/*
* Copyright 2010 INRIA Saclay
*
* Use of this software is governed by the MIT license
*
* Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
* Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
* 91893 Orsay, France
*/
#include <isl_map_private.h>
#include <isl/set.h>
#include <isl_space_private.h>
#include <isl_seq.h>
/*
* Let C be a cone and define
*
* C' := { y | forall x in C : y x >= 0 }
*
* C' contains the coefficients of all linear constraints
* that are valid for C.
* Furthermore, C'' = C.
*
* If C is defined as { x | A x >= 0 }
* then any element in C' must be a non-negative combination
* of the rows of A, i.e., y = t A with t >= 0. That is,
*
* C' = { y | exists t >= 0 : y = t A }
*
* If any of the rows in A actually represents an equality, then
* also negative combinations of this row are allowed and so the
* non-negativity constraint on the corresponding element of t
* can be dropped.
*
* A polyhedron P = { x | b + A x >= 0 } can be represented
* in homogeneous coordinates by the cone
* C = { [z,x] | b z + A x >= and z >= 0 }
* The valid linear constraints on C correspond to the valid affine
* constraints on P.
* This is essentially Farkas' lemma.
*
* Since
* [ 1 0 ]
* [ w y ] = [t_0 t] [ b A ]
*
* we have
*
* C' = { w, y | exists t_0, t >= 0 : y = t A and w = t_0 + t b }
* or
*
* C' = { w, y | exists t >= 0 : y = t A and w - t b >= 0 }
*
* In practice, we introduce an extra variable (w), shifting all
* other variables to the right, and an extra inequality
* (w - t b >= 0) corresponding to the positivity constraint on
* the homogeneous coordinate.
*
* When going back from coefficients to solutions, we immediately
* plug in 1 for z, which corresponds to shifting all variables
* to the left, with the leftmost ending up in the constant position.
*/
/* Add the given prefix to all named isl_dim_set dimensions in "dim".
*/
static __isl_give isl_space *isl_space_prefix(__isl_take isl_space *dim,
const char *prefix)
{
int i;
isl_ctx *ctx;
unsigned nvar;
size_t prefix_len = strlen(prefix);
if (!dim)
return NULL;
ctx = isl_space_get_ctx(dim);
nvar = isl_space_dim(dim, isl_dim_set);
for (i = 0; i < nvar; ++i) {
const char *name;
char *prefix_name;
name = isl_space_get_dim_name(dim, isl_dim_set, i);
if (!name)
continue;
prefix_name = isl_alloc_array(ctx, char,
prefix_len + strlen(name) + 1);
if (!prefix_name)
goto error;
memcpy(prefix_name, prefix, prefix_len);
strcpy(prefix_name + prefix_len, name);
dim = isl_space_set_dim_name(dim, isl_dim_set, i, prefix_name);
free(prefix_name);
}
return dim;
error:
isl_space_free(dim);
return NULL;
}
/* Given a dimension specification of the solutions space, construct
* a dimension specification for the space of coefficients.
*
* In particular transform
*
* [params] -> { S }
*
* to
*
* { coefficients[[cst, params] -> S] }
*
* and prefix each dimension name with "c_".
*/
static __isl_give isl_space *isl_space_coefficients(__isl_take isl_space *dim)
{
isl_space *dim_param;
unsigned nvar;
unsigned nparam;
nvar = isl_space_dim(dim, isl_dim_set);
nparam = isl_space_dim(dim, isl_dim_param);
dim_param = isl_space_copy(dim);
dim_param = isl_space_drop_dims(dim_param, isl_dim_set, 0, nvar);
dim_param = isl_space_move_dims(dim_param, isl_dim_set, 0,
isl_dim_param, 0, nparam);
dim_param = isl_space_prefix(dim_param, "c_");
dim_param = isl_space_insert_dims(dim_param, isl_dim_set, 0, 1);
dim_param = isl_space_set_dim_name(dim_param, isl_dim_set, 0, "c_cst");
dim = isl_space_drop_dims(dim, isl_dim_param, 0, nparam);
dim = isl_space_prefix(dim, "c_");
dim = isl_space_join(isl_space_from_domain(dim_param),
isl_space_from_range(dim));
dim = isl_space_wrap(dim);
dim = isl_space_set_tuple_name(dim, isl_dim_set, "coefficients");
return dim;
}
/* Drop the given prefix from all named dimensions of type "type" in "dim".
*/
static __isl_give isl_space *isl_space_unprefix(__isl_take isl_space *dim,
enum isl_dim_type type, const char *prefix)
{
int i;
unsigned n;
size_t prefix_len = strlen(prefix);
n = isl_space_dim(dim, type);
for (i = 0; i < n; ++i) {
const char *name;
name = isl_space_get_dim_name(dim, type, i);
if (!name)
continue;
if (strncmp(name, prefix, prefix_len))
continue;
dim = isl_space_set_dim_name(dim, type, i, name + prefix_len);
}
return dim;
}
/* Given a dimension specification of the space of coefficients, construct
* a dimension specification for the space of solutions.
*
* In particular transform
*
* { coefficients[[cst, params] -> S] }
*
* to
*
* [params] -> { S }
*
* and drop the "c_" prefix from the dimension names.
*/
static __isl_give isl_space *isl_space_solutions(__isl_take isl_space *dim)
{
unsigned nparam;
dim = isl_space_unwrap(dim);
dim = isl_space_drop_dims(dim, isl_dim_in, 0, 1);
dim = isl_space_unprefix(dim, isl_dim_in, "c_");
dim = isl_space_unprefix(dim, isl_dim_out, "c_");
nparam = isl_space_dim(dim, isl_dim_in);
dim = isl_space_move_dims(dim, isl_dim_param, 0, isl_dim_in, 0, nparam);
dim = isl_space_range(dim);
return dim;
}
/* Return the rational universe basic set in the given space.
*/
static __isl_give isl_basic_set *rational_universe(__isl_take isl_space *space)
{
isl_basic_set *bset;
bset = isl_basic_set_universe(space);
bset = isl_basic_set_set_rational(bset);
return bset;
}
/* Compute the dual of "bset" by applying Farkas' lemma.
* As explained above, we add an extra dimension to represent
* the coefficient of the constant term when going from solutions
* to coefficients (shift == 1) and we drop the extra dimension when going
* in the opposite direction (shift == -1). "dim" is the space in which
* the dual should be created.
*
* If "bset" is (obviously) empty, then the way this emptiness
* is represented by the constraints does not allow for the application
* of the standard farkas algorithm. We therefore handle this case
* specifically and return the universe basic set.
*/
static __isl_give isl_basic_set *farkas(__isl_take isl_space *space,
__isl_take isl_basic_set *bset, int shift)
{
int i, j, k;
isl_basic_set *dual = NULL;
unsigned total;
if (isl_basic_set_plain_is_empty(bset)) {
isl_basic_set_free(bset);
return rational_universe(space);
}
total = isl_basic_set_total_dim(bset);
dual = isl_basic_set_alloc_space(space, bset->n_eq + bset->n_ineq,
total, bset->n_ineq + (shift > 0));
dual = isl_basic_set_set_rational(dual);
for (i = 0; i < bset->n_eq + bset->n_ineq; ++i) {
k = isl_basic_set_alloc_div(dual);
if (k < 0)
goto error;
isl_int_set_si(dual->div[k][0], 0);
}
for (i = 0; i < total; ++i) {
k = isl_basic_set_alloc_equality(dual);
if (k < 0)
goto error;
isl_seq_clr(dual->eq[k], 1 + shift + total);
isl_int_set_si(dual->eq[k][1 + shift + i], -1);
for (j = 0; j < bset->n_eq; ++j)
isl_int_set(dual->eq[k][1 + shift + total + j],
bset->eq[j][1 + i]);
for (j = 0; j < bset->n_ineq; ++j)
isl_int_set(dual->eq[k][1 + shift + total + bset->n_eq + j],
bset->ineq[j][1 + i]);
}
for (i = 0; i < bset->n_ineq; ++i) {
k = isl_basic_set_alloc_inequality(dual);
if (k < 0)
goto error;
isl_seq_clr(dual->ineq[k],
1 + shift + total + bset->n_eq + bset->n_ineq);
isl_int_set_si(dual->ineq[k][1 + shift + total + bset->n_eq + i], 1);
}
if (shift > 0) {
k = isl_basic_set_alloc_inequality(dual);
if (k < 0)
goto error;
isl_seq_clr(dual->ineq[k], 2 + total);
isl_int_set_si(dual->ineq[k][1], 1);
for (j = 0; j < bset->n_eq; ++j)
isl_int_neg(dual->ineq[k][2 + total + j],
bset->eq[j][0]);
for (j = 0; j < bset->n_ineq; ++j)
isl_int_neg(dual->ineq[k][2 + total + bset->n_eq + j],
bset->ineq[j][0]);
}
dual = isl_basic_set_remove_divs(dual);
isl_basic_set_simplify(dual);
isl_basic_set_finalize(dual);
isl_basic_set_free(bset);
return dual;
error:
isl_basic_set_free(bset);
isl_basic_set_free(dual);
return NULL;
}
/* Construct a basic set containing the tuples of coefficients of all
* valid affine constraints on the given basic set.
*/
__isl_give isl_basic_set *isl_basic_set_coefficients(
__isl_take isl_basic_set *bset)
{
isl_space *dim;
if (!bset)
return NULL;
if (bset->n_div)
isl_die(bset->ctx, isl_error_invalid,
"input set not allowed to have local variables",
goto error);
dim = isl_basic_set_get_space(bset);
dim = isl_space_coefficients(dim);
return farkas(dim, bset, 1);
error:
isl_basic_set_free(bset);
return NULL;
}
/* Construct a basic set containing the elements that satisfy all
* affine constraints whose coefficient tuples are
* contained in the given basic set.
*/
__isl_give isl_basic_set *isl_basic_set_solutions(
__isl_take isl_basic_set *bset)
{
isl_space *dim;
if (!bset)
return NULL;
if (bset->n_div)
isl_die(bset->ctx, isl_error_invalid,
"input set not allowed to have local variables",
goto error);
dim = isl_basic_set_get_space(bset);
dim = isl_space_solutions(dim);
return farkas(dim, bset, -1);
error:
isl_basic_set_free(bset);
return NULL;
}
/* Construct a basic set containing the tuples of coefficients of all
* valid affine constraints on the given set.
*/
__isl_give isl_basic_set *isl_set_coefficients(__isl_take isl_set *set)
{
int i;
isl_basic_set *coeff;
if (!set)
return NULL;
if (set->n == 0) {
isl_space *space = isl_set_get_space(set);
space = isl_space_coefficients(space);
isl_set_free(set);
return rational_universe(space);
}
coeff = isl_basic_set_coefficients(isl_basic_set_copy(set->p[0]));
for (i = 1; i < set->n; ++i) {
isl_basic_set *bset, *coeff_i;
bset = isl_basic_set_copy(set->p[i]);
coeff_i = isl_basic_set_coefficients(bset);
coeff = isl_basic_set_intersect(coeff, coeff_i);
}
isl_set_free(set);
return coeff;
}
/* Construct a basic set containing the elements that satisfy all
* affine constraints whose coefficient tuples are
* contained in the given set.
*/
__isl_give isl_basic_set *isl_set_solutions(__isl_take isl_set *set)
{
int i;
isl_basic_set *sol;
if (!set)
return NULL;
if (set->n == 0) {
isl_space *space = isl_set_get_space(set);
space = isl_space_solutions(space);
isl_set_free(set);
return rational_universe(space);
}
sol = isl_basic_set_solutions(isl_basic_set_copy(set->p[0]));
for (i = 1; i < set->n; ++i) {
isl_basic_set *bset, *sol_i;
bset = isl_basic_set_copy(set->p[i]);
sol_i = isl_basic_set_solutions(bset);
sol = isl_basic_set_intersect(sol, sol_i);
}
isl_set_free(set);
return sol;
}