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_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
, isl_basic_map_identity(dim
));
645 isl_basic_set_free(delta
);
646 isl_basic_map_free(path
);
650 /* Given a dimension specification Z^{n+1} -> Z^{n+1} and a parameter "param",
651 * construct a map that equates the parameter to the difference
652 * in the final coordinates and imposes that this difference is positive.
655 * { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
657 static __isl_give isl_map
*equate_parameter_to_length(__isl_take isl_dim
*dim
,
660 struct isl_basic_map
*bmap
;
665 d
= isl_dim_size(dim
, isl_dim_in
);
666 nparam
= isl_dim_size(dim
, isl_dim_param
);
667 bmap
= isl_basic_map_alloc_dim(dim
, 0, 1, 1);
668 k
= isl_basic_map_alloc_equality(bmap
);
671 isl_seq_clr(bmap
->eq
[k
], 1 + isl_basic_map_total_dim(bmap
));
672 isl_int_set_si(bmap
->eq
[k
][1 + param
], -1);
673 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
- 1], -1);
674 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
+ d
- 1], 1);
676 k
= isl_basic_map_alloc_inequality(bmap
);
679 isl_seq_clr(bmap
->ineq
[k
], 1 + isl_basic_map_total_dim(bmap
));
680 isl_int_set_si(bmap
->ineq
[k
][1 + param
], 1);
681 isl_int_set_si(bmap
->ineq
[k
][0], -1);
683 bmap
= isl_basic_map_finalize(bmap
);
684 return isl_map_from_basic_map(bmap
);
686 isl_basic_map_free(bmap
);
690 /* Check whether "path" is acyclic, where the last coordinates of domain
691 * and range of path encode the number of steps taken.
692 * That is, check whether
694 * { d | d = y - x and (x,y) in path }
696 * does not contain any element with positive last coordinate (positive length)
697 * and zero remaining coordinates (cycle).
699 static int is_acyclic(__isl_take isl_map
*path
)
704 struct isl_set
*delta
;
706 delta
= isl_map_deltas(path
);
707 dim
= isl_set_dim(delta
, isl_dim_set
);
708 for (i
= 0; i
< dim
; ++i
) {
710 delta
= isl_set_lower_bound_si(delta
, isl_dim_set
, i
, 1);
712 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
715 acyclic
= isl_set_is_empty(delta
);
721 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
722 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
723 * construct a map that is an overapproximation of the map
724 * that takes an element from the space D \times Z to another
725 * element from the same space, such that the first n coordinates of the
726 * difference between them is a sum of differences between images
727 * and pre-images in one of the R_i and such that the last coordinate
728 * is equal to the number of steps taken.
731 * \Delta_i = { y - x | (x, y) in R_i }
733 * then the constructed map is an overapproximation of
735 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
736 * d = (\sum_i k_i \delta_i, \sum_i k_i) }
738 * The elements of the singleton \Delta_i's are collected as the
739 * rows of the steps matrix. For all these \Delta_i's together,
740 * a single path is constructed.
741 * For each of the other \Delta_i's, we compute an overapproximation
742 * of the paths along elements of \Delta_i.
743 * Since each of these paths performs an addition, composition is
744 * symmetric and we can simply compose all resulting paths in any order.
746 static __isl_give isl_map
*construct_extended_path(__isl_take isl_dim
*dim
,
747 __isl_keep isl_map
*map
, int *project
)
749 struct isl_mat
*steps
= NULL
;
750 struct isl_map
*path
= NULL
;
754 d
= isl_map_dim(map
, isl_dim_in
);
756 path
= isl_map_identity(isl_dim_copy(dim
));
758 steps
= isl_mat_alloc(map
->ctx
, map
->n
, d
);
763 for (i
= 0; i
< map
->n
; ++i
) {
764 struct isl_basic_set
*delta
;
766 delta
= isl_basic_map_deltas(isl_basic_map_copy(map
->p
[i
]));
768 for (j
= 0; j
< d
; ++j
) {
771 fixed
= isl_basic_set_fast_dim_is_fixed(delta
, j
,
774 isl_basic_set_free(delta
);
783 path
= isl_map_apply_range(path
,
784 path_along_delta(isl_dim_copy(dim
), delta
));
785 path
= isl_map_coalesce(path
);
787 isl_basic_set_free(delta
);
794 path
= isl_map_apply_range(path
,
795 path_along_steps(isl_dim_copy(dim
), steps
));
798 if (project
&& *project
) {
799 *project
= is_acyclic(isl_map_copy(path
));
814 static int isl_set_overlaps(__isl_keep isl_set
*set1
, __isl_keep isl_set
*set2
)
819 if (!isl_dim_tuple_match(set1
->dim
, isl_dim_set
, set2
->dim
, isl_dim_set
))
822 i
= isl_set_intersect(isl_set_copy(set1
), isl_set_copy(set2
));
823 no_overlap
= isl_set_is_empty(i
);
826 return no_overlap
< 0 ? -1 : !no_overlap
;
829 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
830 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
831 * construct a map that is an overapproximation of the map
832 * that takes an element from the dom R \times Z to an
833 * element from ran R \times Z, such that the first n coordinates of the
834 * difference between them is a sum of differences between images
835 * and pre-images in one of the R_i and such that the last coordinate
836 * is equal to the number of steps taken.
839 * \Delta_i = { y - x | (x, y) in R_i }
841 * then the constructed map is an overapproximation of
843 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
844 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
845 * x in dom R and x + d in ran R and
848 static __isl_give isl_map
*construct_component(__isl_take isl_dim
*dim
,
849 __isl_keep isl_map
*map
, int *exact
, int project
)
851 struct isl_set
*domain
= NULL
;
852 struct isl_set
*range
= NULL
;
853 struct isl_map
*app
= NULL
;
854 struct isl_map
*path
= NULL
;
856 domain
= isl_map_domain(isl_map_copy(map
));
857 domain
= isl_set_coalesce(domain
);
858 range
= isl_map_range(isl_map_copy(map
));
859 range
= isl_set_coalesce(range
);
860 if (!isl_set_overlaps(domain
, range
)) {
861 isl_set_free(domain
);
865 map
= isl_map_copy(map
);
866 map
= isl_map_add_dims(map
, isl_dim_in
, 1);
867 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
868 map
= set_path_length(map
, 1, 1);
871 app
= isl_map_from_domain_and_range(domain
, range
);
872 app
= isl_map_add_dims(app
, isl_dim_in
, 1);
873 app
= isl_map_add_dims(app
, isl_dim_out
, 1);
875 path
= construct_extended_path(isl_dim_copy(dim
), map
,
876 exact
&& *exact
? &project
: NULL
);
877 app
= isl_map_intersect(app
, path
);
879 if (exact
&& *exact
&&
880 (*exact
= check_exactness(isl_map_copy(map
), isl_map_copy(app
),
885 app
= set_path_length(app
, 0, 1);
893 /* Call construct_component and, if "project" is set, project out
894 * the final coordinates.
896 static __isl_give isl_map
*construct_projected_component(
897 __isl_take isl_dim
*dim
,
898 __isl_keep isl_map
*map
, int *exact
, int project
)
905 d
= isl_dim_size(dim
, isl_dim_in
);
907 app
= construct_component(dim
, map
, exact
, project
);
909 app
= isl_map_project_out(app
, isl_dim_in
, d
- 1, 1);
910 app
= isl_map_project_out(app
, isl_dim_out
, d
- 1, 1);
915 /* Compute an extended version, i.e., with path lengths, of
916 * an overapproximation of the transitive closure of "bmap"
917 * with path lengths greater than or equal to zero and with
918 * domain and range equal to "dom".
920 static __isl_give isl_map
*q_closure(__isl_take isl_dim
*dim
,
921 __isl_take isl_set
*dom
, __isl_keep isl_basic_map
*bmap
, int *exact
)
928 dom
= isl_set_add_dims(dom
, isl_dim_set
, 1);
929 app
= isl_map_from_domain_and_range(dom
, isl_set_copy(dom
));
930 map
= isl_map_from_basic_map(isl_basic_map_copy(bmap
));
931 path
= construct_extended_path(dim
, map
, &project
);
932 app
= isl_map_intersect(app
, path
);
934 if ((*exact
= check_exactness(map
, isl_map_copy(app
), project
)) < 0)
943 /* Check whether qc has any elements of length at least one
944 * with domain and/or range outside of dom and ran.
946 static int has_spurious_elements(__isl_keep isl_map
*qc
,
947 __isl_keep isl_set
*dom
, __isl_keep isl_set
*ran
)
953 if (!qc
|| !dom
|| !ran
)
956 d
= isl_map_dim(qc
, isl_dim_in
);
958 qc
= isl_map_copy(qc
);
959 qc
= set_path_length(qc
, 0, 1);
960 qc
= isl_map_project_out(qc
, isl_dim_in
, d
- 1, 1);
961 qc
= isl_map_project_out(qc
, isl_dim_out
, d
- 1, 1);
963 s
= isl_map_domain(isl_map_copy(qc
));
964 subset
= isl_set_is_subset(s
, dom
);
973 s
= isl_map_range(qc
);
974 subset
= isl_set_is_subset(s
, ran
);
977 return subset
< 0 ? -1 : !subset
;
986 /* For each basic map in "map", except i, check whether it combines
987 * with the transitive closure that is reflexive on C combines
988 * to the left and to the right.
992 * dom map_j \subseteq C
994 * then right[j] is set to 1. Otherwise, if
996 * ran map_i \cap dom map_j = \emptyset
998 * then right[j] is set to 0. Otherwise, composing to the right
1001 * Similar, for composing to the left, we have if
1003 * ran map_j \subseteq C
1005 * then left[j] is set to 1. Otherwise, if
1007 * dom map_i \cap ran map_j = \emptyset
1009 * then left[j] is set to 0. Otherwise, composing to the left
1012 * The return value is or'd with LEFT if composing to the left
1013 * is possible and with RIGHT if composing to the right is possible.
1015 static int composability(__isl_keep isl_set
*C
, int i
,
1016 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
1017 __isl_keep isl_map
*map
)
1023 for (j
= 0; j
< map
->n
&& ok
; ++j
) {
1024 int overlaps
, subset
;
1030 dom
[j
] = isl_set_from_basic_set(
1031 isl_basic_map_domain(
1032 isl_basic_map_copy(map
->p
[j
])));
1035 overlaps
= isl_set_overlaps(ran
[i
], dom
[j
]);
1041 subset
= isl_set_is_subset(dom
[j
], C
);
1053 ran
[j
] = isl_set_from_basic_set(
1054 isl_basic_map_range(
1055 isl_basic_map_copy(map
->p
[j
])));
1058 overlaps
= isl_set_overlaps(dom
[i
], ran
[j
]);
1064 subset
= isl_set_is_subset(ran
[j
], C
);
1078 static __isl_give isl_map
*anonymize(__isl_take isl_map
*map
)
1080 map
= isl_map_reset(map
, isl_dim_in
);
1081 map
= isl_map_reset(map
, isl_dim_out
);
1085 /* Return a map that is a union of the basic maps in "map", except i,
1086 * composed to left and right with qc based on the entries of "left"
1089 static __isl_give isl_map
*compose(__isl_keep isl_map
*map
, int i
,
1090 __isl_take isl_map
*qc
, int *left
, int *right
)
1095 comp
= isl_map_empty(isl_map_get_dim(map
));
1096 for (j
= 0; j
< map
->n
; ++j
) {
1102 map_j
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[j
]));
1103 map_j
= anonymize(map_j
);
1104 if (left
&& left
[j
])
1105 map_j
= isl_map_apply_range(map_j
, isl_map_copy(qc
));
1106 if (right
&& right
[j
])
1107 map_j
= isl_map_apply_range(isl_map_copy(qc
), map_j
);
1108 comp
= isl_map_union(comp
, map_j
);
1111 comp
= isl_map_compute_divs(comp
);
1112 comp
= isl_map_coalesce(comp
);
1119 /* Compute the transitive closure of "map" incrementally by
1126 * map_i^+ \cup ((id \cup map_i^) \circ qc^+)
1130 * map_i^+ \cup (qc^+ \circ (id \cup map_i^))
1132 * depending on whether left or right are NULL.
1134 static __isl_give isl_map
*compute_incremental(
1135 __isl_take isl_dim
*dim
, __isl_keep isl_map
*map
,
1136 int i
, __isl_take isl_map
*qc
, int *left
, int *right
, int *exact
)
1140 isl_map
*rtc
= NULL
;
1144 isl_assert(map
->ctx
, left
|| right
, goto error
);
1146 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
1147 tc
= construct_projected_component(isl_dim_copy(dim
), map_i
,
1149 isl_map_free(map_i
);
1152 qc
= isl_map_transitive_closure(qc
, exact
);
1158 return isl_map_universe(isl_map_get_dim(map
));
1161 if (!left
|| !right
)
1162 rtc
= isl_map_union(isl_map_copy(tc
),
1163 isl_map_identity(isl_map_get_dim(tc
)));
1165 qc
= isl_map_apply_range(rtc
, qc
);
1167 qc
= isl_map_apply_range(qc
, rtc
);
1168 qc
= isl_map_union(tc
, qc
);
1179 /* Given a map "map", try to find a basic map such that
1180 * map^+ can be computed as
1182 * map^+ = map_i^+ \cup
1183 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1185 * with C the simple hull of the domain and range of the input map.
1186 * map_i^ \cup Id_C is computed by allowing the path lengths to be zero
1187 * and by intersecting domain and range with C.
1188 * Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
1189 * Also, we only use the incremental computation if all the transitive
1190 * closures are exact and if the number of basic maps in the union,
1191 * after computing the integer divisions, is smaller than the number
1192 * of basic maps in the input map.
1194 static int incemental_on_entire_domain(__isl_keep isl_dim
*dim
,
1195 __isl_keep isl_map
*map
,
1196 isl_set
**dom
, isl_set
**ran
, int *left
, int *right
,
1197 __isl_give isl_map
**res
)
1205 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
1206 isl_map_range(isl_map_copy(map
)));
1207 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1215 d
= isl_map_dim(map
, isl_dim_in
);
1217 for (i
= 0; i
< map
->n
; ++i
) {
1219 int exact_i
, spurious
;
1221 dom
[i
] = isl_set_from_basic_set(isl_basic_map_domain(
1222 isl_basic_map_copy(map
->p
[i
])));
1223 ran
[i
] = isl_set_from_basic_set(isl_basic_map_range(
1224 isl_basic_map_copy(map
->p
[i
])));
1225 qc
= q_closure(isl_dim_copy(dim
), isl_set_copy(C
),
1226 map
->p
[i
], &exact_i
);
1233 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1240 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1241 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1242 qc
= isl_map_compute_divs(qc
);
1243 for (j
= 0; j
< map
->n
; ++j
)
1244 left
[j
] = right
[j
] = 1;
1245 qc
= compose(map
, i
, qc
, left
, right
);
1248 if (qc
->n
>= map
->n
) {
1252 *res
= compute_incremental(isl_dim_copy(dim
), map
, i
, qc
,
1253 left
, right
, &exact_i
);
1264 return *res
!= NULL
;
1270 /* Try and compute the transitive closure of "map" as
1272 * map^+ = map_i^+ \cup
1273 * \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
1275 * with C either the simple hull of the domain and range of the entire
1276 * map or the simple hull of domain and range of map_i.
1278 static __isl_give isl_map
*incremental_closure(__isl_take isl_dim
*dim
,
1279 __isl_keep isl_map
*map
, int *exact
, int project
)
1282 isl_set
**dom
= NULL
;
1283 isl_set
**ran
= NULL
;
1288 isl_map
*res
= NULL
;
1291 return construct_projected_component(dim
, map
, exact
, project
);
1296 return construct_projected_component(dim
, map
, exact
, project
);
1298 d
= isl_map_dim(map
, isl_dim_in
);
1300 dom
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1301 ran
= isl_calloc_array(map
->ctx
, isl_set
*, map
->n
);
1302 left
= isl_calloc_array(map
->ctx
, int, map
->n
);
1303 right
= isl_calloc_array(map
->ctx
, int, map
->n
);
1304 if (!ran
|| !dom
|| !left
|| !right
)
1307 if (incemental_on_entire_domain(dim
, map
, dom
, ran
, left
, right
, &res
) < 0)
1310 for (i
= 0; !res
&& i
< map
->n
; ++i
) {
1312 int exact_i
, spurious
, comp
;
1314 dom
[i
] = isl_set_from_basic_set(
1315 isl_basic_map_domain(
1316 isl_basic_map_copy(map
->p
[i
])));
1320 ran
[i
] = isl_set_from_basic_set(
1321 isl_basic_map_range(
1322 isl_basic_map_copy(map
->p
[i
])));
1325 C
= isl_set_union(isl_set_copy(dom
[i
]),
1326 isl_set_copy(ran
[i
]));
1327 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
1334 comp
= composability(C
, i
, dom
, ran
, left
, right
, map
);
1335 if (!comp
|| comp
< 0) {
1341 qc
= q_closure(isl_dim_copy(dim
), C
, map
->p
[i
], &exact_i
);
1348 spurious
= has_spurious_elements(qc
, dom
[i
], ran
[i
]);
1355 qc
= isl_map_project_out(qc
, isl_dim_in
, d
, 1);
1356 qc
= isl_map_project_out(qc
, isl_dim_out
, d
, 1);
1357 qc
= isl_map_compute_divs(qc
);
1358 qc
= compose(map
, i
, qc
, (comp
& LEFT
) ? left
: NULL
,
1359 (comp
& RIGHT
) ? right
: NULL
);
1362 if (qc
->n
>= map
->n
) {
1366 res
= compute_incremental(isl_dim_copy(dim
), map
, i
, qc
,
1367 (comp
& LEFT
) ? left
: NULL
,
1368 (comp
& RIGHT
) ? right
: NULL
, &exact_i
);
1377 for (i
= 0; i
< map
->n
; ++i
) {
1378 isl_set_free(dom
[i
]);
1379 isl_set_free(ran
[i
]);
1391 return construct_projected_component(dim
, map
, exact
, project
);
1394 for (i
= 0; i
< map
->n
; ++i
)
1395 isl_set_free(dom
[i
]);
1398 for (i
= 0; i
< map
->n
; ++i
)
1399 isl_set_free(ran
[i
]);
1407 /* Given an array of sets "set", add "dom" at position "pos"
1408 * and search for elements at earlier positions that overlap with "dom".
1409 * If any can be found, then merge all of them, together with "dom", into
1410 * a single set and assign the union to the first in the array,
1411 * which becomes the new group leader for all groups involved in the merge.
1412 * During the search, we only consider group leaders, i.e., those with
1413 * group[i] = i, as the other sets have already been combined
1414 * with one of the group leaders.
1416 static int merge(isl_set
**set
, int *group
, __isl_take isl_set
*dom
, int pos
)
1421 set
[pos
] = isl_set_copy(dom
);
1423 for (i
= pos
- 1; i
>= 0; --i
) {
1429 o
= isl_set_overlaps(set
[i
], dom
);
1435 set
[i
] = isl_set_union(set
[i
], set
[group
[pos
]]);
1436 set
[group
[pos
]] = NULL
;
1439 group
[group
[pos
]] = i
;
1450 /* Replace each entry in the n by n grid of maps by the cross product
1451 * with the relation { [i] -> [i + 1] }.
1453 static int add_length(__isl_keep isl_map
*map
, isl_map
***grid
, int n
)
1457 isl_basic_map
*bstep
;
1464 dim
= isl_map_get_dim(map
);
1465 nparam
= isl_dim_size(dim
, isl_dim_param
);
1466 dim
= isl_dim_drop(dim
, isl_dim_in
, 0, isl_dim_size(dim
, isl_dim_in
));
1467 dim
= isl_dim_drop(dim
, isl_dim_out
, 0, isl_dim_size(dim
, isl_dim_out
));
1468 dim
= isl_dim_add(dim
, isl_dim_in
, 1);
1469 dim
= isl_dim_add(dim
, isl_dim_out
, 1);
1470 bstep
= isl_basic_map_alloc_dim(dim
, 0, 1, 0);
1471 k
= isl_basic_map_alloc_equality(bstep
);
1473 isl_basic_map_free(bstep
);
1476 isl_seq_clr(bstep
->eq
[k
], 1 + isl_basic_map_total_dim(bstep
));
1477 isl_int_set_si(bstep
->eq
[k
][0], 1);
1478 isl_int_set_si(bstep
->eq
[k
][1 + nparam
], 1);
1479 isl_int_set_si(bstep
->eq
[k
][1 + nparam
+ 1], -1);
1480 bstep
= isl_basic_map_finalize(bstep
);
1481 step
= isl_map_from_basic_map(bstep
);
1483 for (i
= 0; i
< n
; ++i
)
1484 for (j
= 0; j
< n
; ++j
)
1485 grid
[i
][j
] = isl_map_product(grid
[i
][j
],
1486 isl_map_copy(step
));
1493 /* The core of the Floyd-Warshall algorithm.
1494 * Updates the given n x x matrix of relations in place.
1496 * The algorithm iterates over all vertices. In each step, the whole
1497 * matrix is updated to include all paths that go to the current vertex,
1498 * possibly stay there a while (including passing through earlier vertices)
1499 * and then come back. At the start of each iteration, the diagonal
1500 * element corresponding to the current vertex is replaced by its
1501 * transitive closure to account for all indirect paths that stay
1502 * in the current vertex.
1504 static void floyd_warshall_iterate(isl_map
***grid
, int n
, int *exact
)
1508 for (r
= 0; r
< n
; ++r
) {
1510 grid
[r
][r
] = isl_map_transitive_closure(grid
[r
][r
],
1511 (exact
&& *exact
) ? &r_exact
: NULL
);
1512 if (exact
&& *exact
&& !r_exact
)
1515 for (p
= 0; p
< n
; ++p
)
1516 for (q
= 0; q
< n
; ++q
) {
1518 if (p
== r
&& q
== r
)
1520 loop
= isl_map_apply_range(
1521 isl_map_copy(grid
[p
][r
]),
1522 isl_map_copy(grid
[r
][q
]));
1523 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1524 loop
= isl_map_apply_range(
1525 isl_map_copy(grid
[p
][r
]),
1526 isl_map_apply_range(
1527 isl_map_copy(grid
[r
][r
]),
1528 isl_map_copy(grid
[r
][q
])));
1529 grid
[p
][q
] = isl_map_union(grid
[p
][q
], loop
);
1530 grid
[p
][q
] = isl_map_coalesce(grid
[p
][q
]);
1535 /* Given a partition of the domains and ranges of the basic maps in "map",
1536 * apply the Floyd-Warshall algorithm with the elements in the partition
1539 * In particular, there are "n" elements in the partition and "group" is
1540 * an array of length 2 * map->n with entries in [0,n-1].
1542 * We first construct a matrix of relations based on the partition information,
1543 * apply Floyd-Warshall on this matrix of relations and then take the
1544 * union of all entries in the matrix as the final result.
1546 * If we are actually computing the power instead of the transitive closure,
1547 * i.e., when "project" is not set, then the result should have the
1548 * path lengths encoded as the difference between an extra pair of
1549 * coordinates. We therefore apply the nested transitive closures
1550 * to relations that include these lengths. In particular, we replace
1551 * the input relation by the cross product with the unit length relation
1552 * { [i] -> [i + 1] }.
1554 static __isl_give isl_map
*floyd_warshall_with_groups(__isl_take isl_dim
*dim
,
1555 __isl_keep isl_map
*map
, int *exact
, int project
, int *group
, int n
)
1558 isl_map
***grid
= NULL
;
1566 return incremental_closure(dim
, map
, exact
, project
);
1569 grid
= isl_calloc_array(map
->ctx
, isl_map
**, n
);
1572 for (i
= 0; i
< n
; ++i
) {
1573 grid
[i
] = isl_calloc_array(map
->ctx
, isl_map
*, n
);
1576 for (j
= 0; j
< n
; ++j
)
1577 grid
[i
][j
] = isl_map_empty(isl_map_get_dim(map
));
1580 for (k
= 0; k
< map
->n
; ++k
) {
1582 j
= group
[2 * k
+ 1];
1583 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
1584 isl_map_from_basic_map(
1585 isl_basic_map_copy(map
->p
[k
])));
1588 if (!project
&& add_length(map
, grid
, n
) < 0)
1591 floyd_warshall_iterate(grid
, n
, exact
);
1593 app
= isl_map_empty(isl_map_get_dim(map
));
1595 for (i
= 0; i
< n
; ++i
) {
1596 for (j
= 0; j
< n
; ++j
)
1597 app
= isl_map_union(app
, grid
[i
][j
]);
1608 for (i
= 0; i
< n
; ++i
) {
1611 for (j
= 0; j
< n
; ++j
)
1612 isl_map_free(grid
[i
][j
]);
1621 /* Partition the domains and ranges of the n basic relations in list
1622 * into disjoint cells.
1624 * To find the partition, we simply consider all of the domains
1625 * and ranges in turn and combine those that overlap.
1626 * "set" contains the partition elements and "group" indicates
1627 * to which partition element a given domain or range belongs.
1628 * The domain of basic map i corresponds to element 2 * i in these arrays,
1629 * while the domain corresponds to element 2 * i + 1.
1630 * During the construction group[k] is either equal to k,
1631 * in which case set[k] contains the union of all the domains and
1632 * ranges in the corresponding group, or is equal to some l < k,
1633 * with l another domain or range in the same group.
1635 static int *setup_groups(isl_ctx
*ctx
, __isl_keep isl_basic_map
**list
, int n
,
1636 isl_set
***set
, int *n_group
)
1642 *set
= isl_calloc_array(ctx
, isl_set
*, 2 * n
);
1643 group
= isl_alloc_array(ctx
, int, 2 * n
);
1645 if (!*set
|| !group
)
1648 for (i
= 0; i
< n
; ++i
) {
1650 dom
= isl_set_from_basic_set(isl_basic_map_domain(
1651 isl_basic_map_copy(list
[i
])));
1652 if (merge(*set
, group
, dom
, 2 * i
) < 0)
1654 dom
= isl_set_from_basic_set(isl_basic_map_range(
1655 isl_basic_map_copy(list
[i
])));
1656 if (merge(*set
, group
, dom
, 2 * i
+ 1) < 0)
1661 for (i
= 0; i
< 2 * n
; ++i
)
1662 if (group
[i
] == i
) {
1664 (*set
)[g
] = (*set
)[i
];
1669 group
[i
] = group
[group
[i
]];
1676 for (i
= 0; i
< 2 * n
; ++i
)
1677 isl_set_free((*set
)[i
]);
1685 /* Check if the domains and ranges of the basic maps in "map" can
1686 * be partitioned, and if so, apply Floyd-Warshall on the elements
1687 * of the partition. Note that we also apply this algorithm
1688 * if we want to compute the power, i.e., when "project" is not set.
1689 * However, the results are unlikely to be exact since the recursive
1690 * calls inside the Floyd-Warshall algorithm typically result in
1691 * non-linear path lengths quite quickly.
1693 static __isl_give isl_map
*floyd_warshall(__isl_take isl_dim
*dim
,
1694 __isl_keep isl_map
*map
, int *exact
, int project
)
1697 isl_set
**set
= NULL
;
1704 return incremental_closure(dim
, map
, exact
, project
);
1706 group
= setup_groups(map
->ctx
, map
->p
, map
->n
, &set
, &n
);
1710 for (i
= 0; i
< 2 * map
->n
; ++i
)
1711 isl_set_free(set
[i
]);
1715 return floyd_warshall_with_groups(dim
, map
, exact
, project
, group
, n
);
1721 /* Structure for representing the nodes in the graph being traversed
1722 * using Tarjan's algorithm.
1723 * index represents the order in which nodes are visited.
1724 * min_index is the index of the root of a (sub)component.
1725 * on_stack indicates whether the node is currently on the stack.
1727 struct basic_map_sort_node
{
1732 /* Structure for representing the graph being traversed
1733 * using Tarjan's algorithm.
1734 * len is the number of nodes
1735 * node is an array of nodes
1736 * stack contains the nodes on the path from the root to the current node
1737 * sp is the stack pointer
1738 * index is the index of the last node visited
1739 * order contains the elements of the components separated by -1
1740 * op represents the current position in order
1742 * check_closed is set if we may have used the fact that
1743 * a pair of basic maps can be interchanged
1745 struct basic_map_sort
{
1747 struct basic_map_sort_node
*node
;
1756 static void basic_map_sort_free(struct basic_map_sort
*s
)
1766 static struct basic_map_sort
*basic_map_sort_alloc(struct isl_ctx
*ctx
, int len
)
1768 struct basic_map_sort
*s
;
1771 s
= isl_calloc_type(ctx
, struct basic_map_sort
);
1775 s
->node
= isl_alloc_array(ctx
, struct basic_map_sort_node
, len
);
1778 for (i
= 0; i
< len
; ++i
)
1779 s
->node
[i
].index
= -1;
1780 s
->stack
= isl_alloc_array(ctx
, int, len
);
1783 s
->order
= isl_alloc_array(ctx
, int, 2 * len
);
1791 s
->check_closed
= 0;
1795 basic_map_sort_free(s
);
1799 /* Check whether in the computation of the transitive closure
1800 * "bmap1" (R_1) should follow (or be part of the same component as)
1803 * That is check whether
1811 * If so, then there is no reason for R_1 to immediately follow R_2
1814 * *check_closed is set if the subset relation holds while
1815 * R_1 \circ R_2 is not empty.
1817 static int basic_map_follows(__isl_keep isl_basic_map
*bmap1
,
1818 __isl_keep isl_basic_map
*bmap2
, int *check_closed
)
1820 struct isl_map
*map12
= NULL
;
1821 struct isl_map
*map21
= NULL
;
1824 if (!isl_dim_tuple_match(bmap1
->dim
, isl_dim_in
, bmap2
->dim
, isl_dim_out
))
1827 map21
= isl_map_from_basic_map(
1828 isl_basic_map_apply_range(
1829 isl_basic_map_copy(bmap2
),
1830 isl_basic_map_copy(bmap1
)));
1831 subset
= isl_map_is_empty(map21
);
1835 isl_map_free(map21
);
1839 if (!isl_dim_tuple_match(bmap1
->dim
, isl_dim_in
, bmap1
->dim
, isl_dim_out
) ||
1840 !isl_dim_tuple_match(bmap2
->dim
, isl_dim_in
, bmap2
->dim
, isl_dim_out
)) {
1841 isl_map_free(map21
);
1845 map12
= isl_map_from_basic_map(
1846 isl_basic_map_apply_range(
1847 isl_basic_map_copy(bmap1
),
1848 isl_basic_map_copy(bmap2
)));
1850 subset
= isl_map_is_subset(map21
, map12
);
1852 isl_map_free(map12
);
1853 isl_map_free(map21
);
1858 return subset
< 0 ? -1 : !subset
;
1860 isl_map_free(map21
);
1864 /* Perform Tarjan's algorithm for computing the strongly connected components
1865 * in the graph with the disjuncts of "map" as vertices and with an
1866 * edge between any pair of disjuncts such that the first has
1867 * to be applied after the second.
1869 static int power_components_tarjan(struct basic_map_sort
*s
,
1870 __isl_keep isl_basic_map
**list
, int i
)
1874 s
->node
[i
].index
= s
->index
;
1875 s
->node
[i
].min_index
= s
->index
;
1876 s
->node
[i
].on_stack
= 1;
1878 s
->stack
[s
->sp
++] = i
;
1880 for (j
= s
->len
- 1; j
>= 0; --j
) {
1885 if (s
->node
[j
].index
>= 0 &&
1886 (!s
->node
[j
].on_stack
||
1887 s
->node
[j
].index
> s
->node
[i
].min_index
))
1890 f
= basic_map_follows(list
[i
], list
[j
], &s
->check_closed
);
1896 if (s
->node
[j
].index
< 0) {
1897 power_components_tarjan(s
, list
, j
);
1898 if (s
->node
[j
].min_index
< s
->node
[i
].min_index
)
1899 s
->node
[i
].min_index
= s
->node
[j
].min_index
;
1900 } else if (s
->node
[j
].index
< s
->node
[i
].min_index
)
1901 s
->node
[i
].min_index
= s
->node
[j
].index
;
1904 if (s
->node
[i
].index
!= s
->node
[i
].min_index
)
1908 j
= s
->stack
[--s
->sp
];
1909 s
->node
[j
].on_stack
= 0;
1910 s
->order
[s
->op
++] = j
;
1912 s
->order
[s
->op
++] = -1;
1917 /* Decompose the "len" basic relations in "list" into strongly connected
1920 static struct basic_map_sort
*basic_map_sort_init(isl_ctx
*ctx
, int len
,
1921 __isl_keep isl_basic_map
**list
)
1924 struct basic_map_sort
*s
= NULL
;
1926 s
= basic_map_sort_alloc(ctx
, len
);
1929 for (i
= len
- 1; i
>= 0; --i
) {
1930 if (s
->node
[i
].index
>= 0)
1932 if (power_components_tarjan(s
, list
, i
) < 0)
1938 basic_map_sort_free(s
);
1942 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
1943 * and a dimension specification (Z^{n+1} -> Z^{n+1}),
1944 * construct a map that is an overapproximation of the map
1945 * that takes an element from the dom R \times Z to an
1946 * element from ran R \times Z, such that the first n coordinates of the
1947 * difference between them is a sum of differences between images
1948 * and pre-images in one of the R_i and such that the last coordinate
1949 * is equal to the number of steps taken.
1950 * If "project" is set, then these final coordinates are not included,
1951 * i.e., a relation of type Z^n -> Z^n is returned.
1954 * \Delta_i = { y - x | (x, y) in R_i }
1956 * then the constructed map is an overapproximation of
1958 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1959 * d = (\sum_i k_i \delta_i, \sum_i k_i) and
1960 * x in dom R and x + d in ran R }
1964 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
1965 * d = (\sum_i k_i \delta_i) and
1966 * x in dom R and x + d in ran R }
1968 * if "project" is set.
1970 * We first split the map into strongly connected components, perform
1971 * the above on each component and then join the results in the correct
1972 * order, at each join also taking in the union of both arguments
1973 * to allow for paths that do not go through one of the two arguments.
1975 static __isl_give isl_map
*construct_power_components(__isl_take isl_dim
*dim
,
1976 __isl_keep isl_map
*map
, int *exact
, int project
)
1979 struct isl_map
*path
= NULL
;
1980 struct basic_map_sort
*s
= NULL
;
1987 return floyd_warshall(dim
, map
, exact
, project
);
1989 s
= basic_map_sort_init(map
->ctx
, map
->n
, map
->p
);
1994 if (s
->check_closed
&& !exact
)
1995 exact
= &local_exact
;
2001 path
= isl_map_empty(isl_map_get_dim(map
));
2003 path
= isl_map_empty(isl_dim_copy(dim
));
2004 path
= anonymize(path
);
2006 struct isl_map
*comp
;
2007 isl_map
*path_comp
, *path_comb
;
2008 comp
= isl_map_alloc_dim(isl_map_get_dim(map
), n
, 0);
2009 while (s
->order
[i
] != -1) {
2010 comp
= isl_map_add_basic_map(comp
,
2011 isl_basic_map_copy(map
->p
[s
->order
[i
]]));
2015 path_comp
= floyd_warshall(isl_dim_copy(dim
),
2016 comp
, exact
, project
);
2017 path_comp
= anonymize(path_comp
);
2018 path_comb
= isl_map_apply_range(isl_map_copy(path
),
2019 isl_map_copy(path_comp
));
2020 path
= isl_map_union(path
, path_comp
);
2021 path
= isl_map_union(path
, path_comb
);
2027 if (c
> 1 && s
->check_closed
&& !*exact
) {
2030 closed
= isl_map_is_transitively_closed(path
);
2034 basic_map_sort_free(s
);
2036 return floyd_warshall(dim
, map
, orig_exact
, project
);
2040 basic_map_sort_free(s
);
2045 basic_map_sort_free(s
);
2051 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
2052 * construct a map that is an overapproximation of the map
2053 * that takes an element from the space D to another
2054 * element from the same space, such that the difference between
2055 * them is a strictly positive sum of differences between images
2056 * and pre-images in one of the R_i.
2057 * The number of differences in the sum is equated to parameter "param".
2060 * \Delta_i = { y - x | (x, y) in R_i }
2062 * then the constructed map is an overapproximation of
2064 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
2065 * d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
2068 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
2069 * d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
2071 * if "project" is set.
2073 * If "project" is not set, then
2074 * we construct an extended mapping with an extra coordinate
2075 * that indicates the number of steps taken. In particular,
2076 * the difference in the last coordinate is equal to the number
2077 * of steps taken to move from a domain element to the corresponding
2080 static __isl_give isl_map
*construct_power(__isl_keep isl_map
*map
,
2081 int *exact
, int project
)
2083 struct isl_map
*app
= NULL
;
2084 struct isl_dim
*dim
= NULL
;
2090 dim
= isl_map_get_dim(map
);
2092 d
= isl_dim_size(dim
, isl_dim_in
);
2093 dim
= isl_dim_add(dim
, isl_dim_in
, 1);
2094 dim
= isl_dim_add(dim
, isl_dim_out
, 1);
2096 app
= construct_power_components(isl_dim_copy(dim
), map
,
2104 /* Compute the positive powers of "map", or an overapproximation.
2105 * If the result is exact, then *exact is set to 1.
2107 * If project is set, then we are actually interested in the transitive
2108 * closure, so we can use a more relaxed exactness check.
2109 * The lengths of the paths are also projected out instead of being
2110 * encoded as the difference between an extra pair of final coordinates.
2112 static __isl_give isl_map
*map_power(__isl_take isl_map
*map
,
2113 int *exact
, int project
)
2115 struct isl_map
*app
= NULL
;
2123 isl_assert(map
->ctx
,
2124 isl_map_dim(map
, isl_dim_in
) == isl_map_dim(map
, isl_dim_out
),
2127 app
= construct_power(map
, exact
, project
);
2137 /* Compute the positive powers of "map", or an overapproximation.
2138 * The result maps the exponent to a nested copy of the corresponding power.
2139 * If the result is exact, then *exact is set to 1.
2140 * map_power constructs an extended relation with the path lengths
2141 * encoded as the difference between the final coordinates.
2142 * In the final step, this difference is equated to an extra parameter
2143 * and made positive. The extra coordinates are subsequently projected out
2144 * and the parameter is turned into the domain of the result.
2146 __isl_give isl_map
*isl_map_power(__isl_take isl_map
*map
, int *exact
)
2148 isl_dim
*target_dim
;
2157 d
= isl_map_dim(map
, isl_dim_in
);
2158 param
= isl_map_dim(map
, isl_dim_param
);
2160 map
= isl_map_compute_divs(map
);
2161 map
= isl_map_coalesce(map
);
2163 if (isl_map_fast_is_empty(map
)) {
2164 map
= isl_map_from_range(isl_map_wrap(map
));
2165 map
= isl_map_add_dims(map
, isl_dim_in
, 1);
2166 map
= isl_map_set_dim_name(map
, isl_dim_in
, 0, "k");
2170 target_dim
= isl_map_get_dim(map
);
2171 target_dim
= isl_dim_from_range(isl_dim_wrap(target_dim
));
2172 target_dim
= isl_dim_add(target_dim
, isl_dim_in
, 1);
2173 target_dim
= isl_dim_set_name(target_dim
, isl_dim_in
, 0, "k");
2175 map
= map_power(map
, exact
, 0);
2177 map
= isl_map_add_dims(map
, isl_dim_param
, 1);
2178 dim
= isl_map_get_dim(map
);
2179 diff
= equate_parameter_to_length(dim
, param
);
2180 map
= isl_map_intersect(map
, diff
);
2181 map
= isl_map_project_out(map
, isl_dim_in
, d
, 1);
2182 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2183 map
= isl_map_from_range(isl_map_wrap(map
));
2184 map
= isl_map_move_dims(map
, isl_dim_in
, 0, isl_dim_param
, param
, 1);
2186 map
= isl_map_reset_dim(map
, target_dim
);
2191 /* Compute a relation that maps each element in the range of the input
2192 * relation to the lengths of all paths composed of edges in the input
2193 * relation that end up in the given range element.
2194 * The result may be an overapproximation, in which case *exact is set to 0.
2195 * The resulting relation is very similar to the power relation.
2196 * The difference are that the domain has been projected out, the
2197 * range has become the domain and the exponent is the range instead
2200 __isl_give isl_map
*isl_map_reaching_path_lengths(__isl_take isl_map
*map
,
2211 d
= isl_map_dim(map
, isl_dim_in
);
2212 param
= isl_map_dim(map
, isl_dim_param
);
2214 map
= isl_map_compute_divs(map
);
2215 map
= isl_map_coalesce(map
);
2217 if (isl_map_fast_is_empty(map
)) {
2220 map
= isl_map_project_out(map
, isl_dim_out
, 0, d
);
2221 map
= isl_map_add_dims(map
, isl_dim_out
, 1);
2225 map
= map_power(map
, exact
, 0);
2227 map
= isl_map_add_dims(map
, isl_dim_param
, 1);
2228 dim
= isl_map_get_dim(map
);
2229 diff
= equate_parameter_to_length(dim
, param
);
2230 map
= isl_map_intersect(map
, diff
);
2231 map
= isl_map_project_out(map
, isl_dim_in
, 0, d
+ 1);
2232 map
= isl_map_project_out(map
, isl_dim_out
, d
, 1);
2233 map
= isl_map_reverse(map
);
2234 map
= isl_map_move_dims(map
, isl_dim_out
, 0, isl_dim_param
, param
, 1);
2239 /* Check whether equality i of bset is a pure stride constraint
2240 * on a single dimensions, i.e., of the form
2244 * with k a constant and e an existentially quantified variable.
2246 static int is_eq_stride(__isl_keep isl_basic_set
*bset
, int i
)
2258 if (!isl_int_is_zero(bset
->eq
[i
][0]))
2261 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
2262 d
= isl_basic_set_dim(bset
, isl_dim_set
);
2263 n_div
= isl_basic_set_dim(bset
, isl_dim_div
);
2265 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1, nparam
) != -1)
2267 pos1
= isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
, d
);
2270 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ pos1
+ 1,
2271 d
- pos1
- 1) != -1)
2274 pos2
= isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ d
, n_div
);
2277 if (isl_seq_first_non_zero(bset
->eq
[i
] + 1 + nparam
+ d
+ pos2
+ 1,
2278 n_div
- pos2
- 1) != -1)
2280 if (!isl_int_is_one(bset
->eq
[i
][1 + nparam
+ pos1
]) &&
2281 !isl_int_is_negone(bset
->eq
[i
][1 + nparam
+ pos1
]))
2287 /* Given a map, compute the smallest superset of this map that is of the form
2289 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2291 * (where p ranges over the (non-parametric) dimensions),
2292 * compute the transitive closure of this map, i.e.,
2294 * { i -> j : exists k > 0:
2295 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2297 * and intersect domain and range of this transitive closure with
2298 * the given domain and range.
2300 * If with_id is set, then try to include as much of the identity mapping
2301 * as possible, by computing
2303 * { i -> j : exists k >= 0:
2304 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2306 * instead (i.e., allow k = 0).
2308 * In practice, we compute the difference set
2310 * delta = { j - i | i -> j in map },
2312 * look for stride constraint on the individual dimensions and compute
2313 * (constant) lower and upper bounds for each individual dimension,
2314 * adding a constraint for each bound not equal to infinity.
2316 static __isl_give isl_map
*box_closure_on_domain(__isl_take isl_map
*map
,
2317 __isl_take isl_set
*dom
, __isl_take isl_set
*ran
, int with_id
)
2326 isl_map
*app
= NULL
;
2327 isl_basic_set
*aff
= NULL
;
2328 isl_basic_map
*bmap
= NULL
;
2329 isl_vec
*obj
= NULL
;
2334 delta
= isl_map_deltas(isl_map_copy(map
));
2336 aff
= isl_set_affine_hull(isl_set_copy(delta
));
2339 dim
= isl_map_get_dim(map
);
2340 d
= isl_dim_size(dim
, isl_dim_in
);
2341 nparam
= isl_dim_size(dim
, isl_dim_param
);
2342 total
= isl_dim_total(dim
);
2343 bmap
= isl_basic_map_alloc_dim(dim
,
2344 aff
->n_div
+ 1, aff
->n_div
, 2 * d
+ 1);
2345 for (i
= 0; i
< aff
->n_div
+ 1; ++i
) {
2346 k
= isl_basic_map_alloc_div(bmap
);
2349 isl_int_set_si(bmap
->div
[k
][0], 0);
2351 for (i
= 0; i
< aff
->n_eq
; ++i
) {
2352 if (!is_eq_stride(aff
, i
))
2354 k
= isl_basic_map_alloc_equality(bmap
);
2357 isl_seq_clr(bmap
->eq
[k
], 1 + nparam
);
2358 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ d
,
2359 aff
->eq
[i
] + 1 + nparam
, d
);
2360 isl_seq_neg(bmap
->eq
[k
] + 1 + nparam
,
2361 aff
->eq
[i
] + 1 + nparam
, d
);
2362 isl_seq_cpy(bmap
->eq
[k
] + 1 + nparam
+ 2 * d
,
2363 aff
->eq
[i
] + 1 + nparam
+ d
, aff
->n_div
);
2364 isl_int_set_si(bmap
->eq
[k
][1 + total
+ aff
->n_div
], 0);
2366 obj
= isl_vec_alloc(map
->ctx
, 1 + nparam
+ d
);
2369 isl_seq_clr(obj
->el
, 1 + nparam
+ d
);
2370 for (i
= 0; i
< d
; ++ i
) {
2371 enum isl_lp_result res
;
2373 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 1);
2375 res
= isl_set_solve_lp(delta
, 0, obj
->el
, map
->ctx
->one
, &opt
,
2377 if (res
== isl_lp_error
)
2379 if (res
== isl_lp_ok
) {
2380 k
= isl_basic_map_alloc_inequality(bmap
);
2383 isl_seq_clr(bmap
->ineq
[k
],
2384 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2385 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], -1);
2386 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], 1);
2387 isl_int_neg(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2390 res
= isl_set_solve_lp(delta
, 1, obj
->el
, map
->ctx
->one
, &opt
,
2392 if (res
== isl_lp_error
)
2394 if (res
== isl_lp_ok
) {
2395 k
= isl_basic_map_alloc_inequality(bmap
);
2398 isl_seq_clr(bmap
->ineq
[k
],
2399 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2400 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ i
], 1);
2401 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ d
+ i
], -1);
2402 isl_int_set(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], opt
);
2405 isl_int_set_si(obj
->el
[1 + nparam
+ i
], 0);
2407 k
= isl_basic_map_alloc_inequality(bmap
);
2410 isl_seq_clr(bmap
->ineq
[k
],
2411 1 + nparam
+ 2 * d
+ bmap
->n_div
);
2413 isl_int_set_si(bmap
->ineq
[k
][0], -1);
2414 isl_int_set_si(bmap
->ineq
[k
][1 + nparam
+ 2 * d
+ aff
->n_div
], 1);
2416 app
= isl_map_from_domain_and_range(dom
, ran
);
2419 isl_basic_set_free(aff
);
2421 bmap
= isl_basic_map_finalize(bmap
);
2422 isl_set_free(delta
);
2425 map
= isl_map_from_basic_map(bmap
);
2426 map
= isl_map_intersect(map
, app
);
2431 isl_basic_map_free(bmap
);
2432 isl_basic_set_free(aff
);
2436 isl_set_free(delta
);
2441 /* Given a map, compute the smallest superset of this map that is of the form
2443 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2445 * (where p ranges over the (non-parametric) dimensions),
2446 * compute the transitive closure of this map, i.e.,
2448 * { i -> j : exists k > 0:
2449 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2451 * and intersect domain and range of this transitive closure with
2452 * domain and range of the original map.
2454 static __isl_give isl_map
*box_closure(__isl_take isl_map
*map
)
2459 domain
= isl_map_domain(isl_map_copy(map
));
2460 domain
= isl_set_coalesce(domain
);
2461 range
= isl_map_range(isl_map_copy(map
));
2462 range
= isl_set_coalesce(range
);
2464 return box_closure_on_domain(map
, domain
, range
, 0);
2467 /* Given a map, compute the smallest superset of this map that is of the form
2469 * { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
2471 * (where p ranges over the (non-parametric) dimensions),
2472 * compute the transitive and partially reflexive closure of this map, i.e.,
2474 * { i -> j : exists k >= 0:
2475 * k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
2477 * and intersect domain and range of this transitive closure with
2480 static __isl_give isl_map
*box_closure_with_identity(__isl_take isl_map
*map
,
2481 __isl_take isl_set
*dom
)
2483 return box_closure_on_domain(map
, dom
, isl_set_copy(dom
), 1);
2486 /* Check whether app is the transitive closure of map.
2487 * In particular, check that app is acyclic and, if so,
2490 * app \subset (map \cup (map \circ app))
2492 static int check_exactness_omega(__isl_keep isl_map
*map
,
2493 __isl_keep isl_map
*app
)
2497 int is_empty
, is_exact
;
2501 delta
= isl_map_deltas(isl_map_copy(app
));
2502 d
= isl_set_dim(delta
, isl_dim_set
);
2503 for (i
= 0; i
< d
; ++i
)
2504 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
2505 is_empty
= isl_set_is_empty(delta
);
2506 isl_set_free(delta
);
2512 test
= isl_map_apply_range(isl_map_copy(app
), isl_map_copy(map
));
2513 test
= isl_map_union(test
, isl_map_copy(map
));
2514 is_exact
= isl_map_is_subset(app
, test
);
2520 /* Check if basic map M_i can be combined with all the other
2521 * basic maps such that
2525 * can be computed as
2527 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2529 * In particular, check if we can compute a compact representation
2532 * M_i^* \circ M_j \circ M_i^*
2535 * Let M_i^? be an extension of M_i^+ that allows paths
2536 * of length zero, i.e., the result of box_closure(., 1).
2537 * The criterion, as proposed by Kelly et al., is that
2538 * id = M_i^? - M_i^+ can be represented as a basic map
2541 * id \circ M_j \circ id = M_j
2545 * If this function returns 1, then tc and qc are set to
2546 * M_i^+ and M_i^?, respectively.
2548 static int can_be_split_off(__isl_keep isl_map
*map
, int i
,
2549 __isl_give isl_map
**tc
, __isl_give isl_map
**qc
)
2551 isl_map
*map_i
, *id
= NULL
;
2558 C
= isl_set_union(isl_map_domain(isl_map_copy(map
)),
2559 isl_map_range(isl_map_copy(map
)));
2560 C
= isl_set_from_basic_set(isl_set_simple_hull(C
));
2564 map_i
= isl_map_from_basic_map(isl_basic_map_copy(map
->p
[i
]));
2565 *tc
= box_closure(isl_map_copy(map_i
));
2566 *qc
= box_closure_with_identity(map_i
, C
);
2567 id
= isl_map_subtract(isl_map_copy(*qc
), isl_map_copy(*tc
));
2571 if (id
->n
!= 1 || (*qc
)->n
!= 1)
2574 for (j
= 0; j
< map
->n
; ++j
) {
2575 isl_map
*map_j
, *test
;
2580 map_j
= isl_map_from_basic_map(
2581 isl_basic_map_copy(map
->p
[j
]));
2582 test
= isl_map_apply_range(isl_map_copy(id
),
2583 isl_map_copy(map_j
));
2584 test
= isl_map_apply_range(test
, isl_map_copy(id
));
2585 is_ok
= isl_map_is_equal(test
, map_j
);
2586 isl_map_free(map_j
);
2614 static __isl_give isl_map
*box_closure_with_check(__isl_take isl_map
*map
,
2619 app
= box_closure(isl_map_copy(map
));
2621 *exact
= check_exactness_omega(map
, app
);
2627 /* Compute an overapproximation of the transitive closure of "map"
2628 * using a variation of the algorithm from
2629 * "Transitive Closure of Infinite Graphs and its Applications"
2632 * We first check whether we can can split of any basic map M_i and
2639 * M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
2641 * using a recursive call on the remaining map.
2643 * If not, we simply call box_closure on the whole map.
2645 static __isl_give isl_map
*transitive_closure_omega(__isl_take isl_map
*map
,
2655 return box_closure_with_check(map
, exact
);
2657 for (i
= 0; i
< map
->n
; ++i
) {
2660 ok
= can_be_split_off(map
, i
, &tc
, &qc
);
2666 app
= isl_map_alloc_dim(isl_map_get_dim(map
), map
->n
- 1, 0);
2668 for (j
= 0; j
< map
->n
; ++j
) {
2671 app
= isl_map_add_basic_map(app
,
2672 isl_basic_map_copy(map
->p
[j
]));
2675 app
= isl_map_apply_range(isl_map_copy(qc
), app
);
2676 app
= isl_map_apply_range(app
, qc
);
2678 app
= isl_map_union(tc
, transitive_closure_omega(app
, NULL
));
2679 exact_i
= check_exactness_omega(map
, app
);
2691 return box_closure_with_check(map
, exact
);
2697 /* Compute the transitive closure of "map", or an overapproximation.
2698 * If the result is exact, then *exact is set to 1.
2699 * Simply use map_power to compute the powers of map, but tell
2700 * it to project out the lengths of the paths instead of equating
2701 * the length to a parameter.
2703 __isl_give isl_map
*isl_map_transitive_closure(__isl_take isl_map
*map
,
2706 isl_dim
*target_dim
;
2712 if (map
->ctx
->opt
->closure
== ISL_CLOSURE_BOX
)
2713 return transitive_closure_omega(map
, exact
);
2715 map
= isl_map_compute_divs(map
);
2716 map
= isl_map_coalesce(map
);
2717 closed
= isl_map_is_transitively_closed(map
);
2726 target_dim
= isl_map_get_dim(map
);
2727 map
= map_power(map
, exact
, 1);
2728 map
= isl_map_reset_dim(map
, target_dim
);
2736 static int inc_count(__isl_take isl_map
*map
, void *user
)
2747 static int collect_basic_map(__isl_take isl_map
*map
, void *user
)
2750 isl_basic_map
***next
= user
;
2752 for (i
= 0; i
< map
->n
; ++i
) {
2753 **next
= isl_basic_map_copy(map
->p
[i
]);
2766 /* Perform Floyd-Warshall on the given list of basic relations.
2767 * The basic relations may live in different dimensions,
2768 * but basic relations that get assigned to the diagonal of the
2769 * grid have domains and ranges of the same dimension and so
2770 * the standard algorithm can be used because the nested transitive
2771 * closures are only applied to diagonal elements and because all
2772 * compositions are peformed on relations with compatible domains and ranges.
2774 static __isl_give isl_union_map
*union_floyd_warshall_on_list(isl_ctx
*ctx
,
2775 __isl_keep isl_basic_map
**list
, int n
, int *exact
)
2780 isl_set
**set
= NULL
;
2781 isl_map
***grid
= NULL
;
2784 group
= setup_groups(ctx
, list
, n
, &set
, &n_group
);
2788 grid
= isl_calloc_array(ctx
, isl_map
**, n_group
);
2791 for (i
= 0; i
< n_group
; ++i
) {
2792 grid
[i
] = isl_calloc_array(map
->ctx
, isl_map
*, n_group
);
2795 for (j
= 0; j
< n_group
; ++j
) {
2796 isl_dim
*dim1
, *dim2
, *dim
;
2797 dim1
= isl_dim_reverse(isl_set_get_dim(set
[i
]));
2798 dim2
= isl_set_get_dim(set
[j
]);
2799 dim
= isl_dim_join(dim1
, dim2
);
2800 grid
[i
][j
] = isl_map_empty(dim
);
2804 for (k
= 0; k
< n
; ++k
) {
2806 j
= group
[2 * k
+ 1];
2807 grid
[i
][j
] = isl_map_union(grid
[i
][j
],
2808 isl_map_from_basic_map(
2809 isl_basic_map_copy(list
[k
])));
2812 floyd_warshall_iterate(grid
, n_group
, exact
);
2814 app
= isl_union_map_empty(isl_map_get_dim(grid
[0][0]));
2816 for (i
= 0; i
< n_group
; ++i
) {
2817 for (j
= 0; j
< n_group
; ++j
)
2818 app
= isl_union_map_add_map(app
, grid
[i
][j
]);
2823 for (i
= 0; i
< 2 * n
; ++i
)
2824 isl_set_free(set
[i
]);
2831 for (i
= 0; i
< n_group
; ++i
) {
2834 for (j
= 0; j
< n_group
; ++j
)
2835 isl_map_free(grid
[i
][j
]);
2840 for (i
= 0; i
< 2 * n
; ++i
)
2841 isl_set_free(set
[i
]);
2848 /* Perform Floyd-Warshall on the given union relation.
2849 * The implementation is very similar to that for non-unions.
2850 * The main difference is that it is applied unconditionally.
2851 * We first extract a list of basic maps from the union map
2852 * and then perform the algorithm on this list.
2854 static __isl_give isl_union_map
*union_floyd_warshall(
2855 __isl_take isl_union_map
*umap
, int *exact
)
2859 isl_basic_map
**list
;
2860 isl_basic_map
**next
;
2864 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2867 ctx
= isl_union_map_get_ctx(umap
);
2868 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2873 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2876 res
= union_floyd_warshall_on_list(ctx
, list
, n
, exact
);
2879 for (i
= 0; i
< n
; ++i
)
2880 isl_basic_map_free(list
[i
]);
2884 isl_union_map_free(umap
);
2888 for (i
= 0; i
< n
; ++i
)
2889 isl_basic_map_free(list
[i
]);
2892 isl_union_map_free(umap
);
2896 /* Decompose the give union relation into strongly connected components.
2897 * The implementation is essentially the same as that of
2898 * construct_power_components with the major difference that all
2899 * operations are performed on union maps.
2901 static __isl_give isl_union_map
*union_components(
2902 __isl_take isl_union_map
*umap
, int *exact
)
2907 isl_basic_map
**list
;
2908 isl_basic_map
**next
;
2909 isl_union_map
*path
= NULL
;
2910 struct basic_map_sort
*s
= NULL
;
2915 if (isl_union_map_foreach_map(umap
, inc_count
, &n
) < 0)
2919 return union_floyd_warshall(umap
, exact
);
2921 ctx
= isl_union_map_get_ctx(umap
);
2922 list
= isl_calloc_array(ctx
, isl_basic_map
*, n
);
2927 if (isl_union_map_foreach_map(umap
, collect_basic_map
, &next
) < 0)
2930 s
= basic_map_sort_init(ctx
, n
, list
);
2937 path
= isl_union_map_empty(isl_union_map_get_dim(umap
));
2939 isl_union_map
*comp
;
2940 isl_union_map
*path_comp
, *path_comb
;
2941 comp
= isl_union_map_empty(isl_union_map_get_dim(umap
));
2942 while (s
->order
[i
] != -1) {
2943 comp
= isl_union_map_add_map(comp
,
2944 isl_map_from_basic_map(
2945 isl_basic_map_copy(list
[s
->order
[i
]])));
2949 path_comp
= union_floyd_warshall(comp
, exact
);
2950 path_comb
= isl_union_map_apply_range(isl_union_map_copy(path
),
2951 isl_union_map_copy(path_comp
));
2952 path
= isl_union_map_union(path
, path_comp
);
2953 path
= isl_union_map_union(path
, path_comb
);
2958 if (c
> 1 && s
->check_closed
&& !*exact
) {
2961 closed
= isl_union_map_is_transitively_closed(path
);
2967 basic_map_sort_free(s
);
2969 for (i
= 0; i
< n
; ++i
)
2970 isl_basic_map_free(list
[i
]);
2974 isl_union_map_free(path
);
2975 return union_floyd_warshall(umap
, exact
);
2978 isl_union_map_free(umap
);
2982 basic_map_sort_free(s
);
2984 for (i
= 0; i
< n
; ++i
)
2985 isl_basic_map_free(list
[i
]);
2988 isl_union_map_free(umap
);
2989 isl_union_map_free(path
);
2993 /* Compute the transitive closure of "umap", or an overapproximation.
2994 * If the result is exact, then *exact is set to 1.
2996 __isl_give isl_union_map
*isl_union_map_transitive_closure(
2997 __isl_take isl_union_map
*umap
, int *exact
)
3007 umap
= isl_union_map_compute_divs(umap
);
3008 umap
= isl_union_map_coalesce(umap
);
3009 closed
= isl_union_map_is_transitively_closed(umap
);
3014 umap
= union_components(umap
, exact
);
3017 isl_union_map_free(umap
);
3021 struct isl_union_power
{
3026 static int power(__isl_take isl_map
*map
, void *user
)
3028 struct isl_union_power
*up
= user
;
3030 map
= isl_map_power(map
, up
->exact
);
3031 up
->pow
= isl_union_map_from_map(map
);
3036 /* Construct a map [x] -> [x+1], with parameters prescribed by "dim".
3038 static __isl_give isl_union_map
*increment(__isl_take isl_dim
*dim
)
3041 isl_basic_map
*bmap
;
3043 dim
= isl_dim_add(dim
, isl_dim_in
, 1);
3044 dim
= isl_dim_add(dim
, isl_dim_out
, 1);
3045 bmap
= isl_basic_map_alloc_dim(dim
, 0, 1, 0);
3046 k
= isl_basic_map_alloc_equality(bmap
);
3049 isl_seq_clr(bmap
->eq
[k
], isl_basic_map_total_dim(bmap
));
3050 isl_int_set_si(bmap
->eq
[k
][0], 1);
3051 isl_int_set_si(bmap
->eq
[k
][isl_basic_map_offset(bmap
, isl_dim_in
)], 1);
3052 isl_int_set_si(bmap
->eq
[k
][isl_basic_map_offset(bmap
, isl_dim_out
)], -1);
3053 return isl_union_map_from_map(isl_map_from_basic_map(bmap
));
3055 isl_basic_map_free(bmap
);
3059 /* Construct a map [[x]->[y]] -> [y-x], with parameters prescribed by "dim".
3061 static __isl_give isl_union_map
*deltas_map(__isl_take isl_dim
*dim
)
3063 isl_basic_map
*bmap
;
3065 dim
= isl_dim_add(dim
, isl_dim_in
, 1);
3066 dim
= isl_dim_add(dim
, isl_dim_out
, 1);
3067 bmap
= isl_basic_map_universe(dim
);
3068 bmap
= isl_basic_map_deltas_map(bmap
);
3070 return isl_union_map_from_map(isl_map_from_basic_map(bmap
));
3073 /* Compute the positive powers of "map", or an overapproximation.
3074 * The result maps the exponent to a nested copy of the corresponding power.
3075 * If the result is exact, then *exact is set to 1.
3077 __isl_give isl_union_map
*isl_union_map_power(__isl_take isl_union_map
*umap
,
3086 n
= isl_union_map_n_map(umap
);
3090 struct isl_union_power up
= { NULL
, exact
};
3091 isl_union_map_foreach_map(umap
, &power
, &up
);
3092 isl_union_map_free(umap
);
3095 inc
= increment(isl_union_map_get_dim(umap
));
3096 umap
= isl_union_map_product(inc
, umap
);
3097 umap
= isl_union_map_transitive_closure(umap
, exact
);
3098 umap
= isl_union_map_zip(umap
);
3099 dm
= deltas_map(isl_union_map_get_dim(umap
));
3100 umap
= isl_union_map_apply_domain(umap
, dm
);