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_ctx
*ctx
,
53 struct isl_mat
*B
, struct isl_vec
*d
)
56 struct isl_mat
*M
= NULL
;
57 struct isl_mat
*C
= NULL
;
58 struct isl_mat
*U
= NULL
;
59 struct isl_mat
*H
= NULL
;
60 struct isl_mat
*cst
= NULL
;
61 struct isl_mat
*T
= NULL
;
63 M
= isl_mat_alloc(ctx
, B
->n_row
, B
->n_row
+ B
->n_col
- 1);
64 C
= isl_mat_alloc(ctx
, 1 + B
->n_row
, 1);
67 isl_int_set_si(C
->row
[0][0], 1);
68 for (i
= 0; i
< B
->n_row
; ++i
) {
69 isl_seq_clr(M
->row
[i
], B
->n_row
);
70 isl_int_set(M
->row
[i
][i
], d
->block
.data
[i
]);
71 isl_int_neg(C
->row
[1 + i
][0], B
->row
[i
][0]);
72 isl_int_fdiv_r(C
->row
[1+i
][0], C
->row
[1+i
][0], M
->row
[i
][i
]);
73 for (j
= 0; j
< B
->n_col
- 1; ++j
)
74 isl_int_fdiv_r(M
->row
[i
][B
->n_row
+ j
],
75 B
->row
[i
][1 + j
], M
->row
[i
][i
]);
77 M
= isl_mat_left_hermite(ctx
, M
, 0, &U
, NULL
);
80 H
= isl_mat_sub_alloc(ctx
, M
->row
, 0, B
->n_row
, 0, B
->n_row
);
81 H
= isl_mat_lin_to_aff(ctx
, H
);
82 C
= isl_mat_inverse_product(ctx
, H
, C
);
85 for (i
= 0; i
< B
->n_row
; ++i
) {
86 if (!isl_int_is_divisible_by(C
->row
[1+i
][0], C
->row
[0][0]))
88 isl_int_divexact(C
->row
[1+i
][0], C
->row
[1+i
][0], C
->row
[0][0]);
91 cst
= isl_mat_alloc(ctx
, B
->n_row
, 0);
93 cst
= isl_mat_sub_alloc(ctx
, C
->row
, 1, B
->n_row
, 0, 1);
94 T
= isl_mat_sub_alloc(ctx
, U
->row
, B
->n_row
, B
->n_col
- 1, 0, B
->n_row
);
95 cst
= isl_mat_product(ctx
, T
, cst
);
101 isl_mat_free(ctx
, M
);
102 isl_mat_free(ctx
, C
);
103 isl_mat_free(ctx
, U
);
107 static struct isl_mat
*unimodular_complete(struct isl_ctx
*ctx
,
108 struct isl_mat
*M
, int row
)
111 struct isl_mat
*H
= NULL
, *Q
= NULL
;
113 isl_assert(ctx
, M
->n_row
== M
->n_col
, goto error
);
115 H
= isl_mat_left_hermite(ctx
, isl_mat_copy(ctx
, M
), 0, NULL
, &Q
);
119 for (r
= 0; r
< row
; ++r
)
120 isl_assert(ctx
, isl_int_is_one(H
->row
[r
][r
]), goto error
);
121 for (r
= row
; r
< M
->n_row
; ++r
)
122 isl_seq_cpy(M
->row
[r
], Q
->row
[r
], M
->n_col
);
123 isl_mat_free(ctx
, H
);
124 isl_mat_free(ctx
, Q
);
127 isl_mat_free(ctx
, H
);
128 isl_mat_free(ctx
, Q
);
129 isl_mat_free(ctx
, M
);
133 /* Compute and return the matrix
135 * U_1^{-1} diag(d_1, 1, ..., 1)
137 * with U_1 the unimodular completion of the first (and only) row of B.
138 * The columns of this matrix generate the lattice that satisfies
139 * the single (linear) modulo constraint.
141 static struct isl_mat
*parameter_compression_1(struct isl_ctx
*ctx
,
142 struct isl_mat
*B
, struct isl_vec
*d
)
146 U
= isl_mat_alloc(ctx
, B
->n_col
- 1, B
->n_col
- 1);
149 isl_seq_cpy(U
->row
[0], B
->row
[0] + 1, B
->n_col
- 1);
150 U
= unimodular_complete(ctx
, U
, 1);
151 U
= isl_mat_right_inverse(ctx
, U
);
154 isl_mat_col_mul(U
, 0, d
->block
.data
[0], 0);
155 U
= isl_mat_lin_to_aff(ctx
, U
);
158 isl_mat_free(ctx
, U
);
162 /* Compute a common lattice of solutions to the linear modulo
163 * constraints specified by B and d.
164 * See also the documentation of isl_mat_parameter_compression.
167 * A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
169 * on a common denominator. This denominator D is the lcm of modulos d.
170 * Since L_i = U_i^{-1} diag(d_i, 1, ... 1), we have
171 * L_i^{-T} = U_i^T diag(d_i, 1, ... 1)^{-T} = U_i^T diag(1/d_i, 1, ..., 1).
172 * Putting this on the common denominator, we have
173 * D * L_i^{-T} = U_i^T diag(D/d_i, D, ..., D).
175 static struct isl_mat
*parameter_compression_multi(struct isl_ctx
*ctx
,
176 struct isl_mat
*B
, struct isl_vec
*d
)
181 struct isl_mat
*A
= NULL
, *U
= NULL
;
187 isl_vec_lcm(ctx
, d
, &D
);
190 A
= isl_mat_alloc(ctx
, size
, B
->n_row
* size
);
191 U
= isl_mat_alloc(ctx
, size
, size
);
194 for (i
= 0; i
< B
->n_row
; ++i
) {
195 isl_seq_cpy(U
->row
[0], B
->row
[i
] + 1, size
);
196 U
= unimodular_complete(ctx
, U
, 1);
199 isl_int_divexact(D
, D
, d
->block
.data
[i
]);
200 for (k
= 0; k
< U
->n_col
; ++k
)
201 isl_int_mul(A
->row
[k
][i
*size
+0], D
, U
->row
[0][k
]);
202 isl_int_mul(D
, D
, d
->block
.data
[i
]);
203 for (j
= 1; j
< U
->n_row
; ++j
)
204 for (k
= 0; k
< U
->n_col
; ++k
)
205 isl_int_mul(A
->row
[k
][i
*size
+j
],
208 A
= isl_mat_left_hermite(ctx
, A
, 0, NULL
, NULL
);
209 T
= isl_mat_sub_alloc(ctx
, A
->row
, 0, A
->n_row
, 0, A
->n_row
);
210 T
= isl_mat_lin_to_aff(ctx
, T
);
211 isl_int_set(T
->row
[0][0], D
);
212 T
= isl_mat_right_inverse(ctx
, T
);
213 isl_assert(ctx
, isl_int_is_one(T
->row
[0][0]), goto error
);
214 T
= isl_mat_transpose(ctx
, T
);
215 isl_mat_free(ctx
, A
);
216 isl_mat_free(ctx
, U
);
221 isl_mat_free(ctx
, A
);
222 isl_mat_free(ctx
, U
);
227 /* Given a set of modulo constraints
231 * this function returns an affine transformation T,
235 * that bijectively maps the integer vectors y' to integer
236 * vectors y that satisfy the modulo constraints.
238 * This function is inspired by Section 2.5.3
239 * of B. Meister, "Stating and Manipulating Periodicity in the Polytope
240 * Model. Applications to Program Analysis and Optimization".
241 * However, the implementation only follows the algorithm of that
242 * section for computing a particular solution and not for computing
243 * a general homogeneous solution. The latter is incomplete and
244 * may remove some valid solutions.
245 * Instead, we use an adaptation of the algorithm in Section 7 of
246 * B. Meister, S. Verdoolaege, "Polynomial Approximations in the Polytope
247 * Model: Bringing the Power of Quasi-Polynomials to the Masses".
249 * The input is given as a matrix B = [ c A ] and a vector d.
250 * Each element of the vector d corresponds to a row in B.
251 * The output is a lower triangular matrix.
252 * If no integer vector y satisfies the given constraints then
253 * a matrix with zero columns is returned.
255 * We first compute a particular solution y_0 to the given set of
256 * modulo constraints in particular_solution. If no such solution
257 * exists, then we return a zero-columned transformation matrix.
258 * Otherwise, we compute the generic solution to
262 * That is we want to compute G such that
266 * with y'' integer, describes the set of solutions.
268 * We first remove the common factors of each row.
269 * In particular if gcd(A_i,d_i) != 1, then we divide the whole
270 * row i (including d_i) by this common factor. If afterwards gcd(A_i) != 1,
271 * then we divide this row of A by the common factor, unless gcd(A_i) = 0.
272 * In the later case, we simply drop the row (in both A and d).
274 * If there are no rows left in A, the G is the identity matrix. Otherwise,
275 * for each row i, we now determine the lattice of integer vectors
276 * that satisfies this row. Let U_i be the unimodular extension of the
277 * row A_i. This unimodular extension exists because gcd(A_i) = 1.
278 * The first component of
282 * needs to be a multiple of d_i. Let y' = diag(d_i, 1, ..., 1) y''.
285 * y = U_i^{-1} diag(d_i, 1, ..., 1) y''
287 * for arbitrary integer vectors y''. That is, y belongs to the lattice
288 * generated by the columns of L_i = U_i^{-1} diag(d_i, 1, ..., 1).
289 * If there is only one row, then G = L_1.
291 * If there is more than one row left, we need to compute the intersection
292 * of the lattices. That is, we need to compute an L such that
294 * L = L_i L_i' for all i
296 * with L_i' some integer matrices. Let A be constructed as follows
298 * A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
300 * and computed the Hermite Normal Form of A = [ H 0 ] U
303 * L_i^{-T} = H U_{1,i}
307 * H^{-T} = L_i U_{1,i}^T
309 * In other words G = L = H^{-T}.
310 * To ensure that G is lower triangular, we compute and use its Hermite
313 * The affine transformation matrix returned is then
318 * as any y = y_0 + G y' with y' integer is a solution to the original
319 * modulo constraints.
321 struct isl_mat
*isl_mat_parameter_compression(struct isl_ctx
*ctx
,
322 struct isl_mat
*B
, struct isl_vec
*d
)
325 struct isl_mat
*cst
= NULL
;
326 struct isl_mat
*T
= NULL
;
331 isl_assert(ctx
, B
->n_row
== d
->size
, goto error
);
332 cst
= particular_solution(ctx
, B
, d
);
335 if (cst
->n_col
== 0) {
336 T
= isl_mat_alloc(ctx
, B
->n_col
, 0);
337 isl_mat_free(ctx
, cst
);
338 isl_mat_free(ctx
, B
);
339 isl_vec_free(ctx
, d
);
343 /* Replace a*g*row = 0 mod g*m by row = 0 mod m */
344 for (i
= 0; i
< B
->n_row
; ++i
) {
345 isl_seq_gcd(B
->row
[i
] + 1, B
->n_col
- 1, &D
);
346 if (isl_int_is_one(D
))
348 if (isl_int_is_zero(D
)) {
349 B
= isl_mat_drop_rows(ctx
, B
, i
, 1);
350 d
= isl_vec_cow(ctx
, d
);
353 isl_seq_cpy(d
->block
.data
+i
, d
->block
.data
+i
+1,
359 B
= isl_mat_cow(ctx
, B
);
362 isl_seq_scale_down(B
->row
[i
] + 1, B
->row
[i
] + 1, D
, B
->n_col
-1);
363 isl_int_gcd(D
, D
, d
->block
.data
[i
]);
364 d
= isl_vec_cow(ctx
, d
);
367 isl_int_divexact(d
->block
.data
[i
], d
->block
.data
[i
], D
);
371 T
= isl_mat_identity(ctx
, B
->n_col
);
372 else if (B
->n_row
== 1)
373 T
= parameter_compression_1(ctx
, B
, d
);
375 T
= parameter_compression_multi(ctx
, B
, d
);
376 T
= isl_mat_left_hermite(ctx
, T
, 0, NULL
, NULL
);
379 isl_mat_sub_copy(ctx
, T
->row
+ 1, cst
->row
, cst
->n_row
, 0, 0, 1);
380 isl_mat_free(ctx
, cst
);
381 isl_mat_free(ctx
, B
);
382 isl_vec_free(ctx
, d
);
387 isl_mat_free(ctx
, cst
);
388 isl_mat_free(ctx
, B
);
389 isl_vec_free(ctx
, d
);
393 /* Given a set of equalities
397 * this function computes unimodular transformation from a lower-dimensional
398 * space to the original space that bijectively maps the integer points x'
399 * in the lower-dimensional space to the integer points x in the original
400 * space that satisfy the equalities.
402 * The input is given as a matrix B = [ -c M ] and the out is a
403 * matrix that maps [1 x'] to [1 x].
404 * If T2 is not NULL, then *T2 is set to a matrix mapping [1 x] to [1 x'].
406 * First compute the (left) Hermite normal form of M,
408 * M [U1 U2] = M U = H = [H1 0]
410 * M = H Q = [H1 0] [Q1]
413 * with U, Q unimodular, Q = U^{-1} (and H lower triangular).
414 * Define the transformed variables as
416 * x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
419 * The equalities then become
421 * H1 x1' - c = 0 or x1' = H1^{-1} c = c'
423 * If any of the c' is non-integer, then the original set has no
424 * integer solutions (since the x' are a unimodular transformation
426 * Otherwise, the transformation is given by
428 * x = U1 H1^{-1} c + U2 x2'
430 * The inverse transformation is simply
434 struct isl_mat
*isl_mat_variable_compression(struct isl_ctx
*ctx
,
435 struct isl_mat
*B
, struct isl_mat
**T2
)
438 struct isl_mat
*H
= NULL
, *C
= NULL
, *H1
, *U
= NULL
, *U1
, *U2
, *TC
;
447 H
= isl_mat_sub_alloc(ctx
, B
->row
, 0, B
->n_row
, 1, dim
);
448 H
= isl_mat_left_hermite(ctx
, H
, 0, &U
, T2
);
449 if (!H
|| !U
|| (T2
&& !*T2
))
452 *T2
= isl_mat_drop_rows(ctx
, *T2
, 0, B
->n_row
);
453 *T2
= isl_mat_lin_to_aff(ctx
, *T2
);
457 C
= isl_mat_alloc(ctx
, 1+B
->n_row
, 1);
460 isl_int_set_si(C
->row
[0][0], 1);
461 isl_mat_sub_neg(ctx
, C
->row
+1, B
->row
, B
->n_row
, 0, 0, 1);
462 H1
= isl_mat_sub_alloc(ctx
, H
->row
, 0, H
->n_row
, 0, H
->n_row
);
463 H1
= isl_mat_lin_to_aff(ctx
, H1
);
464 TC
= isl_mat_inverse_product(ctx
, H1
, C
);
467 isl_mat_free(ctx
, H
);
468 if (!isl_int_is_one(TC
->row
[0][0])) {
469 for (i
= 0; i
< B
->n_row
; ++i
) {
470 if (!isl_int_is_divisible_by(TC
->row
[1+i
][0], TC
->row
[0][0])) {
471 isl_mat_free(ctx
, B
);
472 isl_mat_free(ctx
, TC
);
473 isl_mat_free(ctx
, U
);
475 isl_mat_free(ctx
, *T2
);
478 return isl_mat_alloc(ctx
, 1 + dim
, 0);
480 isl_seq_scale_down(TC
->row
[1+i
], TC
->row
[1+i
], TC
->row
[0][0], 1);
482 isl_int_set_si(TC
->row
[0][0], 1);
484 U1
= isl_mat_sub_alloc(ctx
, U
->row
, 0, U
->n_row
, 0, B
->n_row
);
485 U1
= isl_mat_lin_to_aff(ctx
, U1
);
486 U2
= isl_mat_sub_alloc(ctx
, U
->row
, 0, U
->n_row
,
487 B
->n_row
, U
->n_row
- B
->n_row
);
488 U2
= isl_mat_lin_to_aff(ctx
, U2
);
489 isl_mat_free(ctx
, U
);
490 TC
= isl_mat_product(ctx
, U1
, TC
);
491 TC
= isl_mat_aff_direct_sum(ctx
, TC
, U2
);
493 isl_mat_free(ctx
, B
);
497 isl_mat_free(ctx
, B
);
498 isl_mat_free(ctx
, H
);
499 isl_mat_free(ctx
, U
);
501 isl_mat_free(ctx
, *T2
);
507 /* Use the n equalities of bset to unimodularly transform the
508 * variables x such that n transformed variables x1' have a constant value
509 * and rewrite the constraints of bset in terms of the remaining
510 * transformed variables x2'. The matrix pointed to by T maps
511 * the new variables x2' back to the original variables x, while T2
512 * maps the original variables to the new variables.
514 static struct isl_basic_set
*compress_variables(struct isl_ctx
*ctx
,
515 struct isl_basic_set
*bset
, struct isl_mat
**T
, struct isl_mat
**T2
)
517 struct isl_mat
*B
, *TC
;
526 isl_assert(ctx
, isl_basic_set_n_param(bset
) == 0, goto error
);
527 isl_assert(ctx
, bset
->n_div
== 0, goto error
);
528 dim
= isl_basic_set_n_dim(bset
);
529 isl_assert(ctx
, bset
->n_eq
<= dim
, goto error
);
533 B
= isl_mat_sub_alloc(ctx
, bset
->eq
, 0, bset
->n_eq
, 0, 1 + dim
);
534 TC
= isl_mat_variable_compression(ctx
, B
, T2
);
537 if (TC
->n_col
== 0) {
538 isl_mat_free(ctx
, TC
);
540 isl_mat_free(ctx
, *T2
);
543 return isl_basic_set_set_to_empty(bset
);
546 bset
= isl_basic_set_preimage(bset
, T
? isl_mat_copy(ctx
, TC
) : TC
);
551 isl_basic_set_free(bset
);
555 struct isl_basic_set
*isl_basic_set_remove_equalities(
556 struct isl_basic_set
*bset
, struct isl_mat
**T
, struct isl_mat
**T2
)
564 isl_assert(bset
->ctx
, isl_basic_set_n_param(bset
) == 0, goto error
);
565 bset
= isl_basic_set_gauss(bset
, NULL
);
566 if (ISL_F_ISSET(bset
, ISL_BASIC_SET_EMPTY
))
568 bset
= compress_variables(bset
->ctx
, bset
, T
, T2
);
571 isl_basic_set_free(bset
);
576 /* Check if dimension dim belongs to a residue class
577 * i_dim \equiv r mod m
578 * with m != 1 and if so return m in *modulo and r in *residue.
580 int isl_basic_set_dim_residue_class(struct isl_basic_set
*bset
,
581 int pos
, isl_int
*modulo
, isl_int
*residue
)
584 struct isl_mat
*H
= NULL
, *U
= NULL
, *C
, *H1
, *U1
;
588 if (!bset
|| !modulo
|| !residue
)
592 total
= isl_basic_set_total_dim(bset
);
593 nparam
= isl_basic_set_n_param(bset
);
594 H
= isl_mat_sub_alloc(ctx
, bset
->eq
, 0, bset
->n_eq
, 1, total
);
595 H
= isl_mat_left_hermite(ctx
, H
, 0, &U
, NULL
);
599 isl_seq_gcd(U
->row
[nparam
+ pos
]+bset
->n_eq
,
600 total
-bset
->n_eq
, modulo
);
601 if (isl_int_is_zero(*modulo
) || isl_int_is_one(*modulo
)) {
602 isl_int_set_si(*residue
, 0);
603 isl_mat_free(ctx
, H
);
604 isl_mat_free(ctx
, U
);
608 C
= isl_mat_alloc(ctx
, 1+bset
->n_eq
, 1);
611 isl_int_set_si(C
->row
[0][0], 1);
612 isl_mat_sub_neg(ctx
, C
->row
+1, bset
->eq
, bset
->n_eq
, 0, 0, 1);
613 H1
= isl_mat_sub_alloc(ctx
, H
->row
, 0, H
->n_row
, 0, H
->n_row
);
614 H1
= isl_mat_lin_to_aff(ctx
, H1
);
615 C
= isl_mat_inverse_product(ctx
, H1
, C
);
616 isl_mat_free(ctx
, H
);
617 U1
= isl_mat_sub_alloc(ctx
, U
->row
, nparam
+pos
, 1, 0, bset
->n_eq
);
618 U1
= isl_mat_lin_to_aff(ctx
, U1
);
619 isl_mat_free(ctx
, U
);
620 C
= isl_mat_product(ctx
, U1
, C
);
623 if (!isl_int_is_divisible_by(C
->row
[1][0], C
->row
[0][0])) {
624 bset
= isl_basic_set_copy(bset
);
625 bset
= isl_basic_set_set_to_empty(bset
);
626 isl_basic_set_free(bset
);
627 isl_int_set_si(*modulo
, 0);
628 isl_int_set_si(*residue
, 0);
631 isl_int_divexact(*residue
, C
->row
[1][0], C
->row
[0][0]);
632 isl_int_fdiv_r(*residue
, *residue
, *modulo
);
633 isl_mat_free(ctx
, C
);
636 isl_mat_free(ctx
, H
);
637 isl_mat_free(ctx
, U
);