2 * Copyright 2010 INRIA Saclay
4 * Use of this software is governed by the GNU LGPLv2.1 license
6 * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
7 * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
12 #include "isl_map_private.h"
15 * The transitive closure implementation is based on the paper
16 * "Computing the Transitive Closure of a Union of Affine Integer
17 * Tuple Relations" by Anna Beletska, Denis Barthou, Wlodzimierz Bielecki and
21 /* Given a set of n offsets v_i (the rows of "steps"), construct a relation
22 * of the given dimension specification (Z^{n+1} -> Z^{n+1})
23 * that maps an element x to any element that can be reached
24 * by taking a non-negative number of steps along any of
25 * the extended offsets v'_i = [v_i 1].
28 * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v'_i }
30 * For any element in this relation, the number of steps taken
31 * is equal to the difference in the final coordinates.
33 static __isl_give isl_map
*path_along_steps(__isl_take isl_dim
*dim
,
34 __isl_keep isl_mat
*steps
)
37 struct isl_basic_map
*path
= NULL
;
45 d
= isl_dim_size(dim
, isl_dim_in
);
47 nparam
= isl_dim_size(dim
, isl_dim_param
);
49 path
= isl_basic_map_alloc_dim(isl_dim_copy(dim
), n
, d
, n
);
51 for (i
= 0; i
< n
; ++i
) {
52 k
= isl_basic_map_alloc_div(path
);
55 isl_assert(steps
->ctx
, i
== k
, goto error
);
56 isl_int_set_si(path
->div
[k
][0], 0);
59 for (i
= 0; i
< d
; ++i
) {
60 k
= isl_basic_map_alloc_equality(path
);
63 isl_seq_clr(path
->eq
[k
], 1 + isl_basic_map_total_dim(path
));
64 isl_int_set_si(path
->eq
[k
][1 + nparam
+ i
], 1);
65 isl_int_set_si(path
->eq
[k
][1 + nparam
+ d
+ i
], -1);
67 for (j
= 0; j
< n
; ++j
)
68 isl_int_set_si(path
->eq
[k
][1 + nparam
+ 2 * d
+ j
], 1);
70 for (j
= 0; j
< n
; ++j
)
71 isl_int_set(path
->eq
[k
][1 + nparam
+ 2 * d
+ j
],
75 for (i
= 0; i
< n
; ++i
) {
76 k
= isl_basic_map_alloc_inequality(path
);
79 isl_seq_clr(path
->ineq
[k
], 1 + isl_basic_map_total_dim(path
));
80 isl_int_set_si(path
->ineq
[k
][1 + nparam
+ 2 * d
+ i
], 1);
85 path
= isl_basic_map_simplify(path
);
86 path
= isl_basic_map_finalize(path
);
87 return isl_map_from_basic_map(path
);
90 isl_basic_map_free(path
);
94 /* Given a dimenion specification Z^{n+1} -> Z^{n+1} and a parameter "param",
95 * construct a map that equates the parameter to the difference
96 * in the final coordinates and imposes that this difference is positive.
99 * { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
101 static __isl_give isl_map
*equate_parameter_to_length(__isl_take isl_dim
*dim
,
104 struct isl_basic_map
*bmap
;
109 d
= isl_dim_size(dim
, isl_dim_in
);
110 nparam
= isl_dim_size(dim
, isl_dim_param
);
111 bmap
= isl_basic_map_alloc_dim(dim
, 0, 1, 1);
112 k
= isl_basic_map_alloc_equality(bmap
);
115 isl_seq_clr(bmap
->eq
[k
], 1 + isl_basic_map_total_dim(bmap
));
116 isl_int_set_si(bmap
->eq
[k
][1 + param
], -1);
117 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
- 1], -1);
118 isl_int_set_si(bmap
->eq
[k
][1 + nparam
+ d
+ d
- 1], 1);
120 k
= isl_basic_map_alloc_inequality(bmap
);
123 isl_seq_clr(bmap
->ineq
[k
], 1 + isl_basic_map_total_dim(bmap
));
124 isl_int_set_si(bmap
->ineq
[k
][1 + param
], 1);
125 isl_int_set_si(bmap
->ineq
[k
][0], -1);
127 bmap
= isl_basic_map_finalize(bmap
);
128 return isl_map_from_basic_map(bmap
);
130 isl_basic_map_free(bmap
);
134 /* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
135 * construct a map that is an overapproximation of the map
136 * that takes an element from the space D to another
137 * element from the same space, such that the difference between
138 * them is a strictly positive sum of differences between images
139 * and pre-images in one of the R_i.
140 * The number of differences in the sum is equated to parameter "param".
143 * \Delta_i = { y - x | (x, y) in R_i }
145 * then the constructed map is an overapproximation of
147 * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
148 * d = \sum_i k_i and k = \sum_i k_i > 0 }
150 * We first construct an extended mapping with an extra coordinate
151 * that indicates the number of steps taken. In particular,
152 * the difference in the last coordinate is equal to the number
153 * of steps taken to move from a domain element to the corresponding
155 * In the final step, this difference is equated to the parameter "param"
156 * and made positive. The extra coordinates are subsequently projected out.
158 * The elements of the singleton \Delta_i's are collected as the
159 * rows of the steps matrix. For all these \Delta_i's together,
160 * a single path is constructed.
161 * For each of the other \Delta_i's
162 * we currently simply construct a universal map { (x) -> (y) }.
164 static __isl_give isl_map
*construct_path(__isl_keep isl_map
*map
,
167 struct isl_mat
*steps
= NULL
;
168 struct isl_map
*path
= NULL
;
169 struct isl_map
*diff
;
170 struct isl_dim
*dim
= NULL
;
177 dim
= isl_map_get_dim(map
);
179 d
= isl_dim_size(dim
, isl_dim_in
);
180 dim
= isl_dim_add(dim
, isl_dim_in
, 1);
181 dim
= isl_dim_add(dim
, isl_dim_out
, 1);
183 path
= isl_map_identity(isl_dim_domain(isl_dim_copy(dim
)));
185 steps
= isl_mat_alloc(map
->ctx
, map
->n
, d
);
190 for (i
= 0; i
< map
->n
; ++i
) {
191 struct isl_basic_set
*delta
;
193 delta
= isl_basic_map_deltas(isl_basic_map_copy(map
->p
[i
]));
195 for (j
= 0; j
< d
; ++j
) {
198 fixed
= isl_basic_set_fast_dim_is_fixed(delta
, j
,
201 isl_basic_set_free(delta
);
208 isl_basic_set_free(delta
);
212 path
= isl_map_universe(isl_dim_copy(dim
));
219 path
= isl_map_apply_range(path
,
220 path_along_steps(isl_dim_copy(dim
), steps
));
223 diff
= equate_parameter_to_length(dim
, param
);
224 path
= isl_map_intersect(path
, diff
);
225 path
= isl_map_project_out(path
, isl_dim_in
, d
, 1);
226 path
= isl_map_project_out(path
, isl_dim_out
, d
, 1);
236 /* Check whether "path" is acyclic.
237 * That is, check whether
239 * { d | d = y - x and (x,y) in path }
241 * does not contain the origin.
243 static int is_acyclic(__isl_take isl_map
*path
)
248 struct isl_set
*delta
;
250 delta
= isl_map_deltas(path
);
251 dim
= isl_set_dim(delta
, isl_dim_set
);
252 for (i
= 0; i
< dim
; ++i
)
253 delta
= isl_set_fix_si(delta
, isl_dim_set
, i
, 0);
255 acyclic
= isl_set_is_empty(delta
);
261 /* Shift variable at position "pos" up by one.
262 * That is, replace the corresponding variable v by v - 1.
264 static __isl_give isl_basic_map
*basic_map_shift_pos(
265 __isl_take isl_basic_map
*bmap
, unsigned pos
)
269 bmap
= isl_basic_map_cow(bmap
);
273 for (i
= 0; i
< bmap
->n_eq
; ++i
)
274 isl_int_sub(bmap
->eq
[i
][0], bmap
->eq
[i
][0], bmap
->eq
[i
][pos
]);
276 for (i
= 0; i
< bmap
->n_ineq
; ++i
)
277 isl_int_sub(bmap
->ineq
[i
][0],
278 bmap
->ineq
[i
][0], bmap
->ineq
[i
][pos
]);
280 for (i
= 0; i
< bmap
->n_div
; ++i
) {
281 if (isl_int_is_zero(bmap
->div
[i
][0]))
283 isl_int_sub(bmap
->div
[i
][1],
284 bmap
->div
[i
][1], bmap
->div
[i
][1 + pos
]);
290 /* Shift variable at position "pos" up by one.
291 * That is, replace the corresponding variable v by v - 1.
293 static __isl_give isl_map
*map_shift_pos(__isl_take isl_map
*map
, unsigned pos
)
297 map
= isl_map_cow(map
);
301 for (i
= 0; i
< map
->n
; ++i
) {
302 map
->p
[i
] = basic_map_shift_pos(map
->p
[i
], pos
);
306 ISL_F_CLR(map
, ISL_MAP_NORMALIZED
);
313 /* Check whether the overapproximation of the power of "map" is exactly
314 * the power of "map". Let R be "map" and A_k the overapproximation.
315 * The approximation is exact if
318 * A_k = A_{k-1} \circ R k >= 2
320 * Since A_k is known to be an overapproximation, we only need to check
323 * A_k \subset A_{k-1} \circ R k >= 2
326 static int check_power_exactness(__isl_take isl_map
*map
,
327 __isl_take isl_map
*app
, unsigned param
)
333 app_1
= isl_map_fix_si(isl_map_copy(app
), isl_dim_param
, param
, 1);
335 exact
= isl_map_is_subset(app_1
, map
);
338 if (!exact
|| exact
< 0) {
344 app_2
= isl_map_lower_bound_si(isl_map_copy(app
),
345 isl_dim_param
, param
, 2);
346 app_1
= map_shift_pos(app
, 1 + param
);
347 app_1
= isl_map_apply_range(map
, app_1
);
349 exact
= isl_map_is_subset(app_2
, app_1
);
357 /* Check whether the overapproximation of the power of "map" is exactly
358 * the power of "map", possibly after projecting out the power (if "project"
361 * If "project" is set and if "steps" can only result in acyclic paths,
364 * A = R \cup (A \circ R)
366 * where A is the overapproximation with the power projected out, i.e.,
367 * an overapproximation of the transitive closure.
368 * More specifically, since A is known to be an overapproximation, we check
370 * A \subset R \cup (A \circ R)
372 * Otherwise, we check if the power is exact.
374 static int check_exactness(__isl_take isl_map
*map
, __isl_take isl_map
*app
,
375 __isl_take isl_map
*path
, unsigned param
, int project
)
381 project
= is_acyclic(path
);
388 return check_power_exactness(map
, app
, param
);
390 map
= isl_map_project_out(map
, isl_dim_param
, param
, 1);
391 app
= isl_map_project_out(app
, isl_dim_param
, param
, 1);
393 test
= isl_map_apply_range(isl_map_copy(map
), isl_map_copy(app
));
394 test
= isl_map_union(test
, isl_map_copy(map
));
396 exact
= isl_map_is_subset(app
, test
);
410 /* Compute the positive powers of "map", or an overapproximation.
411 * The power is given by parameter "param". If the result is exact,
412 * then *exact is set to 1.
413 * If project is set, then we are actually interested in the transitive
414 * closure, so we can use a more relaxed exactness check.
416 static __isl_give isl_map
*map_power(__isl_take isl_map
*map
, unsigned param
,
417 int *exact
, int project
)
419 struct isl_set
*domain
= NULL
;
420 struct isl_set
*range
= NULL
;
421 struct isl_map
*app
= NULL
;
422 struct isl_map
*path
= NULL
;
427 map
= isl_map_remove_empty_parts(map
);
431 if (isl_map_fast_is_empty(map
))
434 isl_assert(map
->ctx
, param
< isl_map_dim(map
, isl_dim_param
), goto error
);
436 isl_map_dim(map
, isl_dim_in
) == isl_map_dim(map
, isl_dim_out
),
439 domain
= isl_map_domain(isl_map_copy(map
));
440 domain
= isl_set_coalesce(domain
);
441 range
= isl_map_range(isl_map_copy(map
));
442 range
= isl_set_coalesce(range
);
443 app
= isl_map_from_domain_and_range(isl_set_copy(domain
),
444 isl_set_copy(range
));
446 path
= construct_path(map
, param
);
447 app
= isl_map_intersect(app
, isl_map_copy(path
));
450 (*exact
= check_exactness(isl_map_copy(map
), isl_map_copy(app
),
451 isl_map_copy(path
), param
, project
)) < 0)
454 isl_set_free(domain
);
460 isl_set_free(domain
);
468 /* Compute the positive powers of "map", or an overapproximation.
469 * The power is given by parameter "param". If the result is exact,
470 * then *exact is set to 1.
472 __isl_give isl_map
*isl_map_power(__isl_take isl_map
*map
, unsigned param
,
475 return map_power(map
, param
, exact
, 0);
478 /* Compute the transitive closure of "map", or an overapproximation.
479 * If the result is exact, then *exact is set to 1.
480 * Simply compute the powers of map and then project out the parameter
481 * describing the power.
483 __isl_give isl_map
*isl_map_transitive_closure(__isl_take isl_map
*map
,
491 param
= isl_map_dim(map
, isl_dim_param
);
492 map
= isl_map_add(map
, isl_dim_param
, 1);
493 map
= map_power(map
, param
, exact
, 1);
494 map
= isl_map_project_out(map
, isl_dim_param
, param
, 1);