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,
11 #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 * If div_purity is NULL then we are dealing with a non-parametric set
327 * and so the constraint is obviously PURE_VAR.
329 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
);
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_dim_domain(isl_basic_map_get_dim(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 static __isl_give isl_basic_map
*add_delta_constraints(
452 __isl_take isl_basic_map
*path
,
453 __isl_keep isl_basic_set
*delta
, unsigned off
, unsigned nparam
,
454 unsigned d
, int *div_purity
, int eq
, int *impurity
)
457 int n
= eq
? delta
->n_eq
: delta
->n_ineq
;
458 isl_int
**delta_c
= eq
? delta
->eq
: delta
->ineq
;
461 n_div
= isl_basic_set_dim(delta
, isl_dim_div
);
463 for (i
= 0; i
< n
; ++i
) {
465 int p
= purity(delta
, delta_c
[i
], div_purity
, eq
);
468 if (p
!= PURE_VAR
&& p
!= PURE_PARAM
&& !*impurity
)
472 if (eq
&& p
!= MIXED
) {
473 k
= isl_basic_map_alloc_equality(path
);
474 path_c
= path
->eq
[k
];
476 k
= isl_basic_map_alloc_inequality(path
);
477 path_c
= path
->ineq
[k
];
481 isl_seq_clr(path_c
, 1 + isl_basic_map_total_dim(path
));
483 isl_seq_cpy(path_c
+ off
,
484 delta_c
[i
] + 1 + nparam
, d
);
485 isl_int_set(path_c
[off
+ d
], delta_c
[i
][0]);
486 } else if (p
== PURE_PARAM
) {
487 isl_seq_cpy(path_c
, delta_c
[i
], 1 + nparam
);
489 isl_seq_cpy(path_c
+ off
,
490 delta_c
[i
] + 1 + nparam
, d
);
491 isl_seq_cpy(path_c
, delta_c
[i
], 1 + nparam
);
493 isl_seq_cpy(path_c
+ off
- n_div
,
494 delta_c
[i
] + 1 + nparam
+ d
, n_div
);
499 isl_basic_map_free(path
);
503 /* Given a set of offsets "delta", construct a relation of the
504 * given dimension specification (Z^{n+1} -> Z^{n+1}) that
505 * is an overapproximation of the relations that
506 * maps an element x to any element that can be reached
507 * by taking a non-negative number of steps along any of
508 * the elements in "delta".
509 * That is, construct an approximation of
511 * { [x] -> [y] : exists f \in \delta, k \in Z :
512 * y = x + k [f, 1] and k >= 0 }
514 * For any element in this relation, the number of steps taken
515 * is equal to the difference in the final coordinates.
517 * In particular, let delta be defined as
519 * \delta = [p] -> { [x] : A x + a >= 0 and B p + b >= 0 and
520 * C x + C'p + c >= 0 and
521 * D x + D'p + d >= 0 }
523 * where the constraints C x + C'p + c >= 0 are such that the parametric
524 * constant term of each constraint j, "C_j x + C'_j p + c_j",
525 * can never attain positive values, then the relation is constructed as
527 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
528 * A f + k a >= 0 and B p + b >= 0 and
529 * C f + C'p + c >= 0 and k >= 1 }
530 * union { [x] -> [x] }
532 * If the zero-length paths happen to correspond exactly to the identity
533 * mapping, then we return
535 * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
536 * A f + k a >= 0 and B p + b >= 0 and
537 * C f + C'p + c >= 0 and k >= 0 }
541 * Existentially quantified variables in \delta are handled by
542 * classifying them as independent of the parameters, purely
543 * parameter dependent and others. Constraints containing
544 * any of the other existentially quantified variables are removed.
545 * This is safe, but leads to an additional overapproximation.
547 * If there are any impure constraints, then we also eliminate
548 * the parameters from \delta, resulting in a set
550 * \delta' = { [x] : E x + e >= 0 }
552 * and add the constraints
556 * to the constructed relation.
558 static __isl_give isl_map
*path_along_delta(__isl_take isl_dim
*dim
,
559 __isl_take isl_basic_set
*delta
)
561 isl_basic_map
*path
= NULL
;
568 int *div_purity
= NULL
;
573 n_div
= isl_basic_set_dim(delta
, isl_dim_div
);
574 d
= isl_basic_set_dim(delta
, isl_dim_set
);
575 nparam
= isl_basic_set_dim(delta
, isl_dim_param
);
576 path
= isl_basic_map_alloc_dim(isl_dim_copy(dim
), n_div
+ d
+ 1,
577 d
+ 1 + delta
->n_eq
, delta
->n_eq
+ delta
->n_ineq
+ 1);
578 off
= 1 + nparam
+ 2 * (d
+ 1) + n_div
;
580 for (i
= 0; i
< n_div
+ d
+ 1; ++i
) {
581 k
= isl_basic_map_alloc_div(path
);
584 isl_int_set_si(path
->div
[k
][0], 0);
587 for (i
= 0; i
< d
+ 1; ++i
) {
588 k
= isl_basic_map_alloc_equality(path
);
591 isl_seq_clr(path
->eq
[k
], 1 + isl_basic_map_total_dim(path
));
592 isl_int_set_si(path
->eq
[k
][1 + nparam
+ i
], 1);
593 isl_int_set_si(path
->eq
[k
][1 + nparam
+ d
+ 1 + i
], -1);
594 isl_int_set_si(path
->eq
[k
][off
+ i
], 1);
597 div_purity
= get_div_purity(delta
);
601 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
602 div_purity
, 1, &impurity
);
603 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
604 div_purity
, 0, &impurity
);
606 isl_dim
*dim
= isl_basic_set_get_dim(delta
);
607 delta
= isl_basic_set_project_out(delta
,
608 isl_dim_param
, 0, nparam
);
609 delta
= isl_basic_set_add(delta
, isl_dim_param
, nparam
);
610 delta
= isl_basic_set_reset_dim(delta
, dim
);
613 path
= isl_basic_map_extend_constraints(path
, delta
->n_eq
,
615 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
617 path
= add_delta_constraints(path
, delta
, off
, nparam
, d
,
619 path
= isl_basic_map_gauss(path
, NULL
);
622 is_id
= empty_path_is_identity(path
, off
+ d
);
626 k
= isl_basic_map_alloc_inequality(path
);
629 isl_seq_clr(path
->ineq
[k
], 1 + isl_basic_map_total_dim(path
));
631 isl_int_set_si(path
->ineq
[k
][0], -1);
632 isl_int_set_si(path
->ineq
[k
][off
+ d
], 1);
635 isl_basic_set_free(delta
);
636 path
= isl_basic_map_finalize(path
);
639 return isl_map_from_basic_map(path
);
641 return isl_basic_map_union(path
,
642 isl_basic_map_identity(isl_dim_domain(dim
)));
646 isl_basic_set_free(delta
);
647 isl_basic_map_free(path
);
651 /* Given a dimension specification Z^{n+1} -> Z^{n+1} and a parameter "param",
652 * construct a map that equates the parameter to the difference
653 * in the final coordinates and imposes that this difference is positive.
656 * { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
658 static __isl_give isl_map
*equate_parameter_to_length(__isl_take isl_dim
*dim
,
661 struct isl_basic_map
*bmap
;
666 d
= isl_dim_size(dim
, isl_dim_in
);
667 nparam
= isl_dim_size(dim
, isl_dim_param
);
668 bmap
= isl_basic_map_alloc_dim(dim
, 0, 1, 1);
669 k
= isl_basic_map_alloc_equality(bmap
);
672 isl_seq_clr(bmap
->eq
[k
], 1 + isl_basic_map_total_dim(bmap
));
673 isl_int_set_si(bmap
->eq
[k
][1 + param
], -1);
674 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
- 1], -1);
675 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
+ d
- 1], 1);
677 k
= isl_basic_map_alloc_inequality(bmap
);
680 isl_seq_clr(bmap
->ineq
[k
], 1 + isl_basic_map_total_dim(bmap
));
681 isl_int_set_si(bmap
->ineq
[k
][1 + param
], 1);
682 isl_int_set_si(bmap
->ineq
[k
][0], -1);
684 bmap
= isl_basic_map_finalize(bmap
);
685 return isl_map_from_basic_map(bmap
);
687 isl_basic_map_free(bmap
);
691 /* Check whether "path" is acyclic, where the last coordinates of domain
692 * and range of path encode the number of steps taken.
693 * That is, check whether
695 * { d | d = y - x and (x,y) in path }
697 * does not contain any element with positive last coordinate (positive length)
698 * and zero remaining coordinates (cycle).
700 static int is_acyclic(__isl_take isl_map
*path
)
705 struct isl_set
*delta
;
707 delta
= isl_map_deltas(path
);
708 dim
= isl_set_dim(delta
, isl_dim_set
);
709 for (i
= 0; i
< dim
; ++i
) {
711 delta
= isl_set_lower_bound_si(delta
, isl_dim_set
, i
, 1);
713 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
716 acyclic
= isl_set_is_empty(delta
);
722 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
723 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
724 * construct a map that is an overapproximation of the map
725 * that takes an element from the space D \times Z to another
726 * element from the same space, such that the first n coordinates of the
727 * difference between them is a sum of differences between images
728 * and pre-images in one of the R_i and such that the last coordinate
729 * is equal to the number of steps taken.
732 * \Delta_i = { y - x | (x, y) in R_i }
734 * then the constructed map is an overapproximation of
736 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
737 * d = (\sum_i k_i \delta_i, \sum_i k_i) }
739 * The elements of the singleton \Delta_i's are collected as the
740 * rows of the steps matrix. For all these \Delta_i's together,
741 * a single path is constructed.
742 * For each of the other \Delta_i's, we compute an overapproximation
743 * of the paths along elements of \Delta_i.
744 * Since each of these paths performs an addition, composition is
745 * symmetric and we can simply compose all resulting paths in any order.
747 static __isl_give isl_map
*construct_extended_path(__isl_take isl_dim
*dim
,
748 __isl_keep isl_map
*map
, int *project
)
750 struct isl_mat
*steps
= NULL
;
751 struct isl_map
*path
= NULL
;
755 d
= isl_map_dim(map
, isl_dim_in
);
757 path
= isl_map_identity(isl_dim_domain(isl_dim_copy(dim
)));
759 steps
= isl_mat_alloc(map
->ctx
, map
->n
, d
);
764 for (i
= 0; i
< map
->n
; ++i
) {
765 struct isl_basic_set
*delta
;
767 delta
= isl_basic_map_deltas(isl_basic_map_copy(map
->p
[i
]));
769 for (j
= 0; j
< d
; ++j
) {
772 fixed
= isl_basic_set_fast_dim_is_fixed(delta
, j
,
775 isl_basic_set_free(delta
);
784 path
= isl_map_apply_range(path
,
785 path_along_delta(isl_dim_copy(dim
), delta
));
786 path
= isl_map_coalesce(path
);
788 isl_basic_set_free(delta
);
795 path
= isl_map_apply_range(path
,
796 path_along_steps(isl_dim_copy(dim
), steps
));
799 if (project
&& *project
) {
800 *project
= is_acyclic(isl_map_copy(path
));
815 static int isl_set_overlaps(__isl_keep isl_set
*set1
, __isl_keep isl_set
*set2
)
820 if (!isl_dim_tuple_match(set1
->dim
, isl_dim_set
, set2
->dim
, isl_dim_set
))
823 i
= isl_set_intersect(isl_set_copy(set1
), isl_set_copy(set2
));
824 no_overlap
= isl_set_is_empty(i
);
827 return no_overlap
< 0 ? -1 : !no_overlap
;
830 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
831 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
832 * construct a map that is an overapproximation of the map
833 * that takes an element from the dom R \times Z to an
834 * element from ran R \times Z, such that the first n coordinates of the
835 * difference between them is a sum of differences between images
836 * and pre-images in one of the R_i and such that the last coordinate
837 * is equal to the number of steps taken.
840 * \Delta_i = { y - x | (x, y) in R_i }
842 * then the constructed map is an overapproximation of
844 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
845 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
846 * x in dom R and x + d in ran R and
849 static __isl_give isl_map
*construct_component(__isl_take isl_dim
*dim
,
850 __isl_keep isl_map
*map
, int *exact
, int project
)
852 struct isl_set
*domain
= NULL
;
853 struct isl_set
*range
= NULL
;
854 struct isl_map
*app
= NULL
;
855 struct isl_map
*path
= NULL
;
857 domain
= isl_map_domain(isl_map_copy(map
));
858 domain
= isl_set_coalesce(domain
);
859 range
= isl_map_range(isl_map_copy(map
));
860 range
= isl_set_coalesce(range
);
861 if (!isl_set_overlaps(domain
, range
)) {
862 isl_set_free(domain
);
866 map
= isl_map_copy(map
);
867 map
= isl_map_add_dims(map
, isl_dim_in
, 1);
868 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
869 map
= set_path_length(map
, 1, 1);
872 app
= isl_map_from_domain_and_range(domain
, range
);
873 app
= isl_map_add_dims(app
, isl_dim_in
, 1);
874 app
= isl_map_add_dims(app
, isl_dim_out
, 1);
876 path
= construct_extended_path(isl_dim_copy(dim
), map
,
877 exact
&& *exact
? &project
: NULL
);
878 app
= isl_map_intersect(app
, path
);
880 if (exact
&& *exact
&&
881 (*exact
= check_exactness(isl_map_copy(map
), isl_map_copy(app
),
886 app
= set_path_length(app
, 0, 1);
894 /* Call construct_component and, if "project" is set, project out
895 * the final coordinates.
897 static __isl_give isl_map
*construct_projected_component(
898 __isl_take isl_dim
*dim
,
899 __isl_keep isl_map
*map
, int *exact
, int project
)
906 d
= isl_dim_size(dim
, isl_dim_in
);
908 app
= construct_component(dim
, map
, exact
, project
);
910 app
= isl_map_project_out(app
, isl_dim_in
, d
- 1, 1);
911 app
= isl_map_project_out(app
, isl_dim_out
, d
- 1, 1);
916 /* Compute an extended version, i.e., with path lengths, of
917 * an overapproximation of the transitive closure of "bmap"
918 * with path lengths greater than or equal to zero and with
919 * domain and range equal to "dom".
921 static __isl_give isl_map
*q_closure(__isl_take isl_dim
*dim
,
922 __isl_take isl_set
*dom
, __isl_keep isl_basic_map
*bmap
, int *exact
)
929 dom
= isl_set_add_dims(dom
, isl_dim_set
, 1);
930 app
= isl_map_from_domain_and_range(dom
, isl_set_copy(dom
));
931 map
= isl_map_from_basic_map(isl_basic_map_copy(bmap
));
932 path
= construct_extended_path(dim
, map
, &project
);
933 app
= isl_map_intersect(app
, path
);
935 if ((*exact
= check_exactness(map
, isl_map_copy(app
), project
)) < 0)
944 /* Check whether qc has any elements of length at least one
945 * with domain and/or range outside of dom and ran.
947 static int has_spurious_elements(__isl_keep isl_map
*qc
,
948 __isl_keep isl_set
*dom
, __isl_keep isl_set
*ran
)
954 if (!qc
|| !dom
|| !ran
)
957 d
= isl_map_dim(qc
, isl_dim_in
);
959 qc
= isl_map_copy(qc
);
960 qc
= set_path_length(qc
, 0, 1);
961 qc
= isl_map_project_out(qc
, isl_dim_in
, d
- 1, 1);
962 qc
= isl_map_project_out(qc
, isl_dim_out
, d
- 1, 1);
964 s
= isl_map_domain(isl_map_copy(qc
));
965 subset
= isl_set_is_subset(s
, dom
);
974 s
= isl_map_range(qc
);
975 subset
= isl_set_is_subset(s
, ran
);
978 return subset
< 0 ? -1 : !subset
;
987 /* For each basic map in "map", except i, check whether it combines
988 * with the transitive closure that is reflexive on C combines
989 * to the left and to the right.
993 * dom map_j \subseteq C
995 * then right[j] is set to 1. Otherwise, if
997 * ran map_i \cap dom map_j = \emptyset
999 * then right[j] is set to 0. Otherwise, composing to the right
1002 * Similar, for composing to the left, we have if
1004 * ran map_j \subseteq C
1006 * then left[j] is set to 1. Otherwise, if
1008 * dom map_i \cap ran map_j = \emptyset
1010 * then left[j] is set to 0. Otherwise, composing to the left
1013 * The return value is or'd with LEFT if composing to the left
1014 * is possible and with RIGHT if composing to the right is possible.
1016 static int composability(__isl_keep isl_set
*C
, int i
,
1017 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
1018 __isl_keep isl_map
*map
)
1024 for (j
= 0; j
< map
->n
&& ok
; ++j
) {
1025 int overlaps
, subset
;
1031 dom
[j
] = isl_set_from_basic_set(
1032 isl_basic_map_domain(
1033 isl_basic_map_copy(map
->p
[j
])));
1036 overlaps
= isl_set_overlaps(ran
[i
], dom
[j
]);
1042 subset
= isl_set_is_subset(dom
[j
], C
);
1054 ran
[j
] = isl_set_from_basic_set(
1055 isl_basic_map_range(
1056 isl_basic_map_copy(map
->p
[j
])));
1059 overlaps
= isl_set_overlaps(dom
[i
], ran
[j
]);
1065 subset
= isl_set_is_subset(ran
[j
], C
);
1079 static __isl_give isl_map
*anonymize(__isl_take isl_map
*map
)
1081 map
= isl_map_reset(map
, isl_dim_in
);
1082 map
= isl_map_reset(map
, isl_dim_out
);
1086 /* Return a map that is a union of the basic maps in "map", except i,
1087 * composed to left and right with qc based on the entries of "left"
1090 static __isl_give isl_map
*compose(__isl_keep isl_map
*map
, int i
,
1091 __isl_take isl_map
*qc
, int *left
, int *right
)
1096 comp
= isl_map_empty(isl_map_get_dim(map
));
1097 for (j
= 0; j
< map
->n
; ++j
) {
1103 map_j
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[j
]));
1104 map_j
= anonymize(map_j
);
1105 if (left
&& left
[j
])
1106 map_j
= isl_map_apply_range(map_j
, isl_map_copy(qc
));
1107 if (right
&& right
[j
])
1108 map_j
= isl_map_apply_range(isl_map_copy(qc
), map_j
);
1109 comp
= isl_map_union(comp
, map_j
);
1112 comp
= isl_map_compute_divs(comp
);
1113 comp
= isl_map_coalesce(comp
);
1120 /* Compute the transitive closure of "map" incrementally by
1127 * map_i^+ \cup ((id \cup map_i^) \circ qc^+)
1131 * map_i^+ \cup (qc^+ \circ (id \cup map_i^))
1133 * depending on whether left or right are NULL.
1135 static __isl_give isl_map
*compute_incremental(
1136 __isl_take isl_dim
*dim
, __isl_keep isl_map
*map
,
1137 int i
, __isl_take isl_map
*qc
, int *left
, int *right
, int *exact
)
1141 isl_map
*rtc
= NULL
;
1145 isl_assert(map
->ctx
, left
|| right
, goto error
);
1147 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
1148 tc
= construct_projected_component(isl_dim_copy(dim
), map_i
,
1150 isl_map_free(map_i
);
1153 qc
= isl_map_transitive_closure(qc
, exact
);
1159 return isl_map_universe(isl_map_get_dim(map
));
1162 if (!left
|| !right
)
1163 rtc
= isl_map_union(isl_map_copy(tc
),
1164 isl_map_identity(isl_dim_domain(isl_map_get_dim(tc
))));
1166 qc
= isl_map_apply_range(rtc
, qc
);
1168 qc
= isl_map_apply_range(qc
, rtc
);
1169 qc
= isl_map_union(tc
, qc
);
1180 /* Given a map "map", try to find a basic map such that
1181 * map^+ can be computed as
1183 * map^+ = map_i^+ \cup
1184 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1186 * with C the simple hull of the domain and range of the input map.
1187 * map_i^ \cup Id_C is computed by allowing the path lengths to be zero
1188 * and by intersecting domain and range with C.
1189 * Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
1190 * Also, we only use the incremental computation if all the transitive
1191 * closures are exact and if the number of basic maps in the union,
1192 * after computing the integer divisions, is smaller than the number
1193 * of basic maps in the input map.
1195 static int incemental_on_entire_domain(__isl_keep isl_dim
*dim
,
1196 __isl_keep isl_map
*map
,
1197 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
1198 __isl_give isl_map
**res
)
1206 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
1207 isl_map_range(isl_map_copy(map
)));
1208 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1216 d
= isl_map_dim(map
, isl_dim_in
);
1218 for (i
= 0; i
< map
->n
; ++i
) {
1220 int exact_i
, spurious
;
1222 dom
[i
] = isl_set_from_basic_set(isl_basic_map_domain(
1223 isl_basic_map_copy(map
->p
[i
])));
1224 ran
[i
] = isl_set_from_basic_set(isl_basic_map_range(
1225 isl_basic_map_copy(map
->p
[i
])));
1226 qc
= q_closure(isl_dim_copy(dim
), isl_set_copy(C
),
1227 map
->p
[i
], &exact_i
);
1234 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1241 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1242 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1243 qc
= isl_map_compute_divs(qc
);
1244 for (j
= 0; j
< map
->n
; ++j
)
1245 left
[j
] = right
[j
] = 1;
1246 qc
= compose(map
, i
, qc
, left
, right
);
1249 if (qc
->n
>= map
->n
) {
1253 *res
= compute_incremental(isl_dim_copy(dim
), map
, i
, qc
,
1254 left
, right
, &exact_i
);
1265 return *res
!= NULL
;
1271 /* Try and compute the transitive closure of "map" as
1273 * map^+ = map_i^+ \cup
1274 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1276 * with C either the simple hull of the domain and range of the entire
1277 * map or the simple hull of domain and range of map_i.
1279 static __isl_give isl_map
*incremental_closure(__isl_take isl_dim
*dim
,
1280 __isl_keep isl_map
*map
, int *exact
, int project
)
1283 isl_set
**dom
= NULL
;
1284 isl_set
**ran
= NULL
;
1289 isl_map
*res
= NULL
;
1292 return construct_projected_component(dim
, map
, exact
, project
);
1297 return construct_projected_component(dim
, map
, exact
, project
);
1299 d
= isl_map_dim(map
, isl_dim_in
);
1301 dom
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1302 ran
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1303 left
= isl_calloc_array(map
->ctx
, int, map
->n
);
1304 right
= isl_calloc_array(map
->ctx
, int, map
->n
);
1305 if (!ran
|| !dom
|| !left
|| !right
)
1308 if (incemental_on_entire_domain(dim
, map
, dom
, ran
, left
, right
, &res
) < 0)
1311 for (i
= 0; !res
&& i
< map
->n
; ++i
) {
1313 int exact_i
, spurious
, comp
;
1315 dom
[i
] = isl_set_from_basic_set(
1316 isl_basic_map_domain(
1317 isl_basic_map_copy(map
->p
[i
])));
1321 ran
[i
] = isl_set_from_basic_set(
1322 isl_basic_map_range(
1323 isl_basic_map_copy(map
->p
[i
])));
1326 C
= isl_set_union(isl_set_copy(dom
[i
]),
1327 isl_set_copy(ran
[i
]));
1328 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1335 comp
= composability(C
, i
, dom
, ran
, left
, right
, map
);
1336 if (!comp
|| comp
< 0) {
1342 qc
= q_closure(isl_dim_copy(dim
), C
, map
->p
[i
], &exact_i
);
1349 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1356 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1357 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1358 qc
= isl_map_compute_divs(qc
);
1359 qc
= compose(map
, i
, qc
, (comp
& LEFT
) ? left
: NULL
,
1360 (comp
& RIGHT
) ? right
: NULL
);
1363 if (qc
->n
>= map
->n
) {
1367 res
= compute_incremental(isl_dim_copy(dim
), map
, i
, qc
,
1368 (comp
& LEFT
) ? left
: NULL
,
1369 (comp
& RIGHT
) ? right
: NULL
, &exact_i
);
1378 for (i
= 0; i
< map
->n
; ++i
) {
1379 isl_set_free(dom
[i
]);
1380 isl_set_free(ran
[i
]);
1392 return construct_projected_component(dim
, map
, exact
, project
);
1395 for (i
= 0; i
< map
->n
; ++i
)
1396 isl_set_free(dom
[i
]);
1399 for (i
= 0; i
< map
->n
; ++i
)
1400 isl_set_free(ran
[i
]);
1408 /* Given an array of sets "set", add "dom" at position "pos"
1409 * and search for elements at earlier positions that overlap with "dom".
1410 * If any can be found, then merge all of them, together with "dom", into
1411 * a single set and assign the union to the first in the array,
1412 * which becomes the new group leader for all groups involved in the merge.
1413 * During the search, we only consider group leaders, i.e., those with
1414 * group[i] = i, as the other sets have already been combined
1415 * with one of the group leaders.
1417 static int merge(isl_set
**set
, int *group
, __isl_take isl_set
*dom
, int pos
)
1422 set
[pos
] = isl_set_copy(dom
);
1424 for (i
= pos
- 1; i
>= 0; --i
) {
1430 o
= isl_set_overlaps(set
[i
], dom
);
1436 set
[i
] = isl_set_union(set
[i
], set
[group
[pos
]]);
1437 set
[group
[pos
]] = NULL
;
1440 group
[group
[pos
]] = i
;
1451 /* Replace each entry in the n by n grid of maps by the cross product
1452 * with the relation { [i] -> [i + 1] }.
1454 static int add_length(__isl_keep isl_map
*map
, isl_map
***grid
, int n
)
1458 isl_basic_map
*bstep
;
1465 dim
= isl_map_get_dim(map
);
1466 nparam
= isl_dim_size(dim
, isl_dim_param
);
1467 dim
= isl_dim_drop(dim
, isl_dim_in
, 0, isl_dim_size(dim
, isl_dim_in
));
1468 dim
= isl_dim_drop(dim
, isl_dim_out
, 0, isl_dim_size(dim
, isl_dim_out
));
1469 dim
= isl_dim_add(dim
, isl_dim_in
, 1);
1470 dim
= isl_dim_add(dim
, isl_dim_out
, 1);
1471 bstep
= isl_basic_map_alloc_dim(dim
, 0, 1, 0);
1472 k
= isl_basic_map_alloc_equality(bstep
);
1474 isl_basic_map_free(bstep
);
1477 isl_seq_clr(bstep
->eq
[k
], 1 + isl_basic_map_total_dim(bstep
));
1478 isl_int_set_si(bstep
->eq
[k
][0], 1);
1479 isl_int_set_si(bstep
->eq
[k
][1 + nparam
], 1);
1480 isl_int_set_si(bstep
->eq
[k
][1 + nparam
+ 1], -1);
1481 bstep
= isl_basic_map_finalize(bstep
);
1482 step
= isl_map_from_basic_map(bstep
);
1484 for (i
= 0; i
< n
; ++i
)
1485 for (j
= 0; j
< n
; ++j
)
1486 grid
[i
][j
] = isl_map_product(grid
[i
][j
],
1487 isl_map_copy(step
));
1494 /* The core of the Floyd-Warshall algorithm.
1495 * Updates the given n x x matrix of relations in place.
1497 * The algorithm iterates over all vertices. In each step, the whole
1498 * matrix is updated to include all paths that go to the current vertex,
1499 * possibly stay there a while (including passing through earlier vertices)
1500 * and then come back. At the start of each iteration, the diagonal
1501 * element corresponding to the current vertex is replaced by its
1502 * transitive closure to account for all indirect paths that stay
1503 * in the current vertex.
1505 static void floyd_warshall_iterate(isl_map
***grid
, int n
, int *exact
)
1509 for (r
= 0; r
< n
; ++r
) {
1511 grid
[r
][r
] = isl_map_transitive_closure(grid
[r
][r
],
1512 (exact
&& *exact
) ? &r_exact
: NULL
);
1513 if (exact
&& *exact
&& !r_exact
)
1516 for (p
= 0; p
< n
; ++p
)
1517 for (q
= 0; q
< n
; ++q
) {
1519 if (p
== r
&& q
== r
)
1521 loop
= isl_map_apply_range(
1522 isl_map_copy(grid
[p
][r
]),
1523 isl_map_copy(grid
[r
][q
]));
1524 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1525 loop
= isl_map_apply_range(
1526 isl_map_copy(grid
[p
][r
]),
1527 isl_map_apply_range(
1528 isl_map_copy(grid
[r
][r
]),
1529 isl_map_copy(grid
[r
][q
])));
1530 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1531 grid
[p
][q
] = isl_map_coalesce(grid
[p
][q
]);
1536 /* Given a partition of the domains and ranges of the basic maps in "map",
1537 * apply the Floyd-Warshall algorithm with the elements in the partition
1540 * In particular, there are "n" elements in the partition and "group" is
1541 * an array of length 2 * map->n with entries in [0,n-1].
1543 * We first construct a matrix of relations based on the partition information,
1544 * apply Floyd-Warshall on this matrix of relations and then take the
1545 * union of all entries in the matrix as the final result.
1547 * If we are actually computing the power instead of the transitive closure,
1548 * i.e., when "project" is not set, then the result should have the
1549 * path lengths encoded as the difference between an extra pair of
1550 * coordinates. We therefore apply the nested transitive closures
1551 * to relations that include these lengths. In particular, we replace
1552 * the input relation by the cross product with the unit length relation
1553 * { [i] -> [i + 1] }.
1555 static __isl_give isl_map
*floyd_warshall_with_groups(__isl_take isl_dim
*dim
,
1556 __isl_keep isl_map
*map
, int *exact
, int project
, int *group
, int n
)
1559 isl_map
***grid
= NULL
;
1567 return incremental_closure(dim
, map
, exact
, project
);
1570 grid
= isl_calloc_array(map
->ctx
, isl_map
**, n
);
1573 for (i
= 0; i
< n
; ++i
) {
1574 grid
[i
] = isl_calloc_array(map
->ctx
, isl_map
*, n
);
1577 for (j
= 0; j
< n
; ++j
)
1578 grid
[i
][j
] = isl_map_empty(isl_map_get_dim(map
));
1581 for (k
= 0; k
< map
->n
; ++k
) {
1583 j
= group
[2 * k
+ 1];
1584 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
1585 isl_map_from_basic_map(
1586 isl_basic_map_copy(map
->p
[k
])));
1589 if (!project
&& add_length(map
, grid
, n
) < 0)
1592 floyd_warshall_iterate(grid
, n
, exact
);
1594 app
= isl_map_empty(isl_map_get_dim(map
));
1596 for (i
= 0; i
< n
; ++i
) {
1597 for (j
= 0; j
< n
; ++j
)
1598 app
= isl_map_union(app
, grid
[i
][j
]);
1609 for (i
= 0; i
< n
; ++i
) {
1612 for (j
= 0; j
< n
; ++j
)
1613 isl_map_free(grid
[i
][j
]);
1622 /* Partition the domains and ranges of the n basic relations in list
1623 * into disjoint cells.
1625 * To find the partition, we simply consider all of the domains
1626 * and ranges in turn and combine those that overlap.
1627 * "set" contains the partition elements and "group" indicates
1628 * to which partition element a given domain or range belongs.
1629 * The domain of basic map i corresponds to element 2 * i in these arrays,
1630 * while the domain corresponds to element 2 * i + 1.
1631 * During the construction group[k] is either equal to k,
1632 * in which case set[k] contains the union of all the domains and
1633 * ranges in the corresponding group, or is equal to some l < k,
1634 * with l another domain or range in the same group.
1636 static int *setup_groups(isl_ctx
*ctx
, __isl_keep isl_basic_map
**list
, int n
,
1637 isl_set
***set
, int *n_group
)
1643 *set
= isl_calloc_array(ctx
, isl_set
*, 2 * n
);
1644 group
= isl_alloc_array(ctx
, int, 2 * n
);
1646 if (!*set
|| !group
)
1649 for (i
= 0; i
< n
; ++i
) {
1651 dom
= isl_set_from_basic_set(isl_basic_map_domain(
1652 isl_basic_map_copy(list
[i
])));
1653 if (merge(*set
, group
, dom
, 2 * i
) < 0)
1655 dom
= isl_set_from_basic_set(isl_basic_map_range(
1656 isl_basic_map_copy(list
[i
])));
1657 if (merge(*set
, group
, dom
, 2 * i
+ 1) < 0)
1662 for (i
= 0; i
< 2 * n
; ++i
)
1663 if (group
[i
] == i
) {
1665 (*set
)[g
] = (*set
)[i
];
1670 group
[i
] = group
[group
[i
]];
1677 for (i
= 0; i
< 2 * n
; ++i
)
1678 isl_set_free((*set
)[i
]);
1686 /* Check if the domains and ranges of the basic maps in "map" can
1687 * be partitioned, and if so, apply Floyd-Warshall on the elements
1688 * of the partition. Note that we also apply this algorithm
1689 * if we want to compute the power, i.e., when "project" is not set.
1690 * However, the results are unlikely to be exact since the recursive
1691 * calls inside the Floyd-Warshall algorithm typically result in
1692 * non-linear path lengths quite quickly.
1694 static __isl_give isl_map
*floyd_warshall(__isl_take isl_dim
*dim
,
1695 __isl_keep isl_map
*map
, int *exact
, int project
)
1698 isl_set
**set
= NULL
;
1705 return incremental_closure(dim
, map
, exact
, project
);
1707 group
= setup_groups(map
->ctx
, map
->p
, map
->n
, &set
, &n
);
1711 for (i
= 0; i
< 2 * map
->n
; ++i
)
1712 isl_set_free(set
[i
]);
1716 return floyd_warshall_with_groups(dim
, map
, exact
, project
, group
, n
);
1722 /* Structure for representing the nodes in the graph being traversed
1723 * using Tarjan's algorithm.
1724 * index represents the order in which nodes are visited.
1725 * min_index is the index of the root of a (sub)component.
1726 * on_stack indicates whether the node is currently on the stack.
1728 struct basic_map_sort_node
{
1733 /* Structure for representing the graph being traversed
1734 * using Tarjan's algorithm.
1735 * len is the number of nodes
1736 * node is an array of nodes
1737 * stack contains the nodes on the path from the root to the current node
1738 * sp is the stack pointer
1739 * index is the index of the last node visited
1740 * order contains the elements of the components separated by -1
1741 * op represents the current position in order
1743 * check_closed is set if we may have used the fact that
1744 * a pair of basic maps can be interchanged
1746 struct basic_map_sort
{
1748 struct basic_map_sort_node
*node
;
1757 static void basic_map_sort_free(struct basic_map_sort
*s
)
1767 static struct basic_map_sort
*basic_map_sort_alloc(struct isl_ctx
*ctx
, int len
)
1769 struct basic_map_sort
*s
;
1772 s
= isl_calloc_type(ctx
, struct basic_map_sort
);
1776 s
->node
= isl_alloc_array(ctx
, struct basic_map_sort_node
, len
);
1779 for (i
= 0; i
< len
; ++i
)
1780 s
->node
[i
].index
= -1;
1781 s
->stack
= isl_alloc_array(ctx
, int, len
);
1784 s
->order
= isl_alloc_array(ctx
, int, 2 * len
);
1792 s
->check_closed
= 0;
1796 basic_map_sort_free(s
);
1800 /* Check whether in the computation of the transitive closure
1801 * "bmap1" (R_1) should follow (or be part of the same component as)
1804 * That is check whether
1812 * If so, then there is no reason for R_1 to immediately follow R_2
1815 * *check_closed is set if the subset relation holds while
1816 * R_1 \circ R_2 is not empty.
1818 static int basic_map_follows(__isl_keep isl_basic_map
*bmap1
,
1819 __isl_keep isl_basic_map
*bmap2
, int *check_closed
)
1821 struct isl_map
*map12
= NULL
;
1822 struct isl_map
*map21
= NULL
;
1825 if (!isl_dim_tuple_match(bmap1
->dim
, isl_dim_in
, bmap2
->dim
, isl_dim_out
))
1828 map21
= isl_map_from_basic_map(
1829 isl_basic_map_apply_range(
1830 isl_basic_map_copy(bmap2
),
1831 isl_basic_map_copy(bmap1
)));
1832 subset
= isl_map_is_empty(map21
);
1836 isl_map_free(map21
);
1840 if (!isl_dim_tuple_match(bmap1
->dim
, isl_dim_in
, bmap1
->dim
, isl_dim_out
) ||
1841 !isl_dim_tuple_match(bmap2
->dim
, isl_dim_in
, bmap2
->dim
, isl_dim_out
)) {
1842 isl_map_free(map21
);
1846 map12
= isl_map_from_basic_map(
1847 isl_basic_map_apply_range(
1848 isl_basic_map_copy(bmap1
),
1849 isl_basic_map_copy(bmap2
)));
1851 subset
= isl_map_is_subset(map21
, map12
);
1853 isl_map_free(map12
);
1854 isl_map_free(map21
);
1859 return subset
< 0 ? -1 : !subset
;
1861 isl_map_free(map21
);
1865 /* Perform Tarjan's algorithm for computing the strongly connected components
1866 * in the graph with the disjuncts of "map" as vertices and with an
1867 * edge between any pair of disjuncts such that the first has
1868 * to be applied after the second.
1870 static int power_components_tarjan(struct basic_map_sort
*s
,
1871 __isl_keep isl_basic_map
**list
, int i
)
1875 s
->node
[i
].index
= s
->index
;
1876 s
->node
[i
].min_index
= s
->index
;
1877 s
->node
[i
].on_stack
= 1;
1879 s
->stack
[s
->sp
++] = i
;
1881 for (j
= s
->len
- 1; j
>= 0; --j
) {
1886 if (s
->node
[j
].index
>= 0 &&
1887 (!s
->node
[j
].on_stack
||
1888 s
->node
[j
].index
> s
->node
[i
].min_index
))
1891 f
= basic_map_follows(list
[i
], list
[j
], &s
->check_closed
);
1897 if (s
->node
[j
].index
< 0) {
1898 power_components_tarjan(s
, list
, j
);
1899 if (s
->node
[j
].min_index
< s
->node
[i
].min_index
)
1900 s
->node
[i
].min_index
= s
->node
[j
].min_index
;
1901 } else if (s
->node
[j
].index
< s
->node
[i
].min_index
)
1902 s
->node
[i
].min_index
= s
->node
[j
].index
;
1905 if (s
->node
[i
].index
!= s
->node
[i
].min_index
)
1909 j
= s
->stack
[--s
->sp
];
1910 s
->node
[j
].on_stack
= 0;
1911 s
->order
[s
->op
++] = j
;
1913 s
->order
[s
->op
++] = -1;
1918 /* Decompose the "len" basic relations in "list" into strongly connected
1921 static struct basic_map_sort
*basic_map_sort_init(isl_ctx
*ctx
, int len
,
1922 __isl_keep isl_basic_map
**list
)
1925 struct basic_map_sort
*s
= NULL
;
1927 s
= basic_map_sort_alloc(ctx
, len
);
1930 for (i
= len
- 1; i
>= 0; --i
) {
1931 if (s
->node
[i
].index
>= 0)
1933 if (power_components_tarjan(s
, list
, i
) < 0)
1939 basic_map_sort_free(s
);
1943 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
1944 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
1945 * construct a map that is an overapproximation of the map
1946 * that takes an element from the dom R \times Z to an
1947 * element from ran R \times Z, such that the first n coordinates of the
1948 * difference between them is a sum of differences between images
1949 * and pre-images in one of the R_i and such that the last coordinate
1950 * is equal to the number of steps taken.
1951 * If "project" is set, then these final coordinates are not included,
1952 * i.e., a relation of type Z^n -> Z^n is returned.
1955 * \Delta_i = { y - x | (x, y) in R_i }
1957 * then the constructed map is an overapproximation of
1959 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1960 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
1961 * x in dom R and x + d in ran R }
1965 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1966 * d = (\sum_i k_i \delta_i) and
1967 * x in dom R and x + d in ran R }
1969 * if "project" is set.
1971 * We first split the map into strongly connected components, perform
1972 * the above on each component and then join the results in the correct
1973 * order, at each join also taking in the union of both arguments
1974 * to allow for paths that do not go through one of the two arguments.
1976 static __isl_give isl_map
*construct_power_components(__isl_take isl_dim
*dim
,
1977 __isl_keep isl_map
*map
, int *exact
, int project
)
1980 struct isl_map
*path
= NULL
;
1981 struct basic_map_sort
*s
= NULL
;
1988 return floyd_warshall(dim
, map
, exact
, project
);
1990 s
= basic_map_sort_init(map
->ctx
, map
->n
, map
->p
);
1995 if (s
->check_closed
&& !exact
)
1996 exact
= &local_exact
;
2002 path
= isl_map_empty(isl_map_get_dim(map
));
2004 path
= isl_map_empty(isl_dim_copy(dim
));
2005 path
= anonymize(path
);
2007 struct isl_map
*comp
;
2008 isl_map
*path_comp
, *path_comb
;
2009 comp
= isl_map_alloc_dim(isl_map_get_dim(map
), n
, 0);
2010 while (s
->order
[i
] != -1) {
2011 comp
= isl_map_add_basic_map(comp
,
2012 isl_basic_map_copy(map
->p
[s
->order
[i
]]));
2016 path_comp
= floyd_warshall(isl_dim_copy(dim
),
2017 comp
, exact
, project
);
2018 path_comp
= anonymize(path_comp
);
2019 path_comb
= isl_map_apply_range(isl_map_copy(path
),
2020 isl_map_copy(path_comp
));
2021 path
= isl_map_union(path
, path_comp
);
2022 path
= isl_map_union(path
, path_comb
);
2028 if (c
> 1 && s
->check_closed
&& !*exact
) {
2031 closed
= isl_map_is_transitively_closed(path
);
2035 basic_map_sort_free(s
);
2037 return floyd_warshall(dim
, map
, orig_exact
, project
);
2041 basic_map_sort_free(s
);
2046 basic_map_sort_free(s
);
2052 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
2053 * construct a map that is an overapproximation of the map
2054 * that takes an element from the space D to another
2055 * element from the same space, such that the difference between
2056 * them is a strictly positive sum of differences between images
2057 * and pre-images in one of the R_i.
2058 * The number of differences in the sum is equated to parameter "param".
2061 * \Delta_i = { y - x | (x, y) in R_i }
2063 * then the constructed map is an overapproximation of
2065 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
2066 * d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
2069 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
2070 * d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
2072 * if "project" is set.
2074 * If "project" is not set, then
2075 * we construct an extended mapping with an extra coordinate
2076 * that indicates the number of steps taken. In particular,
2077 * the difference in the last coordinate is equal to the number
2078 * of steps taken to move from a domain element to the corresponding
2081 static __isl_give isl_map
*construct_power(__isl_keep isl_map
*map
,
2082 int *exact
, int project
)
2084 struct isl_map
*app
= NULL
;
2085 struct isl_dim
*dim
= NULL
;
2091 dim
= isl_map_get_dim(map
);
2093 d
= isl_dim_size(dim
, isl_dim_in
);
2094 dim
= isl_dim_add(dim
, isl_dim_in
, 1);
2095 dim
= isl_dim_add(dim
, isl_dim_out
, 1);
2097 app
= construct_power_components(isl_dim_copy(dim
), map
,
2105 /* Compute the positive powers of "map", or an overapproximation.
2106 * If the result is exact, then *exact is set to 1.
2108 * If project is set, then we are actually interested in the transitive
2109 * closure, so we can use a more relaxed exactness check.
2110 * The lengths of the paths are also projected out instead of being
2111 * encoded as the difference between an extra pair of final coordinates.
2113 static __isl_give isl_map
*map_power(__isl_take isl_map
*map
,
2114 int *exact
, int project
)
2116 struct isl_map
*app
= NULL
;
2124 isl_assert(map
->ctx
,
2125 isl_map_dim(map
, isl_dim_in
) == isl_map_dim(map
, isl_dim_out
),
2128 app
= construct_power(map
, exact
, project
);
2138 /* Compute the positive powers of "map", or an overapproximation.
2139 * The power is given by parameter "param". If the result is exact,
2140 * then *exact is set to 1.
2141 * map_power constructs an extended relation with the path lengths
2142 * encoded as the difference between the final coordinates.
2143 * In the final step, this difference is equated to the parameter "param"
2144 * and made positive. The extra coordinates are subsequently projected out.
2146 __isl_give isl_map
*isl_map_power(__isl_take isl_map
*map
, unsigned param
,
2149 isl_dim
*target_dim
;
2157 isl_assert(map
->ctx
, param
< isl_map_dim(map
, isl_dim_param
),
2160 d
= isl_map_dim(map
, isl_dim_in
);
2162 map
= isl_map_compute_divs(map
);
2163 map
= isl_map_coalesce(map
);
2165 if (isl_map_fast_is_empty(map
))
2168 target_dim
= isl_map_get_dim(map
);
2169 map
= map_power(map
, exact
, 0);
2171 dim
= isl_map_get_dim(map
);
2172 diff
= equate_parameter_to_length(dim
, param
);
2173 map
= isl_map_intersect(map
, diff
);
2174 map
= isl_map_project_out(map
, isl_dim_in
, d
, 1);
2175 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2177 map
= isl_map_reset_dim(map
, target_dim
);
2185 /* Compute a relation that maps each element in the range of the input
2186 * relation to the lengths of all paths composed of edges in the input
2187 * relation that end up in the given range element.
2188 * The result may be an overapproximation, in which case *exact is set to 0.
2189 * The resulting relation is very similar to the power relation.
2190 * The difference are that the domain has been projected out, the
2191 * range has become the domain and the exponent is the range instead
2194 __isl_give isl_map
*isl_map_reaching_path_lengths(__isl_take isl_map
*map
,
2205 d
= isl_map_dim(map
, isl_dim_in
);
2206 param
= isl_map_dim(map
, isl_dim_param
);
2208 map
= isl_map_compute_divs(map
);
2209 map
= isl_map_coalesce(map
);
2211 if (isl_map_fast_is_empty(map
)) {
2214 map
= isl_map_project_out(map
, isl_dim_out
, 0, d
);
2215 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
2219 map
= map_power(map
, exact
, 0);
2221 map
= isl_map_add_dims(map
, isl_dim_param
, 1);
2222 dim
= isl_map_get_dim(map
);
2223 diff
= equate_parameter_to_length(dim
, param
);
2224 map
= isl_map_intersect(map
, diff
);
2225 map
= isl_map_project_out(map
, isl_dim_in
, 0, d
+ 1);
2226 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2227 map
= isl_map_reverse(map
);
2228 map
= isl_map_move_dims(map
, isl_dim_out
, 0, isl_dim_param
, param
, 1);
2233 /* Check whether equality i of bset is a pure stride constraint
2234 * on a single dimensions, i.e., of the form
2238 * with k a constant and e an existentially quantified variable.
2240 static int is_eq_stride(__isl_keep isl_basic_set
*bset
, int i
)
2252 if (!isl_int_is_zero(bset
->eq
[i
][0]))
2255 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
2256 d
= isl_basic_set_dim(bset
, isl_dim_set
);
2257 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
2259 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1, nparam
) != -1)
2261 pos1
= isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
, d
);
2264 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ pos1
+ 1,
2265 d
- pos1
- 1) != -1)
2268 pos2
= isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ d
, n_div
);
2271 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ d
+ pos2
+ 1,
2272 n_div
- pos2
- 1) != -1)
2274 if (!isl_int_is_one(bset
->eq
[i
][1 + nparam
+ pos1
]) &&
2275 !isl_int_is_negone(bset
->eq
[i
][1 + nparam
+ pos1
]))
2281 /* Given a map, compute the smallest superset of this map that is of the form
2283 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2285 * (where p ranges over the (non-parametric) dimensions),
2286 * compute the transitive closure of this map, i.e.,
2288 * { i -> j : exists k > 0:
2289 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2291 * and intersect domain and range of this transitive closure with
2292 * the given domain and range.
2294 * If with_id is set, then try to include as much of the identity mapping
2295 * as possible, by computing
2297 * { i -> j : exists k >= 0:
2298 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2300 * instead (i.e., allow k = 0).
2302 * In practice, we compute the difference set
2304 * delta = { j - i | i -> j in map },
2306 * look for stride constraint on the individual dimensions and compute
2307 * (constant) lower and upper bounds for each individual dimension,
2308 * adding a constraint for each bound not equal to infinity.
2310 static __isl_give isl_map
*box_closure_on_domain(__isl_take isl_map
*map
,
2311 __isl_take isl_set
*dom
, __isl_take isl_set
*ran
, int with_id
)
2320 isl_map
*app
= NULL
;
2321 isl_basic_set
*aff
= NULL
;
2322 isl_basic_map
*bmap
= NULL
;
2323 isl_vec
*obj
= NULL
;
2328 delta
= isl_map_deltas(isl_map_copy(map
));
2330 aff
= isl_set_affine_hull(isl_set_copy(delta
));
2333 dim
= isl_map_get_dim(map
);
2334 d
= isl_dim_size(dim
, isl_dim_in
);
2335 nparam
= isl_dim_size(dim
, isl_dim_param
);
2336 total
= isl_dim_total(dim
);
2337 bmap
= isl_basic_map_alloc_dim(dim
,
2338 aff
->n_div
+ 1, aff
->n_div
, 2 * d
+ 1);
2339 for (i
= 0; i
< aff
->n_div
+ 1; ++i
) {
2340 k
= isl_basic_map_alloc_div(bmap
);
2343 isl_int_set_si(bmap
->div
[k
][0], 0);
2345 for (i
= 0; i
< aff
->n_eq
; ++i
) {
2346 if (!is_eq_stride(aff
, i
))
2348 k
= isl_basic_map_alloc_equality(bmap
);
2351 isl_seq_clr(bmap
->eq
[k
], 1 + nparam
);
2352 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ d
,
2353 aff
->eq
[i
] + 1 + nparam
, d
);
2354 isl_seq_neg(bmap
->eq
[k
] + 1 + nparam
,
2355 aff
->eq
[i
] + 1 + nparam
, d
);
2356 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ 2 * d
,
2357 aff
->eq
[i
] + 1 + nparam
+ d
, aff
->n_div
);
2358 isl_int_set_si(bmap
->eq
[k
][1 + total
+ aff
->n_div
], 0);
2360 obj
= isl_vec_alloc(map
->ctx
, 1 + nparam
+ d
);
2363 isl_seq_clr(obj
->el
, 1 + nparam
+ d
);
2364 for (i
= 0; i
< d
; ++ i
) {
2365 enum isl_lp_result res
;
2367 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 1);
2369 res
= isl_set_solve_lp(delta
, 0, obj
->el
, map
->ctx
->one
, &opt
,
2371 if (res
== isl_lp_error
)
2373 if (res
== isl_lp_ok
) {
2374 k
= isl_basic_map_alloc_inequality(bmap
);
2377 isl_seq_clr(bmap
->ineq
[k
],
2378 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2379 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], -1);
2380 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], 1);
2381 isl_int_neg(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2384 res
= isl_set_solve_lp(delta
, 1, obj
->el
, map
->ctx
->one
, &opt
,
2386 if (res
== isl_lp_error
)
2388 if (res
== isl_lp_ok
) {
2389 k
= isl_basic_map_alloc_inequality(bmap
);
2392 isl_seq_clr(bmap
->ineq
[k
],
2393 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2394 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], 1);
2395 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], -1);
2396 isl_int_set(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2399 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 0);
2401 k
= isl_basic_map_alloc_inequality(bmap
);
2404 isl_seq_clr(bmap
->ineq
[k
],
2405 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2407 isl_int_set_si(bmap
->ineq
[k
][0], -1);
2408 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], 1);
2410 app
= isl_map_from_domain_and_range(dom
, ran
);
2413 isl_basic_set_free(aff
);
2415 bmap
= isl_basic_map_finalize(bmap
);
2416 isl_set_free(delta
);
2419 map
= isl_map_from_basic_map(bmap
);
2420 map
= isl_map_intersect(map
, app
);
2425 isl_basic_map_free(bmap
);
2426 isl_basic_set_free(aff
);
2430 isl_set_free(delta
);
2435 /* Given a map, compute the smallest superset of this map that is of the form
2437 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2439 * (where p ranges over the (non-parametric) dimensions),
2440 * compute the transitive closure of this map, i.e.,
2442 * { i -> j : exists k > 0:
2443 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2445 * and intersect domain and range of this transitive closure with
2446 * domain and range of the original map.
2448 static __isl_give isl_map
*box_closure(__isl_take isl_map
*map
)
2453 domain
= isl_map_domain(isl_map_copy(map
));
2454 domain
= isl_set_coalesce(domain
);
2455 range
= isl_map_range(isl_map_copy(map
));
2456 range
= isl_set_coalesce(range
);
2458 return box_closure_on_domain(map
, domain
, range
, 0);
2461 /* Given a map, compute the smallest superset of this map that is of the form
2463 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2465 * (where p ranges over the (non-parametric) dimensions),
2466 * compute the transitive and partially reflexive closure of this map, i.e.,
2468 * { i -> j : exists k >= 0:
2469 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2471 * and intersect domain and range of this transitive closure with
2474 static __isl_give isl_map
*box_closure_with_identity(__isl_take isl_map
*map
,
2475 __isl_take isl_set
*dom
)
2477 return box_closure_on_domain(map
, dom
, isl_set_copy(dom
), 1);
2480 /* Check whether app is the transitive closure of map.
2481 * In particular, check that app is acyclic and, if so,
2484 * app \subset (map \cup (map \circ app))
2486 static int check_exactness_omega(__isl_keep isl_map
*map
,
2487 __isl_keep isl_map
*app
)
2491 int is_empty
, is_exact
;
2495 delta
= isl_map_deltas(isl_map_copy(app
));
2496 d
= isl_set_dim(delta
, isl_dim_set
);
2497 for (i
= 0; i
< d
; ++i
)
2498 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
2499 is_empty
= isl_set_is_empty(delta
);
2500 isl_set_free(delta
);
2506 test
= isl_map_apply_range(isl_map_copy(app
), isl_map_copy(map
));
2507 test
= isl_map_union(test
, isl_map_copy(map
));
2508 is_exact
= isl_map_is_subset(app
, test
);
2514 /* Check if basic map M_i can be combined with all the other
2515 * basic maps such that
2519 * can be computed as
2521 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2523 * In particular, check if we can compute a compact representation
2526 * M_i^* \circ M_j \circ M_i^*
2529 * Let M_i^? be an extension of M_i^+ that allows paths
2530 * of length zero, i.e., the result of box_closure(., 1).
2531 * The criterion, as proposed by Kelly et al., is that
2532 * id = M_i^? - M_i^+ can be represented as a basic map
2535 * id \circ M_j \circ id = M_j
2539 * If this function returns 1, then tc and qc are set to
2540 * M_i^+ and M_i^?, respectively.
2542 static int can_be_split_off(__isl_keep isl_map
*map
, int i
,
2543 __isl_give isl_map
**tc
, __isl_give isl_map
**qc
)
2545 isl_map
*map_i
, *id
= NULL
;
2552 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
2553 isl_map_range(isl_map_copy(map
)));
2554 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
2558 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
2559 *tc
= box_closure(isl_map_copy(map_i
));
2560 *qc
= box_closure_with_identity(map_i
, C
);
2561 id
= isl_map_subtract(isl_map_copy(*qc
), isl_map_copy(*tc
));
2565 if (id
->n
!= 1 || (*qc
)->n
!= 1)
2568 for (j
= 0; j
< map
->n
; ++j
) {
2569 isl_map
*map_j
, *test
;
2574 map_j
= isl_map_from_basic_map(
2575 isl_basic_map_copy(map
->p
[j
]));
2576 test
= isl_map_apply_range(isl_map_copy(id
),
2577 isl_map_copy(map_j
));
2578 test
= isl_map_apply_range(test
, isl_map_copy(id
));
2579 is_ok
= isl_map_is_equal(test
, map_j
);
2580 isl_map_free(map_j
);
2608 static __isl_give isl_map
*box_closure_with_check(__isl_take isl_map
*map
,
2613 app
= box_closure(isl_map_copy(map
));
2615 *exact
= check_exactness_omega(map
, app
);
2621 /* Compute an overapproximation of the transitive closure of "map"
2622 * using a variation of the algorithm from
2623 * "Transitive Closure of Infinite Graphs and its Applications"
2626 * We first check whether we can can split of any basic map M_i and
2633 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2635 * using a recursive call on the remaining map.
2637 * If not, we simply call box_closure on the whole map.
2639 static __isl_give isl_map
*transitive_closure_omega(__isl_take isl_map
*map
,
2649 return box_closure_with_check(map
, exact
);
2651 for (i
= 0; i
< map
->n
; ++i
) {
2654 ok
= can_be_split_off(map
, i
, &tc
, &qc
);
2660 app
= isl_map_alloc_dim(isl_map_get_dim(map
), map
->n
- 1, 0);
2662 for (j
= 0; j
< map
->n
; ++j
) {
2665 app
= isl_map_add_basic_map(app
,
2666 isl_basic_map_copy(map
->p
[j
]));
2669 app
= isl_map_apply_range(isl_map_copy(qc
), app
);
2670 app
= isl_map_apply_range(app
, qc
);
2672 app
= isl_map_union(tc
, transitive_closure_omega(app
, NULL
));
2673 exact_i
= check_exactness_omega(map
, app
);
2685 return box_closure_with_check(map
, exact
);
2691 /* Compute the transitive closure of "map", or an overapproximation.
2692 * If the result is exact, then *exact is set to 1.
2693 * Simply use map_power to compute the powers of map, but tell
2694 * it to project out the lengths of the paths instead of equating
2695 * the length to a parameter.
2697 __isl_give isl_map
*isl_map_transitive_closure(__isl_take isl_map
*map
,
2700 isl_dim
*target_dim
;
2706 if (map
->ctx
->opt
->closure
== ISL_CLOSURE_BOX
)
2707 return transitive_closure_omega(map
, exact
);
2709 map
= isl_map_compute_divs(map
);
2710 map
= isl_map_coalesce(map
);
2711 closed
= isl_map_is_transitively_closed(map
);
2720 target_dim
= isl_map_get_dim(map
);
2721 map
= map_power(map
, exact
, 1);
2722 map
= isl_map_reset_dim(map
, target_dim
);
2730 static int inc_count(__isl_take isl_map
*map
, void *user
)
2741 static int collect_basic_map(__isl_take isl_map
*map
, void *user
)
2744 isl_basic_map
***next
= user
;
2746 for (i
= 0; i
< map
->n
; ++i
) {
2747 **next
= isl_basic_map_copy(map
->p
[i
]);
2760 /* Perform Floyd-Warshall on the given list of basic relations.
2761 * The basic relations may live in different dimensions,
2762 * but basic relations that get assigned to the diagonal of the
2763 * grid have domains and ranges of the same dimension and so
2764 * the standard algorithm can be used because the nested transitive
2765 * closures are only applied to diagonal elements and because all
2766 * compositions are peformed on relations with compatible domains and ranges.
2768 static __isl_give isl_union_map
*union_floyd_warshall_on_list(isl_ctx
*ctx
,
2769 __isl_keep isl_basic_map
**list
, int n
, int *exact
)
2774 isl_set
**set
= NULL
;
2775 isl_map
***grid
= NULL
;
2778 group
= setup_groups(ctx
, list
, n
, &set
, &n_group
);
2782 grid
= isl_calloc_array(ctx
, isl_map
**, n_group
);
2785 for (i
= 0; i
< n_group
; ++i
) {
2786 grid
[i
] = isl_calloc_array(map
->ctx
, isl_map
*, n_group
);
2789 for (j
= 0; j
< n_group
; ++j
) {
2790 isl_dim
*dim1
, *dim2
, *dim
;
2791 dim1
= isl_dim_reverse(isl_set_get_dim(set
[i
]));
2792 dim2
= isl_set_get_dim(set
[j
]);
2793 dim
= isl_dim_join(dim1
, dim2
);
2794 grid
[i
][j
] = isl_map_empty(dim
);
2798 for (k
= 0; k
< n
; ++k
) {
2800 j
= group
[2 * k
+ 1];
2801 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
2802 isl_map_from_basic_map(
2803 isl_basic_map_copy(list
[k
])));
2806 floyd_warshall_iterate(grid
, n_group
, exact
);
2808 app
= isl_union_map_empty(isl_map_get_dim(grid
[0][0]));
2810 for (i
= 0; i
< n_group
; ++i
) {
2811 for (j
= 0; j
< n_group
; ++j
)
2812 app
= isl_union_map_add_map(app
, grid
[i
][j
]);
2817 for (i
= 0; i
< 2 * n
; ++i
)
2818 isl_set_free(set
[i
]);
2825 for (i
= 0; i
< n_group
; ++i
) {
2828 for (j
= 0; j
< n_group
; ++j
)
2829 isl_map_free(grid
[i
][j
]);
2834 for (i
= 0; i
< 2 * n
; ++i
)
2835 isl_set_free(set
[i
]);
2842 /* Perform Floyd-Warshall on the given union relation.
2843 * The implementation is very similar to that for non-unions.
2844 * The main difference is that it is applied unconditionally.
2845 * We first extract a list of basic maps from the union map
2846 * and then perform the algorithm on this list.
2848 static __isl_give isl_union_map
*union_floyd_warshall(
2849 __isl_take isl_union_map
*umap
, int *exact
)
2853 isl_basic_map
**list
;
2854 isl_basic_map
**next
;
2858 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2861 ctx
= isl_union_map_get_ctx(umap
);
2862 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2867 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2870 res
= union_floyd_warshall_on_list(ctx
, list
, n
, exact
);
2873 for (i
= 0; i
< n
; ++i
)
2874 isl_basic_map_free(list
[i
]);
2878 isl_union_map_free(umap
);
2882 for (i
= 0; i
< n
; ++i
)
2883 isl_basic_map_free(list
[i
]);
2886 isl_union_map_free(umap
);
2890 /* Decompose the give union relation into strongly connected components.
2891 * The implementation is essentially the same as that of
2892 * construct_power_components with the major difference that all
2893 * operations are performed on union maps.
2895 static __isl_give isl_union_map
*union_components(
2896 __isl_take isl_union_map
*umap
, int *exact
)
2901 isl_basic_map
**list
;
2902 isl_basic_map
**next
;
2903 isl_union_map
*path
= NULL
;
2904 struct basic_map_sort
*s
= NULL
;
2909 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2913 return union_floyd_warshall(umap
, exact
);
2915 ctx
= isl_union_map_get_ctx(umap
);
2916 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2921 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2924 s
= basic_map_sort_init(ctx
, n
, list
);
2931 path
= isl_union_map_empty(isl_union_map_get_dim(umap
));
2933 isl_union_map
*comp
;
2934 isl_union_map
*path_comp
, *path_comb
;
2935 comp
= isl_union_map_empty(isl_union_map_get_dim(umap
));
2936 while (s
->order
[i
] != -1) {
2937 comp
= isl_union_map_add_map(comp
,
2938 isl_map_from_basic_map(
2939 isl_basic_map_copy(list
[s
->order
[i
]])));
2943 path_comp
= union_floyd_warshall(comp
, exact
);
2944 path_comb
= isl_union_map_apply_range(isl_union_map_copy(path
),
2945 isl_union_map_copy(path_comp
));
2946 path
= isl_union_map_union(path
, path_comp
);
2947 path
= isl_union_map_union(path
, path_comb
);
2952 if (c
> 1 && s
->check_closed
&& !*exact
) {
2955 closed
= isl_union_map_is_transitively_closed(path
);
2961 basic_map_sort_free(s
);
2963 for (i
= 0; i
< n
; ++i
)
2964 isl_basic_map_free(list
[i
]);
2968 isl_union_map_free(path
);
2969 return union_floyd_warshall(umap
, exact
);
2972 isl_union_map_free(umap
);
2976 basic_map_sort_free(s
);
2978 for (i
= 0; i
< n
; ++i
)
2979 isl_basic_map_free(list
[i
]);
2982 isl_union_map_free(umap
);
2983 isl_union_map_free(path
);
2987 /* Compute the transitive closure of "umap", or an overapproximation.
2988 * If the result is exact, then *exact is set to 1.
2990 __isl_give isl_union_map
*isl_union_map_transitive_closure(
2991 __isl_take isl_union_map
*umap
, int *exact
)
3001 umap
= isl_union_map_compute_divs(umap
);
3002 umap
= isl_union_map_coalesce(umap
);
3003 closed
= isl_union_map_is_transitively_closed(umap
);
3008 umap
= union_components(umap
, exact
);
3011 isl_union_map_free(umap
);