copy(data
->list
[j
])));
1793 subset
= isl_map_is_subset(map21
, map12
);
1795 isl_map_free(map12
);
1796 isl_map_free(map21
);
1799 data
->check_closed
= 1;
1801 return subset
< 0 ? -1 : !subset
;
1803 isl_map_free(map21
);
1807 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
1808 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
1809 * construct a map that is an overapproximation of the map
1810 * that takes an element from the dom R \times Z to an
1811 * element from ran R \times Z, such that the first n coordinates of the
1812 * difference between them is a sum of differences between images
1813 * and pre-images in one of the R_i and such that the last coordinate
1814 * is equal to the number of steps taken.
1815 * If "project" is set, then these final coordinates are not included,
1816 * i.e., a relation of type Z^n -> Z^n is returned.
1819 * \Delta_i = { y - x | (x, y) in R_i }
1821 * then the constructed map is an overapproximation of
1823 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1824 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
1825 * x in dom R and x + d in ran R }
1829 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1830 * d = (\sum_i k_i \delta_i) and
1831 * x in dom R and x + d in ran R }
1833 * if "project" is set.
1835 * We first split the map into strongly connected components, perform
1836 * the above on each component and then join the results in the correct
1837 * order, at each join also taking in the union of both arguments
1838 * to allow for paths that do not go through one of the two arguments.
1840 static __isl_give isl_map
*construct_power_components(__isl_take isl_space
*dim
,
1841 __isl_keep isl_map
*map
, int *exact
, int project
)
1844 struct isl_map
*path
= NULL
;
1845 struct isl_tc_follows_data data
;
1846 struct isl_tarjan_graph
*g
= NULL
;
1853 return floyd_warshall(dim
, map
, exact
, project
);
1856 data
.check_closed
= 0;
1857 g
= isl_tarjan_graph_init(map
->ctx
, map
->n
, &basic_map_follows
, &data
);
1862 if (data
.check_closed
&& !exact
)
1863 exact
= &local_exact
;
1869 path
= isl_map_empty(isl_map_get_space(map
));
1871 path
= isl_map_empty(isl_space_copy(dim
));
1872 path
= anonymize(path
);
1874 struct isl_map
*comp
;
1875 isl_map
*path_comp
, *path_comb
;
1876 comp
= isl_map_alloc_space(isl_map_get_space(map
), n
, 0);
1877 while (g
->order
[i
] != -1) {
1878 comp
= isl_map_add_basic_map(comp
,
1879 isl_basic_map_copy(map
->p
[g
->order
[i
]]));
1883 path_comp
= floyd_warshall(isl_space_copy(dim
),
1884 comp
, exact
, project
);
1885 path_comp
= anonymize(path_comp
);
1886 path_comb
= isl_map_apply_range(isl_map_copy(path
),
1887 isl_map_copy(path_comp
));
1888 path
= isl_map_union(path
, path_comp
);
1889 path
= isl_map_union(path
, path_comb
);
1895 if (c
> 1 && data
.check_closed
&& !*exact
) {
1898 closed
= isl_map_is_transitively_closed(path
);
1902 isl_tarjan_graph_free(g
);
1904 return floyd_warshall(dim
, map
, orig_exact
, project
);
1908 isl_tarjan_graph_free(g
);
1909 isl_space_free(dim
);
1913 isl_tarjan_graph_free(g
);
1914 isl_space_free(dim
);
1919 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
1920 * construct a map that is an overapproximation of the map
1921 * that takes an element from the space D to another
1922 * element from the same space, such that the difference between
1923 * them is a strictly positive sum of differences between images
1924 * and pre-images in one of the R_i.
1925 * The number of differences in the sum is equated to parameter "param".
1928 * \Delta_i = { y - x | (x, y) in R_i }
1930 * then the constructed map is an overapproximation of
1932 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1933 * d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
1936 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1937 * d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
1939 * if "project" is set.
1941 * If "project" is not set, then
1942 * we construct an extended mapping with an extra coordinate
1943 * that indicates the number of steps taken. In particular,
1944 * the difference in the last coordinate is equal to the number
1945 * of steps taken to move from a domain element to the corresponding
1948 static __isl_give isl_map
*construct_power(__isl_keep isl_map
*map
,
1949 int *exact
, int project
)
1951 struct isl_map
*app
= NULL
;
1952 isl_space
*dim
= NULL
;
1958 dim
= isl_map_get_space(map
);
1960 d
= isl_space_dim(dim
, isl_dim_in
);
1961 dim
= isl_space_add_dims(dim
, isl_dim_in
, 1);
1962 dim
= isl_space_add_dims(dim
, isl_dim_out
, 1);
1964 app
= construct_power_components(isl_space_copy(dim
), map
,
1967 isl_space_free(dim
);
1972 /* Compute the positive powers of "map", or an overapproximation.
1973 * If the result is exact, then *exact is set to 1.
1975 * If project is set, then we are actually interested in the transitive
1976 * closure, so we can use a more relaxed exactness check.
1977 * The lengths of the paths are also projected out instead of being
1978 * encoded as the difference between an extra pair of final coordinates.
1980 static __isl_give isl_map
*map_power(__isl_take isl_map
*map
,
1981 int *exact
, int project
)
1983 struct isl_map
*app
= NULL
;
1991 isl_assert(map
->ctx
,
1992 isl_map_dim(map
, isl_dim_in
) == isl_map_dim(map
, isl_dim_out
),
1995 app
= construct_power(map
, exact
, project
);
2005 /* Compute the positive powers of "map", or an overapproximation.
2006 * The result maps the exponent to a nested copy of the corresponding power.
2007 * If the result is exact, then *exact is set to 1.
2008 * map_power constructs an extended relation with the path lengths
2009 * encoded as the difference between the final coordinates.
2010 * In the final step, this difference is equated to an extra parameter
2011 * and made positive. The extra coordinates are subsequently projected out
2012 * and the parameter is turned into the domain of the result.
2014 __isl_give isl_map
*isl_map_power(__isl_take isl_map
*map
, int *exact
)
2016 isl_space
*target_dim
;
2025 d
= isl_map_dim(map
, isl_dim_in
);
2026 param
= isl_map_dim(map
, isl_dim_param
);
2028 map
= isl_map_compute_divs(map
);
2029 map
= isl_map_coalesce(map
);
2031 if (isl_map_plain_is_empty(map
)) {
2032 map
= isl_map_from_range(isl_map_wrap(map
));
2033 map
= isl_map_add_dims(map
, isl_dim_in
, 1);
2034 map
= isl_map_set_dim_name(map
, isl_dim_in
, 0, "k");
2038 target_dim
= isl_map_get_space(map
);
2039 target_dim
= isl_space_from_range(isl_space_wrap(target_dim
));
2040 target_dim
= isl_space_add_dims(target_dim
, isl_dim_in
, 1);
2041 target_dim
= isl_space_set_dim_name(target_dim
, isl_dim_in
, 0, "k");
2043 map
= map_power(map
, exact
, 0);
2045 map
= isl_map_add_dims(map
, isl_dim_param
, 1);
2046 dim
= isl_map_get_space(map
);
2047 diff
= equate_parameter_to_length(dim
, param
);
2048 map
= isl_map_intersect(map
, diff
);
2049 map
= isl_map_project_out(map
, isl_dim_in
, d
, 1);
2050 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2051 map
= isl_map_from_range(isl_map_wrap(map
));
2052 map
= isl_map_move_dims(map
, isl_dim_in
, 0, isl_dim_param
, param
, 1);
2054 map
= isl_map_reset_space(map
, target_dim
);
2059 /* Compute a relation that maps each element in the range of the input
2060 * relation to the lengths of all paths composed of edges in the input
2061 * relation that end up in the given range element.
2062 * The result may be an overapproximation, in which case *exact is set to 0.
2063 * The resulting relation is very similar to the power relation.
2064 * The difference are that the domain has been projected out, the
2065 * range has become the domain and the exponent is the range instead
2068 __isl_give isl_map
*isl_map_reaching_path_lengths(__isl_take isl_map
*map
,
2079 d
= isl_map_dim(map
, isl_dim_in
);
2080 param
= isl_map_dim(map
, isl_dim_param
);
2082 map
= isl_map_compute_divs(map
);
2083 map
= isl_map_coalesce(map
);
2085 if (isl_map_plain_is_empty(map
)) {
2088 map
= isl_map_project_out(map
, isl_dim_out
, 0, d
);
2089 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
2093 map
= map_power(map
, exact
, 0);
2095 map
= isl_map_add_dims(map
, isl_dim_param
, 1);
2096 dim
= isl_map_get_space(map
);
2097 diff
= equate_parameter_to_length(dim
, param
);
2098 map
= isl_map_intersect(map
, diff
);
2099 map
= isl_map_project_out(map
, isl_dim_in
, 0, d
+ 1);
2100 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2101 map
= isl_map_reverse(map
);
2102 map
= isl_map_move_dims(map
, isl_dim_out
, 0, isl_dim_param
, param
, 1);
2107 /* Check whether equality i of bset is a pure stride constraint
2108 * on a single dimensions, i.e., of the form
2112 * with k a constant and e an existentially quantified variable.
2114 static int is_eq_stride(__isl_keep isl_basic_set
*bset
, int i
)
2125 if (!isl_int_is_zero(bset
->eq
[i
][0]))
2128 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
2129 d
= isl_basic_set_dim(bset
, isl_dim_set
);
2130 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
2132 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1, nparam
) != -1)
2134 pos1
= isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
, d
);
2137 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ pos1
+ 1,
2138 d
- pos1
- 1) != -1)
2141 pos2
= isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ d
, n_div
);
2144 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ d
+ pos2
+ 1,
2145 n_div
- pos2
- 1) != -1)
2147 if (!isl_int_is_one(bset
->eq
[i
][1 + nparam
+ pos1
]) &&
2148 !isl_int_is_negone(bset
->eq
[i
][1 + nparam
+ pos1
]))
2154 /* Given a map, compute the smallest superset of this map that is of the form
2156 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2158 * (where p ranges over the (non-parametric) dimensions),
2159 * compute the transitive closure of this map, i.e.,
2161 * { i -> j : exists k > 0:
2162 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2164 * and intersect domain and range of this transitive closure with
2165 * the given domain and range.
2167 * If with_id is set, then try to include as much of the identity mapping
2168 * as possible, by computing
2170 * { i -> j : exists k >= 0:
2171 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2173 * instead (i.e., allow k = 0).
2175 * In practice, we compute the difference set
2177 * delta = { j - i | i -> j in map },
2179 * look for stride constraint on the individual dimensions and compute
2180 * (constant) lower and upper bounds for each individual dimension,
2181 * adding a constraint for each bound not equal to infinity.
2183 static __isl_give isl_map
*box_closure_on_domain(__isl_take isl_map
*map
,
2184 __isl_take isl_set
*dom
, __isl_take isl_set
*ran
, int with_id
)
2193 isl_map
*app
= NULL
;
2194 isl_basic_set
*aff
= NULL
;
2195 isl_basic_map
*bmap
= NULL
;
2196 isl_vec
*obj
= NULL
;
2201 delta
= isl_map_deltas(isl_map_copy(map
));
2203 aff
= isl_set_affine_hull(isl_set_copy(delta
));
2206 dim
= isl_map_get_space(map
);
2207 d
= isl_space_dim(dim
, isl_dim_in
);
2208 nparam
= isl_space_dim(dim
, isl_dim_param
);
2209 total
= isl_space_dim(dim
, isl_dim_all
);
2210 bmap
= isl_basic_map_alloc_space(dim
,
2211 aff
->n_div
+ 1, aff
->n_div
, 2 * d
+ 1);
2212 for (i
= 0; i
< aff
->n_div
+ 1; ++i
) {
2213 k
= isl_basic_map_alloc_div(bmap
);
2216 isl_int_set_si(bmap
->div
[k
][0], 0);
2218 for (i
= 0; i
< aff
->n_eq
; ++i
) {
2219 if (!is_eq_stride(aff
, i
))
2221 k
= isl_basic_map_alloc_equality(bmap
);
2224 isl_seq_clr(bmap
->eq
[k
], 1 + nparam
);
2225 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ d
,
2226 aff
->eq
[i
] + 1 + nparam
, d
);
2227 isl_seq_neg(bmap
->eq
[k
] + 1 + nparam
,
2228 aff
->eq
[i
] + 1 + nparam
, d
);
2229 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ 2 * d
,
2230 aff
->eq
[i
] + 1 + nparam
+ d
, aff
->n_div
);
2231 isl_int_set_si(bmap
->eq
[k
][1 + total
+ aff
->n_div
], 0);
2233 obj
= isl_vec_alloc(map
->ctx
, 1 + nparam
+ d
);
2236 isl_seq_clr(obj
->el
, 1 + nparam
+ d
);
2237 for (i
= 0; i
< d
; ++ i
) {
2238 enum isl_lp_result res
;
2240 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 1);
2242 res
= isl_set_solve_lp(delta
, 0, obj
->el
, map
->ctx
->one
, &opt
,
2244 if (res
== isl_lp_error
)
2246 if (res
== isl_lp_ok
) {
2247 k
= isl_basic_map_alloc_inequality(bmap
);
2250 isl_seq_clr(bmap
->ineq
[k
],
2251 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2252 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], -1);
2253 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], 1);
2254 isl_int_neg(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2257 res
= isl_set_solve_lp(delta
, 1, obj
->el
, map
->ctx
->one
, &opt
,
2259 if (res
== isl_lp_error
)
2261 if (res
== isl_lp_ok
) {
2262 k
= isl_basic_map_alloc_inequality(bmap
);
2265 isl_seq_clr(bmap
->ineq
[k
],
2266 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2267 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], 1);
2268 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], -1);
2269 isl_int_set(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2272 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 0);
2274 k
= isl_basic_map_alloc_inequality(bmap
);
2277 isl_seq_clr(bmap
->ineq
[k
],
2278 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2280 isl_int_set_si(bmap
->ineq
[k
][0], -1);
2281 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], 1);
2283 app
= isl_map_from_domain_and_range(dom
, ran
);
2286 isl_basic_set_free(aff
);
2288 bmap
= isl_basic_map_finalize(bmap
);
2289 isl_set_free(delta
);
2292 map
= isl_map_from_basic_map(bmap
);
2293 map
= isl_map_intersect(map
, app
);
2298 isl_basic_map_free(bmap
);
2299 isl_basic_set_free(aff
);
2303 isl_set_free(delta
);
2308 /* Given a map, compute the smallest superset of this map that is of the form
2310 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2312 * (where p ranges over the (non-parametric) dimensions),
2313 * compute the transitive closure of this map, i.e.,
2315 * { i -> j : exists k > 0:
2316 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2318 * and intersect domain and range of this transitive closure with
2319 * domain and range of the original map.
2321 static __isl_give isl_map
*box_closure(__isl_take isl_map
*map
)
2326 domain
= isl_map_domain(isl_map_copy(map
));
2327 domain
= isl_set_coalesce(domain
);
2328 range
= isl_map_range(isl_map_copy(map
));
2329 range
= isl_set_coalesce(range
);
2331 return box_closure_on_domain(map
, domain
, range
, 0);
2334 /* Given a map, compute the smallest superset of this map that is of the form
2336 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2338 * (where p ranges over the (non-parametric) dimensions),
2339 * compute the transitive and partially reflexive closure of this map, i.e.,
2341 * { i -> j : exists k >= 0:
2342 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2344 * and intersect domain and range of this transitive closure with
2347 static __isl_give isl_map
*box_closure_with_identity(__isl_take isl_map
*map
,
2348 __isl_take isl_set
*dom
)
2350 return box_closure_on_domain(map
, dom
, isl_set_copy(dom
), 1);
2353 /* Check whether app is the transitive closure of map.
2354 * In particular, check that app is acyclic and, if so,
2357 * app \subset (map \cup (map \circ app))
2359 static int check_exactness_omega(__isl_keep isl_map
*map
,
2360 __isl_keep isl_map
*app
)
2364 int is_empty
, is_exact
;
2368 delta
= isl_map_deltas(isl_map_copy(app
));
2369 d
= isl_set_dim(delta
, isl_dim_set
);
2370 for (i
= 0; i
< d
; ++i
)
2371 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
2372 is_empty
= isl_set_is_empty(delta
);
2373 isl_set_free(delta
);
2379 test
= isl_map_apply_range(isl_map_copy(app
), isl_map_copy(map
));
2380 test
= isl_map_union(test
, isl_map_copy(map
));
2381 is_exact
= isl_map_is_subset(app
, test
);
2387 /* Check if basic map M_i can be combined with all the other
2388 * basic maps such that
2392 * can be computed as
2394 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2396 * In particular, check if we can compute a compact representation
2399 * M_i^* \circ M_j \circ M_i^*
2402 * Let M_i^? be an extension of M_i^+ that allows paths
2403 * of length zero, i.e., the result of box_closure(., 1).
2404 * The criterion, as proposed by Kelly et al., is that
2405 * id = M_i^? - M_i^+ can be represented as a basic map
2408 * id \circ M_j \circ id = M_j
2412 * If this function returns 1, then tc and qc are set to
2413 * M_i^+ and M_i^?, respectively.
2415 static int can_be_split_off(__isl_keep isl_map
*map
, int i
,
2416 __isl_give isl_map
**tc
, __isl_give isl_map
**qc
)
2418 isl_map
*map_i
, *id
= NULL
;
2425 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
2426 isl_map_range(isl_map_copy(map
)));
2427 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
2431 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
2432 *tc
= box_closure(isl_map_copy(map_i
));
2433 *qc
= box_closure_with_identity(map_i
, C
);
2434 id
= isl_map_subtract(isl_map_copy(*qc
), isl_map_copy(*tc
));
2438 if (id
->n
!= 1 || (*qc
)->n
!= 1)
2441 for (j
= 0; j
< map
->n
; ++j
) {
2442 isl_map
*map_j
, *test
;
2447 map_j
= isl_map_from_basic_map(
2448 isl_basic_map_copy(map
->p
[j
]));
2449 test
= isl_map_apply_range(isl_map_copy(id
),
2450 isl_map_copy(map_j
));
2451 test
= isl_map_apply_range(test
, isl_map_copy(id
));
2452 is_ok
= isl_map_is_equal(test
, map_j
);
2453 isl_map_free(map_j
);
2481 static __isl_give isl_map
*box_closure_with_check(__isl_take isl_map
*map
,
2486 app
= box_closure(isl_map_copy(map
));
2488 *exact
= check_exactness_omega(map
, app
);
2494 /* Compute an overapproximation of the transitive closure of "map"
2495 * using a variation of the algorithm from
2496 * "Transitive Closure of Infinite Graphs and its Applications"
2499 * We first check whether we can can split of any basic map M_i and
2506 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2508 * using a recursive call on the remaining map.
2510 * If not, we simply call box_closure on the whole map.
2512 static __isl_give isl_map
*transitive_closure_omega(__isl_take isl_map
*map
,
2522 return box_closure_with_check(map
, exact
);
2524 for (i
= 0; i
< map
->n
; ++i
) {
2527 ok
= can_be_split_off(map
, i
, &tc
, &qc
);
2533 app
= isl_map_alloc_space(isl_map_get_space(map
), map
->n
- 1, 0);
2535 for (j
= 0; j
< map
->n
; ++j
) {
2538 app
= isl_map_add_basic_map(app
,
2539 isl_basic_map_copy(map
->p
[j
]));
2542 app
= isl_map_apply_range(isl_map_copy(qc
), app
);
2543 app
= isl_map_apply_range(app
, qc
);
2545 app
= isl_map_union(tc
, transitive_closure_omega(app
, NULL
));
2546 exact_i
= check_exactness_omega(map
, app
);
2558 return box_closure_with_check(map
, exact
);
2564 /* Compute the transitive closure of "map", or an overapproximation.
2565 * If the result is exact, then *exact is set to 1.
2566 * Simply use map_power to compute the powers of map, but tell
2567 * it to project out the lengths of the paths instead of equating
2568 * the length to a parameter.
2570 __isl_give isl_map
*isl_map_transitive_closure(__isl_take isl_map
*map
,
2573 isl_space
*target_dim
;
2579 if (map
->ctx
->opt
->closure
== ISL_CLOSURE_BOX
)
2580 return transitive_closure_omega(map
, exact
);
2582 map
= isl_map_compute_divs(map
);
2583 map
= isl_map_coalesce(map
);
2584 closed
= isl_map_is_transitively_closed(map
);
2593 target_dim
= isl_map_get_space(map
);
2594 map
= map_power(map
, exact
, 1);
2595 map
= isl_map_reset_space(map
, target_dim
);
2603 static int inc_count(__isl_take isl_map
*map
, void *user
)
2614 static int collect_basic_map(__isl_take isl_map
*map
, void *user
)
2617 isl_basic_map
***next
= user
;
2619 for (i
= 0; i
< map
->n
; ++i
) {
2620 **next
= isl_basic_map_copy(map
->p
[i
]);
2633 /* Perform Floyd-Warshall on the given list of basic relations.
2634 * The basic relations may live in different dimensions,
2635 * but basic relations that get assigned to the diagonal of the
2636 * grid have domains and ranges of the same dimension and so
2637 * the standard algorithm can be used because the nested transitive
2638 * closures are only applied to diagonal elements and because all
2639 * compositions are peformed on relations with compatible domains and ranges.
2641 static __isl_give isl_union_map
*union_floyd_warshall_on_list(isl_ctx
*ctx
,
2642 __isl_keep isl_basic_map
**list
, int n
, int *exact
)
2647 isl_set
**set
= NULL
;
2648 isl_map
***grid
= NULL
;
2651 group
= setup_groups(ctx
, list
, n
, &set
, &n_group
);
2655 grid
= isl_calloc_array(ctx
, isl_map
**, n_group
);
2658 for (i
= 0; i
< n_group
; ++i
) {
2659 grid
[i
] = isl_calloc_array(ctx
, isl_map
*, n_group
);
2662 for (j
= 0; j
< n_group
; ++j
) {
2663 isl_space
*dim1
, *dim2
, *dim
;
2664 dim1
= isl_space_reverse(isl_set_get_space(set
[i
]));
2665 dim2
= isl_set_get_space(set
[j
]);
2666 dim
= isl_space_join(dim1
, dim2
);
2667 grid
[i
][j
] = isl_map_empty(dim
);
2671 for (k
= 0; k
< n
; ++k
) {
2673 j
= group
[2 * k
+ 1];
2674 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
2675 isl_map_from_basic_map(
2676 isl_basic_map_copy(list
[k
])));
2679 floyd_warshall_iterate(grid
, n_group
, exact
);
2681 app
= isl_union_map_empty(isl_map_get_space(grid
[0][0]));
2683 for (i
= 0; i
< n_group
; ++i
) {
2684 for (j
= 0; j
< n_group
; ++j
)
2685 app
= isl_union_map_add_map(app
, grid
[i
][j
]);
2690 for (i
= 0; i
< 2 * n
; ++i
)
2691 isl_set_free(set
[i
]);
2698 for (i
= 0; i
< n_group
; ++i
) {
2701 for (j
= 0; j
< n_group
; ++j
)
2702 isl_map_free(grid
[i
][j
]);
2707 for (i
= 0; i
< 2 * n
; ++i
)
2708 isl_set_free(set
[i
]);
2715 /* Perform Floyd-Warshall on the given union relation.
2716 * The implementation is very similar to that for non-unions.
2717 * The main difference is that it is applied unconditionally.
2718 * We first extract a list of basic maps from the union map
2719 * and then perform the algorithm on this list.
2721 static __isl_give isl_union_map
*union_floyd_warshall(
2722 __isl_take isl_union_map
*umap
, int *exact
)
2726 isl_basic_map
**list
= NULL
;
2727 isl_basic_map
**next
;
2731 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2734 ctx
= isl_union_map_get_ctx(umap
);
2735 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2740 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2743 res
= union_floyd_warshall_on_list(ctx
, list
, n
, exact
);
2746 for (i
= 0; i
< n
; ++i
)
2747 isl_basic_map_free(list
[i
]);
2751 isl_union_map_free(umap
);
2755 for (i
= 0; i
< n
; ++i
)
2756 isl_basic_map_free(list
[i
]);
2759 isl_union_map_free(umap
);
2763 /* Decompose the give union relation into strongly connected components.
2764 * The implementation is essentially the same as that of
2765 * construct_power_components with the major difference that all
2766 * operations are performed on union maps.
2768 static __isl_give isl_union_map
*union_components(
2769 __isl_take isl_union_map
*umap
, int *exact
)
2774 isl_basic_map
**list
= NULL
;
2775 isl_basic_map
**next
;
2776 isl_union_map
*path
= NULL
;
2777 struct isl_tc_follows_data data
;
2778 struct isl_tarjan_graph
*g
= NULL
;
2783 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2789 return union_floyd_warshall(umap
, exact
);
2791 ctx
= isl_union_map_get_ctx(umap
);
2792 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2797 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2801 data
.check_closed
= 0;
2802 g
= isl_tarjan_graph_init(ctx
, n
, &basic_map_follows
, &data
);
2809 path
= isl_union_map_empty(isl_union_map_get_space(umap
));
2811 isl_union_map
*comp
;
2812 isl_union_map
*path_comp
, *path_comb
;
2813 comp
= isl_union_map_empty(isl_union_map_get_space(umap
));
2814 while (g
->order
[i
] != -1) {
2815 comp
= isl_union_map_add_map(comp
,
2816 isl_map_from_basic_map(
2817 isl_basic_map_copy(list
[g
->order
[i
]])));
2821 path_comp
= union_floyd_warshall(comp
, exact
);
2822 path_comb
= isl_union_map_apply_range(isl_union_map_copy(path
),
2823 isl_union_map_copy(path_comp
));
2824 path
= isl_union_map_union(path
, path_comp
);
2825 path
= isl_union_map_union(path
, path_comb
);
2830 if (c
> 1 && data
.check_closed
&& !*exact
) {
2833 closed
= isl_union_map_is_transitively_closed(path
);
2839 isl_tarjan_graph_free(g
);
2841 for (i
= 0; i
< n
; ++i
)
2842 isl_basic_map_free(list
[i
]);
2846 isl_union_map_free(path
);
2847 return union_floyd_warshall(umap
, exact
);
2850 isl_union_map_free(umap
);
2854 isl_tarjan_graph_free(g
);
2856 for (i
= 0; i
< n
; ++i
)
2857 isl_basic_map_free(list
[i
]);
2860 isl_union_map_free(umap
);
2861 isl_union_map_free(path
);
2865 /* Compute the transitive closure of "umap", or an overapproximation.
2866 * If the result is exact, then *exact is set to 1.
2868 __isl_give isl_union_map
*isl_union_map_transitive_closure(
2869 __isl_take isl_union_map
*umap
, int *exact
)
2879 umap
= isl_union_map_compute_divs(umap
);
2880 umap
= isl_union_map_coalesce(umap
);
2881 closed
= isl_union_map_is_transitively_closed(umap
);
2886 umap
= union_components(umap
, exact
);
2889 isl_union_map_free(umap
);
2893 struct isl_union_power
{
2898 static int power(__isl_take isl_map
*map
, void *user
)
2900 struct isl_union_power
*up
= user
;
2902 map
= isl_map_power(map
, up
->exact
);
2903 up
->pow
= isl_union_map_from_map(map
);
2908 /* Construct a map [x] -> [x+1], with parameters prescribed by "dim".
2910 static __isl_give isl_union_map
*increment(__isl_take isl_space
*dim
)
2913 isl_basic_map
*bmap
;
2915 dim
= isl_space_add_dims(dim
, isl_dim_in
, 1);
2916 dim
= isl_space_add_dims(dim
, isl_dim_out
, 1);
2917 bmap
= isl_basic_map_alloc_space(dim
, 0, 1, 0);
2918 k
= isl_basic_map_alloc_equality(bmap
);
2921 isl_seq_clr(bmap
->eq
[k
], isl_basic_map_total_dim(bmap
));
2922 isl_int_set_si(bmap
->eq
[k
][0], 1);
2923 isl_int_set_si(bmap
->eq
[k
][isl_basic_map_offset(bmap
, isl_dim_in
)], 1);
2924 isl_int_set_si(bmap
->eq
[k
][isl_basic_map_offset(bmap
, isl_dim_out
)], -1);
2925 return isl_union_map_from_map(isl_map_from_basic_map(bmap
));
2927 isl_basic_map_free(bmap
);
2931 /* Construct a map [[x]->[y]] -> [y-x], with parameters prescribed by "dim".
2933 static __isl_give isl_union_map
*deltas_map(__isl_take isl_space
*dim
)
2935 isl_basic_map
*bmap
;
2937 dim
= isl_space_add_dims(dim
, isl_dim_in
, 1);
2938 dim
= isl_space_add_dims(dim
, isl_dim_out
, 1);
2939 bmap
= isl_basic_map_universe(dim
);
2940 bmap
= isl_basic_map_deltas_map(bmap
);
2942 return isl_union_map_from_map(isl_map_from_basic_map(bmap
));
2945 /* Compute the positive powers of "map", or an overapproximation.
2946 * The result maps the exponent to a nested copy of the corresponding power.
2947 * If the result is exact, then *exact is set to 1.
2949 __isl_give isl_union_map
*isl_union_map_power(__isl_take isl_union_map
*umap
,
2958 n
= isl_union_map_n_map(umap
);
2962 struct isl_union_power up
= { NULL
, exact
};
2963 isl_union_map_foreach_map(umap
, &power
, &up
);
2964 isl_union_map_free(umap
);
2967 inc
= increment(isl_union_map_get_space(umap
));
2968 umap
= isl_union_map_product(inc
, umap
);
2969 umap
= isl_union_map_transitive_closure(umap
, exact
);
2970 umap
= isl_union_map_zip(umap
);
2971 dm
= deltas_map(isl_union_map_get_space(umap
));
2972 umap
= isl_union_map_apply_domain(umap
, dm
);
2978 #define TYPE isl_map
2979 #include "isl_power_templ.c"
2982 #define TYPE isl_union_map
2983 #include "isl_power_templ.c"