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_gauss(bmap, NULL);
267 bmap = isl_basic_map_normalize_constraints(bmap);
268 bmap = isl_basic_map_finalize(bmap);
269 sol->map = isl_map_grow(sol->map, 1);
270 sol->map = isl_map_add(sol->map, bmap);
275 isl_basic_map_free(bmap);
280 static struct isl_sol *sol_map_add_wrap(struct isl_sol *sol,
283 return (struct isl_sol *)sol_map_add((struct isl_sol_map *)sol, tab);
287 /* Store the "parametric constant" of row "row" of tableau "tab" in "line",
288 * i.e., the constant term and the coefficients of all variables that
289 * appear in the context tableau.
290 * Note that the coefficient of the big parameter M is NOT copied.
291 * The context tableau may not have a big parameter and even when it
292 * does, it is a different big parameter.
294 static void get_row_parameter_line(struct isl_tab *tab, int row, isl_int *line)
297 unsigned off = 2 + tab->M;
299 isl_int_set(line[0], tab->mat->row[row][1]);
300 for (i = 0; i < tab->n_param; ++i) {
301 if (tab->var[i].is_row)
302 isl_int_set_si(line[1 + i], 0);
304 int col = tab->var[i].index;
305 isl_int_set(line[1 + i], tab->mat->row[row][off + col]);
308 for (i = 0; i < tab->n_div; ++i) {
309 if (tab->var[tab->n_var - tab->n_div + i].is_row)
310 isl_int_set_si(line[1 + tab->n_param + i], 0);
312 int col = tab->var[tab->n_var - tab->n_div + i].index;
313 isl_int_set(line[1 + tab->n_param + i],
314 tab->mat->row[row][off + col]);
319 /* Check if rows "row1" and "row2" have identical "parametric constants",
320 * as explained above.
321 * In this case, we also insist that the coefficients of the big parameter
322 * be the same as the values of the constants will only be the same
323 * if these coefficients are also the same.
325 static int identical_parameter_line(struct isl_tab *tab, int row1, int row2)
328 unsigned off = 2 + tab->M;
330 if (isl_int_ne(tab->mat->row[row1][1], tab->mat->row[row2][1]))
333 if (tab->M && isl_int_ne(tab->mat->row[row1][2],
334 tab->mat->row[row2][2]))
337 for (i = 0; i < tab->n_param + tab->n_div; ++i) {
338 int pos = i < tab->n_param ? i :
339 tab->n_var - tab->n_div + i - tab->n_param;
342 if (tab->var[pos].is_row)
344 col = tab->var[pos].index;
345 if (isl_int_ne(tab->mat->row[row1][off + col],
346 tab->mat->row[row2][off + col]))
352 /* Return an inequality that expresses that the "parametric constant"
353 * should be non-negative.
354 * This function is only called when the coefficient of the big parameter
357 static struct isl_vec *get_row_parameter_ineq(struct isl_tab *tab, int row)
359 struct isl_vec *ineq;
361 ineq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_param + tab->n_div);
365 get_row_parameter_line(tab, row, ineq->el);
367 ineq = isl_vec_normalize(ineq);
372 /* Return a integer division for use in a parametric cut based on the given row.
373 * In particular, let the parametric constant of the row be
377 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
378 * The div returned is equal to
380 * floor(\sum_i {-a_i} y_i) = floor((\sum_i (-a_i mod d) y_i)/d)
382 static struct isl_vec *get_row_parameter_div(struct isl_tab *tab, int row)
386 div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
390 isl_int_set(div->el[0], tab->mat->row[row][0]);
391 get_row_parameter_line(tab, row, div->el + 1);
392 div = isl_vec_normalize(div);
393 isl_seq_neg(div->el + 1, div->el + 1, div->size - 1);
394 isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
399 /* Return a integer division for use in transferring an integrality constraint
401 * In particular, let the parametric constant of the row be
405 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
406 * The the returned div is equal to
408 * floor(\sum_i {a_i} y_i) = floor((\sum_i (a_i mod d) y_i)/d)
410 static struct isl_vec *get_row_split_div(struct isl_tab *tab, int row)
414 div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
418 isl_int_set(div->el[0], tab->mat->row[row][0]);
419 get_row_parameter_line(tab, row, div->el + 1);
420 div = isl_vec_normalize(div);
421 isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
426 /* Construct and return an inequality that expresses an upper bound
428 * In particular, if the div is given by
432 * then the inequality expresses
436 static struct isl_vec *ineq_for_div(struct isl_basic_set *bset, unsigned div)
440 struct isl_vec *ineq;
442 total = isl_basic_set_total_dim(bset);
443 div_pos = 1 + total - bset->n_div + div;
445 ineq = isl_vec_alloc(bset->ctx, 1 + total);
449 isl_seq_cpy(ineq->el, bset->div[div] + 1, 1 + total);
450 isl_int_neg(ineq->el[div_pos], bset->div[div][0]);
454 /* Given a row in the tableau and a div that was created
455 * using get_row_split_div and that been constrained to equality, i.e.,
457 * d = floor(\sum_i {a_i} y_i) = \sum_i {a_i} y_i
459 * replace the expression "\sum_i {a_i} y_i" in the row by d,
460 * i.e., we subtract "\sum_i {a_i} y_i" and add 1 d.
461 * The coefficients of the non-parameters in the tableau have been
462 * verified to be integral. We can therefore simply replace coefficient b
463 * by floor(b). For the coefficients of the parameters we have
464 * floor(a_i) = a_i - {a_i}, while for the other coefficients, we have
467 static struct isl_tab *set_row_cst_to_div(struct isl_tab *tab, int row, int div)
470 unsigned off = 2 + tab->M;
472 isl_seq_fdiv_q(tab->mat->row[row] + 1, tab->mat->row[row] + 1,
473 tab->mat->row[row][0], 1 + tab->M + tab->n_col);
475 isl_int_set_si(tab->mat->row[row][0], 1);
477 isl_assert(tab->mat->ctx,
478 !tab->var[tab->n_var - tab->n_div + div].is_row, goto error);
480 col = tab->var[tab->n_var - tab->n_div + div].index;
481 isl_int_set_si(tab->mat->row[row][off + col], 1);
489 /* Check if the (parametric) constant of the given row is obviously
490 * negative, meaning that we don't need to consult the context tableau.
491 * If there is a big parameter and its coefficient is non-zero,
492 * then this coefficient determines the outcome.
493 * Otherwise, we check whether the constant is negative and
494 * all non-zero coefficients of parameters are negative and
495 * belong to non-negative parameters.
497 static int is_obviously_neg(struct isl_tab *tab, int row)
501 unsigned off = 2 + tab->M;
504 if (isl_int_is_pos(tab->mat->row[row][2]))
506 if (isl_int_is_neg(tab->mat->row[row][2]))
510 if (isl_int_is_nonneg(tab->mat->row[row][1]))
512 for (i = 0; i < tab->n_param; ++i) {
513 /* Eliminated parameter */
514 if (tab->var[i].is_row)
516 col = tab->var[i].index;
517 if (isl_int_is_zero(tab->mat->row[row][off + col]))
519 if (!tab->var[i].is_nonneg)
521 if (isl_int_is_pos(tab->mat->row[row][off + col]))
524 for (i = 0; i < tab->n_div; ++i) {
525 if (tab->var[tab->n_var - tab->n_div + i].is_row)
527 col = tab->var[tab->n_var - tab->n_div + i].index;
528 if (isl_int_is_zero(tab->mat->row[row][off + col]))
530 if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
532 if (isl_int_is_pos(tab->mat->row[row][off + col]))
538 /* Check if the (parametric) constant of the given row is obviously
539 * non-negative, meaning that we don't need to consult the context tableau.
540 * If there is a big parameter and its coefficient is non-zero,
541 * then this coefficient determines the outcome.
542 * Otherwise, we check whether the constant is non-negative and
543 * all non-zero coefficients of parameters are positive and
544 * belong to non-negative parameters.
546 static int is_obviously_nonneg(struct isl_tab *tab, int row)
550 unsigned off = 2 + tab->M;
553 if (isl_int_is_pos(tab->mat->row[row][2]))
555 if (isl_int_is_neg(tab->mat->row[row][2]))
559 if (isl_int_is_neg(tab->mat->row[row][1]))
561 for (i = 0; i < tab->n_param; ++i) {
562 /* Eliminated parameter */
563 if (tab->var[i].is_row)
565 col = tab->var[i].index;
566 if (isl_int_is_zero(tab->mat->row[row][off + col]))
568 if (!tab->var[i].is_nonneg)
570 if (isl_int_is_neg(tab->mat->row[row][off + col]))
573 for (i = 0; i < tab->n_div; ++i) {
574 if (tab->var[tab->n_var - tab->n_div + i].is_row)
576 col = tab->var[tab->n_var - tab->n_div + i].index;
577 if (isl_int_is_zero(tab->mat->row[row][off + col]))
579 if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
581 if (isl_int_is_neg(tab->mat->row[row][off + col]))
587 /* Given a row r and two columns, return the column that would
588 * lead to the lexicographically smallest increment in the sample
589 * solution when leaving the basis in favor of the row.
590 * Pivoting with column c will increment the sample value by a non-negative
591 * constant times a_{V,c}/a_{r,c}, with a_{V,c} the elements of column c
592 * corresponding to the non-parametric variables.
593 * If variable v appears in a column c_v, the a_{v,c} = 1 iff c = c_v,
594 * with all other entries in this virtual row equal to zero.
595 * If variable v appears in a row, then a_{v,c} is the element in column c
598 * Let v be the first variable with a_{v,c1}/a_{r,c1} != a_{v,c2}/a_{r,c2}.
599 * Then if a_{v,c1}/a_{r,c1} < a_{v,c2}/a_{r,c2}, i.e.,
600 * a_{v,c2} a_{r,c1} - a_{v,c1} a_{r,c2} > 0, c1 results in the minimal
601 * increment. Otherwise, it's c2.
603 static int lexmin_col_pair(struct isl_tab *tab,
604 int row, int col1, int col2, isl_int tmp)
609 tr = tab->mat->row[row] + 2 + tab->M;
611 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
615 if (!tab->var[i].is_row) {
616 if (tab->var[i].index == col1)
618 if (tab->var[i].index == col2)
623 if (tab->var[i].index == row)
626 r = tab->mat->row[tab->var[i].index] + 2 + tab->M;
627 s1 = isl_int_sgn(r[col1]);
628 s2 = isl_int_sgn(r[col2]);
629 if (s1 == 0 && s2 == 0)
636 isl_int_mul(tmp, r[col2], tr[col1]);
637 isl_int_submul(tmp, r[col1], tr[col2]);
638 if (isl_int_is_pos(tmp))
640 if (isl_int_is_neg(tmp))
646 /* Given a row in the tableau, find and return the column that would
647 * result in the lexicographically smallest, but positive, increment
648 * in the sample point.
649 * If there is no such column, then return tab->n_col.
650 * If anything goes wrong, return -1.
652 static int lexmin_pivot_col(struct isl_tab *tab, int row)
655 int col = tab->n_col;
659 tr = tab->mat->row[row] + 2 + tab->M;
663 for (j = tab->n_dead; j < tab->n_col; ++j) {
664 if (tab->col_var[j] >= 0 &&
665 (tab->col_var[j] < tab->n_param ||
666 tab->col_var[j] >= tab->n_var - tab->n_div))
669 if (!isl_int_is_pos(tr[j]))
672 if (col == tab->n_col)
675 col = lexmin_col_pair(tab, row, col, j, tmp);
676 isl_assert(tab->mat->ctx, col >= 0, goto error);
686 /* Return the first known violated constraint, i.e., a non-negative
687 * contraint that currently has an either obviously negative value
688 * or a previously determined to be negative value.
690 * If any constraint has a negative coefficient for the big parameter,
691 * if any, then we return one of these first.
693 static int first_neg(struct isl_tab *tab)
698 for (row = tab->n_redundant; row < tab->n_row; ++row) {
699 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
701 if (isl_int_is_neg(tab->mat->row[row][2]))
704 for (row = tab->n_redundant; row < tab->n_row; ++row) {
705 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
708 if (tab->row_sign[row] == 0 &&
709 is_obviously_neg(tab, row))
710 tab->row_sign[row] = isl_tab_row_neg;
711 if (tab->row_sign[row] != isl_tab_row_neg)
713 } else if (!is_obviously_neg(tab, row))
720 /* Resolve all known or obviously violated constraints through pivoting.
721 * In particular, as long as we can find any violated constraint, we
722 * look for a pivoting column that would result in the lexicographicallly
723 * smallest increment in the sample point. If there is no such column
724 * then the tableau is infeasible.
726 static struct isl_tab *restore_lexmin(struct isl_tab *tab)
734 while ((row = first_neg(tab)) != -1) {
735 col = lexmin_pivot_col(tab, row);
736 if (col >= tab->n_col)
737 return isl_tab_mark_empty(tab);
740 isl_tab_pivot(tab, row, col);
748 /* Given a row that represents an equality, look for an appropriate
750 * In particular, if there are any non-zero coefficients among
751 * the non-parameter variables, then we take the last of these
752 * variables. Eliminating this variable in terms of the other
753 * variables and/or parameters does not influence the property
754 * that all column in the initial tableau are lexicographically
755 * positive. The row corresponding to the eliminated variable
756 * will only have non-zero entries below the diagonal of the
757 * initial tableau. That is, we transform
763 * If there is no such non-parameter variable, then we are dealing with
764 * pure parameter equality and we pick any parameter with coefficient 1 or -1
765 * for elimination. This will ensure that the eliminated parameter
766 * always has an integer value whenever all the other parameters are integral.
767 * If there is no such parameter then we return -1.
769 static int last_var_col_or_int_par_col(struct isl_tab *tab, int row)
771 unsigned off = 2 + tab->M;
774 for (i = tab->n_var - tab->n_div - 1; i >= 0 && i >= tab->n_param; --i) {
776 if (tab->var[i].is_row)
778 col = tab->var[i].index;
779 if (col <= tab->n_dead)
781 if (!isl_int_is_zero(tab->mat->row[row][off + col]))
784 for (i = tab->n_dead; i < tab->n_col; ++i) {
785 if (isl_int_is_one(tab->mat->row[row][off + i]))
787 if (isl_int_is_negone(tab->mat->row[row][off + i]))
793 /* Add an equality that is known to be valid to the tableau.
794 * We first check if we can eliminate a variable or a parameter.
795 * If not, we add the equality as two inequalities.
796 * In this case, the equality was a pure parameter equality and there
797 * is no need to resolve any constraint violations.
799 static struct isl_tab *add_lexmin_valid_eq(struct isl_tab *tab, isl_int *eq)
806 r = isl_tab_add_row(tab, eq);
810 r = tab->con[r].index;
811 i = last_var_col_or_int_par_col(tab, r);
813 tab->con[r].is_nonneg = 1;
814 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
815 isl_seq_neg(eq, eq, 1 + tab->n_var);
816 r = isl_tab_add_row(tab, eq);
819 tab->con[r].is_nonneg = 1;
820 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
822 isl_tab_pivot(tab, r, i);
823 isl_tab_kill_col(tab, i);
826 tab = restore_lexmin(tab);
835 /* Check if the given row is a pure constant.
837 static int is_constant(struct isl_tab *tab, int row)
839 unsigned off = 2 + tab->M;
841 return isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
842 tab->n_col - tab->n_dead) == -1;
845 /* Add an equality that may or may not be valid to the tableau.
846 * If the resulting row is a pure constant, then it must be zero.
847 * Otherwise, the resulting tableau is empty.
849 * If the row is not a pure constant, then we add two inequalities,
850 * each time checking that they can be satisfied.
851 * In the end we try to use one of the two constraints to eliminate
854 static struct isl_tab *add_lexmin_eq(struct isl_tab *tab, isl_int *eq)
862 tab->bset = isl_basic_set_add_eq(tab->bset, eq);
863 isl_tab_push(tab, isl_tab_undo_bset_eq);
867 r1 = isl_tab_add_row(tab, eq);
870 tab->con[r1].is_nonneg = 1;
871 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r1]);
873 row = tab->con[r1].index;
874 if (is_constant(tab, row)) {
875 if (!isl_int_is_zero(tab->mat->row[row][1]) ||
876 (tab->M && !isl_int_is_zero(tab->mat->row[row][2])))
877 return isl_tab_mark_empty(tab);
881 tab = restore_lexmin(tab);
882 if (!tab || tab->empty)
885 isl_seq_neg(eq, eq, 1 + tab->n_var);
887 r2 = isl_tab_add_row(tab, eq);
890 tab->con[r2].is_nonneg = 1;
891 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r2]);
893 tab = restore_lexmin(tab);
894 if (!tab || tab->empty)
897 if (!tab->con[r1].is_row)
898 isl_tab_kill_col(tab, tab->con[r1].index);
899 else if (!tab->con[r2].is_row)
900 isl_tab_kill_col(tab, tab->con[r2].index);
901 else if (isl_int_is_zero(tab->mat->row[tab->con[r1].index][1])) {
902 unsigned off = 2 + tab->M;
904 int row = tab->con[r1].index;
905 i = isl_seq_first_non_zero(tab->mat->row[row]+off+tab->n_dead,
906 tab->n_col - tab->n_dead);
908 isl_tab_pivot(tab, row, tab->n_dead + i);
909 isl_tab_kill_col(tab, tab->n_dead + i);
919 /* Add an inequality to the tableau, resolving violations using
922 static struct isl_tab *add_lexmin_ineq(struct isl_tab *tab, isl_int *ineq)
929 tab->bset = isl_basic_set_add_ineq(tab->bset, ineq);
930 isl_tab_push(tab, isl_tab_undo_bset_ineq);
934 r = isl_tab_add_row(tab, ineq);
937 tab->con[r].is_nonneg = 1;
938 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
939 if (isl_tab_row_is_redundant(tab, tab->con[r].index)) {
940 isl_tab_mark_redundant(tab, tab->con[r].index);
944 tab = restore_lexmin(tab);
945 if (tab && !tab->empty && tab->con[r].is_row &&
946 isl_tab_row_is_redundant(tab, tab->con[r].index))
947 isl_tab_mark_redundant(tab, tab->con[r].index);
954 /* Check if the coefficients of the parameters are all integral.
956 static int integer_parameter(struct isl_tab *tab, int row)
960 unsigned off = 2 + tab->M;
962 for (i = 0; i < tab->n_param; ++i) {
963 /* Eliminated parameter */
964 if (tab->var[i].is_row)
966 col = tab->var[i].index;
967 if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
968 tab->mat->row[row][0]))
971 for (i = 0; i < tab->n_div; ++i) {
972 if (tab->var[tab->n_var - tab->n_div + i].is_row)
974 col = tab->var[tab->n_var - tab->n_div + i].index;
975 if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
976 tab->mat->row[row][0]))
982 /* Check if the coefficients of the non-parameter variables are all integral.
984 static int integer_variable(struct isl_tab *tab, int row)
987 unsigned off = 2 + tab->M;
989 for (i = 0; i < tab->n_col; ++i) {
990 if (tab->col_var[i] >= 0 &&
991 (tab->col_var[i] < tab->n_param ||
992 tab->col_var[i] >= tab->n_var - tab->n_div))
994 if (!isl_int_is_divisible_by(tab->mat->row[row][off + i],
995 tab->mat->row[row][0]))
1001 /* Check if the constant term is integral.
1003 static int integer_constant(struct isl_tab *tab, int row)
1005 return isl_int_is_divisible_by(tab->mat->row[row][1],
1006 tab->mat->row[row][0]);
1009 #define I_CST 1 << 0
1010 #define I_PAR 1 << 1
1011 #define I_VAR 1 << 2
1013 /* Check for first (non-parameter) variable that is non-integer and
1014 * therefore requires a cut.
1015 * For parametric tableaus, there are three parts in a row,
1016 * the constant, the coefficients of the parameters and the rest.
1017 * For each part, we check whether the coefficients in that part
1018 * are all integral and if so, set the corresponding flag in *f.
1019 * If the constant and the parameter part are integral, then the
1020 * current sample value is integral and no cut is required
1021 * (irrespective of whether the variable part is integral).
1023 static int first_non_integer(struct isl_tab *tab, int *f)
1027 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
1030 if (!tab->var[i].is_row)
1032 row = tab->var[i].index;
1033 if (integer_constant(tab, row))
1034 ISL_FL_SET(flags, I_CST);
1035 if (integer_parameter(tab, row))
1036 ISL_FL_SET(flags, I_PAR);
1037 if (ISL_FL_ISSET(flags, I_CST) && ISL_FL_ISSET(flags, I_PAR))
1039 if (integer_variable(tab, row))
1040 ISL_FL_SET(flags, I_VAR);
1047 /* Add a (non-parametric) cut to cut away the non-integral sample
1048 * value of the given row.
1050 * If the row is given by
1052 * m r = f + \sum_i a_i y_i
1056 * c = - {-f/m} + \sum_i {a_i/m} y_i >= 0
1058 * The big parameter, if any, is ignored, since it is assumed to be big
1059 * enough to be divisible by any integer.
1060 * If the tableau is actually a parametric tableau, then this function
1061 * is only called when all coefficients of the parameters are integral.
1062 * The cut therefore has zero coefficients for the parameters.
1064 * The current value is known to be negative, so row_sign, if it
1065 * exists, is set accordingly.
1067 * Return the row of the cut or -1.
1069 static int add_cut(struct isl_tab *tab, int row)
1074 unsigned off = 2 + tab->M;
1076 if (isl_tab_extend_cons(tab, 1) < 0)
1078 r = isl_tab_allocate_con(tab);
1082 r_row = tab->mat->row[tab->con[r].index];
1083 isl_int_set(r_row[0], tab->mat->row[row][0]);
1084 isl_int_neg(r_row[1], tab->mat->row[row][1]);
1085 isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
1086 isl_int_neg(r_row[1], r_row[1]);
1088 isl_int_set_si(r_row[2], 0);
1089 for (i = 0; i < tab->n_col; ++i)
1090 isl_int_fdiv_r(r_row[off + i],
1091 tab->mat->row[row][off + i], tab->mat->row[row][0]);
1093 tab->con[r].is_nonneg = 1;
1094 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
1096 tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
1098 return tab->con[r].index;
1101 /* Given a non-parametric tableau, add cuts until an integer
1102 * sample point is obtained or until the tableau is determined
1103 * to be integer infeasible.
1104 * As long as there is any non-integer value in the sample point,
1105 * we add an appropriate cut, if possible and resolve the violated
1106 * cut constraint using restore_lexmin.
1107 * If one of the corresponding rows is equal to an integral
1108 * combination of variables/constraints plus a non-integral constant,
1109 * then there is no way to obtain an integer point an we return
1110 * a tableau that is marked empty.
1112 static struct isl_tab *cut_to_integer_lexmin(struct isl_tab *tab)
1122 while ((row = first_non_integer(tab, &flags)) != -1) {
1123 if (ISL_FL_ISSET(flags, I_VAR))
1124 return isl_tab_mark_empty(tab);
1125 row = add_cut(tab, row);
1128 tab = restore_lexmin(tab);
1129 if (!tab || tab->empty)
1138 static struct isl_tab *drop_sample(struct isl_tab *tab, int s)
1140 if (s != tab->n_outside)
1141 isl_mat_swap_rows(tab->samples, tab->n_outside, s);
1143 isl_tab_push(tab, isl_tab_undo_drop_sample);
1148 /* Check whether all the currently active samples also satisfy the inequality
1149 * "ineq" (treated as an equality if eq is set).
1150 * Remove those samples that do not.
1152 static struct isl_tab *check_samples(struct isl_tab *tab, isl_int *ineq, int eq)
1160 isl_assert(tab->mat->ctx, tab->bset, goto error);
1161 isl_assert(tab->mat->ctx, tab->samples, goto error);
1162 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);
1165 for (i = tab->n_outside; i < tab->n_sample; ++i) {
1167 isl_seq_inner_product(ineq, tab->samples->row[i],
1168 1 + tab->n_var, &v);
1169 sgn = isl_int_sgn(v);
1170 if (eq ? (sgn == 0) : (sgn >= 0))
1172 tab = drop_sample(tab, i);
1184 /* Check whether the sample value of the tableau is finite,
1185 * i.e., either the tableau does not use a big parameter, or
1186 * all values of the variables are equal to the big parameter plus
1187 * some constant. This constant is the actual sample value.
1189 static int sample_is_finite(struct isl_tab *tab)
1196 for (i = 0; i < tab->n_var; ++i) {
1198 if (!tab->var[i].is_row)
1200 row = tab->var[i].index;
1201 if (isl_int_ne(tab->mat->row[row][0], tab->mat->row[row][2]))
1207 /* Check if the context tableau of sol has any integer points.
1208 * Returns -1 if an error occurred.
1209 * If an integer point can be found and if moreover it is finite,
1210 * then it is added to the list of sample values.
1212 * This function is only called when none of the currently active sample
1213 * values satisfies the most recently added constraint.
1215 static int context_is_feasible(struct isl_sol *sol)
1217 struct isl_tab_undo *snap;
1218 struct isl_tab *tab;
1221 if (!sol || !sol->context_tab)
1224 snap = isl_tab_snap(sol->context_tab);
1225 isl_tab_push_basis(sol->context_tab);
1227 sol->context_tab = cut_to_integer_lexmin(sol->context_tab);
1228 if (!sol->context_tab)
1231 tab = sol->context_tab;
1232 if (!tab->empty && sample_is_finite(tab)) {
1233 struct isl_vec *sample;
1235 tab->samples = isl_mat_extend(tab->samples,
1236 tab->n_sample + 1, tab->samples->n_col);
1240 sample = isl_tab_get_sample_value(tab);
1243 isl_seq_cpy(tab->samples->row[tab->n_sample],
1244 sample->el, sample->size);
1245 isl_vec_free(sample);
1249 feasible = !sol->context_tab->empty;
1250 if (isl_tab_rollback(sol->context_tab, snap) < 0)
1255 isl_tab_free(sol->context_tab);
1256 sol->context_tab = NULL;
1260 /* First check if any of the currently active sample values satisfies
1261 * the inequality "ineq" (an equality if eq is set).
1262 * If not, continue with check_integer_feasible.
1264 static int context_valid_sample_or_feasible(struct isl_sol *sol,
1265 isl_int *ineq, int eq)
1269 struct isl_tab *tab;
1271 if (!sol || !sol->context_tab)
1274 tab = sol->context_tab;
1275 isl_assert(tab->mat->ctx, tab->bset, return -1);
1276 isl_assert(tab->mat->ctx, tab->samples, return -1);
1277 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, return -1);
1280 for (i = tab->n_outside; i < tab->n_sample; ++i) {
1282 isl_seq_inner_product(ineq, tab->samples->row[i],
1283 1 + tab->n_var, &v);
1284 sgn = isl_int_sgn(v);
1285 if (eq ? (sgn == 0) : (sgn >= 0))
1290 if (i < tab->n_sample)
1293 return context_is_feasible(sol);
1296 /* For a div d = floor(f/m), add the constraints
1299 * -(f-(m-1)) + m d >= 0
1301 * Note that the second constraint is the negation of
1305 static struct isl_tab *add_div_constraints(struct isl_tab *tab, unsigned div)
1309 struct isl_vec *ineq;
1314 total = isl_basic_set_total_dim(tab->bset);
1315 div_pos = 1 + total - tab->bset->n_div + div;
1317 ineq = ineq_for_div(tab->bset, div);
1321 tab = add_lexmin_ineq(tab, ineq->el);
1323 isl_seq_neg(ineq->el, tab->bset->div[div] + 1, 1 + total);
1324 isl_int_set(ineq->el[div_pos], tab->bset->div[div][0]);
1325 isl_int_add(ineq->el[0], ineq->el[0], ineq->el[div_pos]);
1326 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1327 tab = add_lexmin_ineq(tab, ineq->el);
1337 /* Add a div specified by "div" to both the main tableau and
1338 * the context tableau. In case of the main tableau, we only
1339 * need to add an extra div. In the context tableau, we also
1340 * need to express the meaning of the div.
1341 * Return the index of the div or -1 if anything went wrong.
1343 static int add_div(struct isl_tab *tab, struct isl_tab **context_tab,
1344 struct isl_vec *div)
1349 struct isl_mat *samples;
1351 if (isl_tab_extend_vars(*context_tab, 1) < 0)
1353 r = isl_tab_allocate_var(*context_tab);
1356 (*context_tab)->var[r].is_nonneg = 1;
1357 (*context_tab)->var[r].frozen = 1;
1359 samples = isl_mat_extend((*context_tab)->samples,
1360 (*context_tab)->n_sample, 1 + (*context_tab)->n_var);
1361 (*context_tab)->samples = samples;
1364 for (i = (*context_tab)->n_outside; i < samples->n_row; ++i) {
1365 isl_seq_inner_product(div->el + 1, samples->row[i],
1366 div->size - 1, &samples->row[i][samples->n_col - 1]);
1367 isl_int_fdiv_q(samples->row[i][samples->n_col - 1],
1368 samples->row[i][samples->n_col - 1], div->el[0]);
1371 (*context_tab)->bset = isl_basic_set_extend_dim((*context_tab)->bset,
1372 isl_basic_set_get_dim((*context_tab)->bset), 1, 0, 2);
1373 k = isl_basic_set_alloc_div((*context_tab)->bset);
1376 isl_seq_cpy((*context_tab)->bset->div[k], div->el, div->size);
1377 isl_tab_push((*context_tab), isl_tab_undo_bset_div);
1378 *context_tab = add_div_constraints(*context_tab, k);
1382 if (isl_tab_extend_vars(tab, 1) < 0)
1384 r = isl_tab_allocate_var(tab);
1387 if (!(*context_tab)->M)
1388 tab->var[r].is_nonneg = 1;
1389 tab->var[r].frozen = 1;
1392 return tab->n_div - 1;
1394 isl_tab_free(*context_tab);
1395 *context_tab = NULL;
1399 static int find_div(struct isl_tab *tab, isl_int *div, isl_int denom)
1402 unsigned total = isl_basic_set_total_dim(tab->bset);
1404 for (i = 0; i < tab->bset->n_div; ++i) {
1405 if (isl_int_ne(tab->bset->div[i][0], denom))
1407 if (!isl_seq_eq(tab->bset->div[i] + 1, div, total))
1414 /* Return the index of a div that corresponds to "div".
1415 * We first check if we already have such a div and if not, we create one.
1417 static int get_div(struct isl_tab *tab, struct isl_tab **context_tab,
1418 struct isl_vec *div)
1422 d = find_div(*context_tab, div->el + 1, div->el[0]);
1426 return add_div(tab, context_tab, div);
1429 /* Add a parametric cut to cut away the non-integral sample value
1431 * Let a_i be the coefficients of the constant term and the parameters
1432 * and let b_i be the coefficients of the variables or constraints
1433 * in basis of the tableau.
1434 * Let q be the div q = floor(\sum_i {-a_i} y_i).
1436 * The cut is expressed as
1438 * c = \sum_i -{-a_i} y_i + \sum_i {b_i} x_i + q >= 0
1440 * If q did not already exist in the context tableau, then it is added first.
1441 * If q is in a column of the main tableau then the "+ q" can be accomplished
1442 * by setting the corresponding entry to the denominator of the constraint.
1443 * If q happens to be in a row of the main tableau, then the corresponding
1444 * row needs to be added instead (taking care of the denominators).
1445 * Note that this is very unlikely, but perhaps not entirely impossible.
1447 * The current value of the cut is known to be negative (or at least
1448 * non-positive), so row_sign is set accordingly.
1450 * Return the row of the cut or -1.
1452 static int add_parametric_cut(struct isl_tab *tab, int row,
1453 struct isl_tab **context_tab)
1455 struct isl_vec *div;
1461 unsigned off = 2 + tab->M;
1466 if (isl_tab_extend_cons(*context_tab, 3) < 0)
1469 div = get_row_parameter_div(tab, row);
1473 d = get_div(tab, context_tab, div);
1477 if (isl_tab_extend_cons(tab, 1) < 0)
1479 r = isl_tab_allocate_con(tab);
1483 r_row = tab->mat->row[tab->con[r].index];
1484 isl_int_set(r_row[0], tab->mat->row[row][0]);
1485 isl_int_neg(r_row[1], tab->mat->row[row][1]);
1486 isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
1487 isl_int_neg(r_row[1], r_row[1]);
1489 isl_int_set_si(r_row[2], 0);
1490 for (i = 0; i < tab->n_param; ++i) {
1491 if (tab->var[i].is_row)
1493 col = tab->var[i].index;
1494 isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
1495 isl_int_fdiv_r(r_row[off + col], r_row[off + col],
1496 tab->mat->row[row][0]);
1497 isl_int_neg(r_row[off + col], r_row[off + col]);
1499 for (i = 0; i < tab->n_div; ++i) {
1500 if (tab->var[tab->n_var - tab->n_div + i].is_row)
1502 col = tab->var[tab->n_var - tab->n_div + i].index;
1503 isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
1504 isl_int_fdiv_r(r_row[off + col], r_row[off + col],
1505 tab->mat->row[row][0]);
1506 isl_int_neg(r_row[off + col], r_row[off + col]);
1508 for (i = 0; i < tab->n_col; ++i) {
1509 if (tab->col_var[i] >= 0 &&
1510 (tab->col_var[i] < tab->n_param ||
1511 tab->col_var[i] >= tab->n_var - tab->n_div))
1513 isl_int_fdiv_r(r_row[off + i],
1514 tab->mat->row[row][off + i], tab->mat->row[row][0]);
1516 if (tab->var[tab->n_var - tab->n_div + d].is_row) {
1518 int d_row = tab->var[tab->n_var - tab->n_div + d].index;
1520 isl_int_gcd(gcd, tab->mat->row[d_row][0], r_row[0]);
1521 isl_int_divexact(r_row[0], r_row[0], gcd);
1522 isl_int_divexact(gcd, tab->mat->row[d_row][0], gcd);
1523 isl_seq_combine(r_row + 1, gcd, r_row + 1,
1524 r_row[0], tab->mat->row[d_row] + 1,
1525 off - 1 + tab->n_col);
1526 isl_int_mul(r_row[0], r_row[0], tab->mat->row[d_row][0]);
1529 col = tab->var[tab->n_var - tab->n_div + d].index;
1530 isl_int_set(r_row[off + col], tab->mat->row[row][0]);
1533 tab->con[r].is_nonneg = 1;
1534 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
1536 tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
1540 return tab->con[r].index;
1542 isl_tab_free(*context_tab);
1543 *context_tab = NULL;
1547 /* Construct a tableau for bmap that can be used for computing
1548 * the lexicographic minimum (or maximum) of bmap.
1549 * If not NULL, then dom is the domain where the minimum
1550 * should be computed. In this case, we set up a parametric
1551 * tableau with row signs (initialized to "unknown").
1552 * If M is set, then the tableau will use a big parameter.
1553 * If max is set, then a maximum should be computed instead of a minimum.
1554 * This means that for each variable x, the tableau will contain the variable
1555 * x' = M - x, rather than x' = M + x. This in turn means that the coefficient
1556 * of the variables in all constraints are negated prior to adding them
1559 static struct isl_tab *tab_for_lexmin(struct isl_basic_map *bmap,
1560 struct isl_basic_set *dom, unsigned M, int max)
1563 struct isl_tab *tab;
1565 tab = isl_tab_alloc(bmap->ctx, 2 * bmap->n_eq + bmap->n_ineq + 1,
1566 isl_basic_map_total_dim(bmap), M);
1570 tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
1572 tab->n_param = isl_basic_set_total_dim(dom) - dom->n_div;
1573 tab->n_div = dom->n_div;
1574 tab->row_sign = isl_calloc_array(bmap->ctx,
1575 enum isl_tab_row_sign, tab->mat->n_row);
1579 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
1580 return isl_tab_mark_empty(tab);
1582 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
1583 tab->var[i].is_nonneg = 1;
1584 tab->var[i].frozen = 1;
1586 for (i = 0; i < bmap->n_eq; ++i) {
1588 isl_seq_neg(bmap->eq[i] + 1 + tab->n_param,
1589 bmap->eq[i] + 1 + tab->n_param,
1590 tab->n_var - tab->n_param - tab->n_div);
1591 tab = add_lexmin_valid_eq(tab, bmap->eq[i]);
1593 isl_seq_neg(bmap->eq[i] + 1 + tab->n_param,
1594 bmap->eq[i] + 1 + tab->n_param,
1595 tab->n_var - tab->n_param - tab->n_div);
1596 if (!tab || tab->empty)
1599 for (i = 0; i < bmap->n_ineq; ++i) {
1601 isl_seq_neg(bmap->ineq[i] + 1 + tab->n_param,
1602 bmap->ineq[i] + 1 + tab->n_param,
1603 tab->n_var - tab->n_param - tab->n_div);
1604 tab = add_lexmin_ineq(tab, bmap->ineq[i]);
1606 isl_seq_neg(bmap->ineq[i] + 1 + tab->n_param,
1607 bmap->ineq[i] + 1 + tab->n_param,
1608 tab->n_var - tab->n_param - tab->n_div);
1609 if (!tab || tab->empty)
1618 static struct isl_tab *context_tab_for_lexmin(struct isl_basic_set *bset)
1620 struct isl_tab *tab;
1622 bset = isl_basic_set_cow(bset);
1625 tab = tab_for_lexmin((struct isl_basic_map *)bset, NULL, 1, 0);
1631 tab->samples = isl_mat_alloc(bset->ctx, 1, 1 + tab->n_var);
1636 isl_basic_set_free(bset);
1640 /* Construct an isl_sol_map structure for accumulating the solution.
1641 * If track_empty is set, then we also keep track of the parts
1642 * of the context where there is no solution.
1643 * If max is set, then we are solving a maximization, rather than
1644 * a minimization problem, which means that the variables in the
1645 * tableau have value "M - x" rather than "M + x".
1647 static struct isl_sol_map *sol_map_init(struct isl_basic_map *bmap,
1648 struct isl_basic_set *dom, int track_empty, int max)
1650 struct isl_sol_map *sol_map;
1651 struct isl_tab *context_tab;
1654 sol_map = isl_calloc_type(bset->ctx, struct isl_sol_map);
1659 sol_map->sol.add = &sol_map_add_wrap;
1660 sol_map->sol.free = &sol_map_free_wrap;
1661 sol_map->map = isl_map_alloc_dim(isl_basic_map_get_dim(bmap), 1,
1666 context_tab = context_tab_for_lexmin(isl_basic_set_copy(dom));
1667 context_tab = restore_lexmin(context_tab);
1668 sol_map->sol.context_tab = context_tab;
1669 f = context_is_feasible(&sol_map->sol);
1674 sol_map->empty = isl_set_alloc_dim(isl_basic_set_get_dim(dom),
1675 1, ISL_SET_DISJOINT);
1676 if (!sol_map->empty)
1680 isl_basic_set_free(dom);
1683 isl_basic_set_free(dom);
1684 sol_map_free(sol_map);
1688 /* For each variable in the context tableau, check if the variable can
1689 * only attain non-negative values. If so, mark the parameter as non-negative
1690 * in the main tableau. This allows for a more direct identification of some
1691 * cases of violated constraints.
1693 static struct isl_tab *tab_detect_nonnegative_parameters(struct isl_tab *tab,
1694 struct isl_tab *context_tab)
1697 struct isl_tab_undo *snap, *snap2;
1698 struct isl_vec *ineq = NULL;
1699 struct isl_tab_var *var;
1702 if (context_tab->n_var == 0)
1705 ineq = isl_vec_alloc(tab->mat->ctx, 1 + context_tab->n_var);
1709 if (isl_tab_extend_cons(context_tab, 1) < 0)
1712 snap = isl_tab_snap(context_tab);
1713 isl_tab_push_basis(context_tab);
1715 snap2 = isl_tab_snap(context_tab);
1718 isl_seq_clr(ineq->el, ineq->size);
1719 for (i = 0; i < context_tab->n_var; ++i) {
1720 isl_int_set_si(ineq->el[1 + i], 1);
1721 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
1722 var = &context_tab->con[context_tab->n_con - 1];
1723 if (!context_tab->empty &&
1724 !isl_tab_min_at_most_neg_one(context_tab, var)) {
1726 if (i >= tab->n_param)
1727 j = i - tab->n_param + tab->n_var - tab->n_div;
1728 tab->var[j].is_nonneg = 1;
1731 isl_int_set_si(ineq->el[1 + i], 0);
1732 if (isl_tab_rollback(context_tab, snap2) < 0)
1736 if (isl_tab_rollback(context_tab, snap) < 0)
1739 if (n == context_tab->n_var) {
1740 context_tab->mat = isl_mat_drop_cols(context_tab->mat, 2, 1);
1752 /* Check whether all coefficients of (non-parameter) variables
1753 * are non-positive, meaning that no pivots can be performed on the row.
1755 static int is_critical(struct isl_tab *tab, int row)
1758 unsigned off = 2 + tab->M;
1760 for (j = tab->n_dead; j < tab->n_col; ++j) {
1761 if (tab->col_var[j] >= 0 &&
1762 (tab->col_var[j] < tab->n_param ||
1763 tab->col_var[j] >= tab->n_var - tab->n_div))
1766 if (isl_int_is_pos(tab->mat->row[row][off + j]))
1773 /* Check whether the inequality represented by vec is strict over the integers,
1774 * i.e., there are no integer values satisfying the constraint with
1775 * equality. This happens if the gcd of the coefficients is not a divisor
1776 * of the constant term. If so, scale the constraint down by the gcd
1777 * of the coefficients.
1779 static int is_strict(struct isl_vec *vec)
1785 isl_seq_gcd(vec->el + 1, vec->size - 1, &gcd);
1786 if (!isl_int_is_one(gcd)) {
1787 strict = !isl_int_is_divisible_by(vec->el[0], gcd);
1788 isl_int_fdiv_q(vec->el[0], vec->el[0], gcd);
1789 isl_seq_scale_down(vec->el + 1, vec->el + 1, gcd, vec->size-1);
1796 /* Determine the sign of the given row of the main tableau.
1797 * The result is one of
1798 * isl_tab_row_pos: always non-negative; no pivot needed
1799 * isl_tab_row_neg: always non-positive; pivot
1800 * isl_tab_row_any: can be both positive and negative; split
1802 * We first handle some simple cases
1803 * - the row sign may be known already
1804 * - the row may be obviously non-negative
1805 * - the parametric constant may be equal to that of another row
1806 * for which we know the sign. This sign will be either "pos" or
1807 * "any". If it had been "neg" then we would have pivoted before.
1809 * If none of these cases hold, we check the value of the row for each
1810 * of the currently active samples. Based on the signs of these values
1811 * we make an initial determination of the sign of the row.
1813 * all zero -> unk(nown)
1814 * all non-negative -> pos
1815 * all non-positive -> neg
1816 * both negative and positive -> all
1818 * If we end up with "all", we are done.
1819 * Otherwise, we perform a check for positive and/or negative
1820 * values as follows.
1822 * samples neg unk pos
1828 * There is no special sign for "zero", because we can usually treat zero
1829 * as either non-negative or non-positive, whatever works out best.
1830 * However, if the row is "critical", meaning that pivoting is impossible
1831 * then we don't want to limp zero with the non-positive case, because
1832 * then we we would lose the solution for those values of the parameters
1833 * where the value of the row is zero. Instead, we treat 0 as non-negative
1834 * ensuring a split if the row can attain both zero and negative values.
1835 * The same happens when the original constraint was one that could not
1836 * be satisfied with equality by any integer values of the parameters.
1837 * In this case, we normalize the constraint, but then a value of zero
1838 * for the normalized constraint is actually a positive value for the
1839 * original constraint, so again we need to treat zero as non-negative.
1840 * In both these cases, we have the following decision tree instead:
1842 * all non-negative -> pos
1843 * all negative -> neg
1844 * both negative and non-negative -> all
1852 static int row_sign(struct isl_tab *tab, struct isl_sol *sol, int row)
1855 struct isl_tab_undo *snap = NULL;
1856 struct isl_vec *ineq = NULL;
1857 int res = isl_tab_row_unknown;
1863 struct isl_tab *context_tab = sol->context_tab;
1865 if (tab->row_sign[row] != isl_tab_row_unknown)
1866 return tab->row_sign[row];
1867 if (is_obviously_nonneg(tab, row))
1868 return isl_tab_row_pos;
1869 for (row2 = tab->n_redundant; row2 < tab->n_row; ++row2) {
1870 if (tab->row_sign[row2] == isl_tab_row_unknown)
1872 if (identical_parameter_line(tab, row, row2))
1873 return tab->row_sign[row2];
1876 critical = is_critical(tab, row);
1878 isl_assert(tab->mat->ctx, context_tab->samples, goto error);
1879 isl_assert(tab->mat->ctx, context_tab->samples->n_col == 1 + context_tab->n_var, goto error);
1881 ineq = get_row_parameter_ineq(tab, row);
1885 strict = is_strict(ineq);
1888 for (i = context_tab->n_outside; i < context_tab->n_sample; ++i) {
1889 isl_seq_inner_product(context_tab->samples->row[i], ineq->el,
1891 sgn = isl_int_sgn(tmp);
1892 if (sgn > 0 || (sgn == 0 && (critical || strict))) {
1893 if (res == isl_tab_row_unknown)
1894 res = isl_tab_row_pos;
1895 if (res == isl_tab_row_neg)
1896 res = isl_tab_row_any;
1899 if (res == isl_tab_row_unknown)
1900 res = isl_tab_row_neg;
1901 if (res == isl_tab_row_pos)
1902 res = isl_tab_row_any;
1904 if (res == isl_tab_row_any)
1909 if (res != isl_tab_row_any) {
1910 if (isl_tab_extend_cons(context_tab, 1) < 0)
1913 snap = isl_tab_snap(context_tab);
1914 isl_tab_push_basis(context_tab);
1917 if (res == isl_tab_row_unknown || res == isl_tab_row_pos) {
1918 /* test for negative values */
1920 isl_seq_neg(ineq->el, ineq->el, ineq->size);
1921 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1923 isl_tab_push_basis(context_tab);
1924 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq->el);
1925 feasible = context_is_feasible(sol);
1928 context_tab = sol->context_tab;
1930 res = isl_tab_row_pos;
1932 res = (res == isl_tab_row_unknown) ? isl_tab_row_neg
1934 if (isl_tab_rollback(context_tab, snap) < 0)
1937 if (res == isl_tab_row_neg) {
1938 isl_seq_neg(ineq->el, ineq->el, ineq->size);
1939 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1943 if (res == isl_tab_row_neg) {
1944 /* test for positive values */
1946 if (!critical && !strict)
1947 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1949 isl_tab_push_basis(context_tab);
1950 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq->el);
1951 feasible = context_is_feasible(sol);
1954 context_tab = sol->context_tab;
1956 res = isl_tab_row_any;
1957 if (isl_tab_rollback(context_tab, snap) < 0)
1968 static struct isl_sol *find_solutions(struct isl_sol *sol, struct isl_tab *tab);
1970 /* Find solutions for values of the parameters that satisfy the given
1973 * We currently take a snapshot of the context tableau that is reset
1974 * when we return from this function, while we make a copy of the main
1975 * tableau, leaving the original main tableau untouched.
1976 * These are fairly arbitrary choices. Making a copy also of the context
1977 * tableau would obviate the need to undo any changes made to it later,
1978 * while taking a snapshot of the main tableau could reduce memory usage.
1979 * If we were to switch to taking a snapshot of the main tableau,
1980 * we would have to keep in mind that we need to save the row signs
1981 * and that we need to do this before saving the current basis
1982 * such that the basis has been restore before we restore the row signs.
1984 static struct isl_sol *find_in_pos(struct isl_sol *sol,
1985 struct isl_tab *tab, isl_int *ineq)
1987 struct isl_tab_undo *snap;
1989 snap = isl_tab_snap(sol->context_tab);
1990 isl_tab_push_basis(sol->context_tab);
1991 if (isl_tab_extend_cons(sol->context_tab, 1) < 0)
1994 tab = isl_tab_dup(tab);
1998 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq);
1999 sol->context_tab = check_samples(sol->context_tab, ineq, 0);
2001 sol = find_solutions(sol, tab);
2003 isl_tab_rollback(sol->context_tab, snap);
2006 isl_tab_rollback(sol->context_tab, snap);
2011 /* Record the absence of solutions for those values of the parameters
2012 * that do not satisfy the given inequality with equality.
2014 static struct isl_sol *no_sol_in_strict(struct isl_sol *sol,
2015 struct isl_tab *tab, struct isl_vec *ineq)
2019 struct isl_tab_undo *snap;
2020 snap = isl_tab_snap(sol->context_tab);
2021 isl_tab_push_basis(sol->context_tab);
2022 if (isl_tab_extend_cons(sol->context_tab, 1) < 0)
2025 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
2027 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq->el);
2028 f = context_valid_sample_or_feasible(sol, ineq->el, 0);
2034 sol = sol->add(sol, tab);
2037 isl_int_add_ui(ineq->el[0], ineq->el[0], 1);
2039 if (isl_tab_rollback(sol->context_tab, snap) < 0)
2047 /* Given a main tableau where more than one row requires a split,
2048 * determine and return the "best" row to split on.
2050 * Given two rows in the main tableau, if the inequality corresponding
2051 * to the first row is redundant with respect to that of the second row
2052 * in the current tableau, then it is better to split on the second row,
2053 * since in the positive part, both row will be positive.
2054 * (In the negative part a pivot will have to be performed and just about
2055 * anything can happen to the sign of the other row.)
2057 * As a simple heuristic, we therefore select the row that makes the most
2058 * of the other rows redundant.
2060 * Perhaps it would also be useful to look at the number of constraints
2061 * that conflict with any given constraint.
2063 static int best_split(struct isl_tab *tab, struct isl_tab *context_tab)
2065 struct isl_tab_undo *snap, *snap2;
2071 if (isl_tab_extend_cons(context_tab, 2) < 0)
2074 snap = isl_tab_snap(context_tab);
2075 isl_tab_push_basis(context_tab);
2076 snap2 = isl_tab_snap(context_tab);
2078 for (split = tab->n_redundant; split < tab->n_row; ++split) {
2079 struct isl_tab_undo *snap3;
2080 struct isl_vec *ineq = NULL;
2083 if (!isl_tab_var_from_row(tab, split)->is_nonneg)
2085 if (tab->row_sign[split] != isl_tab_row_any)
2088 ineq = get_row_parameter_ineq(tab, split);
2091 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
2094 snap3 = isl_tab_snap(context_tab);
2096 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2097 struct isl_tab_var *var;
2101 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
2103 if (tab->row_sign[row] != isl_tab_row_any)
2106 ineq = get_row_parameter_ineq(tab, row);
2109 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
2111 var = &context_tab->con[context_tab->n_con - 1];
2112 if (!context_tab->empty &&
2113 !isl_tab_min_at_most_neg_one(context_tab, var))
2115 if (isl_tab_rollback(context_tab, snap3) < 0)
2118 if (best == -1 || r > best_r) {
2122 if (isl_tab_rollback(context_tab, snap2) < 0)
2126 if (isl_tab_rollback(context_tab, snap) < 0)
2132 /* Compute the lexicographic minimum of the set represented by the main
2133 * tableau "tab" within the context "sol->context_tab".
2134 * On entry the sample value of the main tableau is lexicographically
2135 * less than or equal to this lexicographic minimum.
2136 * Pivots are performed until a feasible point is found, which is then
2137 * necessarily equal to the minimum, or until the tableau is found to
2138 * be infeasible. Some pivots may need to be performed for only some
2139 * feasible values of the context tableau. If so, the context tableau
2140 * is split into a part where the pivot is needed and a part where it is not.
2142 * Whenever we enter the main loop, the main tableau is such that no
2143 * "obvious" pivots need to be performed on it, where "obvious" means
2144 * that the given row can be seen to be negative without looking at
2145 * the context tableau. In particular, for non-parametric problems,
2146 * no pivots need to be performed on the main tableau.
2147 * The caller of find_solutions is responsible for making this property
2148 * hold prior to the first iteration of the loop, while restore_lexmin
2149 * is called before every other iteration.
2151 * Inside the main loop, we first examine the signs of the rows of
2152 * the main tableau within the context of the context tableau.
2153 * If we find a row that is always non-positive for all values of
2154 * the parameters satisfying the context tableau and negative for at
2155 * least one value of the parameters, we perform the appropriate pivot
2156 * and start over. An exception is the case where no pivot can be
2157 * performed on the row. In this case, we require that the sign of
2158 * the row is negative for all values of the parameters (rather than just
2159 * non-positive). This special case is handled inside row_sign, which
2160 * will say that the row can have any sign if it determines that it can
2161 * attain both negative and zero values.
2163 * If we can't find a row that always requires a pivot, but we can find
2164 * one or more rows that require a pivot for some values of the parameters
2165 * (i.e., the row can attain both positive and negative signs), then we split
2166 * the context tableau into two parts, one where we force the sign to be
2167 * non-negative and one where we force is to be negative.
2168 * The non-negative part is handled by a recursive call (through find_in_pos).
2169 * Upon returning from this call, we continue with the negative part and
2170 * perform the required pivot.
2172 * If no such rows can be found, all rows are non-negative and we have
2173 * found a (rational) feasible point. If we only wanted a rational point
2175 * Otherwise, we check if all values of the sample point of the tableau
2176 * are integral for the variables. If so, we have found the minimal
2177 * integral point and we are done.
2178 * If the sample point is not integral, then we need to make a distinction
2179 * based on whether the constant term is non-integral or the coefficients
2180 * of the parameters. Furthermore, in order to decide how to handle
2181 * the non-integrality, we also need to know whether the coefficients
2182 * of the other columns in the tableau are integral. This leads
2183 * to the following table. The first two rows do not correspond
2184 * to a non-integral sample point and are only mentioned for completeness.
2186 * constant parameters other
2189 * int int rat | -> no problem
2191 * rat int int -> fail
2193 * rat int rat -> cut
2196 * rat rat rat | -> parametric cut
2199 * rat rat int | -> split context
2201 * If the parametric constant is completely integral, then there is nothing
2202 * to be done. If the constant term is non-integral, but all the other
2203 * coefficient are integral, then there is nothing that can be done
2204 * and the tableau has no integral solution.
2205 * If, on the other hand, one or more of the other columns have rational
2206 * coeffcients, but the parameter coefficients are all integral, then
2207 * we can perform a regular (non-parametric) cut.
2208 * Finally, if there is any parameter coefficient that is non-integral,
2209 * then we need to involve the context tableau. There are two cases here.
2210 * If at least one other column has a rational coefficient, then we
2211 * can perform a parametric cut in the main tableau by adding a new
2212 * integer division in the context tableau.
2213 * If all other columns have integral coefficients, then we need to
2214 * enforce that the rational combination of parameters (c + \sum a_i y_i)/m
2215 * is always integral. We do this by introducing an integer division
2216 * q = floor((c + \sum a_i y_i)/m) and stipulating that its argument should
2217 * always be integral in the context tableau, i.e., m q = c + \sum a_i y_i.
2218 * Since q is expressed in the tableau as
2219 * c + \sum a_i y_i - m q >= 0
2220 * -c - \sum a_i y_i + m q + m - 1 >= 0
2221 * it is sufficient to add the inequality
2222 * -c - \sum a_i y_i + m q >= 0
2223 * In the part of the context where this inequality does not hold, the
2224 * main tableau is marked as being empty.
2226 static struct isl_sol *find_solutions(struct isl_sol *sol, struct isl_tab *tab)
2228 struct isl_tab **context_tab;
2233 context_tab = &sol->context_tab;
2237 if ((*context_tab)->empty)
2240 for (; tab && !tab->empty; tab = restore_lexmin(tab)) {
2247 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2248 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
2250 sgn = row_sign(tab, sol, row);
2253 tab->row_sign[row] = sgn;
2254 if (sgn == isl_tab_row_any)
2256 if (sgn == isl_tab_row_any && split == -1)
2258 if (sgn == isl_tab_row_neg)
2261 if (row < tab->n_row)
2264 struct isl_vec *ineq;
2266 split = best_split(tab, *context_tab);
2269 ineq = get_row_parameter_ineq(tab, split);
2273 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2274 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
2276 if (tab->row_sign[row] == isl_tab_row_any)
2277 tab->row_sign[row] = isl_tab_row_unknown;
2279 tab->row_sign[split] = isl_tab_row_pos;
2280 sol = find_in_pos(sol, tab, ineq->el);
2281 tab->row_sign[split] = isl_tab_row_neg;
2283 isl_seq_neg(ineq->el, ineq->el, ineq->size);
2284 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
2285 *context_tab = add_lexmin_ineq(*context_tab, ineq->el);
2286 *context_tab = check_samples(*context_tab, ineq->el, 0);
2294 row = first_non_integer(tab, &flags);
2297 if (ISL_FL_ISSET(flags, I_PAR)) {
2298 if (ISL_FL_ISSET(flags, I_VAR)) {
2299 tab = isl_tab_mark_empty(tab);
2302 row = add_cut(tab, row);
2303 } else if (ISL_FL_ISSET(flags, I_VAR)) {
2304 struct isl_vec *div;
2305 struct isl_vec *ineq;
2307 if (isl_tab_extend_cons(*context_tab, 3) < 0)
2309 div = get_row_split_div(tab, row);
2312 d = get_div(tab, context_tab, div);
2316 ineq = ineq_for_div((*context_tab)->bset, d);
2317 sol = no_sol_in_strict(sol, tab, ineq);
2318 isl_seq_neg(ineq->el, ineq->el, ineq->size);
2319 *context_tab = add_lexmin_ineq(*context_tab, ineq->el);
2320 *context_tab = check_samples(*context_tab, ineq->el, 0);
2324 tab = set_row_cst_to_div(tab, row, d);
2326 row = add_parametric_cut(tab, row, context_tab);
2331 sol = sol->add(sol, tab);
2340 /* Compute the lexicographic minimum of the set represented by the main
2341 * tableau "tab" within the context "sol->context_tab".
2343 * As a preprocessing step, we first transfer all the purely parametric
2344 * equalities from the main tableau to the context tableau, i.e.,
2345 * parameters that have been pivoted to a row.
2346 * These equalities are ignored by the main algorithm, because the
2347 * corresponding rows may not be marked as being non-negative.
2348 * In parts of the context where the added equality does not hold,
2349 * the main tableau is marked as being empty.
2351 static struct isl_sol *find_solutions_main(struct isl_sol *sol,
2352 struct isl_tab *tab)
2356 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2360 if (tab->row_var[row] < 0)
2362 if (tab->row_var[row] >= tab->n_param &&
2363 tab->row_var[row] < tab->n_var - tab->n_div)
2365 if (tab->row_var[row] < tab->n_param)
2366 p = tab->row_var[row];
2368 p = tab->row_var[row]
2369 + tab->n_param - (tab->n_var - tab->n_div);
2371 if (isl_tab_extend_cons(sol->context_tab, 2) < 0)
2374 eq = isl_vec_alloc(tab->mat->ctx, 1+tab->n_param+tab->n_div);
2375 get_row_parameter_line(tab, row, eq->el);
2376 isl_int_neg(eq->el[1 + p], tab->mat->row[row][0]);
2377 eq = isl_vec_normalize(eq);
2379 sol = no_sol_in_strict(sol, tab, eq);
2381 isl_seq_neg(eq->el, eq->el, eq->size);
2382 sol = no_sol_in_strict(sol, tab, eq);
2383 isl_seq_neg(eq->el, eq->el, eq->size);
2385 sol->context_tab = add_lexmin_eq(sol->context_tab, eq->el);
2386 context_valid_sample_or_feasible(sol, eq->el, 1);
2387 sol->context_tab = check_samples(sol->context_tab, eq->el, 1);
2391 isl_tab_mark_redundant(tab, row);
2393 if (!sol->context_tab)
2395 if (sol->context_tab->empty)
2398 row = tab->n_redundant - 1;
2401 return find_solutions(sol, tab);
2408 static struct isl_sol_map *sol_map_find_solutions(struct isl_sol_map *sol_map,
2409 struct isl_tab *tab)
2411 return (struct isl_sol_map *)find_solutions_main(&sol_map->sol, tab);
2414 /* Check if integer division "div" of "dom" also occurs in "bmap".
2415 * If so, return its position within the divs.
2416 * If not, return -1.
2418 static int find_context_div(struct isl_basic_map *bmap,
2419 struct isl_basic_set *dom, unsigned div)
2422 unsigned b_dim = isl_dim_total(bmap->dim);
2423 unsigned d_dim = isl_dim_total(dom->dim);
2425 if (isl_int_is_zero(dom->div[div][0]))
2427 if (isl_seq_first_non_zero(dom->div[div] + 2 + d_dim, dom->n_div) != -1)
2430 for (i = 0; i < bmap->n_div; ++i) {
2431 if (isl_int_is_zero(bmap->div[i][0]))
2433 if (isl_seq_first_non_zero(bmap->div[i] + 2 + d_dim,
2434 (b_dim - d_dim) + bmap->n_div) != -1)
2436 if (isl_seq_eq(bmap->div[i], dom->div[div], 2 + d_dim))
2442 /* The correspondence between the variables in the main tableau,
2443 * the context tableau, and the input map and domain is as follows.
2444 * The first n_param and the last n_div variables of the main tableau
2445 * form the variables of the context tableau.
2446 * In the basic map, these n_param variables correspond to the
2447 * parameters and the input dimensions. In the domain, they correspond
2448 * to the parameters and the set dimensions.
2449 * The n_div variables correspond to the integer divisions in the domain.
2450 * To ensure that everything lines up, we may need to copy some of the
2451 * integer divisions of the domain to the map. These have to be placed
2452 * in the same order as those in the context and they have to be placed
2453 * after any other integer divisions that the map may have.
2454 * This function performs the required reordering.
2456 static struct isl_basic_map *align_context_divs(struct isl_basic_map *bmap,
2457 struct isl_basic_set *dom)
2463 for (i = 0; i < dom->n_div; ++i)
2464 if (find_context_div(bmap, dom, i) != -1)
2466 other = bmap->n_div - common;
2467 if (dom->n_div - common > 0) {
2468 bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim),
2469 dom->n_div - common, 0, 0);
2473 for (i = 0; i < dom->n_div; ++i) {
2474 int pos = find_context_div(bmap, dom, i);
2476 pos = isl_basic_map_alloc_div(bmap);
2479 isl_int_set_si(bmap->div[pos][0], 0);
2481 if (pos != other + i)
2482 isl_basic_map_swap_div(bmap, pos, other + i);
2486 isl_basic_map_free(bmap);
2490 /* Compute the lexicographic minimum (or maximum if "max" is set)
2491 * of "bmap" over the domain "dom" and return the result as a map.
2492 * If "empty" is not NULL, then *empty is assigned a set that
2493 * contains those parts of the domain where there is no solution.
2494 * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL),
2495 * then we compute the rational optimum. Otherwise, we compute
2496 * the integral optimum.
2498 * We perform some preprocessing. As the PILP solver does not
2499 * handle implicit equalities very well, we first make sure all
2500 * the equalities are explicitly available.
2501 * We also make sure the divs in the domain are properly order,
2502 * because they will be added one by one in the given order
2503 * during the construction of the solution map.
2505 struct isl_map *isl_tab_basic_map_partial_lexopt(
2506 struct isl_basic_map *bmap, struct isl_basic_set *dom,
2507 struct isl_set **empty, int max)
2509 struct isl_tab *tab;
2510 struct isl_map *result = NULL;
2511 struct isl_sol_map *sol_map = NULL;
2518 isl_assert(bmap->ctx,
2519 isl_basic_map_compatible_domain(bmap, dom), goto error);
2521 bmap = isl_basic_map_detect_equalities(bmap);
2524 dom = isl_basic_set_order_divs(dom);
2525 bmap = align_context_divs(bmap, dom);
2527 sol_map = sol_map_init(bmap, dom, !!empty, max);
2531 if (isl_basic_set_fast_is_empty(sol_map->sol.context_tab->bset))
2533 else if (isl_basic_map_fast_is_empty(bmap))
2534 sol_map = add_empty(sol_map);
2536 tab = tab_for_lexmin(bmap,
2537 sol_map->sol.context_tab->bset, 1, max);
2538 tab = tab_detect_nonnegative_parameters(tab,
2539 sol_map->sol.context_tab);
2540 sol_map = sol_map_find_solutions(sol_map, tab);
2545 result = isl_map_copy(sol_map->map);
2547 *empty = isl_set_copy(sol_map->empty);
2548 sol_map_free(sol_map);
2549 isl_basic_map_free(bmap);
2552 sol_map_free(sol_map);
2553 isl_basic_map_free(bmap);
2557 struct isl_sol_for {
2559 int (*fn)(__isl_take isl_basic_set *dom,
2560 __isl_take isl_mat *map, void *user);
2565 static void sol_for_free(struct isl_sol_for *sol_for)
2567 isl_tab_free(sol_for->sol.context_tab);
2571 static void sol_for_free_wrap(struct isl_sol *sol)
2573 sol_for_free((struct isl_sol_for *)sol);
2576 /* Add the solution identified by the tableau and the context tableau.
2578 * See documentation of sol_map_add for more details.
2580 * Instead of constructing a basic map, this function calls a user
2581 * defined function with the current context as a basic set and
2582 * an affine matrix reprenting the relation between the input and output.
2583 * The number of rows in this matrix is equal to one plus the number
2584 * of output variables. The number of columns is equal to one plus
2585 * the total dimension of the context, i.e., the number of parameters,
2586 * input variables and divs. Since some of the columns in the matrix
2587 * may refer to the divs, the basic set is not simplified.
2588 * (Simplification may reorder or remove divs.)
2590 static struct isl_sol_for *sol_for_add(struct isl_sol_for *sol,
2591 struct isl_tab *tab)
2593 struct isl_tab *context_tab;
2594 struct isl_basic_set *bset;
2595 struct isl_mat *mat = NULL;
2607 context_tab = sol->sol.context_tab;
2609 n_out = tab->n_var - tab->n_param - tab->n_div;
2610 mat = isl_mat_alloc(tab->mat->ctx, 1 + n_out, 1 + tab->n_param + tab->n_div);
2614 isl_seq_clr(mat->row[0] + 1, mat->n_col - 1);
2615 isl_int_set_si(mat->row[0][0], 1);
2616 for (row = 0; row < n_out; ++row) {
2617 int i = tab->n_param + row;
2620 isl_seq_clr(mat->row[1 + row], mat->n_col);
2621 if (!tab->var[i].is_row)
2624 r = tab->var[i].index;
2627 isl_assert(mat->ctx, isl_int_eq(tab->mat->row[r][2],
2628 tab->mat->row[r][0]),
2630 isl_int_set(mat->row[1 + row][0], tab->mat->row[r][1]);
2631 for (j = 0; j < tab->n_param; ++j) {
2633 if (tab->var[j].is_row)
2635 col = tab->var[j].index;
2636 isl_int_set(mat->row[1 + row][1 + j],
2637 tab->mat->row[r][off + col]);
2639 for (j = 0; j < tab->n_div; ++j) {
2641 if (tab->var[tab->n_var - tab->n_div+j].is_row)
2643 col = tab->var[tab->n_var - tab->n_div+j].index;
2644 isl_int_set(mat->row[1 + row][1 + tab->n_param + j],
2645 tab->mat->row[r][off + col]);
2647 if (!isl_int_is_one(tab->mat->row[r][0]))
2648 isl_seq_scale_down(mat->row[1 + row], mat->row[1 + row],
2649 tab->mat->row[r][0], mat->n_col);
2651 isl_seq_neg(mat->row[1 + row], mat->row[1 + row],
2655 bset = isl_basic_set_dup(context_tab->bset);
2656 bset = isl_basic_set_finalize(bset);
2658 if (sol->fn(bset, isl_mat_copy(mat), sol->user) < 0)
2665 sol_free(&sol->sol);
2669 static struct isl_sol *sol_for_add_wrap(struct isl_sol *sol,
2670 struct isl_tab *tab)
2672 return (struct isl_sol *)sol_for_add((struct isl_sol_for *)sol, tab);
2675 static struct isl_sol_for *sol_for_init(struct isl_basic_map *bmap, int max,
2676 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_mat *map,
2680 struct isl_sol_for *sol_for = NULL;
2681 struct isl_dim *dom_dim;
2682 struct isl_basic_set *dom = NULL;
2683 struct isl_tab *context_tab;
2686 sol_for = isl_calloc_type(bset->ctx, struct isl_sol_for);
2690 dom_dim = isl_dim_domain(isl_dim_copy(bmap->dim));
2691 dom = isl_basic_set_universe(dom_dim);
2694 sol_for->user = user;
2696 sol_for->sol.add = &sol_for_add_wrap;
2697 sol_for->sol.free = &sol_for_free_wrap;
2699 context_tab = context_tab_for_lexmin(isl_basic_set_copy(dom));
2700 context_tab = restore_lexmin(context_tab);
2701 sol_for->sol.context_tab = context_tab;
2702 f = context_is_feasible(&sol_for->sol);
2706 isl_basic_set_free(dom);
2709 isl_basic_set_free(dom);
2710 sol_for_free(sol_for);
2714 static struct isl_sol_for *sol_for_find_solutions(struct isl_sol_for *sol_for,
2715 struct isl_tab *tab)
2717 return (struct isl_sol_for *)find_solutions_main(&sol_for->sol, tab);
2720 int isl_basic_map_foreach_lexopt(__isl_keep isl_basic_map *bmap, int max,
2721 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_mat *map,
2725 struct isl_sol_for *sol_for = NULL;
2727 bmap = isl_basic_map_copy(bmap);
2731 bmap = isl_basic_map_detect_equalities(bmap);
2732 sol_for = sol_for_init(bmap, max, fn, user);
2734 if (isl_basic_map_fast_is_empty(bmap))
2737 struct isl_tab *tab;
2738 tab = tab_for_lexmin(bmap,
2739 sol_for->sol.context_tab->bset, 1, max);
2740 tab = tab_detect_nonnegative_parameters(tab,
2741 sol_for->sol.context_tab);
2742 sol_for = sol_for_find_solutions(sol_for, tab);
2747 sol_for_free(sol_for);
2748 isl_basic_map_free(bmap);
2751 sol_for_free(sol_for);
2752 isl_basic_map_free(bmap);
2756 int isl_basic_map_foreach_lexmin(__isl_keep isl_basic_map *bmap,
2757 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_mat *map,
2761 return isl_basic_map_foreach_lexopt(bmap, 0, fn, user);
2764 int isl_basic_map_foreach_lexmax(__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, 1, fn, user);