/* 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;
}
/* 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 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;
}
/* 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;
}
