2 * Copyright 2008-2009 Katholieke Universiteit Leuven
4 * Use of this software is governed by the MIT license
6 * Written by Sven Verdoolaege, K.U.Leuven, Departement
7 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
10 #include <isl_ctx_private.h>
11 #include <isl_map_private.h>
12 #include "isl_sample.h"
13 #include "isl_sample_piplib.h"
17 #include "isl_equalities.h"
19 #include "isl_basis_reduction.h"
20 #include <isl_factorization.h>
21 #include <isl_point_private.h>
22 #include <isl_options_private.h>
24 static struct isl_vec
*empty_sample(struct isl_basic_set
*bset
)
28 vec
= isl_vec_alloc(bset
->ctx
, 0);
29 isl_basic_set_free(bset
);
33 /* Construct a zero sample of the same dimension as bset.
34 * As a special case, if bset is zero-dimensional, this
35 * function creates a zero-dimensional sample point.
37 static struct isl_vec
*zero_sample(struct isl_basic_set
*bset
)
40 struct isl_vec
*sample
;
42 dim
= isl_basic_set_total_dim(bset
);
43 sample
= isl_vec_alloc(bset
->ctx
, 1 + dim
);
45 isl_int_set_si(sample
->el
[0], 1);
46 isl_seq_clr(sample
->el
+ 1, dim
);
48 isl_basic_set_free(bset
);
52 static struct isl_vec
*interval_sample(struct isl_basic_set
*bset
)
56 struct isl_vec
*sample
;
58 bset
= isl_basic_set_simplify(bset
);
61 if (isl_basic_set_plain_is_empty(bset
))
62 return empty_sample(bset
);
63 if (bset
->n_eq
== 0 && bset
->n_ineq
== 0)
64 return zero_sample(bset
);
66 sample
= isl_vec_alloc(bset
->ctx
, 2);
71 isl_int_set_si(sample
->block
.data
[0], 1);
74 isl_assert(bset
->ctx
, bset
->n_eq
== 1, goto error
);
75 isl_assert(bset
->ctx
, bset
->n_ineq
== 0, goto error
);
76 if (isl_int_is_one(bset
->eq
[0][1]))
77 isl_int_neg(sample
->el
[1], bset
->eq
[0][0]);
79 isl_assert(bset
->ctx
, isl_int_is_negone(bset
->eq
[0][1]),
81 isl_int_set(sample
->el
[1], bset
->eq
[0][0]);
83 isl_basic_set_free(bset
);
88 if (isl_int_is_one(bset
->ineq
[0][1]))
89 isl_int_neg(sample
->block
.data
[1], bset
->ineq
[0][0]);
91 isl_int_set(sample
->block
.data
[1], bset
->ineq
[0][0]);
92 for (i
= 1; i
< bset
->n_ineq
; ++i
) {
93 isl_seq_inner_product(sample
->block
.data
,
94 bset
->ineq
[i
], 2, &t
);
95 if (isl_int_is_neg(t
))
99 if (i
< bset
->n_ineq
) {
100 isl_vec_free(sample
);
101 return empty_sample(bset
);
104 isl_basic_set_free(bset
);
107 isl_basic_set_free(bset
);
108 isl_vec_free(sample
);
112 static struct isl_mat
*independent_bounds(struct isl_basic_set
*bset
)
115 struct isl_mat
*dirs
= NULL
;
116 struct isl_mat
*bounds
= NULL
;
122 dim
= isl_basic_set_n_dim(bset
);
123 bounds
= isl_mat_alloc(bset
->ctx
, 1+dim
, 1+dim
);
127 isl_int_set_si(bounds
->row
[0][0], 1);
128 isl_seq_clr(bounds
->row
[0]+1, dim
);
131 if (bset
->n_ineq
== 0)
134 dirs
= isl_mat_alloc(bset
->ctx
, dim
, dim
);
136 isl_mat_free(bounds
);
139 isl_seq_cpy(dirs
->row
[0], bset
->ineq
[0]+1, dirs
->n_col
);
140 isl_seq_cpy(bounds
->row
[1], bset
->ineq
[0], bounds
->n_col
);
141 for (j
= 1, n
= 1; n
< dim
&& j
< bset
->n_ineq
; ++j
) {
144 isl_seq_cpy(dirs
->row
[n
], bset
->ineq
[j
]+1, dirs
->n_col
);
146 pos
= isl_seq_first_non_zero(dirs
->row
[n
], dirs
->n_col
);
149 for (i
= 0; i
< n
; ++i
) {
151 pos_i
= isl_seq_first_non_zero(dirs
->row
[i
], dirs
->n_col
);
156 isl_seq_elim(dirs
->row
[n
], dirs
->row
[i
], pos
,
158 pos
= isl_seq_first_non_zero(dirs
->row
[n
], dirs
->n_col
);
166 isl_int
*t
= dirs
->row
[n
];
167 for (k
= n
; k
> i
; --k
)
168 dirs
->row
[k
] = dirs
->row
[k
-1];
172 isl_seq_cpy(bounds
->row
[n
], bset
->ineq
[j
], bounds
->n_col
);
179 static void swap_inequality(struct isl_basic_set
*bset
, int a
, int b
)
181 isl_int
*t
= bset
->ineq
[a
];
182 bset
->ineq
[a
] = bset
->ineq
[b
];
186 /* Skew into positive orthant and project out lineality space.
188 * We perform a unimodular transformation that turns a selected
189 * maximal set of linearly independent bounds into constraints
190 * on the first dimensions that impose that these first dimensions
191 * are non-negative. In particular, the constraint matrix is lower
192 * triangular with positive entries on the diagonal and negative
194 * If "bset" has a lineality space then these constraints (and therefore
195 * all constraints in bset) only involve the first dimensions.
196 * The remaining dimensions then do not appear in any constraints and
197 * we can select any value for them, say zero. We therefore project
198 * out this final dimensions and plug in the value zero later. This
199 * is accomplished by simply dropping the final columns of
200 * the unimodular transformation.
202 static struct isl_basic_set
*isl_basic_set_skew_to_positive_orthant(
203 struct isl_basic_set
*bset
, struct isl_mat
**T
)
205 struct isl_mat
*U
= NULL
;
206 struct isl_mat
*bounds
= NULL
;
208 unsigned old_dim
, new_dim
;
214 isl_assert(bset
->ctx
, isl_basic_set_n_param(bset
) == 0, goto error
);
215 isl_assert(bset
->ctx
, bset
->n_div
== 0, goto error
);
216 isl_assert(bset
->ctx
, bset
->n_eq
== 0, goto error
);
218 old_dim
= isl_basic_set_n_dim(bset
);
219 /* Try to move (multiples of) unit rows up. */
220 for (i
= 0, j
= 0; i
< bset
->n_ineq
; ++i
) {
221 int pos
= isl_seq_first_non_zero(bset
->ineq
[i
]+1, old_dim
);
224 if (isl_seq_first_non_zero(bset
->ineq
[i
]+1+pos
+1,
228 swap_inequality(bset
, i
, j
);
231 bounds
= independent_bounds(bset
);
234 new_dim
= bounds
->n_row
- 1;
235 bounds
= isl_mat_left_hermite(bounds
, 1, &U
, NULL
);
238 U
= isl_mat_drop_cols(U
, 1 + new_dim
, old_dim
- new_dim
);
239 bset
= isl_basic_set_preimage(bset
, isl_mat_copy(U
));
243 isl_mat_free(bounds
);
246 isl_mat_free(bounds
);
248 isl_basic_set_free(bset
);
252 /* Find a sample integer point, if any, in bset, which is known
253 * to have equalities. If bset contains no integer points, then
254 * return a zero-length vector.
255 * We simply remove the known equalities, compute a sample
256 * in the resulting bset, using the specified recurse function,
257 * and then transform the sample back to the original space.
259 static struct isl_vec
*sample_eq(struct isl_basic_set
*bset
,
260 struct isl_vec
*(*recurse
)(struct isl_basic_set
*))
263 struct isl_vec
*sample
;
268 bset
= isl_basic_set_remove_equalities(bset
, &T
, NULL
);
269 sample
= recurse(bset
);
270 if (!sample
|| sample
->size
== 0)
273 sample
= isl_mat_vec_product(T
, sample
);
277 /* Return a matrix containing the equalities of the tableau
278 * in constraint form. The tableau is assumed to have
279 * an associated bset that has been kept up-to-date.
281 static struct isl_mat
*tab_equalities(struct isl_tab
*tab
)
286 struct isl_basic_set
*bset
;
291 bset
= isl_tab_peek_bset(tab
);
292 isl_assert(tab
->mat
->ctx
, bset
, return NULL
);
294 n_eq
= tab
->n_var
- tab
->n_col
+ tab
->n_dead
;
295 if (tab
->empty
|| n_eq
== 0)
296 return isl_mat_alloc(tab
->mat
->ctx
, 0, tab
->n_var
);
297 if (n_eq
== tab
->n_var
)
298 return isl_mat_identity(tab
->mat
->ctx
, tab
->n_var
);
300 eq
= isl_mat_alloc(tab
->mat
->ctx
, n_eq
, tab
->n_var
);
303 for (i
= 0, j
= 0; i
< tab
->n_con
; ++i
) {
304 if (tab
->con
[i
].is_row
)
306 if (tab
->con
[i
].index
>= 0 && tab
->con
[i
].index
>= tab
->n_dead
)
309 isl_seq_cpy(eq
->row
[j
], bset
->eq
[i
] + 1, tab
->n_var
);
311 isl_seq_cpy(eq
->row
[j
],
312 bset
->ineq
[i
- bset
->n_eq
] + 1, tab
->n_var
);
315 isl_assert(bset
->ctx
, j
== n_eq
, goto error
);
322 /* Compute and return an initial basis for the bounded tableau "tab".
324 * If the tableau is either full-dimensional or zero-dimensional,
325 * the we simply return an identity matrix.
326 * Otherwise, we construct a basis whose first directions correspond
329 static struct isl_mat
*initial_basis(struct isl_tab
*tab
)
335 tab
->n_unbounded
= 0;
336 tab
->n_zero
= n_eq
= tab
->n_var
- tab
->n_col
+ tab
->n_dead
;
337 if (tab
->empty
|| n_eq
== 0 || n_eq
== tab
->n_var
)
338 return isl_mat_identity(tab
->mat
->ctx
, 1 + tab
->n_var
);
340 eq
= tab_equalities(tab
);
341 eq
= isl_mat_left_hermite(eq
, 0, NULL
, &Q
);
346 Q
= isl_mat_lin_to_aff(Q
);
350 /* Compute the minimum of the current ("level") basis row over "tab"
351 * and store the result in position "level" of "min".
353 static enum isl_lp_result
compute_min(isl_ctx
*ctx
, struct isl_tab
*tab
,
354 __isl_keep isl_vec
*min
, int level
)
356 return isl_tab_min(tab
, tab
->basis
->row
[1 + level
],
357 ctx
->one
, &min
->el
[level
], NULL
, 0);
360 /* Compute the maximum of the current ("level") basis row over "tab"
361 * and store the result in position "level" of "max".
363 static enum isl_lp_result
compute_max(isl_ctx
*ctx
, struct isl_tab
*tab
,
364 __isl_keep isl_vec
*max
, int level
)
366 enum isl_lp_result res
;
367 unsigned dim
= tab
->n_var
;
369 isl_seq_neg(tab
->basis
->row
[1 + level
] + 1,
370 tab
->basis
->row
[1 + level
] + 1, dim
);
371 res
= isl_tab_min(tab
, tab
->basis
->row
[1 + level
],
372 ctx
->one
, &max
->el
[level
], NULL
, 0);
373 isl_seq_neg(tab
->basis
->row
[1 + level
] + 1,
374 tab
->basis
->row
[1 + level
] + 1, dim
);
375 isl_int_neg(max
->el
[level
], max
->el
[level
]);
380 /* Given a tableau representing a set, find and return
381 * an integer point in the set, if there is any.
383 * We perform a depth first search
384 * for an integer point, by scanning all possible values in the range
385 * attained by a basis vector, where an initial basis may have been set
386 * by the calling function. Otherwise an initial basis that exploits
387 * the equalities in the tableau is created.
388 * tab->n_zero is currently ignored and is clobbered by this function.
390 * The tableau is allowed to have unbounded direction, but then
391 * the calling function needs to set an initial basis, with the
392 * unbounded directions last and with tab->n_unbounded set
393 * to the number of unbounded directions.
394 * Furthermore, the calling functions needs to add shifted copies
395 * of all constraints involving unbounded directions to ensure
396 * that any feasible rational value in these directions can be rounded
397 * up to yield a feasible integer value.
398 * In particular, let B define the given basis x' = B x
399 * and let T be the inverse of B, i.e., X = T x'.
400 * Let a x + c >= 0 be a constraint of the set represented by the tableau,
401 * or a T x' + c >= 0 in terms of the given basis. Assume that
402 * the bounded directions have an integer value, then we can safely
403 * round up the values for the unbounded directions if we make sure
404 * that x' not only satisfies the original constraint, but also
405 * the constraint "a T x' + c + s >= 0" with s the sum of all
406 * negative values in the last n_unbounded entries of "a T".
407 * The calling function therefore needs to add the constraint
408 * a x + c + s >= 0. The current function then scans the first
409 * directions for an integer value and once those have been found,
410 * it can compute "T ceil(B x)" to yield an integer point in the set.
411 * Note that during the search, the first rows of B may be changed
412 * by a basis reduction, but the last n_unbounded rows of B remain
413 * unaltered and are also not mixed into the first rows.
415 * The search is implemented iteratively. "level" identifies the current
416 * basis vector. "init" is true if we want the first value at the current
417 * level and false if we want the next value.
419 * The initial basis is the identity matrix. If the range in some direction
420 * contains more than one integer value, we perform basis reduction based
421 * on the value of ctx->opt->gbr
422 * - ISL_GBR_NEVER: never perform basis reduction
423 * - ISL_GBR_ONCE: only perform basis reduction the first
424 * time such a range is encountered
425 * - ISL_GBR_ALWAYS: always perform basis reduction when
426 * such a range is encountered
428 * When ctx->opt->gbr is set to ISL_GBR_ALWAYS, then we allow the basis
429 * reduction computation to return early. That is, as soon as it
430 * finds a reasonable first direction.
432 struct isl_vec
*isl_tab_sample(struct isl_tab
*tab
)
437 struct isl_vec
*sample
;
440 enum isl_lp_result res
;
444 struct isl_tab_undo
**snap
;
449 return isl_vec_alloc(tab
->mat
->ctx
, 0);
452 tab
->basis
= initial_basis(tab
);
455 isl_assert(tab
->mat
->ctx
, tab
->basis
->n_row
== tab
->n_var
+ 1,
457 isl_assert(tab
->mat
->ctx
, tab
->basis
->n_col
== tab
->n_var
+ 1,
464 if (tab
->n_unbounded
== tab
->n_var
) {
465 sample
= isl_tab_get_sample_value(tab
);
466 sample
= isl_mat_vec_product(isl_mat_copy(tab
->basis
), sample
);
467 sample
= isl_vec_ceil(sample
);
468 sample
= isl_mat_vec_inverse_product(isl_mat_copy(tab
->basis
),
473 if (isl_tab_extend_cons(tab
, dim
+ 1) < 0)
476 min
= isl_vec_alloc(ctx
, dim
);
477 max
= isl_vec_alloc(ctx
, dim
);
478 snap
= isl_alloc_array(ctx
, struct isl_tab_undo
*, dim
);
480 if (!min
|| !max
|| !snap
)
489 res
= compute_min(ctx
, tab
, min
, level
);
490 if (res
== isl_lp_error
)
492 if (res
!= isl_lp_ok
)
493 isl_die(ctx
, isl_error_internal
,
494 "expecting bounded rational solution",
496 if (isl_tab_sample_is_integer(tab
))
498 res
= compute_max(ctx
, tab
, max
, level
);
499 if (res
== isl_lp_error
)
501 if (res
!= isl_lp_ok
)
502 isl_die(ctx
, isl_error_internal
,
503 "expecting bounded rational solution",
505 if (isl_tab_sample_is_integer(tab
))
507 if (!reduced
&& ctx
->opt
->gbr
!= ISL_GBR_NEVER
&&
508 isl_int_lt(min
->el
[level
], max
->el
[level
])) {
509 unsigned gbr_only_first
;
510 if (ctx
->opt
->gbr
== ISL_GBR_ONCE
)
511 ctx
->opt
->gbr
= ISL_GBR_NEVER
;
513 gbr_only_first
= ctx
->opt
->gbr_only_first
;
514 ctx
->opt
->gbr_only_first
=
515 ctx
->opt
->gbr
== ISL_GBR_ALWAYS
;
516 tab
= isl_tab_compute_reduced_basis(tab
);
517 ctx
->opt
->gbr_only_first
= gbr_only_first
;
518 if (!tab
|| !tab
->basis
)
524 snap
[level
] = isl_tab_snap(tab
);
526 isl_int_add_ui(min
->el
[level
], min
->el
[level
], 1);
528 if (isl_int_gt(min
->el
[level
], max
->el
[level
])) {
532 if (isl_tab_rollback(tab
, snap
[level
]) < 0)
536 isl_int_neg(tab
->basis
->row
[1 + level
][0], min
->el
[level
]);
537 if (isl_tab_add_valid_eq(tab
, tab
->basis
->row
[1 + level
]) < 0)
539 isl_int_set_si(tab
->basis
->row
[1 + level
][0], 0);
540 if (level
+ tab
->n_unbounded
< dim
- 1) {
549 sample
= isl_tab_get_sample_value(tab
);
552 if (tab
->n_unbounded
&& !isl_int_is_one(sample
->el
[0])) {
553 sample
= isl_mat_vec_product(isl_mat_copy(tab
->basis
),
555 sample
= isl_vec_ceil(sample
);
556 sample
= isl_mat_vec_inverse_product(
557 isl_mat_copy(tab
->basis
), sample
);
560 sample
= isl_vec_alloc(ctx
, 0);
575 static struct isl_vec
*sample_bounded(struct isl_basic_set
*bset
);
577 /* Compute a sample point of the given basic set, based on the given,
578 * non-trivial factorization.
580 static __isl_give isl_vec
*factored_sample(__isl_take isl_basic_set
*bset
,
581 __isl_take isl_factorizer
*f
)
584 isl_vec
*sample
= NULL
;
589 ctx
= isl_basic_set_get_ctx(bset
);
593 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
594 nvar
= isl_basic_set_dim(bset
, isl_dim_set
);
596 sample
= isl_vec_alloc(ctx
, 1 + isl_basic_set_total_dim(bset
));
599 isl_int_set_si(sample
->el
[0], 1);
601 bset
= isl_morph_basic_set(isl_morph_copy(f
->morph
), bset
);
603 for (i
= 0, n
= 0; i
< f
->n_group
; ++i
) {
604 isl_basic_set
*bset_i
;
607 bset_i
= isl_basic_set_copy(bset
);
608 bset_i
= isl_basic_set_drop_constraints_involving(bset_i
,
609 nparam
+ n
+ f
->len
[i
], nvar
- n
- f
->len
[i
]);
610 bset_i
= isl_basic_set_drop_constraints_involving(bset_i
,
612 bset_i
= isl_basic_set_drop(bset_i
, isl_dim_set
,
613 n
+ f
->len
[i
], nvar
- n
- f
->len
[i
]);
614 bset_i
= isl_basic_set_drop(bset_i
, isl_dim_set
, 0, n
);
616 sample_i
= sample_bounded(bset_i
);
619 if (sample_i
->size
== 0) {
620 isl_basic_set_free(bset
);
621 isl_factorizer_free(f
);
622 isl_vec_free(sample
);
625 isl_seq_cpy(sample
->el
+ 1 + nparam
+ n
,
626 sample_i
->el
+ 1, f
->len
[i
]);
627 isl_vec_free(sample_i
);
632 f
->morph
= isl_morph_inverse(f
->morph
);
633 sample
= isl_morph_vec(isl_morph_copy(f
->morph
), sample
);
635 isl_basic_set_free(bset
);
636 isl_factorizer_free(f
);
639 isl_basic_set_free(bset
);
640 isl_factorizer_free(f
);
641 isl_vec_free(sample
);
645 /* Given a basic set that is known to be bounded, find and return
646 * an integer point in the basic set, if there is any.
648 * After handling some trivial cases, we construct a tableau
649 * and then use isl_tab_sample to find a sample, passing it
650 * the identity matrix as initial basis.
652 static struct isl_vec
*sample_bounded(struct isl_basic_set
*bset
)
656 struct isl_vec
*sample
;
657 struct isl_tab
*tab
= NULL
;
663 if (isl_basic_set_plain_is_empty(bset
))
664 return empty_sample(bset
);
666 dim
= isl_basic_set_total_dim(bset
);
668 return zero_sample(bset
);
670 return interval_sample(bset
);
672 return sample_eq(bset
, sample_bounded
);
674 f
= isl_basic_set_factorizer(bset
);
678 return factored_sample(bset
, f
);
679 isl_factorizer_free(f
);
683 tab
= isl_tab_from_basic_set(bset
, 1);
684 if (tab
&& tab
->empty
) {
686 ISL_F_SET(bset
, ISL_BASIC_SET_EMPTY
);
687 sample
= isl_vec_alloc(bset
->ctx
, 0);
688 isl_basic_set_free(bset
);
692 if (!ISL_F_ISSET(bset
, ISL_BASIC_SET_NO_IMPLICIT
))
693 if (isl_tab_detect_implicit_equalities(tab
) < 0)
696 sample
= isl_tab_sample(tab
);
700 if (sample
->size
> 0) {
701 isl_vec_free(bset
->sample
);
702 bset
->sample
= isl_vec_copy(sample
);
705 isl_basic_set_free(bset
);
709 isl_basic_set_free(bset
);
714 /* Given a basic set "bset" and a value "sample" for the first coordinates
715 * of bset, plug in these values and drop the corresponding coordinates.
717 * We do this by computing the preimage of the transformation
723 * where [1 s] is the sample value and I is the identity matrix of the
724 * appropriate dimension.
726 static struct isl_basic_set
*plug_in(struct isl_basic_set
*bset
,
727 struct isl_vec
*sample
)
733 if (!bset
|| !sample
)
736 total
= isl_basic_set_total_dim(bset
);
737 T
= isl_mat_alloc(bset
->ctx
, 1 + total
, 1 + total
- (sample
->size
- 1));
741 for (i
= 0; i
< sample
->size
; ++i
) {
742 isl_int_set(T
->row
[i
][0], sample
->el
[i
]);
743 isl_seq_clr(T
->row
[i
] + 1, T
->n_col
- 1);
745 for (i
= 0; i
< T
->n_col
- 1; ++i
) {
746 isl_seq_clr(T
->row
[sample
->size
+ i
], T
->n_col
);
747 isl_int_set_si(T
->row
[sample
->size
+ i
][1 + i
], 1);
749 isl_vec_free(sample
);
751 bset
= isl_basic_set_preimage(bset
, T
);
754 isl_basic_set_free(bset
);
755 isl_vec_free(sample
);
759 /* Given a basic set "bset", return any (possibly non-integer) point
762 static struct isl_vec
*rational_sample(struct isl_basic_set
*bset
)
765 struct isl_vec
*sample
;
770 tab
= isl_tab_from_basic_set(bset
, 0);
771 sample
= isl_tab_get_sample_value(tab
);
774 isl_basic_set_free(bset
);
779 /* Given a linear cone "cone" and a rational point "vec",
780 * construct a polyhedron with shifted copies of the constraints in "cone",
781 * i.e., a polyhedron with "cone" as its recession cone, such that each
782 * point x in this polyhedron is such that the unit box positioned at x
783 * lies entirely inside the affine cone 'vec + cone'.
784 * Any rational point in this polyhedron may therefore be rounded up
785 * to yield an integer point that lies inside said affine cone.
787 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
788 * point "vec" by v/d.
789 * Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given
790 * by <a_i, x> - b/d >= 0.
791 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
792 * We prefer this polyhedron over the actual affine cone because it doesn't
793 * require a scaling of the constraints.
794 * If each of the vertices of the unit cube positioned at x lies inside
795 * this polyhedron, then the whole unit cube at x lies inside the affine cone.
796 * We therefore impose that x' = x + \sum e_i, for any selection of unit
797 * vectors lies inside the polyhedron, i.e.,
799 * <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
801 * The most stringent of these constraints is the one that selects
802 * all negative a_i, so the polyhedron we are looking for has constraints
804 * <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
806 * Note that if cone were known to have only non-negative rays
807 * (which can be accomplished by a unimodular transformation),
808 * then we would only have to check the points x' = x + e_i
809 * and we only have to add the smallest negative a_i (if any)
810 * instead of the sum of all negative a_i.
812 static struct isl_basic_set
*shift_cone(struct isl_basic_set
*cone
,
818 struct isl_basic_set
*shift
= NULL
;
823 isl_assert(cone
->ctx
, cone
->n_eq
== 0, goto error
);
825 total
= isl_basic_set_total_dim(cone
);
827 shift
= isl_basic_set_alloc_space(isl_basic_set_get_space(cone
),
830 for (i
= 0; i
< cone
->n_ineq
; ++i
) {
831 k
= isl_basic_set_alloc_inequality(shift
);
834 isl_seq_cpy(shift
->ineq
[k
] + 1, cone
->ineq
[i
] + 1, total
);
835 isl_seq_inner_product(shift
->ineq
[k
] + 1, vec
->el
+ 1, total
,
837 isl_int_cdiv_q(shift
->ineq
[k
][0],
838 shift
->ineq
[k
][0], vec
->el
[0]);
839 isl_int_neg(shift
->ineq
[k
][0], shift
->ineq
[k
][0]);
840 for (j
= 0; j
< total
; ++j
) {
841 if (isl_int_is_nonneg(shift
->ineq
[k
][1 + j
]))
843 isl_int_add(shift
->ineq
[k
][0],
844 shift
->ineq
[k
][0], shift
->ineq
[k
][1 + j
]);
848 isl_basic_set_free(cone
);
851 return isl_basic_set_finalize(shift
);
853 isl_basic_set_free(shift
);
854 isl_basic_set_free(cone
);
859 /* Given a rational point vec in a (transformed) basic set,
860 * such that cone is the recession cone of the original basic set,
861 * "round up" the rational point to an integer point.
863 * We first check if the rational point just happens to be integer.
864 * If not, we transform the cone in the same way as the basic set,
865 * pick a point x in this cone shifted to the rational point such that
866 * the whole unit cube at x is also inside this affine cone.
867 * Then we simply round up the coordinates of x and return the
868 * resulting integer point.
870 static struct isl_vec
*round_up_in_cone(struct isl_vec
*vec
,
871 struct isl_basic_set
*cone
, struct isl_mat
*U
)
875 if (!vec
|| !cone
|| !U
)
878 isl_assert(vec
->ctx
, vec
->size
!= 0, goto error
);
879 if (isl_int_is_one(vec
->el
[0])) {
881 isl_basic_set_free(cone
);
885 total
= isl_basic_set_total_dim(cone
);
886 cone
= isl_basic_set_preimage(cone
, U
);
887 cone
= isl_basic_set_remove_dims(cone
, isl_dim_set
,
888 0, total
- (vec
->size
- 1));
890 cone
= shift_cone(cone
, vec
);
892 vec
= rational_sample(cone
);
893 vec
= isl_vec_ceil(vec
);
898 isl_basic_set_free(cone
);
902 /* Concatenate two integer vectors, i.e., two vectors with denominator
903 * (stored in element 0) equal to 1.
905 static struct isl_vec
*vec_concat(struct isl_vec
*vec1
, struct isl_vec
*vec2
)
911 isl_assert(vec1
->ctx
, vec1
->size
> 0, goto error
);
912 isl_assert(vec2
->ctx
, vec2
->size
> 0, goto error
);
913 isl_assert(vec1
->ctx
, isl_int_is_one(vec1
->el
[0]), goto error
);
914 isl_assert(vec2
->ctx
, isl_int_is_one(vec2
->el
[0]), goto error
);
916 vec
= isl_vec_alloc(vec1
->ctx
, vec1
->size
+ vec2
->size
- 1);
920 isl_seq_cpy(vec
->el
, vec1
->el
, vec1
->size
);
921 isl_seq_cpy(vec
->el
+ vec1
->size
, vec2
->el
+ 1, vec2
->size
- 1);
933 /* Give a basic set "bset" with recession cone "cone", compute and
934 * return an integer point in bset, if any.
936 * If the recession cone is full-dimensional, then we know that
937 * bset contains an infinite number of integer points and it is
938 * fairly easy to pick one of them.
939 * If the recession cone is not full-dimensional, then we first
940 * transform bset such that the bounded directions appear as
941 * the first dimensions of the transformed basic set.
942 * We do this by using a unimodular transformation that transforms
943 * the equalities in the recession cone to equalities on the first
946 * The transformed set is then projected onto its bounded dimensions.
947 * Note that to compute this projection, we can simply drop all constraints
948 * involving any of the unbounded dimensions since these constraints
949 * cannot be combined to produce a constraint on the bounded dimensions.
950 * To see this, assume that there is such a combination of constraints
951 * that produces a constraint on the bounded dimensions. This means
952 * that some combination of the unbounded dimensions has both an upper
953 * bound and a lower bound in terms of the bounded dimensions, but then
954 * this combination would be a bounded direction too and would have been
955 * transformed into a bounded dimensions.
957 * We then compute a sample value in the bounded dimensions.
958 * If no such value can be found, then the original set did not contain
959 * any integer points and we are done.
960 * Otherwise, we plug in the value we found in the bounded dimensions,
961 * project out these bounded dimensions and end up with a set with
962 * a full-dimensional recession cone.
963 * A sample point in this set is computed by "rounding up" any
964 * rational point in the set.
966 * The sample points in the bounded and unbounded dimensions are
967 * then combined into a single sample point and transformed back
968 * to the original space.
970 __isl_give isl_vec
*isl_basic_set_sample_with_cone(
971 __isl_take isl_basic_set
*bset
, __isl_take isl_basic_set
*cone
)
975 struct isl_mat
*M
, *U
;
976 struct isl_vec
*sample
;
977 struct isl_vec
*cone_sample
;
979 struct isl_basic_set
*bounded
;
985 total
= isl_basic_set_total_dim(cone
);
986 cone_dim
= total
- cone
->n_eq
;
988 M
= isl_mat_sub_alloc6(bset
->ctx
, cone
->eq
, 0, cone
->n_eq
, 1, total
);
989 M
= isl_mat_left_hermite(M
, 0, &U
, NULL
);
994 U
= isl_mat_lin_to_aff(U
);
995 bset
= isl_basic_set_preimage(bset
, isl_mat_copy(U
));
997 bounded
= isl_basic_set_copy(bset
);
998 bounded
= isl_basic_set_drop_constraints_involving(bounded
,
999 total
- cone_dim
, cone_dim
);
1000 bounded
= isl_basic_set_drop_dims(bounded
, total
- cone_dim
, cone_dim
);
1001 sample
= sample_bounded(bounded
);
1002 if (!sample
|| sample
->size
== 0) {
1003 isl_basic_set_free(bset
);
1004 isl_basic_set_free(cone
);
1008 bset
= plug_in(bset
, isl_vec_copy(sample
));
1009 cone_sample
= rational_sample(bset
);
1010 cone_sample
= round_up_in_cone(cone_sample
, cone
, isl_mat_copy(U
));
1011 sample
= vec_concat(sample
, cone_sample
);
1012 sample
= isl_mat_vec_product(U
, sample
);
1015 isl_basic_set_free(cone
);
1016 isl_basic_set_free(bset
);
1020 static void vec_sum_of_neg(struct isl_vec
*v
, isl_int
*s
)
1024 isl_int_set_si(*s
, 0);
1026 for (i
= 0; i
< v
->size
; ++i
)
1027 if (isl_int_is_neg(v
->el
[i
]))
1028 isl_int_add(*s
, *s
, v
->el
[i
]);
1031 /* Given a tableau "tab", a tableau "tab_cone" that corresponds
1032 * to the recession cone and the inverse of a new basis U = inv(B),
1033 * with the unbounded directions in B last,
1034 * add constraints to "tab" that ensure any rational value
1035 * in the unbounded directions can be rounded up to an integer value.
1037 * The new basis is given by x' = B x, i.e., x = U x'.
1038 * For any rational value of the last tab->n_unbounded coordinates
1039 * in the update tableau, the value that is obtained by rounding
1040 * up this value should be contained in the original tableau.
1041 * For any constraint "a x + c >= 0", we therefore need to add
1042 * a constraint "a x + c + s >= 0", with s the sum of all negative
1043 * entries in the last elements of "a U".
1045 * Since we are not interested in the first entries of any of the "a U",
1046 * we first drop the columns of U that correpond to bounded directions.
1048 static int tab_shift_cone(struct isl_tab
*tab
,
1049 struct isl_tab
*tab_cone
, struct isl_mat
*U
)
1053 struct isl_basic_set
*bset
= NULL
;
1055 if (tab
&& tab
->n_unbounded
== 0) {
1060 if (!tab
|| !tab_cone
|| !U
)
1062 bset
= isl_tab_peek_bset(tab_cone
);
1063 U
= isl_mat_drop_cols(U
, 0, tab
->n_var
- tab
->n_unbounded
);
1064 for (i
= 0; i
< bset
->n_ineq
; ++i
) {
1066 struct isl_vec
*row
= NULL
;
1067 if (isl_tab_is_equality(tab_cone
, tab_cone
->n_eq
+ i
))
1069 row
= isl_vec_alloc(bset
->ctx
, tab_cone
->n_var
);
1072 isl_seq_cpy(row
->el
, bset
->ineq
[i
] + 1, tab_cone
->n_var
);
1073 row
= isl_vec_mat_product(row
, isl_mat_copy(U
));
1076 vec_sum_of_neg(row
, &v
);
1078 if (isl_int_is_zero(v
))
1080 tab
= isl_tab_extend(tab
, 1);
1081 isl_int_add(bset
->ineq
[i
][0], bset
->ineq
[i
][0], v
);
1082 ok
= isl_tab_add_ineq(tab
, bset
->ineq
[i
]) >= 0;
1083 isl_int_sub(bset
->ineq
[i
][0], bset
->ineq
[i
][0], v
);
1097 /* Compute and return an initial basis for the possibly
1098 * unbounded tableau "tab". "tab_cone" is a tableau
1099 * for the corresponding recession cone.
1100 * Additionally, add constraints to "tab" that ensure
1101 * that any rational value for the unbounded directions
1102 * can be rounded up to an integer value.
1104 * If the tableau is bounded, i.e., if the recession cone
1105 * is zero-dimensional, then we just use inital_basis.
1106 * Otherwise, we construct a basis whose first directions
1107 * correspond to equalities, followed by bounded directions,
1108 * i.e., equalities in the recession cone.
1109 * The remaining directions are then unbounded.
1111 int isl_tab_set_initial_basis_with_cone(struct isl_tab
*tab
,
1112 struct isl_tab
*tab_cone
)
1115 struct isl_mat
*cone_eq
;
1116 struct isl_mat
*U
, *Q
;
1118 if (!tab
|| !tab_cone
)
1121 if (tab_cone
->n_col
== tab_cone
->n_dead
) {
1122 tab
->basis
= initial_basis(tab
);
1123 return tab
->basis
? 0 : -1;
1126 eq
= tab_equalities(tab
);
1129 tab
->n_zero
= eq
->n_row
;
1130 cone_eq
= tab_equalities(tab_cone
);
1131 eq
= isl_mat_concat(eq
, cone_eq
);
1134 tab
->n_unbounded
= tab
->n_var
- (eq
->n_row
- tab
->n_zero
);
1135 eq
= isl_mat_left_hermite(eq
, 0, &U
, &Q
);
1139 tab
->basis
= isl_mat_lin_to_aff(Q
);
1140 if (tab_shift_cone(tab
, tab_cone
, U
) < 0)
1147 /* Compute and return a sample point in bset using generalized basis
1148 * reduction. We first check if the input set has a non-trivial
1149 * recession cone. If so, we perform some extra preprocessing in
1150 * sample_with_cone. Otherwise, we directly perform generalized basis
1153 static struct isl_vec
*gbr_sample(struct isl_basic_set
*bset
)
1156 struct isl_basic_set
*cone
;
1158 dim
= isl_basic_set_total_dim(bset
);
1160 cone
= isl_basic_set_recession_cone(isl_basic_set_copy(bset
));
1164 if (cone
->n_eq
< dim
)
1165 return isl_basic_set_sample_with_cone(bset
, cone
);
1167 isl_basic_set_free(cone
);
1168 return sample_bounded(bset
);
1170 isl_basic_set_free(bset
);
1174 static struct isl_vec
*pip_sample(struct isl_basic_set
*bset
)
1177 struct isl_ctx
*ctx
;
1178 struct isl_vec
*sample
;
1180 bset
= isl_basic_set_skew_to_positive_orthant(bset
, &T
);
1185 sample
= isl_pip_basic_set_sample(bset
);
1187 if (sample
&& sample
->size
!= 0)
1188 sample
= isl_mat_vec_product(T
, sample
);
1195 static struct isl_vec
*basic_set_sample(struct isl_basic_set
*bset
, int bounded
)
1197 struct isl_ctx
*ctx
;
1203 if (isl_basic_set_plain_is_empty(bset
))
1204 return empty_sample(bset
);
1206 dim
= isl_basic_set_n_dim(bset
);
1207 isl_assert(ctx
, isl_basic_set_n_param(bset
) == 0, goto error
);
1208 isl_assert(ctx
, bset
->n_div
== 0, goto error
);
1210 if (bset
->sample
&& bset
->sample
->size
== 1 + dim
) {
1211 int contains
= isl_basic_set_contains(bset
, bset
->sample
);
1215 struct isl_vec
*sample
= isl_vec_copy(bset
->sample
);
1216 isl_basic_set_free(bset
);
1220 isl_vec_free(bset
->sample
);
1221 bset
->sample
= NULL
;
1224 return sample_eq(bset
, bounded
? isl_basic_set_sample_bounded
1225 : isl_basic_set_sample_vec
);
1227 return zero_sample(bset
);
1229 return interval_sample(bset
);
1231 switch (bset
->ctx
->opt
->ilp_solver
) {
1233 return pip_sample(bset
);
1235 return bounded
? sample_bounded(bset
) : gbr_sample(bset
);
1237 isl_assert(bset
->ctx
, 0, );
1239 isl_basic_set_free(bset
);
1243 __isl_give isl_vec
*isl_basic_set_sample_vec(__isl_take isl_basic_set
*bset
)
1245 return basic_set_sample(bset
, 0);
1248 /* Compute an integer sample in "bset", where the caller guarantees
1249 * that "bset" is bounded.
1251 struct isl_vec
*isl_basic_set_sample_bounded(struct isl_basic_set
*bset
)
1253 return basic_set_sample(bset
, 1);
1256 __isl_give isl_basic_set
*isl_basic_set_from_vec(__isl_take isl_vec
*vec
)
1260 struct isl_basic_set
*bset
= NULL
;
1261 struct isl_ctx
*ctx
;
1267 isl_assert(ctx
, vec
->size
!= 0, goto error
);
1269 bset
= isl_basic_set_alloc(ctx
, 0, vec
->size
- 1, 0, vec
->size
- 1, 0);
1272 dim
= isl_basic_set_n_dim(bset
);
1273 for (i
= dim
- 1; i
>= 0; --i
) {
1274 k
= isl_basic_set_alloc_equality(bset
);
1277 isl_seq_clr(bset
->eq
[k
], 1 + dim
);
1278 isl_int_neg(bset
->eq
[k
][0], vec
->el
[1 + i
]);
1279 isl_int_set(bset
->eq
[k
][1 + i
], vec
->el
[0]);
1285 isl_basic_set_free(bset
);
1290 __isl_give isl_basic_map
*isl_basic_map_sample(__isl_take isl_basic_map
*bmap
)
1292 struct isl_basic_set
*bset
;
1293 struct isl_vec
*sample_vec
;
1295 bset
= isl_basic_map_underlying_set(isl_basic_map_copy(bmap
));
1296 sample_vec
= isl_basic_set_sample_vec(bset
);
1299 if (sample_vec
->size
== 0) {
1300 struct isl_basic_map
*sample
;
1301 sample
= isl_basic_map_empty_like(bmap
);
1302 isl_vec_free(sample_vec
);
1303 isl_basic_map_free(bmap
);
1306 bset
= isl_basic_set_from_vec(sample_vec
);
1307 return isl_basic_map_overlying_set(bset
, bmap
);
1309 isl_basic_map_free(bmap
);
1313 __isl_give isl_basic_set
*isl_basic_set_sample(__isl_take isl_basic_set
*bset
)
1315 return isl_basic_map_sample(bset
);
1318 __isl_give isl_basic_map
*isl_map_sample(__isl_take isl_map
*map
)
1321 isl_basic_map
*sample
= NULL
;
1326 for (i
= 0; i
< map
->n
; ++i
) {
1327 sample
= isl_basic_map_sample(isl_basic_map_copy(map
->p
[i
]));
1330 if (!ISL_F_ISSET(sample
, ISL_BASIC_MAP_EMPTY
))
1332 isl_basic_map_free(sample
);
1335 sample
= isl_basic_map_empty_like_map(map
);
1343 __isl_give isl_basic_set
*isl_set_sample(__isl_take isl_set
*set
)
1345 return (isl_basic_set
*) isl_map_sample((isl_map
*)set
);
1348 __isl_give isl_point
*isl_basic_set_sample_point(__isl_take isl_basic_set
*bset
)
1353 dim
= isl_basic_set_get_space(bset
);
1354 bset
= isl_basic_set_underlying_set(bset
);
1355 vec
= isl_basic_set_sample_vec(bset
);
1357 return isl_point_alloc(dim
, vec
);
1360 __isl_give isl_point
*isl_set_sample_point(__isl_take isl_set
*set
)
1368 for (i
= 0; i
< set
->n
; ++i
) {
1369 pnt
= isl_basic_set_sample_point(isl_basic_set_copy(set
->p
[i
]));
1372 if (!isl_point_is_void(pnt
))
1374 isl_point_free(pnt
);
1377 pnt
= isl_point_void(isl_set_get_space(set
));