2 * Searches for "good" ways to divide n matchsticks up and reassemble them
3 * into m matchsticks. "Good" means the smallest fragment is as big
8 * The algorithm is faster if the arguments are ordered so that n > m.
12 * matchsticks/main.c Copyright 2014 Ian Jackson
14 * This program is free software: you can redistribute it and/or modify
15 * it under the terms of the GNU General Public License as published by
16 * the Free Software Foundation, either version 3 of the License, or
17 * (at your option) any later version.
19 * This program is distributed in the hope that it will be useful,
20 * but WITHOUT ANY WARRANTY; without even the implied warranty of
21 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
22 * GNU General Public License for more details.
38 #include <sys/types.h>
41 #include <sys/fcntl.h>
46 #define VERSION "(unknown-version)"
52 * Each input match contributes, or does not contribute, to each
53 * output match; we do not need to consider multiple fragments
54 * relating to the same input/output pair this gives an n*m adjacency
55 * matrix (bitmap). Given such an adjacency matrix, the problem of
56 * finding the best sizes for the fragments can be expressed as a
57 * linear programming problem.
59 * We search all possible adjacency matrices, and for each one we run
60 * GLPK's simplex solver. We represent the adjacency matrix as an
61 * array of bitmaps: one word per input stick, with one bit per output
64 * However, there are a couple of wrinkles:
66 * To best represent the problem as a standard LP problem, we separate
67 * out the size of each fragment into a common minimum size variable,
68 * plus a fragment-specific extra size variable. This reduces the LP
69 * problem size at the cost of making the problem construction, and
70 * interpretation of the results, a bit fiddly.
72 * Many of the adjacency matrices are equivalent. In particular,
73 * permutations of the columns, or of the rows, do not change the
74 * meaning. It is only necessasry to consider any one permutation.
75 * We make use of this by considering only adjacency matrices whose
76 * bitmap array contains bitmap words whose numerical values are
77 * nondecreasing in array order.
79 * Once we have a solution, we also avoid considering any candidate
80 * which involves dividing one of the input sticks into so many
81 * fragment that the smallest fragment would necessarily be no bigger
82 * than our best solution. That is, we reject candidates where any of
83 * the hamming weights of the adjacency bitmap words are too large.
85 * And, we want to do the search in order of increasing maximum
86 * hamming weight. This is because in practice optimal solutions tend
87 * to have low hamming weight, and having found a reasonable solution
88 * early allows us to eliminate a lot of candidates without doing the
92 typedef uint32_t AdjWord;
93 #define PRADJ "08"PRIx32
95 #define FOR_BITS(j,m) for (j=0, j##bit=1; j < (m); j++, j##bit<<=1)
97 static int n, m, maxhamweight;
98 static AdjWord *adjmatrix;
99 static AdjWord adjall;
102 static glp_prob *best_prob;
103 static AdjWord *best_adjmatrix;
105 static int n_max_frags, m_max_frags;
108 static unsigned printcounter;
110 static void iterate(void);
111 static void iterate_recurse(int i, AdjWord min);
112 static bool preconsider_ok(int nwords, bool doprint);
113 static bool maxhamweight_ok(void);
114 static void optimise(bool doprint);
116 static void progress_eol(void) {
117 fprintf(stderr," \r");
121 static void set_best(double new_best) {
124 * When computing n_max_frags, we want to set a value that will skip
125 * anything that won't provide strictly better solutions. So we
129 * <=> frags < | n / best |
131 * <=> frags <= | n / best | - 1
133 n_max_frags = ceil(n / best) - 1;
134 m_max_frags = ceil(m / best) - 1;
137 /*----- multicore support -----*/
148 * - one pipe ("work") from generator to workers
149 * - ever-extending file ("bus") containing new "best" values
150 * - one file for each worker giving maxhamweight and adjmatrix for best
152 * generator runs iterate_recurse to a certain depth and writes the
153 * candidates to a pipe
155 * workers read candidates from the pipe and resume iterate_recurse
156 * halfway through the recursion
158 * whenever a worker does a doprint, it checks the bus for new best
159 * value; actual best values are appended
161 * master waits for generator and all workers to finish and then
162 * runs optimise() for each worker's best, then prints
165 static int ncpus = 0, multicore_iteration_boundary = INT_MAX;
167 static int mc_bus, mc_work[2];
168 static off_t mc_bus_read;
175 static Worker *mc_us;
176 static bool mc_am_generator;
178 static void multicore_check_for_new_best(void);
181 static AdjWord mc_iter_min;
183 static size_t mc_iovlen;
184 static struct iovec mc_iov[MAX_NIOVS];
186 #define IOV0 (mc_niovs = mc_iovlen = 0)
188 #define IOV(obj, count) ({ \
189 assert(mc_niovs < MAX_NIOVS); \
190 mc_iov[mc_niovs].iov_base = &(obj); \
191 mc_iov[mc_niovs].iov_len = sizeof(obj) * (count); \
192 mc_iovlen += mc_iov[mc_niovs].iov_len; \
196 static void mc_rwvsetup_outer(void) {
198 IOV(maxhamweight, 1);
200 IOV(*adjmatrix, multicore_iteration_boundary);
204 static void mc_rwvsetup_full(void) {
209 static void vlprintf(const char *fmt, va_list al) {
210 vfprintf(stderr,fmt,al);
214 static void LPRINTF(const char *fmt, ...) {
221 static void mc_awaitpid(int wnum, pid_t pid) {
222 LPRINTF("master awaiting %2d [%ld]",wnum,(long)pid);
224 pid_t got = waitpid(pid, &status, 0);
227 fprintf(stderr,"\nFAILED SUBPROC %2d [%ld] %d\n",
228 wnum, (long)pid, status);
233 static void multicore_outer_iteration(int i, AdjWord min) {
234 static unsigned check_counter;
236 assert(i == multicore_iteration_boundary);
239 ssize_t r = writev(mc_work[1], mc_iov, mc_niovs);
240 assert(r == mc_iovlen);
241 /* effectively, this writev arranges to transfers control
242 * to some worker's instance of iterate_recurse via mc_iterate_worker */
244 if (!(check_counter++ & 0xff))
245 multicore_check_for_new_best();
248 static void mc_iterate_worker(void) {
251 ssize_t r = readv(mc_work[0], mc_iov, mc_niovs);
253 assert(r == mc_iovlen);
255 bool ok = maxhamweight_ok();
258 ok = preconsider_ok(multicore_iteration_boundary, 1);
262 /* stop iterate_recurse from trying to run multicore_outer_iteration */
263 int mc_org_it_bound = multicore_iteration_boundary;
264 multicore_iteration_boundary = INT_MAX;
265 iterate_recurse(mc_org_it_bound, mc_iter_min);
266 multicore_iteration_boundary = mc_org_it_bound;
268 if (best_adjmatrix) {
269 LPRINTF("worker %2d reporting",mc_us->w);
270 adjmatrix = best_adjmatrix;
272 ssize_t r = writev(fileno(mc_us->results), mc_iov, mc_niovs);
273 assert(r == mc_iovlen);
275 LPRINTF("worker %2d ending",mc_us->w);
279 static void multicore(void) {
284 multicore_iteration_boundary = n / 2;
286 FILE *busf = tmpfile(); assert(busf);
287 mc_bus = fileno(busf);
288 int r = fcntl(mc_bus, F_GETFL); assert(r >= 0);
290 r = fcntl(mc_bus, F_SETFL, r); assert(r >= 0);
292 r = pipe(mc_work); assert(!r);
294 mc_workers = xmalloc(sizeof(*mc_workers) * ncpus);
295 for (w=0; w<ncpus; w++) {
297 mc_workers[w].results = tmpfile(); assert(mc_workers[w].results);
298 mc_workers[w].pid = fork(); assert(mc_workers[w].pid >= 0);
299 if (!mc_workers[w].pid) {
300 mc_us = &mc_workers[w];
302 LPRINTF("worker %2d running", w);
310 genpid = fork(); assert(genpid >= 0);
313 LPRINTF("generator running");
319 mc_awaitpid(-1, genpid);
320 for (w=0; w<ncpus; w++)
321 mc_awaitpid(w, mc_workers[w].pid);
323 for (w=0; w<ncpus; w++) {
325 LPRINTF("reading report from %2d",w);
326 ssize_t sr = preadv(fileno(mc_workers[w].results), mc_iov, mc_niovs, 0);
328 LPRINTF("got report from %2d",w);
334 static void multicore_check_for_new_best(void) {
335 if (!(mc_us || mc_am_generator))
340 ssize_t got = pread(mc_bus, &msg, sizeof(msg), mc_bus_read);
342 assert(got == sizeof(msg));
345 mc_bus_read += sizeof(msg);
349 static void multicore_found_new_best(void) {
353 if (mc_us /* might be master */) fprintf(stderr," w%-2d ",mc_us->w);
354 ssize_t wrote = write(mc_bus, &best, sizeof(best));
355 assert(wrote == sizeof(best));
358 /*----- end of multicore support -----*/
360 static AdjWord *xalloc_adjmatrix(void) {
361 return xmalloc(sizeof(*adjmatrix)*n);
364 static void prep(void) {
365 adjall = ~((~(AdjWord)0) << m);
366 adjmatrix = xalloc_adjmatrix();
367 glp_term_out(GLP_OFF);
369 weight = calloc(sizeof(*weight), m); assert(weight);
370 n_max_frags = INT_MAX;
371 m_max_frags = INT_MAX;
375 static AdjWord one_adj_bit(int bitnum) {
376 return (AdjWord)1 << bitnum;
380 static int count_set_adj_bits(AdjWord w) {
384 total += !!(w & jbit);
388 #define PRINTF(...) if (!doprint) ; else fprintf(stderr, __VA_ARGS__)
390 static int totalfrags;
392 static bool maxhamweight_ok(void) {
393 return maxhamweight <= m_max_frags;
396 static bool preconsider_ok(int nwords, bool doprint) {
399 PRINTF("%2d ", maxhamweight);
402 for (i=0, totalfrags=0; i<nwords; i++) {
403 int frags = count_set_adj_bits(adjmatrix[i]);
404 PRINTF("%"PRADJ" ", adjmatrix[i]);
405 if (frags > m_max_frags) {
409 had_max += (frags >= maxhamweight);
413 /* Skip this candidate as its max hamming weight is lower than
414 * we're currently looking for (which means we must have done it
415 * already). (The recursive iteration ensures that none of the
416 * words have more than the max hamming weight.) */
426 static void optimise(bool doprint) {
427 /* Consider the best answer (if any) for a given adjacency matrix */
433 * Up to a certain point, optimise() can be restarted. We use this
434 * to go back and print the debugging output if it turns out that we
435 * have an interesting case. The HAVE_PRINTED macro does this: its
436 * semantics are to go back in time and make sure that we have
437 * printed the description of the search case.
439 #define HAVE_PRINTED ({ \
440 if (!doprint) { doprint = 1; goto retry_with_print; } \
444 glp_delete_prob(prob);
448 bool ok = preconsider_ok(n, doprint);
453 * We formulate our problem as an LP problem as follows.
454 * In this file "n" and "m" are the matchstick numbers.
456 * Each set bit in the adjacency matrix corresponds to taking a
457 * fragment from old match i and making it part of new match j.
459 * The structural variables (columns) are:
460 * x_minimum minimum size of any fragment (bounded below by 0)
461 * x_morefrag_i_j the amount by which the size of the fragment
462 * i,j exceeds the minimum size (bounded below by 0)
464 * The auxiliary variables (rows) are:
465 * x_total_i total length for each input match (fixed variable)
466 * x_total_j total length for each output match (fixed variable)
468 * The objective function is simply
471 * We use X_ and Y_ to refer to GLPK's (1-based) column and row indices.
472 * ME_ refers to entries in the list of constraint matrix elements
473 * which we build up as we go.
476 prob = glp_create_prob();
478 int Y_totals_i = glp_add_rows(prob, n);
479 int Y_totals_j = glp_add_rows(prob, m);
480 int X_minimum = glp_add_cols(prob, 1);
483 int next_matrix_entry = 1; /* wtf GLPK! */
484 int matrix_entries_size = next_matrix_entry + n + m + totalfrags*2;
485 double matrix_entries[matrix_entries_size];
486 int matrix_entries_XY[2][matrix_entries_size];
488 #define ADD_MATRIX_ENTRY(Y,X) ({ \
489 assert(next_matrix_entry < matrix_entries_size); \
490 matrix_entries_XY[0][next_matrix_entry] = (X); \
491 matrix_entries_XY[1][next_matrix_entry] = (Y); \
492 matrix_entries[next_matrix_entry] = 0; \
493 next_matrix_entry++; \
496 int ME_totals_i__minimum = next_matrix_entry;
497 for (i=0; i<n; i++) ADD_MATRIX_ENTRY(Y_totals_i+i, X_minimum);
499 int ME_totals_j__minimum = next_matrix_entry;
500 for (j=0; j<m; j++) ADD_MATRIX_ENTRY(Y_totals_j+j, X_minimum);
502 /* \forall_i x_total_i = m */
503 /* \forall_i x_total_j = n */
504 for (i=0; i<n; i++) glp_set_row_bnds(prob, Y_totals_i+i, GLP_FX, m,m);
505 for (j=0; j<m; j++) glp_set_row_bnds(prob, Y_totals_j+j, GLP_FX, n,n);
508 glp_set_col_bnds(prob, X_minimum, GLP_LO, 0, 0);
509 glp_set_col_name(prob, X_minimum, "minimum");
511 /* objective is maximising x_minimum */
512 glp_set_obj_dir(prob, GLP_MAX);
513 glp_set_obj_coef(prob, X_minimum, 1);
515 for (i=0; i<n; i++) {
517 if (!(adjmatrix[i] & jbit))
519 /* x_total_i += x_minimum */
520 /* x_total_j += x_minimum */
521 matrix_entries[ ME_totals_i__minimum + i ] ++;
522 matrix_entries[ ME_totals_j__minimum + j ] ++;
524 /* x_morefrag_i_j >= 0 */
525 int X_morefrag_i_j = glp_add_cols(prob, 1);
526 glp_set_col_bnds(prob, X_morefrag_i_j, GLP_LO, 0, 0);
529 snprintf(buf,sizeof(buf),"mf %d,%d",i,j);
530 glp_set_col_name(prob, X_morefrag_i_j, buf);
533 /* x_total_i += x_morefrag_i_j */
534 /* x_total_j += x_morefrag_i_j */
535 int ME_totals_i__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_i+i, X_morefrag_i_j);
536 int ME_totals_j__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_j+j, X_morefrag_i_j);
537 matrix_entries[ME_totals_i__mf_i_j] = 1;
538 matrix_entries[ME_totals_j__mf_i_j] = 1;
542 assert(next_matrix_entry == matrix_entries_size);
544 glp_load_matrix(prob, matrix_entries_size-1,
545 matrix_entries_XY[1], matrix_entries_XY[0],
548 int r = glp_simplex(prob, NULL);
549 PRINTF(" glp=%d", r);
552 case e: PRINTF(" " #e ); goto out;
554 case e: HAVE_PRINTED; printf(" " #e " CRASHING\n"); exit(-1);
556 default: HAVE_PRINTED; printf(" ! CRASHING\n"); exit(-1);
576 r = glp_get_status(prob);
577 PRINTF(" status=%d", r);
589 double got = glp_get_obj_val(prob);
597 multicore_found_new_best();
599 if (best_prob) glp_delete_prob(best_prob);
602 free(best_adjmatrix);
603 best_adjmatrix = xalloc_adjmatrix();
604 memcpy(best_adjmatrix, adjmatrix, sizeof(*adjmatrix)*n);
612 glp_delete_prob(prob);
613 if (doprint) progress_eol();
614 if (doprint) multicore_check_for_new_best();
617 static void iterate_recurse(int i, AdjWord min) {
623 optimise(!(printcounter & 0xfff));
626 if (i >= multicore_iteration_boundary) {
627 multicore_outer_iteration(i, min);
630 for (adjmatrix[i] = min;
633 if (count_set_adj_bits(adjmatrix[i]) > maxhamweight)
635 if (i == 0 && (adjmatrix[i] & (1+adjmatrix[i])))
639 if (adjmatrix[i] & jbit)
641 for (int j = 0; j < m; j++)
642 if (weight[j] >= n_max_frags)
645 iterate_recurse(i+1, adjmatrix[i]);
649 if (adjmatrix[i] & jbit)
653 if (adjmatrix[i] == adjall)
658 static void iterate(void) {
659 for (maxhamweight=1; maxhamweight<=m; maxhamweight++) {
660 if (!maxhamweight_ok())
663 iterate_recurse(0, 1);
667 static void report(void) {
668 fprintf(stderr, "\n");
670 double min = glp_get_obj_val(best_prob);
673 for (i = 0; i < n; i++)
674 for (j = 0; j < m; j++)
676 cols = glp_get_num_cols(best_prob);
677 for (i = 1; i <= cols; i++) {
679 if (2 != sscanf(glp_get_col_name(best_prob, i), "mf %d,%d", &x, &y))
681 a[x][y] = min + glp_get_col_prim(best_prob, i);
683 printf("%d into %d: min fragment %g [%s]\n", n, m, min, VERSION);
684 for (i = 0; i < n; i++) {
685 for (j = 0; j < m; j++) {
687 printf(" %9.3f", a[i][j]);
694 if (ferror(stdout) || fclose(stdout)) { perror("stdout"); exit(-1); }
697 int main(int argc, char **argv) {
699 while ((opt = getopt(argc,argv,"j:")) >= 0) {
701 case 'j': ncpus = atoi(optarg); break;
702 case '+': assert(!"bad option");
714 if (ncpus) multicore();