/* Make first row entries in column col of bset1 identical to
* those of bset2, using the fact that entry bset1->eq[row][col]=a
* is non-zero. Initially, these elements of bset1 are all zero.
* For each row i < row, we set
* A[i] = a * A[i] + B[i][col] * A[row]
* B[i] = a * B[i]
* so that
* A[i][col] = B[i][col] = a * old(B[i][col])
*/
static void construct_column(
struct isl_basic_set *bset1, struct isl_basic_set *bset2,
unsigned row, unsigned col)
{
int r;
isl_int a;
isl_int b;
unsigned total;
isl_int_init(a);
isl_int_init(b);
total = 1 + isl_basic_set_n_dim(bset1);
for (r = 0; r < row; ++r) {
if (isl_int_is_zero(bset2->eq[r][col]))
continue;
isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]);
isl_int_divexact(a, bset1->eq[row][col], b);
isl_int_divexact(b, bset2->eq[r][col], b);
isl_seq_combine(bset1->eq[r], a, bset1->eq[r],
b, bset1->eq[row], total);
isl_seq_scale(bset2->eq[r], bset2->eq[r], a, total);
}
isl_int_clear(a);
isl_int_clear(b);
delete_row(bset1, row);
}
/* Make eq[row][col] of both bmaps equal so we can add the row
* add the column to the common matrix.
* Note that because of the echelon form, the columns of row row
* after column col are zero.
*/
static void set_common_multiple(
struct isl_basic_set *bset1, struct isl_basic_set *bset2,
unsigned row, unsigned col)
{
isl_int m, c;
if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col]))
return;
isl_int_init(c);
isl_int_init(m);
isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]);
isl_int_divexact(c, m, bset1->eq[row][col]);
isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1);
isl_int_divexact(c, m, bset2->eq[row][col]);
isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1);
isl_int_clear(c);
isl_int_clear(m);
}
/* 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;
}
/* 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;
}
/* Make first row entries in column col of bset1 identical to
* those of bset2, using only these entries of the two matrices.
* Let t be the last row with different entries.
* For each row i < t, we set
* A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
* B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
* so that
* A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
*/
static int transform_column(
struct isl_basic_set *bset1, struct isl_basic_set *bset2,
unsigned row, unsigned col)
{
int i, t;
isl_int a, b, g;
unsigned total;
for (t = row-1; t >= 0; --t)
if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))
break;
if (t < 0)
return 0;
total = 1 + isl_basic_set_n_dim(bset1);
isl_int_init(a);
isl_int_init(b);
isl_int_init(g);
isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]);
for (i = 0; i < t; ++i) {
isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]);
isl_int_gcd(g, a, b);
isl_int_divexact(a, a, g);
isl_int_divexact(g, b, g);
isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t],
total);
isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],
total);
}
isl_int_clear(a);
isl_int_clear(b);
isl_int_clear(g);
delete_row(bset1, t);
delete_row(bset2, t);
return 1;
}
static void copy_solution(struct isl_vec *vec, int maximize, isl_int *opt,
isl_int *opt_denom, PipQuast *sol)
{
int i;
PipList *list;
isl_int tmp;
if (opt) {
if (opt_denom) {
isl_seq_cpy_from_pip(opt,
&sol->list->vector->the_vector[0], 1);
isl_seq_cpy_from_pip(opt_denom,
&sol->list->vector->the_deno[0], 1);
} else if (maximize)
mpz_fdiv_q(*opt, sol->list->vector->the_vector[0],
sol->list->vector->the_deno[0]);
else
mpz_cdiv_q(*opt, sol->list->vector->the_vector[0],
sol->list->vector->the_deno[0]);
}
if (!vec)
return;
isl_int_init(tmp);
isl_int_set_si(vec->el[0], 1);
for (i = 0, list = sol->list->next; list; ++i, list = list->next) {
isl_seq_cpy_from_pip(&vec->el[1 + i],
&list->vector->the_deno[0], 1);
isl_int_lcm(vec->el[0], vec->el[0], vec->el[1 + i]);
}
for (i = 0, list = sol->list->next; list; ++i, list = list->next) {
isl_seq_cpy_from_pip(&tmp, &list->vector->the_deno[0], 1);
isl_int_divexact(tmp, vec->el[0], tmp);
isl_seq_cpy_from_pip(&vec->el[1 + i],
&list->vector->the_vector[0], 1);
isl_int_mul(vec->el[1 + i], vec->el[1 + i], tmp);
}
isl_int_clear(tmp);
}
/* 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 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;
}
