1 /* glpini01.c */
2
3 /***********************************************************************
4 * This code is part of GLPK (GNU Linear Programming Kit).
5 *
6 * Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,
7 * 2009, 2010 Andrew Makhorin, Department for Applied Informatics,
8 * Moscow Aviation Institute, Moscow, Russia. All rights reserved.
9 * E-mail: <mao@gnu.org>.
10 *
11 * GLPK is free software: you can redistribute it and/or modify it
12 * under the terms of the GNU General Public License as published by
13 * the Free Software Foundation, either version 3 of the License, or
14 * (at your option) any later version.
15 *
16 * GLPK is distributed in the hope that it will be useful, but WITHOUT
17 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
18 * or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
19 * License for more details.
20 *
21 * You should have received a copy of the GNU General Public License
22 * along with GLPK. If not, see <http://www.gnu.org/licenses/>.
23 ***********************************************************************/
24
25 #include "glpapi.h"
26
27 /*----------------------------------------------------------------------
28 -- triang - find maximal triangular part of a rectangular matrix.
29 --
30 -- *Synopsis*
31 --
32 -- int triang(int m, int n,
33 -- void *info, int (*mat)(void *info, int k, int ndx[]),
34 -- int rn[], int cn[]);
35 --
36 -- *Description*
37 --
38 -- For a given rectangular (sparse) matrix A with m rows and n columns
39 -- the routine triang tries to find such permutation matrices P and Q
40 -- that the first rows and columns of the matrix B = P*A*Q form a lower
41 -- triangular submatrix of as greatest size as possible:
42 --
43 -- 1 n
44 -- 1 * . . . . . . x x x x x x
45 -- * * . . . . . x x x x x x
46 -- * * * . . . . x x x x x x
47 -- * * * * . . . x x x x x x
48 -- B = P*A*Q = * * * * * . . x x x x x x
49 -- * * * * * * . x x x x x x
50 -- * * * * * * * x x x x x x
51 -- x x x x x x x x x x x x x
52 -- x x x x x x x x x x x x x
53 -- m x x x x x x x x x x x x x
54 --
55 -- where: '*' - elements of the lower triangular part, '.' - structural
56 -- zeros, 'x' - other (either non-zero or zero) elements.
57 --
58 -- The parameter info is a transit pointer passed to the formal routine
59 -- mat (see below).
60 --
61 -- The formal routine mat specifies the given matrix A in both row- and
62 -- column-wise formats. In order to obtain an i-th row of the matrix A
63 -- the routine triang calls the routine mat with the parameter k = +i,
64 -- 1 <= i <= m. In response the routine mat should store column indices
65 -- of (non-zero) elements of the i-th row to the locations ndx[1], ...,
66 -- ndx[len], where len is number of non-zeros in the i-th row returned
67 -- on exit. Analogously, in order to obtain a j-th column of the matrix
68 -- A, the routine mat is called with the parameter k = -j, 1 <= j <= n,
69 -- and should return pattern of the j-th column in the same way as for
70 -- row patterns. Note that the routine mat may be called more than once
71 -- for the same rows and columns.
72 --
73 -- On exit the routine computes two resultant arrays rn and cn, which
74 -- define the permutation matrices P and Q, respectively. The array rn
75 -- should have at least 1+m locations, where rn[i] = i' (1 <= i <= m)
76 -- means that i-th row of the original matrix A corresponds to i'-th row
77 -- of the matrix B = P*A*Q. Similarly, the array cn should have at least
78 -- 1+n locations, where cn[j] = j' (1 <= j <= n) means that j-th column
79 -- of the matrix A corresponds to j'-th column of the matrix B.
80 --
81 -- *Returns*
82 --
83 -- The routine triang returns the size of the lower tringular part of
84 -- the matrix B = P*A*Q (see the figure above).
85 --
86 -- *Complexity*
87 --
88 -- The time complexity of the routine triang is O(nnz), where nnz is
89 -- number of non-zeros in the given matrix A.
90 --
91 -- *Algorithm*
92 --
93 -- The routine triang starts from the matrix B = P*Q*A, where P and Q
94 -- are unity matrices, so initially B = A.
95 --
96 -- Before the next iteration B = (B1 | B2 | B3), where B1 is partially
97 -- built a lower triangular submatrix, B2 is the active submatrix, and
98 -- B3 is a submatrix that contains rejected columns. Thus, the current
99 -- matrix B looks like follows (initially k1 = 1 and k2 = n):
100 --
101 -- 1 k1 k2 n
102 -- 1 x . . . . . . . . . . . . . # # #
103 -- x x . . . . . . . . . . . . # # #
104 -- x x x . . . . . . . . . . # # # #
105 -- x x x x . . . . . . . . . # # # #
106 -- x x x x x . . . . . . . # # # # #
107 -- k1 x x x x x * * * * * * * # # # # #
108 -- x x x x x * * * * * * * # # # # #
109 -- x x x x x * * * * * * * # # # # #
110 -- x x x x x * * * * * * * # # # # #
111 -- m x x x x x * * * * * * * # # # # #
112 -- <--B1---> <----B2-----> <---B3-->
113 --
114 -- On each iteartion the routine looks for a singleton row, i.e. some
115 -- row that has the only non-zero in the active submatrix B2. If such
116 -- row exists and the corresponding non-zero is b[i,j], where (by the
117 -- definition) k1 <= i <= m and k1 <= j <= k2, the routine permutes
118 -- k1-th and i-th rows and k1-th and j-th columns of the matrix B (in
119 -- order to place the element in the position b[k1,k1]), removes the
120 -- k1-th column from the active submatrix B2, and adds this column to
121 -- the submatrix B1. If no row singletons exist, but B2 is not empty
122 -- yet, the routine chooses a j-th column, which has maximal number of
123 -- non-zeros among other columns of B2, removes this column from B2 and
124 -- adds it to the submatrix B3 in the hope that new row singletons will
125 -- appear in the active submatrix. */
126
triang(int m,int n,void * info,int (* mat)(void * info,int k,int ndx[]),int rn[],int cn[])127 static int triang(int m, int n,
128 void *info, int (*mat)(void *info, int k, int ndx[]),
129 int rn[], int cn[])
130 { int *ndx; /* int ndx[1+max(m,n)]; */
131 /* this array is used for querying row and column patterns of the
132 given matrix A (the third parameter to the routine mat) */
133 int *rs_len; /* int rs_len[1+m]; */
134 /* rs_len[0] is not used;
135 rs_len[i], 1 <= i <= m, is number of non-zeros in the i-th row
136 of the matrix A, which (non-zeros) belong to the current active
137 submatrix */
138 int *rs_head; /* int rs_head[1+n]; */
139 /* rs_head[len], 0 <= len <= n, is the number i of the first row
140 of the matrix A, for which rs_len[i] = len */
141 int *rs_prev; /* int rs_prev[1+m]; */
142 /* rs_prev[0] is not used;
143 rs_prev[i], 1 <= i <= m, is a number i' of the previous row of
144 the matrix A, for which rs_len[i] = rs_len[i'] (zero marks the
145 end of this linked list) */
146 int *rs_next; /* int rs_next[1+m]; */
147 /* rs_next[0] is not used;
148 rs_next[i], 1 <= i <= m, is a number i' of the next row of the
149 matrix A, for which rs_len[i] = rs_len[i'] (zero marks the end
150 this linked list) */
151 int cs_head;
152 /* is a number j of the first column of the matrix A, which has
153 maximal number of non-zeros among other columns */
154 int *cs_prev; /* cs_prev[1+n]; */
155 /* cs_prev[0] is not used;
156 cs_prev[j], 1 <= j <= n, is a number of the previous column of
157 the matrix A with the same or greater number of non-zeros than
158 in the j-th column (zero marks the end of this linked list) */
159 int *cs_next; /* cs_next[1+n]; */
160 /* cs_next[0] is not used;
161 cs_next[j], 1 <= j <= n, is a number of the next column of
162 the matrix A with the same or lesser number of non-zeros than
163 in the j-th column (zero marks the end of this linked list) */
164 int i, j, ii, jj, k1, k2, len, t, size = 0;
165 int *head, *rn_inv, *cn_inv;
166 if (!(m > 0 && n > 0))
167 xerror("triang: m = %d; n = %d; invalid dimension\n", m, n);
168 /* allocate working arrays */
169 ndx = xcalloc(1+(m >= n ? m : n), sizeof(int));
170 rs_len = xcalloc(1+m, sizeof(int));
171 rs_head = xcalloc(1+n, sizeof(int));
172 rs_prev = xcalloc(1+m, sizeof(int));
173 rs_next = xcalloc(1+m, sizeof(int));
174 cs_prev = xcalloc(1+n, sizeof(int));
175 cs_next = xcalloc(1+n, sizeof(int));
176 /* build linked lists of columns of the matrix A with the same
177 number of non-zeros */
178 head = rs_len; /* currently rs_len is used as working array */
179 for (len = 0; len <= m; len ++) head[len] = 0;
180 for (j = 1; j <= n; j++)
181 { /* obtain length of the j-th column */
182 len = mat(info, -j, ndx);
183 xassert(0 <= len && len <= m);
184 /* include the j-th column in the corresponding linked list */
185 cs_prev[j] = head[len];
186 head[len] = j;
187 }
188 /* merge all linked lists of columns in one linked list, where
189 columns are ordered by descending of their lengths */
190 cs_head = 0;
191 for (len = 0; len <= m; len++)
192 { for (j = head[len]; j != 0; j = cs_prev[j])
193 { cs_next[j] = cs_head;
194 cs_head = j;
195 }
196 }
197 jj = 0;
198 for (j = cs_head; j != 0; j = cs_next[j])
199 { cs_prev[j] = jj;
200 jj = j;
201 }
202 /* build initial doubly linked lists of rows of the matrix A with
203 the same number of non-zeros */
204 for (len = 0; len <= n; len++) rs_head[len] = 0;
205 for (i = 1; i <= m; i++)
206 { /* obtain length of the i-th row */
207 rs_len[i] = len = mat(info, +i, ndx);
208 xassert(0 <= len && len <= n);
209 /* include the i-th row in the correspondng linked list */
210 rs_prev[i] = 0;
211 rs_next[i] = rs_head[len];
212 if (rs_next[i] != 0) rs_prev[rs_next[i]] = i;
213 rs_head[len] = i;
214 }
215 /* initially all rows and columns of the matrix A are active */
216 for (i = 1; i <= m; i++) rn[i] = 0;
217 for (j = 1; j <= n; j++) cn[j] = 0;
218 /* set initial bounds of the active submatrix */
219 k1 = 1, k2 = n;
220 /* main loop starts here */
221 while (k1 <= k2)
222 { i = rs_head[1];
223 if (i != 0)
224 { /* the i-th row of the matrix A is a row singleton, since
225 it has the only non-zero in the active submatrix */
226 xassert(rs_len[i] == 1);
227 /* determine the number j of an active column of the matrix
228 A, in which this non-zero is placed */
229 j = 0;
230 t = mat(info, +i, ndx);
231 xassert(0 <= t && t <= n);
232 for (t = t; t >= 1; t--)
233 { jj = ndx[t];
234 xassert(1 <= jj && jj <= n);
235 if (cn[jj] == 0)
236 { xassert(j == 0);
237 j = jj;
238 }
239 }
240 xassert(j != 0);
241 /* the singleton is a[i,j]; move a[i,j] to the position
242 b[k1,k1] of the matrix B */
243 rn[i] = cn[j] = k1;
244 /* shift the left bound of the active submatrix */
245 k1++;
246 /* increase the size of the lower triangular part */
247 size++;
248 }
249 else
250 { /* the current active submatrix has no row singletons */
251 /* remove an active column with maximal number of non-zeros
252 from the active submatrix */
253 j = cs_head;
254 xassert(j != 0);
255 cn[j] = k2;
256 /* shift the right bound of the active submatrix */
257 k2--;
258 }
259 /* the j-th column of the matrix A has been removed from the
260 active submatrix */
261 /* remove the j-th column from the linked list */
262 if (cs_prev[j] == 0)
263 cs_head = cs_next[j];
264 else
265 cs_next[cs_prev[j]] = cs_next[j];
266 if (cs_next[j] == 0)
267 /* nop */;
268 else
269 cs_prev[cs_next[j]] = cs_prev[j];
270 /* go through non-zeros of the j-th columns and update active
271 lengths of the corresponding rows */
272 t = mat(info, -j, ndx);
273 xassert(0 <= t && t <= m);
274 for (t = t; t >= 1; t--)
275 { i = ndx[t];
276 xassert(1 <= i && i <= m);
277 /* the non-zero a[i,j] has left the active submatrix */
278 len = rs_len[i];
279 xassert(len >= 1);
280 /* remove the i-th row from the linked list of rows with
281 active length len */
282 if (rs_prev[i] == 0)
283 rs_head[len] = rs_next[i];
284 else
285 rs_next[rs_prev[i]] = rs_next[i];
286 if (rs_next[i] == 0)
287 /* nop */;
288 else
289 rs_prev[rs_next[i]] = rs_prev[i];
290 /* decrease the active length of the i-th row */
291 rs_len[i] = --len;
292 /* return the i-th row to the corresponding linked list */
293 rs_prev[i] = 0;
294 rs_next[i] = rs_head[len];
295 if (rs_next[i] != 0) rs_prev[rs_next[i]] = i;
296 rs_head[len] = i;
297 }
298 }
299 /* other rows of the matrix A, which are still active, correspond
300 to rows k1, ..., m of the matrix B (in arbitrary order) */
301 for (i = 1; i <= m; i++) if (rn[i] == 0) rn[i] = k1++;
302 /* but for columns this is not needed, because now the submatrix
303 B2 has no columns */
304 for (j = 1; j <= n; j++) xassert(cn[j] != 0);
305 /* perform some optional checks */
306 /* make sure that rn is a permutation of {1, ..., m} and cn is a
307 permutation of {1, ..., n} */
308 rn_inv = rs_len; /* used as working array */
309 for (ii = 1; ii <= m; ii++) rn_inv[ii] = 0;
310 for (i = 1; i <= m; i++)
311 { ii = rn[i];
312 xassert(1 <= ii && ii <= m);
313 xassert(rn_inv[ii] == 0);
314 rn_inv[ii] = i;
315 }
316 cn_inv = rs_head; /* used as working array */
317 for (jj = 1; jj <= n; jj++) cn_inv[jj] = 0;
318 for (j = 1; j <= n; j++)
319 { jj = cn[j];
320 xassert(1 <= jj && jj <= n);
321 xassert(cn_inv[jj] == 0);
322 cn_inv[jj] = j;
323 }
324 /* make sure that the matrix B = P*A*Q really has the form, which
325 was declared */
326 for (ii = 1; ii <= size; ii++)
327 { int diag = 0;
328 i = rn_inv[ii];
329 t = mat(info, +i, ndx);
330 xassert(0 <= t && t <= n);
331 for (t = t; t >= 1; t--)
332 { j = ndx[t];
333 xassert(1 <= j && j <= n);
334 jj = cn[j];
335 if (jj <= size) xassert(jj <= ii);
336 if (jj == ii)
337 { xassert(!diag);
338 diag = 1;
339 }
340 }
341 xassert(diag);
342 }
343 /* free working arrays */
344 xfree(ndx);
345 xfree(rs_len);
346 xfree(rs_head);
347 xfree(rs_prev);
348 xfree(rs_next);
349 xfree(cs_prev);
350 xfree(cs_next);
351 /* return to the calling program */
352 return size;
353 }
354
355 /*----------------------------------------------------------------------
356 -- adv_basis - construct advanced initial LP basis.
357 --
358 -- *Synopsis*
359 --
360 -- #include "glpini.h"
361 -- void adv_basis(glp_prob *lp);
362 --
363 -- *Description*
364 --
365 -- The routine adv_basis constructs an advanced initial basis for an LP
366 -- problem object, which the parameter lp points to.
367 --
368 -- In order to build the initial basis the routine does the following:
369 --
370 -- 1) includes in the basis all non-fixed auxiliary variables;
371 --
372 -- 2) includes in the basis as many as possible non-fixed structural
373 -- variables preserving triangular form of the basis matrix;
374 --
375 -- 3) includes in the basis appropriate (fixed) auxiliary variables
376 -- in order to complete the basis.
377 --
378 -- As a result the initial basis has minimum of fixed variables and the
379 -- corresponding basis matrix is triangular. */
380
mat(void * info,int k,int ndx[])381 static int mat(void *info, int k, int ndx[])
382 { /* this auxiliary routine returns the pattern of a given row or
383 a given column of the augmented constraint matrix A~ = (I|-A),
384 in which columns of fixed variables are implicitly cleared */
385 LPX *lp = info;
386 int m = lpx_get_num_rows(lp);
387 int n = lpx_get_num_cols(lp);
388 int typx, i, j, lll, len = 0;
389 if (k > 0)
390 { /* the pattern of the i-th row is required */
391 i = +k;
392 xassert(1 <= i && i <= m);
393 #if 0 /* 22/XII-2003 */
394 /* if the auxiliary variable x[i] is non-fixed, include its
395 element (placed in the i-th column) in the pattern */
396 lpx_get_row_bnds(lp, i, &typx, NULL, NULL);
397 if (typx != LPX_FX) ndx[++len] = i;
398 /* include in the pattern elements placed in columns, which
399 correspond to non-fixed structural varables */
400 i_beg = aa_ptr[i];
401 i_end = i_beg + aa_len[i] - 1;
402 for (i_ptr = i_beg; i_ptr <= i_end; i_ptr++)
403 { j = m + sv_ndx[i_ptr];
404 lpx_get_col_bnds(lp, j-m, &typx, NULL, NULL);
405 if (typx != LPX_FX) ndx[++len] = j;
406 }
407 #else
408 lll = lpx_get_mat_row(lp, i, ndx, NULL);
409 for (k = 1; k <= lll; k++)
410 { lpx_get_col_bnds(lp, ndx[k], &typx, NULL, NULL);
411 if (typx != LPX_FX) ndx[++len] = m + ndx[k];
412 }
413 lpx_get_row_bnds(lp, i, &typx, NULL, NULL);
414 if (typx != LPX_FX) ndx[++len] = i;
415 #endif
416 }
417 else
418 { /* the pattern of the j-th column is required */
419 j = -k;
420 xassert(1 <= j && j <= m+n);
421 /* if the (auxiliary or structural) variable x[j] is fixed,
422 the pattern of its column is empty */
423 if (j <= m)
424 lpx_get_row_bnds(lp, j, &typx, NULL, NULL);
425 else
426 lpx_get_col_bnds(lp, j-m, &typx, NULL, NULL);
427 if (typx != LPX_FX)
428 { if (j <= m)
429 { /* x[j] is non-fixed auxiliary variable */
430 ndx[++len] = j;
431 }
432 else
433 { /* x[j] is non-fixed structural variables */
434 #if 0 /* 22/XII-2003 */
435 j_beg = aa_ptr[j];
436 j_end = j_beg + aa_len[j] - 1;
437 for (j_ptr = j_beg; j_ptr <= j_end; j_ptr++)
438 ndx[++len] = sv_ndx[j_ptr];
439 #else
440 len = lpx_get_mat_col(lp, j-m, ndx, NULL);
441 #endif
442 }
443 }
444 }
445 /* return the length of the row/column pattern */
446 return len;
447 }
448
adv_basis(glp_prob * lp)449 static void adv_basis(glp_prob *lp)
450 { int m = lpx_get_num_rows(lp);
451 int n = lpx_get_num_cols(lp);
452 int i, j, jj, k, size;
453 int *rn, *cn, *rn_inv, *cn_inv;
454 int typx, *tagx = xcalloc(1+m+n, sizeof(int));
455 double lb, ub;
456 xprintf("Constructing initial basis...\n");
457 #if 0 /* 13/V-2009 */
458 if (m == 0)
459 xerror("glp_adv_basis: problem has no rows\n");
460 if (n == 0)
461 xerror("glp_adv_basis: problem has no columns\n");
462 #else
463 if (m == 0 || n == 0)
464 { glp_std_basis(lp);
465 return;
466 }
467 #endif
468 /* use the routine triang (see above) to find maximal triangular
469 part of the augmented constraint matrix A~ = (I|-A); in order
470 to prevent columns of fixed variables to be included in the
471 triangular part, such columns are implictly removed from the
472 matrix A~ by the routine adv_mat */
473 rn = xcalloc(1+m, sizeof(int));
474 cn = xcalloc(1+m+n, sizeof(int));
475 size = triang(m, m+n, lp, mat, rn, cn);
476 if (lpx_get_int_parm(lp, LPX_K_MSGLEV) >= 3)
477 xprintf("Size of triangular part = %d\n", size);
478 /* the first size rows and columns of the matrix P*A~*Q (where
479 P and Q are permutation matrices defined by the arrays rn and
480 cn) form a lower triangular matrix; build the arrays (rn_inv
481 and cn_inv), which define the matrices inv(P) and inv(Q) */
482 rn_inv = xcalloc(1+m, sizeof(int));
483 cn_inv = xcalloc(1+m+n, sizeof(int));
484 for (i = 1; i <= m; i++) rn_inv[rn[i]] = i;
485 for (j = 1; j <= m+n; j++) cn_inv[cn[j]] = j;
486 /* include the columns of the matrix A~, which correspond to the
487 first size columns of the matrix P*A~*Q, in the basis */
488 for (k = 1; k <= m+n; k++) tagx[k] = -1;
489 for (jj = 1; jj <= size; jj++)
490 { j = cn_inv[jj];
491 /* the j-th column of A~ is the jj-th column of P*A~*Q */
492 tagx[j] = LPX_BS;
493 }
494 /* if size < m, we need to add appropriate columns of auxiliary
495 variables to the basis */
496 for (jj = size + 1; jj <= m; jj++)
497 { /* the jj-th column of P*A~*Q should be replaced by the column
498 of the auxiliary variable, for which the only unity element
499 is placed in the position [jj,jj] */
500 i = rn_inv[jj];
501 /* the jj-th row of P*A~*Q is the i-th row of A~, but in the
502 i-th row of A~ the unity element belongs to the i-th column
503 of A~; therefore the disired column corresponds to the i-th
504 auxiliary variable (note that this column doesn't belong to
505 the triangular part found by the routine triang) */
506 xassert(1 <= i && i <= m);
507 xassert(cn[i] > size);
508 tagx[i] = LPX_BS;
509 }
510 /* free working arrays */
511 xfree(rn);
512 xfree(cn);
513 xfree(rn_inv);
514 xfree(cn_inv);
515 /* build tags of non-basic variables */
516 for (k = 1; k <= m+n; k++)
517 { if (tagx[k] != LPX_BS)
518 { if (k <= m)
519 lpx_get_row_bnds(lp, k, &typx, &lb, &ub);
520 else
521 lpx_get_col_bnds(lp, k-m, &typx, &lb, &ub);
522 switch (typx)
523 { case LPX_FR:
524 tagx[k] = LPX_NF; break;
525 case LPX_LO:
526 tagx[k] = LPX_NL; break;
527 case LPX_UP:
528 tagx[k] = LPX_NU; break;
529 case LPX_DB:
530 tagx[k] =
531 (fabs(lb) <= fabs(ub) ? LPX_NL : LPX_NU);
532 break;
533 case LPX_FX:
534 tagx[k] = LPX_NS; break;
535 default:
536 xassert(typx != typx);
537 }
538 }
539 }
540 for (k = 1; k <= m+n; k++)
541 { if (k <= m)
542 lpx_set_row_stat(lp, k, tagx[k]);
543 else
544 lpx_set_col_stat(lp, k-m, tagx[k]);
545 }
546 xfree(tagx);
547 return;
548 }
549
550 /***********************************************************************
551 * NAME
552 *
553 * glp_adv_basis - construct advanced initial LP basis
554 *
555 * SYNOPSIS
556 *
557 * void glp_adv_basis(glp_prob *lp, int flags);
558 *
559 * DESCRIPTION
560 *
561 * The routine glp_adv_basis constructs an advanced initial basis for
562 * the specified problem object.
563 *
564 * The parameter flags is reserved for use in the future and must be
565 * specified as zero. */
566
glp_adv_basis(glp_prob * lp,int flags)567 void glp_adv_basis(glp_prob *lp, int flags)
568 { if (flags != 0)
569 xerror("glp_adv_basis: flags = %d; invalid flags\n", flags);
570 if (lp->m == 0 || lp->n == 0)
571 glp_std_basis(lp);
572 else
573 adv_basis(lp);
574 return;
575 }
576
577 /* eof */
578