nexmon – Rev 1
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/*
* Copyright 2008-2009 Katholieke Universiteit Leuven
*
* Use of this software is governed by the GNU LGPLv2.1 license
*
* Written by Sven Verdoolaege, K.U.Leuven, Departement
* Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
*/
#include <isl_mat_private.h>
#include <isl/seq.h>
#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_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;
}
/* 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_mat *B, struct isl_vec *d)
{
struct isl_mat *U;
U = isl_mat_alloc(B->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(U, 1);
U = isl_mat_right_inverse(U);
if (!U)
return NULL;
isl_mat_col_mul(U, 0, d->block.data[0], 0);
U = isl_mat_lin_to_aff(U);
return U;
}
/* 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 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;
}
/* 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
*/
struct isl_mat *isl_mat_variable_compression(struct isl_mat *B,
struct 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;
}
/* Use the n equalities of bset to unimodularly transform the
* variables x such that n transformed variables x1' have a constant value
* and rewrite the constraints of bset in terms of the remaining
* transformed variables x2'. The matrix pointed to by T maps
* the new variables x2' back to the original variables x, while T2
* maps the original variables to the new variables.
*/
static struct isl_basic_set *compress_variables(
struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
{
struct isl_mat *B, *TC;
unsigned dim;
if (T)
*T = NULL;
if (T2)
*T2 = NULL;
if (!bset)
goto error;
isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
isl_assert(bset->ctx, bset->n_div == 0, goto error);
dim = isl_basic_set_n_dim(bset);
isl_assert(bset->ctx, bset->n_eq <= dim, goto error);
if (bset->n_eq == 0)
return bset;
B = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq, 0, 1 + dim);
TC = isl_mat_variable_compression(B, T2);
if (!TC)
goto error;
if (TC->n_col == 0) {
isl_mat_free(TC);
if (T2) {
isl_mat_free(*T2);
*T2 = NULL;
}
return isl_basic_set_set_to_empty(bset);
}
bset = isl_basic_set_preimage(bset, T ? isl_mat_copy(TC) : TC);
if (T)
*T = TC;
return bset;
error:
isl_basic_set_free(bset);
return NULL;
}
struct isl_basic_set *isl_basic_set_remove_equalities(
struct isl_basic_set *bset, struct isl_mat **T, struct isl_mat **T2)
{
if (T)
*T = NULL;
if (T2)
*T2 = NULL;
if (!bset)
return NULL;
isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
bset = isl_basic_set_gauss(bset, NULL);
if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
return bset;
bset = compress_variables(bset, T, T2);
return bset;
error:
isl_basic_set_free(bset);
*T = NULL;
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 = bset->ctx;
total = isl_basic_set_total_dim(bset);
nparam = isl_basic_set_n_param(bset);
H = isl_mat_sub_alloc6(bset->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(bset->ctx, 1+bset->n_eq, 1);
if (!C)
goto error;
isl_int_set_si(C->row[0][0], 1);
isl_mat_sub_neg(C->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)
goto error;
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;
}
/* 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[0], pos, &m, &r) < 0)
goto error;
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;
if (!isl_int_is_zero(*modulo))
isl_int_fdiv_r(r, r, *modulo);
if (isl_int_ne(*residue, r)) {
isl_int_set_si(*modulo, 1);
isl_int_set_si(*residue, 0);
break;
}
}
isl_int_clear(m);
isl_int_clear(r);
return 0;
error:
isl_int_clear(m);
isl_int_clear(r);
return -1;
}