2 #include "isl_map_private.h"
6 * The implementation of tableaus in this file was inspired by Section 8
7 * of David Detlefs, Greg Nelson and James B. Saxe, "Simplify: a theorem
8 * prover for program checking".
11 struct isl_tab
*isl_tab_alloc(struct isl_ctx
*ctx
,
12 unsigned n_row
, unsigned n_var
)
17 tab
= isl_calloc_type(ctx
, struct isl_tab
);
20 tab
->mat
= isl_mat_alloc(ctx
, n_row
, 2 + n_var
);
23 tab
->var
= isl_alloc_array(ctx
, struct isl_tab_var
, n_var
);
26 tab
->con
= isl_alloc_array(ctx
, struct isl_tab_var
, n_row
);
29 tab
->col_var
= isl_alloc_array(ctx
, int, n_var
);
32 tab
->row_var
= isl_alloc_array(ctx
, int, n_row
);
35 for (i
= 0; i
< n_var
; ++i
) {
36 tab
->var
[i
].index
= i
;
37 tab
->var
[i
].is_row
= 0;
38 tab
->var
[i
].is_nonneg
= 0;
39 tab
->var
[i
].is_zero
= 0;
40 tab
->var
[i
].is_redundant
= 0;
41 tab
->var
[i
].frozen
= 0;
58 tab
->bottom
.type
= isl_tab_undo_bottom
;
59 tab
->bottom
.next
= NULL
;
60 tab
->top
= &tab
->bottom
;
67 int isl_tab_extend_cons(struct isl_tab
*tab
, unsigned n_new
)
69 if (tab
->max_con
< tab
->n_con
+ n_new
) {
70 struct isl_tab_var
*con
;
72 con
= isl_realloc_array(tab
->mat
->ctx
, tab
->con
,
73 struct isl_tab_var
, tab
->max_con
+ n_new
);
77 tab
->max_con
+= n_new
;
79 if (tab
->mat
->n_row
< tab
->n_row
+ n_new
) {
82 tab
->mat
= isl_mat_extend(tab
->mat
,
83 tab
->n_row
+ n_new
, tab
->n_col
);
86 row_var
= isl_realloc_array(tab
->mat
->ctx
, tab
->row_var
,
87 int, tab
->mat
->n_row
);
90 tab
->row_var
= row_var
;
95 struct isl_tab
*isl_tab_extend(struct isl_tab
*tab
, unsigned n_new
)
97 if (isl_tab_extend_cons(tab
, n_new
) >= 0)
104 static void free_undo(struct isl_tab
*tab
)
106 struct isl_tab_undo
*undo
, *next
;
108 for (undo
= tab
->top
; undo
&& undo
!= &tab
->bottom
; undo
= next
) {
115 void isl_tab_free(struct isl_tab
*tab
)
120 isl_mat_free(tab
->mat
);
121 isl_vec_free(tab
->dual
);
129 struct isl_tab
*isl_tab_dup(struct isl_tab
*tab
)
137 dup
= isl_calloc_type(tab
->ctx
, struct isl_tab
);
140 dup
->mat
= isl_mat_dup(tab
->mat
);
143 dup
->var
= isl_alloc_array(tab
->ctx
, struct isl_tab_var
, tab
->n_var
);
146 for (i
= 0; i
< tab
->n_var
; ++i
)
147 dup
->var
[i
] = tab
->var
[i
];
148 dup
->con
= isl_alloc_array(tab
->ctx
, struct isl_tab_var
, tab
->max_con
);
151 for (i
= 0; i
< tab
->n_con
; ++i
)
152 dup
->con
[i
] = tab
->con
[i
];
153 dup
->col_var
= isl_alloc_array(tab
->ctx
, int, tab
->mat
->n_col
);
156 for (i
= 0; i
< tab
->n_var
; ++i
)
157 dup
->col_var
[i
] = tab
->col_var
[i
];
158 dup
->row_var
= isl_alloc_array(tab
->ctx
, int, tab
->mat
->n_row
);
161 for (i
= 0; i
< tab
->n_row
; ++i
)
162 dup
->row_var
[i
] = tab
->row_var
[i
];
163 dup
->n_row
= tab
->n_row
;
164 dup
->n_con
= tab
->n_con
;
165 dup
->n_eq
= tab
->n_eq
;
166 dup
->max_con
= tab
->max_con
;
167 dup
->n_col
= tab
->n_col
;
168 dup
->n_var
= tab
->n_var
;
169 dup
->n_param
= tab
->n_param
;
170 dup
->n_div
= tab
->n_div
;
171 dup
->n_dead
= tab
->n_dead
;
172 dup
->n_redundant
= tab
->n_redundant
;
173 dup
->rational
= tab
->rational
;
174 dup
->empty
= tab
->empty
;
177 dup
->bottom
.type
= isl_tab_undo_bottom
;
178 dup
->bottom
.next
= NULL
;
179 dup
->top
= &dup
->bottom
;
186 static struct isl_tab_var
*var_from_index(struct isl_tab
*tab
, int i
)
191 return &tab
->con
[~i
];
194 struct isl_tab_var
*isl_tab_var_from_row(struct isl_tab
*tab
, int i
)
196 return var_from_index(tab
, tab
->row_var
[i
]);
199 static struct isl_tab_var
*var_from_col(struct isl_tab
*tab
, int i
)
201 return var_from_index(tab
, tab
->col_var
[i
]);
204 /* Check if there are any upper bounds on column variable "var",
205 * i.e., non-negative rows where var appears with a negative coefficient.
206 * Return 1 if there are no such bounds.
208 static int max_is_manifestly_unbounded(struct isl_tab
*tab
,
209 struct isl_tab_var
*var
)
215 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
216 if (!isl_int_is_neg(tab
->mat
->row
[i
][2 + var
->index
]))
218 if (isl_tab_var_from_row(tab
, i
)->is_nonneg
)
224 /* Check if there are any lower bounds on column variable "var",
225 * i.e., non-negative rows where var appears with a positive coefficient.
226 * Return 1 if there are no such bounds.
228 static int min_is_manifestly_unbounded(struct isl_tab
*tab
,
229 struct isl_tab_var
*var
)
235 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
236 if (!isl_int_is_pos(tab
->mat
->row
[i
][2 + var
->index
]))
238 if (isl_tab_var_from_row(tab
, i
)->is_nonneg
)
244 /* Given the index of a column "c", return the index of a row
245 * that can be used to pivot the column in, with either an increase
246 * (sgn > 0) or a decrease (sgn < 0) of the corresponding variable.
247 * If "var" is not NULL, then the row returned will be different from
248 * the one associated with "var".
250 * Each row in the tableau is of the form
252 * x_r = a_r0 + \sum_i a_ri x_i
254 * Only rows with x_r >= 0 and with the sign of a_ri opposite to "sgn"
255 * impose any limit on the increase or decrease in the value of x_c
256 * and this bound is equal to a_r0 / |a_rc|. We are therefore looking
257 * for the row with the smallest (most stringent) such bound.
258 * Note that the common denominator of each row drops out of the fraction.
259 * To check if row j has a smaller bound than row r, i.e.,
260 * a_j0 / |a_jc| < a_r0 / |a_rc| or a_j0 |a_rc| < a_r0 |a_jc|,
261 * we check if -sign(a_jc) (a_j0 a_rc - a_r0 a_jc) < 0,
262 * where -sign(a_jc) is equal to "sgn".
264 static int pivot_row(struct isl_tab
*tab
,
265 struct isl_tab_var
*var
, int sgn
, int c
)
272 for (j
= tab
->n_redundant
; j
< tab
->n_row
; ++j
) {
273 if (var
&& j
== var
->index
)
275 if (!isl_tab_var_from_row(tab
, j
)->is_nonneg
)
277 if (sgn
* isl_int_sgn(tab
->mat
->row
[j
][2 + c
]) >= 0)
283 isl_int_mul(t
, tab
->mat
->row
[r
][1], tab
->mat
->row
[j
][2 + c
]);
284 isl_int_submul(t
, tab
->mat
->row
[j
][1], tab
->mat
->row
[r
][2 + c
]);
285 tsgn
= sgn
* isl_int_sgn(t
);
286 if (tsgn
< 0 || (tsgn
== 0 &&
287 tab
->row_var
[j
] < tab
->row_var
[r
]))
294 /* Find a pivot (row and col) that will increase (sgn > 0) or decrease
295 * (sgn < 0) the value of row variable var.
296 * If not NULL, then skip_var is a row variable that should be ignored
297 * while looking for a pivot row. It is usually equal to var.
299 * As the given row in the tableau is of the form
301 * x_r = a_r0 + \sum_i a_ri x_i
303 * we need to find a column such that the sign of a_ri is equal to "sgn"
304 * (such that an increase in x_i will have the desired effect) or a
305 * column with a variable that may attain negative values.
306 * If a_ri is positive, then we need to move x_i in the same direction
307 * to obtain the desired effect. Otherwise, x_i has to move in the
308 * opposite direction.
310 static void find_pivot(struct isl_tab
*tab
,
311 struct isl_tab_var
*var
, struct isl_tab_var
*skip_var
,
312 int sgn
, int *row
, int *col
)
319 isl_assert(tab
->mat
->ctx
, var
->is_row
, return);
320 tr
= tab
->mat
->row
[var
->index
];
323 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
324 if (isl_int_is_zero(tr
[2 + j
]))
326 if (isl_int_sgn(tr
[2 + j
]) != sgn
&&
327 var_from_col(tab
, j
)->is_nonneg
)
329 if (c
< 0 || tab
->col_var
[j
] < tab
->col_var
[c
])
335 sgn
*= isl_int_sgn(tr
[2 + c
]);
336 r
= pivot_row(tab
, skip_var
, sgn
, c
);
337 *row
= r
< 0 ? var
->index
: r
;
341 /* Return 1 if row "row" represents an obviously redundant inequality.
343 * - it represents an inequality or a variable
344 * - that is the sum of a non-negative sample value and a positive
345 * combination of zero or more non-negative variables.
347 int isl_tab_row_is_redundant(struct isl_tab
*tab
, int row
)
351 if (tab
->row_var
[row
] < 0 && !isl_tab_var_from_row(tab
, row
)->is_nonneg
)
354 if (isl_int_is_neg(tab
->mat
->row
[row
][1]))
357 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
358 if (isl_int_is_zero(tab
->mat
->row
[row
][2 + i
]))
360 if (isl_int_is_neg(tab
->mat
->row
[row
][2 + i
]))
362 if (!var_from_col(tab
, i
)->is_nonneg
)
368 static void swap_rows(struct isl_tab
*tab
, int row1
, int row2
)
371 t
= tab
->row_var
[row1
];
372 tab
->row_var
[row1
] = tab
->row_var
[row2
];
373 tab
->row_var
[row2
] = t
;
374 isl_tab_var_from_row(tab
, row1
)->index
= row1
;
375 isl_tab_var_from_row(tab
, row2
)->index
= row2
;
376 tab
->mat
= isl_mat_swap_rows(tab
->mat
, row1
, row2
);
379 static void push_union(struct isl_tab
*tab
,
380 enum isl_tab_undo_type type
, union isl_tab_undo_val u
)
382 struct isl_tab_undo
*undo
;
387 undo
= isl_alloc_type(tab
->mat
->ctx
, struct isl_tab_undo
);
395 undo
->next
= tab
->top
;
399 void isl_tab_push_var(struct isl_tab
*tab
,
400 enum isl_tab_undo_type type
, struct isl_tab_var
*var
)
402 union isl_tab_undo_val u
;
404 u
.var_index
= tab
->row_var
[var
->index
];
406 u
.var_index
= tab
->col_var
[var
->index
];
407 push_union(tab
, type
, u
);
410 void isl_tab_push(struct isl_tab
*tab
, enum isl_tab_undo_type type
)
412 union isl_tab_undo_val u
= { 0 };
413 push_union(tab
, type
, u
);
416 /* Push a record on the undo stack describing the current basic
417 * variables, so that the this state can be restored during rollback.
419 void isl_tab_push_basis(struct isl_tab
*tab
)
422 union isl_tab_undo_val u
;
424 u
.col_var
= isl_alloc_array(tab
->mat
->ctx
, int, tab
->n_col
);
430 for (i
= 0; i
< tab
->n_col
; ++i
)
431 u
.col_var
[i
] = tab
->col_var
[i
];
432 push_union(tab
, isl_tab_undo_saved_basis
, u
);
435 /* Mark row with index "row" as being redundant.
436 * If we may need to undo the operation or if the row represents
437 * a variable of the original problem, the row is kept,
438 * but no longer considered when looking for a pivot row.
439 * Otherwise, the row is simply removed.
441 * The row may be interchanged with some other row. If it
442 * is interchanged with a later row, return 1. Otherwise return 0.
443 * If the rows are checked in order in the calling function,
444 * then a return value of 1 means that the row with the given
445 * row number may now contain a different row that hasn't been checked yet.
447 int isl_tab_mark_redundant(struct isl_tab
*tab
, int row
)
449 struct isl_tab_var
*var
= isl_tab_var_from_row(tab
, row
);
450 var
->is_redundant
= 1;
451 isl_assert(tab
->mat
->ctx
, row
>= tab
->n_redundant
, return);
452 if (tab
->need_undo
|| tab
->row_var
[row
] >= 0) {
453 if (tab
->row_var
[row
] >= 0 && !var
->is_nonneg
) {
455 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, var
);
457 if (row
!= tab
->n_redundant
)
458 swap_rows(tab
, row
, tab
->n_redundant
);
459 isl_tab_push_var(tab
, isl_tab_undo_redundant
, var
);
463 if (row
!= tab
->n_row
- 1)
464 swap_rows(tab
, row
, tab
->n_row
- 1);
465 isl_tab_var_from_row(tab
, tab
->n_row
- 1)->index
= -1;
471 struct isl_tab
*isl_tab_mark_empty(struct isl_tab
*tab
)
473 if (!tab
->empty
&& tab
->need_undo
)
474 isl_tab_push(tab
, isl_tab_undo_empty
);
479 /* Given a row number "row" and a column number "col", pivot the tableau
480 * such that the associated variables are interchanged.
481 * The given row in the tableau expresses
483 * x_r = a_r0 + \sum_i a_ri x_i
487 * x_c = 1/a_rc x_r - a_r0/a_rc + sum_{i \ne r} -a_ri/a_rc
489 * Substituting this equality into the other rows
491 * x_j = a_j0 + \sum_i a_ji x_i
493 * with a_jc \ne 0, we obtain
495 * 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
502 * where i is any other column and j is any other row,
503 * is therefore transformed into
505 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
506 * 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)
508 * The transformation is performed along the following steps
513 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
516 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
517 * n_jc/(|n_rc| d_j) n_ji/(|n_rc| d_j)
519 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
520 * n_jc/(|n_rc| d_j) (n_ji |n_rc|)/(|n_rc| d_j)
522 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
523 * n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
525 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
526 * 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)
529 void isl_tab_pivot(struct isl_tab
*tab
, int row
, int col
)
534 struct isl_mat
*mat
= tab
->mat
;
535 struct isl_tab_var
*var
;
537 isl_int_swap(mat
->row
[row
][0], mat
->row
[row
][2 + col
]);
538 sgn
= isl_int_sgn(mat
->row
[row
][0]);
540 isl_int_neg(mat
->row
[row
][0], mat
->row
[row
][0]);
541 isl_int_neg(mat
->row
[row
][2 + col
], mat
->row
[row
][2 + col
]);
543 for (j
= 0; j
< 1 + tab
->n_col
; ++j
) {
546 isl_int_neg(mat
->row
[row
][1 + j
], mat
->row
[row
][1 + j
]);
548 if (!isl_int_is_one(mat
->row
[row
][0]))
549 isl_seq_normalize(mat
->row
[row
], 2 + tab
->n_col
);
550 for (i
= 0; i
< tab
->n_row
; ++i
) {
553 if (isl_int_is_zero(mat
->row
[i
][2 + col
]))
555 isl_int_mul(mat
->row
[i
][0], mat
->row
[i
][0], mat
->row
[row
][0]);
556 for (j
= 0; j
< 1 + tab
->n_col
; ++j
) {
559 isl_int_mul(mat
->row
[i
][1 + j
],
560 mat
->row
[i
][1 + j
], mat
->row
[row
][0]);
561 isl_int_addmul(mat
->row
[i
][1 + j
],
562 mat
->row
[i
][2 + col
], mat
->row
[row
][1 + j
]);
564 isl_int_mul(mat
->row
[i
][2 + col
],
565 mat
->row
[i
][2 + col
], mat
->row
[row
][2 + col
]);
566 if (!isl_int_is_one(mat
->row
[i
][0]))
567 isl_seq_normalize(mat
->row
[i
], 2 + tab
->n_col
);
569 t
= tab
->row_var
[row
];
570 tab
->row_var
[row
] = tab
->col_var
[col
];
571 tab
->col_var
[col
] = t
;
572 var
= isl_tab_var_from_row(tab
, row
);
575 var
= var_from_col(tab
, col
);
580 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
581 if (isl_int_is_zero(mat
->row
[i
][2 + col
]))
583 if (!isl_tab_var_from_row(tab
, i
)->frozen
&&
584 isl_tab_row_is_redundant(tab
, i
))
585 if (isl_tab_mark_redundant(tab
, i
))
590 /* If "var" represents a column variable, then pivot is up (sgn > 0)
591 * or down (sgn < 0) to a row. The variable is assumed not to be
592 * unbounded in the specified direction.
593 * If sgn = 0, then the variable is unbounded in both directions,
594 * and we pivot with any row we can find.
596 static void to_row(struct isl_tab
*tab
, struct isl_tab_var
*var
, int sign
)
604 for (r
= tab
->n_redundant
; r
< tab
->n_row
; ++r
)
605 if (!isl_int_is_zero(tab
->mat
->row
[r
][2 + var
->index
]))
607 isl_assert(tab
->mat
->ctx
, r
< tab
->n_row
, return);
609 r
= pivot_row(tab
, NULL
, sign
, var
->index
);
610 isl_assert(tab
->mat
->ctx
, r
>= 0, return);
613 isl_tab_pivot(tab
, r
, var
->index
);
616 static void check_table(struct isl_tab
*tab
)
622 for (i
= 0; i
< tab
->n_row
; ++i
) {
623 if (!isl_tab_var_from_row(tab
, i
)->is_nonneg
)
625 assert(!isl_int_is_neg(tab
->mat
->row
[i
][1]));
629 /* Return the sign of the maximal value of "var".
630 * If the sign is not negative, then on return from this function,
631 * the sample value will also be non-negative.
633 * If "var" is manifestly unbounded wrt positive values, we are done.
634 * Otherwise, we pivot the variable up to a row if needed
635 * Then we continue pivoting down until either
636 * - no more down pivots can be performed
637 * - the sample value is positive
638 * - the variable is pivoted into a manifestly unbounded column
640 static int sign_of_max(struct isl_tab
*tab
, struct isl_tab_var
*var
)
644 if (max_is_manifestly_unbounded(tab
, var
))
647 while (!isl_int_is_pos(tab
->mat
->row
[var
->index
][1])) {
648 find_pivot(tab
, var
, var
, 1, &row
, &col
);
650 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
651 isl_tab_pivot(tab
, row
, col
);
652 if (!var
->is_row
) /* manifestly unbounded */
658 /* Perform pivots until the row variable "var" has a non-negative
659 * sample value or until no more upward pivots can be performed.
660 * Return the sign of the sample value after the pivots have been
663 static int restore_row(struct isl_tab
*tab
, struct isl_tab_var
*var
)
667 while (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
668 find_pivot(tab
, var
, var
, 1, &row
, &col
);
671 isl_tab_pivot(tab
, row
, col
);
672 if (!var
->is_row
) /* manifestly unbounded */
675 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
678 /* Perform pivots until we are sure that the row variable "var"
679 * can attain non-negative values. After return from this
680 * function, "var" is still a row variable, but its sample
681 * value may not be non-negative, even if the function returns 1.
683 static int at_least_zero(struct isl_tab
*tab
, struct isl_tab_var
*var
)
687 while (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
688 find_pivot(tab
, var
, var
, 1, &row
, &col
);
691 if (row
== var
->index
) /* manifestly unbounded */
693 isl_tab_pivot(tab
, row
, col
);
695 return !isl_int_is_neg(tab
->mat
->row
[var
->index
][1]);
698 /* Return a negative value if "var" can attain negative values.
699 * Return a non-negative value otherwise.
701 * If "var" is manifestly unbounded wrt negative values, we are done.
702 * Otherwise, if var is in a column, we can pivot it down to a row.
703 * Then we continue pivoting down until either
704 * - the pivot would result in a manifestly unbounded column
705 * => we don't perform the pivot, but simply return -1
706 * - no more down pivots can be performed
707 * - the sample value is negative
708 * If the sample value becomes negative and the variable is supposed
709 * to be nonnegative, then we undo the last pivot.
710 * However, if the last pivot has made the pivoting variable
711 * obviously redundant, then it may have moved to another row.
712 * In that case we look for upward pivots until we reach a non-negative
715 static int sign_of_min(struct isl_tab
*tab
, struct isl_tab_var
*var
)
718 struct isl_tab_var
*pivot_var
;
720 if (min_is_manifestly_unbounded(tab
, var
))
724 row
= pivot_row(tab
, NULL
, -1, col
);
725 pivot_var
= var_from_col(tab
, col
);
726 isl_tab_pivot(tab
, row
, col
);
727 if (var
->is_redundant
)
729 if (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
730 if (var
->is_nonneg
) {
731 if (!pivot_var
->is_redundant
&&
732 pivot_var
->index
== row
)
733 isl_tab_pivot(tab
, row
, col
);
735 restore_row(tab
, var
);
740 if (var
->is_redundant
)
742 while (!isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
743 find_pivot(tab
, var
, var
, -1, &row
, &col
);
744 if (row
== var
->index
)
747 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
748 pivot_var
= var_from_col(tab
, col
);
749 isl_tab_pivot(tab
, row
, col
);
750 if (var
->is_redundant
)
753 if (var
->is_nonneg
) {
754 /* pivot back to non-negative value */
755 if (!pivot_var
->is_redundant
&& pivot_var
->index
== row
)
756 isl_tab_pivot(tab
, row
, col
);
758 restore_row(tab
, var
);
763 /* Return 1 if "var" can attain values <= -1.
764 * Return 0 otherwise.
766 * The sample value of "var" is assumed to be non-negative when the
767 * the function is called and will be made non-negative again before
768 * the function returns.
770 int isl_tab_min_at_most_neg_one(struct isl_tab
*tab
, struct isl_tab_var
*var
)
773 struct isl_tab_var
*pivot_var
;
775 if (min_is_manifestly_unbounded(tab
, var
))
779 row
= pivot_row(tab
, NULL
, -1, col
);
780 pivot_var
= var_from_col(tab
, col
);
781 isl_tab_pivot(tab
, row
, col
);
782 if (var
->is_redundant
)
784 if (isl_int_is_neg(tab
->mat
->row
[var
->index
][1]) &&
785 isl_int_abs_ge(tab
->mat
->row
[var
->index
][1],
786 tab
->mat
->row
[var
->index
][0])) {
787 if (var
->is_nonneg
) {
788 if (!pivot_var
->is_redundant
&&
789 pivot_var
->index
== row
)
790 isl_tab_pivot(tab
, row
, col
);
792 restore_row(tab
, var
);
797 if (var
->is_redundant
)
800 find_pivot(tab
, var
, var
, -1, &row
, &col
);
801 if (row
== var
->index
)
805 pivot_var
= var_from_col(tab
, col
);
806 isl_tab_pivot(tab
, row
, col
);
807 if (var
->is_redundant
)
809 } while (!isl_int_is_neg(tab
->mat
->row
[var
->index
][1]) ||
810 isl_int_abs_lt(tab
->mat
->row
[var
->index
][1],
811 tab
->mat
->row
[var
->index
][0]));
812 if (var
->is_nonneg
) {
813 /* pivot back to non-negative value */
814 if (!pivot_var
->is_redundant
&& pivot_var
->index
== row
)
815 isl_tab_pivot(tab
, row
, col
);
816 restore_row(tab
, var
);
821 /* Return 1 if "var" can attain values >= 1.
822 * Return 0 otherwise.
824 static int at_least_one(struct isl_tab
*tab
, struct isl_tab_var
*var
)
829 if (max_is_manifestly_unbounded(tab
, var
))
832 r
= tab
->mat
->row
[var
->index
];
833 while (isl_int_lt(r
[1], r
[0])) {
834 find_pivot(tab
, var
, var
, 1, &row
, &col
);
836 return isl_int_ge(r
[1], r
[0]);
837 if (row
== var
->index
) /* manifestly unbounded */
839 isl_tab_pivot(tab
, row
, col
);
844 static void swap_cols(struct isl_tab
*tab
, int col1
, int col2
)
847 t
= tab
->col_var
[col1
];
848 tab
->col_var
[col1
] = tab
->col_var
[col2
];
849 tab
->col_var
[col2
] = t
;
850 var_from_col(tab
, col1
)->index
= col1
;
851 var_from_col(tab
, col2
)->index
= col2
;
852 tab
->mat
= isl_mat_swap_cols(tab
->mat
, 2 + col1
, 2 + col2
);
855 /* Mark column with index "col" as representing a zero variable.
856 * If we may need to undo the operation the column is kept,
857 * but no longer considered.
858 * Otherwise, the column is simply removed.
860 * The column may be interchanged with some other column. If it
861 * is interchanged with a later column, return 1. Otherwise return 0.
862 * If the columns are checked in order in the calling function,
863 * then a return value of 1 means that the column with the given
864 * column number may now contain a different column that
865 * hasn't been checked yet.
867 int isl_tab_kill_col(struct isl_tab
*tab
, int col
)
869 var_from_col(tab
, col
)->is_zero
= 1;
870 if (tab
->need_undo
) {
871 isl_tab_push_var(tab
, isl_tab_undo_zero
, var_from_col(tab
, col
));
872 if (col
!= tab
->n_dead
)
873 swap_cols(tab
, col
, tab
->n_dead
);
877 if (col
!= tab
->n_col
- 1)
878 swap_cols(tab
, col
, tab
->n_col
- 1);
879 var_from_col(tab
, tab
->n_col
- 1)->index
= -1;
885 /* Row variable "var" is non-negative and cannot attain any values
886 * larger than zero. This means that the coefficients of the unrestricted
887 * column variables are zero and that the coefficients of the non-negative
888 * column variables are zero or negative.
889 * Each of the non-negative variables with a negative coefficient can
890 * then also be written as the negative sum of non-negative variables
891 * and must therefore also be zero.
893 static void close_row(struct isl_tab
*tab
, struct isl_tab_var
*var
)
896 struct isl_mat
*mat
= tab
->mat
;
898 isl_assert(tab
->mat
->ctx
, var
->is_nonneg
, return);
900 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
901 if (isl_int_is_zero(mat
->row
[var
->index
][2 + j
]))
903 isl_assert(tab
->mat
->ctx
,
904 isl_int_is_neg(mat
->row
[var
->index
][2 + j
]), return);
905 if (isl_tab_kill_col(tab
, j
))
908 isl_tab_mark_redundant(tab
, var
->index
);
911 /* Add a constraint to the tableau and allocate a row for it.
912 * Return the index into the constraint array "con".
914 int isl_tab_allocate_con(struct isl_tab
*tab
)
918 isl_assert(tab
->mat
->ctx
, tab
->n_row
< tab
->mat
->n_row
, return -1);
921 tab
->con
[r
].index
= tab
->n_row
;
922 tab
->con
[r
].is_row
= 1;
923 tab
->con
[r
].is_nonneg
= 0;
924 tab
->con
[r
].is_zero
= 0;
925 tab
->con
[r
].is_redundant
= 0;
926 tab
->con
[r
].frozen
= 0;
927 tab
->row_var
[tab
->n_row
] = ~r
;
931 isl_tab_push_var(tab
, isl_tab_undo_allocate
, &tab
->con
[r
]);
936 /* Add a row to the tableau. The row is given as an affine combination
937 * of the original variables and needs to be expressed in terms of the
940 * We add each term in turn.
941 * If r = n/d_r is the current sum and we need to add k x, then
942 * if x is a column variable, we increase the numerator of
943 * this column by k d_r
944 * if x = f/d_x is a row variable, then the new representation of r is
946 * n k f d_x/g n + d_r/g k f m/d_r n + m/d_g k f
947 * --- + --- = ------------------- = -------------------
948 * d_r d_r d_r d_x/g m
950 * with g the gcd of d_r and d_x and m the lcm of d_r and d_x.
952 int isl_tab_add_row(struct isl_tab
*tab
, isl_int
*line
)
959 r
= isl_tab_allocate_con(tab
);
965 row
= tab
->mat
->row
[tab
->con
[r
].index
];
966 isl_int_set_si(row
[0], 1);
967 isl_int_set(row
[1], line
[0]);
968 isl_seq_clr(row
+ 2, tab
->n_col
);
969 for (i
= 0; i
< tab
->n_var
; ++i
) {
970 if (tab
->var
[i
].is_zero
)
972 if (tab
->var
[i
].is_row
) {
974 row
[0], tab
->mat
->row
[tab
->var
[i
].index
][0]);
975 isl_int_swap(a
, row
[0]);
976 isl_int_divexact(a
, row
[0], a
);
978 row
[0], tab
->mat
->row
[tab
->var
[i
].index
][0]);
979 isl_int_mul(b
, b
, line
[1 + i
]);
980 isl_seq_combine(row
+ 1, a
, row
+ 1,
981 b
, tab
->mat
->row
[tab
->var
[i
].index
] + 1,
984 isl_int_addmul(row
[2 + tab
->var
[i
].index
],
985 line
[1 + i
], row
[0]);
987 isl_seq_normalize(row
, 2 + tab
->n_col
);
994 static int drop_row(struct isl_tab
*tab
, int row
)
996 isl_assert(tab
->mat
->ctx
, ~tab
->row_var
[row
] == tab
->n_con
- 1, return -1);
997 if (row
!= tab
->n_row
- 1)
998 swap_rows(tab
, row
, tab
->n_row
- 1);
1004 /* Add inequality "ineq" and check if it conflicts with the
1005 * previously added constraints or if it is obviously redundant.
1007 struct isl_tab
*isl_tab_add_ineq(struct isl_tab
*tab
, isl_int
*ineq
)
1014 r
= isl_tab_add_row(tab
, ineq
);
1017 tab
->con
[r
].is_nonneg
= 1;
1018 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1019 if (isl_tab_row_is_redundant(tab
, tab
->con
[r
].index
)) {
1020 isl_tab_mark_redundant(tab
, tab
->con
[r
].index
);
1024 sgn
= restore_row(tab
, &tab
->con
[r
]);
1026 return isl_tab_mark_empty(tab
);
1027 if (tab
->con
[r
].is_row
&& isl_tab_row_is_redundant(tab
, tab
->con
[r
].index
))
1028 isl_tab_mark_redundant(tab
, tab
->con
[r
].index
);
1035 /* Pivot a non-negative variable down until it reaches the value zero
1036 * and then pivot the variable into a column position.
1038 int to_col(struct isl_tab
*tab
, struct isl_tab_var
*var
)
1046 while (isl_int_is_pos(tab
->mat
->row
[var
->index
][1])) {
1047 find_pivot(tab
, var
, NULL
, -1, &row
, &col
);
1048 isl_assert(tab
->mat
->ctx
, row
!= -1, return -1);
1049 isl_tab_pivot(tab
, row
, col
);
1054 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
)
1055 if (!isl_int_is_zero(tab
->mat
->row
[var
->index
][2 + i
]))
1058 isl_assert(tab
->mat
->ctx
, i
< tab
->n_col
, return -1);
1059 isl_tab_pivot(tab
, var
->index
, i
);
1064 /* We assume Gaussian elimination has been performed on the equalities.
1065 * The equalities can therefore never conflict.
1066 * Adding the equalities is currently only really useful for a later call
1067 * to isl_tab_ineq_type.
1069 static struct isl_tab
*add_eq(struct isl_tab
*tab
, isl_int
*eq
)
1076 r
= isl_tab_add_row(tab
, eq
);
1080 r
= tab
->con
[r
].index
;
1081 i
= isl_seq_first_non_zero(tab
->mat
->row
[r
] + 2 + tab
->n_dead
,
1082 tab
->n_col
- tab
->n_dead
);
1083 isl_assert(tab
->mat
->ctx
, i
>= 0, goto error
);
1085 isl_tab_pivot(tab
, r
, i
);
1086 isl_tab_kill_col(tab
, i
);
1095 /* Add an equality that is known to be valid for the given tableau.
1097 struct isl_tab
*isl_tab_add_valid_eq(struct isl_tab
*tab
, isl_int
*eq
)
1099 struct isl_tab_var
*var
;
1105 r
= isl_tab_add_row(tab
, eq
);
1111 if (isl_int_is_neg(tab
->mat
->row
[r
][1]))
1112 isl_seq_neg(tab
->mat
->row
[r
] + 1, tab
->mat
->row
[r
] + 1,
1115 if (to_col(tab
, var
) < 0)
1118 isl_tab_kill_col(tab
, var
->index
);
1126 struct isl_tab
*isl_tab_from_basic_map(struct isl_basic_map
*bmap
)
1129 struct isl_tab
*tab
;
1133 tab
= isl_tab_alloc(bmap
->ctx
,
1134 isl_basic_map_total_dim(bmap
) + bmap
->n_ineq
+ 1,
1135 isl_basic_map_total_dim(bmap
));
1138 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1139 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
))
1140 return isl_tab_mark_empty(tab
);
1141 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1142 tab
= add_eq(tab
, bmap
->eq
[i
]);
1146 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1147 tab
= isl_tab_add_ineq(tab
, bmap
->ineq
[i
]);
1148 if (!tab
|| tab
->empty
)
1154 struct isl_tab
*isl_tab_from_basic_set(struct isl_basic_set
*bset
)
1156 return isl_tab_from_basic_map((struct isl_basic_map
*)bset
);
1159 /* Construct a tableau corresponding to the recession cone of "bmap".
1161 struct isl_tab
*isl_tab_from_recession_cone(struct isl_basic_map
*bmap
)
1165 struct isl_tab
*tab
;
1169 tab
= isl_tab_alloc(bmap
->ctx
, bmap
->n_eq
+ bmap
->n_ineq
,
1170 isl_basic_map_total_dim(bmap
));
1173 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1176 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1177 isl_int_swap(bmap
->eq
[i
][0], cst
);
1178 tab
= add_eq(tab
, bmap
->eq
[i
]);
1179 isl_int_swap(bmap
->eq
[i
][0], cst
);
1183 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1185 isl_int_swap(bmap
->ineq
[i
][0], cst
);
1186 r
= isl_tab_add_row(tab
, bmap
->ineq
[i
]);
1187 isl_int_swap(bmap
->ineq
[i
][0], cst
);
1190 tab
->con
[r
].is_nonneg
= 1;
1191 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1202 /* Assuming "tab" is the tableau of a cone, check if the cone is
1203 * bounded, i.e., if it is empty or only contains the origin.
1205 int isl_tab_cone_is_bounded(struct isl_tab
*tab
)
1213 if (tab
->n_dead
== tab
->n_col
)
1217 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1218 struct isl_tab_var
*var
;
1219 var
= isl_tab_var_from_row(tab
, i
);
1220 if (!var
->is_nonneg
)
1222 if (sign_of_max(tab
, var
) != 0)
1224 close_row(tab
, var
);
1227 if (tab
->n_dead
== tab
->n_col
)
1229 if (i
== tab
->n_row
)
1234 int isl_tab_sample_is_integer(struct isl_tab
*tab
)
1241 for (i
= 0; i
< tab
->n_var
; ++i
) {
1243 if (!tab
->var
[i
].is_row
)
1245 row
= tab
->var
[i
].index
;
1246 if (!isl_int_is_divisible_by(tab
->mat
->row
[row
][1],
1247 tab
->mat
->row
[row
][0]))
1253 static struct isl_vec
*extract_integer_sample(struct isl_tab
*tab
)
1256 struct isl_vec
*vec
;
1258 vec
= isl_vec_alloc(tab
->mat
->ctx
, 1 + tab
->n_var
);
1262 isl_int_set_si(vec
->block
.data
[0], 1);
1263 for (i
= 0; i
< tab
->n_var
; ++i
) {
1264 if (!tab
->var
[i
].is_row
)
1265 isl_int_set_si(vec
->block
.data
[1 + i
], 0);
1267 int row
= tab
->var
[i
].index
;
1268 isl_int_divexact(vec
->block
.data
[1 + i
],
1269 tab
->mat
->row
[row
][1], tab
->mat
->row
[row
][0]);
1276 struct isl_vec
*isl_tab_get_sample_value(struct isl_tab
*tab
)
1279 struct isl_vec
*vec
;
1285 vec
= isl_vec_alloc(tab
->mat
->ctx
, 1 + tab
->n_var
);
1291 isl_int_set_si(vec
->block
.data
[0], 1);
1292 for (i
= 0; i
< tab
->n_var
; ++i
) {
1294 if (!tab
->var
[i
].is_row
) {
1295 isl_int_set_si(vec
->block
.data
[1 + i
], 0);
1298 row
= tab
->var
[i
].index
;
1299 isl_int_gcd(m
, vec
->block
.data
[0], tab
->mat
->row
[row
][0]);
1300 isl_int_divexact(m
, tab
->mat
->row
[row
][0], m
);
1301 isl_seq_scale(vec
->block
.data
, vec
->block
.data
, m
, 1 + i
);
1302 isl_int_divexact(m
, vec
->block
.data
[0], tab
->mat
->row
[row
][0]);
1303 isl_int_mul(vec
->block
.data
[1 + i
], m
, tab
->mat
->row
[row
][1]);
1305 isl_seq_normalize(vec
->block
.data
, vec
->size
);
1311 /* Update "bmap" based on the results of the tableau "tab".
1312 * In particular, implicit equalities are made explicit, redundant constraints
1313 * are removed and if the sample value happens to be integer, it is stored
1314 * in "bmap" (unless "bmap" already had an integer sample).
1316 * The tableau is assumed to have been created from "bmap" using
1317 * isl_tab_from_basic_map.
1319 struct isl_basic_map
*isl_basic_map_update_from_tab(struct isl_basic_map
*bmap
,
1320 struct isl_tab
*tab
)
1332 bmap
= isl_basic_map_set_to_empty(bmap
);
1334 for (i
= bmap
->n_ineq
- 1; i
>= 0; --i
) {
1335 if (isl_tab_is_equality(tab
, n_eq
+ i
))
1336 isl_basic_map_inequality_to_equality(bmap
, i
);
1337 else if (isl_tab_is_redundant(tab
, n_eq
+ i
))
1338 isl_basic_map_drop_inequality(bmap
, i
);
1340 if (!tab
->rational
&&
1341 !bmap
->sample
&& isl_tab_sample_is_integer(tab
))
1342 bmap
->sample
= extract_integer_sample(tab
);
1346 struct isl_basic_set
*isl_basic_set_update_from_tab(struct isl_basic_set
*bset
,
1347 struct isl_tab
*tab
)
1349 return (struct isl_basic_set
*)isl_basic_map_update_from_tab(
1350 (struct isl_basic_map
*)bset
, tab
);
1353 /* Given a non-negative variable "var", add a new non-negative variable
1354 * that is the opposite of "var", ensuring that var can only attain the
1356 * If var = n/d is a row variable, then the new variable = -n/d.
1357 * If var is a column variables, then the new variable = -var.
1358 * If the new variable cannot attain non-negative values, then
1359 * the resulting tableau is empty.
1360 * Otherwise, we know the value will be zero and we close the row.
1362 static struct isl_tab
*cut_to_hyperplane(struct isl_tab
*tab
,
1363 struct isl_tab_var
*var
)
1369 if (isl_tab_extend_cons(tab
, 1) < 0)
1373 tab
->con
[r
].index
= tab
->n_row
;
1374 tab
->con
[r
].is_row
= 1;
1375 tab
->con
[r
].is_nonneg
= 0;
1376 tab
->con
[r
].is_zero
= 0;
1377 tab
->con
[r
].is_redundant
= 0;
1378 tab
->con
[r
].frozen
= 0;
1379 tab
->row_var
[tab
->n_row
] = ~r
;
1380 row
= tab
->mat
->row
[tab
->n_row
];
1383 isl_int_set(row
[0], tab
->mat
->row
[var
->index
][0]);
1384 isl_seq_neg(row
+ 1,
1385 tab
->mat
->row
[var
->index
] + 1, 1 + tab
->n_col
);
1387 isl_int_set_si(row
[0], 1);
1388 isl_seq_clr(row
+ 1, 1 + tab
->n_col
);
1389 isl_int_set_si(row
[2 + var
->index
], -1);
1394 isl_tab_push_var(tab
, isl_tab_undo_allocate
, &tab
->con
[r
]);
1396 sgn
= sign_of_max(tab
, &tab
->con
[r
]);
1398 return isl_tab_mark_empty(tab
);
1399 tab
->con
[r
].is_nonneg
= 1;
1400 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1402 close_row(tab
, &tab
->con
[r
]);
1410 /* Given a tableau "tab" and an inequality constraint "con" of the tableau,
1411 * relax the inequality by one. That is, the inequality r >= 0 is replaced
1412 * by r' = r + 1 >= 0.
1413 * If r is a row variable, we simply increase the constant term by one
1414 * (taking into account the denominator).
1415 * If r is a column variable, then we need to modify each row that
1416 * refers to r = r' - 1 by substituting this equality, effectively
1417 * subtracting the coefficient of the column from the constant.
1419 struct isl_tab
*isl_tab_relax(struct isl_tab
*tab
, int con
)
1421 struct isl_tab_var
*var
;
1425 var
= &tab
->con
[con
];
1427 if (!var
->is_row
&& !max_is_manifestly_unbounded(tab
, var
))
1428 to_row(tab
, var
, 1);
1431 isl_int_add(tab
->mat
->row
[var
->index
][1],
1432 tab
->mat
->row
[var
->index
][1], tab
->mat
->row
[var
->index
][0]);
1436 for (i
= 0; i
< tab
->n_row
; ++i
) {
1437 if (isl_int_is_zero(tab
->mat
->row
[i
][2 + var
->index
]))
1439 isl_int_sub(tab
->mat
->row
[i
][1], tab
->mat
->row
[i
][1],
1440 tab
->mat
->row
[i
][2 + var
->index
]);
1445 isl_tab_push_var(tab
, isl_tab_undo_relax
, var
);
1450 struct isl_tab
*isl_tab_select_facet(struct isl_tab
*tab
, int con
)
1455 return cut_to_hyperplane(tab
, &tab
->con
[con
]);
1458 static int may_be_equality(struct isl_tab
*tab
, int row
)
1460 return (tab
->rational
? isl_int_is_zero(tab
->mat
->row
[row
][1])
1461 : isl_int_lt(tab
->mat
->row
[row
][1],
1462 tab
->mat
->row
[row
][0])) &&
1463 isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1464 tab
->n_col
- tab
->n_dead
) != -1;
1467 /* Check for (near) equalities among the constraints.
1468 * A constraint is an equality if it is non-negative and if
1469 * its maximal value is either
1470 * - zero (in case of rational tableaus), or
1471 * - strictly less than 1 (in case of integer tableaus)
1473 * We first mark all non-redundant and non-dead variables that
1474 * are not frozen and not obviously not an equality.
1475 * Then we iterate over all marked variables if they can attain
1476 * any values larger than zero or at least one.
1477 * If the maximal value is zero, we mark any column variables
1478 * that appear in the row as being zero and mark the row as being redundant.
1479 * Otherwise, if the maximal value is strictly less than one (and the
1480 * tableau is integer), then we restrict the value to being zero
1481 * by adding an opposite non-negative variable.
1483 struct isl_tab
*isl_tab_detect_equalities(struct isl_tab
*tab
)
1492 if (tab
->n_dead
== tab
->n_col
)
1496 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1497 struct isl_tab_var
*var
= isl_tab_var_from_row(tab
, i
);
1498 var
->marked
= !var
->frozen
&& var
->is_nonneg
&&
1499 may_be_equality(tab
, i
);
1503 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1504 struct isl_tab_var
*var
= var_from_col(tab
, i
);
1505 var
->marked
= !var
->frozen
&& var
->is_nonneg
;
1510 struct isl_tab_var
*var
;
1511 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1512 var
= isl_tab_var_from_row(tab
, i
);
1516 if (i
== tab
->n_row
) {
1517 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1518 var
= var_from_col(tab
, i
);
1522 if (i
== tab
->n_col
)
1527 if (sign_of_max(tab
, var
) == 0)
1528 close_row(tab
, var
);
1529 else if (!tab
->rational
&& !at_least_one(tab
, var
)) {
1530 tab
= cut_to_hyperplane(tab
, var
);
1531 return isl_tab_detect_equalities(tab
);
1533 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1534 var
= isl_tab_var_from_row(tab
, i
);
1537 if (may_be_equality(tab
, i
))
1547 /* Check for (near) redundant constraints.
1548 * A constraint is redundant if it is non-negative and if
1549 * its minimal value (temporarily ignoring the non-negativity) is either
1550 * - zero (in case of rational tableaus), or
1551 * - strictly larger than -1 (in case of integer tableaus)
1553 * We first mark all non-redundant and non-dead variables that
1554 * are not frozen and not obviously negatively unbounded.
1555 * Then we iterate over all marked variables if they can attain
1556 * any values smaller than zero or at most negative one.
1557 * If not, we mark the row as being redundant (assuming it hasn't
1558 * been detected as being obviously redundant in the mean time).
1560 struct isl_tab
*isl_tab_detect_redundant(struct isl_tab
*tab
)
1569 if (tab
->n_redundant
== tab
->n_row
)
1573 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1574 struct isl_tab_var
*var
= isl_tab_var_from_row(tab
, i
);
1575 var
->marked
= !var
->frozen
&& var
->is_nonneg
;
1579 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1580 struct isl_tab_var
*var
= var_from_col(tab
, i
);
1581 var
->marked
= !var
->frozen
&& var
->is_nonneg
&&
1582 !min_is_manifestly_unbounded(tab
, var
);
1587 struct isl_tab_var
*var
;
1588 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1589 var
= isl_tab_var_from_row(tab
, i
);
1593 if (i
== tab
->n_row
) {
1594 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1595 var
= var_from_col(tab
, i
);
1599 if (i
== tab
->n_col
)
1604 if ((tab
->rational
? (sign_of_min(tab
, var
) >= 0)
1605 : !isl_tab_min_at_most_neg_one(tab
, var
)) &&
1607 isl_tab_mark_redundant(tab
, var
->index
);
1608 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1609 var
= var_from_col(tab
, i
);
1612 if (!min_is_manifestly_unbounded(tab
, var
))
1622 int isl_tab_is_equality(struct isl_tab
*tab
, int con
)
1628 if (tab
->con
[con
].is_zero
)
1630 if (tab
->con
[con
].is_redundant
)
1632 if (!tab
->con
[con
].is_row
)
1633 return tab
->con
[con
].index
< tab
->n_dead
;
1635 row
= tab
->con
[con
].index
;
1637 return isl_int_is_zero(tab
->mat
->row
[row
][1]) &&
1638 isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1639 tab
->n_col
- tab
->n_dead
) == -1;
1642 /* Return the minimial value of the affine expression "f" with denominator
1643 * "denom" in *opt, *opt_denom, assuming the tableau is not empty and
1644 * the expression cannot attain arbitrarily small values.
1645 * If opt_denom is NULL, then *opt is rounded up to the nearest integer.
1646 * The return value reflects the nature of the result (empty, unbounded,
1647 * minmimal value returned in *opt).
1649 enum isl_lp_result
isl_tab_min(struct isl_tab
*tab
,
1650 isl_int
*f
, isl_int denom
, isl_int
*opt
, isl_int
*opt_denom
,
1654 enum isl_lp_result res
= isl_lp_ok
;
1655 struct isl_tab_var
*var
;
1656 struct isl_tab_undo
*snap
;
1659 return isl_lp_empty
;
1661 snap
= isl_tab_snap(tab
);
1662 r
= isl_tab_add_row(tab
, f
);
1664 return isl_lp_error
;
1666 isl_int_mul(tab
->mat
->row
[var
->index
][0],
1667 tab
->mat
->row
[var
->index
][0], denom
);
1670 find_pivot(tab
, var
, var
, -1, &row
, &col
);
1671 if (row
== var
->index
) {
1672 res
= isl_lp_unbounded
;
1677 isl_tab_pivot(tab
, row
, col
);
1679 if (isl_tab_rollback(tab
, snap
) < 0)
1680 return isl_lp_error
;
1681 if (ISL_FL_ISSET(flags
, ISL_TAB_SAVE_DUAL
)) {
1684 isl_vec_free(tab
->dual
);
1685 tab
->dual
= isl_vec_alloc(tab
->mat
->ctx
, 1 + tab
->n_con
);
1687 return isl_lp_error
;
1688 isl_int_set(tab
->dual
->el
[0], tab
->mat
->row
[var
->index
][0]);
1689 for (i
= 0; i
< tab
->n_con
; ++i
) {
1690 if (tab
->con
[i
].is_row
)
1691 isl_int_set_si(tab
->dual
->el
[1 + i
], 0);
1693 int pos
= 2 + tab
->con
[i
].index
;
1694 isl_int_set(tab
->dual
->el
[1 + i
],
1695 tab
->mat
->row
[var
->index
][pos
]);
1699 if (res
== isl_lp_ok
) {
1701 isl_int_set(*opt
, tab
->mat
->row
[var
->index
][1]);
1702 isl_int_set(*opt_denom
, tab
->mat
->row
[var
->index
][0]);
1704 isl_int_cdiv_q(*opt
, tab
->mat
->row
[var
->index
][1],
1705 tab
->mat
->row
[var
->index
][0]);
1710 int isl_tab_is_redundant(struct isl_tab
*tab
, int con
)
1717 if (tab
->con
[con
].is_zero
)
1719 if (tab
->con
[con
].is_redundant
)
1721 return tab
->con
[con
].is_row
&& tab
->con
[con
].index
< tab
->n_redundant
;
1724 /* Take a snapshot of the tableau that can be restored by s call to
1727 struct isl_tab_undo
*isl_tab_snap(struct isl_tab
*tab
)
1735 /* Undo the operation performed by isl_tab_relax.
1737 static void unrelax(struct isl_tab
*tab
, struct isl_tab_var
*var
)
1739 if (!var
->is_row
&& !max_is_manifestly_unbounded(tab
, var
))
1740 to_row(tab
, var
, 1);
1743 isl_int_sub(tab
->mat
->row
[var
->index
][1],
1744 tab
->mat
->row
[var
->index
][1], tab
->mat
->row
[var
->index
][0]);
1748 for (i
= 0; i
< tab
->n_row
; ++i
) {
1749 if (isl_int_is_zero(tab
->mat
->row
[i
][2 + var
->index
]))
1751 isl_int_add(tab
->mat
->row
[i
][1], tab
->mat
->row
[i
][1],
1752 tab
->mat
->row
[i
][2 + var
->index
]);
1758 static void perform_undo_var(struct isl_tab
*tab
, struct isl_tab_undo
*undo
)
1760 struct isl_tab_var
*var
= var_from_index(tab
, undo
->u
.var_index
);
1761 switch(undo
->type
) {
1762 case isl_tab_undo_nonneg
:
1765 case isl_tab_undo_redundant
:
1766 var
->is_redundant
= 0;
1769 case isl_tab_undo_zero
:
1773 case isl_tab_undo_allocate
:
1775 if (!max_is_manifestly_unbounded(tab
, var
))
1776 to_row(tab
, var
, 1);
1777 else if (!min_is_manifestly_unbounded(tab
, var
))
1778 to_row(tab
, var
, -1);
1780 to_row(tab
, var
, 0);
1782 drop_row(tab
, var
->index
);
1784 case isl_tab_undo_relax
:
1790 /* Restore the tableau to the state where the basic variables
1791 * are those in "col_var".
1792 * We first construct a list of variables that are currently in
1793 * the basis, but shouldn't. Then we iterate over all variables
1794 * that should be in the basis and for each one that is currently
1795 * not in the basis, we exchange it with one of the elements of the
1796 * list constructed before.
1797 * We can always find an appropriate variable to pivot with because
1798 * the current basis is mapped to the old basis by a non-singular
1799 * matrix and so we can never end up with a zero row.
1801 static int restore_basis(struct isl_tab
*tab
, int *col_var
)
1805 int *extra
= NULL
; /* current columns that contain bad stuff */
1808 extra
= isl_alloc_array(tab
->mat
->ctx
, int, tab
->n_col
);
1811 for (i
= 0; i
< tab
->n_col
; ++i
) {
1812 for (j
= 0; j
< tab
->n_col
; ++j
)
1813 if (tab
->col_var
[i
] == col_var
[j
])
1817 extra
[n_extra
++] = i
;
1819 for (i
= 0; i
< tab
->n_col
&& n_extra
> 0; ++i
) {
1820 struct isl_tab_var
*var
;
1823 for (j
= 0; j
< tab
->n_col
; ++j
)
1824 if (col_var
[i
] == tab
->col_var
[j
])
1828 var
= var_from_index(tab
, col_var
[i
]);
1830 for (j
= 0; j
< n_extra
; ++j
)
1831 if (!isl_int_is_zero(tab
->mat
->row
[row
][off
+extra
[j
]]))
1833 isl_assert(tab
->mat
->ctx
, j
< n_extra
, goto error
);
1834 isl_tab_pivot(tab
, row
, extra
[j
]);
1835 extra
[j
] = extra
[--n_extra
];
1847 static int perform_undo(struct isl_tab
*tab
, struct isl_tab_undo
*undo
)
1849 switch (undo
->type
) {
1850 case isl_tab_undo_empty
:
1853 case isl_tab_undo_nonneg
:
1854 case isl_tab_undo_redundant
:
1855 case isl_tab_undo_zero
:
1856 case isl_tab_undo_allocate
:
1857 case isl_tab_undo_relax
:
1858 perform_undo_var(tab
, undo
);
1860 case isl_tab_undo_saved_basis
:
1861 if (restore_basis(tab
, undo
->u
.col_var
) < 0)
1865 isl_assert(tab
->mat
->ctx
, 0, return -1);
1870 /* Return the tableau to the state it was in when the snapshot "snap"
1873 int isl_tab_rollback(struct isl_tab
*tab
, struct isl_tab_undo
*snap
)
1875 struct isl_tab_undo
*undo
, *next
;
1881 for (undo
= tab
->top
; undo
&& undo
!= &tab
->bottom
; undo
= next
) {
1885 if (perform_undo(tab
, undo
) < 0) {
1899 /* The given row "row" represents an inequality violated by all
1900 * points in the tableau. Check for some special cases of such
1901 * separating constraints.
1902 * In particular, if the row has been reduced to the constant -1,
1903 * then we know the inequality is adjacent (but opposite) to
1904 * an equality in the tableau.
1905 * If the row has been reduced to r = -1 -r', with r' an inequality
1906 * of the tableau, then the inequality is adjacent (but opposite)
1907 * to the inequality r'.
1909 static enum isl_ineq_type
separation_type(struct isl_tab
*tab
, unsigned row
)
1914 return isl_ineq_separate
;
1916 if (!isl_int_is_one(tab
->mat
->row
[row
][0]))
1917 return isl_ineq_separate
;
1918 if (!isl_int_is_negone(tab
->mat
->row
[row
][1]))
1919 return isl_ineq_separate
;
1921 pos
= isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1922 tab
->n_col
- tab
->n_dead
);
1924 return isl_ineq_adj_eq
;
1926 if (!isl_int_is_negone(tab
->mat
->row
[row
][2 + tab
->n_dead
+ pos
]))
1927 return isl_ineq_separate
;
1929 pos
= isl_seq_first_non_zero(
1930 tab
->mat
->row
[row
] + 2 + tab
->n_dead
+ pos
+ 1,
1931 tab
->n_col
- tab
->n_dead
- pos
- 1);
1933 return pos
== -1 ? isl_ineq_adj_ineq
: isl_ineq_separate
;
1936 /* Check the effect of inequality "ineq" on the tableau "tab".
1938 * isl_ineq_redundant: satisfied by all points in the tableau
1939 * isl_ineq_separate: satisfied by no point in the tableau
1940 * isl_ineq_cut: satisfied by some by not all points
1941 * isl_ineq_adj_eq: adjacent to an equality
1942 * isl_ineq_adj_ineq: adjacent to an inequality.
1944 enum isl_ineq_type
isl_tab_ineq_type(struct isl_tab
*tab
, isl_int
*ineq
)
1946 enum isl_ineq_type type
= isl_ineq_error
;
1947 struct isl_tab_undo
*snap
= NULL
;
1952 return isl_ineq_error
;
1954 if (isl_tab_extend_cons(tab
, 1) < 0)
1955 return isl_ineq_error
;
1957 snap
= isl_tab_snap(tab
);
1959 con
= isl_tab_add_row(tab
, ineq
);
1963 row
= tab
->con
[con
].index
;
1964 if (isl_tab_row_is_redundant(tab
, row
))
1965 type
= isl_ineq_redundant
;
1966 else if (isl_int_is_neg(tab
->mat
->row
[row
][1]) &&
1968 isl_int_abs_ge(tab
->mat
->row
[row
][1],
1969 tab
->mat
->row
[row
][0]))) {
1970 if (at_least_zero(tab
, &tab
->con
[con
]))
1971 type
= isl_ineq_cut
;
1973 type
= separation_type(tab
, row
);
1974 } else if (tab
->rational
? (sign_of_min(tab
, &tab
->con
[con
]) < 0)
1975 : isl_tab_min_at_most_neg_one(tab
, &tab
->con
[con
]))
1976 type
= isl_ineq_cut
;
1978 type
= isl_ineq_redundant
;
1980 if (isl_tab_rollback(tab
, snap
))
1981 return isl_ineq_error
;
1984 isl_tab_rollback(tab
, snap
);
1985 return isl_ineq_error
;
1988 void isl_tab_dump(struct isl_tab
*tab
, FILE *out
, int indent
)
1994 fprintf(out
, "%*snull tab\n", indent
, "");
1997 fprintf(out
, "%*sn_redundant: %d, n_dead: %d", indent
, "",
1998 tab
->n_redundant
, tab
->n_dead
);
2000 fprintf(out
, ", rational");
2002 fprintf(out
, ", empty");
2004 fprintf(out
, "%*s[", indent
, "");
2005 for (i
= 0; i
< tab
->n_var
; ++i
) {
2007 fprintf(out
, (i
== tab
->n_param
||
2008 i
== tab
->n_var
- tab
->n_div
) ? "; "
2010 fprintf(out
, "%c%d%s", tab
->var
[i
].is_row
? 'r' : 'c',
2012 tab
->var
[i
].is_zero
? " [=0]" :
2013 tab
->var
[i
].is_redundant
? " [R]" : "");
2015 fprintf(out
, "]\n");
2016 fprintf(out
, "%*s[", indent
, "");
2017 for (i
= 0; i
< tab
->n_con
; ++i
) {
2020 fprintf(out
, "%c%d%s", tab
->con
[i
].is_row
? 'r' : 'c',
2022 tab
->con
[i
].is_zero
? " [=0]" :
2023 tab
->con
[i
].is_redundant
? " [R]" : "");
2025 fprintf(out
, "]\n");
2026 fprintf(out
, "%*s[", indent
, "");
2027 for (i
= 0; i
< tab
->n_row
; ++i
) {
2030 fprintf(out
, "r%d: %d%s", i
, tab
->row_var
[i
],
2031 isl_tab_var_from_row(tab
, i
)->is_nonneg
? " [>=0]" : "");
2033 fprintf(out
, "]\n");
2034 fprintf(out
, "%*s[", indent
, "");
2035 for (i
= 0; i
< tab
->n_col
; ++i
) {
2038 fprintf(out
, "c%d: %d%s", i
, tab
->col_var
[i
],
2039 var_from_col(tab
, i
)->is_nonneg
? " [>=0]" : "");
2041 fprintf(out
, "]\n");
2042 r
= tab
->mat
->n_row
;
2043 tab
->mat
->n_row
= tab
->n_row
;
2044 c
= tab
->mat
->n_col
;
2045 tab
->mat
->n_col
= 2 + tab
->n_col
;
2046 isl_mat_dump(tab
->mat
, out
, indent
);
2047 tab
->mat
->n_row
= r
;
2048 tab
->mat
->n_col
= c
;