1 #include "isl_map_private.h"
6 * The implementation of parametric integer linear programming in this file
7 * was inspired by the paper "Parametric Integer Programming" and the
8 * report "Solving systems of affine (in)equalities" by Paul Feautrier
11 * The strategy used for obtaining a feasible solution is different
12 * from the one used in isl_tab.c. In particular, in isl_tab.c,
13 * upon finding a constraint that is not yet satisfied, we pivot
14 * in a row that increases the constant term of row holding the
15 * constraint, making sure the sample solution remains feasible
16 * for all the constraints it already satisfied.
17 * Here, we always pivot in the row holding the constraint,
18 * choosing a column that induces the lexicographically smallest
19 * increment to the sample solution.
21 * By starting out from a sample value that is lexicographically
22 * smaller than any integer point in the problem space, the first
23 * feasible integer sample point we find will also be the lexicographically
24 * smallest. If all variables can be assumed to be non-negative,
25 * then the initial sample value may be chosen equal to zero.
26 * However, we will not make this assumption. Instead, we apply
27 * the "big parameter" trick. Any variable x is then not directly
28 * used in the tableau, but instead it its represented by another
29 * variable x' = M + x, where M is an arbitrarily large (positive)
30 * value. x' is therefore always non-negative, whatever the value of x.
31 * Taking as initial smaple value x' = 0 corresponds to x = -M,
32 * which is always smaller than any possible value of x.
34 * We use the big parameter trick both in the main tableau and
35 * the context tableau, each of course having its own big parameter.
36 * Before doing any real work, we check if all the parameters
37 * happen to be non-negative. If so, we drop the column corresponding
38 * to M from the initial context tableau.
41 /* isl_sol is an interface for constructing a solution to
42 * a parametric integer linear programming problem.
43 * Every time the algorithm reaches a state where a solution
44 * can be read off from the tableau (including cases where the tableau
45 * is empty), the function "add" is called on the isl_sol passed
46 * to find_solutions_main.
48 * The context tableau is owned by isl_sol and is updated incrementally.
50 * There is currently only one implementation of this interface,
51 * isl_sol_map, which simply collects the solutions in an isl_map
52 * and (optionally) the parts of the context where there is no solution
56 struct isl_tab
*context_tab
;
57 struct isl_sol
*(*add
)(struct isl_sol
*sol
, struct isl_tab
*tab
);
58 void (*free
)(struct isl_sol
*sol
);
61 static void sol_free(struct isl_sol
*sol
)
71 struct isl_set
*empty
;
75 static void sol_map_free(struct isl_sol_map
*sol_map
)
77 isl_tab_free(sol_map
->sol
.context_tab
);
78 isl_map_free(sol_map
->map
);
79 isl_set_free(sol_map
->empty
);
83 static void sol_map_free_wrap(struct isl_sol
*sol
)
85 sol_map_free((struct isl_sol_map
*)sol
);
88 static struct isl_sol_map
*add_empty(struct isl_sol_map
*sol
)
90 struct isl_basic_set
*bset
;
94 sol
->empty
= isl_set_grow(sol
->empty
, 1);
95 bset
= isl_basic_set_copy(sol
->sol
.context_tab
->bset
);
96 bset
= isl_basic_set_simplify(bset
);
97 bset
= isl_basic_set_finalize(bset
);
98 sol
->empty
= isl_set_add(sol
->empty
, bset
);
107 /* Add the solution identified by the tableau and the context tableau.
109 * The layout of the variables is as follows.
110 * tab->n_var is equal to the total number of variables in the input
111 * map (including divs that were copied from the context)
112 * + the number of extra divs constructed
113 * Of these, the first tab->n_param and the last tab->n_div variables
114 * correspond to the variables in the context, i.e.,
115 tab->n_param + tab->n_div = context_tab->n_var
116 * tab->n_param is equal to the number of parameters and input
117 * dimensions in the input map
118 * tab->n_div is equal to the number of divs in the context
120 * If there is no solution, then the basic set corresponding to the
121 * context tableau is added to the set "empty".
123 * Otherwise, a basic map is constructed with the same parameters
124 * and divs as the context, the dimensions of the context as input
125 * dimensions and a number of output dimensions that is equal to
126 * the number of output dimensions in the input map.
127 * The divs in the input map (if any) that do not correspond to any
128 * div in the context do not appear in the solution.
129 * The algorithm will make sure that they have an integer value,
130 * but these values themselves are of no interest.
132 * The constraints and divs of the context are simply copied
133 * fron context_tab->bset.
134 * To extract the value of the output variables, it should be noted
135 * that we always use a big parameter M and so the variable stored
136 * in the tableau is not an output variable x itself, but
137 * x' = M + x (in case of minimization)
139 * x' = M - x (in case of maximization)
140 * If x' appears in a column, then its optimal value is zero,
141 * which means that the optimal value of x is an unbounded number
142 * (-M for minimization and M for maximization).
143 * We currently assume that the output dimensions in the original map
144 * are bounded, so this cannot occur.
145 * Similarly, when x' appears in a row, then the coefficient of M in that
146 * row is necessarily 1.
147 * If the row represents
148 * d x' = c + d M + e(y)
149 * then, in case of minimization, an equality
150 * c + e(y) - d x' = 0
151 * is added, and in case of maximization,
152 * c + e(y) + d x' = 0
154 static struct isl_sol_map
*sol_map_add(struct isl_sol_map
*sol
,
158 struct isl_basic_map
*bmap
= NULL
;
159 struct isl_tab
*context_tab
;
172 return add_empty(sol
);
174 context_tab
= sol
->sol
.context_tab
;
176 n_out
= isl_map_dim(sol
->map
, isl_dim_out
);
177 n_eq
= context_tab
->bset
->n_eq
+ n_out
;
178 n_ineq
= context_tab
->bset
->n_ineq
;
179 nparam
= tab
->n_param
;
180 total
= isl_map_dim(sol
->map
, isl_dim_all
);
181 bmap
= isl_basic_map_alloc_dim(isl_map_get_dim(sol
->map
),
182 tab
->n_div
, n_eq
, 2 * tab
->n_div
+ n_ineq
);
187 ISL_F_SET(bmap
, ISL_BASIC_MAP_RATIONAL
);
188 for (i
= 0; i
< context_tab
->bset
->n_div
; ++i
) {
189 int k
= isl_basic_map_alloc_div(bmap
);
192 isl_seq_cpy(bmap
->div
[k
],
193 context_tab
->bset
->div
[i
], 1 + 1 + nparam
);
194 isl_seq_clr(bmap
->div
[k
] + 1 + 1 + nparam
, total
- nparam
);
195 isl_seq_cpy(bmap
->div
[k
] + 1 + 1 + total
,
196 context_tab
->bset
->div
[i
] + 1 + 1 + nparam
, i
);
198 for (i
= 0; i
< context_tab
->bset
->n_eq
; ++i
) {
199 int k
= isl_basic_map_alloc_equality(bmap
);
202 isl_seq_cpy(bmap
->eq
[k
], context_tab
->bset
->eq
[i
], 1 + nparam
);
203 isl_seq_clr(bmap
->eq
[k
] + 1 + nparam
, total
- nparam
);
204 isl_seq_cpy(bmap
->eq
[k
] + 1 + total
,
205 context_tab
->bset
->eq
[i
] + 1 + nparam
, n_div
);
207 for (i
= 0; i
< context_tab
->bset
->n_ineq
; ++i
) {
208 int k
= isl_basic_map_alloc_inequality(bmap
);
211 isl_seq_cpy(bmap
->ineq
[k
],
212 context_tab
->bset
->ineq
[i
], 1 + nparam
);
213 isl_seq_clr(bmap
->ineq
[k
] + 1 + nparam
, total
- nparam
);
214 isl_seq_cpy(bmap
->ineq
[k
] + 1 + total
,
215 context_tab
->bset
->ineq
[i
] + 1 + nparam
, n_div
);
217 for (i
= tab
->n_param
; i
< total
; ++i
) {
218 int k
= isl_basic_map_alloc_equality(bmap
);
221 isl_seq_clr(bmap
->eq
[k
] + 1, isl_basic_map_total_dim(bmap
));
222 if (!tab
->var
[i
].is_row
) {
224 isl_assert(bmap
->ctx
, !tab
->M
, goto error
);
225 isl_int_set_si(bmap
->eq
[k
][0], 0);
227 isl_int_set_si(bmap
->eq
[k
][1 + i
], 1);
229 isl_int_set_si(bmap
->eq
[k
][1 + i
], -1);
232 row
= tab
->var
[i
].index
;
235 isl_assert(bmap
->ctx
,
236 isl_int_eq(tab
->mat
->row
[row
][2],
237 tab
->mat
->row
[row
][0]),
239 isl_int_set(bmap
->eq
[k
][0], tab
->mat
->row
[row
][1]);
240 for (j
= 0; j
< tab
->n_param
; ++j
) {
242 if (tab
->var
[j
].is_row
)
244 col
= tab
->var
[j
].index
;
245 isl_int_set(bmap
->eq
[k
][1 + j
],
246 tab
->mat
->row
[row
][off
+ col
]);
248 for (j
= 0; j
< tab
->n_div
; ++j
) {
250 if (tab
->var
[tab
->n_var
- tab
->n_div
+j
].is_row
)
252 col
= tab
->var
[tab
->n_var
- tab
->n_div
+j
].index
;
253 isl_int_set(bmap
->eq
[k
][1 + total
+ j
],
254 tab
->mat
->row
[row
][off
+ col
]);
257 isl_int_set(bmap
->eq
[k
][1 + i
],
258 tab
->mat
->row
[row
][0]);
260 isl_int_neg(bmap
->eq
[k
][1 + i
],
261 tab
->mat
->row
[row
][0]);
264 bmap
= isl_basic_map_gauss(bmap
, NULL
);
265 bmap
= isl_basic_map_normalize_constraints(bmap
);
266 bmap
= isl_basic_map_finalize(bmap
);
267 sol
->map
= isl_map_grow(sol
->map
, 1);
268 sol
->map
= isl_map_add(sol
->map
, bmap
);
273 isl_basic_map_free(bmap
);
278 static struct isl_sol
*sol_map_add_wrap(struct isl_sol
*sol
,
281 return (struct isl_sol
*)sol_map_add((struct isl_sol_map
*)sol
, tab
);
285 static struct isl_basic_set
*isl_basic_set_add_ineq(struct isl_basic_set
*bset
,
290 bset
= isl_basic_set_extend_constraints(bset
, 0, 1);
293 k
= isl_basic_set_alloc_inequality(bset
);
296 isl_seq_cpy(bset
->ineq
[k
], ineq
, 1 + isl_basic_set_total_dim(bset
));
299 isl_basic_set_free(bset
);
303 static struct isl_basic_set
*isl_basic_set_add_eq(struct isl_basic_set
*bset
,
308 bset
= isl_basic_set_extend_constraints(bset
, 1, 0);
311 k
= isl_basic_set_alloc_equality(bset
);
314 isl_seq_cpy(bset
->eq
[k
], eq
, 1 + isl_basic_set_total_dim(bset
));
317 isl_basic_set_free(bset
);
322 /* Store the "parametric constant" of row "row" of tableau "tab" in "line",
323 * i.e., the constant term and the coefficients of all variables that
324 * appear in the context tableau.
325 * Note that the coefficient of the big parameter M is NOT copied.
326 * The context tableau may not have a big parameter and even when it
327 * does, it is a different big parameter.
329 static void get_row_parameter_line(struct isl_tab
*tab
, int row
, isl_int
*line
)
332 unsigned off
= 2 + tab
->M
;
334 isl_int_set(line
[0], tab
->mat
->row
[row
][1]);
335 for (i
= 0; i
< tab
->n_param
; ++i
) {
336 if (tab
->var
[i
].is_row
)
337 isl_int_set_si(line
[1 + i
], 0);
339 int col
= tab
->var
[i
].index
;
340 isl_int_set(line
[1 + i
], tab
->mat
->row
[row
][off
+ col
]);
343 for (i
= 0; i
< tab
->n_div
; ++i
) {
344 if (tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_row
)
345 isl_int_set_si(line
[1 + tab
->n_param
+ i
], 0);
347 int col
= tab
->var
[tab
->n_var
- tab
->n_div
+ i
].index
;
348 isl_int_set(line
[1 + tab
->n_param
+ i
],
349 tab
->mat
->row
[row
][off
+ col
]);
354 /* Check if rows "row1" and "row2" have identical "parametric constants",
355 * as explained above.
356 * In this case, we also insist that the coefficients of the big parameter
357 * be the same as the values of the constants will only be the same
358 * if these coefficients are also the same.
360 static int identical_parameter_line(struct isl_tab
*tab
, int row1
, int row2
)
363 unsigned off
= 2 + tab
->M
;
365 if (isl_int_ne(tab
->mat
->row
[row1
][1], tab
->mat
->row
[row2
][1]))
368 if (tab
->M
&& isl_int_ne(tab
->mat
->row
[row1
][2],
369 tab
->mat
->row
[row2
][2]))
372 for (i
= 0; i
< tab
->n_param
+ tab
->n_div
; ++i
) {
373 int pos
= i
< tab
->n_param
? i
:
374 tab
->n_var
- tab
->n_div
+ i
- tab
->n_param
;
377 if (tab
->var
[pos
].is_row
)
379 col
= tab
->var
[pos
].index
;
380 if (isl_int_ne(tab
->mat
->row
[row1
][off
+ col
],
381 tab
->mat
->row
[row2
][off
+ col
]))
387 /* Return an inequality that expresses that the "parametric constant"
388 * should be non-negative.
389 * This function is only called when the coefficient of the big parameter
392 static struct isl_vec
*get_row_parameter_ineq(struct isl_tab
*tab
, int row
)
394 struct isl_vec
*ineq
;
396 ineq
= isl_vec_alloc(tab
->mat
->ctx
, 1 + tab
->n_param
+ tab
->n_div
);
400 get_row_parameter_line(tab
, row
, ineq
->el
);
402 ineq
= isl_vec_normalize(ineq
);
407 /* Return a integer division for use in a parametric cut based on the given row.
408 * In particular, let the parametric constant of the row be
412 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
413 * The div returned is equal to
415 * floor(\sum_i {-a_i} y_i) = floor((\sum_i (-a_i mod d) y_i)/d)
417 static struct isl_vec
*get_row_parameter_div(struct isl_tab
*tab
, int row
)
421 div
= isl_vec_alloc(tab
->mat
->ctx
, 1 + 1 + tab
->n_param
+ tab
->n_div
);
425 isl_int_set(div
->el
[0], tab
->mat
->row
[row
][0]);
426 get_row_parameter_line(tab
, row
, div
->el
+ 1);
427 div
= isl_vec_normalize(div
);
428 isl_seq_neg(div
->el
+ 1, div
->el
+ 1, div
->size
- 1);
429 isl_seq_fdiv_r(div
->el
+ 1, div
->el
+ 1, div
->el
[0], div
->size
- 1);
434 /* Return a integer division for use in transferring an integrality constraint
436 * In particular, let the parametric constant of the row be
440 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
441 * The the returned div is equal to
443 * floor(\sum_i {a_i} y_i) = floor((\sum_i (a_i mod d) y_i)/d)
445 static struct isl_vec
*get_row_split_div(struct isl_tab
*tab
, int row
)
449 div
= isl_vec_alloc(tab
->mat
->ctx
, 1 + 1 + tab
->n_param
+ tab
->n_div
);
453 isl_int_set(div
->el
[0], tab
->mat
->row
[row
][0]);
454 get_row_parameter_line(tab
, row
, div
->el
+ 1);
455 div
= isl_vec_normalize(div
);
456 isl_seq_fdiv_r(div
->el
+ 1, div
->el
+ 1, div
->el
[0], div
->size
- 1);
461 /* Construct and return an inequality that expresses an upper bound
463 * In particular, if the div is given by
467 * then the inequality expresses
471 static struct isl_vec
*ineq_for_div(struct isl_basic_set
*bset
, unsigned div
)
475 struct isl_vec
*ineq
;
477 total
= isl_basic_set_total_dim(bset
);
478 div_pos
= 1 + total
- bset
->n_div
+ div
;
480 ineq
= isl_vec_alloc(bset
->ctx
, 1 + total
);
484 isl_seq_cpy(ineq
->el
, bset
->div
[div
] + 1, 1 + total
);
485 isl_int_neg(ineq
->el
[div_pos
], bset
->div
[div
][0]);
489 /* Given a row in the tableau and a div that was created
490 * using get_row_split_div and that been constrained to equality, i.e.,
492 * d = floor(\sum_i {a_i} y_i) = \sum_i {a_i} y_i
494 * replace the expression "\sum_i {a_i} y_i" in the row by d,
495 * i.e., we subtract "\sum_i {a_i} y_i" and add 1 d.
496 * The coefficients of the non-parameters in the tableau have been
497 * verified to be integral. We can therefore simply replace coefficient b
498 * by floor(b). For the coefficients of the parameters we have
499 * floor(a_i) = a_i - {a_i}, while for the other coefficients, we have
502 static struct isl_tab
*set_row_cst_to_div(struct isl_tab
*tab
, int row
, int div
)
506 unsigned off
= 2 + tab
->M
;
508 isl_seq_fdiv_q(tab
->mat
->row
[row
] + 1, tab
->mat
->row
[row
] + 1,
509 tab
->mat
->row
[row
][0], 1 + tab
->M
+ tab
->n_col
);
511 isl_int_set_si(tab
->mat
->row
[row
][0], 1);
513 isl_assert(tab
->mat
->ctx
,
514 !tab
->var
[tab
->n_var
- tab
->n_div
+ div
].is_row
, goto error
);
516 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ div
].index
;
517 isl_int_set_si(tab
->mat
->row
[row
][off
+ col
], 1);
525 /* Check if the (parametric) constant of the given row is obviously
526 * negative, meaning that we don't need to consult the context tableau.
527 * If there is a big parameter and its coefficient is non-zero,
528 * then this coefficient determines the outcome.
529 * Otherwise, we check whether the constant is negative and
530 * all non-zero coefficients of parameters are negative and
531 * belong to non-negative parameters.
533 static int is_obviously_neg(struct isl_tab
*tab
, int row
)
537 unsigned off
= 2 + tab
->M
;
540 if (isl_int_is_pos(tab
->mat
->row
[row
][2]))
542 if (isl_int_is_neg(tab
->mat
->row
[row
][2]))
546 if (isl_int_is_nonneg(tab
->mat
->row
[row
][1]))
548 for (i
= 0; i
< tab
->n_param
; ++i
) {
549 /* Eliminated parameter */
550 if (tab
->var
[i
].is_row
)
552 col
= tab
->var
[i
].index
;
553 if (isl_int_is_zero(tab
->mat
->row
[row
][off
+ col
]))
555 if (!tab
->var
[i
].is_nonneg
)
557 if (isl_int_is_pos(tab
->mat
->row
[row
][off
+ col
]))
560 for (i
= 0; i
< tab
->n_div
; ++i
) {
561 if (tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_row
)
563 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ i
].index
;
564 if (isl_int_is_zero(tab
->mat
->row
[row
][off
+ col
]))
566 if (!tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_nonneg
)
568 if (isl_int_is_pos(tab
->mat
->row
[row
][off
+ col
]))
574 /* Check if the (parametric) constant of the given row is obviously
575 * non-negative, meaning that we don't need to consult the context tableau.
576 * If there is a big parameter and its coefficient is non-zero,
577 * then this coefficient determines the outcome.
578 * Otherwise, we check whether the constant is non-negative and
579 * all non-zero coefficients of parameters are positive and
580 * belong to non-negative parameters.
582 static int is_obviously_nonneg(struct isl_tab
*tab
, int row
)
586 unsigned off
= 2 + tab
->M
;
589 if (isl_int_is_pos(tab
->mat
->row
[row
][2]))
591 if (isl_int_is_neg(tab
->mat
->row
[row
][2]))
595 if (isl_int_is_neg(tab
->mat
->row
[row
][1]))
597 for (i
= 0; i
< tab
->n_param
; ++i
) {
598 /* Eliminated parameter */
599 if (tab
->var
[i
].is_row
)
601 col
= tab
->var
[i
].index
;
602 if (isl_int_is_zero(tab
->mat
->row
[row
][off
+ col
]))
604 if (!tab
->var
[i
].is_nonneg
)
606 if (isl_int_is_neg(tab
->mat
->row
[row
][off
+ col
]))
609 for (i
= 0; i
< tab
->n_div
; ++i
) {
610 if (tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_row
)
612 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ i
].index
;
613 if (isl_int_is_zero(tab
->mat
->row
[row
][off
+ col
]))
615 if (!tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_nonneg
)
617 if (isl_int_is_neg(tab
->mat
->row
[row
][off
+ col
]))
623 /* Given a row r and two columns, return the column that would
624 * lead to the lexicographically smallest increment in the sample
625 * solution when leaving the basis in favor of the row.
626 * Pivoting with column c will increment the sample value by a non-negative
627 * constant times a_{V,c}/a_{r,c}, with a_{V,c} the elements of column c
628 * corresponding to the non-parametric variables.
629 * If variable v appears in a column c_v, the a_{v,c} = 1 iff c = c_v,
630 * with all other entries in this virtual row equal to zero.
631 * If variable v appears in a row, then a_{v,c} is the element in column c
634 * Let v be the first variable with a_{v,c1}/a_{r,c1} != a_{v,c2}/a_{r,c2}.
635 * Then if a_{v,c1}/a_{r,c1} < a_{v,c2}/a_{r,c2}, i.e.,
636 * a_{v,c2} a_{r,c1} - a_{v,c1} a_{r,c2} > 0, c1 results in the minimal
637 * increment. Otherwise, it's c2.
639 static int lexmin_col_pair(struct isl_tab
*tab
,
640 int row
, int col1
, int col2
, isl_int tmp
)
645 tr
= tab
->mat
->row
[row
] + 2 + tab
->M
;
647 for (i
= tab
->n_param
; i
< tab
->n_var
- tab
->n_div
; ++i
) {
651 if (!tab
->var
[i
].is_row
) {
652 if (tab
->var
[i
].index
== col1
)
654 if (tab
->var
[i
].index
== col2
)
659 if (tab
->var
[i
].index
== row
)
662 r
= tab
->mat
->row
[tab
->var
[i
].index
] + 2 + tab
->M
;
663 s1
= isl_int_sgn(r
[col1
]);
664 s2
= isl_int_sgn(r
[col2
]);
665 if (s1
== 0 && s2
== 0)
672 isl_int_mul(tmp
, r
[col2
], tr
[col1
]);
673 isl_int_submul(tmp
, r
[col1
], tr
[col2
]);
674 if (isl_int_is_pos(tmp
))
676 if (isl_int_is_neg(tmp
))
682 /* Given a row in the tableau, find and return the column that would
683 * result in the lexicographically smallest, but positive, increment
684 * in the sample point.
685 * If there is no such column, then return tab->n_col.
686 * If anything goes wrong, return -1.
688 static int lexmin_pivot_col(struct isl_tab
*tab
, int row
)
691 int col
= tab
->n_col
;
695 tr
= tab
->mat
->row
[row
] + 2 + tab
->M
;
699 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
700 if (tab
->col_var
[j
] >= 0 &&
701 (tab
->col_var
[j
] < tab
->n_param
||
702 tab
->col_var
[j
] >= tab
->n_var
- tab
->n_div
))
705 if (!isl_int_is_pos(tr
[j
]))
708 if (col
== tab
->n_col
)
711 col
= lexmin_col_pair(tab
, row
, col
, j
, tmp
);
712 isl_assert(tab
->mat
->ctx
, col
>= 0, goto error
);
722 /* Return the first known violated constraint, i.e., a non-negative
723 * contraint that currently has an either obviously negative value
724 * or a previously determined to be negative value.
726 * If any constraint has a negative coefficient for the big parameter,
727 * if any, then we return one of these first.
729 static int first_neg(struct isl_tab
*tab
)
734 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
735 if (!isl_tab_var_from_row(tab
, row
)->is_nonneg
)
737 if (isl_int_is_neg(tab
->mat
->row
[row
][2]))
740 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
741 if (!isl_tab_var_from_row(tab
, row
)->is_nonneg
)
744 if (tab
->row_sign
[row
] == 0 &&
745 is_obviously_neg(tab
, row
))
746 tab
->row_sign
[row
] = isl_tab_row_neg
;
747 if (tab
->row_sign
[row
] != isl_tab_row_neg
)
749 } else if (!is_obviously_neg(tab
, row
))
756 /* Resolve all known or obviously violated constraints through pivoting.
757 * In particular, as long as we can find any violated constraint, we
758 * look for a pivoting column that would result in the lexicographicallly
759 * smallest increment in the sample point. If there is no such column
760 * then the tableau is infeasible.
762 static struct isl_tab
*restore_lexmin(struct isl_tab
*tab
)
770 while ((row
= first_neg(tab
)) != -1) {
771 col
= lexmin_pivot_col(tab
, row
);
772 if (col
>= tab
->n_col
)
773 return isl_tab_mark_empty(tab
);
776 isl_tab_pivot(tab
, row
, col
);
784 /* Given a row that represents an equality, look for an appropriate
786 * In particular, if there are any non-zero coefficients among
787 * the non-parameter variables, then we take the last of these
788 * variables. Eliminating this variable in terms of the other
789 * variables and/or parameters does not influence the property
790 * that all column in the initial tableau are lexicographically
791 * positive. The row corresponding to the eliminated variable
792 * will only have non-zero entries below the diagonal of the
793 * initial tableau. That is, we transform
799 * If there is no such non-parameter variable, then we are dealing with
800 * pure parameter equality and we pick any parameter with coefficient 1 or -1
801 * for elimination. This will ensure that the eliminated parameter
802 * always has an integer value whenever all the other parameters are integral.
803 * If there is no such parameter then we return -1.
805 static int last_var_col_or_int_par_col(struct isl_tab
*tab
, int row
)
807 unsigned off
= 2 + tab
->M
;
810 for (i
= tab
->n_var
- tab
->n_div
- 1; i
>= 0 && i
>= tab
->n_param
; --i
) {
812 if (tab
->var
[i
].is_row
)
814 col
= tab
->var
[i
].index
;
815 if (col
<= tab
->n_dead
)
817 if (!isl_int_is_zero(tab
->mat
->row
[row
][off
+ col
]))
820 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
821 if (isl_int_is_one(tab
->mat
->row
[row
][off
+ i
]))
823 if (isl_int_is_negone(tab
->mat
->row
[row
][off
+ i
]))
829 /* Add an equality that is known to be valid to the tableau.
830 * We first check if we can eliminate a variable or a parameter.
831 * If not, we add the equality as two inequalities.
832 * In this case, the equality was a pure parameter equality and there
833 * is no need to resolve any constraint violations.
835 static struct isl_tab
*add_lexmin_valid_eq(struct isl_tab
*tab
, isl_int
*eq
)
842 r
= isl_tab_add_row(tab
, eq
);
846 r
= tab
->con
[r
].index
;
847 i
= last_var_col_or_int_par_col(tab
, r
);
849 tab
->con
[r
].is_nonneg
= 1;
850 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
851 isl_seq_neg(eq
, eq
, 1 + tab
->n_var
);
852 r
= isl_tab_add_row(tab
, eq
);
855 tab
->con
[r
].is_nonneg
= 1;
856 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
858 isl_tab_pivot(tab
, r
, i
);
859 isl_tab_kill_col(tab
, i
);
862 tab
= restore_lexmin(tab
);
871 /* Check if the given row is a pure constant.
873 static int is_constant(struct isl_tab
*tab
, int row
)
875 unsigned off
= 2 + tab
->M
;
877 return isl_seq_first_non_zero(tab
->mat
->row
[row
] + off
+ tab
->n_dead
,
878 tab
->n_col
- tab
->n_dead
) == -1;
881 /* Add an equality that may or may not be valid to the tableau.
882 * If the resulting row is a pure constant, then it must be zero.
883 * Otherwise, the resulting tableau is empty.
885 * If the row is not a pure constant, then we add two inequalities,
886 * each time checking that they can be satisfied.
887 * In the end we try to use one of the two constraints to eliminate
890 static struct isl_tab
*add_lexmin_eq(struct isl_tab
*tab
, isl_int
*eq
)
899 tab
->bset
= isl_basic_set_add_eq(tab
->bset
, eq
);
900 isl_tab_push(tab
, isl_tab_undo_bset_eq
);
904 r1
= isl_tab_add_row(tab
, eq
);
907 tab
->con
[r1
].is_nonneg
= 1;
908 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r1
]);
910 row
= tab
->con
[r1
].index
;
911 if (is_constant(tab
, row
)) {
912 if (!isl_int_is_zero(tab
->mat
->row
[row
][1]) ||
913 (tab
->M
&& !isl_int_is_zero(tab
->mat
->row
[row
][2])))
914 return isl_tab_mark_empty(tab
);
918 tab
= restore_lexmin(tab
);
919 if (!tab
|| tab
->empty
)
922 isl_seq_neg(eq
, eq
, 1 + tab
->n_var
);
924 r2
= isl_tab_add_row(tab
, eq
);
927 tab
->con
[r2
].is_nonneg
= 1;
928 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r2
]);
930 tab
= restore_lexmin(tab
);
931 if (!tab
|| tab
->empty
)
934 if (!tab
->con
[r1
].is_row
)
935 isl_tab_kill_col(tab
, tab
->con
[r1
].index
);
936 else if (!tab
->con
[r2
].is_row
)
937 isl_tab_kill_col(tab
, tab
->con
[r2
].index
);
938 else if (isl_int_is_zero(tab
->mat
->row
[tab
->con
[r1
].index
][1])) {
939 unsigned off
= 2 + tab
->M
;
941 int row
= tab
->con
[r1
].index
;
942 i
= isl_seq_first_non_zero(tab
->mat
->row
[row
]+off
+tab
->n_dead
,
943 tab
->n_col
- tab
->n_dead
);
945 isl_tab_pivot(tab
, row
, tab
->n_dead
+ i
);
946 isl_tab_kill_col(tab
, tab
->n_dead
+ i
);
956 /* Add an inequality to the tableau, resolving violations using
959 static struct isl_tab
*add_lexmin_ineq(struct isl_tab
*tab
, isl_int
*ineq
)
967 tab
->bset
= isl_basic_set_add_ineq(tab
->bset
, ineq
);
968 isl_tab_push(tab
, isl_tab_undo_bset_ineq
);
972 r
= isl_tab_add_row(tab
, ineq
);
975 tab
->con
[r
].is_nonneg
= 1;
976 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
977 if (isl_tab_row_is_redundant(tab
, tab
->con
[r
].index
)) {
978 isl_tab_mark_redundant(tab
, tab
->con
[r
].index
);
982 tab
= restore_lexmin(tab
);
983 if (tab
&& !tab
->empty
&& tab
->con
[r
].is_row
&&
984 isl_tab_row_is_redundant(tab
, tab
->con
[r
].index
))
985 isl_tab_mark_redundant(tab
, tab
->con
[r
].index
);
992 /* Check if the coefficients of the parameters are all integral.
994 static int integer_parameter(struct isl_tab
*tab
, int row
)
998 unsigned off
= 2 + tab
->M
;
1000 for (i
= 0; i
< tab
->n_param
; ++i
) {
1001 /* Eliminated parameter */
1002 if (tab
->var
[i
].is_row
)
1004 col
= tab
->var
[i
].index
;
1005 if (!isl_int_is_divisible_by(tab
->mat
->row
[row
][off
+ col
],
1006 tab
->mat
->row
[row
][0]))
1009 for (i
= 0; i
< tab
->n_div
; ++i
) {
1010 if (tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_row
)
1012 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ i
].index
;
1013 if (!isl_int_is_divisible_by(tab
->mat
->row
[row
][off
+ col
],
1014 tab
->mat
->row
[row
][0]))
1020 /* Check if the coefficients of the non-parameter variables are all integral.
1022 static int integer_variable(struct isl_tab
*tab
, int row
)
1025 unsigned off
= 2 + tab
->M
;
1027 for (i
= 0; i
< tab
->n_col
; ++i
) {
1028 if (tab
->col_var
[i
] >= 0 &&
1029 (tab
->col_var
[i
] < tab
->n_param
||
1030 tab
->col_var
[i
] >= tab
->n_var
- tab
->n_div
))
1032 if (!isl_int_is_divisible_by(tab
->mat
->row
[row
][off
+ i
],
1033 tab
->mat
->row
[row
][0]))
1039 /* Check if the constant term is integral.
1041 static int integer_constant(struct isl_tab
*tab
, int row
)
1043 return isl_int_is_divisible_by(tab
->mat
->row
[row
][1],
1044 tab
->mat
->row
[row
][0]);
1047 #define I_CST 1 << 0
1048 #define I_PAR 1 << 1
1049 #define I_VAR 1 << 2
1051 /* Check for first (non-parameter) variable that is non-integer and
1052 * therefore requires a cut.
1053 * For parametric tableaus, there are three parts in a row,
1054 * the constant, the coefficients of the parameters and the rest.
1055 * For each part, we check whether the coefficients in that part
1056 * are all integral and if so, set the corresponding flag in *f.
1057 * If the constant and the parameter part are integral, then the
1058 * current sample value is integral and no cut is required
1059 * (irrespective of whether the variable part is integral).
1061 static int first_non_integer(struct isl_tab
*tab
, int *f
)
1065 for (i
= tab
->n_param
; i
< tab
->n_var
- tab
->n_div
; ++i
) {
1068 if (!tab
->var
[i
].is_row
)
1070 row
= tab
->var
[i
].index
;
1071 if (integer_constant(tab
, row
))
1072 ISL_FL_SET(flags
, I_CST
);
1073 if (integer_parameter(tab
, row
))
1074 ISL_FL_SET(flags
, I_PAR
);
1075 if (ISL_FL_ISSET(flags
, I_CST
) && ISL_FL_ISSET(flags
, I_PAR
))
1077 if (integer_variable(tab
, row
))
1078 ISL_FL_SET(flags
, I_VAR
);
1085 /* Add a (non-parametric) cut to cut away the non-integral sample
1086 * value of the given row.
1088 * If the row is given by
1090 * m r = f + \sum_i a_i y_i
1094 * c = - {-f/m} + \sum_i {a_i/m} y_i >= 0
1096 * The big parameter, if any, is ignored, since it is assumed to be big
1097 * enough to be divisible by any integer.
1098 * If the tableau is actually a parametric tableau, then this function
1099 * is only called when all coefficients of the parameters are integral.
1100 * The cut therefore has zero coefficients for the parameters.
1102 * The current value is known to be negative, so row_sign, if it
1103 * exists, is set accordingly.
1105 * Return the row of the cut or -1.
1107 static int add_cut(struct isl_tab
*tab
, int row
)
1112 unsigned off
= 2 + tab
->M
;
1114 if (isl_tab_extend_cons(tab
, 1) < 0)
1116 r
= isl_tab_allocate_con(tab
);
1120 r_row
= tab
->mat
->row
[tab
->con
[r
].index
];
1121 isl_int_set(r_row
[0], tab
->mat
->row
[row
][0]);
1122 isl_int_neg(r_row
[1], tab
->mat
->row
[row
][1]);
1123 isl_int_fdiv_r(r_row
[1], r_row
[1], tab
->mat
->row
[row
][0]);
1124 isl_int_neg(r_row
[1], r_row
[1]);
1126 isl_int_set_si(r_row
[2], 0);
1127 for (i
= 0; i
< tab
->n_col
; ++i
)
1128 isl_int_fdiv_r(r_row
[off
+ i
],
1129 tab
->mat
->row
[row
][off
+ i
], tab
->mat
->row
[row
][0]);
1131 tab
->con
[r
].is_nonneg
= 1;
1132 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1134 tab
->row_sign
[tab
->con
[r
].index
] = isl_tab_row_neg
;
1136 return tab
->con
[r
].index
;
1139 /* Given a non-parametric tableau, add cuts until an integer
1140 * sample point is obtained or until the tableau is determined
1141 * to be integer infeasible.
1142 * As long as there is any non-integer value in the sample point,
1143 * we add an appropriate cut, if possible and resolve the violated
1144 * cut constraint using restore_lexmin.
1145 * If one of the corresponding rows is equal to an integral
1146 * combination of variables/constraints plus a non-integral constant,
1147 * then there is no way to obtain an integer point an we return
1148 * a tableau that is marked empty.
1150 static struct isl_tab
*cut_to_integer_lexmin(struct isl_tab
*tab
)
1160 while ((row
= first_non_integer(tab
, &flags
)) != -1) {
1161 if (ISL_FL_ISSET(flags
, I_VAR
))
1162 return isl_tab_mark_empty(tab
);
1163 row
= add_cut(tab
, row
);
1166 tab
= restore_lexmin(tab
);
1167 if (!tab
|| tab
->empty
)
1176 static struct isl_tab
*drop_sample(struct isl_tab
*tab
, int s
)
1178 if (s
!= tab
->n_outside
)
1179 isl_mat_swap_rows(tab
->samples
, tab
->n_outside
, s
);
1181 isl_tab_push(tab
, isl_tab_undo_drop_sample
);
1186 /* Check whether all the currently active samples also satisfy the inequality
1187 * "ineq" (treated as an equality if eq is set).
1188 * Remove those samples that do not.
1190 static struct isl_tab
*check_samples(struct isl_tab
*tab
, isl_int
*ineq
, int eq
)
1198 isl_assert(tab
->mat
->ctx
, tab
->bset
, goto error
);
1199 isl_assert(tab
->mat
->ctx
, tab
->samples
, goto error
);
1200 isl_assert(tab
->mat
->ctx
, tab
->samples
->n_col
== 1 + tab
->n_var
, goto error
);
1203 for (i
= tab
->n_outside
; i
< tab
->n_sample
; ++i
) {
1205 isl_seq_inner_product(ineq
, tab
->samples
->row
[i
],
1206 1 + tab
->n_var
, &v
);
1207 sgn
= isl_int_sgn(v
);
1208 if (eq
? (sgn
== 0) : (sgn
>= 0))
1210 tab
= drop_sample(tab
, i
);
1222 /* Check whether the sample value of the tableau is finite,
1223 * i.e., either the tableau does not use a big parameter, or
1224 * all values of the variables are equal to the big parameter plus
1225 * some constant. This constant is the actual sample value.
1227 int sample_is_finite(struct isl_tab
*tab
)
1234 for (i
= 0; i
< tab
->n_var
; ++i
) {
1236 if (!tab
->var
[i
].is_row
)
1238 row
= tab
->var
[i
].index
;
1239 if (isl_int_ne(tab
->mat
->row
[row
][0], tab
->mat
->row
[row
][2]))
1245 /* Check if the context tableau of sol has any integer points.
1246 * Returns -1 if an error occurred.
1247 * If an integer point can be found and if moreover it is finite,
1248 * then it is added to the list of sample values.
1250 * This function is only called when none of the currently active sample
1251 * values satisfies the most recently added constraint.
1253 static int context_is_feasible(struct isl_sol
*sol
)
1255 struct isl_tab_undo
*snap
;
1256 struct isl_tab
*tab
;
1259 if (!sol
|| !sol
->context_tab
)
1262 snap
= isl_tab_snap(sol
->context_tab
);
1263 isl_tab_push_basis(sol
->context_tab
);
1265 sol
->context_tab
= cut_to_integer_lexmin(sol
->context_tab
);
1266 if (!sol
->context_tab
)
1269 tab
= sol
->context_tab
;
1270 if (!tab
->empty
&& sample_is_finite(tab
)) {
1271 struct isl_vec
*sample
;
1273 tab
->samples
= isl_mat_extend(tab
->samples
,
1274 tab
->n_sample
+ 1, tab
->samples
->n_col
);
1278 sample
= isl_tab_get_sample_value(tab
);
1281 isl_seq_cpy(tab
->samples
->row
[tab
->n_sample
],
1282 sample
->el
, sample
->size
);
1283 isl_vec_free(sample
);
1287 feasible
= !sol
->context_tab
->empty
;
1288 if (isl_tab_rollback(sol
->context_tab
, snap
) < 0)
1293 isl_tab_free(sol
->context_tab
);
1294 sol
->context_tab
= NULL
;
1298 /* First check if any of the currently active sample values satisfies
1299 * the inequality "ineq" (an equality if eq is set).
1300 * If not, continue with check_integer_feasible.
1302 static int context_valid_sample_or_feasible(struct isl_sol
*sol
,
1303 isl_int
*ineq
, int eq
)
1307 struct isl_tab
*tab
;
1309 if (!sol
|| !sol
->context_tab
)
1312 tab
= sol
->context_tab
;
1313 isl_assert(tab
->mat
->ctx
, tab
->bset
, return -1);
1314 isl_assert(tab
->mat
->ctx
, tab
->samples
, return -1);
1315 isl_assert(tab
->mat
->ctx
, tab
->samples
->n_col
== 1 + tab
->n_var
, return -1);
1318 for (i
= tab
->n_outside
; i
< tab
->n_sample
; ++i
) {
1320 isl_seq_inner_product(ineq
, tab
->samples
->row
[i
],
1321 1 + tab
->n_var
, &v
);
1322 sgn
= isl_int_sgn(v
);
1323 if (eq
? (sgn
== 0) : (sgn
>= 0))
1328 if (i
< tab
->n_sample
)
1331 return context_is_feasible(sol
);
1334 /* For a div d = floor(f/m), add the constraints
1337 * -(f-(m-1)) + m d >= 0
1339 * Note that the second constraint is the negation of
1343 static struct isl_tab
*add_div_constraints(struct isl_tab
*tab
, unsigned div
)
1348 struct isl_vec
*ineq
;
1353 total
= isl_basic_set_total_dim(tab
->bset
);
1354 div_pos
= 1 + total
- tab
->bset
->n_div
+ div
;
1356 ineq
= ineq_for_div(tab
->bset
, div
);
1360 tab
= add_lexmin_ineq(tab
, ineq
->el
);
1362 isl_seq_neg(ineq
->el
, tab
->bset
->div
[div
] + 1, 1 + total
);
1363 isl_int_set(ineq
->el
[div_pos
], tab
->bset
->div
[div
][0]);
1364 isl_int_add(ineq
->el
[0], ineq
->el
[0], ineq
->el
[div_pos
]);
1365 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
1366 tab
= add_lexmin_ineq(tab
, ineq
->el
);
1376 /* Add a div specified by "div" to both the main tableau and
1377 * the context tableau. In case of the main tableau, we only
1378 * need to add an extra div. In the context tableau, we also
1379 * need to express the meaning of the div.
1380 * Return the index of the div or -1 if anything went wrong.
1382 static int add_div(struct isl_tab
*tab
, struct isl_tab
**context_tab
,
1383 struct isl_vec
*div
)
1388 struct isl_mat
*samples
;
1390 if (isl_tab_extend_vars(*context_tab
, 1) < 0)
1392 r
= isl_tab_allocate_var(*context_tab
);
1395 (*context_tab
)->var
[r
].is_nonneg
= 1;
1396 (*context_tab
)->var
[r
].frozen
= 1;
1398 samples
= isl_mat_extend((*context_tab
)->samples
,
1399 (*context_tab
)->n_sample
, 1 + (*context_tab
)->n_var
);
1400 (*context_tab
)->samples
= samples
;
1403 for (i
= (*context_tab
)->n_outside
; i
< samples
->n_row
; ++i
) {
1404 isl_seq_inner_product(div
->el
+ 1, samples
->row
[i
],
1405 div
->size
- 1, &samples
->row
[i
][samples
->n_col
- 1]);
1406 isl_int_fdiv_q(samples
->row
[i
][samples
->n_col
- 1],
1407 samples
->row
[i
][samples
->n_col
- 1], div
->el
[0]);
1410 (*context_tab
)->bset
= isl_basic_set_extend_dim((*context_tab
)->bset
,
1411 isl_basic_set_get_dim((*context_tab
)->bset
), 1, 0, 2);
1412 k
= isl_basic_set_alloc_div((*context_tab
)->bset
);
1415 isl_seq_cpy((*context_tab
)->bset
->div
[k
], div
->el
, div
->size
);
1416 isl_tab_push((*context_tab
), isl_tab_undo_bset_div
);
1417 *context_tab
= add_div_constraints(*context_tab
, k
);
1421 if (isl_tab_extend_vars(tab
, 1) < 0)
1423 r
= isl_tab_allocate_var(tab
);
1426 if (!(*context_tab
)->M
)
1427 tab
->var
[r
].is_nonneg
= 1;
1428 tab
->var
[r
].frozen
= 1;
1431 return tab
->n_div
- 1;
1433 isl_tab_free(*context_tab
);
1434 *context_tab
= NULL
;
1438 static int find_div(struct isl_tab
*tab
, isl_int
*div
, isl_int denom
)
1441 unsigned total
= isl_basic_set_total_dim(tab
->bset
);
1443 for (i
= 0; i
< tab
->bset
->n_div
; ++i
) {
1444 if (isl_int_ne(tab
->bset
->div
[i
][0], denom
))
1446 if (!isl_seq_eq(tab
->bset
->div
[i
] + 1, div
, total
))
1453 /* Return the index of a div that corresponds to "div".
1454 * We first check if we already have such a div and if not, we create one.
1456 static int get_div(struct isl_tab
*tab
, struct isl_tab
**context_tab
,
1457 struct isl_vec
*div
)
1461 d
= find_div(*context_tab
, div
->el
+ 1, div
->el
[0]);
1465 return add_div(tab
, context_tab
, div
);
1468 /* Add a parametric cut to cut away the non-integral sample value
1470 * Let a_i be the coefficients of the constant term and the parameters
1471 * and let b_i be the coefficients of the variables or constraints
1472 * in basis of the tableau.
1473 * Let q be the div q = floor(\sum_i {-a_i} y_i).
1475 * The cut is expressed as
1477 * c = \sum_i -{-a_i} y_i + \sum_i {b_i} x_i + q >= 0
1479 * If q did not already exist in the context tableau, then it is added first.
1480 * If q is in a column of the main tableau then the "+ q" can be accomplished
1481 * by setting the corresponding entry to the denominator of the constraint.
1482 * If q happens to be in a row of the main tableau, then the corresponding
1483 * row needs to be added instead (taking care of the denominators).
1484 * Note that this is very unlikely, but perhaps not entirely impossible.
1486 * The current value of the cut is known to be negative (or at least
1487 * non-positive), so row_sign is set accordingly.
1489 * Return the row of the cut or -1.
1491 static int add_parametric_cut(struct isl_tab
*tab
, int row
,
1492 struct isl_tab
**context_tab
)
1494 struct isl_vec
*div
;
1500 unsigned off
= 2 + tab
->M
;
1505 if (isl_tab_extend_cons(*context_tab
, 3) < 0)
1508 div
= get_row_parameter_div(tab
, row
);
1512 d
= get_div(tab
, context_tab
, div
);
1516 if (isl_tab_extend_cons(tab
, 1) < 0)
1518 r
= isl_tab_allocate_con(tab
);
1522 r_row
= tab
->mat
->row
[tab
->con
[r
].index
];
1523 isl_int_set(r_row
[0], tab
->mat
->row
[row
][0]);
1524 isl_int_neg(r_row
[1], tab
->mat
->row
[row
][1]);
1525 isl_int_fdiv_r(r_row
[1], r_row
[1], tab
->mat
->row
[row
][0]);
1526 isl_int_neg(r_row
[1], r_row
[1]);
1528 isl_int_set_si(r_row
[2], 0);
1529 for (i
= 0; i
< tab
->n_param
; ++i
) {
1530 if (tab
->var
[i
].is_row
)
1532 col
= tab
->var
[i
].index
;
1533 isl_int_neg(r_row
[off
+ col
], tab
->mat
->row
[row
][off
+ col
]);
1534 isl_int_fdiv_r(r_row
[off
+ col
], r_row
[off
+ col
],
1535 tab
->mat
->row
[row
][0]);
1536 isl_int_neg(r_row
[off
+ col
], r_row
[off
+ col
]);
1538 for (i
= 0; i
< tab
->n_div
; ++i
) {
1539 if (tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_row
)
1541 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ i
].index
;
1542 isl_int_neg(r_row
[off
+ col
], tab
->mat
->row
[row
][off
+ col
]);
1543 isl_int_fdiv_r(r_row
[off
+ col
], r_row
[off
+ col
],
1544 tab
->mat
->row
[row
][0]);
1545 isl_int_neg(r_row
[off
+ col
], r_row
[off
+ col
]);
1547 for (i
= 0; i
< tab
->n_col
; ++i
) {
1548 if (tab
->col_var
[i
] >= 0 &&
1549 (tab
->col_var
[i
] < tab
->n_param
||
1550 tab
->col_var
[i
] >= tab
->n_var
- tab
->n_div
))
1552 isl_int_fdiv_r(r_row
[off
+ i
],
1553 tab
->mat
->row
[row
][off
+ i
], tab
->mat
->row
[row
][0]);
1555 if (tab
->var
[tab
->n_var
- tab
->n_div
+ d
].is_row
) {
1557 int d_row
= tab
->var
[tab
->n_var
- tab
->n_div
+ d
].index
;
1559 isl_int_gcd(gcd
, tab
->mat
->row
[d_row
][0], r_row
[0]);
1560 isl_int_divexact(r_row
[0], r_row
[0], gcd
);
1561 isl_int_divexact(gcd
, tab
->mat
->row
[d_row
][0], gcd
);
1562 isl_seq_combine(r_row
+ 1, gcd
, r_row
+ 1,
1563 r_row
[0], tab
->mat
->row
[d_row
] + 1,
1564 off
- 1 + tab
->n_col
);
1565 isl_int_mul(r_row
[0], r_row
[0], tab
->mat
->row
[d_row
][0]);
1568 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ d
].index
;
1569 isl_int_set(r_row
[off
+ col
], tab
->mat
->row
[row
][0]);
1572 tab
->con
[r
].is_nonneg
= 1;
1573 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1575 tab
->row_sign
[tab
->con
[r
].index
] = isl_tab_row_neg
;
1579 return tab
->con
[r
].index
;
1581 isl_tab_free(*context_tab
);
1582 *context_tab
= NULL
;
1586 /* Construct a tableau for bmap that can be used for computing
1587 * the lexicographic minimum (or maximum) of bmap.
1588 * If not NULL, then dom is the domain where the minimum
1589 * should be computed. In this case, we set up a parametric
1590 * tableau with row signs (initialized to "unknown").
1591 * If M is set, then the tableau will use a big parameter.
1592 * If max is set, then a maximum should be computed instead of a minimum.
1593 * This means that for each variable x, the tableau will contain the variable
1594 * x' = M - x, rather than x' = M + x. This in turn means that the coefficient
1595 * of the variables in all constraints are negated prior to adding them
1598 static struct isl_tab
*tab_for_lexmin(struct isl_basic_map
*bmap
,
1599 struct isl_basic_set
*dom
, unsigned M
, int max
)
1602 struct isl_tab
*tab
;
1604 tab
= isl_tab_alloc(bmap
->ctx
, 2 * bmap
->n_eq
+ bmap
->n_ineq
+ 1,
1605 isl_basic_map_total_dim(bmap
), M
);
1609 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1611 tab
->n_param
= isl_basic_set_total_dim(dom
) - dom
->n_div
;
1612 tab
->n_div
= dom
->n_div
;
1613 tab
->row_sign
= isl_calloc_array(bmap
->ctx
,
1614 enum isl_tab_row_sign
, tab
->mat
->n_row
);
1618 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
))
1619 return isl_tab_mark_empty(tab
);
1621 for (i
= tab
->n_param
; i
< tab
->n_var
- tab
->n_div
; ++i
) {
1622 tab
->var
[i
].is_nonneg
= 1;
1623 tab
->var
[i
].frozen
= 1;
1625 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1627 isl_seq_neg(bmap
->eq
[i
] + 1 + tab
->n_param
,
1628 bmap
->eq
[i
] + 1 + tab
->n_param
,
1629 tab
->n_var
- tab
->n_param
- tab
->n_div
);
1630 tab
= add_lexmin_valid_eq(tab
, bmap
->eq
[i
]);
1632 isl_seq_neg(bmap
->eq
[i
] + 1 + tab
->n_param
,
1633 bmap
->eq
[i
] + 1 + tab
->n_param
,
1634 tab
->n_var
- tab
->n_param
- tab
->n_div
);
1635 if (!tab
|| tab
->empty
)
1638 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1640 isl_seq_neg(bmap
->ineq
[i
] + 1 + tab
->n_param
,
1641 bmap
->ineq
[i
] + 1 + tab
->n_param
,
1642 tab
->n_var
- tab
->n_param
- tab
->n_div
);
1643 tab
= add_lexmin_ineq(tab
, bmap
->ineq
[i
]);
1645 isl_seq_neg(bmap
->ineq
[i
] + 1 + tab
->n_param
,
1646 bmap
->ineq
[i
] + 1 + tab
->n_param
,
1647 tab
->n_var
- tab
->n_param
- tab
->n_div
);
1648 if (!tab
|| tab
->empty
)
1657 static struct isl_tab
*context_tab_for_lexmin(struct isl_basic_set
*bset
)
1659 struct isl_tab
*tab
;
1661 bset
= isl_basic_set_cow(bset
);
1664 tab
= tab_for_lexmin((struct isl_basic_map
*)bset
, NULL
, 1, 0);
1670 tab
->samples
= isl_mat_alloc(bset
->ctx
, 1, 1 + tab
->n_var
);
1675 isl_basic_set_free(bset
);
1679 /* Construct an isl_sol_map structure for accumulating the solution.
1680 * If track_empty is set, then we also keep track of the parts
1681 * of the context where there is no solution.
1682 * If max is set, then we are solving a maximization, rather than
1683 * a minimization problem, which means that the variables in the
1684 * tableau have value "M - x" rather than "M + x".
1686 static struct isl_sol_map
*sol_map_init(struct isl_basic_map
*bmap
,
1687 struct isl_basic_set
*dom
, int track_empty
, int max
)
1689 struct isl_sol_map
*sol_map
;
1690 struct isl_tab
*context_tab
;
1693 sol_map
= isl_calloc_type(bset
->ctx
, struct isl_sol_map
);
1698 sol_map
->sol
.add
= &sol_map_add_wrap
;
1699 sol_map
->sol
.free
= &sol_map_free_wrap
;
1700 sol_map
->map
= isl_map_alloc_dim(isl_basic_map_get_dim(bmap
), 1,
1705 context_tab
= context_tab_for_lexmin(isl_basic_set_copy(dom
));
1706 context_tab
= restore_lexmin(context_tab
);
1707 sol_map
->sol
.context_tab
= context_tab
;
1708 f
= context_is_feasible(&sol_map
->sol
);
1713 sol_map
->empty
= isl_set_alloc_dim(isl_basic_set_get_dim(dom
),
1714 1, ISL_SET_DISJOINT
);
1715 if (!sol_map
->empty
)
1719 isl_basic_set_free(dom
);
1722 isl_basic_set_free(dom
);
1723 sol_map_free(sol_map
);
1727 /* For each variable in the context tableau, check if the variable can
1728 * only attain non-negative values. If so, mark the parameter as non-negative
1729 * in the main tableau. This allows for a more direct identification of some
1730 * cases of violated constraints.
1732 static struct isl_tab
*tab_detect_nonnegative_parameters(struct isl_tab
*tab
,
1733 struct isl_tab
*context_tab
)
1736 struct isl_tab_undo
*snap
, *snap2
;
1737 struct isl_vec
*ineq
= NULL
;
1738 struct isl_tab_var
*var
;
1741 if (context_tab
->n_var
== 0)
1744 ineq
= isl_vec_alloc(tab
->mat
->ctx
, 1 + context_tab
->n_var
);
1748 if (isl_tab_extend_cons(context_tab
, 1) < 0)
1751 snap
= isl_tab_snap(context_tab
);
1752 isl_tab_push_basis(context_tab
);
1754 snap2
= isl_tab_snap(context_tab
);
1757 isl_seq_clr(ineq
->el
, ineq
->size
);
1758 for (i
= 0; i
< context_tab
->n_var
; ++i
) {
1759 isl_int_set_si(ineq
->el
[1 + i
], 1);
1760 context_tab
= isl_tab_add_ineq(context_tab
, ineq
->el
);
1761 var
= &context_tab
->con
[context_tab
->n_con
- 1];
1762 if (!context_tab
->empty
&&
1763 !isl_tab_min_at_most_neg_one(context_tab
, var
)) {
1765 if (i
>= tab
->n_param
)
1766 j
= i
- tab
->n_param
+ tab
->n_var
- tab
->n_div
;
1767 tab
->var
[j
].is_nonneg
= 1;
1770 isl_int_set_si(ineq
->el
[1 + i
], 0);
1771 if (isl_tab_rollback(context_tab
, snap2
) < 0)
1775 if (isl_tab_rollback(context_tab
, snap
) < 0)
1778 if (n
== context_tab
->n_var
) {
1779 context_tab
->mat
= isl_mat_drop_cols(context_tab
->mat
, 2, 1);
1791 /* Check whether all coefficients of (non-parameter) variables
1792 * are non-positive, meaning that no pivots can be performed on the row.
1794 static int is_critical(struct isl_tab
*tab
, int row
)
1797 unsigned off
= 2 + tab
->M
;
1799 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
1800 if (tab
->col_var
[j
] >= 0 &&
1801 (tab
->col_var
[j
] < tab
->n_param
||
1802 tab
->col_var
[j
] >= tab
->n_var
- tab
->n_div
))
1805 if (isl_int_is_pos(tab
->mat
->row
[row
][off
+ j
]))
1812 /* Check whether the inequality represented by vec is strict over the integers,
1813 * i.e., there are no integer values satisfying the constraint with
1814 * equality. This happens if the gcd of the coefficients is not a divisor
1815 * of the constant term. If so, scale the constraint down by the gcd
1816 * of the coefficients.
1818 static int is_strict(struct isl_vec
*vec
)
1824 isl_seq_gcd(vec
->el
+ 1, vec
->size
- 1, &gcd
);
1825 if (!isl_int_is_one(gcd
)) {
1826 strict
= !isl_int_is_divisible_by(vec
->el
[0], gcd
);
1827 isl_int_fdiv_q(vec
->el
[0], vec
->el
[0], gcd
);
1828 isl_seq_scale_down(vec
->el
+ 1, vec
->el
+ 1, gcd
, vec
->size
-1);
1835 /* Determine the sign of the given row of the main tableau.
1836 * The result is one of
1837 * isl_tab_row_pos: always non-negative; no pivot needed
1838 * isl_tab_row_neg: always non-positive; pivot
1839 * isl_tab_row_any: can be both positive and negative; split
1841 * We first handle some simple cases
1842 * - the row sign may be known already
1843 * - the row may be obviously non-negative
1844 * - the parametric constant may be equal to that of another row
1845 * for which we know the sign. This sign will be either "pos" or
1846 * "any". If it had been "neg" then we would have pivoted before.
1848 * If none of these cases hold, we check the value of the row for each
1849 * of the currently active samples. Based on the signs of these values
1850 * we make an initial determination of the sign of the row.
1852 * all zero -> unk(nown)
1853 * all non-negative -> pos
1854 * all non-positive -> neg
1855 * both negative and positive -> all
1857 * If we end up with "all", we are done.
1858 * Otherwise, we perform a check for positive and/or negative
1859 * values as follows.
1861 * samples neg unk pos
1867 * There is no special sign for "zero", because we can usually treat zero
1868 * as either non-negative or non-positive, whatever works out best.
1869 * However, if the row is "critical", meaning that pivoting is impossible
1870 * then we don't want to limp zero with the non-positive case, because
1871 * then we we would lose the solution for those values of the parameters
1872 * where the value of the row is zero. Instead, we treat 0 as non-negative
1873 * ensuring a split if the row can attain both zero and negative values.
1874 * The same happens when the original constraint was one that could not
1875 * be satisfied with equality by any integer values of the parameters.
1876 * In this case, we normalize the constraint, but then a value of zero
1877 * for the normalized constraint is actually a positive value for the
1878 * original constraint, so again we need to treat zero as non-negative.
1879 * In both these cases, we have the following decision tree instead:
1881 * all non-negative -> pos
1882 * all negative -> neg
1883 * both negative and non-negative -> all
1891 static int row_sign(struct isl_tab
*tab
, struct isl_sol
*sol
, int row
)
1894 struct isl_tab_undo
*snap
= NULL
;
1895 struct isl_vec
*ineq
= NULL
;
1896 int res
= isl_tab_row_unknown
;
1904 struct isl_tab
*context_tab
= sol
->context_tab
;
1906 if (tab
->row_sign
[row
] != isl_tab_row_unknown
)
1907 return tab
->row_sign
[row
];
1908 if (is_obviously_nonneg(tab
, row
))
1909 return isl_tab_row_pos
;
1910 for (row2
= tab
->n_redundant
; row2
< tab
->n_row
; ++row2
) {
1911 if (tab
->row_sign
[row2
] == isl_tab_row_unknown
)
1913 if (identical_parameter_line(tab
, row
, row2
))
1914 return tab
->row_sign
[row2
];
1917 critical
= is_critical(tab
, row
);
1919 isl_assert(tab
->mat
->ctx
, context_tab
->samples
, goto error
);
1920 isl_assert(tab
->mat
->ctx
, context_tab
->samples
->n_col
== 1 + context_tab
->n_var
, goto error
);
1922 ineq
= get_row_parameter_ineq(tab
, row
);
1926 strict
= is_strict(ineq
);
1929 for (i
= context_tab
->n_outside
; i
< context_tab
->n_sample
; ++i
) {
1930 isl_seq_inner_product(context_tab
->samples
->row
[i
], ineq
->el
,
1932 sgn
= isl_int_sgn(tmp
);
1933 if (sgn
> 0 || (sgn
== 0 && (critical
|| strict
))) {
1934 if (res
== isl_tab_row_unknown
)
1935 res
= isl_tab_row_pos
;
1936 if (res
== isl_tab_row_neg
)
1937 res
= isl_tab_row_any
;
1940 if (res
== isl_tab_row_unknown
)
1941 res
= isl_tab_row_neg
;
1942 if (res
== isl_tab_row_pos
)
1943 res
= isl_tab_row_any
;
1945 if (res
== isl_tab_row_any
)
1950 if (res
!= isl_tab_row_any
) {
1951 if (isl_tab_extend_cons(context_tab
, 1) < 0)
1954 snap
= isl_tab_snap(context_tab
);
1955 isl_tab_push_basis(context_tab
);
1958 if (res
== isl_tab_row_unknown
|| res
== isl_tab_row_pos
) {
1959 /* test for negative values */
1961 isl_seq_neg(ineq
->el
, ineq
->el
, ineq
->size
);
1962 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
1964 isl_tab_push_basis(context_tab
);
1965 sol
->context_tab
= add_lexmin_ineq(sol
->context_tab
, ineq
->el
);
1966 feasible
= context_is_feasible(sol
);
1969 context_tab
= sol
->context_tab
;
1971 res
= isl_tab_row_pos
;
1973 res
= (res
== isl_tab_row_unknown
) ? isl_tab_row_neg
1975 if (isl_tab_rollback(context_tab
, snap
) < 0)
1978 if (res
== isl_tab_row_neg
) {
1979 isl_seq_neg(ineq
->el
, ineq
->el
, ineq
->size
);
1980 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
1984 if (res
== isl_tab_row_neg
) {
1985 /* test for positive values */
1987 if (!critical
&& !strict
)
1988 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
1990 isl_tab_push_basis(context_tab
);
1991 sol
->context_tab
= add_lexmin_ineq(sol
->context_tab
, ineq
->el
);
1992 feasible
= context_is_feasible(sol
);
1995 context_tab
= sol
->context_tab
;
1997 res
= isl_tab_row_any
;
1998 if (isl_tab_rollback(context_tab
, snap
) < 0)
2009 static struct isl_sol
*find_solutions(struct isl_sol
*sol
, struct isl_tab
*tab
);
2011 /* Find solutions for values of the parameters that satisfy the given
2014 * We currently take a snapshot of the context tableau that is reset
2015 * when we return from this function, while we make a copy of the main
2016 * tableau, leaving the original main tableau untouched.
2017 * These are fairly arbitrary choices. Making a copy also of the context
2018 * tableau would obviate the need to undo any changes made to it later,
2019 * while taking a snapshot of the main tableau could reduce memory usage.
2020 * If we were to switch to taking a snapshot of the main tableau,
2021 * we would have to keep in mind that we need to save the row signs
2022 * and that we need to do this before saving the current basis
2023 * such that the basis has been restore before we restore the row signs.
2025 static struct isl_sol
*find_in_pos(struct isl_sol
*sol
,
2026 struct isl_tab
*tab
, isl_int
*ineq
)
2028 struct isl_tab_undo
*snap
;
2030 snap
= isl_tab_snap(sol
->context_tab
);
2031 isl_tab_push_basis(sol
->context_tab
);
2032 if (isl_tab_extend_cons(sol
->context_tab
, 1) < 0)
2035 tab
= isl_tab_dup(tab
);
2039 sol
->context_tab
= add_lexmin_ineq(sol
->context_tab
, ineq
);
2040 sol
->context_tab
= check_samples(sol
->context_tab
, ineq
, 0);
2042 sol
= find_solutions(sol
, tab
);
2044 isl_tab_rollback(sol
->context_tab
, snap
);
2047 isl_tab_rollback(sol
->context_tab
, snap
);
2052 /* Record the absence of solutions for those values of the parameters
2053 * that do not satisfy the given inequality with equality.
2055 static struct isl_sol
*no_sol_in_strict(struct isl_sol
*sol
,
2056 struct isl_tab
*tab
, struct isl_vec
*ineq
)
2060 struct isl_tab_undo
*snap
;
2061 snap
= isl_tab_snap(sol
->context_tab
);
2062 isl_tab_push_basis(sol
->context_tab
);
2063 if (isl_tab_extend_cons(sol
->context_tab
, 1) < 0)
2066 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
2068 sol
->context_tab
= add_lexmin_ineq(sol
->context_tab
, ineq
->el
);
2069 f
= context_valid_sample_or_feasible(sol
, ineq
->el
, 0);
2075 sol
= sol
->add(sol
, tab
);
2078 isl_int_add_ui(ineq
->el
[0], ineq
->el
[0], 1);
2080 if (isl_tab_rollback(sol
->context_tab
, snap
) < 0)
2088 /* Given a main tableau where more than one row requires a split,
2089 * determine and return the "best" row to split on.
2091 * Given two rows in the main tableau, if the inequality corresponding
2092 * to the first row is redundant with respect to that of the second row
2093 * in the current tableau, then it is better to split on the second row,
2094 * since in the positive part, both row will be positive.
2095 * (In the negative part a pivot will have to be performed and just about
2096 * anything can happen to the sign of the other row.)
2098 * As a simple heuristic, we therefore select the row that makes the most
2099 * of the other rows redundant.
2101 * Perhaps it would also be useful to look at the number of constraints
2102 * that conflict with any given constraint.
2104 static int best_split(struct isl_tab
*tab
, struct isl_tab
*context_tab
)
2106 struct isl_tab_undo
*snap
, *snap2
;
2112 if (isl_tab_extend_cons(context_tab
, 2) < 0)
2115 snap
= isl_tab_snap(context_tab
);
2116 isl_tab_push_basis(context_tab
);
2117 snap2
= isl_tab_snap(context_tab
);
2119 for (split
= tab
->n_redundant
; split
< tab
->n_row
; ++split
) {
2120 struct isl_tab_undo
*snap3
;
2121 struct isl_vec
*ineq
= NULL
;
2124 if (!isl_tab_var_from_row(tab
, split
)->is_nonneg
)
2126 if (tab
->row_sign
[split
] != isl_tab_row_any
)
2129 ineq
= get_row_parameter_ineq(tab
, split
);
2132 context_tab
= isl_tab_add_ineq(context_tab
, ineq
->el
);
2135 snap3
= isl_tab_snap(context_tab
);
2137 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
2138 struct isl_tab_var
*var
;
2142 if (!isl_tab_var_from_row(tab
, row
)->is_nonneg
)
2144 if (tab
->row_sign
[row
] != isl_tab_row_any
)
2147 ineq
= get_row_parameter_ineq(tab
, row
);
2150 context_tab
= isl_tab_add_ineq(context_tab
, ineq
->el
);
2152 var
= &context_tab
->con
[context_tab
->n_con
- 1];
2153 if (!context_tab
->empty
&&
2154 !isl_tab_min_at_most_neg_one(context_tab
, var
))
2156 if (isl_tab_rollback(context_tab
, snap3
) < 0)
2159 if (best
== -1 || r
> best_r
) {
2163 if (isl_tab_rollback(context_tab
, snap2
) < 0)
2167 if (isl_tab_rollback(context_tab
, snap
) < 0)
2173 /* Compute the lexicographic minimum of the set represented by the main
2174 * tableau "tab" within the context "sol->context_tab".
2175 * On entry the sample value of the main tableau is lexicographically
2176 * less than or equal to this lexicographic minimum.
2177 * Pivots are performed until a feasible point is found, which is then
2178 * necessarily equal to the minimum, or until the tableau is found to
2179 * be infeasible. Some pivots may need to be performed for only some
2180 * feasible values of the context tableau. If so, the context tableau
2181 * is split into a part where the pivot is needed and a part where it is not.
2183 * Whenever we enter the main loop, the main tableau is such that no
2184 * "obvious" pivots need to be performed on it, where "obvious" means
2185 * that the given row can be seen to be negative without looking at
2186 * the context tableau. In particular, for non-parametric problems,
2187 * no pivots need to be performed on the main tableau.
2188 * The caller of find_solutions is responsible for making this property
2189 * hold prior to the first iteration of the loop, while restore_lexmin
2190 * is called before every other iteration.
2192 * Inside the main loop, we first examine the signs of the rows of
2193 * the main tableau within the context of the context tableau.
2194 * If we find a row that is always non-positive for all values of
2195 * the parameters satisfying the context tableau and negative for at
2196 * least one value of the parameters, we perform the appropriate pivot
2197 * and start over. An exception is the case where no pivot can be
2198 * performed on the row. In this case, we require that the sign of
2199 * the row is negative for all values of the parameters (rather than just
2200 * non-positive). This special case is handled inside row_sign, which
2201 * will say that the row can have any sign if it determines that it can
2202 * attain both negative and zero values.
2204 * If we can't find a row that always requires a pivot, but we can find
2205 * one or more rows that require a pivot for some values of the parameters
2206 * (i.e., the row can attain both positive and negative signs), then we split
2207 * the context tableau into two parts, one where we force the sign to be
2208 * non-negative and one where we force is to be negative.
2209 * The non-negative part is handled by a recursive call (through find_in_pos).
2210 * Upon returning from this call, we continue with the negative part and
2211 * perform the required pivot.
2213 * If no such rows can be found, all rows are non-negative and we have
2214 * found a (rational) feasible point. If we only wanted a rational point
2216 * Otherwise, we check if all values of the sample point of the tableau
2217 * are integral for the variables. If so, we have found the minimal
2218 * integral point and we are done.
2219 * If the sample point is not integral, then we need to make a distinction
2220 * based on whether the constant term is non-integral or the coefficients
2221 * of the parameters. Furthermore, in order to decide how to handle
2222 * the non-integrality, we also need to know whether the coefficients
2223 * of the other columns in the tableau are integral. This leads
2224 * to the following table. The first two rows do not correspond
2225 * to a non-integral sample point and are only mentioned for completeness.
2227 * constant parameters other
2230 * int int rat | -> no problem
2232 * rat int int -> fail
2234 * rat int rat -> cut
2237 * rat rat rat | -> parametric cut
2240 * rat rat int | -> split context
2242 * If the parametric constant is completely integral, then there is nothing
2243 * to be done. If the constant term is non-integral, but all the other
2244 * coefficient are integral, then there is nothing that can be done
2245 * and the tableau has no integral solution.
2246 * If, on the other hand, one or more of the other columns have rational
2247 * coeffcients, but the parameter coefficients are all integral, then
2248 * we can perform a regular (non-parametric) cut.
2249 * Finally, if there is any parameter coefficient that is non-integral,
2250 * then we need to involve the context tableau. There are two cases here.
2251 * If at least one other column has a rational coefficient, then we
2252 * can perform a parametric cut in the main tableau by adding a new
2253 * integer division in the context tableau.
2254 * If all other columns have integral coefficients, then we need to
2255 * enforce that the rational combination of parameters (c + \sum a_i y_i)/m
2256 * is always integral. We do this by introducing an integer division
2257 * q = floor((c + \sum a_i y_i)/m) and stipulating that its argument should
2258 * always be integral in the context tableau, i.e., m q = c + \sum a_i y_i.
2259 * Since q is expressed in the tableau as
2260 * c + \sum a_i y_i - m q >= 0
2261 * -c - \sum a_i y_i + m q + m - 1 >= 0
2262 * it is sufficient to add the inequality
2263 * -c - \sum a_i y_i + m q >= 0
2264 * In the part of the context where this inequality does not hold, the
2265 * main tableau is marked as being empty.
2267 static struct isl_sol
*find_solutions(struct isl_sol
*sol
, struct isl_tab
*tab
)
2269 struct isl_tab
**context_tab
;
2274 context_tab
= &sol
->context_tab
;
2278 if ((*context_tab
)->empty
)
2281 for (; tab
&& !tab
->empty
; tab
= restore_lexmin(tab
)) {
2288 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
2289 if (!isl_tab_var_from_row(tab
, row
)->is_nonneg
)
2291 sgn
= row_sign(tab
, sol
, row
);
2294 tab
->row_sign
[row
] = sgn
;
2295 if (sgn
== isl_tab_row_any
)
2297 if (sgn
== isl_tab_row_any
&& split
== -1)
2299 if (sgn
== isl_tab_row_neg
)
2302 if (row
< tab
->n_row
)
2305 struct isl_vec
*ineq
;
2307 split
= best_split(tab
, *context_tab
);
2310 ineq
= get_row_parameter_ineq(tab
, split
);
2314 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
2315 if (!isl_tab_var_from_row(tab
, row
)->is_nonneg
)
2317 if (tab
->row_sign
[row
] == isl_tab_row_any
)
2318 tab
->row_sign
[row
] = isl_tab_row_unknown
;
2320 tab
->row_sign
[split
] = isl_tab_row_pos
;
2321 sol
= find_in_pos(sol
, tab
, ineq
->el
);
2322 tab
->row_sign
[split
] = isl_tab_row_neg
;
2324 isl_seq_neg(ineq
->el
, ineq
->el
, ineq
->size
);
2325 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
2326 *context_tab
= add_lexmin_ineq(*context_tab
, ineq
->el
);
2327 *context_tab
= check_samples(*context_tab
, ineq
->el
, 0);
2335 row
= first_non_integer(tab
, &flags
);
2338 if (ISL_FL_ISSET(flags
, I_PAR
)) {
2339 if (ISL_FL_ISSET(flags
, I_VAR
)) {
2340 tab
= isl_tab_mark_empty(tab
);
2343 row
= add_cut(tab
, row
);
2344 } else if (ISL_FL_ISSET(flags
, I_VAR
)) {
2345 struct isl_vec
*div
;
2346 struct isl_vec
*ineq
;
2348 if (isl_tab_extend_cons(*context_tab
, 3) < 0)
2350 div
= get_row_split_div(tab
, row
);
2353 d
= get_div(tab
, context_tab
, div
);
2357 ineq
= ineq_for_div((*context_tab
)->bset
, d
);
2358 sol
= no_sol_in_strict(sol
, tab
, ineq
);
2359 isl_seq_neg(ineq
->el
, ineq
->el
, ineq
->size
);
2360 *context_tab
= add_lexmin_ineq(*context_tab
, ineq
->el
);
2361 *context_tab
= check_samples(*context_tab
, ineq
->el
, 0);
2365 tab
= set_row_cst_to_div(tab
, row
, d
);
2367 row
= add_parametric_cut(tab
, row
, context_tab
);
2372 sol
= sol
->add(sol
, tab
);
2381 /* Compute the lexicographic minimum of the set represented by the main
2382 * tableau "tab" within the context "sol->context_tab".
2384 * As a preprocessing step, we first transfer all the purely parametric
2385 * equalities from the main tableau to the context tableau, i.e.,
2386 * parameters that have been pivoted to a row.
2387 * These equalities are ignored by the main algorithm, because the
2388 * corresponding rows may not be marked as being non-negative.
2389 * In parts of the context where the added equality does not hold,
2390 * the main tableau is marked as being empty.
2392 static struct isl_sol
*find_solutions_main(struct isl_sol
*sol
,
2393 struct isl_tab
*tab
)
2397 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
2401 if (tab
->row_var
[row
] < 0)
2403 if (tab
->row_var
[row
] >= tab
->n_param
&&
2404 tab
->row_var
[row
] < tab
->n_var
- tab
->n_div
)
2406 if (tab
->row_var
[row
] < tab
->n_param
)
2407 p
= tab
->row_var
[row
];
2409 p
= tab
->row_var
[row
]
2410 + tab
->n_param
- (tab
->n_var
- tab
->n_div
);
2412 if (isl_tab_extend_cons(sol
->context_tab
, 2) < 0)
2415 eq
= isl_vec_alloc(tab
->mat
->ctx
, 1+tab
->n_param
+tab
->n_div
);
2416 get_row_parameter_line(tab
, row
, eq
->el
);
2417 isl_int_neg(eq
->el
[1 + p
], tab
->mat
->row
[row
][0]);
2418 eq
= isl_vec_normalize(eq
);
2420 sol
= no_sol_in_strict(sol
, tab
, eq
);
2422 isl_seq_neg(eq
->el
, eq
->el
, eq
->size
);
2423 sol
= no_sol_in_strict(sol
, tab
, eq
);
2424 isl_seq_neg(eq
->el
, eq
->el
, eq
->size
);
2426 sol
->context_tab
= add_lexmin_eq(sol
->context_tab
, eq
->el
);
2427 context_valid_sample_or_feasible(sol
, eq
->el
, 1);
2428 sol
->context_tab
= check_samples(sol
->context_tab
, eq
->el
, 1);
2432 isl_tab_mark_redundant(tab
, row
);
2434 if (!sol
->context_tab
)
2436 if (sol
->context_tab
->empty
)
2439 row
= tab
->n_redundant
- 1;
2442 return find_solutions(sol
, tab
);
2449 static struct isl_sol_map
*sol_map_find_solutions(struct isl_sol_map
*sol_map
,
2450 struct isl_tab
*tab
)
2452 return (struct isl_sol_map
*)find_solutions_main(&sol_map
->sol
, tab
);
2455 /* Check if integer division "div" of "dom" also occurs in "bmap".
2456 * If so, return its position within the divs.
2457 * If not, return -1.
2459 static int find_context_div(struct isl_basic_map
*bmap
,
2460 struct isl_basic_set
*dom
, unsigned div
)
2463 unsigned b_dim
= isl_dim_total(bmap
->dim
);
2464 unsigned d_dim
= isl_dim_total(dom
->dim
);
2466 if (isl_int_is_zero(dom
->div
[div
][0]))
2468 if (isl_seq_first_non_zero(dom
->div
[div
] + 2 + d_dim
, dom
->n_div
) != -1)
2471 for (i
= 0; i
< bmap
->n_div
; ++i
) {
2472 if (isl_int_is_zero(bmap
->div
[i
][0]))
2474 if (isl_seq_first_non_zero(bmap
->div
[i
] + 2 + d_dim
,
2475 (b_dim
- d_dim
) + bmap
->n_div
) != -1)
2477 if (isl_seq_eq(bmap
->div
[i
], dom
->div
[div
], 2 + d_dim
))
2483 /* The correspondence between the variables in the main tableau,
2484 * the context tableau, and the input map and domain is as follows.
2485 * The first n_param and the last n_div variables of the main tableau
2486 * form the variables of the context tableau.
2487 * In the basic map, these n_param variables correspond to the
2488 * parameters and the input dimensions. In the domain, they correspond
2489 * to the parameters and the set dimensions.
2490 * The n_div variables correspond to the integer divisions in the domain.
2491 * To ensure that everything lines up, we may need to copy some of the
2492 * integer divisions of the domain to the map. These have to be placed
2493 * in the same order as those in the context and they have to be placed
2494 * after any other integer divisions that the map may have.
2495 * This function performs the required reordering.
2497 static struct isl_basic_map
*align_context_divs(struct isl_basic_map
*bmap
,
2498 struct isl_basic_set
*dom
)
2504 for (i
= 0; i
< dom
->n_div
; ++i
)
2505 if (find_context_div(bmap
, dom
, i
) != -1)
2507 other
= bmap
->n_div
- common
;
2508 if (dom
->n_div
- common
> 0) {
2509 bmap
= isl_basic_map_extend_dim(bmap
, isl_dim_copy(bmap
->dim
),
2510 dom
->n_div
- common
, 0, 0);
2514 for (i
= 0; i
< dom
->n_div
; ++i
) {
2515 int pos
= find_context_div(bmap
, dom
, i
);
2517 pos
= isl_basic_map_alloc_div(bmap
);
2520 isl_int_set_si(bmap
->div
[pos
][0], 0);
2522 if (pos
!= other
+ i
)
2523 isl_basic_map_swap_div(bmap
, pos
, other
+ i
);
2527 isl_basic_map_free(bmap
);
2531 /* Compute the lexicographic minimum (or maximum if "max" is set)
2532 * of "bmap" over the domain "dom" and return the result as a map.
2533 * If "empty" is not NULL, then *empty is assigned a set that
2534 * contains those parts of the domain where there is no solution.
2535 * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL),
2536 * then we compute the rational optimum. Otherwise, we compute
2537 * the integral optimum.
2539 * We perform some preprocessing. As the PILP solver does not
2540 * handle implicit equalities very well, we first make sure all
2541 * the equalities are explicitly available.
2542 * We also make sure the divs in the domain are properly order,
2543 * because they will be added one by one in the given order
2544 * during the construction of the solution map.
2546 struct isl_map
*isl_tab_basic_map_partial_lexopt(
2547 struct isl_basic_map
*bmap
, struct isl_basic_set
*dom
,
2548 struct isl_set
**empty
, int max
)
2550 struct isl_tab
*tab
;
2551 struct isl_map
*result
= NULL
;
2552 struct isl_sol_map
*sol_map
= NULL
;
2559 isl_assert(bmap
->ctx
,
2560 isl_basic_map_compatible_domain(bmap
, dom
), goto error
);
2562 bmap
= isl_basic_map_detect_equalities(bmap
);
2565 dom
= isl_basic_set_order_divs(dom
);
2566 bmap
= align_context_divs(bmap
, dom
);
2568 sol_map
= sol_map_init(bmap
, dom
, !!empty
, max
);
2572 if (isl_basic_set_fast_is_empty(sol_map
->sol
.context_tab
->bset
))
2574 else if (isl_basic_map_fast_is_empty(bmap
))
2575 sol_map
= add_empty(sol_map
);
2577 tab
= tab_for_lexmin(bmap
,
2578 sol_map
->sol
.context_tab
->bset
, 1, max
);
2579 tab
= tab_detect_nonnegative_parameters(tab
,
2580 sol_map
->sol
.context_tab
);
2581 sol_map
= sol_map_find_solutions(sol_map
, tab
);
2586 result
= isl_map_copy(sol_map
->map
);
2588 *empty
= isl_set_copy(sol_map
->empty
);
2589 sol_map_free(sol_map
);
2590 isl_basic_map_free(bmap
);
2593 sol_map_free(sol_map
);
2594 isl_basic_map_free(bmap
);