#include "isl_map_private.h"
#include "isl_equalities.h"
+/* 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_ctx *ctx,
+ 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(ctx, B->n_row, B->n_row + B->n_col - 1);
+ C = isl_mat_alloc(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(ctx, M, 0, &U, NULL);
+ if (!M || !U)
+ goto error;
+ H = isl_mat_sub_alloc(ctx, M->row, 0, B->n_row, 0, B->n_row);
+ H = isl_mat_lin_to_aff(ctx, H);
+ C = isl_mat_inverse_product(ctx, 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(ctx, B->n_row, 0);
+ else
+ cst = isl_mat_sub_alloc(ctx, C->row, 1, B->n_row, 0, 1);
+ T = isl_mat_sub_alloc(ctx, U->row, B->n_row, B->n_col - 1, 0, B->n_row);
+ cst = isl_mat_product(ctx, T, cst);
+ isl_mat_free(ctx, M);
+ isl_mat_free(ctx, C);
+ isl_mat_free(ctx, U);
+ return cst;
+error:
+ isl_mat_free(ctx, M);
+ isl_mat_free(ctx, C);
+ isl_mat_free(ctx, U);
+ return NULL;
+}
+
+/* Compute and return the matrix
+ *
+ * U_1^{-1} diag(d_1, 1, ..., 1)
+ *
+ * with U_1 the unimodular completion of the first (and only) row of B.
+ * The columns of this matrix generate the lattice that satisfies
+ * the single (linear) modulo constraint.
+ */
+static struct isl_mat *parameter_compression_1(struct isl_ctx *ctx,
+ struct isl_mat *B, struct isl_vec *d)
+{
+ struct isl_mat *U;
+
+ U = isl_mat_alloc(ctx, B->n_col - 1, B->n_col - 1);
+ if (!U)
+ return NULL;
+ isl_seq_cpy(U->row[0], B->row[0] + 1, B->n_col - 1);
+ U = isl_mat_unimodular_complete(ctx, U, 1);
+ U = isl_mat_right_inverse(ctx, U);
+ if (!U)
+ return NULL;
+ isl_mat_col_mul(U, 0, d->block.data[0], 0);
+ U = isl_mat_lin_to_aff(ctx, U);
+ return U;
+error:
+ isl_mat_free(ctx, U);
+ return NULL;
+}
+
+/* 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_ctx *ctx,
+ struct isl_mat *B, struct isl_vec *d)
+{
+ int i, j, k;
+ int ok;
+ isl_int D;
+ struct isl_mat *A = NULL, *U = NULL;
+ struct isl_mat *T;
+ unsigned size;
+
+ isl_int_init(D);
+
+ isl_vec_lcm(ctx, d, &D);
+
+ size = B->n_col - 1;
+ A = isl_mat_alloc(ctx, size, B->n_row * size);
+ U = isl_mat_alloc(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(ctx, 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(ctx, A, 0, NULL, NULL);
+ T = isl_mat_sub_alloc(ctx, A->row, 0, A->n_row, 0, A->n_row);
+ T = isl_mat_lin_to_aff(ctx, T);
+ isl_int_set(T->row[0][0], D);
+ T = isl_mat_right_inverse(ctx, T);
+ isl_assert(ctx, isl_int_is_one(T->row[0][0]), goto error);
+ T = isl_mat_transpose(ctx, T);
+ isl_mat_free(ctx, A);
+ isl_mat_free(ctx, U);
+
+ isl_int_clear(D);
+ return T;
+error:
+ isl_mat_free(ctx, A);
+ isl_mat_free(ctx, U);
+ isl_int_clear(D);
+ 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, the 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_ctx *ctx,
+ 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(ctx, B->n_row == d->size, goto error);
+ cst = particular_solution(ctx, B, d);
+ if (!cst)
+ goto error;
+ if (cst->n_col == 0) {
+ T = isl_mat_alloc(ctx, B->n_col, 0);
+ isl_mat_free(ctx, cst);
+ isl_mat_free(ctx, B);
+ isl_vec_free(ctx, 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(ctx, B, i, 1);
+ d = isl_vec_cow(ctx, 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(ctx, 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(ctx, 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(ctx, B->n_col);
+ else if (B->n_row == 1)
+ T = parameter_compression_1(ctx, B, d);
+ else
+ T = parameter_compression_multi(ctx, B, d);
+ T = isl_mat_left_hermite(ctx, T, 0, NULL, NULL);
+ if (!T)
+ goto error;
+ isl_mat_sub_copy(ctx, T->row + 1, cst->row, cst->n_row, 0, 0, 1);
+ isl_mat_free(ctx, cst);
+ isl_mat_free(ctx, B);
+ isl_vec_free(ctx, d);
+ return T;
+error2:
+ isl_int_clear(D);
+error:
+ isl_mat_free(ctx, cst);
+ isl_mat_free(ctx, B);
+ isl_vec_free(ctx, d);
+ return NULL;
+}
+
/* Given a set of equalities
*
* M x - c = 0
isl_mat_free(ctx, *T2);
*T2 = NULL;
}
- return isl_mat_alloc(ctx, 1 + B->n_col, 0);
+ 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);
}
return isl_basic_set_set_to_empty(bset);
}
- bset = isl_basic_set_preimage(ctx, bset, T ? isl_mat_copy(ctx, TC) : TC);
+ bset = isl_basic_set_preimage(bset, T ? isl_mat_copy(ctx, TC) : TC);
if (T)
*T = TC;
return bset;
return NULL;
isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
bset = isl_basic_set_gauss(bset, NULL);
- if (F_ISSET(bset, ISL_BASIC_SET_EMPTY))
+ if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
return bset;
bset = compress_variables(bset->ctx, bset, T, T2);
return bset;
projects / platform / upstream / isl.git / blobdiff