2 * Copyright 2008-2009 Katholieke Universiteit Leuven
4 * Use of this software is governed by the MIT license
6 * Written by Sven Verdoolaege, K.U.Leuven, Departement
7 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
10 #include <isl_ctx_private.h>
11 #include <isl_map_private.h>
13 #include "isl_sample.h"
15 #include "isl_equalities.h"
16 #include <isl_aff_private.h>
17 #include <isl_local_space_private.h>
18 #include <isl_mat_private.h>
19 #include <isl_val_private.h>
21 /* Given a basic set "bset", construct a basic set U such that for
22 * each element x in U, the whole unit box positioned at x is inside
23 * the given basic set.
24 * Note that U may not contain all points that satisfy this property.
26 * We simply add the sum of all negative coefficients to the constant
27 * term. This ensures that if x satisfies the resulting constraints,
28 * then x plus any sum of unit vectors satisfies the original constraints.
30 static struct isl_basic_set *unit_box_base_points(struct isl_basic_set *bset)
33 struct isl_basic_set *unit_box = NULL;
39 if (bset->n_eq != 0) {
40 unit_box = isl_basic_set_empty_like(bset);
41 isl_basic_set_free(bset);
45 total = isl_basic_set_total_dim(bset);
46 unit_box = isl_basic_set_alloc_space(isl_basic_set_get_space(bset),
49 for (i = 0; i < bset->n_ineq; ++i) {
50 k = isl_basic_set_alloc_inequality(unit_box);
53 isl_seq_cpy(unit_box->ineq[k], bset->ineq[i], 1 + total);
54 for (j = 0; j < total; ++j) {
55 if (isl_int_is_nonneg(unit_box->ineq[k][1 + j]))
57 isl_int_add(unit_box->ineq[k][0],
58 unit_box->ineq[k][0], unit_box->ineq[k][1 + j]);
62 isl_basic_set_free(bset);
65 isl_basic_set_free(bset);
66 isl_basic_set_free(unit_box);
70 /* Find an integer point in "bset", preferably one that is
71 * close to minimizing "f".
73 * We first check if we can easily put unit boxes inside bset.
74 * If so, we take the best base point of any of the unit boxes we can find
75 * and round it up to the nearest integer.
76 * If not, we simply pick any integer point in "bset".
78 static struct isl_vec *initial_solution(struct isl_basic_set *bset, isl_int *f)
80 enum isl_lp_result res;
81 struct isl_basic_set *unit_box;
84 unit_box = unit_box_base_points(isl_basic_set_copy(bset));
86 res = isl_basic_set_solve_lp(unit_box, 0, f, bset->ctx->one,
88 if (res == isl_lp_ok) {
89 isl_basic_set_free(unit_box);
90 return isl_vec_ceil(sol);
93 isl_basic_set_free(unit_box);
95 return isl_basic_set_sample_vec(isl_basic_set_copy(bset));
98 /* Restrict "bset" to those points with values for f in the interval [l, u].
100 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
101 isl_int *f, isl_int l, isl_int u)
106 total = isl_basic_set_total_dim(bset);
107 bset = isl_basic_set_extend_constraints(bset, 0, 2);
109 k = isl_basic_set_alloc_inequality(bset);
112 isl_seq_cpy(bset->ineq[k], f, 1 + total);
113 isl_int_sub(bset->ineq[k][0], bset->ineq[k][0], l);
115 k = isl_basic_set_alloc_inequality(bset);
118 isl_seq_neg(bset->ineq[k], f, 1 + total);
119 isl_int_add(bset->ineq[k][0], bset->ineq[k][0], u);
123 isl_basic_set_free(bset);
127 /* Find an integer point in "bset" that minimizes f (in any) such that
128 * the value of f lies inside the interval [l, u].
129 * Return this integer point if it can be found.
130 * Otherwise, return sol.
132 * We perform a number of steps until l > u.
133 * In each step, we look for an integer point with value in either
134 * the whole interval [l, u] or half of the interval [l, l+floor(u-l-1/2)].
135 * The choice depends on whether we have found an integer point in the
136 * previous step. If so, we look for the next point in half of the remaining
138 * If we find a point, the current solution is updated and u is set
139 * to its value minus 1.
140 * If no point can be found, we update l to the upper bound of the interval
141 * we checked (u or l+floor(u-l-1/2)) plus 1.
143 static struct isl_vec *solve_ilp_search(struct isl_basic_set *bset,
144 isl_int *f, isl_int *opt, struct isl_vec *sol, isl_int l, isl_int u)
151 while (isl_int_le(l, u)) {
152 struct isl_basic_set *slice;
153 struct isl_vec *sample;
158 isl_int_sub(tmp, u, l);
159 isl_int_fdiv_q_ui(tmp, tmp, 2);
160 isl_int_add(tmp, tmp, l);
162 slice = add_bounds(isl_basic_set_copy(bset), f, l, tmp);
163 sample = isl_basic_set_sample_vec(slice);
169 if (sample->size > 0) {
172 isl_seq_inner_product(f, sol->el, sol->size, opt);
173 isl_int_sub_ui(u, *opt, 1);
176 isl_vec_free(sample);
179 isl_int_add_ui(l, tmp, 1);
189 /* Find an integer point in "bset" that minimizes f (if any).
190 * If sol_p is not NULL then the integer point is returned in *sol_p.
191 * The optimal value of f is returned in *opt.
193 * The algorithm maintains a currently best solution and an interval [l, u]
194 * of values of f for which integer solutions could potentially still be found.
195 * The initial value of the best solution so far is any solution.
196 * The initial value of l is minimal value of f over the rationals
197 * (rounded up to the nearest integer).
198 * The initial value of u is the value of f at the initial solution minus 1.
200 * We then call solve_ilp_search to perform a binary search on the interval.
202 static enum isl_lp_result solve_ilp(struct isl_basic_set *bset,
203 isl_int *f, isl_int *opt,
204 struct isl_vec **sol_p)
206 enum isl_lp_result res;
210 res = isl_basic_set_solve_lp(bset, 0, f, bset->ctx->one,
212 if (res == isl_lp_ok && isl_int_is_one(sol->el[0])) {
220 if (res == isl_lp_error || res == isl_lp_empty)
223 sol = initial_solution(bset, f);
226 if (sol->size == 0) {
230 if (res == isl_lp_unbounded) {
232 return isl_lp_unbounded;
238 isl_int_set(l, *opt);
240 isl_seq_inner_product(f, sol->el, sol->size, opt);
241 isl_int_sub_ui(u, *opt, 1);
243 sol = solve_ilp_search(bset, f, opt, sol, l, u);
258 static enum isl_lp_result solve_ilp_with_eq(struct isl_basic_set *bset, int max,
259 isl_int *f, isl_int *opt,
260 struct isl_vec **sol_p)
263 enum isl_lp_result res;
264 struct isl_mat *T = NULL;
267 bset = isl_basic_set_copy(bset);
268 dim = isl_basic_set_total_dim(bset);
269 v = isl_vec_alloc(bset->ctx, 1 + dim);
272 isl_seq_cpy(v->el, f, 1 + dim);
273 bset = isl_basic_set_remove_equalities(bset, &T, NULL);
274 v = isl_vec_mat_product(v, isl_mat_copy(T));
277 res = isl_basic_set_solve_ilp(bset, max, v->el, opt, sol_p);
279 if (res == isl_lp_ok && sol_p) {
280 *sol_p = isl_mat_vec_product(T, *sol_p);
285 isl_basic_set_free(bset);
289 isl_basic_set_free(bset);
293 /* Find an integer point in "bset" that minimizes (or maximizes if max is set)
295 * If sol_p is not NULL then the integer point is returned in *sol_p.
296 * The optimal value of f is returned in *opt.
298 * If there is any equality among the points in "bset", then we first
299 * project it out. Otherwise, we continue with solve_ilp above.
301 enum isl_lp_result isl_basic_set_solve_ilp(struct isl_basic_set *bset, int max,
302 isl_int *f, isl_int *opt,
303 struct isl_vec **sol_p)
306 enum isl_lp_result res;
313 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
315 if (isl_basic_set_plain_is_empty(bset))
319 return solve_ilp_with_eq(bset, max, f, opt, sol_p);
321 dim = isl_basic_set_total_dim(bset);
324 isl_seq_neg(f, f, 1 + dim);
326 res = solve_ilp(bset, f, opt, sol_p);
329 isl_seq_neg(f, f, 1 + dim);
330 isl_int_neg(*opt, *opt);
335 isl_basic_set_free(bset);
339 static enum isl_lp_result basic_set_opt(__isl_keep isl_basic_set *bset, int max,
340 __isl_keep isl_aff *obj, isl_int *opt)
342 enum isl_lp_result res;
346 bset = isl_basic_set_copy(bset);
347 bset = isl_basic_set_underlying_set(bset);
348 res = isl_basic_set_solve_ilp(bset, max, obj->v->el + 1, opt, NULL);
349 isl_basic_set_free(bset);
353 static __isl_give isl_mat *extract_divs(__isl_keep isl_basic_set *bset)
356 isl_ctx *ctx = isl_basic_set_get_ctx(bset);
359 div = isl_mat_alloc(ctx, bset->n_div,
360 1 + 1 + isl_basic_set_total_dim(bset));
364 for (i = 0; i < bset->n_div; ++i)
365 isl_seq_cpy(div->row[i], bset->div[i], div->n_col);
370 enum isl_lp_result isl_basic_set_opt(__isl_keep isl_basic_set *bset, int max,
371 __isl_keep isl_aff *obj, isl_int *opt)
376 isl_mat *bset_div = NULL;
378 enum isl_lp_result res;
383 ctx = isl_aff_get_ctx(obj);
384 if (!isl_space_is_equal(bset->dim, obj->ls->dim))
385 isl_die(ctx, isl_error_invalid,
386 "spaces don't match", return isl_lp_error);
387 if (!isl_int_is_one(obj->v->el[0]))
388 isl_die(ctx, isl_error_unsupported,
389 "expecting integer affine expression",
390 return isl_lp_error);
392 if (bset->n_div == 0 && obj->ls->div->n_row == 0)
393 return basic_set_opt(bset, max, obj, opt);
395 bset = isl_basic_set_copy(bset);
396 obj = isl_aff_copy(obj);
398 bset_div = extract_divs(bset);
399 exp1 = isl_alloc_array(ctx, int, bset_div->n_row);
400 exp2 = isl_alloc_array(ctx, int, obj->ls->div->n_row);
401 if (!bset_div || !exp1 || !exp2)
404 div = isl_merge_divs(bset_div, obj->ls->div, exp1, exp2);
406 bset = isl_basic_set_expand_divs(bset, isl_mat_copy(div), exp1);
407 obj = isl_aff_expand_divs(obj, isl_mat_copy(div), exp2);
409 res = basic_set_opt(bset, max, obj, opt);
411 isl_mat_free(bset_div);
415 isl_basic_set_free(bset);
421 isl_mat_free(bset_div);
424 isl_basic_set_free(bset);
429 /* Compute the minimum (maximum if max is set) of the integer affine
430 * expression obj over the points in set and put the result in *opt.
432 * The parameters are assumed to have been aligned.
434 static enum isl_lp_result isl_set_opt_aligned(__isl_keep isl_set *set, int max,
435 __isl_keep isl_aff *obj, isl_int *opt)
438 enum isl_lp_result res;
447 res = isl_basic_set_opt(set->p[0], max, obj, opt);
448 if (res == isl_lp_error || res == isl_lp_unbounded)
452 if (res == isl_lp_ok)
456 for (i = 1; i < set->n; ++i) {
457 res = isl_basic_set_opt(set->p[i], max, obj, &opt_i);
458 if (res == isl_lp_error || res == isl_lp_unbounded) {
459 isl_int_clear(opt_i);
462 if (res == isl_lp_ok)
464 if (isl_int_gt(opt_i, *opt))
465 isl_int_set(*opt, opt_i);
467 isl_int_clear(opt_i);
469 return empty ? isl_lp_empty : isl_lp_ok;
472 /* Compute the minimum (maximum if max is set) of the integer affine
473 * expression obj over the points in set and put the result in *opt.
475 enum isl_lp_result isl_set_opt(__isl_keep isl_set *set, int max,
476 __isl_keep isl_aff *obj, isl_int *opt)
478 enum isl_lp_result res;
483 if (isl_space_match(set->dim, isl_dim_param,
484 obj->ls->dim, isl_dim_param))
485 return isl_set_opt_aligned(set, max, obj, opt);
487 set = isl_set_copy(set);
488 obj = isl_aff_copy(obj);
489 set = isl_set_align_params(set, isl_aff_get_domain_space(obj));
490 obj = isl_aff_align_params(obj, isl_set_get_space(set));
492 res = isl_set_opt_aligned(set, max, obj, opt);
500 enum isl_lp_result isl_basic_set_max(__isl_keep isl_basic_set *bset,
501 __isl_keep isl_aff *obj, isl_int *opt)
503 return isl_basic_set_opt(bset, 1, obj, opt);
506 enum isl_lp_result isl_set_max(__isl_keep isl_set *set,
507 __isl_keep isl_aff *obj, isl_int *opt)
509 return isl_set_opt(set, 1, obj, opt);
512 enum isl_lp_result isl_set_min(__isl_keep isl_set *set,
513 __isl_keep isl_aff *obj, isl_int *opt)
515 return isl_set_opt(set, 0, obj, opt);
518 /* Convert the result of a function that returns an isl_lp_result
519 * to an isl_val. The numerator of "v" is set to the optimal value
520 * if lp_res is isl_lp_ok. "max" is set if a maximum was computed.
522 * Return "v" with denominator set to 1 if lp_res is isl_lp_ok.
523 * Return NULL on error.
524 * Return a NaN if lp_res is isl_lp_empty.
525 * Return infinity or negative infinity if lp_res is isl_lp_unbounded,
526 * depending on "max".
528 static __isl_give isl_val *convert_lp_result(enum isl_lp_result lp_res,
529 __isl_take isl_val *v, int max)
533 if (lp_res == isl_lp_ok) {
534 isl_int_set_si(v->d, 1);
535 return isl_val_normalize(v);
537 ctx = isl_val_get_ctx(v);
539 if (lp_res == isl_lp_error)
541 if (lp_res == isl_lp_empty)
542 return isl_val_nan(ctx);
544 return isl_val_infty(ctx);
546 return isl_val_neginfty(ctx);
549 /* Return the minimum (maximum if max is set) of the integer affine
550 * expression "obj" over the points in "bset".
552 * Return infinity or negative infinity if the optimal value is unbounded and
553 * NaN if "bset" is empty.
555 * Call isl_basic_set_opt and translate the results.
557 __isl_give isl_val *isl_basic_set_opt_val(__isl_keep isl_basic_set *bset,
558 int max, __isl_keep isl_aff *obj)
562 enum isl_lp_result lp_res;
567 ctx = isl_aff_get_ctx(obj);
568 res = isl_val_alloc(ctx);
571 lp_res = isl_basic_set_opt(bset, max, obj, &res->n);
572 return convert_lp_result(lp_res, res, max);
575 /* Return the maximum of the integer affine
576 * expression "obj" over the points in "bset".
578 * Return infinity or negative infinity if the optimal value is unbounded and
579 * NaN if "bset" is empty.
581 __isl_give isl_val *isl_basic_set_max_val(__isl_keep isl_basic_set *bset,
582 __isl_keep isl_aff *obj)
584 return isl_basic_set_opt_val(bset, 1, obj);