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
)
505 unsigned off
= 2 + tab
->M
;
507 isl_seq_fdiv_q(tab
->mat
->row
[row
] + 1, tab
->mat
->row
[row
] + 1,
508 tab
->mat
->row
[row
][0], 1 + tab
->M
+ tab
->n_col
);
510 isl_int_set_si(tab
->mat
->row
[row
][0], 1);
512 isl_assert(tab
->mat
->ctx
,
513 !tab
->var
[tab
->n_var
- tab
->n_div
+ div
].is_row
, goto error
);
515 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ div
].index
;
516 isl_int_set_si(tab
->mat
->row
[row
][off
+ col
], 1);
524 /* Check if the (parametric) constant of the given row is obviously
525 * negative, meaning that we don't need to consult the context tableau.
526 * If there is a big parameter and its coefficient is non-zero,
527 * then this coefficient determines the outcome.
528 * Otherwise, we check whether the constant is negative and
529 * all non-zero coefficients of parameters are negative and
530 * belong to non-negative parameters.
532 static int is_obviously_neg(struct isl_tab
*tab
, int row
)
536 unsigned off
= 2 + tab
->M
;
539 if (isl_int_is_pos(tab
->mat
->row
[row
][2]))
541 if (isl_int_is_neg(tab
->mat
->row
[row
][2]))
545 if (isl_int_is_nonneg(tab
->mat
->row
[row
][1]))
547 for (i
= 0; i
< tab
->n_param
; ++i
) {
548 /* Eliminated parameter */
549 if (tab
->var
[i
].is_row
)
551 col
= tab
->var
[i
].index
;
552 if (isl_int_is_zero(tab
->mat
->row
[row
][off
+ col
]))
554 if (!tab
->var
[i
].is_nonneg
)
556 if (isl_int_is_pos(tab
->mat
->row
[row
][off
+ col
]))
559 for (i
= 0; i
< tab
->n_div
; ++i
) {
560 if (tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_row
)
562 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ i
].index
;
563 if (isl_int_is_zero(tab
->mat
->row
[row
][off
+ col
]))
565 if (!tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_nonneg
)
567 if (isl_int_is_pos(tab
->mat
->row
[row
][off
+ col
]))
573 /* Check if the (parametric) constant of the given row is obviously
574 * non-negative, meaning that we don't need to consult the context tableau.
575 * If there is a big parameter and its coefficient is non-zero,
576 * then this coefficient determines the outcome.
577 * Otherwise, we check whether the constant is non-negative and
578 * all non-zero coefficients of parameters are positive and
579 * belong to non-negative parameters.
581 static int is_obviously_nonneg(struct isl_tab
*tab
, int row
)
585 unsigned off
= 2 + tab
->M
;
588 if (isl_int_is_pos(tab
->mat
->row
[row
][2]))
590 if (isl_int_is_neg(tab
->mat
->row
[row
][2]))
594 if (isl_int_is_neg(tab
->mat
->row
[row
][1]))
596 for (i
= 0; i
< tab
->n_param
; ++i
) {
597 /* Eliminated parameter */
598 if (tab
->var
[i
].is_row
)
600 col
= tab
->var
[i
].index
;
601 if (isl_int_is_zero(tab
->mat
->row
[row
][off
+ col
]))
603 if (!tab
->var
[i
].is_nonneg
)
605 if (isl_int_is_neg(tab
->mat
->row
[row
][off
+ col
]))
608 for (i
= 0; i
< tab
->n_div
; ++i
) {
609 if (tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_row
)
611 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ i
].index
;
612 if (isl_int_is_zero(tab
->mat
->row
[row
][off
+ col
]))
614 if (!tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_nonneg
)
616 if (isl_int_is_neg(tab
->mat
->row
[row
][off
+ col
]))
622 /* Given a row r and two columns, return the column that would
623 * lead to the lexicographically smallest increment in the sample
624 * solution when leaving the basis in favor of the row.
625 * Pivoting with column c will increment the sample value by a non-negative
626 * constant times a_{V,c}/a_{r,c}, with a_{V,c} the elements of column c
627 * corresponding to the non-parametric variables.
628 * If variable v appears in a column c_v, the a_{v,c} = 1 iff c = c_v,
629 * with all other entries in this virtual row equal to zero.
630 * If variable v appears in a row, then a_{v,c} is the element in column c
633 * Let v be the first variable with a_{v,c1}/a_{r,c1} != a_{v,c2}/a_{r,c2}.
634 * Then if a_{v,c1}/a_{r,c1} < a_{v,c2}/a_{r,c2}, i.e.,
635 * a_{v,c2} a_{r,c1} - a_{v,c1} a_{r,c2} > 0, c1 results in the minimal
636 * increment. Otherwise, it's c2.
638 static int lexmin_col_pair(struct isl_tab
*tab
,
639 int row
, int col1
, int col2
, isl_int tmp
)
644 tr
= tab
->mat
->row
[row
] + 2 + tab
->M
;
646 for (i
= tab
->n_param
; i
< tab
->n_var
- tab
->n_div
; ++i
) {
650 if (!tab
->var
[i
].is_row
) {
651 if (tab
->var
[i
].index
== col1
)
653 if (tab
->var
[i
].index
== col2
)
658 if (tab
->var
[i
].index
== row
)
661 r
= tab
->mat
->row
[tab
->var
[i
].index
] + 2 + tab
->M
;
662 s1
= isl_int_sgn(r
[col1
]);
663 s2
= isl_int_sgn(r
[col2
]);
664 if (s1
== 0 && s2
== 0)
671 isl_int_mul(tmp
, r
[col2
], tr
[col1
]);
672 isl_int_submul(tmp
, r
[col1
], tr
[col2
]);
673 if (isl_int_is_pos(tmp
))
675 if (isl_int_is_neg(tmp
))
681 /* Given a row in the tableau, find and return the column that would
682 * result in the lexicographically smallest, but positive, increment
683 * in the sample point.
684 * If there is no such column, then return tab->n_col.
685 * If anything goes wrong, return -1.
687 static int lexmin_pivot_col(struct isl_tab
*tab
, int row
)
690 int col
= tab
->n_col
;
694 tr
= tab
->mat
->row
[row
] + 2 + tab
->M
;
698 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
699 if (tab
->col_var
[j
] >= 0 &&
700 (tab
->col_var
[j
] < tab
->n_param
||
701 tab
->col_var
[j
] >= tab
->n_var
- tab
->n_div
))
704 if (!isl_int_is_pos(tr
[j
]))
707 if (col
== tab
->n_col
)
710 col
= lexmin_col_pair(tab
, row
, col
, j
, tmp
);
711 isl_assert(tab
->mat
->ctx
, col
>= 0, goto error
);
721 /* Return the first known violated constraint, i.e., a non-negative
722 * contraint that currently has an either obviously negative value
723 * or a previously determined to be negative value.
725 * If any constraint has a negative coefficient for the big parameter,
726 * if any, then we return one of these first.
728 static int first_neg(struct isl_tab
*tab
)
733 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
734 if (!isl_tab_var_from_row(tab
, row
)->is_nonneg
)
736 if (isl_int_is_neg(tab
->mat
->row
[row
][2]))
739 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
740 if (!isl_tab_var_from_row(tab
, row
)->is_nonneg
)
743 if (tab
->row_sign
[row
] == 0 &&
744 is_obviously_neg(tab
, row
))
745 tab
->row_sign
[row
] = isl_tab_row_neg
;
746 if (tab
->row_sign
[row
] != isl_tab_row_neg
)
748 } else if (!is_obviously_neg(tab
, row
))
755 /* Resolve all known or obviously violated constraints through pivoting.
756 * In particular, as long as we can find any violated constraint, we
757 * look for a pivoting column that would result in the lexicographicallly
758 * smallest increment in the sample point. If there is no such column
759 * then the tableau is infeasible.
761 static struct isl_tab
*restore_lexmin(struct isl_tab
*tab
)
769 while ((row
= first_neg(tab
)) != -1) {
770 col
= lexmin_pivot_col(tab
, row
);
771 if (col
>= tab
->n_col
)
772 return isl_tab_mark_empty(tab
);
775 isl_tab_pivot(tab
, row
, col
);
783 /* Given a row that represents an equality, look for an appropriate
785 * In particular, if there are any non-zero coefficients among
786 * the non-parameter variables, then we take the last of these
787 * variables. Eliminating this variable in terms of the other
788 * variables and/or parameters does not influence the property
789 * that all column in the initial tableau are lexicographically
790 * positive. The row corresponding to the eliminated variable
791 * will only have non-zero entries below the diagonal of the
792 * initial tableau. That is, we transform
798 * If there is no such non-parameter variable, then we are dealing with
799 * pure parameter equality and we pick any parameter with coefficient 1 or -1
800 * for elimination. This will ensure that the eliminated parameter
801 * always has an integer value whenever all the other parameters are integral.
802 * If there is no such parameter then we return -1.
804 static int last_var_col_or_int_par_col(struct isl_tab
*tab
, int row
)
806 unsigned off
= 2 + tab
->M
;
809 for (i
= tab
->n_var
- tab
->n_div
- 1; i
>= 0 && i
>= tab
->n_param
; --i
) {
811 if (tab
->var
[i
].is_row
)
813 col
= tab
->var
[i
].index
;
814 if (col
<= tab
->n_dead
)
816 if (!isl_int_is_zero(tab
->mat
->row
[row
][off
+ col
]))
819 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
820 if (isl_int_is_one(tab
->mat
->row
[row
][off
+ i
]))
822 if (isl_int_is_negone(tab
->mat
->row
[row
][off
+ i
]))
828 /* Add an equality that is known to be valid to the tableau.
829 * We first check if we can eliminate a variable or a parameter.
830 * If not, we add the equality as two inequalities.
831 * In this case, the equality was a pure parameter equality and there
832 * is no need to resolve any constraint violations.
834 static struct isl_tab
*add_lexmin_valid_eq(struct isl_tab
*tab
, isl_int
*eq
)
841 r
= isl_tab_add_row(tab
, eq
);
845 r
= tab
->con
[r
].index
;
846 i
= last_var_col_or_int_par_col(tab
, r
);
848 tab
->con
[r
].is_nonneg
= 1;
849 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
850 isl_seq_neg(eq
, eq
, 1 + tab
->n_var
);
851 r
= isl_tab_add_row(tab
, eq
);
854 tab
->con
[r
].is_nonneg
= 1;
855 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
857 isl_tab_pivot(tab
, r
, i
);
858 isl_tab_kill_col(tab
, i
);
861 tab
= restore_lexmin(tab
);
870 /* Check if the given row is a pure constant.
872 static int is_constant(struct isl_tab
*tab
, int row
)
874 unsigned off
= 2 + tab
->M
;
876 return isl_seq_first_non_zero(tab
->mat
->row
[row
] + off
+ tab
->n_dead
,
877 tab
->n_col
- tab
->n_dead
) == -1;
880 /* Add an equality that may or may not be valid to the tableau.
881 * If the resulting row is a pure constant, then it must be zero.
882 * Otherwise, the resulting tableau is empty.
884 * If the row is not a pure constant, then we add two inequalities,
885 * each time checking that they can be satisfied.
886 * In the end we try to use one of the two constraints to eliminate
889 static struct isl_tab
*add_lexmin_eq(struct isl_tab
*tab
, isl_int
*eq
)
897 tab
->bset
= isl_basic_set_add_eq(tab
->bset
, eq
);
898 isl_tab_push(tab
, isl_tab_undo_bset_eq
);
902 r1
= isl_tab_add_row(tab
, eq
);
905 tab
->con
[r1
].is_nonneg
= 1;
906 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r1
]);
908 row
= tab
->con
[r1
].index
;
909 if (is_constant(tab
, row
)) {
910 if (!isl_int_is_zero(tab
->mat
->row
[row
][1]) ||
911 (tab
->M
&& !isl_int_is_zero(tab
->mat
->row
[row
][2])))
912 return isl_tab_mark_empty(tab
);
916 tab
= restore_lexmin(tab
);
917 if (!tab
|| tab
->empty
)
920 isl_seq_neg(eq
, eq
, 1 + tab
->n_var
);
922 r2
= isl_tab_add_row(tab
, eq
);
925 tab
->con
[r2
].is_nonneg
= 1;
926 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r2
]);
928 tab
= restore_lexmin(tab
);
929 if (!tab
|| tab
->empty
)
932 if (!tab
->con
[r1
].is_row
)
933 isl_tab_kill_col(tab
, tab
->con
[r1
].index
);
934 else if (!tab
->con
[r2
].is_row
)
935 isl_tab_kill_col(tab
, tab
->con
[r2
].index
);
936 else if (isl_int_is_zero(tab
->mat
->row
[tab
->con
[r1
].index
][1])) {
937 unsigned off
= 2 + tab
->M
;
939 int row
= tab
->con
[r1
].index
;
940 i
= isl_seq_first_non_zero(tab
->mat
->row
[row
]+off
+tab
->n_dead
,
941 tab
->n_col
- tab
->n_dead
);
943 isl_tab_pivot(tab
, row
, tab
->n_dead
+ i
);
944 isl_tab_kill_col(tab
, tab
->n_dead
+ i
);
954 /* Add an inequality to the tableau, resolving violations using
957 static struct isl_tab
*add_lexmin_ineq(struct isl_tab
*tab
, isl_int
*ineq
)
964 tab
->bset
= isl_basic_set_add_ineq(tab
->bset
, ineq
);
965 isl_tab_push(tab
, isl_tab_undo_bset_ineq
);
969 r
= isl_tab_add_row(tab
, ineq
);
972 tab
->con
[r
].is_nonneg
= 1;
973 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
974 if (isl_tab_row_is_redundant(tab
, tab
->con
[r
].index
)) {
975 isl_tab_mark_redundant(tab
, tab
->con
[r
].index
);
979 tab
= restore_lexmin(tab
);
980 if (tab
&& !tab
->empty
&& tab
->con
[r
].is_row
&&
981 isl_tab_row_is_redundant(tab
, tab
->con
[r
].index
))
982 isl_tab_mark_redundant(tab
, tab
->con
[r
].index
);
989 /* Check if the coefficients of the parameters are all integral.
991 static int integer_parameter(struct isl_tab
*tab
, int row
)
995 unsigned off
= 2 + tab
->M
;
997 for (i
= 0; i
< tab
->n_param
; ++i
) {
998 /* Eliminated parameter */
999 if (tab
->var
[i
].is_row
)
1001 col
= tab
->var
[i
].index
;
1002 if (!isl_int_is_divisible_by(tab
->mat
->row
[row
][off
+ col
],
1003 tab
->mat
->row
[row
][0]))
1006 for (i
= 0; i
< tab
->n_div
; ++i
) {
1007 if (tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_row
)
1009 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ i
].index
;
1010 if (!isl_int_is_divisible_by(tab
->mat
->row
[row
][off
+ col
],
1011 tab
->mat
->row
[row
][0]))
1017 /* Check if the coefficients of the non-parameter variables are all integral.
1019 static int integer_variable(struct isl_tab
*tab
, int row
)
1022 unsigned off
= 2 + tab
->M
;
1024 for (i
= 0; i
< tab
->n_col
; ++i
) {
1025 if (tab
->col_var
[i
] >= 0 &&
1026 (tab
->col_var
[i
] < tab
->n_param
||
1027 tab
->col_var
[i
] >= tab
->n_var
- tab
->n_div
))
1029 if (!isl_int_is_divisible_by(tab
->mat
->row
[row
][off
+ i
],
1030 tab
->mat
->row
[row
][0]))
1036 /* Check if the constant term is integral.
1038 static int integer_constant(struct isl_tab
*tab
, int row
)
1040 return isl_int_is_divisible_by(tab
->mat
->row
[row
][1],
1041 tab
->mat
->row
[row
][0]);
1044 #define I_CST 1 << 0
1045 #define I_PAR 1 << 1
1046 #define I_VAR 1 << 2
1048 /* Check for first (non-parameter) variable that is non-integer and
1049 * therefore requires a cut.
1050 * For parametric tableaus, there are three parts in a row,
1051 * the constant, the coefficients of the parameters and the rest.
1052 * For each part, we check whether the coefficients in that part
1053 * are all integral and if so, set the corresponding flag in *f.
1054 * If the constant and the parameter part are integral, then the
1055 * current sample value is integral and no cut is required
1056 * (irrespective of whether the variable part is integral).
1058 static int first_non_integer(struct isl_tab
*tab
, int *f
)
1062 for (i
= tab
->n_param
; i
< tab
->n_var
- tab
->n_div
; ++i
) {
1065 if (!tab
->var
[i
].is_row
)
1067 row
= tab
->var
[i
].index
;
1068 if (integer_constant(tab
, row
))
1069 ISL_FL_SET(flags
, I_CST
);
1070 if (integer_parameter(tab
, row
))
1071 ISL_FL_SET(flags
, I_PAR
);
1072 if (ISL_FL_ISSET(flags
, I_CST
) && ISL_FL_ISSET(flags
, I_PAR
))
1074 if (integer_variable(tab
, row
))
1075 ISL_FL_SET(flags
, I_VAR
);
1082 /* Add a (non-parametric) cut to cut away the non-integral sample
1083 * value of the given row.
1085 * If the row is given by
1087 * m r = f + \sum_i a_i y_i
1091 * c = - {-f/m} + \sum_i {a_i/m} y_i >= 0
1093 * The big parameter, if any, is ignored, since it is assumed to be big
1094 * enough to be divisible by any integer.
1095 * If the tableau is actually a parametric tableau, then this function
1096 * is only called when all coefficients of the parameters are integral.
1097 * The cut therefore has zero coefficients for the parameters.
1099 * The current value is known to be negative, so row_sign, if it
1100 * exists, is set accordingly.
1102 * Return the row of the cut or -1.
1104 static int add_cut(struct isl_tab
*tab
, int row
)
1109 unsigned off
= 2 + tab
->M
;
1111 if (isl_tab_extend_cons(tab
, 1) < 0)
1113 r
= isl_tab_allocate_con(tab
);
1117 r_row
= tab
->mat
->row
[tab
->con
[r
].index
];
1118 isl_int_set(r_row
[0], tab
->mat
->row
[row
][0]);
1119 isl_int_neg(r_row
[1], tab
->mat
->row
[row
][1]);
1120 isl_int_fdiv_r(r_row
[1], r_row
[1], tab
->mat
->row
[row
][0]);
1121 isl_int_neg(r_row
[1], r_row
[1]);
1123 isl_int_set_si(r_row
[2], 0);
1124 for (i
= 0; i
< tab
->n_col
; ++i
)
1125 isl_int_fdiv_r(r_row
[off
+ i
],
1126 tab
->mat
->row
[row
][off
+ i
], tab
->mat
->row
[row
][0]);
1128 tab
->con
[r
].is_nonneg
= 1;
1129 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1131 tab
->row_sign
[tab
->con
[r
].index
] = isl_tab_row_neg
;
1133 return tab
->con
[r
].index
;
1136 /* Given a non-parametric tableau, add cuts until an integer
1137 * sample point is obtained or until the tableau is determined
1138 * to be integer infeasible.
1139 * As long as there is any non-integer value in the sample point,
1140 * we add an appropriate cut, if possible and resolve the violated
1141 * cut constraint using restore_lexmin.
1142 * If one of the corresponding rows is equal to an integral
1143 * combination of variables/constraints plus a non-integral constant,
1144 * then there is no way to obtain an integer point an we return
1145 * a tableau that is marked empty.
1147 static struct isl_tab
*cut_to_integer_lexmin(struct isl_tab
*tab
)
1157 while ((row
= first_non_integer(tab
, &flags
)) != -1) {
1158 if (ISL_FL_ISSET(flags
, I_VAR
))
1159 return isl_tab_mark_empty(tab
);
1160 row
= add_cut(tab
, row
);
1163 tab
= restore_lexmin(tab
);
1164 if (!tab
|| tab
->empty
)
1173 static struct isl_tab
*drop_sample(struct isl_tab
*tab
, int s
)
1175 if (s
!= tab
->n_outside
)
1176 isl_mat_swap_rows(tab
->samples
, tab
->n_outside
, s
);
1178 isl_tab_push(tab
, isl_tab_undo_drop_sample
);
1183 /* Check whether all the currently active samples also satisfy the inequality
1184 * "ineq" (treated as an equality if eq is set).
1185 * Remove those samples that do not.
1187 static struct isl_tab
*check_samples(struct isl_tab
*tab
, isl_int
*ineq
, int eq
)
1195 isl_assert(tab
->mat
->ctx
, tab
->bset
, goto error
);
1196 isl_assert(tab
->mat
->ctx
, tab
->samples
, goto error
);
1197 isl_assert(tab
->mat
->ctx
, tab
->samples
->n_col
== 1 + tab
->n_var
, goto error
);
1200 for (i
= tab
->n_outside
; i
< tab
->n_sample
; ++i
) {
1202 isl_seq_inner_product(ineq
, tab
->samples
->row
[i
],
1203 1 + tab
->n_var
, &v
);
1204 sgn
= isl_int_sgn(v
);
1205 if (eq
? (sgn
== 0) : (sgn
>= 0))
1207 tab
= drop_sample(tab
, i
);
1219 /* Check whether the sample value of the tableau is finite,
1220 * i.e., either the tableau does not use a big parameter, or
1221 * all values of the variables are equal to the big parameter plus
1222 * some constant. This constant is the actual sample value.
1224 int sample_is_finite(struct isl_tab
*tab
)
1231 for (i
= 0; i
< tab
->n_var
; ++i
) {
1233 if (!tab
->var
[i
].is_row
)
1235 row
= tab
->var
[i
].index
;
1236 if (isl_int_ne(tab
->mat
->row
[row
][0], tab
->mat
->row
[row
][2]))
1242 /* Check if the context tableau of sol has any integer points.
1243 * Returns -1 if an error occurred.
1244 * If an integer point can be found and if moreover it is finite,
1245 * then it is added to the list of sample values.
1247 * This function is only called when none of the currently active sample
1248 * values satisfies the most recently added constraint.
1250 static int context_is_feasible(struct isl_sol
*sol
)
1252 struct isl_tab_undo
*snap
;
1253 struct isl_tab
*tab
;
1256 if (!sol
|| !sol
->context_tab
)
1259 snap
= isl_tab_snap(sol
->context_tab
);
1260 isl_tab_push_basis(sol
->context_tab
);
1262 sol
->context_tab
= cut_to_integer_lexmin(sol
->context_tab
);
1263 if (!sol
->context_tab
)
1266 tab
= sol
->context_tab
;
1267 if (!tab
->empty
&& sample_is_finite(tab
)) {
1268 struct isl_vec
*sample
;
1270 tab
->samples
= isl_mat_extend(tab
->samples
,
1271 tab
->n_sample
+ 1, tab
->samples
->n_col
);
1275 sample
= isl_tab_get_sample_value(tab
);
1278 isl_seq_cpy(tab
->samples
->row
[tab
->n_sample
],
1279 sample
->el
, sample
->size
);
1280 isl_vec_free(sample
);
1284 feasible
= !sol
->context_tab
->empty
;
1285 if (isl_tab_rollback(sol
->context_tab
, snap
) < 0)
1290 isl_tab_free(sol
->context_tab
);
1291 sol
->context_tab
= NULL
;
1295 /* First check if any of the currently active sample values satisfies
1296 * the inequality "ineq" (an equality if eq is set).
1297 * If not, continue with check_integer_feasible.
1299 static int context_valid_sample_or_feasible(struct isl_sol
*sol
,
1300 isl_int
*ineq
, int eq
)
1304 struct isl_tab
*tab
;
1306 if (!sol
|| !sol
->context_tab
)
1309 tab
= sol
->context_tab
;
1310 isl_assert(tab
->mat
->ctx
, tab
->bset
, return -1);
1311 isl_assert(tab
->mat
->ctx
, tab
->samples
, return -1);
1312 isl_assert(tab
->mat
->ctx
, tab
->samples
->n_col
== 1 + tab
->n_var
, return -1);
1315 for (i
= tab
->n_outside
; i
< tab
->n_sample
; ++i
) {
1317 isl_seq_inner_product(ineq
, tab
->samples
->row
[i
],
1318 1 + tab
->n_var
, &v
);
1319 sgn
= isl_int_sgn(v
);
1320 if (eq
? (sgn
== 0) : (sgn
>= 0))
1325 if (i
< tab
->n_sample
)
1328 return context_is_feasible(sol
);
1331 /* For a div d = floor(f/m), add the constraints
1334 * -(f-(m-1)) + m d >= 0
1336 * Note that the second constraint is the negation of
1340 static struct isl_tab
*add_div_constraints(struct isl_tab
*tab
, unsigned div
)
1344 struct isl_vec
*ineq
;
1349 total
= isl_basic_set_total_dim(tab
->bset
);
1350 div_pos
= 1 + total
- tab
->bset
->n_div
+ div
;
1352 ineq
= ineq_for_div(tab
->bset
, div
);
1356 tab
= add_lexmin_ineq(tab
, ineq
->el
);
1358 isl_seq_neg(ineq
->el
, tab
->bset
->div
[div
] + 1, 1 + total
);
1359 isl_int_set(ineq
->el
[div_pos
], tab
->bset
->div
[div
][0]);
1360 isl_int_add(ineq
->el
[0], ineq
->el
[0], ineq
->el
[div_pos
]);
1361 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
1362 tab
= add_lexmin_ineq(tab
, ineq
->el
);
1372 /* Add a div specified by "div" to both the main tableau and
1373 * the context tableau. In case of the main tableau, we only
1374 * need to add an extra div. In the context tableau, we also
1375 * need to express the meaning of the div.
1376 * Return the index of the div or -1 if anything went wrong.
1378 static int add_div(struct isl_tab
*tab
, struct isl_tab
**context_tab
,
1379 struct isl_vec
*div
)
1384 struct isl_mat
*samples
;
1386 if (isl_tab_extend_vars(*context_tab
, 1) < 0)
1388 r
= isl_tab_allocate_var(*context_tab
);
1391 (*context_tab
)->var
[r
].is_nonneg
= 1;
1392 (*context_tab
)->var
[r
].frozen
= 1;
1394 samples
= isl_mat_extend((*context_tab
)->samples
,
1395 (*context_tab
)->n_sample
, 1 + (*context_tab
)->n_var
);
1396 (*context_tab
)->samples
= samples
;
1399 for (i
= (*context_tab
)->n_outside
; i
< samples
->n_row
; ++i
) {
1400 isl_seq_inner_product(div
->el
+ 1, samples
->row
[i
],
1401 div
->size
- 1, &samples
->row
[i
][samples
->n_col
- 1]);
1402 isl_int_fdiv_q(samples
->row
[i
][samples
->n_col
- 1],
1403 samples
->row
[i
][samples
->n_col
- 1], div
->el
[0]);
1406 (*context_tab
)->bset
= isl_basic_set_extend_dim((*context_tab
)->bset
,
1407 isl_basic_set_get_dim((*context_tab
)->bset
), 1, 0, 2);
1408 k
= isl_basic_set_alloc_div((*context_tab
)->bset
);
1411 isl_seq_cpy((*context_tab
)->bset
->div
[k
], div
->el
, div
->size
);
1412 isl_tab_push((*context_tab
), isl_tab_undo_bset_div
);
1413 *context_tab
= add_div_constraints(*context_tab
, k
);
1417 if (isl_tab_extend_vars(tab
, 1) < 0)
1419 r
= isl_tab_allocate_var(tab
);
1422 if (!(*context_tab
)->M
)
1423 tab
->var
[r
].is_nonneg
= 1;
1424 tab
->var
[r
].frozen
= 1;
1427 return tab
->n_div
- 1;
1429 isl_tab_free(*context_tab
);
1430 *context_tab
= NULL
;
1434 static int find_div(struct isl_tab
*tab
, isl_int
*div
, isl_int denom
)
1437 unsigned total
= isl_basic_set_total_dim(tab
->bset
);
1439 for (i
= 0; i
< tab
->bset
->n_div
; ++i
) {
1440 if (isl_int_ne(tab
->bset
->div
[i
][0], denom
))
1442 if (!isl_seq_eq(tab
->bset
->div
[i
] + 1, div
, total
))
1449 /* Return the index of a div that corresponds to "div".
1450 * We first check if we already have such a div and if not, we create one.
1452 static int get_div(struct isl_tab
*tab
, struct isl_tab
**context_tab
,
1453 struct isl_vec
*div
)
1457 d
= find_div(*context_tab
, div
->el
+ 1, div
->el
[0]);
1461 return add_div(tab
, context_tab
, div
);
1464 /* Add a parametric cut to cut away the non-integral sample value
1466 * Let a_i be the coefficients of the constant term and the parameters
1467 * and let b_i be the coefficients of the variables or constraints
1468 * in basis of the tableau.
1469 * Let q be the div q = floor(\sum_i {-a_i} y_i).
1471 * The cut is expressed as
1473 * c = \sum_i -{-a_i} y_i + \sum_i {b_i} x_i + q >= 0
1475 * If q did not already exist in the context tableau, then it is added first.
1476 * If q is in a column of the main tableau then the "+ q" can be accomplished
1477 * by setting the corresponding entry to the denominator of the constraint.
1478 * If q happens to be in a row of the main tableau, then the corresponding
1479 * row needs to be added instead (taking care of the denominators).
1480 * Note that this is very unlikely, but perhaps not entirely impossible.
1482 * The current value of the cut is known to be negative (or at least
1483 * non-positive), so row_sign is set accordingly.
1485 * Return the row of the cut or -1.
1487 static int add_parametric_cut(struct isl_tab
*tab
, int row
,
1488 struct isl_tab
**context_tab
)
1490 struct isl_vec
*div
;
1496 unsigned off
= 2 + tab
->M
;
1501 if (isl_tab_extend_cons(*context_tab
, 3) < 0)
1504 div
= get_row_parameter_div(tab
, row
);
1508 d
= get_div(tab
, context_tab
, div
);
1512 if (isl_tab_extend_cons(tab
, 1) < 0)
1514 r
= isl_tab_allocate_con(tab
);
1518 r_row
= tab
->mat
->row
[tab
->con
[r
].index
];
1519 isl_int_set(r_row
[0], tab
->mat
->row
[row
][0]);
1520 isl_int_neg(r_row
[1], tab
->mat
->row
[row
][1]);
1521 isl_int_fdiv_r(r_row
[1], r_row
[1], tab
->mat
->row
[row
][0]);
1522 isl_int_neg(r_row
[1], r_row
[1]);
1524 isl_int_set_si(r_row
[2], 0);
1525 for (i
= 0; i
< tab
->n_param
; ++i
) {
1526 if (tab
->var
[i
].is_row
)
1528 col
= tab
->var
[i
].index
;
1529 isl_int_neg(r_row
[off
+ col
], tab
->mat
->row
[row
][off
+ col
]);
1530 isl_int_fdiv_r(r_row
[off
+ col
], r_row
[off
+ col
],
1531 tab
->mat
->row
[row
][0]);
1532 isl_int_neg(r_row
[off
+ col
], r_row
[off
+ col
]);
1534 for (i
= 0; i
< tab
->n_div
; ++i
) {
1535 if (tab
->var
[tab
->n_var
- tab
->n_div
+ i
].is_row
)
1537 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ i
].index
;
1538 isl_int_neg(r_row
[off
+ col
], tab
->mat
->row
[row
][off
+ col
]);
1539 isl_int_fdiv_r(r_row
[off
+ col
], r_row
[off
+ col
],
1540 tab
->mat
->row
[row
][0]);
1541 isl_int_neg(r_row
[off
+ col
], r_row
[off
+ col
]);
1543 for (i
= 0; i
< tab
->n_col
; ++i
) {
1544 if (tab
->col_var
[i
] >= 0 &&
1545 (tab
->col_var
[i
] < tab
->n_param
||
1546 tab
->col_var
[i
] >= tab
->n_var
- tab
->n_div
))
1548 isl_int_fdiv_r(r_row
[off
+ i
],
1549 tab
->mat
->row
[row
][off
+ i
], tab
->mat
->row
[row
][0]);
1551 if (tab
->var
[tab
->n_var
- tab
->n_div
+ d
].is_row
) {
1553 int d_row
= tab
->var
[tab
->n_var
- tab
->n_div
+ d
].index
;
1555 isl_int_gcd(gcd
, tab
->mat
->row
[d_row
][0], r_row
[0]);
1556 isl_int_divexact(r_row
[0], r_row
[0], gcd
);
1557 isl_int_divexact(gcd
, tab
->mat
->row
[d_row
][0], gcd
);
1558 isl_seq_combine(r_row
+ 1, gcd
, r_row
+ 1,
1559 r_row
[0], tab
->mat
->row
[d_row
] + 1,
1560 off
- 1 + tab
->n_col
);
1561 isl_int_mul(r_row
[0], r_row
[0], tab
->mat
->row
[d_row
][0]);
1564 col
= tab
->var
[tab
->n_var
- tab
->n_div
+ d
].index
;
1565 isl_int_set(r_row
[off
+ col
], tab
->mat
->row
[row
][0]);
1568 tab
->con
[r
].is_nonneg
= 1;
1569 isl_tab_push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1571 tab
->row_sign
[tab
->con
[r
].index
] = isl_tab_row_neg
;
1575 return tab
->con
[r
].index
;
1577 isl_tab_free(*context_tab
);
1578 *context_tab
= NULL
;
1582 /* Construct a tableau for bmap that can be used for computing
1583 * the lexicographic minimum (or maximum) of bmap.
1584 * If not NULL, then dom is the domain where the minimum
1585 * should be computed. In this case, we set up a parametric
1586 * tableau with row signs (initialized to "unknown").
1587 * If M is set, then the tableau will use a big parameter.
1588 * If max is set, then a maximum should be computed instead of a minimum.
1589 * This means that for each variable x, the tableau will contain the variable
1590 * x' = M - x, rather than x' = M + x. This in turn means that the coefficient
1591 * of the variables in all constraints are negated prior to adding them
1594 static struct isl_tab
*tab_for_lexmin(struct isl_basic_map
*bmap
,
1595 struct isl_basic_set
*dom
, unsigned M
, int max
)
1598 struct isl_tab
*tab
;
1600 tab
= isl_tab_alloc(bmap
->ctx
, 2 * bmap
->n_eq
+ bmap
->n_ineq
+ 1,
1601 isl_basic_map_total_dim(bmap
), M
);
1605 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1607 tab
->n_param
= isl_basic_set_total_dim(dom
) - dom
->n_div
;
1608 tab
->n_div
= dom
->n_div
;
1609 tab
->row_sign
= isl_calloc_array(bmap
->ctx
,
1610 enum isl_tab_row_sign
, tab
->mat
->n_row
);
1614 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
))
1615 return isl_tab_mark_empty(tab
);
1617 for (i
= tab
->n_param
; i
< tab
->n_var
- tab
->n_div
; ++i
) {
1618 tab
->var
[i
].is_nonneg
= 1;
1619 tab
->var
[i
].frozen
= 1;
1621 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1623 isl_seq_neg(bmap
->eq
[i
] + 1 + tab
->n_param
,
1624 bmap
->eq
[i
] + 1 + tab
->n_param
,
1625 tab
->n_var
- tab
->n_param
- tab
->n_div
);
1626 tab
= add_lexmin_valid_eq(tab
, bmap
->eq
[i
]);
1628 isl_seq_neg(bmap
->eq
[i
] + 1 + tab
->n_param
,
1629 bmap
->eq
[i
] + 1 + tab
->n_param
,
1630 tab
->n_var
- tab
->n_param
- tab
->n_div
);
1631 if (!tab
|| tab
->empty
)
1634 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1636 isl_seq_neg(bmap
->ineq
[i
] + 1 + tab
->n_param
,
1637 bmap
->ineq
[i
] + 1 + tab
->n_param
,
1638 tab
->n_var
- tab
->n_param
- tab
->n_div
);
1639 tab
= add_lexmin_ineq(tab
, bmap
->ineq
[i
]);
1641 isl_seq_neg(bmap
->ineq
[i
] + 1 + tab
->n_param
,
1642 bmap
->ineq
[i
] + 1 + tab
->n_param
,
1643 tab
->n_var
- tab
->n_param
- tab
->n_div
);
1644 if (!tab
|| tab
->empty
)
1653 static struct isl_tab
*context_tab_for_lexmin(struct isl_basic_set
*bset
)
1655 struct isl_tab
*tab
;
1657 bset
= isl_basic_set_cow(bset
);
1660 tab
= tab_for_lexmin((struct isl_basic_map
*)bset
, NULL
, 1, 0);
1666 tab
->samples
= isl_mat_alloc(bset
->ctx
, 1, 1 + tab
->n_var
);
1671 isl_basic_set_free(bset
);
1675 /* Construct an isl_sol_map structure for accumulating the solution.
1676 * If track_empty is set, then we also keep track of the parts
1677 * of the context where there is no solution.
1678 * If max is set, then we are solving a maximization, rather than
1679 * a minimization problem, which means that the variables in the
1680 * tableau have value "M - x" rather than "M + x".
1682 static struct isl_sol_map
*sol_map_init(struct isl_basic_map
*bmap
,
1683 struct isl_basic_set
*dom
, int track_empty
, int max
)
1685 struct isl_sol_map
*sol_map
;
1686 struct isl_tab
*context_tab
;
1689 sol_map
= isl_calloc_type(bset
->ctx
, struct isl_sol_map
);
1694 sol_map
->sol
.add
= &sol_map_add_wrap
;
1695 sol_map
->sol
.free
= &sol_map_free_wrap
;
1696 sol_map
->map
= isl_map_alloc_dim(isl_basic_map_get_dim(bmap
), 1,
1701 context_tab
= context_tab_for_lexmin(isl_basic_set_copy(dom
));
1702 context_tab
= restore_lexmin(context_tab
);
1703 sol_map
->sol
.context_tab
= context_tab
;
1704 f
= context_is_feasible(&sol_map
->sol
);
1709 sol_map
->empty
= isl_set_alloc_dim(isl_basic_set_get_dim(dom
),
1710 1, ISL_SET_DISJOINT
);
1711 if (!sol_map
->empty
)
1715 isl_basic_set_free(dom
);
1718 isl_basic_set_free(dom
);
1719 sol_map_free(sol_map
);
1723 /* For each variable in the context tableau, check if the variable can
1724 * only attain non-negative values. If so, mark the parameter as non-negative
1725 * in the main tableau. This allows for a more direct identification of some
1726 * cases of violated constraints.
1728 static struct isl_tab
*tab_detect_nonnegative_parameters(struct isl_tab
*tab
,
1729 struct isl_tab
*context_tab
)
1732 struct isl_tab_undo
*snap
, *snap2
;
1733 struct isl_vec
*ineq
= NULL
;
1734 struct isl_tab_var
*var
;
1737 if (context_tab
->n_var
== 0)
1740 ineq
= isl_vec_alloc(tab
->mat
->ctx
, 1 + context_tab
->n_var
);
1744 if (isl_tab_extend_cons(context_tab
, 1) < 0)
1747 snap
= isl_tab_snap(context_tab
);
1748 isl_tab_push_basis(context_tab
);
1750 snap2
= isl_tab_snap(context_tab
);
1753 isl_seq_clr(ineq
->el
, ineq
->size
);
1754 for (i
= 0; i
< context_tab
->n_var
; ++i
) {
1755 isl_int_set_si(ineq
->el
[1 + i
], 1);
1756 context_tab
= isl_tab_add_ineq(context_tab
, ineq
->el
);
1757 var
= &context_tab
->con
[context_tab
->n_con
- 1];
1758 if (!context_tab
->empty
&&
1759 !isl_tab_min_at_most_neg_one(context_tab
, var
)) {
1761 if (i
>= tab
->n_param
)
1762 j
= i
- tab
->n_param
+ tab
->n_var
- tab
->n_div
;
1763 tab
->var
[j
].is_nonneg
= 1;
1766 isl_int_set_si(ineq
->el
[1 + i
], 0);
1767 if (isl_tab_rollback(context_tab
, snap2
) < 0)
1771 if (isl_tab_rollback(context_tab
, snap
) < 0)
1774 if (n
== context_tab
->n_var
) {
1775 context_tab
->mat
= isl_mat_drop_cols(context_tab
->mat
, 2, 1);
1787 /* Check whether all coefficients of (non-parameter) variables
1788 * are non-positive, meaning that no pivots can be performed on the row.
1790 static int is_critical(struct isl_tab
*tab
, int row
)
1793 unsigned off
= 2 + tab
->M
;
1795 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
1796 if (tab
->col_var
[j
] >= 0 &&
1797 (tab
->col_var
[j
] < tab
->n_param
||
1798 tab
->col_var
[j
] >= tab
->n_var
- tab
->n_div
))
1801 if (isl_int_is_pos(tab
->mat
->row
[row
][off
+ j
]))
1808 /* Check whether the inequality represented by vec is strict over the integers,
1809 * i.e., there are no integer values satisfying the constraint with
1810 * equality. This happens if the gcd of the coefficients is not a divisor
1811 * of the constant term. If so, scale the constraint down by the gcd
1812 * of the coefficients.
1814 static int is_strict(struct isl_vec
*vec
)
1820 isl_seq_gcd(vec
->el
+ 1, vec
->size
- 1, &gcd
);
1821 if (!isl_int_is_one(gcd
)) {
1822 strict
= !isl_int_is_divisible_by(vec
->el
[0], gcd
);
1823 isl_int_fdiv_q(vec
->el
[0], vec
->el
[0], gcd
);
1824 isl_seq_scale_down(vec
->el
+ 1, vec
->el
+ 1, gcd
, vec
->size
-1);
1831 /* Determine the sign of the given row of the main tableau.
1832 * The result is one of
1833 * isl_tab_row_pos: always non-negative; no pivot needed
1834 * isl_tab_row_neg: always non-positive; pivot
1835 * isl_tab_row_any: can be both positive and negative; split
1837 * We first handle some simple cases
1838 * - the row sign may be known already
1839 * - the row may be obviously non-negative
1840 * - the parametric constant may be equal to that of another row
1841 * for which we know the sign. This sign will be either "pos" or
1842 * "any". If it had been "neg" then we would have pivoted before.
1844 * If none of these cases hold, we check the value of the row for each
1845 * of the currently active samples. Based on the signs of these values
1846 * we make an initial determination of the sign of the row.
1848 * all zero -> unk(nown)
1849 * all non-negative -> pos
1850 * all non-positive -> neg
1851 * both negative and positive -> all
1853 * If we end up with "all", we are done.
1854 * Otherwise, we perform a check for positive and/or negative
1855 * values as follows.
1857 * samples neg unk pos
1863 * There is no special sign for "zero", because we can usually treat zero
1864 * as either non-negative or non-positive, whatever works out best.
1865 * However, if the row is "critical", meaning that pivoting is impossible
1866 * then we don't want to limp zero with the non-positive case, because
1867 * then we we would lose the solution for those values of the parameters
1868 * where the value of the row is zero. Instead, we treat 0 as non-negative
1869 * ensuring a split if the row can attain both zero and negative values.
1870 * The same happens when the original constraint was one that could not
1871 * be satisfied with equality by any integer values of the parameters.
1872 * In this case, we normalize the constraint, but then a value of zero
1873 * for the normalized constraint is actually a positive value for the
1874 * original constraint, so again we need to treat zero as non-negative.
1875 * In both these cases, we have the following decision tree instead:
1877 * all non-negative -> pos
1878 * all negative -> neg
1879 * both negative and non-negative -> all
1887 static int row_sign(struct isl_tab
*tab
, struct isl_sol
*sol
, int row
)
1890 struct isl_tab_undo
*snap
= NULL
;
1891 struct isl_vec
*ineq
= NULL
;
1892 int res
= isl_tab_row_unknown
;
1898 struct isl_tab
*context_tab
= sol
->context_tab
;
1900 if (tab
->row_sign
[row
] != isl_tab_row_unknown
)
1901 return tab
->row_sign
[row
];
1902 if (is_obviously_nonneg(tab
, row
))
1903 return isl_tab_row_pos
;
1904 for (row2
= tab
->n_redundant
; row2
< tab
->n_row
; ++row2
) {
1905 if (tab
->row_sign
[row2
] == isl_tab_row_unknown
)
1907 if (identical_parameter_line(tab
, row
, row2
))
1908 return tab
->row_sign
[row2
];
1911 critical
= is_critical(tab
, row
);
1913 isl_assert(tab
->mat
->ctx
, context_tab
->samples
, goto error
);
1914 isl_assert(tab
->mat
->ctx
, context_tab
->samples
->n_col
== 1 + context_tab
->n_var
, goto error
);
1916 ineq
= get_row_parameter_ineq(tab
, row
);
1920 strict
= is_strict(ineq
);
1923 for (i
= context_tab
->n_outside
; i
< context_tab
->n_sample
; ++i
) {
1924 isl_seq_inner_product(context_tab
->samples
->row
[i
], ineq
->el
,
1926 sgn
= isl_int_sgn(tmp
);
1927 if (sgn
> 0 || (sgn
== 0 && (critical
|| strict
))) {
1928 if (res
== isl_tab_row_unknown
)
1929 res
= isl_tab_row_pos
;
1930 if (res
== isl_tab_row_neg
)
1931 res
= isl_tab_row_any
;
1934 if (res
== isl_tab_row_unknown
)
1935 res
= isl_tab_row_neg
;
1936 if (res
== isl_tab_row_pos
)
1937 res
= isl_tab_row_any
;
1939 if (res
== isl_tab_row_any
)
1944 if (res
!= isl_tab_row_any
) {
1945 if (isl_tab_extend_cons(context_tab
, 1) < 0)
1948 snap
= isl_tab_snap(context_tab
);
1949 isl_tab_push_basis(context_tab
);
1952 if (res
== isl_tab_row_unknown
|| res
== isl_tab_row_pos
) {
1953 /* test for negative values */
1955 isl_seq_neg(ineq
->el
, ineq
->el
, ineq
->size
);
1956 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
1958 isl_tab_push_basis(context_tab
);
1959 sol
->context_tab
= add_lexmin_ineq(sol
->context_tab
, ineq
->el
);
1960 feasible
= context_is_feasible(sol
);
1963 context_tab
= sol
->context_tab
;
1965 res
= isl_tab_row_pos
;
1967 res
= (res
== isl_tab_row_unknown
) ? isl_tab_row_neg
1969 if (isl_tab_rollback(context_tab
, snap
) < 0)
1972 if (res
== isl_tab_row_neg
) {
1973 isl_seq_neg(ineq
->el
, ineq
->el
, ineq
->size
);
1974 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
1978 if (res
== isl_tab_row_neg
) {
1979 /* test for positive values */
1981 if (!critical
&& !strict
)
1982 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
1984 isl_tab_push_basis(context_tab
);
1985 sol
->context_tab
= add_lexmin_ineq(sol
->context_tab
, ineq
->el
);
1986 feasible
= context_is_feasible(sol
);
1989 context_tab
= sol
->context_tab
;
1991 res
= isl_tab_row_any
;
1992 if (isl_tab_rollback(context_tab
, snap
) < 0)
2003 static struct isl_sol
*find_solutions(struct isl_sol
*sol
, struct isl_tab
*tab
);
2005 /* Find solutions for values of the parameters that satisfy the given
2008 * We currently take a snapshot of the context tableau that is reset
2009 * when we return from this function, while we make a copy of the main
2010 * tableau, leaving the original main tableau untouched.
2011 * These are fairly arbitrary choices. Making a copy also of the context
2012 * tableau would obviate the need to undo any changes made to it later,
2013 * while taking a snapshot of the main tableau could reduce memory usage.
2014 * If we were to switch to taking a snapshot of the main tableau,
2015 * we would have to keep in mind that we need to save the row signs
2016 * and that we need to do this before saving the current basis
2017 * such that the basis has been restore before we restore the row signs.
2019 static struct isl_sol
*find_in_pos(struct isl_sol
*sol
,
2020 struct isl_tab
*tab
, isl_int
*ineq
)
2022 struct isl_tab_undo
*snap
;
2024 snap
= isl_tab_snap(sol
->context_tab
);
2025 isl_tab_push_basis(sol
->context_tab
);
2026 if (isl_tab_extend_cons(sol
->context_tab
, 1) < 0)
2029 tab
= isl_tab_dup(tab
);
2033 sol
->context_tab
= add_lexmin_ineq(sol
->context_tab
, ineq
);
2034 sol
->context_tab
= check_samples(sol
->context_tab
, ineq
, 0);
2036 sol
= find_solutions(sol
, tab
);
2038 isl_tab_rollback(sol
->context_tab
, snap
);
2041 isl_tab_rollback(sol
->context_tab
, snap
);
2046 /* Record the absence of solutions for those values of the parameters
2047 * that do not satisfy the given inequality with equality.
2049 static struct isl_sol
*no_sol_in_strict(struct isl_sol
*sol
,
2050 struct isl_tab
*tab
, struct isl_vec
*ineq
)
2054 struct isl_tab_undo
*snap
;
2055 snap
= isl_tab_snap(sol
->context_tab
);
2056 isl_tab_push_basis(sol
->context_tab
);
2057 if (isl_tab_extend_cons(sol
->context_tab
, 1) < 0)
2060 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
2062 sol
->context_tab
= add_lexmin_ineq(sol
->context_tab
, ineq
->el
);
2063 f
= context_valid_sample_or_feasible(sol
, ineq
->el
, 0);
2069 sol
= sol
->add(sol
, tab
);
2072 isl_int_add_ui(ineq
->el
[0], ineq
->el
[0], 1);
2074 if (isl_tab_rollback(sol
->context_tab
, snap
) < 0)
2082 /* Given a main tableau where more than one row requires a split,
2083 * determine and return the "best" row to split on.
2085 * Given two rows in the main tableau, if the inequality corresponding
2086 * to the first row is redundant with respect to that of the second row
2087 * in the current tableau, then it is better to split on the second row,
2088 * since in the positive part, both row will be positive.
2089 * (In the negative part a pivot will have to be performed and just about
2090 * anything can happen to the sign of the other row.)
2092 * As a simple heuristic, we therefore select the row that makes the most
2093 * of the other rows redundant.
2095 * Perhaps it would also be useful to look at the number of constraints
2096 * that conflict with any given constraint.
2098 static int best_split(struct isl_tab
*tab
, struct isl_tab
*context_tab
)
2100 struct isl_tab_undo
*snap
, *snap2
;
2106 if (isl_tab_extend_cons(context_tab
, 2) < 0)
2109 snap
= isl_tab_snap(context_tab
);
2110 isl_tab_push_basis(context_tab
);
2111 snap2
= isl_tab_snap(context_tab
);
2113 for (split
= tab
->n_redundant
; split
< tab
->n_row
; ++split
) {
2114 struct isl_tab_undo
*snap3
;
2115 struct isl_vec
*ineq
= NULL
;
2118 if (!isl_tab_var_from_row(tab
, split
)->is_nonneg
)
2120 if (tab
->row_sign
[split
] != isl_tab_row_any
)
2123 ineq
= get_row_parameter_ineq(tab
, split
);
2126 context_tab
= isl_tab_add_ineq(context_tab
, ineq
->el
);
2129 snap3
= isl_tab_snap(context_tab
);
2131 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
2132 struct isl_tab_var
*var
;
2136 if (!isl_tab_var_from_row(tab
, row
)->is_nonneg
)
2138 if (tab
->row_sign
[row
] != isl_tab_row_any
)
2141 ineq
= get_row_parameter_ineq(tab
, row
);
2144 context_tab
= isl_tab_add_ineq(context_tab
, ineq
->el
);
2146 var
= &context_tab
->con
[context_tab
->n_con
- 1];
2147 if (!context_tab
->empty
&&
2148 !isl_tab_min_at_most_neg_one(context_tab
, var
))
2150 if (isl_tab_rollback(context_tab
, snap3
) < 0)
2153 if (best
== -1 || r
> best_r
) {
2157 if (isl_tab_rollback(context_tab
, snap2
) < 0)
2161 if (isl_tab_rollback(context_tab
, snap
) < 0)
2167 /* Compute the lexicographic minimum of the set represented by the main
2168 * tableau "tab" within the context "sol->context_tab".
2169 * On entry the sample value of the main tableau is lexicographically
2170 * less than or equal to this lexicographic minimum.
2171 * Pivots are performed until a feasible point is found, which is then
2172 * necessarily equal to the minimum, or until the tableau is found to
2173 * be infeasible. Some pivots may need to be performed for only some
2174 * feasible values of the context tableau. If so, the context tableau
2175 * is split into a part where the pivot is needed and a part where it is not.
2177 * Whenever we enter the main loop, the main tableau is such that no
2178 * "obvious" pivots need to be performed on it, where "obvious" means
2179 * that the given row can be seen to be negative without looking at
2180 * the context tableau. In particular, for non-parametric problems,
2181 * no pivots need to be performed on the main tableau.
2182 * The caller of find_solutions is responsible for making this property
2183 * hold prior to the first iteration of the loop, while restore_lexmin
2184 * is called before every other iteration.
2186 * Inside the main loop, we first examine the signs of the rows of
2187 * the main tableau within the context of the context tableau.
2188 * If we find a row that is always non-positive for all values of
2189 * the parameters satisfying the context tableau and negative for at
2190 * least one value of the parameters, we perform the appropriate pivot
2191 * and start over. An exception is the case where no pivot can be
2192 * performed on the row. In this case, we require that the sign of
2193 * the row is negative for all values of the parameters (rather than just
2194 * non-positive). This special case is handled inside row_sign, which
2195 * will say that the row can have any sign if it determines that it can
2196 * attain both negative and zero values.
2198 * If we can't find a row that always requires a pivot, but we can find
2199 * one or more rows that require a pivot for some values of the parameters
2200 * (i.e., the row can attain both positive and negative signs), then we split
2201 * the context tableau into two parts, one where we force the sign to be
2202 * non-negative and one where we force is to be negative.
2203 * The non-negative part is handled by a recursive call (through find_in_pos).
2204 * Upon returning from this call, we continue with the negative part and
2205 * perform the required pivot.
2207 * If no such rows can be found, all rows are non-negative and we have
2208 * found a (rational) feasible point. If we only wanted a rational point
2210 * Otherwise, we check if all values of the sample point of the tableau
2211 * are integral for the variables. If so, we have found the minimal
2212 * integral point and we are done.
2213 * If the sample point is not integral, then we need to make a distinction
2214 * based on whether the constant term is non-integral or the coefficients
2215 * of the parameters. Furthermore, in order to decide how to handle
2216 * the non-integrality, we also need to know whether the coefficients
2217 * of the other columns in the tableau are integral. This leads
2218 * to the following table. The first two rows do not correspond
2219 * to a non-integral sample point and are only mentioned for completeness.
2221 * constant parameters other
2224 * int int rat | -> no problem
2226 * rat int int -> fail
2228 * rat int rat -> cut
2231 * rat rat rat | -> parametric cut
2234 * rat rat int | -> split context
2236 * If the parametric constant is completely integral, then there is nothing
2237 * to be done. If the constant term is non-integral, but all the other
2238 * coefficient are integral, then there is nothing that can be done
2239 * and the tableau has no integral solution.
2240 * If, on the other hand, one or more of the other columns have rational
2241 * coeffcients, but the parameter coefficients are all integral, then
2242 * we can perform a regular (non-parametric) cut.
2243 * Finally, if there is any parameter coefficient that is non-integral,
2244 * then we need to involve the context tableau. There are two cases here.
2245 * If at least one other column has a rational coefficient, then we
2246 * can perform a parametric cut in the main tableau by adding a new
2247 * integer division in the context tableau.
2248 * If all other columns have integral coefficients, then we need to
2249 * enforce that the rational combination of parameters (c + \sum a_i y_i)/m
2250 * is always integral. We do this by introducing an integer division
2251 * q = floor((c + \sum a_i y_i)/m) and stipulating that its argument should
2252 * always be integral in the context tableau, i.e., m q = c + \sum a_i y_i.
2253 * Since q is expressed in the tableau as
2254 * c + \sum a_i y_i - m q >= 0
2255 * -c - \sum a_i y_i + m q + m - 1 >= 0
2256 * it is sufficient to add the inequality
2257 * -c - \sum a_i y_i + m q >= 0
2258 * In the part of the context where this inequality does not hold, the
2259 * main tableau is marked as being empty.
2261 static struct isl_sol
*find_solutions(struct isl_sol
*sol
, struct isl_tab
*tab
)
2263 struct isl_tab
**context_tab
;
2268 context_tab
= &sol
->context_tab
;
2272 if ((*context_tab
)->empty
)
2275 for (; tab
&& !tab
->empty
; tab
= restore_lexmin(tab
)) {
2282 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
2283 if (!isl_tab_var_from_row(tab
, row
)->is_nonneg
)
2285 sgn
= row_sign(tab
, sol
, row
);
2288 tab
->row_sign
[row
] = sgn
;
2289 if (sgn
== isl_tab_row_any
)
2291 if (sgn
== isl_tab_row_any
&& split
== -1)
2293 if (sgn
== isl_tab_row_neg
)
2296 if (row
< tab
->n_row
)
2299 struct isl_vec
*ineq
;
2301 split
= best_split(tab
, *context_tab
);
2304 ineq
= get_row_parameter_ineq(tab
, split
);
2308 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
2309 if (!isl_tab_var_from_row(tab
, row
)->is_nonneg
)
2311 if (tab
->row_sign
[row
] == isl_tab_row_any
)
2312 tab
->row_sign
[row
] = isl_tab_row_unknown
;
2314 tab
->row_sign
[split
] = isl_tab_row_pos
;
2315 sol
= find_in_pos(sol
, tab
, ineq
->el
);
2316 tab
->row_sign
[split
] = isl_tab_row_neg
;
2318 isl_seq_neg(ineq
->el
, ineq
->el
, ineq
->size
);
2319 isl_int_sub_ui(ineq
->el
[0], ineq
->el
[0], 1);
2320 *context_tab
= add_lexmin_ineq(*context_tab
, ineq
->el
);
2321 *context_tab
= check_samples(*context_tab
, ineq
->el
, 0);
2329 row
= first_non_integer(tab
, &flags
);
2332 if (ISL_FL_ISSET(flags
, I_PAR
)) {
2333 if (ISL_FL_ISSET(flags
, I_VAR
)) {
2334 tab
= isl_tab_mark_empty(tab
);
2337 row
= add_cut(tab
, row
);
2338 } else if (ISL_FL_ISSET(flags
, I_VAR
)) {
2339 struct isl_vec
*div
;
2340 struct isl_vec
*ineq
;
2342 if (isl_tab_extend_cons(*context_tab
, 3) < 0)
2344 div
= get_row_split_div(tab
, row
);
2347 d
= get_div(tab
, context_tab
, div
);
2351 ineq
= ineq_for_div((*context_tab
)->bset
, d
);
2352 sol
= no_sol_in_strict(sol
, tab
, ineq
);
2353 isl_seq_neg(ineq
->el
, ineq
->el
, ineq
->size
);
2354 *context_tab
= add_lexmin_ineq(*context_tab
, ineq
->el
);
2355 *context_tab
= check_samples(*context_tab
, ineq
->el
, 0);
2359 tab
= set_row_cst_to_div(tab
, row
, d
);
2361 row
= add_parametric_cut(tab
, row
, context_tab
);
2366 sol
= sol
->add(sol
, tab
);
2375 /* Compute the lexicographic minimum of the set represented by the main
2376 * tableau "tab" within the context "sol->context_tab".
2378 * As a preprocessing step, we first transfer all the purely parametric
2379 * equalities from the main tableau to the context tableau, i.e.,
2380 * parameters that have been pivoted to a row.
2381 * These equalities are ignored by the main algorithm, because the
2382 * corresponding rows may not be marked as being non-negative.
2383 * In parts of the context where the added equality does not hold,
2384 * the main tableau is marked as being empty.
2386 static struct isl_sol
*find_solutions_main(struct isl_sol
*sol
,
2387 struct isl_tab
*tab
)
2391 for (row
= tab
->n_redundant
; row
< tab
->n_row
; ++row
) {
2395 if (tab
->row_var
[row
] < 0)
2397 if (tab
->row_var
[row
] >= tab
->n_param
&&
2398 tab
->row_var
[row
] < tab
->n_var
- tab
->n_div
)
2400 if (tab
->row_var
[row
] < tab
->n_param
)
2401 p
= tab
->row_var
[row
];
2403 p
= tab
->row_var
[row
]
2404 + tab
->n_param
- (tab
->n_var
- tab
->n_div
);
2406 if (isl_tab_extend_cons(sol
->context_tab
, 2) < 0)
2409 eq
= isl_vec_alloc(tab
->mat
->ctx
, 1+tab
->n_param
+tab
->n_div
);
2410 get_row_parameter_line(tab
, row
, eq
->el
);
2411 isl_int_neg(eq
->el
[1 + p
], tab
->mat
->row
[row
][0]);
2412 eq
= isl_vec_normalize(eq
);
2414 sol
= no_sol_in_strict(sol
, tab
, eq
);
2416 isl_seq_neg(eq
->el
, eq
->el
, eq
->size
);
2417 sol
= no_sol_in_strict(sol
, tab
, eq
);
2418 isl_seq_neg(eq
->el
, eq
->el
, eq
->size
);
2420 sol
->context_tab
= add_lexmin_eq(sol
->context_tab
, eq
->el
);
2421 context_valid_sample_or_feasible(sol
, eq
->el
, 1);
2422 sol
->context_tab
= check_samples(sol
->context_tab
, eq
->el
, 1);
2426 isl_tab_mark_redundant(tab
, row
);
2428 if (!sol
->context_tab
)
2430 if (sol
->context_tab
->empty
)
2433 row
= tab
->n_redundant
- 1;
2436 return find_solutions(sol
, tab
);
2443 static struct isl_sol_map
*sol_map_find_solutions(struct isl_sol_map
*sol_map
,
2444 struct isl_tab
*tab
)
2446 return (struct isl_sol_map
*)find_solutions_main(&sol_map
->sol
, tab
);
2449 /* Check if integer division "div" of "dom" also occurs in "bmap".
2450 * If so, return its position within the divs.
2451 * If not, return -1.
2453 static int find_context_div(struct isl_basic_map
*bmap
,
2454 struct isl_basic_set
*dom
, unsigned div
)
2457 unsigned b_dim
= isl_dim_total(bmap
->dim
);
2458 unsigned d_dim
= isl_dim_total(dom
->dim
);
2460 if (isl_int_is_zero(dom
->div
[div
][0]))
2462 if (isl_seq_first_non_zero(dom
->div
[div
] + 2 + d_dim
, dom
->n_div
) != -1)
2465 for (i
= 0; i
< bmap
->n_div
; ++i
) {
2466 if (isl_int_is_zero(bmap
->div
[i
][0]))
2468 if (isl_seq_first_non_zero(bmap
->div
[i
] + 2 + d_dim
,
2469 (b_dim
- d_dim
) + bmap
->n_div
) != -1)
2471 if (isl_seq_eq(bmap
->div
[i
], dom
->div
[div
], 2 + d_dim
))
2477 /* The correspondence between the variables in the main tableau,
2478 * the context tableau, and the input map and domain is as follows.
2479 * The first n_param and the last n_div variables of the main tableau
2480 * form the variables of the context tableau.
2481 * In the basic map, these n_param variables correspond to the
2482 * parameters and the input dimensions. In the domain, they correspond
2483 * to the parameters and the set dimensions.
2484 * The n_div variables correspond to the integer divisions in the domain.
2485 * To ensure that everything lines up, we may need to copy some of the
2486 * integer divisions of the domain to the map. These have to be placed
2487 * in the same order as those in the context and they have to be placed
2488 * after any other integer divisions that the map may have.
2489 * This function performs the required reordering.
2491 static struct isl_basic_map
*align_context_divs(struct isl_basic_map
*bmap
,
2492 struct isl_basic_set
*dom
)
2498 for (i
= 0; i
< dom
->n_div
; ++i
)
2499 if (find_context_div(bmap
, dom
, i
) != -1)
2501 other
= bmap
->n_div
- common
;
2502 if (dom
->n_div
- common
> 0) {
2503 bmap
= isl_basic_map_extend_dim(bmap
, isl_dim_copy(bmap
->dim
),
2504 dom
->n_div
- common
, 0, 0);
2508 for (i
= 0; i
< dom
->n_div
; ++i
) {
2509 int pos
= find_context_div(bmap
, dom
, i
);
2511 pos
= isl_basic_map_alloc_div(bmap
);
2514 isl_int_set_si(bmap
->div
[pos
][0], 0);
2516 if (pos
!= other
+ i
)
2517 isl_basic_map_swap_div(bmap
, pos
, other
+ i
);
2521 isl_basic_map_free(bmap
);
2525 /* Compute the lexicographic minimum (or maximum if "max" is set)
2526 * of "bmap" over the domain "dom" and return the result as a map.
2527 * If "empty" is not NULL, then *empty is assigned a set that
2528 * contains those parts of the domain where there is no solution.
2529 * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL),
2530 * then we compute the rational optimum. Otherwise, we compute
2531 * the integral optimum.
2533 * We perform some preprocessing. As the PILP solver does not
2534 * handle implicit equalities very well, we first make sure all
2535 * the equalities are explicitly available.
2536 * We also make sure the divs in the domain are properly order,
2537 * because they will be added one by one in the given order
2538 * during the construction of the solution map.
2540 struct isl_map
*isl_tab_basic_map_partial_lexopt(
2541 struct isl_basic_map
*bmap
, struct isl_basic_set
*dom
,
2542 struct isl_set
**empty
, int max
)
2544 struct isl_tab
*tab
;
2545 struct isl_map
*result
= NULL
;
2546 struct isl_sol_map
*sol_map
= NULL
;
2553 isl_assert(bmap
->ctx
,
2554 isl_basic_map_compatible_domain(bmap
, dom
), goto error
);
2556 bmap
= isl_basic_map_detect_equalities(bmap
);
2559 dom
= isl_basic_set_order_divs(dom
);
2560 bmap
= align_context_divs(bmap
, dom
);
2562 sol_map
= sol_map_init(bmap
, dom
, !!empty
, max
);
2566 if (isl_basic_set_fast_is_empty(sol_map
->sol
.context_tab
->bset
))
2568 else if (isl_basic_map_fast_is_empty(bmap
))
2569 sol_map
= add_empty(sol_map
);
2571 tab
= tab_for_lexmin(bmap
,
2572 sol_map
->sol
.context_tab
->bset
, 1, max
);
2573 tab
= tab_detect_nonnegative_parameters(tab
,
2574 sol_map
->sol
.context_tab
);
2575 sol_map
= sol_map_find_solutions(sol_map
, tab
);
2580 result
= isl_map_copy(sol_map
->map
);
2582 *empty
= isl_set_copy(sol_map
->empty
);
2583 sol_map_free(sol_map
);
2584 isl_basic_map_free(bmap
);
2587 sol_map_free(sol_map
);
2588 isl_basic_map_free(bmap
);