1 #include "isl_map_private.h"
5 * The implementation of tableaus in this file was inspired by Section 8
6 * of David Detlefs, Greg Nelson and James B. Saxe, "Simplify: a theorem
7 * prover for program checking".
10 struct isl_tab
*isl_tab_alloc(struct isl_ctx
*ctx
,
11 unsigned n_row
, unsigned n_var
)
16 tab
= isl_calloc_type(ctx
, struct isl_tab
);
19 tab
->mat
= isl_mat_alloc(ctx
, n_row
, 2 + n_var
);
22 tab
->var
= isl_alloc_array(ctx
, struct isl_tab_var
, n_var
);
25 tab
->con
= isl_alloc_array(ctx
, struct isl_tab_var
, n_row
);
28 tab
->col_var
= isl_alloc_array(ctx
, int, n_var
);
31 tab
->row_var
= isl_alloc_array(ctx
, int, n_row
);
34 for (i
= 0; i
< n_var
; ++i
) {
35 tab
->var
[i
].index
= i
;
36 tab
->var
[i
].is_row
= 0;
37 tab
->var
[i
].is_nonneg
= 0;
38 tab
->var
[i
].is_zero
= 0;
39 tab
->var
[i
].is_redundant
= 0;
40 tab
->var
[i
].frozen
= 0;
54 tab
->bottom
.type
= isl_tab_undo_bottom
;
55 tab
->bottom
.next
= NULL
;
56 tab
->top
= &tab
->bottom
;
59 isl_tab_free(ctx
, tab
);
63 static int extend_cons(struct isl_ctx
*ctx
, struct isl_tab
*tab
, unsigned n_new
)
65 if (tab
->max_con
< tab
->n_con
+ n_new
) {
66 struct isl_tab_var
*con
;
68 con
= isl_realloc_array(ctx
, tab
->con
,
69 struct isl_tab_var
, tab
->max_con
+ n_new
);
73 tab
->max_con
+= n_new
;
75 if (tab
->mat
->n_row
< tab
->n_row
+ n_new
) {
78 tab
->mat
= isl_mat_extend(ctx
, tab
->mat
,
79 tab
->n_row
+ n_new
, tab
->n_col
);
82 row_var
= isl_realloc_array(ctx
, tab
->row_var
,
83 int, tab
->mat
->n_row
);
86 tab
->row_var
= row_var
;
91 struct isl_tab
*isl_tab_extend(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
94 if (extend_cons(ctx
, tab
, n_new
) >= 0)
97 isl_tab_free(ctx
, tab
);
101 static void free_undo(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
103 struct isl_tab_undo
*undo
, *next
;
105 for (undo
= tab
->top
; undo
&& undo
!= &tab
->bottom
; undo
= next
) {
112 void isl_tab_free(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
117 isl_mat_free(ctx
, tab
->mat
);
118 isl_vec_free(tab
->dual
);
126 static struct isl_tab_var
*var_from_index(struct isl_ctx
*ctx
,
127 struct isl_tab
*tab
, int i
)
132 return &tab
->con
[~i
];
135 static struct isl_tab_var
*var_from_row(struct isl_ctx
*ctx
,
136 struct isl_tab
*tab
, int i
)
138 return var_from_index(ctx
, tab
, tab
->row_var
[i
]);
141 static struct isl_tab_var
*var_from_col(struct isl_ctx
*ctx
,
142 struct isl_tab
*tab
, int i
)
144 return var_from_index(ctx
, tab
, tab
->col_var
[i
]);
147 /* Check if there are any upper bounds on column variable "var",
148 * i.e., non-negative rows where var appears with a negative coefficient.
149 * Return 1 if there are no such bounds.
151 static int max_is_manifestly_unbounded(struct isl_ctx
*ctx
,
152 struct isl_tab
*tab
, struct isl_tab_var
*var
)
158 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
159 if (!isl_int_is_neg(tab
->mat
->row
[i
][2 + var
->index
]))
161 if (var_from_row(ctx
, tab
, i
)->is_nonneg
)
167 /* Check if there are any lower bounds on column variable "var",
168 * i.e., non-negative rows where var appears with a positive coefficient.
169 * Return 1 if there are no such bounds.
171 static int min_is_manifestly_unbounded(struct isl_ctx
*ctx
,
172 struct isl_tab
*tab
, struct isl_tab_var
*var
)
178 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
179 if (!isl_int_is_pos(tab
->mat
->row
[i
][2 + var
->index
]))
181 if (var_from_row(ctx
, tab
, i
)->is_nonneg
)
187 /* Given the index of a column "c", return the index of a row
188 * that can be used to pivot the column in, with either an increase
189 * (sgn > 0) or a decrease (sgn < 0) of the corresponding variable.
190 * If "var" is not NULL, then the row returned will be different from
191 * the one associated with "var".
193 * Each row in the tableau is of the form
195 * x_r = a_r0 + \sum_i a_ri x_i
197 * Only rows with x_r >= 0 and with the sign of a_ri opposite to "sgn"
198 * impose any limit on the increase or decrease in the value of x_c
199 * and this bound is equal to a_r0 / |a_rc|. We are therefore looking
200 * for the row with the smallest (most stringent) such bound.
201 * Note that the common denominator of each row drops out of the fraction.
202 * To check if row j has a smaller bound than row r, i.e.,
203 * a_j0 / |a_jc| < a_r0 / |a_rc| or a_j0 |a_rc| < a_r0 |a_jc|,
204 * we check if -sign(a_jc) (a_j0 a_rc - a_r0 a_jc) < 0,
205 * where -sign(a_jc) is equal to "sgn".
207 static int pivot_row(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
208 struct isl_tab_var
*var
, int sgn
, int c
)
215 for (j
= tab
->n_redundant
; j
< tab
->n_row
; ++j
) {
216 if (var
&& j
== var
->index
)
218 if (!var_from_row(ctx
, tab
, j
)->is_nonneg
)
220 if (sgn
* isl_int_sgn(tab
->mat
->row
[j
][2 + c
]) >= 0)
226 isl_int_mul(t
, tab
->mat
->row
[r
][1], tab
->mat
->row
[j
][2 + c
]);
227 isl_int_submul(t
, tab
->mat
->row
[j
][1], tab
->mat
->row
[r
][2 + c
]);
228 tsgn
= sgn
* isl_int_sgn(t
);
229 if (tsgn
< 0 || (tsgn
== 0 &&
230 tab
->row_var
[j
] < tab
->row_var
[r
]))
237 /* Find a pivot (row and col) that will increase (sgn > 0) or decrease
238 * (sgn < 0) the value of row variable var.
239 * If not NULL, then skip_var is a row variable that should be ignored
240 * while looking for a pivot row. It is usually equal to var.
242 * As the given row in the tableau is of the form
244 * x_r = a_r0 + \sum_i a_ri x_i
246 * we need to find a column such that the sign of a_ri is equal to "sgn"
247 * (such that an increase in x_i will have the desired effect) or a
248 * column with a variable that may attain negative values.
249 * If a_ri is positive, then we need to move x_i in the same direction
250 * to obtain the desired effect. Otherwise, x_i has to move in the
251 * opposite direction.
253 static void find_pivot(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
254 struct isl_tab_var
*var
, struct isl_tab_var
*skip_var
,
255 int sgn
, int *row
, int *col
)
262 isl_assert(ctx
, var
->is_row
, return);
263 tr
= tab
->mat
->row
[var
->index
];
266 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
267 if (isl_int_is_zero(tr
[2 + j
]))
269 if (isl_int_sgn(tr
[2 + j
]) != sgn
&&
270 var_from_col(ctx
, tab
, j
)->is_nonneg
)
272 if (c
< 0 || tab
->col_var
[j
] < tab
->col_var
[c
])
278 sgn
*= isl_int_sgn(tr
[2 + c
]);
279 r
= pivot_row(ctx
, tab
, skip_var
, sgn
, c
);
280 *row
= r
< 0 ? var
->index
: r
;
284 /* Return 1 if row "row" represents an obviously redundant inequality.
286 * - it represents an inequality or a variable
287 * - that is the sum of a non-negative sample value and a positive
288 * combination of zero or more non-negative variables.
290 static int is_redundant(struct isl_ctx
*ctx
, struct isl_tab
*tab
, int row
)
294 if (tab
->row_var
[row
] < 0 && !var_from_row(ctx
, tab
, row
)->is_nonneg
)
297 if (isl_int_is_neg(tab
->mat
->row
[row
][1]))
300 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
301 if (isl_int_is_zero(tab
->mat
->row
[row
][2 + i
]))
303 if (isl_int_is_neg(tab
->mat
->row
[row
][2 + i
]))
305 if (!var_from_col(ctx
, tab
, i
)->is_nonneg
)
311 static void swap_rows(struct isl_ctx
*ctx
,
312 struct isl_tab
*tab
, int row1
, int row2
)
315 t
= tab
->row_var
[row1
];
316 tab
->row_var
[row1
] = tab
->row_var
[row2
];
317 tab
->row_var
[row2
] = t
;
318 var_from_row(ctx
, tab
, row1
)->index
= row1
;
319 var_from_row(ctx
, tab
, row2
)->index
= row2
;
320 tab
->mat
= isl_mat_swap_rows(ctx
, tab
->mat
, row1
, row2
);
323 static void push(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
324 enum isl_tab_undo_type type
, struct isl_tab_var
*var
)
326 struct isl_tab_undo
*undo
;
331 undo
= isl_alloc_type(ctx
, struct isl_tab_undo
);
339 undo
->next
= tab
->top
;
343 /* Mark row with index "row" as being redundant.
344 * If we may need to undo the operation or if the row represents
345 * a variable of the original problem, the row is kept,
346 * but no longer considered when looking for a pivot row.
347 * Otherwise, the row is simply removed.
349 * The row may be interchanged with some other row. If it
350 * is interchanged with a later row, return 1. Otherwise return 0.
351 * If the rows are checked in order in the calling function,
352 * then a return value of 1 means that the row with the given
353 * row number may now contain a different row that hasn't been checked yet.
355 static int mark_redundant(struct isl_ctx
*ctx
,
356 struct isl_tab
*tab
, int row
)
358 struct isl_tab_var
*var
= var_from_row(ctx
, tab
, row
);
359 var
->is_redundant
= 1;
360 isl_assert(ctx
, row
>= tab
->n_redundant
, return);
361 if (tab
->need_undo
|| tab
->row_var
[row
] >= 0) {
362 if (tab
->row_var
[row
] >= 0) {
364 push(ctx
, tab
, isl_tab_undo_nonneg
, var
);
366 if (row
!= tab
->n_redundant
)
367 swap_rows(ctx
, tab
, row
, tab
->n_redundant
);
368 push(ctx
, tab
, isl_tab_undo_redundant
, var
);
372 if (row
!= tab
->n_row
- 1)
373 swap_rows(ctx
, tab
, row
, tab
->n_row
- 1);
374 var_from_row(ctx
, tab
, tab
->n_row
- 1)->index
= -1;
380 static void mark_empty(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
382 if (!tab
->empty
&& tab
->need_undo
)
383 push(ctx
, tab
, isl_tab_undo_empty
, NULL
);
387 /* Given a row number "row" and a column number "col", pivot the tableau
388 * such that the associated variable are interchanged.
389 * The given row in the tableau expresses
391 * x_r = a_r0 + \sum_i a_ri x_i
395 * x_c = 1/a_rc x_r - a_r0/a_rc + sum_{i \ne r} -a_ri/a_rc
397 * Substituting this equality into the other rows
399 * x_j = a_j0 + \sum_i a_ji x_i
401 * with a_jc \ne 0, we obtain
403 * x_j = a_jc/a_rc x_r + a_j0 - a_jc a_r0/a_rc + sum a_ji - a_jc a_ri/a_rc
410 * where i is any other column and j is any other row,
411 * is therefore transformed into
413 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
414 * s(n_rc)d_r n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
416 * The transformation is performed along the following steps
421 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
424 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
425 * n_jc/(|n_rc| d_j) n_ji/(|n_rc| d_j)
427 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
428 * n_jc/(|n_rc| d_j) (n_ji |n_rc|)/(|n_rc| d_j)
430 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
431 * n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
433 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
434 * s(n_rc)d_r n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
437 static void pivot(struct isl_ctx
*ctx
,
438 struct isl_tab
*tab
, int row
, int col
)
443 struct isl_mat
*mat
= tab
->mat
;
444 struct isl_tab_var
*var
;
446 isl_int_swap(mat
->row
[row
][0], mat
->row
[row
][2 + col
]);
447 sgn
= isl_int_sgn(mat
->row
[row
][0]);
449 isl_int_neg(mat
->row
[row
][0], mat
->row
[row
][0]);
450 isl_int_neg(mat
->row
[row
][2 + col
], mat
->row
[row
][2 + col
]);
452 for (j
= 0; j
< 1 + tab
->n_col
; ++j
) {
455 isl_int_neg(mat
->row
[row
][1 + j
], mat
->row
[row
][1 + j
]);
457 if (!isl_int_is_one(mat
->row
[row
][0]))
458 isl_seq_normalize(mat
->row
[row
], 2 + tab
->n_col
);
459 for (i
= 0; i
< tab
->n_row
; ++i
) {
462 if (isl_int_is_zero(mat
->row
[i
][2 + col
]))
464 isl_int_mul(mat
->row
[i
][0], mat
->row
[i
][0], mat
->row
[row
][0]);
465 for (j
= 0; j
< 1 + tab
->n_col
; ++j
) {
468 isl_int_mul(mat
->row
[i
][1 + j
],
469 mat
->row
[i
][1 + j
], mat
->row
[row
][0]);
470 isl_int_addmul(mat
->row
[i
][1 + j
],
471 mat
->row
[i
][2 + col
], mat
->row
[row
][1 + j
]);
473 isl_int_mul(mat
->row
[i
][2 + col
],
474 mat
->row
[i
][2 + col
], mat
->row
[row
][2 + col
]);
475 if (!isl_int_is_one(mat
->row
[row
][0]))
476 isl_seq_normalize(mat
->row
[i
], 2 + tab
->n_col
);
478 t
= tab
->row_var
[row
];
479 tab
->row_var
[row
] = tab
->col_var
[col
];
480 tab
->col_var
[col
] = t
;
481 var
= var_from_row(ctx
, tab
, row
);
484 var
= var_from_col(ctx
, tab
, col
);
487 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
488 if (isl_int_is_zero(mat
->row
[i
][2 + col
]))
490 if (!var_from_row(ctx
, tab
, i
)->frozen
&&
491 is_redundant(ctx
, tab
, i
))
492 if (mark_redundant(ctx
, tab
, i
))
497 /* If "var" represents a column variable, then pivot is up (sgn > 0)
498 * or down (sgn < 0) to a row. The variable is assumed not to be
499 * unbounded in the specified direction.
501 static void to_row(struct isl_ctx
*ctx
,
502 struct isl_tab
*tab
, struct isl_tab_var
*var
, int sign
)
509 r
= pivot_row(ctx
, tab
, NULL
, sign
, var
->index
);
510 isl_assert(ctx
, r
>= 0, return);
511 pivot(ctx
, tab
, r
, var
->index
);
514 static void check_table(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
520 for (i
= 0; i
< tab
->n_row
; ++i
) {
521 if (!var_from_row(ctx
, tab
, i
)->is_nonneg
)
523 assert(!isl_int_is_neg(tab
->mat
->row
[i
][1]));
527 /* Return the sign of the maximal value of "var".
528 * If the sign is not negative, then on return from this function,
529 * the sample value will also be non-negative.
531 * If "var" is manifestly unbounded wrt positive values, we are done.
532 * Otherwise, we pivot the variable up to a row if needed
533 * Then we continue pivoting down until either
534 * - no more down pivots can be performed
535 * - the sample value is positive
536 * - the variable is pivoted into a manifestly unbounded column
538 static int sign_of_max(struct isl_ctx
*ctx
,
539 struct isl_tab
*tab
, struct isl_tab_var
*var
)
543 if (max_is_manifestly_unbounded(ctx
, tab
, var
))
545 to_row(ctx
, tab
, var
, 1);
546 while (!isl_int_is_pos(tab
->mat
->row
[var
->index
][1])) {
547 find_pivot(ctx
, tab
, var
, var
, 1, &row
, &col
);
549 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
550 pivot(ctx
, tab
, row
, col
);
551 if (!var
->is_row
) /* manifestly unbounded */
557 /* Perform pivots until the row variable "var" has a non-negative
558 * sample value or until no more upward pivots can be performed.
559 * Return the sign of the sample value after the pivots have been
562 static int restore_row(struct isl_ctx
*ctx
,
563 struct isl_tab
*tab
, struct isl_tab_var
*var
)
567 while (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
568 find_pivot(ctx
, tab
, var
, var
, 1, &row
, &col
);
571 pivot(ctx
, tab
, row
, col
);
572 if (!var
->is_row
) /* manifestly unbounded */
575 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
578 /* Perform pivots until we are sure that the row variable "var"
579 * can attain non-negative values. After return from this
580 * function, "var" is still a row variable, but its sample
581 * value may not be non-negative, even if the function returns 1.
583 static int at_least_zero(struct isl_ctx
*ctx
,
584 struct isl_tab
*tab
, struct isl_tab_var
*var
)
588 while (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
589 find_pivot(ctx
, tab
, var
, var
, 1, &row
, &col
);
592 if (row
== var
->index
) /* manifestly unbounded */
594 pivot(ctx
, tab
, row
, col
);
596 return !isl_int_is_neg(tab
->mat
->row
[var
->index
][1]);
599 /* Return a negative value if "var" can attain negative values.
600 * Return a non-negative value otherwise.
602 * If "var" is manifestly unbounded wrt negative values, we are done.
603 * Otherwise, if var is in a column, we can pivot it down to a row.
604 * Then we continue pivoting down until either
605 * - the pivot would result in a manifestly unbounded column
606 * => we don't perform the pivot, but simply return -1
607 * - no more down pivots can be performed
608 * - the sample value is negative
609 * If the sample value becomes negative and the variable is supposed
610 * to be nonnegative, then we undo the last pivot.
611 * However, if the last pivot has made the pivoting variable
612 * obviously redundant, then it may have moved to another row.
613 * In that case we look for upward pivots until we reach a non-negative
616 static int sign_of_min(struct isl_ctx
*ctx
,
617 struct isl_tab
*tab
, struct isl_tab_var
*var
)
620 struct isl_tab_var
*pivot_var
;
622 if (min_is_manifestly_unbounded(ctx
, tab
, var
))
626 row
= pivot_row(ctx
, tab
, NULL
, -1, col
);
627 pivot_var
= var_from_col(ctx
, tab
, col
);
628 pivot(ctx
, tab
, row
, col
);
629 if (var
->is_redundant
)
631 if (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
632 if (var
->is_nonneg
) {
633 if (!pivot_var
->is_redundant
&&
634 pivot_var
->index
== row
)
635 pivot(ctx
, tab
, row
, col
);
637 restore_row(ctx
, tab
, var
);
642 if (var
->is_redundant
)
644 while (!isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
645 find_pivot(ctx
, tab
, var
, var
, -1, &row
, &col
);
646 if (row
== var
->index
)
649 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
650 pivot_var
= var_from_col(ctx
, tab
, col
);
651 pivot(ctx
, tab
, row
, col
);
652 if (var
->is_redundant
)
655 if (var
->is_nonneg
) {
656 /* pivot back to non-negative value */
657 if (!pivot_var
->is_redundant
&& pivot_var
->index
== row
)
658 pivot(ctx
, tab
, row
, col
);
660 restore_row(ctx
, tab
, var
);
665 /* Return 1 if "var" can attain values <= -1.
666 * Return 0 otherwise.
668 * The sample value of "var" is assumed to be non-negative when the
669 * the function is called and will be made non-negative again before
670 * the function returns.
672 static int min_at_most_neg_one(struct isl_ctx
*ctx
,
673 struct isl_tab
*tab
, struct isl_tab_var
*var
)
676 struct isl_tab_var
*pivot_var
;
678 if (min_is_manifestly_unbounded(ctx
, tab
, var
))
682 row
= pivot_row(ctx
, tab
, NULL
, -1, col
);
683 pivot_var
= var_from_col(ctx
, tab
, col
);
684 pivot(ctx
, tab
, row
, col
);
685 if (var
->is_redundant
)
687 if (isl_int_is_neg(tab
->mat
->row
[var
->index
][1]) &&
688 isl_int_abs_ge(tab
->mat
->row
[var
->index
][1],
689 tab
->mat
->row
[var
->index
][0])) {
690 if (var
->is_nonneg
) {
691 if (!pivot_var
->is_redundant
&&
692 pivot_var
->index
== row
)
693 pivot(ctx
, tab
, row
, col
);
695 restore_row(ctx
, tab
, var
);
700 if (var
->is_redundant
)
703 find_pivot(ctx
, tab
, var
, var
, -1, &row
, &col
);
704 if (row
== var
->index
)
708 pivot_var
= var_from_col(ctx
, tab
, col
);
709 pivot(ctx
, tab
, row
, col
);
710 if (var
->is_redundant
)
712 } while (!isl_int_is_neg(tab
->mat
->row
[var
->index
][1]) ||
713 isl_int_abs_lt(tab
->mat
->row
[var
->index
][1],
714 tab
->mat
->row
[var
->index
][0]));
715 if (var
->is_nonneg
) {
716 /* pivot back to non-negative value */
717 if (!pivot_var
->is_redundant
&& pivot_var
->index
== row
)
718 pivot(ctx
, tab
, row
, col
);
719 restore_row(ctx
, tab
, var
);
724 /* Return 1 if "var" can attain values >= 1.
725 * Return 0 otherwise.
727 static int at_least_one(struct isl_ctx
*ctx
,
728 struct isl_tab
*tab
, struct isl_tab_var
*var
)
733 if (max_is_manifestly_unbounded(ctx
, tab
, var
))
735 to_row(ctx
, tab
, var
, 1);
736 r
= tab
->mat
->row
[var
->index
];
737 while (isl_int_lt(r
[1], r
[0])) {
738 find_pivot(ctx
, tab
, var
, var
, 1, &row
, &col
);
740 return isl_int_ge(r
[1], r
[0]);
741 if (row
== var
->index
) /* manifestly unbounded */
743 pivot(ctx
, tab
, row
, col
);
748 static void swap_cols(struct isl_ctx
*ctx
,
749 struct isl_tab
*tab
, int col1
, int col2
)
752 t
= tab
->col_var
[col1
];
753 tab
->col_var
[col1
] = tab
->col_var
[col2
];
754 tab
->col_var
[col2
] = t
;
755 var_from_col(ctx
, tab
, col1
)->index
= col1
;
756 var_from_col(ctx
, tab
, col2
)->index
= col2
;
757 tab
->mat
= isl_mat_swap_cols(ctx
, tab
->mat
, 2 + col1
, 2 + col2
);
760 /* Mark column with index "col" as representing a zero variable.
761 * If we may need to undo the operation the column is kept,
762 * but no longer considered.
763 * Otherwise, the column is simply removed.
765 * The column may be interchanged with some other column. If it
766 * is interchanged with a later column, return 1. Otherwise return 0.
767 * If the columns are checked in order in the calling function,
768 * then a return value of 1 means that the column with the given
769 * column number may now contain a different column that
770 * hasn't been checked yet.
772 static int kill_col(struct isl_ctx
*ctx
,
773 struct isl_tab
*tab
, int col
)
775 var_from_col(ctx
, tab
, col
)->is_zero
= 1;
776 if (tab
->need_undo
) {
777 push(ctx
, tab
, isl_tab_undo_zero
, var_from_col(ctx
, tab
, col
));
778 if (col
!= tab
->n_dead
)
779 swap_cols(ctx
, tab
, col
, tab
->n_dead
);
783 if (col
!= tab
->n_col
- 1)
784 swap_cols(ctx
, tab
, col
, tab
->n_col
- 1);
785 var_from_col(ctx
, tab
, tab
->n_col
- 1)->index
= -1;
791 /* Row variable "var" is non-negative and cannot attain any values
792 * larger than zero. This means that the coefficients of the unrestricted
793 * column variables are zero and that the coefficients of the non-negative
794 * column variables are zero or negative.
795 * Each of the non-negative variables with a negative coefficient can
796 * then also be written as the negative sum of non-negative variables
797 * and must therefore also be zero.
799 static void close_row(struct isl_ctx
*ctx
,
800 struct isl_tab
*tab
, struct isl_tab_var
*var
)
803 struct isl_mat
*mat
= tab
->mat
;
805 isl_assert(ctx
, var
->is_nonneg
, return);
807 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
808 if (isl_int_is_zero(mat
->row
[var
->index
][2 + j
]))
810 isl_assert(ctx
, isl_int_is_neg(mat
->row
[var
->index
][2 + j
]),
812 if (kill_col(ctx
, tab
, j
))
815 mark_redundant(ctx
, tab
, var
->index
);
818 /* Add a row to the tableau. The row is given as an affine combination
819 * of the original variables and needs to be expressed in terms of the
822 * We add each term in turn.
823 * If r = n/d_r is the current sum and we need to add k x, then
824 * if x is a column variable, we increase the numerator of
825 * this column by k d_r
826 * if x = f/d_x is a row variable, then the new representation of r is
828 * n k f d_x/g n + d_r/g k f m/d_r n + m/d_g k f
829 * --- + --- = ------------------- = -------------------
830 * d_r d_r d_r d_x/g m
832 * with g the gcd of d_r and d_x and m the lcm of d_r and d_x.
834 static int add_row(struct isl_ctx
*ctx
, struct isl_tab
*tab
, isl_int
*line
)
841 isl_assert(ctx
, tab
->n_row
< tab
->mat
->n_row
, return -1);
846 tab
->con
[r
].index
= tab
->n_row
;
847 tab
->con
[r
].is_row
= 1;
848 tab
->con
[r
].is_nonneg
= 0;
849 tab
->con
[r
].is_zero
= 0;
850 tab
->con
[r
].is_redundant
= 0;
851 tab
->con
[r
].frozen
= 0;
852 tab
->row_var
[tab
->n_row
] = ~r
;
853 row
= tab
->mat
->row
[tab
->n_row
];
854 isl_int_set_si(row
[0], 1);
855 isl_int_set(row
[1], line
[0]);
856 isl_seq_clr(row
+ 2, tab
->n_col
);
857 for (i
= 0; i
< tab
->n_var
; ++i
) {
858 if (tab
->var
[i
].is_zero
)
860 if (tab
->var
[i
].is_row
) {
862 row
[0], tab
->mat
->row
[tab
->var
[i
].index
][0]);
863 isl_int_swap(a
, row
[0]);
864 isl_int_divexact(a
, row
[0], a
);
866 row
[0], tab
->mat
->row
[tab
->var
[i
].index
][0]);
867 isl_int_mul(b
, b
, line
[1 + i
]);
868 isl_seq_combine(row
+ 1, a
, row
+ 1,
869 b
, tab
->mat
->row
[tab
->var
[i
].index
] + 1,
872 isl_int_addmul(row
[2 + tab
->var
[i
].index
],
873 line
[1 + i
], row
[0]);
875 isl_seq_normalize(row
, 2 + tab
->n_col
);
878 push(ctx
, tab
, isl_tab_undo_allocate
, &tab
->con
[r
]);
885 static int drop_row(struct isl_ctx
*ctx
, struct isl_tab
*tab
, int row
)
887 isl_assert(ctx
, ~tab
->row_var
[row
] == tab
->n_con
- 1, return -1);
888 if (row
!= tab
->n_row
- 1)
889 swap_rows(ctx
, tab
, row
, tab
->n_row
- 1);
895 /* Add inequality "ineq" and check if it conflicts with the
896 * previously added constraints or if it is obviously redundant.
898 struct isl_tab
*isl_tab_add_ineq(struct isl_ctx
*ctx
,
899 struct isl_tab
*tab
, isl_int
*ineq
)
906 r
= add_row(ctx
, tab
, ineq
);
909 tab
->con
[r
].is_nonneg
= 1;
910 push(ctx
, tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
911 if (is_redundant(ctx
, tab
, tab
->con
[r
].index
)) {
912 mark_redundant(ctx
, tab
, tab
->con
[r
].index
);
916 sgn
= restore_row(ctx
, tab
, &tab
->con
[r
]);
918 mark_empty(ctx
, tab
);
919 else if (tab
->con
[r
].is_row
&&
920 is_redundant(ctx
, tab
, tab
->con
[r
].index
))
921 mark_redundant(ctx
, tab
, tab
->con
[r
].index
);
924 isl_tab_free(ctx
, tab
);
928 /* Pivot a non-negative variable down until it reaches the value zero
929 * and then pivot the variable into a column position.
931 static int to_col(struct isl_ctx
*ctx
,
932 struct isl_tab
*tab
, struct isl_tab_var
*var
)
940 while (isl_int_is_pos(tab
->mat
->row
[var
->index
][1])) {
941 find_pivot(ctx
, tab
, var
, NULL
, -1, &row
, &col
);
942 isl_assert(ctx
, row
!= -1, return -1);
943 pivot(ctx
, tab
, row
, col
);
948 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
)
949 if (!isl_int_is_zero(tab
->mat
->row
[var
->index
][2 + i
]))
952 isl_assert(ctx
, i
< tab
->n_col
, return -1);
953 pivot(ctx
, tab
, var
->index
, i
);
958 /* We assume Gaussian elimination has been performed on the equalities.
959 * The equalities can therefore never conflict.
960 * Adding the equalities is currently only really useful for a later call
961 * to isl_tab_ineq_type.
963 static struct isl_tab
*add_eq(struct isl_ctx
*ctx
,
964 struct isl_tab
*tab
, isl_int
*eq
)
971 r
= add_row(ctx
, tab
, eq
);
975 r
= tab
->con
[r
].index
;
976 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
977 if (isl_int_is_zero(tab
->mat
->row
[r
][2 + i
]))
979 pivot(ctx
, tab
, r
, i
);
980 kill_col(ctx
, tab
, i
);
987 isl_tab_free(ctx
, tab
);
991 /* Add an equality that is known to be valid for the given tableau.
993 struct isl_tab
*isl_tab_add_valid_eq(struct isl_ctx
*ctx
,
994 struct isl_tab
*tab
, isl_int
*eq
)
996 struct isl_tab_var
*var
;
1002 r
= add_row(ctx
, tab
, eq
);
1008 if (isl_int_is_neg(tab
->mat
->row
[r
][1]))
1009 isl_seq_neg(tab
->mat
->row
[r
] + 1, tab
->mat
->row
[r
] + 1,
1012 if (to_col(ctx
, tab
, var
) < 0)
1015 kill_col(ctx
, tab
, var
->index
);
1019 isl_tab_free(ctx
, tab
);
1023 struct isl_tab
*isl_tab_from_basic_map(struct isl_basic_map
*bmap
)
1026 struct isl_tab
*tab
;
1030 tab
= isl_tab_alloc(bmap
->ctx
,
1031 isl_basic_map_total_dim(bmap
) + bmap
->n_ineq
+ 1,
1032 isl_basic_map_total_dim(bmap
));
1035 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1036 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
)) {
1037 mark_empty(bmap
->ctx
, tab
);
1040 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1041 tab
= add_eq(bmap
->ctx
, tab
, bmap
->eq
[i
]);
1045 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1046 tab
= isl_tab_add_ineq(bmap
->ctx
, tab
, bmap
->ineq
[i
]);
1047 if (!tab
|| tab
->empty
)
1053 struct isl_tab
*isl_tab_from_basic_set(struct isl_basic_set
*bset
)
1055 return isl_tab_from_basic_map((struct isl_basic_map
*)bset
);
1058 /* Construct a tableau corresponding to the recession cone of "bmap".
1060 struct isl_tab
*isl_tab_from_recession_cone(struct isl_basic_map
*bmap
)
1064 struct isl_tab
*tab
;
1068 tab
= isl_tab_alloc(bmap
->ctx
, bmap
->n_eq
+ bmap
->n_ineq
,
1069 isl_basic_map_total_dim(bmap
));
1072 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1075 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1076 isl_int_swap(bmap
->eq
[i
][0], cst
);
1077 tab
= add_eq(bmap
->ctx
, tab
, bmap
->eq
[i
]);
1078 isl_int_swap(bmap
->eq
[i
][0], cst
);
1082 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1084 isl_int_swap(bmap
->ineq
[i
][0], cst
);
1085 r
= add_row(bmap
->ctx
, tab
, bmap
->ineq
[i
]);
1086 isl_int_swap(bmap
->ineq
[i
][0], cst
);
1089 tab
->con
[r
].is_nonneg
= 1;
1090 push(bmap
->ctx
, tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1097 isl_tab_free(bmap
->ctx
, tab
);
1101 /* Assuming "tab" is the tableau of a cone, check if the cone is
1102 * bounded, i.e., if it is empty or only contains the origin.
1104 int isl_tab_cone_is_bounded(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
1112 if (tab
->n_dead
== tab
->n_col
)
1115 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1116 struct isl_tab_var
*var
;
1117 var
= var_from_row(ctx
, tab
, i
);
1118 if (!var
->is_nonneg
)
1120 if (sign_of_max(ctx
, tab
, var
) == 0)
1121 close_row(ctx
, tab
, var
);
1124 if (tab
->n_dead
== tab
->n_col
)
1130 static int sample_is_integer(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
1134 for (i
= 0; i
< tab
->n_var
; ++i
) {
1136 if (!tab
->var
[i
].is_row
)
1138 row
= tab
->var
[i
].index
;
1139 if (!isl_int_is_divisible_by(tab
->mat
->row
[row
][1],
1140 tab
->mat
->row
[row
][0]))
1146 static struct isl_vec
*extract_integer_sample(struct isl_ctx
*ctx
,
1147 struct isl_tab
*tab
)
1150 struct isl_vec
*vec
;
1152 vec
= isl_vec_alloc(ctx
, 1 + tab
->n_var
);
1156 isl_int_set_si(vec
->block
.data
[0], 1);
1157 for (i
= 0; i
< tab
->n_var
; ++i
) {
1158 if (!tab
->var
[i
].is_row
)
1159 isl_int_set_si(vec
->block
.data
[1 + i
], 0);
1161 int row
= tab
->var
[i
].index
;
1162 isl_int_divexact(vec
->block
.data
[1 + i
],
1163 tab
->mat
->row
[row
][1], tab
->mat
->row
[row
][0]);
1170 struct isl_vec
*isl_tab_get_sample_value(struct isl_ctx
*ctx
,
1171 struct isl_tab
*tab
)
1174 struct isl_vec
*vec
;
1180 vec
= isl_vec_alloc(ctx
, 1 + tab
->n_var
);
1186 isl_int_set_si(vec
->block
.data
[0], 1);
1187 for (i
= 0; i
< tab
->n_var
; ++i
) {
1189 if (!tab
->var
[i
].is_row
) {
1190 isl_int_set_si(vec
->block
.data
[1 + i
], 0);
1193 row
= tab
->var
[i
].index
;
1194 isl_int_gcd(m
, vec
->block
.data
[0], tab
->mat
->row
[row
][0]);
1195 isl_int_divexact(m
, tab
->mat
->row
[row
][0], m
);
1196 isl_seq_scale(vec
->block
.data
, vec
->block
.data
, m
, 1 + i
);
1197 isl_int_divexact(m
, vec
->block
.data
[0], tab
->mat
->row
[row
][0]);
1198 isl_int_mul(vec
->block
.data
[1 + i
], m
, tab
->mat
->row
[row
][1]);
1200 isl_seq_normalize(vec
->block
.data
, vec
->size
);
1206 /* Update "bmap" based on the results of the tableau "tab".
1207 * In particular, implicit equalities are made explicit, redundant constraints
1208 * are removed and if the sample value happens to be integer, it is stored
1209 * in "bmap" (unless "bmap" already had an integer sample).
1211 * The tableau is assumed to have been created from "bmap" using
1212 * isl_tab_from_basic_map.
1214 struct isl_basic_map
*isl_basic_map_update_from_tab(struct isl_basic_map
*bmap
,
1215 struct isl_tab
*tab
)
1227 bmap
= isl_basic_map_set_to_empty(bmap
);
1229 for (i
= bmap
->n_ineq
- 1; i
>= 0; --i
) {
1230 if (isl_tab_is_equality(bmap
->ctx
, tab
, n_eq
+ i
))
1231 isl_basic_map_inequality_to_equality(bmap
, i
);
1232 else if (isl_tab_is_redundant(bmap
->ctx
, tab
, n_eq
+ i
))
1233 isl_basic_map_drop_inequality(bmap
, i
);
1235 if (!tab
->rational
&&
1236 !bmap
->sample
&& sample_is_integer(bmap
->ctx
, tab
))
1237 bmap
->sample
= extract_integer_sample(bmap
->ctx
, tab
);
1241 struct isl_basic_set
*isl_basic_set_update_from_tab(struct isl_basic_set
*bset
,
1242 struct isl_tab
*tab
)
1244 return (struct isl_basic_set
*)isl_basic_map_update_from_tab(
1245 (struct isl_basic_map
*)bset
, tab
);
1248 /* Given a non-negative variable "var", add a new non-negative variable
1249 * that is the opposite of "var", ensuring that var can only attain the
1251 * If var = n/d is a row variable, then the new variable = -n/d.
1252 * If var is a column variables, then the new variable = -var.
1253 * If the new variable cannot attain non-negative values, then
1254 * the resulting tableau is empty.
1255 * Otherwise, we know the value will be zero and we close the row.
1257 static struct isl_tab
*cut_to_hyperplane(struct isl_ctx
*ctx
,
1258 struct isl_tab
*tab
, struct isl_tab_var
*var
)
1264 if (extend_cons(ctx
, tab
, 1) < 0)
1268 tab
->con
[r
].index
= tab
->n_row
;
1269 tab
->con
[r
].is_row
= 1;
1270 tab
->con
[r
].is_nonneg
= 0;
1271 tab
->con
[r
].is_zero
= 0;
1272 tab
->con
[r
].is_redundant
= 0;
1273 tab
->con
[r
].frozen
= 0;
1274 tab
->row_var
[tab
->n_row
] = ~r
;
1275 row
= tab
->mat
->row
[tab
->n_row
];
1278 isl_int_set(row
[0], tab
->mat
->row
[var
->index
][0]);
1279 isl_seq_neg(row
+ 1,
1280 tab
->mat
->row
[var
->index
] + 1, 1 + tab
->n_col
);
1282 isl_int_set_si(row
[0], 1);
1283 isl_seq_clr(row
+ 1, 1 + tab
->n_col
);
1284 isl_int_set_si(row
[2 + var
->index
], -1);
1289 push(ctx
, tab
, isl_tab_undo_allocate
, &tab
->con
[r
]);
1291 sgn
= sign_of_max(ctx
, tab
, &tab
->con
[r
]);
1293 mark_empty(ctx
, tab
);
1295 tab
->con
[r
].is_nonneg
= 1;
1296 push(ctx
, tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1298 close_row(ctx
, tab
, &tab
->con
[r
]);
1303 isl_tab_free(ctx
, tab
);
1307 /* Given a tableau "tab" and an inequality constraint "con" of the tableau,
1308 * relax the inequality by one. That is, the inequality r >= 0 is replaced
1309 * by r' = r + 1 >= 0.
1310 * If r is a row variable, we simply increase the constant term by one
1311 * (taking into account the denominator).
1312 * If r is a column variable, then we need to modify each row that
1313 * refers to r = r' - 1 by substituting this equality, effectively
1314 * subtracting the coefficient of the column from the constant.
1316 struct isl_tab
*isl_tab_relax(struct isl_ctx
*ctx
,
1317 struct isl_tab
*tab
, int con
)
1319 struct isl_tab_var
*var
;
1323 var
= &tab
->con
[con
];
1325 if (!var
->is_row
&& !max_is_manifestly_unbounded(ctx
, tab
, var
))
1326 to_row(ctx
, tab
, var
, 1);
1329 isl_int_add(tab
->mat
->row
[var
->index
][1],
1330 tab
->mat
->row
[var
->index
][1], tab
->mat
->row
[var
->index
][0]);
1334 for (i
= 0; i
< tab
->n_row
; ++i
) {
1335 if (isl_int_is_zero(tab
->mat
->row
[i
][2 + var
->index
]))
1337 isl_int_sub(tab
->mat
->row
[i
][1], tab
->mat
->row
[i
][1],
1338 tab
->mat
->row
[i
][2 + var
->index
]);
1343 push(ctx
, tab
, isl_tab_undo_relax
, var
);
1348 struct isl_tab
*isl_tab_select_facet(struct isl_ctx
*ctx
,
1349 struct isl_tab
*tab
, int con
)
1354 return cut_to_hyperplane(ctx
, tab
, &tab
->con
[con
]);
1357 static int may_be_equality(struct isl_tab
*tab
, int row
)
1359 return (tab
->rational
? isl_int_is_zero(tab
->mat
->row
[row
][1])
1360 : isl_int_lt(tab
->mat
->row
[row
][1],
1361 tab
->mat
->row
[row
][0])) &&
1362 isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1363 tab
->n_col
- tab
->n_dead
) != -1;
1366 /* Check for (near) equalities among the constraints.
1367 * A constraint is an equality if it is non-negative and if
1368 * its maximal value is either
1369 * - zero (in case of rational tableaus), or
1370 * - strictly less than 1 (in case of integer tableaus)
1372 * We first mark all non-redundant and non-dead variables that
1373 * are not frozen and not obviously not an equality.
1374 * Then we iterate over all marked variables if they can attain
1375 * any values larger than zero or at least one.
1376 * If the maximal value is zero, we mark any column variables
1377 * that appear in the row as being zero and mark the row as being redundant.
1378 * Otherwise, if the maximal value is strictly less than one (and the
1379 * tableau is integer), then we restrict the value to being zero
1380 * by adding an opposite non-negative variable.
1382 struct isl_tab
*isl_tab_detect_equalities(struct isl_ctx
*ctx
,
1383 struct isl_tab
*tab
)
1392 if (tab
->n_dead
== tab
->n_col
)
1396 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1397 struct isl_tab_var
*var
= var_from_row(ctx
, tab
, i
);
1398 var
->marked
= !var
->frozen
&& var
->is_nonneg
&&
1399 may_be_equality(tab
, i
);
1403 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1404 struct isl_tab_var
*var
= var_from_col(ctx
, tab
, i
);
1405 var
->marked
= !var
->frozen
&& var
->is_nonneg
;
1410 struct isl_tab_var
*var
;
1411 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1412 var
= var_from_row(ctx
, tab
, i
);
1416 if (i
== tab
->n_row
) {
1417 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1418 var
= var_from_col(ctx
, tab
, i
);
1422 if (i
== tab
->n_col
)
1427 if (sign_of_max(ctx
, tab
, var
) == 0)
1428 close_row(ctx
, tab
, var
);
1429 else if (!tab
->rational
&& !at_least_one(ctx
, tab
, var
)) {
1430 tab
= cut_to_hyperplane(ctx
, tab
, var
);
1431 return isl_tab_detect_equalities(ctx
, tab
);
1433 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1434 var
= var_from_row(ctx
, tab
, i
);
1437 if (may_be_equality(tab
, i
))
1447 /* Check for (near) redundant constraints.
1448 * A constraint is redundant if it is non-negative and if
1449 * its minimal value (temporarily ignoring the non-negativity) is either
1450 * - zero (in case of rational tableaus), or
1451 * - strictly larger than -1 (in case of integer tableaus)
1453 * We first mark all non-redundant and non-dead variables that
1454 * are not frozen and not obviously negatively unbounded.
1455 * Then we iterate over all marked variables if they can attain
1456 * any values smaller than zero or at most negative one.
1457 * If not, we mark the row as being redundant (assuming it hasn't
1458 * been detected as being obviously redundant in the mean time).
1460 struct isl_tab
*isl_tab_detect_redundant(struct isl_ctx
*ctx
,
1461 struct isl_tab
*tab
)
1470 if (tab
->n_redundant
== tab
->n_row
)
1474 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1475 struct isl_tab_var
*var
= var_from_row(ctx
, tab
, i
);
1476 var
->marked
= !var
->frozen
&& var
->is_nonneg
;
1480 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1481 struct isl_tab_var
*var
= var_from_col(ctx
, tab
, i
);
1482 var
->marked
= !var
->frozen
&& var
->is_nonneg
&&
1483 !min_is_manifestly_unbounded(ctx
, tab
, var
);
1488 struct isl_tab_var
*var
;
1489 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1490 var
= var_from_row(ctx
, tab
, i
);
1494 if (i
== tab
->n_row
) {
1495 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1496 var
= var_from_col(ctx
, tab
, i
);
1500 if (i
== tab
->n_col
)
1505 if ((tab
->rational
? (sign_of_min(ctx
, tab
, var
) >= 0)
1506 : !min_at_most_neg_one(ctx
, tab
, var
)) &&
1508 mark_redundant(ctx
, tab
, var
->index
);
1509 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1510 var
= var_from_col(ctx
, tab
, i
);
1513 if (!min_is_manifestly_unbounded(ctx
, tab
, var
))
1523 int isl_tab_is_equality(struct isl_ctx
*ctx
, struct isl_tab
*tab
, int con
)
1529 if (tab
->con
[con
].is_zero
)
1531 if (tab
->con
[con
].is_redundant
)
1533 if (!tab
->con
[con
].is_row
)
1534 return tab
->con
[con
].index
< tab
->n_dead
;
1536 row
= tab
->con
[con
].index
;
1538 return isl_int_is_zero(tab
->mat
->row
[row
][1]) &&
1539 isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1540 tab
->n_col
- tab
->n_dead
) == -1;
1543 /* Return the minimial value of the affine expression "f" with denominator
1544 * "denom" in *opt, *opt_denom, assuming the tableau is not empty and
1545 * the expression cannot attain arbitrarily small values.
1546 * If opt_denom is NULL, then *opt is rounded up to the nearest integer.
1547 * The return value reflects the nature of the result (empty, unbounded,
1548 * minmimal value returned in *opt).
1550 enum isl_lp_result
isl_tab_min(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
1551 isl_int
*f
, isl_int denom
, isl_int
*opt
, isl_int
*opt_denom
,
1555 enum isl_lp_result res
= isl_lp_ok
;
1556 struct isl_tab_var
*var
;
1559 return isl_lp_empty
;
1561 r
= add_row(ctx
, tab
, f
);
1563 return isl_lp_error
;
1565 isl_int_mul(tab
->mat
->row
[var
->index
][0],
1566 tab
->mat
->row
[var
->index
][0], denom
);
1569 find_pivot(ctx
, tab
, var
, var
, -1, &row
, &col
);
1570 if (row
== var
->index
) {
1571 res
= isl_lp_unbounded
;
1576 pivot(ctx
, tab
, row
, col
);
1578 if (drop_row(ctx
, tab
, var
->index
) < 0)
1579 return isl_lp_error
;
1580 if (ISL_FL_ISSET(flags
, ISL_TAB_SAVE_DUAL
)) {
1583 isl_vec_free(tab
->dual
);
1584 tab
->dual
= isl_vec_alloc(ctx
, 1 + tab
->n_con
);
1586 return isl_lp_error
;
1587 isl_int_set(tab
->dual
->el
[0], tab
->mat
->row
[var
->index
][0]);
1588 for (i
= 0; i
< tab
->n_con
; ++i
) {
1589 if (tab
->con
[i
].is_row
)
1590 isl_int_set_si(tab
->dual
->el
[1 + i
], 0);
1592 int pos
= 2 + tab
->con
[i
].index
;
1593 isl_int_set(tab
->dual
->el
[1 + i
],
1594 tab
->mat
->row
[var
->index
][pos
]);
1598 if (res
== isl_lp_ok
) {
1600 isl_int_set(*opt
, tab
->mat
->row
[var
->index
][1]);
1601 isl_int_set(*opt_denom
, tab
->mat
->row
[var
->index
][0]);
1603 isl_int_cdiv_q(*opt
, tab
->mat
->row
[var
->index
][1],
1604 tab
->mat
->row
[var
->index
][0]);
1609 int isl_tab_is_redundant(struct isl_ctx
*ctx
, struct isl_tab
*tab
, int con
)
1616 if (tab
->con
[con
].is_zero
)
1618 if (tab
->con
[con
].is_redundant
)
1620 return tab
->con
[con
].is_row
&& tab
->con
[con
].index
< tab
->n_redundant
;
1623 /* Take a snapshot of the tableau that can be restored by s call to
1626 struct isl_tab_undo
*isl_tab_snap(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
1634 /* Undo the operation performed by isl_tab_relax.
1636 static void unrelax(struct isl_ctx
*ctx
,
1637 struct isl_tab
*tab
, struct isl_tab_var
*var
)
1639 if (!var
->is_row
&& !max_is_manifestly_unbounded(ctx
, tab
, var
))
1640 to_row(ctx
, tab
, var
, 1);
1643 isl_int_sub(tab
->mat
->row
[var
->index
][1],
1644 tab
->mat
->row
[var
->index
][1], tab
->mat
->row
[var
->index
][0]);
1648 for (i
= 0; i
< tab
->n_row
; ++i
) {
1649 if (isl_int_is_zero(tab
->mat
->row
[i
][2 + var
->index
]))
1651 isl_int_add(tab
->mat
->row
[i
][1], tab
->mat
->row
[i
][1],
1652 tab
->mat
->row
[i
][2 + var
->index
]);
1658 static void perform_undo(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
1659 struct isl_tab_undo
*undo
)
1661 switch(undo
->type
) {
1662 case isl_tab_undo_empty
:
1665 case isl_tab_undo_nonneg
:
1666 undo
->var
->is_nonneg
= 0;
1668 case isl_tab_undo_redundant
:
1669 undo
->var
->is_redundant
= 0;
1672 case isl_tab_undo_zero
:
1673 undo
->var
->is_zero
= 0;
1676 case isl_tab_undo_allocate
:
1677 if (!undo
->var
->is_row
) {
1678 if (max_is_manifestly_unbounded(ctx
, tab
, undo
->var
))
1679 to_row(ctx
, tab
, undo
->var
, -1);
1681 to_row(ctx
, tab
, undo
->var
, 1);
1683 drop_row(ctx
, tab
, undo
->var
->index
);
1685 case isl_tab_undo_relax
:
1686 unrelax(ctx
, tab
, undo
->var
);
1691 /* Return the tableau to the state it was in when the snapshot "snap"
1694 int isl_tab_rollback(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
1695 struct isl_tab_undo
*snap
)
1697 struct isl_tab_undo
*undo
, *next
;
1702 for (undo
= tab
->top
; undo
&& undo
!= &tab
->bottom
; undo
= next
) {
1706 perform_undo(ctx
, tab
, undo
);
1715 /* The given row "row" represents an inequality violated by all
1716 * points in the tableau. Check for some special cases of such
1717 * separating constraints.
1718 * In particular, if the row has been reduced to the constant -1,
1719 * then we know the inequality is adjacent (but opposite) to
1720 * an equality in the tableau.
1721 * If the row has been reduced to r = -1 -r', with r' an inequality
1722 * of the tableau, then the inequality is adjacent (but opposite)
1723 * to the inequality r'.
1725 static enum isl_ineq_type
separation_type(struct isl_ctx
*ctx
,
1726 struct isl_tab
*tab
, unsigned row
)
1731 return isl_ineq_separate
;
1733 if (!isl_int_is_one(tab
->mat
->row
[row
][0]))
1734 return isl_ineq_separate
;
1735 if (!isl_int_is_negone(tab
->mat
->row
[row
][1]))
1736 return isl_ineq_separate
;
1738 pos
= isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1739 tab
->n_col
- tab
->n_dead
);
1741 return isl_ineq_adj_eq
;
1743 if (!isl_int_is_negone(tab
->mat
->row
[row
][2 + tab
->n_dead
+ pos
]))
1744 return isl_ineq_separate
;
1746 pos
= isl_seq_first_non_zero(
1747 tab
->mat
->row
[row
] + 2 + tab
->n_dead
+ pos
+ 1,
1748 tab
->n_col
- tab
->n_dead
- pos
- 1);
1750 return pos
== -1 ? isl_ineq_adj_ineq
: isl_ineq_separate
;
1753 /* Check the effect of inequality "ineq" on the tableau "tab".
1755 * isl_ineq_redundant: satisfied by all points in the tableau
1756 * isl_ineq_separate: satisfied by no point in the tableau
1757 * isl_ineq_cut: satisfied by some by not all points
1758 * isl_ineq_adj_eq: adjacent to an equality
1759 * isl_ineq_adj_ineq: adjacent to an inequality.
1761 enum isl_ineq_type
isl_tab_ineq_type(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
1764 enum isl_ineq_type type
= isl_ineq_error
;
1765 struct isl_tab_undo
*snap
= NULL
;
1770 return isl_ineq_error
;
1772 if (extend_cons(ctx
, tab
, 1) < 0)
1773 return isl_ineq_error
;
1775 snap
= isl_tab_snap(ctx
, tab
);
1777 con
= add_row(ctx
, tab
, ineq
);
1781 row
= tab
->con
[con
].index
;
1782 if (is_redundant(ctx
, tab
, row
))
1783 type
= isl_ineq_redundant
;
1784 else if (isl_int_is_neg(tab
->mat
->row
[row
][1]) &&
1786 isl_int_abs_ge(tab
->mat
->row
[row
][1],
1787 tab
->mat
->row
[row
][0]))) {
1788 if (at_least_zero(ctx
, tab
, &tab
->con
[con
]))
1789 type
= isl_ineq_cut
;
1791 type
= separation_type(ctx
, tab
, row
);
1792 } else if (tab
->rational
? (sign_of_min(ctx
, tab
, &tab
->con
[con
]) < 0)
1793 : min_at_most_neg_one(ctx
, tab
, &tab
->con
[con
]))
1794 type
= isl_ineq_cut
;
1796 type
= isl_ineq_redundant
;
1798 if (isl_tab_rollback(ctx
, tab
, snap
))
1799 return isl_ineq_error
;
1802 isl_tab_rollback(ctx
, tab
, snap
);
1803 return isl_ineq_error
;
1806 void isl_tab_dump(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
1807 FILE *out
, int indent
)
1813 fprintf(out
, "%*snull tab\n", indent
, "");
1816 fprintf(out
, "%*sn_redundant: %d, n_dead: %d", indent
, "",
1817 tab
->n_redundant
, tab
->n_dead
);
1819 fprintf(out
, ", rational");
1821 fprintf(out
, ", empty");
1823 fprintf(out
, "%*s[", indent
, "");
1824 for (i
= 0; i
< tab
->n_var
; ++i
) {
1827 fprintf(out
, "%c%d%s", tab
->var
[i
].is_row
? 'r' : 'c',
1829 tab
->var
[i
].is_zero
? " [=0]" :
1830 tab
->var
[i
].is_redundant
? " [R]" : "");
1832 fprintf(out
, "]\n");
1833 fprintf(out
, "%*s[", indent
, "");
1834 for (i
= 0; i
< tab
->n_con
; ++i
) {
1837 fprintf(out
, "%c%d%s", tab
->con
[i
].is_row
? 'r' : 'c',
1839 tab
->con
[i
].is_zero
? " [=0]" :
1840 tab
->con
[i
].is_redundant
? " [R]" : "");
1842 fprintf(out
, "]\n");
1843 fprintf(out
, "%*s[", indent
, "");
1844 for (i
= 0; i
< tab
->n_row
; ++i
) {
1847 fprintf(out
, "r%d: %d%s", i
, tab
->row_var
[i
],
1848 var_from_row(ctx
, tab
, i
)->is_nonneg
? " [>=0]" : "");
1850 fprintf(out
, "]\n");
1851 fprintf(out
, "%*s[", indent
, "");
1852 for (i
= 0; i
< tab
->n_col
; ++i
) {
1855 fprintf(out
, "c%d: %d%s", i
, tab
->col_var
[i
],
1856 var_from_col(ctx
, tab
, i
)->is_nonneg
? " [>=0]" : "");
1858 fprintf(out
, "]\n");
1859 r
= tab
->mat
->n_row
;
1860 tab
->mat
->n_row
= tab
->n_row
;
1861 c
= tab
->mat
->n_col
;
1862 tab
->mat
->n_col
= 2 + tab
->n_col
;
1863 isl_mat_dump(ctx
, tab
->mat
, out
, indent
);
1864 tab
->mat
->n_row
= r
;
1865 tab
->mat
->n_col
= c
;