/* Construct a morphism that first does morph2 and then morph1.
*/
__isl_give isl_morph *isl_morph_compose(__isl_take isl_morph *morph1,
__isl_take isl_morph *morph2)
{
isl_mat *map, *inv;
isl_basic_set *dom, *ran;
if (!morph1 || !morph2)
goto error;
map = isl_mat_product(isl_mat_copy(morph1->map), isl_mat_copy(morph2->map));
inv = isl_mat_product(isl_mat_copy(morph2->inv), isl_mat_copy(morph1->inv));
dom = isl_morph_basic_set(isl_morph_inverse(isl_morph_copy(morph2)),
isl_basic_set_copy(morph1->dom));
dom = isl_basic_set_intersect(dom, isl_basic_set_copy(morph2->dom));
ran = isl_morph_basic_set(isl_morph_copy(morph1),
isl_basic_set_copy(morph2->ran));
ran = isl_basic_set_intersect(ran, isl_basic_set_copy(morph1->ran));
isl_morph_free(morph1);
isl_morph_free(morph2);
return isl_morph_alloc(dom, ran, map, inv);
error:
isl_morph_free(morph1);
isl_morph_free(morph2);
return NULL;
}
/* Given a set of equalities
*
* B(y) + A x = 0 (*)
*
* compute and return an affine transformation T,
*
* y = T y'
*
* that bijectively maps the integer vectors y' to integer
* vectors y that satisfy the modulo constraints for some value of x.
*
* Let [H 0] be the Hermite Normal Form of A, i.e.,
*
* A = [H 0] Q
*
* Then y is a solution of (*) iff
*
* H^-1 B(y) (= - [I 0] Q x)
*
* is an integer vector. Let d be the common denominator of H^-1.
* We impose
*
* d H^-1 B(y) = 0 mod d
*
* and compute the solution using isl_mat_parameter_compression.
*/
__isl_give isl_mat *isl_mat_parameter_compression_ext(__isl_take isl_mat *B,
__isl_take isl_mat *A)
{
isl_ctx *ctx;
isl_vec *d;
int n_row, n_col;
if (!A)
return isl_mat_free(B);
ctx = isl_mat_get_ctx(A);
n_row = A->n_row;
n_col = A->n_col;
A = isl_mat_left_hermite(A, 0, NULL, NULL);
A = isl_mat_drop_cols(A, n_row, n_col - n_row);
A = isl_mat_lin_to_aff(A);
A = isl_mat_right_inverse(A);
d = isl_vec_alloc(ctx, n_row);
if (A)
d = isl_vec_set(d, A->row[0][0]);
A = isl_mat_drop_rows(A, 0, 1);
A = isl_mat_drop_cols(A, 0, 1);
B = isl_mat_product(A, B);
return isl_mat_parameter_compression(B, d);
}
/* Construct a parameter compression for "bset".
* We basically just call isl_mat_parameter_compression with the right input
* and then extend the resulting matrix to include the variables.
*
* Let the equalities be given as
*
* B(p) + A x = 0
*
* and let [H 0] be the Hermite Normal Form of A, then
*
* H^-1 B(p)
*
* needs to be integer, so we impose that each row is divisible by
* the denominator.
*/
__isl_give isl_morph *isl_basic_set_parameter_compression(
__isl_keep isl_basic_set *bset)
{
unsigned nparam;
unsigned nvar;
int n_eq;
isl_mat *H, *B;
isl_vec *d;
isl_mat *map, *inv;
isl_basic_set *dom, *ran;
if (!bset)
return NULL;
if (isl_basic_set_plain_is_empty(bset))
return isl_morph_empty(bset);
if (bset->n_eq == 0)
return isl_morph_identity(bset);
isl_assert(bset->ctx, bset->n_div == 0, return NULL);
n_eq = bset->n_eq;
nparam = isl_basic_set_dim(bset, isl_dim_param);
nvar = isl_basic_set_dim(bset, isl_dim_set);
isl_assert(bset->ctx, n_eq <= nvar, return NULL);
d = isl_vec_alloc(bset->ctx, n_eq);
B = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, n_eq, 0, 1 + nparam);
H = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, n_eq, 1 + nparam, nvar);
H = isl_mat_left_hermite(H, 0, NULL, NULL);
H = isl_mat_drop_cols(H, n_eq, nvar - n_eq);
H = isl_mat_lin_to_aff(H);
H = isl_mat_right_inverse(H);
if (!H || !d)
goto error;
isl_seq_set(d->el, H->row[0][0], d->size);
H = isl_mat_drop_rows(H, 0, 1);
H = isl_mat_drop_cols(H, 0, 1);
B = isl_mat_product(H, B);
inv = isl_mat_parameter_compression(B, d);
inv = isl_mat_diagonal(inv, isl_mat_identity(bset->ctx, nvar));
map = isl_mat_right_inverse(isl_mat_copy(inv));
dom = isl_basic_set_universe(isl_space_copy(bset->dim));
ran = isl_basic_set_universe(isl_space_copy(bset->dim));
return isl_morph_alloc(dom, ran, map, inv);
error:
isl_mat_free(H);
isl_mat_free(B);
isl_vec_free(d);
return NULL;
}
/* Given a set of modulo constraints
*
* c + A y = 0 mod d
*
* this function computes a particular solution y_0
*
* The input is given as a matrix B = [ c A ] and a vector d.
*
* The output is matrix containing the solution y_0 or
* a zero-column matrix if the constraints admit no integer solution.
*
* The given set of constrains is equivalent to
*
* c + A y = -D x
*
* with D = diag d and x a fresh set of variables.
* Reducing both c and A modulo d does not change the
* value of y in the solution and may lead to smaller coefficients.
* Let M = [ D A ] and [ H 0 ] = M U, the Hermite normal form of M.
* Then
* [ x ]
* M [ y ] = - c
* and so
* [ x ]
* [ H 0 ] U^{-1} [ y ] = - c
* Let
* [ A ] [ x ]
* [ B ] = U^{-1} [ y ]
* then
* H A + 0 B = -c
*
* so B may be chosen arbitrarily, e.g., B = 0, and then
*
* [ x ] = [ -c ]
* U^{-1} [ y ] = [ 0 ]
* or
* [ x ] [ -c ]
* [ y ] = U [ 0 ]
* specifically,
*
* y = U_{2,1} (-c)
*
* If any of the coordinates of this y are non-integer
* then the constraints admit no integer solution and
* a zero-column matrix is returned.
*/
static struct isl_mat *particular_solution(struct isl_mat *B, struct isl_vec *d)
{
int i, j;
struct isl_mat *M = NULL;
struct isl_mat *C = NULL;
struct isl_mat *U = NULL;
struct isl_mat *H = NULL;
struct isl_mat *cst = NULL;
struct isl_mat *T = NULL;
M = isl_mat_alloc(B->ctx, B->n_row, B->n_row + B->n_col - 1);
C = isl_mat_alloc(B->ctx, 1 + B->n_row, 1);
if (!M || !C)
goto error;
isl_int_set_si(C->row[0][0], 1);
for (i = 0; i < B->n_row; ++i) {
isl_seq_clr(M->row[i], B->n_row);
isl_int_set(M->row[i][i], d->block.data[i]);
isl_int_neg(C->row[1 + i][0], B->row[i][0]);
isl_int_fdiv_r(C->row[1+i][0], C->row[1+i][0], M->row[i][i]);
for (j = 0; j < B->n_col - 1; ++j)
isl_int_fdiv_r(M->row[i][B->n_row + j],
B->row[i][1 + j], M->row[i][i]);
}
M = isl_mat_left_hermite(M, 0, &U, NULL);
if (!M || !U)
goto error;
H = isl_mat_sub_alloc(M, 0, B->n_row, 0, B->n_row);
H = isl_mat_lin_to_aff(H);
C = isl_mat_inverse_product(H, C);
if (!C)
goto error;
for (i = 0; i < B->n_row; ++i) {
if (!isl_int_is_divisible_by(C->row[1+i][0], C->row[0][0]))
break;
isl_int_divexact(C->row[1+i][0], C->row[1+i][0], C->row[0][0]);
}
if (i < B->n_row)
cst = isl_mat_alloc(B->ctx, B->n_row, 0);
else
cst = isl_mat_sub_alloc(C, 1, B->n_row, 0, 1);
T = isl_mat_sub_alloc(U, B->n_row, B->n_col - 1, 0, B->n_row);
cst = isl_mat_product(T, cst);
isl_mat_free(M);
isl_mat_free(C);
isl_mat_free(U);
return cst;
error:
isl_mat_free(M);
isl_mat_free(C);
isl_mat_free(U);
return NULL;
}
static struct isl_mat *isl_basic_set_samples(struct isl_basic_set *bset)
{
struct isl_mat *T;
struct isl_mat *samples;
if (!bset)
return NULL;
if (bset->n_eq == 0)
return isl_basic_set_scan_samples(bset);
bset = isl_basic_set_remove_equalities(bset, &T, NULL);
samples = isl_basic_set_scan_samples(bset);
return isl_mat_product(samples, isl_mat_transpose(T));
}
/* 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;
}