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 are currently two implementations 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
54 * isl_sol_for, which calls a user-defined function for each part of
58 struct isl_tab *context_tab;
59 struct isl_sol *(*add)(struct isl_sol *sol, struct isl_tab *tab);
60 void (*free)(struct isl_sol *sol);
63 static void sol_free(struct isl_sol *sol)
73 struct isl_set *empty;
77 static void sol_map_free(struct isl_sol_map *sol_map)
79 isl_tab_free(sol_map->sol.context_tab);
80 isl_map_free(sol_map->map);
81 isl_set_free(sol_map->empty);
85 static void sol_map_free_wrap(struct isl_sol *sol)
87 sol_map_free((struct isl_sol_map *)sol);
90 static struct isl_sol_map *add_empty(struct isl_sol_map *sol)
92 struct isl_basic_set *bset;
96 sol->empty = isl_set_grow(sol->empty, 1);
97 bset = isl_basic_set_copy(sol->sol.context_tab->bset);
98 bset = isl_basic_set_simplify(bset);
99 bset = isl_basic_set_finalize(bset);
100 sol->empty = isl_set_add(sol->empty, bset);
109 /* Add the solution identified by the tableau and the context tableau.
111 * The layout of the variables is as follows.
112 * tab->n_var is equal to the total number of variables in the input
113 * map (including divs that were copied from the context)
114 * + the number of extra divs constructed
115 * Of these, the first tab->n_param and the last tab->n_div variables
116 * correspond to the variables in the context, i.e.,
117 * tab->n_param + tab->n_div = context_tab->n_var
118 * tab->n_param is equal to the number of parameters and input
119 * dimensions in the input map
120 * tab->n_div is equal to the number of divs in the context
122 * If there is no solution, then the basic set corresponding to the
123 * context tableau is added to the set "empty".
125 * Otherwise, a basic map is constructed with the same parameters
126 * and divs as the context, the dimensions of the context as input
127 * dimensions and a number of output dimensions that is equal to
128 * the number of output dimensions in the input map.
129 * The divs in the input map (if any) that do not correspond to any
130 * div in the context do not appear in the solution.
131 * The algorithm will make sure that they have an integer value,
132 * but these values themselves are of no interest.
134 * The constraints and divs of the context are simply copied
135 * fron context_tab->bset.
136 * To extract the value of the output variables, it should be noted
137 * that we always use a big parameter M and so the variable stored
138 * in the tableau is not an output variable x itself, but
139 * x' = M + x (in case of minimization)
141 * x' = M - x (in case of maximization)
142 * If x' appears in a column, then its optimal value is zero,
143 * which means that the optimal value of x is an unbounded number
144 * (-M for minimization and M for maximization).
145 * We currently assume that the output dimensions in the original map
146 * are bounded, so this cannot occur.
147 * Similarly, when x' appears in a row, then the coefficient of M in that
148 * row is necessarily 1.
149 * If the row represents
150 * d x' = c + d M + e(y)
151 * then, in case of minimization, an equality
152 * c + e(y) - d x' = 0
153 * is added, and in case of maximization,
154 * c + e(y) + d x' = 0
156 static struct isl_sol_map *sol_map_add(struct isl_sol_map *sol,
160 struct isl_basic_map *bmap = NULL;
161 struct isl_tab *context_tab;
174 return add_empty(sol);
176 context_tab = sol->sol.context_tab;
178 n_out = isl_map_dim(sol->map, isl_dim_out);
179 n_eq = context_tab->bset->n_eq + n_out;
180 n_ineq = context_tab->bset->n_ineq;
181 nparam = tab->n_param;
182 total = isl_map_dim(sol->map, isl_dim_all);
183 bmap = isl_basic_map_alloc_dim(isl_map_get_dim(sol->map),
184 tab->n_div, n_eq, 2 * tab->n_div + n_ineq);
189 ISL_F_SET(bmap, ISL_BASIC_MAP_RATIONAL);
190 for (i = 0; i < context_tab->bset->n_div; ++i) {
191 int k = isl_basic_map_alloc_div(bmap);
194 isl_seq_cpy(bmap->div[k],
195 context_tab->bset->div[i], 1 + 1 + nparam);
196 isl_seq_clr(bmap->div[k] + 1 + 1 + nparam, total - nparam);
197 isl_seq_cpy(bmap->div[k] + 1 + 1 + total,
198 context_tab->bset->div[i] + 1 + 1 + nparam, i);
200 for (i = 0; i < context_tab->bset->n_eq; ++i) {
201 int k = isl_basic_map_alloc_equality(bmap);
204 isl_seq_cpy(bmap->eq[k], context_tab->bset->eq[i], 1 + nparam);
205 isl_seq_clr(bmap->eq[k] + 1 + nparam, total - nparam);
206 isl_seq_cpy(bmap->eq[k] + 1 + total,
207 context_tab->bset->eq[i] + 1 + nparam, n_div);
209 for (i = 0; i < context_tab->bset->n_ineq; ++i) {
210 int k = isl_basic_map_alloc_inequality(bmap);
213 isl_seq_cpy(bmap->ineq[k],
214 context_tab->bset->ineq[i], 1 + nparam);
215 isl_seq_clr(bmap->ineq[k] + 1 + nparam, total - nparam);
216 isl_seq_cpy(bmap->ineq[k] + 1 + total,
217 context_tab->bset->ineq[i] + 1 + nparam, n_div);
219 for (i = tab->n_param; i < total; ++i) {
220 int k = isl_basic_map_alloc_equality(bmap);
223 isl_seq_clr(bmap->eq[k] + 1, isl_basic_map_total_dim(bmap));
224 if (!tab->var[i].is_row) {
226 isl_assert(bmap->ctx, !tab->M, goto error);
227 isl_int_set_si(bmap->eq[k][0], 0);
229 isl_int_set_si(bmap->eq[k][1 + i], 1);
231 isl_int_set_si(bmap->eq[k][1 + i], -1);
234 row = tab->var[i].index;
237 isl_assert(bmap->ctx,
238 isl_int_eq(tab->mat->row[row][2],
239 tab->mat->row[row][0]),
241 isl_int_set(bmap->eq[k][0], tab->mat->row[row][1]);
242 for (j = 0; j < tab->n_param; ++j) {
244 if (tab->var[j].is_row)
246 col = tab->var[j].index;
247 isl_int_set(bmap->eq[k][1 + j],
248 tab->mat->row[row][off + col]);
250 for (j = 0; j < tab->n_div; ++j) {
252 if (tab->var[tab->n_var - tab->n_div+j].is_row)
254 col = tab->var[tab->n_var - tab->n_div+j].index;
255 isl_int_set(bmap->eq[k][1 + total + j],
256 tab->mat->row[row][off + col]);
259 isl_int_set(bmap->eq[k][1 + i],
260 tab->mat->row[row][0]);
262 isl_int_neg(bmap->eq[k][1 + i],
263 tab->mat->row[row][0]);
266 bmap = isl_basic_map_simplify(bmap);
267 bmap = isl_basic_map_finalize(bmap);
268 sol->map = isl_map_grow(sol->map, 1);
269 sol->map = isl_map_add(sol->map, bmap);
274 isl_basic_map_free(bmap);
279 static struct isl_sol *sol_map_add_wrap(struct isl_sol *sol,
282 return (struct isl_sol *)sol_map_add((struct isl_sol_map *)sol, tab);
286 /* Store the "parametric constant" of row "row" of tableau "tab" in "line",
287 * i.e., the constant term and the coefficients of all variables that
288 * appear in the context tableau.
289 * Note that the coefficient of the big parameter M is NOT copied.
290 * The context tableau may not have a big parameter and even when it
291 * does, it is a different big parameter.
293 static void get_row_parameter_line(struct isl_tab *tab, int row, isl_int *line)
296 unsigned off = 2 + tab->M;
298 isl_int_set(line[0], tab->mat->row[row][1]);
299 for (i = 0; i < tab->n_param; ++i) {
300 if (tab->var[i].is_row)
301 isl_int_set_si(line[1 + i], 0);
303 int col = tab->var[i].index;
304 isl_int_set(line[1 + i], tab->mat->row[row][off + col]);
307 for (i = 0; i < tab->n_div; ++i) {
308 if (tab->var[tab->n_var - tab->n_div + i].is_row)
309 isl_int_set_si(line[1 + tab->n_param + i], 0);
311 int col = tab->var[tab->n_var - tab->n_div + i].index;
312 isl_int_set(line[1 + tab->n_param + i],
313 tab->mat->row[row][off + col]);
318 /* Check if rows "row1" and "row2" have identical "parametric constants",
319 * as explained above.
320 * In this case, we also insist that the coefficients of the big parameter
321 * be the same as the values of the constants will only be the same
322 * if these coefficients are also the same.
324 static int identical_parameter_line(struct isl_tab *tab, int row1, int row2)
327 unsigned off = 2 + tab->M;
329 if (isl_int_ne(tab->mat->row[row1][1], tab->mat->row[row2][1]))
332 if (tab->M && isl_int_ne(tab->mat->row[row1][2],
333 tab->mat->row[row2][2]))
336 for (i = 0; i < tab->n_param + tab->n_div; ++i) {
337 int pos = i < tab->n_param ? i :
338 tab->n_var - tab->n_div + i - tab->n_param;
341 if (tab->var[pos].is_row)
343 col = tab->var[pos].index;
344 if (isl_int_ne(tab->mat->row[row1][off + col],
345 tab->mat->row[row2][off + col]))
351 /* Return an inequality that expresses that the "parametric constant"
352 * should be non-negative.
353 * This function is only called when the coefficient of the big parameter
356 static struct isl_vec *get_row_parameter_ineq(struct isl_tab *tab, int row)
358 struct isl_vec *ineq;
360 ineq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_param + tab->n_div);
364 get_row_parameter_line(tab, row, ineq->el);
366 ineq = isl_vec_normalize(ineq);
371 /* Return a integer division for use in a parametric cut based on the given row.
372 * In particular, let the parametric constant of the row be
376 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
377 * The div returned is equal to
379 * floor(\sum_i {-a_i} y_i) = floor((\sum_i (-a_i mod d) y_i)/d)
381 static struct isl_vec *get_row_parameter_div(struct isl_tab *tab, int row)
385 div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
389 isl_int_set(div->el[0], tab->mat->row[row][0]);
390 get_row_parameter_line(tab, row, div->el + 1);
391 div = isl_vec_normalize(div);
392 isl_seq_neg(div->el + 1, div->el + 1, div->size - 1);
393 isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
398 /* Return a integer division for use in transferring an integrality constraint
400 * In particular, let the parametric constant of the row be
404 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
405 * The the returned div is equal to
407 * floor(\sum_i {a_i} y_i) = floor((\sum_i (a_i mod d) y_i)/d)
409 static struct isl_vec *get_row_split_div(struct isl_tab *tab, int row)
413 div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
417 isl_int_set(div->el[0], tab->mat->row[row][0]);
418 get_row_parameter_line(tab, row, div->el + 1);
419 div = isl_vec_normalize(div);
420 isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
425 /* Construct and return an inequality that expresses an upper bound
427 * In particular, if the div is given by
431 * then the inequality expresses
435 static struct isl_vec *ineq_for_div(struct isl_basic_set *bset, unsigned div)
439 struct isl_vec *ineq;
441 total = isl_basic_set_total_dim(bset);
442 div_pos = 1 + total - bset->n_div + div;
444 ineq = isl_vec_alloc(bset->ctx, 1 + total);
448 isl_seq_cpy(ineq->el, bset->div[div] + 1, 1 + total);
449 isl_int_neg(ineq->el[div_pos], bset->div[div][0]);
453 /* Given a row in the tableau and a div that was created
454 * using get_row_split_div and that been constrained to equality, i.e.,
456 * d = floor(\sum_i {a_i} y_i) = \sum_i {a_i} y_i
458 * replace the expression "\sum_i {a_i} y_i" in the row by d,
459 * i.e., we subtract "\sum_i {a_i} y_i" and add 1 d.
460 * The coefficients of the non-parameters in the tableau have been
461 * verified to be integral. We can therefore simply replace coefficient b
462 * by floor(b). For the coefficients of the parameters we have
463 * floor(a_i) = a_i - {a_i}, while for the other coefficients, we have
466 static struct isl_tab *set_row_cst_to_div(struct isl_tab *tab, int row, int div)
469 unsigned off = 2 + tab->M;
471 isl_seq_fdiv_q(tab->mat->row[row] + 1, tab->mat->row[row] + 1,
472 tab->mat->row[row][0], 1 + tab->M + tab->n_col);
474 isl_int_set_si(tab->mat->row[row][0], 1);
476 isl_assert(tab->mat->ctx,
477 !tab->var[tab->n_var - tab->n_div + div].is_row, goto error);
479 col = tab->var[tab->n_var - tab->n_div + div].index;
480 isl_int_set_si(tab->mat->row[row][off + col], 1);
488 /* Check if the (parametric) constant of the given row is obviously
489 * negative, meaning that we don't need to consult the context tableau.
490 * If there is a big parameter and its coefficient is non-zero,
491 * then this coefficient determines the outcome.
492 * Otherwise, we check whether the constant is negative and
493 * all non-zero coefficients of parameters are negative and
494 * belong to non-negative parameters.
496 static int is_obviously_neg(struct isl_tab *tab, int row)
500 unsigned off = 2 + tab->M;
503 if (isl_int_is_pos(tab->mat->row[row][2]))
505 if (isl_int_is_neg(tab->mat->row[row][2]))
509 if (isl_int_is_nonneg(tab->mat->row[row][1]))
511 for (i = 0; i < tab->n_param; ++i) {
512 /* Eliminated parameter */
513 if (tab->var[i].is_row)
515 col = tab->var[i].index;
516 if (isl_int_is_zero(tab->mat->row[row][off + col]))
518 if (!tab->var[i].is_nonneg)
520 if (isl_int_is_pos(tab->mat->row[row][off + col]))
523 for (i = 0; i < tab->n_div; ++i) {
524 if (tab->var[tab->n_var - tab->n_div + i].is_row)
526 col = tab->var[tab->n_var - tab->n_div + i].index;
527 if (isl_int_is_zero(tab->mat->row[row][off + col]))
529 if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
531 if (isl_int_is_pos(tab->mat->row[row][off + col]))
537 /* Check if the (parametric) constant of the given row is obviously
538 * non-negative, meaning that we don't need to consult the context tableau.
539 * If there is a big parameter and its coefficient is non-zero,
540 * then this coefficient determines the outcome.
541 * Otherwise, we check whether the constant is non-negative and
542 * all non-zero coefficients of parameters are positive and
543 * belong to non-negative parameters.
545 static int is_obviously_nonneg(struct isl_tab *tab, int row)
549 unsigned off = 2 + tab->M;
552 if (isl_int_is_pos(tab->mat->row[row][2]))
554 if (isl_int_is_neg(tab->mat->row[row][2]))
558 if (isl_int_is_neg(tab->mat->row[row][1]))
560 for (i = 0; i < tab->n_param; ++i) {
561 /* Eliminated parameter */
562 if (tab->var[i].is_row)
564 col = tab->var[i].index;
565 if (isl_int_is_zero(tab->mat->row[row][off + col]))
567 if (!tab->var[i].is_nonneg)
569 if (isl_int_is_neg(tab->mat->row[row][off + col]))
572 for (i = 0; i < tab->n_div; ++i) {
573 if (tab->var[tab->n_var - tab->n_div + i].is_row)
575 col = tab->var[tab->n_var - tab->n_div + i].index;
576 if (isl_int_is_zero(tab->mat->row[row][off + col]))
578 if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
580 if (isl_int_is_neg(tab->mat->row[row][off + col]))
586 /* Given a row r and two columns, return the column that would
587 * lead to the lexicographically smallest increment in the sample
588 * solution when leaving the basis in favor of the row.
589 * Pivoting with column c will increment the sample value by a non-negative
590 * constant times a_{V,c}/a_{r,c}, with a_{V,c} the elements of column c
591 * corresponding to the non-parametric variables.
592 * If variable v appears in a column c_v, the a_{v,c} = 1 iff c = c_v,
593 * with all other entries in this virtual row equal to zero.
594 * If variable v appears in a row, then a_{v,c} is the element in column c
597 * Let v be the first variable with a_{v,c1}/a_{r,c1} != a_{v,c2}/a_{r,c2}.
598 * Then if a_{v,c1}/a_{r,c1} < a_{v,c2}/a_{r,c2}, i.e.,
599 * a_{v,c2} a_{r,c1} - a_{v,c1} a_{r,c2} > 0, c1 results in the minimal
600 * increment. Otherwise, it's c2.
602 static int lexmin_col_pair(struct isl_tab *tab,
603 int row, int col1, int col2, isl_int tmp)
608 tr = tab->mat->row[row] + 2 + tab->M;
610 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
614 if (!tab->var[i].is_row) {
615 if (tab->var[i].index == col1)
617 if (tab->var[i].index == col2)
622 if (tab->var[i].index == row)
625 r = tab->mat->row[tab->var[i].index] + 2 + tab->M;
626 s1 = isl_int_sgn(r[col1]);
627 s2 = isl_int_sgn(r[col2]);
628 if (s1 == 0 && s2 == 0)
635 isl_int_mul(tmp, r[col2], tr[col1]);
636 isl_int_submul(tmp, r[col1], tr[col2]);
637 if (isl_int_is_pos(tmp))
639 if (isl_int_is_neg(tmp))
645 /* Given a row in the tableau, find and return the column that would
646 * result in the lexicographically smallest, but positive, increment
647 * in the sample point.
648 * If there is no such column, then return tab->n_col.
649 * If anything goes wrong, return -1.
651 static int lexmin_pivot_col(struct isl_tab *tab, int row)
654 int col = tab->n_col;
658 tr = tab->mat->row[row] + 2 + tab->M;
662 for (j = tab->n_dead; j < tab->n_col; ++j) {
663 if (tab->col_var[j] >= 0 &&
664 (tab->col_var[j] < tab->n_param ||
665 tab->col_var[j] >= tab->n_var - tab->n_div))
668 if (!isl_int_is_pos(tr[j]))
671 if (col == tab->n_col)
674 col = lexmin_col_pair(tab, row, col, j, tmp);
675 isl_assert(tab->mat->ctx, col >= 0, goto error);
685 /* Return the first known violated constraint, i.e., a non-negative
686 * contraint that currently has an either obviously negative value
687 * or a previously determined to be negative value.
689 * If any constraint has a negative coefficient for the big parameter,
690 * if any, then we return one of these first.
692 static int first_neg(struct isl_tab *tab)
697 for (row = tab->n_redundant; row < tab->n_row; ++row) {
698 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
700 if (isl_int_is_neg(tab->mat->row[row][2]))
703 for (row = tab->n_redundant; row < tab->n_row; ++row) {
704 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
707 if (tab->row_sign[row] == 0 &&
708 is_obviously_neg(tab, row))
709 tab->row_sign[row] = isl_tab_row_neg;
710 if (tab->row_sign[row] != isl_tab_row_neg)
712 } else if (!is_obviously_neg(tab, row))
719 /* Resolve all known or obviously violated constraints through pivoting.
720 * In particular, as long as we can find any violated constraint, we
721 * look for a pivoting column that would result in the lexicographicallly
722 * smallest increment in the sample point. If there is no such column
723 * then the tableau is infeasible.
725 static struct isl_tab *restore_lexmin(struct isl_tab *tab)
733 while ((row = first_neg(tab)) != -1) {
734 col = lexmin_pivot_col(tab, row);
735 if (col >= tab->n_col)
736 return isl_tab_mark_empty(tab);
739 isl_tab_pivot(tab, row, col);
747 /* Given a row that represents an equality, look for an appropriate
749 * In particular, if there are any non-zero coefficients among
750 * the non-parameter variables, then we take the last of these
751 * variables. Eliminating this variable in terms of the other
752 * variables and/or parameters does not influence the property
753 * that all column in the initial tableau are lexicographically
754 * positive. The row corresponding to the eliminated variable
755 * will only have non-zero entries below the diagonal of the
756 * initial tableau. That is, we transform
762 * If there is no such non-parameter variable, then we are dealing with
763 * pure parameter equality and we pick any parameter with coefficient 1 or -1
764 * for elimination. This will ensure that the eliminated parameter
765 * always has an integer value whenever all the other parameters are integral.
766 * If there is no such parameter then we return -1.
768 static int last_var_col_or_int_par_col(struct isl_tab *tab, int row)
770 unsigned off = 2 + tab->M;
773 for (i = tab->n_var - tab->n_div - 1; i >= 0 && i >= tab->n_param; --i) {
775 if (tab->var[i].is_row)
777 col = tab->var[i].index;
778 if (col <= tab->n_dead)
780 if (!isl_int_is_zero(tab->mat->row[row][off + col]))
783 for (i = tab->n_dead; i < tab->n_col; ++i) {
784 if (isl_int_is_one(tab->mat->row[row][off + i]))
786 if (isl_int_is_negone(tab->mat->row[row][off + i]))
792 /* Add an equality that is known to be valid to the tableau.
793 * We first check if we can eliminate a variable or a parameter.
794 * If not, we add the equality as two inequalities.
795 * In this case, the equality was a pure parameter equality and there
796 * is no need to resolve any constraint violations.
798 static struct isl_tab *add_lexmin_valid_eq(struct isl_tab *tab, isl_int *eq)
805 r = isl_tab_add_row(tab, eq);
809 r = tab->con[r].index;
810 i = last_var_col_or_int_par_col(tab, r);
812 tab->con[r].is_nonneg = 1;
813 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
814 isl_seq_neg(eq, eq, 1 + tab->n_var);
815 r = isl_tab_add_row(tab, eq);
818 tab->con[r].is_nonneg = 1;
819 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
821 isl_tab_pivot(tab, r, i);
822 isl_tab_kill_col(tab, i);
825 tab = restore_lexmin(tab);
834 /* Check if the given row is a pure constant.
836 static int is_constant(struct isl_tab *tab, int row)
838 unsigned off = 2 + tab->M;
840 return isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
841 tab->n_col - tab->n_dead) == -1;
844 /* Add an equality that may or may not be valid to the tableau.
845 * If the resulting row is a pure constant, then it must be zero.
846 * Otherwise, the resulting tableau is empty.
848 * If the row is not a pure constant, then we add two inequalities,
849 * each time checking that they can be satisfied.
850 * In the end we try to use one of the two constraints to eliminate
853 static struct isl_tab *add_lexmin_eq(struct isl_tab *tab, isl_int *eq)
857 struct isl_tab_undo *snap;
861 snap = isl_tab_snap(tab);
862 r1 = isl_tab_add_row(tab, eq);
865 tab->con[r1].is_nonneg = 1;
866 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r1]);
868 row = tab->con[r1].index;
869 if (is_constant(tab, row)) {
870 if (!isl_int_is_zero(tab->mat->row[row][1]) ||
871 (tab->M && !isl_int_is_zero(tab->mat->row[row][2])))
872 return isl_tab_mark_empty(tab);
873 if (isl_tab_rollback(tab, snap) < 0)
878 tab = restore_lexmin(tab);
879 if (!tab || tab->empty)
882 isl_seq_neg(eq, eq, 1 + tab->n_var);
884 r2 = isl_tab_add_row(tab, eq);
887 tab->con[r2].is_nonneg = 1;
888 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r2]);
890 tab = restore_lexmin(tab);
891 if (!tab || tab->empty)
894 if (!tab->con[r1].is_row)
895 isl_tab_kill_col(tab, tab->con[r1].index);
896 else if (!tab->con[r2].is_row)
897 isl_tab_kill_col(tab, tab->con[r2].index);
898 else if (isl_int_is_zero(tab->mat->row[tab->con[r1].index][1])) {
899 unsigned off = 2 + tab->M;
901 int row = tab->con[r1].index;
902 i = isl_seq_first_non_zero(tab->mat->row[row]+off+tab->n_dead,
903 tab->n_col - tab->n_dead);
905 isl_tab_pivot(tab, row, tab->n_dead + i);
906 isl_tab_kill_col(tab, tab->n_dead + i);
911 tab->bset = isl_basic_set_add_ineq(tab->bset, eq);
912 isl_tab_push(tab, isl_tab_undo_bset_ineq);
913 isl_seq_neg(eq, eq, 1 + tab->n_var);
914 tab->bset = isl_basic_set_add_ineq(tab->bset, eq);
915 isl_seq_neg(eq, eq, 1 + tab->n_var);
916 isl_tab_push(tab, isl_tab_undo_bset_ineq);
927 /* Add an inequality to the tableau, resolving violations using
930 static struct isl_tab *add_lexmin_ineq(struct isl_tab *tab, isl_int *ineq)
937 tab->bset = isl_basic_set_add_ineq(tab->bset, ineq);
938 isl_tab_push(tab, isl_tab_undo_bset_ineq);
942 r = isl_tab_add_row(tab, ineq);
945 tab->con[r].is_nonneg = 1;
946 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
947 if (isl_tab_row_is_redundant(tab, tab->con[r].index)) {
948 isl_tab_mark_redundant(tab, tab->con[r].index);
952 tab = restore_lexmin(tab);
953 if (tab && !tab->empty && tab->con[r].is_row &&
954 isl_tab_row_is_redundant(tab, tab->con[r].index))
955 isl_tab_mark_redundant(tab, tab->con[r].index);
962 /* Check if the coefficients of the parameters are all integral.
964 static int integer_parameter(struct isl_tab *tab, int row)
968 unsigned off = 2 + tab->M;
970 for (i = 0; i < tab->n_param; ++i) {
971 /* Eliminated parameter */
972 if (tab->var[i].is_row)
974 col = tab->var[i].index;
975 if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
976 tab->mat->row[row][0]))
979 for (i = 0; i < tab->n_div; ++i) {
980 if (tab->var[tab->n_var - tab->n_div + i].is_row)
982 col = tab->var[tab->n_var - tab->n_div + i].index;
983 if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
984 tab->mat->row[row][0]))
990 /* Check if the coefficients of the non-parameter variables are all integral.
992 static int integer_variable(struct isl_tab *tab, int row)
995 unsigned off = 2 + tab->M;
997 for (i = 0; i < tab->n_col; ++i) {
998 if (tab->col_var[i] >= 0 &&
999 (tab->col_var[i] < tab->n_param ||
1000 tab->col_var[i] >= tab->n_var - tab->n_div))
1002 if (!isl_int_is_divisible_by(tab->mat->row[row][off + i],
1003 tab->mat->row[row][0]))
1009 /* Check if the constant term is integral.
1011 static int integer_constant(struct isl_tab *tab, int row)
1013 return isl_int_is_divisible_by(tab->mat->row[row][1],
1014 tab->mat->row[row][0]);
1017 #define I_CST 1 << 0
1018 #define I_PAR 1 << 1
1019 #define I_VAR 1 << 2
1021 /* Check for first (non-parameter) variable that is non-integer and
1022 * therefore requires a cut.
1023 * For parametric tableaus, there are three parts in a row,
1024 * the constant, the coefficients of the parameters and the rest.
1025 * For each part, we check whether the coefficients in that part
1026 * are all integral and if so, set the corresponding flag in *f.
1027 * If the constant and the parameter part are integral, then the
1028 * current sample value is integral and no cut is required
1029 * (irrespective of whether the variable part is integral).
1031 static int first_non_integer(struct isl_tab *tab, int *f)
1035 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
1038 if (!tab->var[i].is_row)
1040 row = tab->var[i].index;
1041 if (integer_constant(tab, row))
1042 ISL_FL_SET(flags, I_CST);
1043 if (integer_parameter(tab, row))
1044 ISL_FL_SET(flags, I_PAR);
1045 if (ISL_FL_ISSET(flags, I_CST) && ISL_FL_ISSET(flags, I_PAR))
1047 if (integer_variable(tab, row))
1048 ISL_FL_SET(flags, I_VAR);
1055 /* Add a (non-parametric) cut to cut away the non-integral sample
1056 * value of the given row.
1058 * If the row is given by
1060 * m r = f + \sum_i a_i y_i
1064 * c = - {-f/m} + \sum_i {a_i/m} y_i >= 0
1066 * The big parameter, if any, is ignored, since it is assumed to be big
1067 * enough to be divisible by any integer.
1068 * If the tableau is actually a parametric tableau, then this function
1069 * is only called when all coefficients of the parameters are integral.
1070 * The cut therefore has zero coefficients for the parameters.
1072 * The current value is known to be negative, so row_sign, if it
1073 * exists, is set accordingly.
1075 * Return the row of the cut or -1.
1077 static int add_cut(struct isl_tab *tab, int row)
1082 unsigned off = 2 + tab->M;
1084 if (isl_tab_extend_cons(tab, 1) < 0)
1086 r = isl_tab_allocate_con(tab);
1090 r_row = tab->mat->row[tab->con[r].index];
1091 isl_int_set(r_row[0], tab->mat->row[row][0]);
1092 isl_int_neg(r_row[1], tab->mat->row[row][1]);
1093 isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
1094 isl_int_neg(r_row[1], r_row[1]);
1096 isl_int_set_si(r_row[2], 0);
1097 for (i = 0; i < tab->n_col; ++i)
1098 isl_int_fdiv_r(r_row[off + i],
1099 tab->mat->row[row][off + i], tab->mat->row[row][0]);
1101 tab->con[r].is_nonneg = 1;
1102 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
1104 tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
1106 return tab->con[r].index;
1109 /* Given a non-parametric tableau, add cuts until an integer
1110 * sample point is obtained or until the tableau is determined
1111 * to be integer infeasible.
1112 * As long as there is any non-integer value in the sample point,
1113 * we add an appropriate cut, if possible and resolve the violated
1114 * cut constraint using restore_lexmin.
1115 * If one of the corresponding rows is equal to an integral
1116 * combination of variables/constraints plus a non-integral constant,
1117 * then there is no way to obtain an integer point an we return
1118 * a tableau that is marked empty.
1120 static struct isl_tab *cut_to_integer_lexmin(struct isl_tab *tab)
1130 while ((row = first_non_integer(tab, &flags)) != -1) {
1131 if (ISL_FL_ISSET(flags, I_VAR))
1132 return isl_tab_mark_empty(tab);
1133 row = add_cut(tab, row);
1136 tab = restore_lexmin(tab);
1137 if (!tab || tab->empty)
1146 static struct isl_tab *drop_sample(struct isl_tab *tab, int s)
1148 if (s != tab->n_outside)
1149 isl_mat_swap_rows(tab->samples, tab->n_outside, s);
1151 isl_tab_push(tab, isl_tab_undo_drop_sample);
1156 /* Check whether all the currently active samples also satisfy the inequality
1157 * "ineq" (treated as an equality if eq is set).
1158 * Remove those samples that do not.
1160 static struct isl_tab *check_samples(struct isl_tab *tab, isl_int *ineq, int eq)
1168 isl_assert(tab->mat->ctx, tab->bset, goto error);
1169 isl_assert(tab->mat->ctx, tab->samples, goto error);
1170 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);
1173 for (i = tab->n_outside; i < tab->n_sample; ++i) {
1175 isl_seq_inner_product(ineq, tab->samples->row[i],
1176 1 + tab->n_var, &v);
1177 sgn = isl_int_sgn(v);
1178 if (eq ? (sgn == 0) : (sgn >= 0))
1180 tab = drop_sample(tab, i);
1192 /* Check whether the sample value of the tableau is finite,
1193 * i.e., either the tableau does not use a big parameter, or
1194 * all values of the variables are equal to the big parameter plus
1195 * some constant. This constant is the actual sample value.
1197 static int sample_is_finite(struct isl_tab *tab)
1204 for (i = 0; i < tab->n_var; ++i) {
1206 if (!tab->var[i].is_row)
1208 row = tab->var[i].index;
1209 if (isl_int_ne(tab->mat->row[row][0], tab->mat->row[row][2]))
1215 /* Check if the context tableau of sol has any integer points.
1216 * Returns -1 if an error occurred.
1217 * If an integer point can be found and if moreover it is finite,
1218 * then it is added to the list of sample values.
1220 * This function is only called when none of the currently active sample
1221 * values satisfies the most recently added constraint.
1223 static int context_is_feasible(struct isl_sol *sol)
1225 struct isl_tab_undo *snap;
1226 struct isl_tab *tab;
1229 if (!sol || !sol->context_tab)
1232 snap = isl_tab_snap(sol->context_tab);
1233 isl_tab_push_basis(sol->context_tab);
1235 sol->context_tab = cut_to_integer_lexmin(sol->context_tab);
1236 if (!sol->context_tab)
1239 tab = sol->context_tab;
1240 if (!tab->empty && sample_is_finite(tab)) {
1241 struct isl_vec *sample;
1243 tab->samples = isl_mat_extend(tab->samples,
1244 tab->n_sample + 1, tab->samples->n_col);
1248 sample = isl_tab_get_sample_value(tab);
1251 isl_seq_cpy(tab->samples->row[tab->n_sample],
1252 sample->el, sample->size);
1253 isl_vec_free(sample);
1257 feasible = !sol->context_tab->empty;
1258 if (isl_tab_rollback(sol->context_tab, snap) < 0)
1263 isl_tab_free(sol->context_tab);
1264 sol->context_tab = NULL;
1268 /* First check if any of the currently active sample values satisfies
1269 * the inequality "ineq" (an equality if eq is set).
1270 * If not, continue with check_integer_feasible.
1272 static int context_valid_sample_or_feasible(struct isl_sol *sol,
1273 isl_int *ineq, int eq)
1277 struct isl_tab *tab;
1279 if (!sol || !sol->context_tab)
1282 tab = sol->context_tab;
1283 isl_assert(tab->mat->ctx, tab->bset, return -1);
1284 isl_assert(tab->mat->ctx, tab->samples, return -1);
1285 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, return -1);
1288 for (i = tab->n_outside; i < tab->n_sample; ++i) {
1290 isl_seq_inner_product(ineq, tab->samples->row[i],
1291 1 + tab->n_var, &v);
1292 sgn = isl_int_sgn(v);
1293 if (eq ? (sgn == 0) : (sgn >= 0))
1298 if (i < tab->n_sample)
1301 return context_is_feasible(sol);
1304 /* For a div d = floor(f/m), add the constraints
1307 * -(f-(m-1)) + m d >= 0
1309 * Note that the second constraint is the negation of
1313 static struct isl_tab *add_div_constraints(struct isl_tab *tab, unsigned div)
1317 struct isl_vec *ineq;
1322 total = isl_basic_set_total_dim(tab->bset);
1323 div_pos = 1 + total - tab->bset->n_div + div;
1325 ineq = ineq_for_div(tab->bset, div);
1329 tab = add_lexmin_ineq(tab, ineq->el);
1331 isl_seq_neg(ineq->el, tab->bset->div[div] + 1, 1 + total);
1332 isl_int_set(ineq->el[div_pos], tab->bset->div[div][0]);
1333 isl_int_add(ineq->el[0], ineq->el[0], ineq->el[div_pos]);
1334 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1335 tab = add_lexmin_ineq(tab, ineq->el);
1345 /* Add a div specified by "div" to both the main tableau and
1346 * the context tableau. In case of the main tableau, we only
1347 * need to add an extra div. In the context tableau, we also
1348 * need to express the meaning of the div.
1349 * Return the index of the div or -1 if anything went wrong.
1351 static int add_div(struct isl_tab *tab, struct isl_tab **context_tab,
1352 struct isl_vec *div)
1357 struct isl_mat *samples;
1359 if (isl_tab_extend_vars(*context_tab, 1) < 0)
1361 r = isl_tab_allocate_var(*context_tab);
1364 (*context_tab)->var[r].is_nonneg = 1;
1365 (*context_tab)->var[r].frozen = 1;
1367 samples = isl_mat_extend((*context_tab)->samples,
1368 (*context_tab)->n_sample, 1 + (*context_tab)->n_var);
1369 (*context_tab)->samples = samples;
1372 for (i = (*context_tab)->n_outside; i < samples->n_row; ++i) {
1373 isl_seq_inner_product(div->el + 1, samples->row[i],
1374 div->size - 1, &samples->row[i][samples->n_col - 1]);
1375 isl_int_fdiv_q(samples->row[i][samples->n_col - 1],
1376 samples->row[i][samples->n_col - 1], div->el[0]);
1379 (*context_tab)->bset = isl_basic_set_extend_dim((*context_tab)->bset,
1380 isl_basic_set_get_dim((*context_tab)->bset), 1, 0, 2);
1381 k = isl_basic_set_alloc_div((*context_tab)->bset);
1384 isl_seq_cpy((*context_tab)->bset->div[k], div->el, div->size);
1385 isl_tab_push((*context_tab), isl_tab_undo_bset_div);
1386 *context_tab = add_div_constraints(*context_tab, k);
1390 if (isl_tab_extend_vars(tab, 1) < 0)
1392 r = isl_tab_allocate_var(tab);
1395 if (!(*context_tab)->M)
1396 tab->var[r].is_nonneg = 1;
1397 tab->var[r].frozen = 1;
1400 return tab->n_div - 1;
1402 isl_tab_free(*context_tab);
1403 *context_tab = NULL;
1407 static int find_div(struct isl_tab *tab, isl_int *div, isl_int denom)
1410 unsigned total = isl_basic_set_total_dim(tab->bset);
1412 for (i = 0; i < tab->bset->n_div; ++i) {
1413 if (isl_int_ne(tab->bset->div[i][0], denom))
1415 if (!isl_seq_eq(tab->bset->div[i] + 1, div, total))
1422 /* Return the index of a div that corresponds to "div".
1423 * We first check if we already have such a div and if not, we create one.
1425 static int get_div(struct isl_tab *tab, struct isl_tab **context_tab,
1426 struct isl_vec *div)
1430 d = find_div(*context_tab, div->el + 1, div->el[0]);
1434 return add_div(tab, context_tab, div);
1437 /* Add a parametric cut to cut away the non-integral sample value
1439 * Let a_i be the coefficients of the constant term and the parameters
1440 * and let b_i be the coefficients of the variables or constraints
1441 * in basis of the tableau.
1442 * Let q be the div q = floor(\sum_i {-a_i} y_i).
1444 * The cut is expressed as
1446 * c = \sum_i -{-a_i} y_i + \sum_i {b_i} x_i + q >= 0
1448 * If q did not already exist in the context tableau, then it is added first.
1449 * If q is in a column of the main tableau then the "+ q" can be accomplished
1450 * by setting the corresponding entry to the denominator of the constraint.
1451 * If q happens to be in a row of the main tableau, then the corresponding
1452 * row needs to be added instead (taking care of the denominators).
1453 * Note that this is very unlikely, but perhaps not entirely impossible.
1455 * The current value of the cut is known to be negative (or at least
1456 * non-positive), so row_sign is set accordingly.
1458 * Return the row of the cut or -1.
1460 static int add_parametric_cut(struct isl_tab *tab, int row,
1461 struct isl_tab **context_tab)
1463 struct isl_vec *div;
1469 unsigned off = 2 + tab->M;
1474 if (isl_tab_extend_cons(*context_tab, 3) < 0)
1477 div = get_row_parameter_div(tab, row);
1481 d = get_div(tab, context_tab, div);
1485 if (isl_tab_extend_cons(tab, 1) < 0)
1487 r = isl_tab_allocate_con(tab);
1491 r_row = tab->mat->row[tab->con[r].index];
1492 isl_int_set(r_row[0], tab->mat->row[row][0]);
1493 isl_int_neg(r_row[1], tab->mat->row[row][1]);
1494 isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
1495 isl_int_neg(r_row[1], r_row[1]);
1497 isl_int_set_si(r_row[2], 0);
1498 for (i = 0; i < tab->n_param; ++i) {
1499 if (tab->var[i].is_row)
1501 col = tab->var[i].index;
1502 isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
1503 isl_int_fdiv_r(r_row[off + col], r_row[off + col],
1504 tab->mat->row[row][0]);
1505 isl_int_neg(r_row[off + col], r_row[off + col]);
1507 for (i = 0; i < tab->n_div; ++i) {
1508 if (tab->var[tab->n_var - tab->n_div + i].is_row)
1510 col = tab->var[tab->n_var - tab->n_div + i].index;
1511 isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
1512 isl_int_fdiv_r(r_row[off + col], r_row[off + col],
1513 tab->mat->row[row][0]);
1514 isl_int_neg(r_row[off + col], r_row[off + col]);
1516 for (i = 0; i < tab->n_col; ++i) {
1517 if (tab->col_var[i] >= 0 &&
1518 (tab->col_var[i] < tab->n_param ||
1519 tab->col_var[i] >= tab->n_var - tab->n_div))
1521 isl_int_fdiv_r(r_row[off + i],
1522 tab->mat->row[row][off + i], tab->mat->row[row][0]);
1524 if (tab->var[tab->n_var - tab->n_div + d].is_row) {
1526 int d_row = tab->var[tab->n_var - tab->n_div + d].index;
1528 isl_int_gcd(gcd, tab->mat->row[d_row][0], r_row[0]);
1529 isl_int_divexact(r_row[0], r_row[0], gcd);
1530 isl_int_divexact(gcd, tab->mat->row[d_row][0], gcd);
1531 isl_seq_combine(r_row + 1, gcd, r_row + 1,
1532 r_row[0], tab->mat->row[d_row] + 1,
1533 off - 1 + tab->n_col);
1534 isl_int_mul(r_row[0], r_row[0], tab->mat->row[d_row][0]);
1537 col = tab->var[tab->n_var - tab->n_div + d].index;
1538 isl_int_set(r_row[off + col], tab->mat->row[row][0]);
1541 tab->con[r].is_nonneg = 1;
1542 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
1544 tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
1548 return tab->con[r].index;
1550 isl_tab_free(*context_tab);
1551 *context_tab = NULL;
1555 /* Construct a tableau for bmap that can be used for computing
1556 * the lexicographic minimum (or maximum) of bmap.
1557 * If not NULL, then dom is the domain where the minimum
1558 * should be computed. In this case, we set up a parametric
1559 * tableau with row signs (initialized to "unknown").
1560 * If M is set, then the tableau will use a big parameter.
1561 * If max is set, then a maximum should be computed instead of a minimum.
1562 * This means that for each variable x, the tableau will contain the variable
1563 * x' = M - x, rather than x' = M + x. This in turn means that the coefficient
1564 * of the variables in all constraints are negated prior to adding them
1567 static struct isl_tab *tab_for_lexmin(struct isl_basic_map *bmap,
1568 struct isl_basic_set *dom, unsigned M, int max)
1571 struct isl_tab *tab;
1573 tab = isl_tab_alloc(bmap->ctx, 2 * bmap->n_eq + bmap->n_ineq + 1,
1574 isl_basic_map_total_dim(bmap), M);
1578 tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
1580 tab->n_param = isl_basic_set_total_dim(dom) - dom->n_div;
1581 tab->n_div = dom->n_div;
1582 tab->row_sign = isl_calloc_array(bmap->ctx,
1583 enum isl_tab_row_sign, tab->mat->n_row);
1587 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
1588 return isl_tab_mark_empty(tab);
1590 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
1591 tab->var[i].is_nonneg = 1;
1592 tab->var[i].frozen = 1;
1594 for (i = 0; i < bmap->n_eq; ++i) {
1596 isl_seq_neg(bmap->eq[i] + 1 + tab->n_param,
1597 bmap->eq[i] + 1 + tab->n_param,
1598 tab->n_var - tab->n_param - tab->n_div);
1599 tab = add_lexmin_valid_eq(tab, bmap->eq[i]);
1601 isl_seq_neg(bmap->eq[i] + 1 + tab->n_param,
1602 bmap->eq[i] + 1 + tab->n_param,
1603 tab->n_var - tab->n_param - tab->n_div);
1604 if (!tab || tab->empty)
1607 for (i = 0; i < bmap->n_ineq; ++i) {
1609 isl_seq_neg(bmap->ineq[i] + 1 + tab->n_param,
1610 bmap->ineq[i] + 1 + tab->n_param,
1611 tab->n_var - tab->n_param - tab->n_div);
1612 tab = add_lexmin_ineq(tab, bmap->ineq[i]);
1614 isl_seq_neg(bmap->ineq[i] + 1 + tab->n_param,
1615 bmap->ineq[i] + 1 + tab->n_param,
1616 tab->n_var - tab->n_param - tab->n_div);
1617 if (!tab || tab->empty)
1626 static struct isl_tab *context_tab_for_lexmin(struct isl_basic_set *bset)
1628 struct isl_tab *tab;
1630 bset = isl_basic_set_cow(bset);
1633 tab = tab_for_lexmin((struct isl_basic_map *)bset, NULL, 1, 0);
1639 tab->samples = isl_mat_alloc(bset->ctx, 1, 1 + tab->n_var);
1644 isl_basic_set_free(bset);
1648 /* Construct an isl_sol_map structure for accumulating the solution.
1649 * If track_empty is set, then we also keep track of the parts
1650 * of the context where there is no solution.
1651 * If max is set, then we are solving a maximization, rather than
1652 * a minimization problem, which means that the variables in the
1653 * tableau have value "M - x" rather than "M + x".
1655 static struct isl_sol_map *sol_map_init(struct isl_basic_map *bmap,
1656 struct isl_basic_set *dom, int track_empty, int max)
1658 struct isl_sol_map *sol_map;
1659 struct isl_tab *context_tab;
1662 sol_map = isl_calloc_type(bset->ctx, struct isl_sol_map);
1667 sol_map->sol.add = &sol_map_add_wrap;
1668 sol_map->sol.free = &sol_map_free_wrap;
1669 sol_map->map = isl_map_alloc_dim(isl_basic_map_get_dim(bmap), 1,
1674 context_tab = context_tab_for_lexmin(isl_basic_set_copy(dom));
1675 context_tab = restore_lexmin(context_tab);
1676 sol_map->sol.context_tab = context_tab;
1677 f = context_is_feasible(&sol_map->sol);
1682 sol_map->empty = isl_set_alloc_dim(isl_basic_set_get_dim(dom),
1683 1, ISL_SET_DISJOINT);
1684 if (!sol_map->empty)
1688 isl_basic_set_free(dom);
1691 isl_basic_set_free(dom);
1692 sol_map_free(sol_map);
1696 /* For each variable in the context tableau, check if the variable can
1697 * only attain non-negative values. If so, mark the parameter as non-negative
1698 * in the main tableau. This allows for a more direct identification of some
1699 * cases of violated constraints.
1701 static struct isl_tab *tab_detect_nonnegative_parameters(struct isl_tab *tab,
1702 struct isl_tab *context_tab)
1705 struct isl_tab_undo *snap, *snap2;
1706 struct isl_vec *ineq = NULL;
1707 struct isl_tab_var *var;
1710 if (context_tab->n_var == 0)
1713 ineq = isl_vec_alloc(tab->mat->ctx, 1 + context_tab->n_var);
1717 if (isl_tab_extend_cons(context_tab, 1) < 0)
1720 snap = isl_tab_snap(context_tab);
1721 isl_tab_push_basis(context_tab);
1723 snap2 = isl_tab_snap(context_tab);
1726 isl_seq_clr(ineq->el, ineq->size);
1727 for (i = 0; i < context_tab->n_var; ++i) {
1728 isl_int_set_si(ineq->el[1 + i], 1);
1729 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
1730 var = &context_tab->con[context_tab->n_con - 1];
1731 if (!context_tab->empty &&
1732 !isl_tab_min_at_most_neg_one(context_tab, var)) {
1734 if (i >= tab->n_param)
1735 j = i - tab->n_param + tab->n_var - tab->n_div;
1736 tab->var[j].is_nonneg = 1;
1739 isl_int_set_si(ineq->el[1 + i], 0);
1740 if (isl_tab_rollback(context_tab, snap2) < 0)
1744 if (isl_tab_rollback(context_tab, snap) < 0)
1747 if (n == context_tab->n_var) {
1748 context_tab->mat = isl_mat_drop_cols(context_tab->mat, 2, 1);
1760 /* Check whether all coefficients of (non-parameter) variables
1761 * are non-positive, meaning that no pivots can be performed on the row.
1763 static int is_critical(struct isl_tab *tab, int row)
1766 unsigned off = 2 + tab->M;
1768 for (j = tab->n_dead; j < tab->n_col; ++j) {
1769 if (tab->col_var[j] >= 0 &&
1770 (tab->col_var[j] < tab->n_param ||
1771 tab->col_var[j] >= tab->n_var - tab->n_div))
1774 if (isl_int_is_pos(tab->mat->row[row][off + j]))
1781 /* Check whether the inequality represented by vec is strict over the integers,
1782 * i.e., there are no integer values satisfying the constraint with
1783 * equality. This happens if the gcd of the coefficients is not a divisor
1784 * of the constant term. If so, scale the constraint down by the gcd
1785 * of the coefficients.
1787 static int is_strict(struct isl_vec *vec)
1793 isl_seq_gcd(vec->el + 1, vec->size - 1, &gcd);
1794 if (!isl_int_is_one(gcd)) {
1795 strict = !isl_int_is_divisible_by(vec->el[0], gcd);
1796 isl_int_fdiv_q(vec->el[0], vec->el[0], gcd);
1797 isl_seq_scale_down(vec->el + 1, vec->el + 1, gcd, vec->size-1);
1804 /* Determine the sign of the given row of the main tableau.
1805 * The result is one of
1806 * isl_tab_row_pos: always non-negative; no pivot needed
1807 * isl_tab_row_neg: always non-positive; pivot
1808 * isl_tab_row_any: can be both positive and negative; split
1810 * We first handle some simple cases
1811 * - the row sign may be known already
1812 * - the row may be obviously non-negative
1813 * - the parametric constant may be equal to that of another row
1814 * for which we know the sign. This sign will be either "pos" or
1815 * "any". If it had been "neg" then we would have pivoted before.
1817 * If none of these cases hold, we check the value of the row for each
1818 * of the currently active samples. Based on the signs of these values
1819 * we make an initial determination of the sign of the row.
1821 * all zero -> unk(nown)
1822 * all non-negative -> pos
1823 * all non-positive -> neg
1824 * both negative and positive -> all
1826 * If we end up with "all", we are done.
1827 * Otherwise, we perform a check for positive and/or negative
1828 * values as follows.
1830 * samples neg unk pos
1836 * There is no special sign for "zero", because we can usually treat zero
1837 * as either non-negative or non-positive, whatever works out best.
1838 * However, if the row is "critical", meaning that pivoting is impossible
1839 * then we don't want to limp zero with the non-positive case, because
1840 * then we we would lose the solution for those values of the parameters
1841 * where the value of the row is zero. Instead, we treat 0 as non-negative
1842 * ensuring a split if the row can attain both zero and negative values.
1843 * The same happens when the original constraint was one that could not
1844 * be satisfied with equality by any integer values of the parameters.
1845 * In this case, we normalize the constraint, but then a value of zero
1846 * for the normalized constraint is actually a positive value for the
1847 * original constraint, so again we need to treat zero as non-negative.
1848 * In both these cases, we have the following decision tree instead:
1850 * all non-negative -> pos
1851 * all negative -> neg
1852 * both negative and non-negative -> all
1860 static int row_sign(struct isl_tab *tab, struct isl_sol *sol, int row)
1863 struct isl_tab_undo *snap = NULL;
1864 struct isl_vec *ineq = NULL;
1865 int res = isl_tab_row_unknown;
1871 struct isl_tab *context_tab = sol->context_tab;
1873 if (tab->row_sign[row] != isl_tab_row_unknown)
1874 return tab->row_sign[row];
1875 if (is_obviously_nonneg(tab, row))
1876 return isl_tab_row_pos;
1877 for (row2 = tab->n_redundant; row2 < tab->n_row; ++row2) {
1878 if (tab->row_sign[row2] == isl_tab_row_unknown)
1880 if (identical_parameter_line(tab, row, row2))
1881 return tab->row_sign[row2];
1884 critical = is_critical(tab, row);
1886 isl_assert(tab->mat->ctx, context_tab->samples, goto error);
1887 isl_assert(tab->mat->ctx, context_tab->samples->n_col == 1 + context_tab->n_var, goto error);
1889 ineq = get_row_parameter_ineq(tab, row);
1893 strict = is_strict(ineq);
1896 for (i = context_tab->n_outside; i < context_tab->n_sample; ++i) {
1897 isl_seq_inner_product(context_tab->samples->row[i], ineq->el,
1899 sgn = isl_int_sgn(tmp);
1900 if (sgn > 0 || (sgn == 0 && (critical || strict))) {
1901 if (res == isl_tab_row_unknown)
1902 res = isl_tab_row_pos;
1903 if (res == isl_tab_row_neg)
1904 res = isl_tab_row_any;
1907 if (res == isl_tab_row_unknown)
1908 res = isl_tab_row_neg;
1909 if (res == isl_tab_row_pos)
1910 res = isl_tab_row_any;
1912 if (res == isl_tab_row_any)
1917 if (res != isl_tab_row_any) {
1918 if (isl_tab_extend_cons(context_tab, 1) < 0)
1921 snap = isl_tab_snap(context_tab);
1922 isl_tab_push_basis(context_tab);
1925 if (res == isl_tab_row_unknown || res == isl_tab_row_pos) {
1926 /* test for negative values */
1928 isl_seq_neg(ineq->el, ineq->el, ineq->size);
1929 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1931 isl_tab_push_basis(context_tab);
1932 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq->el);
1933 feasible = context_is_feasible(sol);
1936 context_tab = sol->context_tab;
1938 res = isl_tab_row_pos;
1940 res = (res == isl_tab_row_unknown) ? isl_tab_row_neg
1942 if (isl_tab_rollback(context_tab, snap) < 0)
1945 if (res == isl_tab_row_neg) {
1946 isl_seq_neg(ineq->el, ineq->el, ineq->size);
1947 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1951 if (res == isl_tab_row_neg) {
1952 /* test for positive values */
1954 if (!critical && !strict)
1955 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1957 isl_tab_push_basis(context_tab);
1958 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq->el);
1959 feasible = context_is_feasible(sol);
1962 context_tab = sol->context_tab;
1964 res = isl_tab_row_any;
1965 if (isl_tab_rollback(context_tab, snap) < 0)
1976 static struct isl_sol *find_solutions(struct isl_sol *sol, struct isl_tab *tab);
1978 /* Find solutions for values of the parameters that satisfy the given
1981 * We currently take a snapshot of the context tableau that is reset
1982 * when we return from this function, while we make a copy of the main
1983 * tableau, leaving the original main tableau untouched.
1984 * These are fairly arbitrary choices. Making a copy also of the context
1985 * tableau would obviate the need to undo any changes made to it later,
1986 * while taking a snapshot of the main tableau could reduce memory usage.
1987 * If we were to switch to taking a snapshot of the main tableau,
1988 * we would have to keep in mind that we need to save the row signs
1989 * and that we need to do this before saving the current basis
1990 * such that the basis has been restore before we restore the row signs.
1992 static struct isl_sol *find_in_pos(struct isl_sol *sol,
1993 struct isl_tab *tab, isl_int *ineq)
1995 struct isl_tab_undo *snap;
1997 snap = isl_tab_snap(sol->context_tab);
1998 isl_tab_push_basis(sol->context_tab);
1999 if (isl_tab_extend_cons(sol->context_tab, 1) < 0)
2002 tab = isl_tab_dup(tab);
2006 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq);
2007 sol->context_tab = check_samples(sol->context_tab, ineq, 0);
2009 sol = find_solutions(sol, tab);
2011 isl_tab_rollback(sol->context_tab, snap);
2014 isl_tab_rollback(sol->context_tab, snap);
2019 /* Record the absence of solutions for those values of the parameters
2020 * that do not satisfy the given inequality with equality.
2022 static struct isl_sol *no_sol_in_strict(struct isl_sol *sol,
2023 struct isl_tab *tab, struct isl_vec *ineq)
2027 struct isl_tab_undo *snap;
2028 snap = isl_tab_snap(sol->context_tab);
2029 isl_tab_push_basis(sol->context_tab);
2030 if (isl_tab_extend_cons(sol->context_tab, 1) < 0)
2033 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
2035 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq->el);
2036 f = context_valid_sample_or_feasible(sol, ineq->el, 0);
2042 sol = sol->add(sol, tab);
2045 isl_int_add_ui(ineq->el[0], ineq->el[0], 1);
2047 if (isl_tab_rollback(sol->context_tab, snap) < 0)
2055 /* Given a main tableau where more than one row requires a split,
2056 * determine and return the "best" row to split on.
2058 * Given two rows in the main tableau, if the inequality corresponding
2059 * to the first row is redundant with respect to that of the second row
2060 * in the current tableau, then it is better to split on the second row,
2061 * since in the positive part, both row will be positive.
2062 * (In the negative part a pivot will have to be performed and just about
2063 * anything can happen to the sign of the other row.)
2065 * As a simple heuristic, we therefore select the row that makes the most
2066 * of the other rows redundant.
2068 * Perhaps it would also be useful to look at the number of constraints
2069 * that conflict with any given constraint.
2071 static int best_split(struct isl_tab *tab, struct isl_tab *context_tab)
2073 struct isl_tab_undo *snap, *snap2;
2079 if (isl_tab_extend_cons(context_tab, 2) < 0)
2082 snap = isl_tab_snap(context_tab);
2083 isl_tab_push_basis(context_tab);
2084 snap2 = isl_tab_snap(context_tab);
2086 for (split = tab->n_redundant; split < tab->n_row; ++split) {
2087 struct isl_tab_undo *snap3;
2088 struct isl_vec *ineq = NULL;
2091 if (!isl_tab_var_from_row(tab, split)->is_nonneg)
2093 if (tab->row_sign[split] != isl_tab_row_any)
2096 ineq = get_row_parameter_ineq(tab, split);
2099 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
2102 snap3 = isl_tab_snap(context_tab);
2104 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2105 struct isl_tab_var *var;
2109 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
2111 if (tab->row_sign[row] != isl_tab_row_any)
2114 ineq = get_row_parameter_ineq(tab, row);
2117 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
2119 var = &context_tab->con[context_tab->n_con - 1];
2120 if (!context_tab->empty &&
2121 !isl_tab_min_at_most_neg_one(context_tab, var))
2123 if (isl_tab_rollback(context_tab, snap3) < 0)
2126 if (best == -1 || r > best_r) {
2130 if (isl_tab_rollback(context_tab, snap2) < 0)
2134 if (isl_tab_rollback(context_tab, snap) < 0)
2140 /* Compute the lexicographic minimum of the set represented by the main
2141 * tableau "tab" within the context "sol->context_tab".
2142 * On entry the sample value of the main tableau is lexicographically
2143 * less than or equal to this lexicographic minimum.
2144 * Pivots are performed until a feasible point is found, which is then
2145 * necessarily equal to the minimum, or until the tableau is found to
2146 * be infeasible. Some pivots may need to be performed for only some
2147 * feasible values of the context tableau. If so, the context tableau
2148 * is split into a part where the pivot is needed and a part where it is not.
2150 * Whenever we enter the main loop, the main tableau is such that no
2151 * "obvious" pivots need to be performed on it, where "obvious" means
2152 * that the given row can be seen to be negative without looking at
2153 * the context tableau. In particular, for non-parametric problems,
2154 * no pivots need to be performed on the main tableau.
2155 * The caller of find_solutions is responsible for making this property
2156 * hold prior to the first iteration of the loop, while restore_lexmin
2157 * is called before every other iteration.
2159 * Inside the main loop, we first examine the signs of the rows of
2160 * the main tableau within the context of the context tableau.
2161 * If we find a row that is always non-positive for all values of
2162 * the parameters satisfying the context tableau and negative for at
2163 * least one value of the parameters, we perform the appropriate pivot
2164 * and start over. An exception is the case where no pivot can be
2165 * performed on the row. In this case, we require that the sign of
2166 * the row is negative for all values of the parameters (rather than just
2167 * non-positive). This special case is handled inside row_sign, which
2168 * will say that the row can have any sign if it determines that it can
2169 * attain both negative and zero values.
2171 * If we can't find a row that always requires a pivot, but we can find
2172 * one or more rows that require a pivot for some values of the parameters
2173 * (i.e., the row can attain both positive and negative signs), then we split
2174 * the context tableau into two parts, one where we force the sign to be
2175 * non-negative and one where we force is to be negative.
2176 * The non-negative part is handled by a recursive call (through find_in_pos).
2177 * Upon returning from this call, we continue with the negative part and
2178 * perform the required pivot.
2180 * If no such rows can be found, all rows are non-negative and we have
2181 * found a (rational) feasible point. If we only wanted a rational point
2183 * Otherwise, we check if all values of the sample point of the tableau
2184 * are integral for the variables. If so, we have found the minimal
2185 * integral point and we are done.
2186 * If the sample point is not integral, then we need to make a distinction
2187 * based on whether the constant term is non-integral or the coefficients
2188 * of the parameters. Furthermore, in order to decide how to handle
2189 * the non-integrality, we also need to know whether the coefficients
2190 * of the other columns in the tableau are integral. This leads
2191 * to the following table. The first two rows do not correspond
2192 * to a non-integral sample point and are only mentioned for completeness.
2194 * constant parameters other
2197 * int int rat | -> no problem
2199 * rat int int -> fail
2201 * rat int rat -> cut
2204 * rat rat rat | -> parametric cut
2207 * rat rat int | -> split context
2209 * If the parametric constant is completely integral, then there is nothing
2210 * to be done. If the constant term is non-integral, but all the other
2211 * coefficient are integral, then there is nothing that can be done
2212 * and the tableau has no integral solution.
2213 * If, on the other hand, one or more of the other columns have rational
2214 * coeffcients, but the parameter coefficients are all integral, then
2215 * we can perform a regular (non-parametric) cut.
2216 * Finally, if there is any parameter coefficient that is non-integral,
2217 * then we need to involve the context tableau. There are two cases here.
2218 * If at least one other column has a rational coefficient, then we
2219 * can perform a parametric cut in the main tableau by adding a new
2220 * integer division in the context tableau.
2221 * If all other columns have integral coefficients, then we need to
2222 * enforce that the rational combination of parameters (c + \sum a_i y_i)/m
2223 * is always integral. We do this by introducing an integer division
2224 * q = floor((c + \sum a_i y_i)/m) and stipulating that its argument should
2225 * always be integral in the context tableau, i.e., m q = c + \sum a_i y_i.
2226 * Since q is expressed in the tableau as
2227 * c + \sum a_i y_i - m q >= 0
2228 * -c - \sum a_i y_i + m q + m - 1 >= 0
2229 * it is sufficient to add the inequality
2230 * -c - \sum a_i y_i + m q >= 0
2231 * In the part of the context where this inequality does not hold, the
2232 * main tableau is marked as being empty.
2234 static struct isl_sol *find_solutions(struct isl_sol *sol, struct isl_tab *tab)
2236 struct isl_tab **context_tab;
2241 context_tab = &sol->context_tab;
2245 if ((*context_tab)->empty)
2248 for (; tab && !tab->empty; tab = restore_lexmin(tab)) {
2255 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2256 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
2258 sgn = row_sign(tab, sol, row);
2261 tab->row_sign[row] = sgn;
2262 if (sgn == isl_tab_row_any)
2264 if (sgn == isl_tab_row_any && split == -1)
2266 if (sgn == isl_tab_row_neg)
2269 if (row < tab->n_row)
2272 struct isl_vec *ineq;
2274 split = best_split(tab, *context_tab);
2277 ineq = get_row_parameter_ineq(tab, split);
2281 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2282 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
2284 if (tab->row_sign[row] == isl_tab_row_any)
2285 tab->row_sign[row] = isl_tab_row_unknown;
2287 tab->row_sign[split] = isl_tab_row_pos;
2288 sol = find_in_pos(sol, tab, ineq->el);
2289 tab->row_sign[split] = isl_tab_row_neg;
2291 isl_seq_neg(ineq->el, ineq->el, ineq->size);
2292 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
2293 *context_tab = add_lexmin_ineq(*context_tab, ineq->el);
2294 *context_tab = check_samples(*context_tab, ineq->el, 0);
2302 row = first_non_integer(tab, &flags);
2305 if (ISL_FL_ISSET(flags, I_PAR)) {
2306 if (ISL_FL_ISSET(flags, I_VAR)) {
2307 tab = isl_tab_mark_empty(tab);
2310 row = add_cut(tab, row);
2311 } else if (ISL_FL_ISSET(flags, I_VAR)) {
2312 struct isl_vec *div;
2313 struct isl_vec *ineq;
2315 if (isl_tab_extend_cons(*context_tab, 3) < 0)
2317 div = get_row_split_div(tab, row);
2320 d = get_div(tab, context_tab, div);
2324 ineq = ineq_for_div((*context_tab)->bset, d);
2325 sol = no_sol_in_strict(sol, tab, ineq);
2326 isl_seq_neg(ineq->el, ineq->el, ineq->size);
2327 *context_tab = add_lexmin_ineq(*context_tab, ineq->el);
2328 *context_tab = check_samples(*context_tab, ineq->el, 0);
2332 tab = set_row_cst_to_div(tab, row, d);
2334 row = add_parametric_cut(tab, row, context_tab);
2339 sol = sol->add(sol, tab);
2348 /* Compute the lexicographic minimum of the set represented by the main
2349 * tableau "tab" within the context "sol->context_tab".
2351 * As a preprocessing step, we first transfer all the purely parametric
2352 * equalities from the main tableau to the context tableau, i.e.,
2353 * parameters that have been pivoted to a row.
2354 * These equalities are ignored by the main algorithm, because the
2355 * corresponding rows may not be marked as being non-negative.
2356 * In parts of the context where the added equality does not hold,
2357 * the main tableau is marked as being empty.
2359 static struct isl_sol *find_solutions_main(struct isl_sol *sol,
2360 struct isl_tab *tab)
2364 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2368 if (tab->row_var[row] < 0)
2370 if (tab->row_var[row] >= tab->n_param &&
2371 tab->row_var[row] < tab->n_var - tab->n_div)
2373 if (tab->row_var[row] < tab->n_param)
2374 p = tab->row_var[row];
2376 p = tab->row_var[row]
2377 + tab->n_param - (tab->n_var - tab->n_div);
2379 if (isl_tab_extend_cons(sol->context_tab, 2) < 0)
2382 eq = isl_vec_alloc(tab->mat->ctx, 1+tab->n_param+tab->n_div);
2383 get_row_parameter_line(tab, row, eq->el);
2384 isl_int_neg(eq->el[1 + p], tab->mat->row[row][0]);
2385 eq = isl_vec_normalize(eq);
2387 sol = no_sol_in_strict(sol, tab, eq);
2389 isl_seq_neg(eq->el, eq->el, eq->size);
2390 sol = no_sol_in_strict(sol, tab, eq);
2391 isl_seq_neg(eq->el, eq->el, eq->size);
2393 sol->context_tab = add_lexmin_eq(sol->context_tab, eq->el);
2394 context_valid_sample_or_feasible(sol, eq->el, 1);
2395 sol->context_tab = check_samples(sol->context_tab, eq->el, 1);
2399 isl_tab_mark_redundant(tab, row);
2401 if (!sol->context_tab)
2403 if (sol->context_tab->empty)
2406 row = tab->n_redundant - 1;
2409 return find_solutions(sol, tab);
2416 static struct isl_sol_map *sol_map_find_solutions(struct isl_sol_map *sol_map,
2417 struct isl_tab *tab)
2419 return (struct isl_sol_map *)find_solutions_main(&sol_map->sol, tab);
2422 /* Check if integer division "div" of "dom" also occurs in "bmap".
2423 * If so, return its position within the divs.
2424 * If not, return -1.
2426 static int find_context_div(struct isl_basic_map *bmap,
2427 struct isl_basic_set *dom, unsigned div)
2430 unsigned b_dim = isl_dim_total(bmap->dim);
2431 unsigned d_dim = isl_dim_total(dom->dim);
2433 if (isl_int_is_zero(dom->div[div][0]))
2435 if (isl_seq_first_non_zero(dom->div[div] + 2 + d_dim, dom->n_div) != -1)
2438 for (i = 0; i < bmap->n_div; ++i) {
2439 if (isl_int_is_zero(bmap->div[i][0]))
2441 if (isl_seq_first_non_zero(bmap->div[i] + 2 + d_dim,
2442 (b_dim - d_dim) + bmap->n_div) != -1)
2444 if (isl_seq_eq(bmap->div[i], dom->div[div], 2 + d_dim))
2450 /* The correspondence between the variables in the main tableau,
2451 * the context tableau, and the input map and domain is as follows.
2452 * The first n_param and the last n_div variables of the main tableau
2453 * form the variables of the context tableau.
2454 * In the basic map, these n_param variables correspond to the
2455 * parameters and the input dimensions. In the domain, they correspond
2456 * to the parameters and the set dimensions.
2457 * The n_div variables correspond to the integer divisions in the domain.
2458 * To ensure that everything lines up, we may need to copy some of the
2459 * integer divisions of the domain to the map. These have to be placed
2460 * in the same order as those in the context and they have to be placed
2461 * after any other integer divisions that the map may have.
2462 * This function performs the required reordering.
2464 static struct isl_basic_map *align_context_divs(struct isl_basic_map *bmap,
2465 struct isl_basic_set *dom)
2471 for (i = 0; i < dom->n_div; ++i)
2472 if (find_context_div(bmap, dom, i) != -1)
2474 other = bmap->n_div - common;
2475 if (dom->n_div - common > 0) {
2476 bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim),
2477 dom->n_div - common, 0, 0);
2481 for (i = 0; i < dom->n_div; ++i) {
2482 int pos = find_context_div(bmap, dom, i);
2484 pos = isl_basic_map_alloc_div(bmap);
2487 isl_int_set_si(bmap->div[pos][0], 0);
2489 if (pos != other + i)
2490 isl_basic_map_swap_div(bmap, pos, other + i);
2494 isl_basic_map_free(bmap);
2498 /* Compute the lexicographic minimum (or maximum if "max" is set)
2499 * of "bmap" over the domain "dom" and return the result as a map.
2500 * If "empty" is not NULL, then *empty is assigned a set that
2501 * contains those parts of the domain where there is no solution.
2502 * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL),
2503 * then we compute the rational optimum. Otherwise, we compute
2504 * the integral optimum.
2506 * We perform some preprocessing. As the PILP solver does not
2507 * handle implicit equalities very well, we first make sure all
2508 * the equalities are explicitly available.
2509 * We also make sure the divs in the domain are properly order,
2510 * because they will be added one by one in the given order
2511 * during the construction of the solution map.
2513 struct isl_map *isl_tab_basic_map_partial_lexopt(
2514 struct isl_basic_map *bmap, struct isl_basic_set *dom,
2515 struct isl_set **empty, int max)
2517 struct isl_tab *tab;
2518 struct isl_map *result = NULL;
2519 struct isl_sol_map *sol_map = NULL;
2526 isl_assert(bmap->ctx,
2527 isl_basic_map_compatible_domain(bmap, dom), goto error);
2529 bmap = isl_basic_map_detect_equalities(bmap);
2532 dom = isl_basic_set_order_divs(dom);
2533 bmap = align_context_divs(bmap, dom);
2535 sol_map = sol_map_init(bmap, dom, !!empty, max);
2539 if (isl_basic_set_fast_is_empty(sol_map->sol.context_tab->bset))
2541 else if (isl_basic_map_fast_is_empty(bmap))
2542 sol_map = add_empty(sol_map);
2544 tab = tab_for_lexmin(bmap,
2545 sol_map->sol.context_tab->bset, 1, max);
2546 tab = tab_detect_nonnegative_parameters(tab,
2547 sol_map->sol.context_tab);
2548 sol_map = sol_map_find_solutions(sol_map, tab);
2553 result = isl_map_copy(sol_map->map);
2555 *empty = isl_set_copy(sol_map->empty);
2556 sol_map_free(sol_map);
2557 isl_basic_map_free(bmap);
2560 sol_map_free(sol_map);
2561 isl_basic_map_free(bmap);
2565 struct isl_sol_for {
2567 int (*fn)(__isl_take isl_basic_set *dom,
2568 __isl_take isl_mat *map, void *user);
2573 static void sol_for_free(struct isl_sol_for *sol_for)
2575 isl_tab_free(sol_for->sol.context_tab);
2579 static void sol_for_free_wrap(struct isl_sol *sol)
2581 sol_for_free((struct isl_sol_for *)sol);
2584 /* Add the solution identified by the tableau and the context tableau.
2586 * See documentation of sol_map_add for more details.
2588 * Instead of constructing a basic map, this function calls a user
2589 * defined function with the current context as a basic set and
2590 * an affine matrix reprenting the relation between the input and output.
2591 * The number of rows in this matrix is equal to one plus the number
2592 * of output variables. The number of columns is equal to one plus
2593 * the total dimension of the context, i.e., the number of parameters,
2594 * input variables and divs. Since some of the columns in the matrix
2595 * may refer to the divs, the basic set is not simplified.
2596 * (Simplification may reorder or remove divs.)
2598 static struct isl_sol_for *sol_for_add(struct isl_sol_for *sol,
2599 struct isl_tab *tab)
2601 struct isl_tab *context_tab;
2602 struct isl_basic_set *bset;
2603 struct isl_mat *mat = NULL;
2615 context_tab = sol->sol.context_tab;
2617 n_out = tab->n_var - tab->n_param - tab->n_div;
2618 mat = isl_mat_alloc(tab->mat->ctx, 1 + n_out, 1 + tab->n_param + tab->n_div);
2622 isl_seq_clr(mat->row[0] + 1, mat->n_col - 1);
2623 isl_int_set_si(mat->row[0][0], 1);
2624 for (row = 0; row < n_out; ++row) {
2625 int i = tab->n_param + row;
2628 isl_seq_clr(mat->row[1 + row], mat->n_col);
2629 if (!tab->var[i].is_row)
2632 r = tab->var[i].index;
2635 isl_assert(mat->ctx, isl_int_eq(tab->mat->row[r][2],
2636 tab->mat->row[r][0]),
2638 isl_int_set(mat->row[1 + row][0], tab->mat->row[r][1]);
2639 for (j = 0; j < tab->n_param; ++j) {
2641 if (tab->var[j].is_row)
2643 col = tab->var[j].index;
2644 isl_int_set(mat->row[1 + row][1 + j],
2645 tab->mat->row[r][off + col]);
2647 for (j = 0; j < tab->n_div; ++j) {
2649 if (tab->var[tab->n_var - tab->n_div+j].is_row)
2651 col = tab->var[tab->n_var - tab->n_div+j].index;
2652 isl_int_set(mat->row[1 + row][1 + tab->n_param + j],
2653 tab->mat->row[r][off + col]);
2655 if (!isl_int_is_one(tab->mat->row[r][0]))
2656 isl_seq_scale_down(mat->row[1 + row], mat->row[1 + row],
2657 tab->mat->row[r][0], mat->n_col);
2659 isl_seq_neg(mat->row[1 + row], mat->row[1 + row],
2663 bset = isl_basic_set_dup(context_tab->bset);
2664 bset = isl_basic_set_finalize(bset);
2666 if (sol->fn(bset, isl_mat_copy(mat), sol->user) < 0)
2673 sol_free(&sol->sol);
2677 static struct isl_sol *sol_for_add_wrap(struct isl_sol *sol,
2678 struct isl_tab *tab)
2680 return (struct isl_sol *)sol_for_add((struct isl_sol_for *)sol, tab);
2683 static struct isl_sol_for *sol_for_init(struct isl_basic_map *bmap, int max,
2684 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_mat *map,
2688 struct isl_sol_for *sol_for = NULL;
2689 struct isl_dim *dom_dim;
2690 struct isl_basic_set *dom = NULL;
2691 struct isl_tab *context_tab;
2694 sol_for = isl_calloc_type(bset->ctx, struct isl_sol_for);
2698 dom_dim = isl_dim_domain(isl_dim_copy(bmap->dim));
2699 dom = isl_basic_set_universe(dom_dim);
2702 sol_for->user = user;
2704 sol_for->sol.add = &sol_for_add_wrap;
2705 sol_for->sol.free = &sol_for_free_wrap;
2707 context_tab = context_tab_for_lexmin(isl_basic_set_copy(dom));
2708 context_tab = restore_lexmin(context_tab);
2709 sol_for->sol.context_tab = context_tab;
2710 f = context_is_feasible(&sol_for->sol);
2714 isl_basic_set_free(dom);
2717 isl_basic_set_free(dom);
2718 sol_for_free(sol_for);
2722 static struct isl_sol_for *sol_for_find_solutions(struct isl_sol_for *sol_for,
2723 struct isl_tab *tab)
2725 return (struct isl_sol_for *)find_solutions_main(&sol_for->sol, tab);
2728 int isl_basic_map_foreach_lexopt(__isl_keep isl_basic_map *bmap, int max,
2729 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_mat *map,
2733 struct isl_sol_for *sol_for = NULL;
2735 bmap = isl_basic_map_copy(bmap);
2739 bmap = isl_basic_map_detect_equalities(bmap);
2740 sol_for = sol_for_init(bmap, max, fn, user);
2742 if (isl_basic_map_fast_is_empty(bmap))
2745 struct isl_tab *tab;
2746 tab = tab_for_lexmin(bmap,
2747 sol_for->sol.context_tab->bset, 1, max);
2748 tab = tab_detect_nonnegative_parameters(tab,
2749 sol_for->sol.context_tab);
2750 sol_for = sol_for_find_solutions(sol_for, tab);
2755 sol_for_free(sol_for);
2756 isl_basic_map_free(bmap);
2759 sol_for_free(sol_for);
2760 isl_basic_map_free(bmap);
2764 int isl_basic_map_foreach_lexmin(__isl_keep isl_basic_map *bmap,
2765 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_mat *map,
2769 return isl_basic_map_foreach_lexopt(bmap, 0, fn, user);
2772 int isl_basic_map_foreach_lexmax(__isl_keep isl_basic_map *bmap,
2773 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_mat *map,
2777 return isl_basic_map_foreach_lexopt(bmap, 1, fn, user);