/**
* cloog_constraint_equal_type function :
* This function returns the type of the equality in the constraint (line) of
* (constraints) for the element (level). An equality is 'constant' iff all
* other factors are null except the constant one. It is a 'pure item' iff
* it is equal or opposite to a single variable or parameter.
* Otherwise it is an 'affine expression'.
* For instance:
* i = -13 is constant, i = j, j = -M are pure items,
* j = 2*M, i = j+1, 2*j = M are affine expressions.
*
* - constraints is the matrix of constraints,
* - level is the column number in equal of the element which is 'equal to',
*/
static int cloog_constraint_equal_type(CloogConstraint *cc, int level)
{
int i;
isl_int c;
int type = EQTYPE_NONE;
struct isl_constraint *constraint = cloog_constraint_to_isl(cc);
isl_int_init(c);
isl_constraint_get_constant(constraint, &c);
if (!isl_int_is_zero(c))
type = EQTYPE_CONSTANT;
isl_constraint_get_coefficient(constraint, isl_dim_set, level - 1, &c);
if (!isl_int_is_one(c) && !isl_int_is_negone(c))
type = EQTYPE_EXAFFINE;
for (i = 0; i < isl_constraint_dim(constraint, isl_dim_param); ++i) {
isl_constraint_get_coefficient(constraint, isl_dim_param, i, &c);
if (isl_int_is_zero(c))
continue;
if ((!isl_int_is_one(c) && !isl_int_is_negone(c)) ||
type != EQTYPE_NONE) {
type = EQTYPE_EXAFFINE;
break;
}
type = EQTYPE_PUREITEM;
}
for (i = 0; i < isl_constraint_dim(constraint, isl_dim_set); ++i) {
if (i == level - 1)
continue;
isl_constraint_get_coefficient(constraint, isl_dim_set, i, &c);
if (isl_int_is_zero(c))
continue;
if ((!isl_int_is_one(c) && !isl_int_is_negone(c)) ||
type != EQTYPE_NONE) {
type = EQTYPE_EXAFFINE;
break;
}
type = EQTYPE_PUREITEM;
}
for (i = 0; i < isl_constraint_dim(constraint, isl_dim_div); ++i) {
isl_constraint_get_coefficient(constraint, isl_dim_div, i, &c);
if (isl_int_is_zero(c))
continue;
if ((!isl_int_is_one(c) && !isl_int_is_negone(c)) ||
type != EQTYPE_NONE) {
type = EQTYPE_EXAFFINE;
break;
}
type = EQTYPE_PUREITEM;
}
isl_int_clear(c);
if (type == EQTYPE_NONE)
type = EQTYPE_CONSTANT;
return type;
}
/// Add an isl constraint to an ScopLib matrix.
///
/// @param user The matrix
/// @param c The constraint
int ScopLib::accessToMatrix_constraint(isl_constraint *c, void *user) {
scoplib_matrix_p m = (scoplib_matrix_p) user;
int nb_params = isl_constraint_dim(c, isl_dim_param);
int nb_in = isl_constraint_dim(c, isl_dim_in);
int nb_div = isl_constraint_dim(c, isl_dim_div);
assert(!nb_div && "Existentially quantified variables not yet supported");
scoplib_vector_p vec =
scoplib_vector_malloc(nb_params + nb_in + 2);
isl_int v;
isl_int_init(v);
// The access dimension has to be one.
isl_constraint_get_coefficient(c, isl_dim_out, 0, &v);
assert((isl_int_is_one(v) || isl_int_is_negone(v))
&& "Access relations not supported in scoplib");
bool inverse = isl_int_is_one(v);
// Assign variables
for (int i = 0; i < nb_in; ++i) {
isl_constraint_get_coefficient(c, isl_dim_in, i, &v);
if (inverse) isl_int_neg(v,v);
isl_int_set(vec->p[i + 1], v);
}
// Assign parameters
for (int i = 0; i < nb_params; ++i) {
isl_constraint_get_coefficient(c, isl_dim_param, i, &v);
if (inverse) isl_int_neg(v,v);
isl_int_set(vec->p[nb_in + i + 1], v);
}
// Assign constant
isl_constraint_get_constant(c, &v);
if (inverse) isl_int_neg(v,v);
isl_int_set(vec->p[nb_in + nb_params + 1], v);
scoplib_matrix_insert_vector(m, vec, m->NbRows);
isl_constraint_free(c);
isl_int_clear(v);
return 0;
}
/* Check if dimension dim belongs to a residue class
* i_dim \equiv r mod m
* with m != 1 and if so return m in *modulo and r in *residue.
* As a special case, when i_dim has a fixed value v, then
* *modulo is set to 0 and *residue to v.
*
* If i_dim does not belong to such a residue class, then *modulo
* is set to 1 and *residue is set to 0.
*/
int isl_set_dim_residue_class(struct isl_set *set,
int pos, isl_int *modulo, isl_int *residue)
{
isl_int m;
isl_int r;
int i;
if (!set || !modulo || !residue)
return -1;
if (set->n == 0) {
isl_int_set_si(*modulo, 0);
isl_int_set_si(*residue, 0);
return 0;
}
if (isl_basic_set_dim_residue_class(set->p[0], pos, modulo, residue)<0)
return -1;
if (set->n == 1)
return 0;
if (isl_int_is_one(*modulo))
return 0;
isl_int_init(m);
isl_int_init(r);
for (i = 1; i < set->n; ++i) {
if (isl_basic_set_dim_residue_class(set->p[i], pos, &m, &r) < 0)
goto error;
isl_int_gcd(*modulo, *modulo, m);
isl_int_sub(m, *residue, r);
isl_int_gcd(*modulo, *modulo, m);
if (!isl_int_is_zero(*modulo))
isl_int_fdiv_r(*residue, *residue, *modulo);
if (isl_int_is_one(*modulo))
break;
}
isl_int_clear(m);
isl_int_clear(r);
return 0;
error:
isl_int_clear(m);
isl_int_clear(r);
return -1;
}
static struct isl_vec *interval_sample(struct isl_basic_set *bset)
{
int i;
isl_int t;
struct isl_vec *sample;
bset = isl_basic_set_simplify(bset);
if (!bset)
return NULL;
if (isl_basic_set_plain_is_empty(bset))
return empty_sample(bset);
if (bset->n_eq == 0 && bset->n_ineq == 0)
return zero_sample(bset);
sample = isl_vec_alloc(bset->ctx, 2);
if (!sample)
goto error;
if (!bset)
return NULL;
isl_int_set_si(sample->block.data[0], 1);
if (bset->n_eq > 0) {
isl_assert(bset->ctx, bset->n_eq == 1, goto error);
isl_assert(bset->ctx, bset->n_ineq == 0, goto error);
if (isl_int_is_one(bset->eq[0][1]))
isl_int_neg(sample->el[1], bset->eq[0][0]);
else {
isl_assert(bset->ctx, isl_int_is_negone(bset->eq[0][1]),
goto error);
isl_int_set(sample->el[1], bset->eq[0][0]);
}
isl_basic_set_free(bset);
return sample;
}
void isl_point_print(__isl_keep isl_point *pnt, FILE *out)
{
int i;
unsigned nparam;
unsigned dim;
if (!pnt)
return;
if (isl_point_is_void(pnt)) {
fprintf(out, "void\n");
return;
}
nparam = isl_dim_size(pnt->dim, isl_dim_param);
dim = isl_dim_size(pnt->dim, isl_dim_set);
if (nparam > 0) {
fprintf(out, "[");
for (i = 0; i < nparam; ++i) {
const char *name;
if (i)
fprintf(out, ", ");
name = isl_dim_get_name(pnt->dim, isl_dim_param, i);
if (name)
fprintf(out, "%s = ", name);
isl_int_print(out, pnt->vec->el[1 + i], 0);
if (!isl_int_is_one(pnt->vec->el[0])) {
fprintf(out, "/");
isl_int_print(out, pnt->vec->el[0], 0);
}
}
fprintf(out, "] -> ");
}
fprintf(out, "[");
for (i = 0; i < dim; ++i) {
if (i)
fprintf(out, ", ");
isl_int_print(out, pnt->vec->el[1 + nparam + i], 0);
if (!isl_int_is_one(pnt->vec->el[0])) {
fprintf(out, "/");
isl_int_print(out, pnt->vec->el[0], 0);
}
}
fprintf(out, "]\n");
}
/* Compute a common lattice of solutions to the linear modulo
* constraints specified by B and d.
* See also the documentation of isl_mat_parameter_compression.
* We put the matrix
*
* A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
*
* on a common denominator. This denominator D is the lcm of modulos d.
* Since L_i = U_i^{-1} diag(d_i, 1, ... 1), we have
* L_i^{-T} = U_i^T diag(d_i, 1, ... 1)^{-T} = U_i^T diag(1/d_i, 1, ..., 1).
* Putting this on the common denominator, we have
* D * L_i^{-T} = U_i^T diag(D/d_i, D, ..., D).
*/
static struct isl_mat *parameter_compression_multi(
struct isl_mat *B, struct isl_vec *d)
{
int i, j, k;
isl_int D;
struct isl_mat *A = NULL, *U = NULL;
struct isl_mat *T;
unsigned size;
isl_int_init(D);
isl_vec_lcm(d, &D);
size = B->n_col - 1;
A = isl_mat_alloc(B->ctx, size, B->n_row * size);
U = isl_mat_alloc(B->ctx, size, size);
if (!U || !A)
goto error;
for (i = 0; i < B->n_row; ++i) {
isl_seq_cpy(U->row[0], B->row[i] + 1, size);
U = isl_mat_unimodular_complete(U, 1);
if (!U)
goto error;
isl_int_divexact(D, D, d->block.data[i]);
for (k = 0; k < U->n_col; ++k)
isl_int_mul(A->row[k][i*size+0], D, U->row[0][k]);
isl_int_mul(D, D, d->block.data[i]);
for (j = 1; j < U->n_row; ++j)
for (k = 0; k < U->n_col; ++k)
isl_int_mul(A->row[k][i*size+j],
D, U->row[j][k]);
}
A = isl_mat_left_hermite(A, 0, NULL, NULL);
T = isl_mat_sub_alloc(A, 0, A->n_row, 0, A->n_row);
T = isl_mat_lin_to_aff(T);
if (!T)
goto error;
isl_int_set(T->row[0][0], D);
T = isl_mat_right_inverse(T);
if (!T)
goto error;
isl_assert(T->ctx, isl_int_is_one(T->row[0][0]), goto error);
T = isl_mat_transpose(T);
isl_mat_free(A);
isl_mat_free(U);
isl_int_clear(D);
return T;
error:
isl_mat_free(A);
isl_mat_free(U);
isl_int_clear(D);
return NULL;
}
static int bound_is_integer(__isl_take isl_constraint *bound, unsigned pos)
{
isl_int c;
int is_int;
if (!bound)
return 1;
isl_int_init(c);
isl_constraint_get_coefficient(bound, isl_dim_set, pos, &c);
is_int = isl_int_is_one(c) || isl_int_is_negone(c);
isl_int_clear(c);
return is_int;
}
/* Add stride constraints to "bset" based on the inverse mapping
* that was plugged in. In particular, if morph maps x' to x,
* the the constraints of the original input
*
* A x' + b >= 0
*
* have been rewritten to
*
* A inv x + b >= 0
*
* However, this substitution may loose information on the integrality of x',
* so we need to impose that
*
* inv x
*
* is integral. If inv = B/d, this means that we need to impose that
*
* B x = 0 mod d
*
* or
*
* exists alpha in Z^m: B x = d alpha
*
*/
static __isl_give isl_basic_set *add_strides(__isl_take isl_basic_set *bset,
__isl_keep isl_morph *morph)
{
int i, div, k;
isl_int gcd;
if (isl_int_is_one(morph->inv->row[0][0]))
return bset;
isl_int_init(gcd);
for (i = 0; 1 + i < morph->inv->n_row; ++i) {
isl_seq_gcd(morph->inv->row[1 + i], morph->inv->n_col, &gcd);
if (isl_int_is_divisible_by(gcd, morph->inv->row[0][0]))
continue;
div = isl_basic_set_alloc_div(bset);
if (div < 0)
goto error;
k = isl_basic_set_alloc_equality(bset);
if (k < 0)
goto error;
isl_seq_cpy(bset->eq[k], morph->inv->row[1 + i],
morph->inv->n_col);
isl_seq_clr(bset->eq[k] + morph->inv->n_col, bset->n_div);
isl_int_set(bset->eq[k][morph->inv->n_col + div],
morph->inv->row[0][0]);
}
isl_int_clear(gcd);
return bset;
error:
isl_int_clear(gcd);
isl_basic_set_free(bset);
return NULL;
}
/* Apply the morphism to the basic set.
* We basically just compute the preimage of "bset" under the inverse mapping
* in morph, add in stride constraints and intersect with the range
* of the morphism.
*/
__isl_give isl_basic_set *isl_morph_basic_set(__isl_take isl_morph *morph,
__isl_take isl_basic_set *bset)
{
isl_basic_set *res = NULL;
isl_mat *mat = NULL;
int i, k;
int max_stride;
if (!morph || !bset)
goto error;
isl_assert(bset->ctx, isl_space_is_equal(bset->dim, morph->dom->dim),
goto error);
max_stride = morph->inv->n_row - 1;
if (isl_int_is_one(morph->inv->row[0][0]))
max_stride = 0;
res = isl_basic_set_alloc_space(isl_space_copy(morph->ran->dim),
bset->n_div + max_stride, bset->n_eq + max_stride, bset->n_ineq);
for (i = 0; i < bset->n_div; ++i)
if (isl_basic_set_alloc_div(res) < 0)
goto error;
mat = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq,
0, morph->inv->n_row);
mat = isl_mat_product(mat, isl_mat_copy(morph->inv));
if (!mat)
goto error;
for (i = 0; i < bset->n_eq; ++i) {
k = isl_basic_set_alloc_equality(res);
if (k < 0)
goto error;
isl_seq_cpy(res->eq[k], mat->row[i], mat->n_col);
isl_seq_scale(res->eq[k] + mat->n_col, bset->eq[i] + mat->n_col,
morph->inv->row[0][0], bset->n_div);
}
isl_mat_free(mat);
mat = isl_mat_sub_alloc6(bset->ctx, bset->ineq, 0, bset->n_ineq,
0, morph->inv->n_row);
mat = isl_mat_product(mat, isl_mat_copy(morph->inv));
if (!mat)
goto error;
for (i = 0; i < bset->n_ineq; ++i) {
k = isl_basic_set_alloc_inequality(res);
if (k < 0)
goto error;
isl_seq_cpy(res->ineq[k], mat->row[i], mat->n_col);
isl_seq_scale(res->ineq[k] + mat->n_col,
bset->ineq[i] + mat->n_col,
morph->inv->row[0][0], bset->n_div);
}
isl_mat_free(mat);
mat = isl_mat_sub_alloc6(bset->ctx, bset->div, 0, bset->n_div,
1, morph->inv->n_row);
mat = isl_mat_product(mat, isl_mat_copy(morph->inv));
if (!mat)
goto error;
for (i = 0; i < bset->n_div; ++i) {
isl_int_mul(res->div[i][0],
morph->inv->row[0][0], bset->div[i][0]);
isl_seq_cpy(res->div[i] + 1, mat->row[i], mat->n_col);
isl_seq_scale(res->div[i] + 1 + mat->n_col,
bset->div[i] + 1 + mat->n_col,
morph->inv->row[0][0], bset->n_div);
}
isl_mat_free(mat);
res = add_strides(res, morph);
if (isl_basic_set_is_rational(bset))
res = isl_basic_set_set_rational(res);
res = isl_basic_set_simplify(res);
res = isl_basic_set_finalize(res);
res = isl_basic_set_intersect(res, isl_basic_set_copy(morph->ran));
isl_morph_free(morph);
isl_basic_set_free(bset);
return res;
error:
isl_mat_free(mat);
isl_morph_free(morph);
isl_basic_set_free(bset);
isl_basic_set_free(res);
return NULL;
}
/* Given a basic set, exploit the equalties in the a basic set to construct
* a morphishm that maps the basic set to a lower-dimensional space.
* Specifically, the morphism reduces the number of dimensions of type "type".
*
* This function is a slight generalization of isl_mat_variable_compression
* in that it allows the input to be parametric and that it allows for the
* compression of either parameters or set variables.
*
* We first select the equalities of interest, that is those that involve
* variables of type "type" and no later variables.
* Denote those equalities as
*
* -C(p) + M x = 0
*
* where C(p) depends on the parameters if type == isl_dim_set and
* is a constant if type == isl_dim_param.
*
* First compute the (left) Hermite normal form of M,
*
* M [U1 U2] = M U = H = [H1 0]
* or
* M = H Q = [H1 0] [Q1]
* [Q2]
*
* with U, Q unimodular, Q = U^{-1} (and H lower triangular).
* Define the transformed variables as
*
* x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
* [ x2' ] [Q2]
*
* The equalities then become
*
* -C(p) + H1 x1' = 0 or x1' = H1^{-1} C(p) = C'(p)
*
* If the denominator of the constant term does not divide the
* the common denominator of the parametric terms, then every
* integer point is mapped to a non-integer point and then the original set has no
* integer solutions (since the x' are a unimodular transformation
* of the x). In this case, an empty morphism is returned.
* Otherwise, the transformation is given by
*
* x = U1 H1^{-1} C(p) + U2 x2'
*
* The inverse transformation is simply
*
* x2' = Q2 x
*
* Both matrices are extended to map the full original space to the full
* compressed space.
*/
__isl_give isl_morph *isl_basic_set_variable_compression(
__isl_keep isl_basic_set *bset, enum isl_dim_type type)
{
unsigned otype;
unsigned ntype;
unsigned orest;
unsigned nrest;
int f_eq, n_eq;
isl_space *dim;
isl_mat *H, *U, *Q, *C = NULL, *H1, *U1, *U2;
isl_basic_set *dom, *ran;
if (!bset)
return NULL;
if (isl_basic_set_plain_is_empty(bset))
return isl_morph_empty(bset);
isl_assert(bset->ctx, bset->n_div == 0, return NULL);
otype = 1 + isl_space_offset(bset->dim, type);
ntype = isl_basic_set_dim(bset, type);
orest = otype + ntype;
nrest = isl_basic_set_total_dim(bset) - (orest - 1);
for (f_eq = 0; f_eq < bset->n_eq; ++f_eq)
if (isl_seq_first_non_zero(bset->eq[f_eq] + orest, nrest) == -1)
break;
for (n_eq = 0; f_eq + n_eq < bset->n_eq; ++n_eq)
if (isl_seq_first_non_zero(bset->eq[f_eq + n_eq] + otype, ntype) == -1)
break;
if (n_eq == 0)
return isl_morph_identity(bset);
H = isl_mat_sub_alloc6(bset->ctx, bset->eq, f_eq, n_eq, otype, ntype);
H = isl_mat_left_hermite(H, 0, &U, &Q);
if (!H || !U || !Q)
goto error;
Q = isl_mat_drop_rows(Q, 0, n_eq);
Q = isl_mat_diagonal(isl_mat_identity(bset->ctx, otype), Q);
Q = isl_mat_diagonal(Q, isl_mat_identity(bset->ctx, nrest));
C = isl_mat_alloc(bset->ctx, 1 + n_eq, otype);
if (!C)
goto error;
isl_int_set_si(C->row[0][0], 1);
isl_seq_clr(C->row[0] + 1, otype - 1);
isl_mat_sub_neg(C->ctx, C->row + 1, bset->eq + f_eq, n_eq, 0, 0, otype);
H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
H1 = isl_mat_lin_to_aff(H1);
C = isl_mat_inverse_product(H1, C);
if (!C)
goto error;
isl_mat_free(H);
if (!isl_int_is_one(C->row[0][0])) {
int i;
isl_int g;
isl_int_init(g);
for (i = 0; i < n_eq; ++i) {
isl_seq_gcd(C->row[1 + i] + 1, otype - 1, &g);
isl_int_gcd(g, g, C->row[0][0]);
if (!isl_int_is_divisible_by(C->row[1 + i][0], g))
break;
}
isl_int_clear(g);
if (i < n_eq) {
isl_mat_free(C);
isl_mat_free(U);
isl_mat_free(Q);
return isl_morph_empty(bset);
}
C = isl_mat_normalize(C);
}
U1 = isl_mat_sub_alloc(U, 0, U->n_row, 0, n_eq);
U1 = isl_mat_lin_to_aff(U1);
U2 = isl_mat_sub_alloc(U, 0, U->n_row, n_eq, U->n_row - n_eq);
U2 = isl_mat_lin_to_aff(U2);
isl_mat_free(U);
C = isl_mat_product(U1, C);
C = isl_mat_aff_direct_sum(C, U2);
C = insert_parameter_rows(C, otype - 1);
C = isl_mat_diagonal(C, isl_mat_identity(bset->ctx, nrest));
dim = isl_space_copy(bset->dim);
dim = isl_space_drop_dims(dim, type, 0, ntype);
dim = isl_space_add_dims(dim, type, ntype - n_eq);
ran = isl_basic_set_universe(dim);
dom = copy_equalities(bset, f_eq, n_eq);
return isl_morph_alloc(dom, ran, Q, C);
error:
isl_mat_free(C);
isl_mat_free(H);
isl_mat_free(U);
isl_mat_free(Q);
return NULL;
}
/* Check if dimension dim belongs to a residue class
* i_dim \equiv r mod m
* with m != 1 and if so return m in *modulo and r in *residue.
* As a special case, when i_dim has a fixed value v, then
* *modulo is set to 0 and *residue to v.
*
* If i_dim does not belong to such a residue class, then *modulo
* is set to 1 and *residue is set to 0.
*/
int isl_basic_set_dim_residue_class(struct isl_basic_set *bset,
int pos, isl_int *modulo, isl_int *residue)
{
struct isl_ctx *ctx;
struct isl_mat *H = NULL, *U = NULL, *C, *H1, *U1;
unsigned total;
unsigned nparam;
if (!bset || !modulo || !residue)
return -1;
if (isl_basic_set_plain_dim_is_fixed(bset, pos, residue)) {
isl_int_set_si(*modulo, 0);
return 0;
}
ctx = isl_basic_set_get_ctx(bset);
total = isl_basic_set_total_dim(bset);
nparam = isl_basic_set_n_param(bset);
H = isl_mat_sub_alloc6(ctx, bset->eq, 0, bset->n_eq, 1, total);
H = isl_mat_left_hermite(H, 0, &U, NULL);
if (!H)
return -1;
isl_seq_gcd(U->row[nparam + pos]+bset->n_eq,
total-bset->n_eq, modulo);
if (isl_int_is_zero(*modulo))
isl_int_set_si(*modulo, 1);
if (isl_int_is_one(*modulo)) {
isl_int_set_si(*residue, 0);
isl_mat_free(H);
isl_mat_free(U);
return 0;
}
C = isl_mat_alloc(ctx, 1 + bset->n_eq, 1);
if (!C)
goto error;
isl_int_set_si(C->row[0][0], 1);
isl_mat_sub_neg(ctx, C->row + 1, bset->eq, bset->n_eq, 0, 0, 1);
H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
H1 = isl_mat_lin_to_aff(H1);
C = isl_mat_inverse_product(H1, C);
isl_mat_free(H);
U1 = isl_mat_sub_alloc(U, nparam+pos, 1, 0, bset->n_eq);
U1 = isl_mat_lin_to_aff(U1);
isl_mat_free(U);
C = isl_mat_product(U1, C);
if (!C)
return -1;
if (!isl_int_is_divisible_by(C->row[1][0], C->row[0][0])) {
bset = isl_basic_set_copy(bset);
bset = isl_basic_set_set_to_empty(bset);
isl_basic_set_free(bset);
isl_int_set_si(*modulo, 1);
isl_int_set_si(*residue, 0);
return 0;
}
isl_int_divexact(*residue, C->row[1][0], C->row[0][0]);
isl_int_fdiv_r(*residue, *residue, *modulo);
isl_mat_free(C);
return 0;
error:
isl_mat_free(H);
isl_mat_free(U);
return -1;
}
/* Given a set of equalities
*
* M x - c = 0
*
* this function computes a unimodular transformation from a lower-dimensional
* space to the original space that bijectively maps the integer points x'
* in the lower-dimensional space to the integer points x in the original
* space that satisfy the equalities.
*
* The input is given as a matrix B = [ -c M ] and the output is a
* matrix that maps [1 x'] to [1 x].
* If T2 is not NULL, then *T2 is set to a matrix mapping [1 x] to [1 x'].
*
* First compute the (left) Hermite normal form of M,
*
* M [U1 U2] = M U = H = [H1 0]
* or
* M = H Q = [H1 0] [Q1]
* [Q2]
*
* with U, Q unimodular, Q = U^{-1} (and H lower triangular).
* Define the transformed variables as
*
* x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
* [ x2' ] [Q2]
*
* The equalities then become
*
* H1 x1' - c = 0 or x1' = H1^{-1} c = c'
*
* If any of the c' is non-integer, then the original set has no
* integer solutions (since the x' are a unimodular transformation
* of the x) and a zero-column matrix is returned.
* Otherwise, the transformation is given by
*
* x = U1 H1^{-1} c + U2 x2'
*
* The inverse transformation is simply
*
* x2' = Q2 x
*/
__isl_give isl_mat *isl_mat_variable_compression(__isl_take isl_mat *B,
__isl_give isl_mat **T2)
{
int i;
struct isl_mat *H = NULL, *C = NULL, *H1, *U = NULL, *U1, *U2, *TC;
unsigned dim;
if (T2)
*T2 = NULL;
if (!B)
goto error;
dim = B->n_col - 1;
H = isl_mat_sub_alloc(B, 0, B->n_row, 1, dim);
H = isl_mat_left_hermite(H, 0, &U, T2);
if (!H || !U || (T2 && !*T2))
goto error;
if (T2) {
*T2 = isl_mat_drop_rows(*T2, 0, B->n_row);
*T2 = isl_mat_lin_to_aff(*T2);
if (!*T2)
goto error;
}
C = isl_mat_alloc(B->ctx, 1+B->n_row, 1);
if (!C)
goto error;
isl_int_set_si(C->row[0][0], 1);
isl_mat_sub_neg(C->ctx, C->row+1, B->row, B->n_row, 0, 0, 1);
H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
H1 = isl_mat_lin_to_aff(H1);
TC = isl_mat_inverse_product(H1, C);
if (!TC)
goto error;
isl_mat_free(H);
if (!isl_int_is_one(TC->row[0][0])) {
for (i = 0; i < B->n_row; ++i) {
if (!isl_int_is_divisible_by(TC->row[1+i][0], TC->row[0][0])) {
struct isl_ctx *ctx = B->ctx;
isl_mat_free(B);
isl_mat_free(TC);
isl_mat_free(U);
if (T2) {
isl_mat_free(*T2);
*T2 = NULL;
}
return isl_mat_alloc(ctx, 1 + dim, 0);
}
isl_seq_scale_down(TC->row[1+i], TC->row[1+i], TC->row[0][0], 1);
}
isl_int_set_si(TC->row[0][0], 1);
}
U1 = isl_mat_sub_alloc(U, 0, U->n_row, 0, B->n_row);
U1 = isl_mat_lin_to_aff(U1);
U2 = isl_mat_sub_alloc(U, 0, U->n_row, B->n_row, U->n_row - B->n_row);
U2 = isl_mat_lin_to_aff(U2);
isl_mat_free(U);
TC = isl_mat_product(U1, TC);
TC = isl_mat_aff_direct_sum(TC, U2);
isl_mat_free(B);
return TC;
error:
isl_mat_free(B);
isl_mat_free(H);
isl_mat_free(U);
if (T2) {
isl_mat_free(*T2);
*T2 = NULL;
}
return NULL;
}
/* Given a set of modulo constraints
*
* c + A y = 0 mod d
*
* this function returns an affine transformation T,
*
* y = T y'
*
* that bijectively maps the integer vectors y' to integer
* vectors y that satisfy the modulo constraints.
*
* This function is inspired by Section 2.5.3
* of B. Meister, "Stating and Manipulating Periodicity in the Polytope
* Model. Applications to Program Analysis and Optimization".
* However, the implementation only follows the algorithm of that
* section for computing a particular solution and not for computing
* a general homogeneous solution. The latter is incomplete and
* may remove some valid solutions.
* Instead, we use an adaptation of the algorithm in Section 7 of
* B. Meister, S. Verdoolaege, "Polynomial Approximations in the Polytope
* Model: Bringing the Power of Quasi-Polynomials to the Masses".
*
* The input is given as a matrix B = [ c A ] and a vector d.
* Each element of the vector d corresponds to a row in B.
* The output is a lower triangular matrix.
* If no integer vector y satisfies the given constraints then
* a matrix with zero columns is returned.
*
* We first compute a particular solution y_0 to the given set of
* modulo constraints in particular_solution. If no such solution
* exists, then we return a zero-columned transformation matrix.
* Otherwise, we compute the generic solution to
*
* A y = 0 mod d
*
* That is we want to compute G such that
*
* y = G y''
*
* with y'' integer, describes the set of solutions.
*
* We first remove the common factors of each row.
* In particular if gcd(A_i,d_i) != 1, then we divide the whole
* row i (including d_i) by this common factor. If afterwards gcd(A_i) != 1,
* then we divide this row of A by the common factor, unless gcd(A_i) = 0.
* In the later case, we simply drop the row (in both A and d).
*
* If there are no rows left in A, then G is the identity matrix. Otherwise,
* for each row i, we now determine the lattice of integer vectors
* that satisfies this row. Let U_i be the unimodular extension of the
* row A_i. This unimodular extension exists because gcd(A_i) = 1.
* The first component of
*
* y' = U_i y
*
* needs to be a multiple of d_i. Let y' = diag(d_i, 1, ..., 1) y''.
* Then,
*
* y = U_i^{-1} diag(d_i, 1, ..., 1) y''
*
* for arbitrary integer vectors y''. That is, y belongs to the lattice
* generated by the columns of L_i = U_i^{-1} diag(d_i, 1, ..., 1).
* If there is only one row, then G = L_1.
*
* If there is more than one row left, we need to compute the intersection
* of the lattices. That is, we need to compute an L such that
*
* L = L_i L_i' for all i
*
* with L_i' some integer matrices. Let A be constructed as follows
*
* A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
*
* and computed the Hermite Normal Form of A = [ H 0 ] U
* Then,
*
* L_i^{-T} = H U_{1,i}
*
* or
*
* H^{-T} = L_i U_{1,i}^T
*
* In other words G = L = H^{-T}.
* To ensure that G is lower triangular, we compute and use its Hermite
* normal form.
*
* The affine transformation matrix returned is then
*
* [ 1 0 ]
* [ y_0 G ]
*
* as any y = y_0 + G y' with y' integer is a solution to the original
* modulo constraints.
*/
struct isl_mat *isl_mat_parameter_compression(
struct isl_mat *B, struct isl_vec *d)
{
int i;
struct isl_mat *cst = NULL;
struct isl_mat *T = NULL;
isl_int D;
if (!B || !d)
goto error;
isl_assert(B->ctx, B->n_row == d->size, goto error);
cst = particular_solution(B, d);
if (!cst)
goto error;
if (cst->n_col == 0) {
T = isl_mat_alloc(B->ctx, B->n_col, 0);
isl_mat_free(cst);
isl_mat_free(B);
isl_vec_free(d);
return T;
}
isl_int_init(D);
/* Replace a*g*row = 0 mod g*m by row = 0 mod m */
for (i = 0; i < B->n_row; ++i) {
isl_seq_gcd(B->row[i] + 1, B->n_col - 1, &D);
if (isl_int_is_one(D))
continue;
if (isl_int_is_zero(D)) {
B = isl_mat_drop_rows(B, i, 1);
d = isl_vec_cow(d);
if (!B || !d)
goto error2;
isl_seq_cpy(d->block.data+i, d->block.data+i+1,
d->size - (i+1));
d->size--;
i--;
continue;
}
B = isl_mat_cow(B);
if (!B)
goto error2;
isl_seq_scale_down(B->row[i] + 1, B->row[i] + 1, D, B->n_col-1);
isl_int_gcd(D, D, d->block.data[i]);
d = isl_vec_cow(d);
if (!d)
goto error2;
isl_int_divexact(d->block.data[i], d->block.data[i], D);
}
isl_int_clear(D);
if (B->n_row == 0)
T = isl_mat_identity(B->ctx, B->n_col);
else if (B->n_row == 1)
T = parameter_compression_1(B, d);
else
T = parameter_compression_multi(B, d);
T = isl_mat_left_hermite(T, 0, NULL, NULL);
if (!T)
goto error;
isl_mat_sub_copy(T->ctx, T->row + 1, cst->row, cst->n_row, 0, 0, 1);
isl_mat_free(cst);
isl_mat_free(B);
isl_vec_free(d);
return T;
error2:
isl_int_clear(D);
error:
isl_mat_free(cst);
isl_mat_free(B);
isl_vec_free(d);
return NULL;
}