3 #include "isl_map_private.h"
4 #include "isl_equalities.h"
6 /* Given a set of modulo constraints
10 * this function computes a particular solution y_0
12 * The input is given as a matrix B = [ c A ] and a vector d.
14 * The output is matrix containing the solution y_0 or
15 * a zero-column matrix if the constraints admit no integer solution.
17 * The given set of constrains is equivalent to
21 * with D = diag d and x a fresh set of variables.
22 * Reducing both c and A modulo d does not change the
23 * value of y in the solution and may lead to smaller coefficients.
24 * Let M = [ D A ] and [ H 0 ] = M U, the Hermite normal form of M.
30 * [ H 0 ] U^{-1} [ y ] = - c
33 * [ B ] = U^{-1} [ y ]
37 * so B may be chosen arbitrarily, e.g., B = 0, and then
40 * U^{-1} [ y ] = [ 0 ]
48 * If any of the coordinates of this y are non-integer
49 * then the constraints admit no integer solution and
50 * a zero-column matrix is returned.
52 static struct isl_mat
*particular_solution(struct isl_mat
*B
, struct isl_vec
*d
)
55 struct isl_mat
*M
= NULL
;
56 struct isl_mat
*C
= NULL
;
57 struct isl_mat
*U
= NULL
;
58 struct isl_mat
*H
= NULL
;
59 struct isl_mat
*cst
= NULL
;
60 struct isl_mat
*T
= NULL
;
62 M
= isl_mat_alloc(B
->ctx
, B
->n_row
, B
->n_row
+ B
->n_col
- 1);
63 C
= isl_mat_alloc(B
->ctx
, 1 + B
->n_row
, 1);
66 isl_int_set_si(C
->row
[0][0], 1);
67 for (i
= 0; i
< B
->n_row
; ++i
) {
68 isl_seq_clr(M
->row
[i
], B
->n_row
);
69 isl_int_set(M
->row
[i
][i
], d
->block
.data
[i
]);
70 isl_int_neg(C
->row
[1 + i
][0], B
->row
[i
][0]);
71 isl_int_fdiv_r(C
->row
[1+i
][0], C
->row
[1+i
][0], M
->row
[i
][i
]);
72 for (j
= 0; j
< B
->n_col
- 1; ++j
)
73 isl_int_fdiv_r(M
->row
[i
][B
->n_row
+ j
],
74 B
->row
[i
][1 + j
], M
->row
[i
][i
]);
76 M
= isl_mat_left_hermite(M
, 0, &U
, NULL
);
79 H
= isl_mat_sub_alloc(B
->ctx
, M
->row
, 0, B
->n_row
, 0, B
->n_row
);
80 H
= isl_mat_lin_to_aff(H
);
81 C
= isl_mat_inverse_product(H
, C
);
84 for (i
= 0; i
< B
->n_row
; ++i
) {
85 if (!isl_int_is_divisible_by(C
->row
[1+i
][0], C
->row
[0][0]))
87 isl_int_divexact(C
->row
[1+i
][0], C
->row
[1+i
][0], C
->row
[0][0]);
90 cst
= isl_mat_alloc(B
->ctx
, B
->n_row
, 0);
92 cst
= isl_mat_sub_alloc(C
->ctx
, C
->row
, 1, B
->n_row
, 0, 1);
93 T
= isl_mat_sub_alloc(U
->ctx
, U
->row
, B
->n_row
, B
->n_col
- 1, 0, B
->n_row
);
94 cst
= isl_mat_product(T
, cst
);
106 /* Compute and return the matrix
108 * U_1^{-1} diag(d_1, 1, ..., 1)
110 * with U_1 the unimodular completion of the first (and only) row of B.
111 * The columns of this matrix generate the lattice that satisfies
112 * the single (linear) modulo constraint.
114 static struct isl_mat
*parameter_compression_1(
115 struct isl_mat
*B
, struct isl_vec
*d
)
119 U
= isl_mat_alloc(B
->ctx
, B
->n_col
- 1, B
->n_col
- 1);
122 isl_seq_cpy(U
->row
[0], B
->row
[0] + 1, B
->n_col
- 1);
123 U
= isl_mat_unimodular_complete(U
, 1);
124 U
= isl_mat_right_inverse(U
);
127 isl_mat_col_mul(U
, 0, d
->block
.data
[0], 0);
128 U
= isl_mat_lin_to_aff(U
);
132 /* Compute a common lattice of solutions to the linear modulo
133 * constraints specified by B and d.
134 * See also the documentation of isl_mat_parameter_compression.
137 * A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
139 * on a common denominator. This denominator D is the lcm of modulos d.
140 * Since L_i = U_i^{-1} diag(d_i, 1, ... 1), we have
141 * L_i^{-T} = U_i^T diag(d_i, 1, ... 1)^{-T} = U_i^T diag(1/d_i, 1, ..., 1).
142 * Putting this on the common denominator, we have
143 * D * L_i^{-T} = U_i^T diag(D/d_i, D, ..., D).
145 static struct isl_mat
*parameter_compression_multi(
146 struct isl_mat
*B
, struct isl_vec
*d
)
150 struct isl_mat
*A
= NULL
, *U
= NULL
;
159 A
= isl_mat_alloc(B
->ctx
, size
, B
->n_row
* size
);
160 U
= isl_mat_alloc(B
->ctx
, size
, size
);
163 for (i
= 0; i
< B
->n_row
; ++i
) {
164 isl_seq_cpy(U
->row
[0], B
->row
[i
] + 1, size
);
165 U
= isl_mat_unimodular_complete(U
, 1);
168 isl_int_divexact(D
, D
, d
->block
.data
[i
]);
169 for (k
= 0; k
< U
->n_col
; ++k
)
170 isl_int_mul(A
->row
[k
][i
*size
+0], D
, U
->row
[0][k
]);
171 isl_int_mul(D
, D
, d
->block
.data
[i
]);
172 for (j
= 1; j
< U
->n_row
; ++j
)
173 for (k
= 0; k
< U
->n_col
; ++k
)
174 isl_int_mul(A
->row
[k
][i
*size
+j
],
177 A
= isl_mat_left_hermite(A
, 0, NULL
, NULL
);
178 T
= isl_mat_sub_alloc(A
->ctx
, A
->row
, 0, A
->n_row
, 0, A
->n_row
);
179 T
= isl_mat_lin_to_aff(T
);
180 isl_int_set(T
->row
[0][0], D
);
181 T
= isl_mat_right_inverse(T
);
182 isl_assert(T
->ctx
, isl_int_is_one(T
->row
[0][0]), goto error
);
183 T
= isl_mat_transpose(T
);
196 /* Given a set of modulo constraints
200 * this function returns an affine transformation T,
204 * that bijectively maps the integer vectors y' to integer
205 * vectors y that satisfy the modulo constraints.
207 * This function is inspired by Section 2.5.3
208 * of B. Meister, "Stating and Manipulating Periodicity in the Polytope
209 * Model. Applications to Program Analysis and Optimization".
210 * However, the implementation only follows the algorithm of that
211 * section for computing a particular solution and not for computing
212 * a general homogeneous solution. The latter is incomplete and
213 * may remove some valid solutions.
214 * Instead, we use an adaptation of the algorithm in Section 7 of
215 * B. Meister, S. Verdoolaege, "Polynomial Approximations in the Polytope
216 * Model: Bringing the Power of Quasi-Polynomials to the Masses".
218 * The input is given as a matrix B = [ c A ] and a vector d.
219 * Each element of the vector d corresponds to a row in B.
220 * The output is a lower triangular matrix.
221 * If no integer vector y satisfies the given constraints then
222 * a matrix with zero columns is returned.
224 * We first compute a particular solution y_0 to the given set of
225 * modulo constraints in particular_solution. If no such solution
226 * exists, then we return a zero-columned transformation matrix.
227 * Otherwise, we compute the generic solution to
231 * That is we want to compute G such that
235 * with y'' integer, describes the set of solutions.
237 * We first remove the common factors of each row.
238 * In particular if gcd(A_i,d_i) != 1, then we divide the whole
239 * row i (including d_i) by this common factor. If afterwards gcd(A_i) != 1,
240 * then we divide this row of A by the common factor, unless gcd(A_i) = 0.
241 * In the later case, we simply drop the row (in both A and d).
243 * If there are no rows left in A, the G is the identity matrix. Otherwise,
244 * for each row i, we now determine the lattice of integer vectors
245 * that satisfies this row. Let U_i be the unimodular extension of the
246 * row A_i. This unimodular extension exists because gcd(A_i) = 1.
247 * The first component of
251 * needs to be a multiple of d_i. Let y' = diag(d_i, 1, ..., 1) y''.
254 * y = U_i^{-1} diag(d_i, 1, ..., 1) y''
256 * for arbitrary integer vectors y''. That is, y belongs to the lattice
257 * generated by the columns of L_i = U_i^{-1} diag(d_i, 1, ..., 1).
258 * If there is only one row, then G = L_1.
260 * If there is more than one row left, we need to compute the intersection
261 * of the lattices. That is, we need to compute an L such that
263 * L = L_i L_i' for all i
265 * with L_i' some integer matrices. Let A be constructed as follows
267 * A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
269 * and computed the Hermite Normal Form of A = [ H 0 ] U
272 * L_i^{-T} = H U_{1,i}
276 * H^{-T} = L_i U_{1,i}^T
278 * In other words G = L = H^{-T}.
279 * To ensure that G is lower triangular, we compute and use its Hermite
282 * The affine transformation matrix returned is then
287 * as any y = y_0 + G y' with y' integer is a solution to the original
288 * modulo constraints.
290 struct isl_mat
*isl_mat_parameter_compression(
291 struct isl_mat
*B
, struct isl_vec
*d
)
294 struct isl_mat
*cst
= NULL
;
295 struct isl_mat
*T
= NULL
;
300 isl_assert(B
->ctx
, B
->n_row
== d
->size
, goto error
);
301 cst
= particular_solution(B
, d
);
304 if (cst
->n_col
== 0) {
305 T
= isl_mat_alloc(B
->ctx
, B
->n_col
, 0);
312 /* Replace a*g*row = 0 mod g*m by row = 0 mod m */
313 for (i
= 0; i
< B
->n_row
; ++i
) {
314 isl_seq_gcd(B
->row
[i
] + 1, B
->n_col
- 1, &D
);
315 if (isl_int_is_one(D
))
317 if (isl_int_is_zero(D
)) {
318 B
= isl_mat_drop_rows(B
, i
, 1);
322 isl_seq_cpy(d
->block
.data
+i
, d
->block
.data
+i
+1,
331 isl_seq_scale_down(B
->row
[i
] + 1, B
->row
[i
] + 1, D
, B
->n_col
-1);
332 isl_int_gcd(D
, D
, d
->block
.data
[i
]);
336 isl_int_divexact(d
->block
.data
[i
], d
->block
.data
[i
], D
);
340 T
= isl_mat_identity(B
->ctx
, B
->n_col
);
341 else if (B
->n_row
== 1)
342 T
= parameter_compression_1(B
, d
);
344 T
= parameter_compression_multi(B
, d
);
345 T
= isl_mat_left_hermite(T
, 0, NULL
, NULL
);
348 isl_mat_sub_copy(T
->ctx
, T
->row
+ 1, cst
->row
, cst
->n_row
, 0, 0, 1);
362 /* Given a set of equalities
366 * this function computes unimodular transformation from a lower-dimensional
367 * space to the original space that bijectively maps the integer points x'
368 * in the lower-dimensional space to the integer points x in the original
369 * space that satisfy the equalities.
371 * The input is given as a matrix B = [ -c M ] and the out is a
372 * matrix that maps [1 x'] to [1 x].
373 * If T2 is not NULL, then *T2 is set to a matrix mapping [1 x] to [1 x'].
375 * First compute the (left) Hermite normal form of M,
377 * M [U1 U2] = M U = H = [H1 0]
379 * M = H Q = [H1 0] [Q1]
382 * with U, Q unimodular, Q = U^{-1} (and H lower triangular).
383 * Define the transformed variables as
385 * x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
388 * The equalities then become
390 * H1 x1' - c = 0 or x1' = H1^{-1} c = c'
392 * If any of the c' is non-integer, then the original set has no
393 * integer solutions (since the x' are a unimodular transformation
395 * Otherwise, the transformation is given by
397 * x = U1 H1^{-1} c + U2 x2'
399 * The inverse transformation is simply
403 struct isl_mat
*isl_mat_variable_compression(struct isl_mat
*B
,
407 struct isl_mat
*H
= NULL
, *C
= NULL
, *H1
, *U
= NULL
, *U1
, *U2
, *TC
;
416 H
= isl_mat_sub_alloc(B
->ctx
, B
->row
, 0, B
->n_row
, 1, dim
);
417 H
= isl_mat_left_hermite(H
, 0, &U
, T2
);
418 if (!H
|| !U
|| (T2
&& !*T2
))
421 *T2
= isl_mat_drop_rows(*T2
, 0, B
->n_row
);
422 *T2
= isl_mat_lin_to_aff(*T2
);
426 C
= isl_mat_alloc(B
->ctx
, 1+B
->n_row
, 1);
429 isl_int_set_si(C
->row
[0][0], 1);
430 isl_mat_sub_neg(C
->ctx
, C
->row
+1, B
->row
, B
->n_row
, 0, 0, 1);
431 H1
= isl_mat_sub_alloc(H
->ctx
, H
->row
, 0, H
->n_row
, 0, H
->n_row
);
432 H1
= isl_mat_lin_to_aff(H1
);
433 TC
= isl_mat_inverse_product(H1
, C
);
437 if (!isl_int_is_one(TC
->row
[0][0])) {
438 for (i
= 0; i
< B
->n_row
; ++i
) {
439 if (!isl_int_is_divisible_by(TC
->row
[1+i
][0], TC
->row
[0][0])) {
440 struct isl_ctx
*ctx
= B
->ctx
;
448 return isl_mat_alloc(ctx
, 1 + dim
, 0);
450 isl_seq_scale_down(TC
->row
[1+i
], TC
->row
[1+i
], TC
->row
[0][0], 1);
452 isl_int_set_si(TC
->row
[0][0], 1);
454 U1
= isl_mat_sub_alloc(U
->ctx
, U
->row
, 0, U
->n_row
, 0, B
->n_row
);
455 U1
= isl_mat_lin_to_aff(U1
);
456 U2
= isl_mat_sub_alloc(U
->ctx
, U
->row
, 0, U
->n_row
,
457 B
->n_row
, U
->n_row
- B
->n_row
);
458 U2
= isl_mat_lin_to_aff(U2
);
460 TC
= isl_mat_product(U1
, TC
);
461 TC
= isl_mat_aff_direct_sum(TC
, U2
);
477 /* Use the n equalities of bset to unimodularly transform the
478 * variables x such that n transformed variables x1' have a constant value
479 * and rewrite the constraints of bset in terms of the remaining
480 * transformed variables x2'. The matrix pointed to by T maps
481 * the new variables x2' back to the original variables x, while T2
482 * maps the original variables to the new variables.
484 static struct isl_basic_set
*compress_variables(
485 struct isl_basic_set
*bset
, struct isl_mat
**T
, struct isl_mat
**T2
)
487 struct isl_mat
*B
, *TC
;
496 isl_assert(bset
->ctx
, isl_basic_set_n_param(bset
) == 0, goto error
);
497 isl_assert(bset
->ctx
, bset
->n_div
== 0, goto error
);
498 dim
= isl_basic_set_n_dim(bset
);
499 isl_assert(bset
->ctx
, bset
->n_eq
<= dim
, goto error
);
503 B
= isl_mat_sub_alloc(bset
->ctx
, bset
->eq
, 0, bset
->n_eq
, 0, 1 + dim
);
504 TC
= isl_mat_variable_compression(B
, T2
);
507 if (TC
->n_col
== 0) {
513 return isl_basic_set_set_to_empty(bset
);
516 bset
= isl_basic_set_preimage(bset
, T
? isl_mat_copy(TC
) : TC
);
521 isl_basic_set_free(bset
);
525 struct isl_basic_set
*isl_basic_set_remove_equalities(
526 struct isl_basic_set
*bset
, struct isl_mat
**T
, struct isl_mat
**T2
)
534 isl_assert(bset
->ctx
, isl_basic_set_n_param(bset
) == 0, goto error
);
535 bset
= isl_basic_set_gauss(bset
, NULL
);
536 if (ISL_F_ISSET(bset
, ISL_BASIC_SET_EMPTY
))
538 bset
= compress_variables(bset
, T
, T2
);
541 isl_basic_set_free(bset
);
546 /* Check if dimension dim belongs to a residue class
547 * i_dim \equiv r mod m
548 * with m != 1 and if so return m in *modulo and r in *residue.
549 * As a special case, when i_dim has a fixed value v, then
550 * *modulo is set to 0 and *residue to v.
552 * If i_dim does not belong to such a residue class, then *modulo
553 * is set to 1 and *residue is set to 0.
555 int isl_basic_set_dim_residue_class(struct isl_basic_set
*bset
,
556 int pos
, isl_int
*modulo
, isl_int
*residue
)
559 struct isl_mat
*H
= NULL
, *U
= NULL
, *C
, *H1
, *U1
;
563 if (!bset
|| !modulo
|| !residue
)
566 if (isl_basic_set_fast_dim_is_fixed(bset
, pos
, residue
)) {
567 isl_int_set_si(*modulo
, 0);
572 total
= isl_basic_set_total_dim(bset
);
573 nparam
= isl_basic_set_n_param(bset
);
574 H
= isl_mat_sub_alloc(bset
->ctx
, bset
->eq
, 0, bset
->n_eq
, 1, total
);
575 H
= isl_mat_left_hermite(H
, 0, &U
, NULL
);
579 isl_seq_gcd(U
->row
[nparam
+ pos
]+bset
->n_eq
,
580 total
-bset
->n_eq
, modulo
);
581 if (isl_int_is_zero(*modulo
))
582 isl_int_set_si(*modulo
, 1);
583 if (isl_int_is_one(*modulo
)) {
584 isl_int_set_si(*residue
, 0);
590 C
= isl_mat_alloc(bset
->ctx
, 1+bset
->n_eq
, 1);
593 isl_int_set_si(C
->row
[0][0], 1);
594 isl_mat_sub_neg(C
->ctx
, C
->row
+1, bset
->eq
, bset
->n_eq
, 0, 0, 1);
595 H1
= isl_mat_sub_alloc(H
->ctx
, H
->row
, 0, H
->n_row
, 0, H
->n_row
);
596 H1
= isl_mat_lin_to_aff(H1
);
597 C
= isl_mat_inverse_product(H1
, C
);
599 U1
= isl_mat_sub_alloc(U
->ctx
, U
->row
, nparam
+pos
, 1, 0, bset
->n_eq
);
600 U1
= isl_mat_lin_to_aff(U1
);
602 C
= isl_mat_product(U1
, C
);
605 if (!isl_int_is_divisible_by(C
->row
[1][0], C
->row
[0][0])) {
606 bset
= isl_basic_set_copy(bset
);
607 bset
= isl_basic_set_set_to_empty(bset
);
608 isl_basic_set_free(bset
);
609 isl_int_set_si(*modulo
, 1);
610 isl_int_set_si(*residue
, 0);
613 isl_int_divexact(*residue
, C
->row
[1][0], C
->row
[0][0]);
614 isl_int_fdiv_r(*residue
, *residue
, *modulo
);
623 /* Check if dimension dim belongs to a residue class
624 * i_dim \equiv r mod m
625 * with m != 1 and if so return m in *modulo and r in *residue.
626 * As a special case, when i_dim has a fixed value v, then
627 * *modulo is set to 0 and *residue to v.
629 * If i_dim does not belong to such a residue class, then *modulo
630 * is set to 1 and *residue is set to 0.
632 int isl_set_dim_residue_class(struct isl_set
*set
,
633 int pos
, isl_int
*modulo
, isl_int
*residue
)
639 if (!set
|| !modulo
|| !residue
)
643 isl_int_set_si(*modulo
, 0);
644 isl_int_set_si(*residue
, 0);
648 if (isl_basic_set_dim_residue_class(set
->p
[0], pos
, modulo
, residue
)<0)
654 if (isl_int_is_one(*modulo
))
660 for (i
= 1; i
< set
->n
; ++i
) {
661 if (isl_basic_set_dim_residue_class(set
->p
[0], pos
, &m
, &r
) < 0)
663 isl_int_gcd(*modulo
, *modulo
, m
);
664 if (!isl_int_is_zero(*modulo
))
665 isl_int_fdiv_r(*residue
, *residue
, *modulo
);
666 if (isl_int_is_one(*modulo
))
668 if (!isl_int_is_zero(*modulo
))
669 isl_int_fdiv_r(r
, r
, *modulo
);
670 if (isl_int_ne(*residue
, r
)) {
671 isl_int_set_si(*modulo
, 1);
672 isl_int_set_si(*residue
, 0);