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
63 * However, there are a couple of wrinkles:
65 * To best represent the problem as a standard LP problem, we separate
66 * out the size of each fragment into a common minimum size variable,
67 * plus a fragment-specific extra size variable. This reduces the LP
68 * problem size at the cost of making the problem construction, and
69 * interpretation of the results, a bit fiddly.
71 * Many of the adjacency matrices are equivalent. In particular,
72 * permutations of the columns, or of the rows, do not change the
73 * meaning. It is only necessasry to consider any one permutation.
74 * We make use of this by considering only adjacency matrices whose
75 * bitmap array contains bitmap words whose numerical values are
76 * nondecreasing in array order.
78 * Once we have a solution, we also avoid considering any candidate
79 * which involves dividing one of the output sticks into so many
80 * fragment that the smallest fragment would necessarily be no bigger
81 * than our best solution. That is, we reject candidates where any of
82 * the hamming weights of the adjacency bitmap words are too large.
84 * And, we want to do the search in order of increasing maximum
85 * hamming weight. This is because in practice optimal solutions tend
86 * to have low hamming weight, and having found a reasonable solution
87 * early allows us to eliminate a lot of candidates without doing the
91 typedef uint32_t AdjWord;
92 #define PRADJ "08"PRIx32
94 static int n, m, maxhamweight;
95 static AdjWord *adjmatrix;
96 static AdjWord adjall;
99 static glp_prob *best_prob;
100 static AdjWord *best_adjmatrix;
102 static int n_over_best;
105 static unsigned printcounter;
107 static void iterate(void);
108 static void iterate_recurse(int i, AdjWord min);
109 static bool preconsider_ok(int nwords, bool doprint);
110 static bool maxhamweight_ok(void);
111 static void optimise(bool doprint);
113 static void progress_eol(void) {
114 fprintf(stderr," \r");
118 static void set_best(double new_best) {
120 n_over_best = floor(n / best);
123 /*----- multicore support -----*/
134 * - one pipe ("work") from generator to workers
135 * - ever-extending file ("bus") containing new "best" values
136 * - one file for each worker giving maxhamweight and adjmatrix for best
138 * generator runs iterate_recurse to a certain depth and writes the
139 * candidates to a pipe
141 * workers read candidates from the pipe and resume iterate_recurse
142 * halfway through the recursion
144 * whenever a worker does a doprint, it checks the bus for new best
145 * value; actual best values are appended
147 * master waits for generator and all workers to finish and then
148 * runs optimise() for each worker's best, then prints
151 static int ncpus = 0, multicore_iteration_boundary = INT_MAX;
153 static int mc_bus, mc_work[2];
154 static off_t mc_bus_read;
161 static Worker *mc_us;
163 static void multicore_check_for_new_best(void);
166 static AdjWord mc_iter_min;
168 static size_t mc_iovlen;
169 static struct iovec mc_iov[MAX_NIOVS];
171 #define IOV0 (mc_niovs = mc_iovlen = 0)
173 #define IOV(obj, count) ({ \
174 assert(mc_niovs < MAX_NIOVS); \
175 mc_iov[mc_niovs].iov_base = &(obj); \
176 mc_iov[mc_niovs].iov_len = sizeof(obj) * (count); \
177 mc_iovlen += mc_iov[mc_niovs].iov_len; \
181 static void mc_rwvsetup_outer(void) {
183 IOV(maxhamweight, 1);
185 IOV(*adjmatrix, multicore_iteration_boundary);
189 static void mc_rwvsetup_full(void) {
194 static void vlprintf(const char *fmt, va_list al) {
195 vfprintf(stderr,fmt,al);
199 static void LPRINTF(const char *fmt, ...) {
206 static void mc_awaitpid(int wnum, pid_t pid) {
207 LPRINTF("master awaiting %2d [%ld]",wnum,(long)pid);
209 pid_t got = waitpid(pid, &status, 0);
212 fprintf(stderr,"\nFAILED SUBPROC %2d [%ld] %d\n",
213 wnum, (long)pid, status);
218 static void multicore_outer_iteration(int i, AdjWord min) {
219 static unsigned check_counter;
221 assert(i == multicore_iteration_boundary);
224 ssize_t r = writev(mc_work[1], mc_iov, mc_niovs);
225 assert(r == mc_iovlen);
226 /* effectively, this writev arranges to transfers control
227 * to some worker's instance of iterate_recurse via mc_iterate_worker */
229 if (!(check_counter++ & 0xff))
230 multicore_check_for_new_best();
233 static void mc_iterate_worker(void) {
236 ssize_t r = readv(mc_work[0], mc_iov, mc_niovs);
238 assert(r == mc_iovlen);
240 bool ok = maxhamweight_ok();
243 ok = preconsider_ok(multicore_iteration_boundary, 1);
247 /* stop iterate_recurse from trying to run multicore_outer_iteration */
248 int mc_org_it_bound = multicore_iteration_boundary;
249 multicore_iteration_boundary = INT_MAX;
250 iterate_recurse(mc_org_it_bound, mc_iter_min);
251 multicore_iteration_boundary = mc_org_it_bound;
253 if (best_adjmatrix) {
254 LPRINTF("worker %2d reporting",mc_us->w);
255 adjmatrix = best_adjmatrix;
257 ssize_t r = writev(fileno(mc_us->results), mc_iov, mc_niovs);
258 assert(r == mc_iovlen);
260 LPRINTF("worker %2d ending",mc_us->w);
264 static void multicore(void) {
269 multicore_iteration_boundary = n / 2;
271 FILE *busf = tmpfile(); assert(busf);
272 mc_bus = fileno(busf);
273 int r = fcntl(mc_bus, F_GETFL); assert(r >= 0);
275 r = fcntl(mc_bus, F_SETFL, r); assert(r >= 0);
277 r = pipe(mc_work); assert(!r);
279 mc_workers = xmalloc(sizeof(*mc_workers) * ncpus);
280 for (w=0; w<ncpus; w++) {
282 mc_workers[w].results = tmpfile(); assert(mc_workers[w].results);
283 mc_workers[w].pid = fork(); assert(mc_workers[w].pid >= 0);
284 if (!mc_workers[w].pid) {
285 mc_us = &mc_workers[w];
287 LPRINTF("worker %2d running", w);
295 genpid = fork(); assert(genpid >= 0);
297 LPRINTF("generator running");
303 mc_awaitpid(-1, genpid);
304 for (w=0; w<ncpus; w++)
305 mc_awaitpid(w, mc_workers[w].pid);
307 for (w=0; w<ncpus; w++) {
309 LPRINTF("reading report from %2d",w);
310 ssize_t sr = preadv(fileno(mc_workers[w].results), mc_iov, mc_niovs, 0);
317 static void multicore_check_for_new_best(void) {
322 ssize_t got = pread(mc_bus, &msg, sizeof(msg), mc_bus_read);
324 assert(got == sizeof(msg));
327 mc_bus_read += sizeof(msg);
331 static void multicore_found_new_best(void) {
334 if (mc_us /* might be master */) fprintf(stderr," w%-2d ",mc_us->w);
335 ssize_t wrote = write(mc_bus, &best, sizeof(best));
336 assert(wrote == sizeof(best));
339 /*----- end of multicore support -----*/
341 static AdjWord *xalloc_adjmatrix(void) {
342 return xmalloc(sizeof(*adjmatrix)*n);
345 static void prep(void) {
346 adjall = ~((~(AdjWord)0) << m);
347 adjmatrix = xalloc_adjmatrix();
348 glp_term_out(GLP_OFF);
350 weight = calloc(sizeof(*weight), m); assert(weight);
351 n_over_best = INT_MAX;
354 static AdjWord one_adj_bit(int bitnum) {
355 return (AdjWord)1 << bitnum;
358 static int count_set_adj_bits(AdjWord w) {
360 for (j=0, total=0; j<m; j++)
361 total += !!(w & one_adj_bit(j));
365 #define PRINTF(...) if (!doprint) ; else fprintf(stderr, __VA_ARGS__)
367 static int totalfrags;
369 static bool maxhamweight_ok(void) {
370 double maxminsize = (double)m / maxhamweight;
371 return maxminsize > best;
374 static bool preconsider_ok(int nwords, bool doprint) {
377 PRINTF("%2d ", maxhamweight);
380 for (i=0, totalfrags=0; i<nwords; i++) {
381 int frags = count_set_adj_bits(adjmatrix[i]);
382 had_max += (frags >= maxhamweight);
384 PRINTF("%"PRADJ" ", adjmatrix[i]);
385 double maxminsize = (double)m / frags;
386 if (maxminsize <= best) {
392 /* Skip this candidate as its max hamming weight is lower than
393 * we're currently looking for (which means we must have done it
394 * already). (The recursive iteration ensures that none of the
395 * words have more than the max hamming weight.) */
405 static void optimise(bool doprint) {
406 /* Consider the best answer (if any) for a given adjacency matrix */
411 * Up to a certain point, optimise() can be restarted. We use this
412 * to go back and print the debugging output if it turns out that we
413 * have an interesting case. The HAVE_PRINTED macro does this: its
414 * semantics are to go back in time and make sure that we have
415 * printed the description of the search case.
417 #define HAVE_PRINTED ({ \
418 if (!doprint) { doprint = 1; goto retry_with_print; } \
422 glp_delete_prob(prob);
426 bool ok = preconsider_ok(n, doprint);
431 * We formulate our problem as an LP problem as follows.
432 * In this file "n" and "m" are the matchstick numbers.
434 * Each set bit in the adjacency matrix corresponds to taking a
435 * fragment from old match i and making it part of new match j.
437 * The structural variables (columns) are:
438 * x_minimum minimum size of any fragment (bounded below by 0)
439 * x_morefrag_i_j the amount by which the size of the fragment
440 * i,j exceeds the minimum size (bounded below by 0)
442 * The auxiliary variables (rows) are:
443 * x_total_i total length for each input match (fixed variable)
444 * x_total_j total length for each output match (fixed variable)
446 * The objective function is simply
449 * We use X_ and Y_ to refer to GLPK's (1-based) column and row indices.
450 * ME_ refers to entries in the list of constraint matrix elements
451 * which we build up as we go.
454 prob = glp_create_prob();
456 int Y_totals_i = glp_add_rows(prob, n);
457 int Y_totals_j = glp_add_rows(prob, m);
458 int X_minimum = glp_add_cols(prob, 1);
461 int next_matrix_entry = 1; /* wtf GLPK! */
462 int matrix_entries_size = next_matrix_entry + n + m + totalfrags*2;
463 double matrix_entries[matrix_entries_size];
464 int matrix_entries_XY[2][matrix_entries_size];
466 #define ADD_MATRIX_ENTRY(Y,X) ({ \
467 assert(next_matrix_entry < matrix_entries_size); \
468 matrix_entries_XY[0][next_matrix_entry] = (X); \
469 matrix_entries_XY[1][next_matrix_entry] = (Y); \
470 matrix_entries[next_matrix_entry] = 0; \
471 next_matrix_entry++; \
474 int ME_totals_i__minimum = next_matrix_entry;
475 for (i=0; i<n; i++) ADD_MATRIX_ENTRY(Y_totals_i+i, X_minimum);
477 int ME_totals_j__minimum = next_matrix_entry;
478 for (j=0; j<m; j++) ADD_MATRIX_ENTRY(Y_totals_j+j, X_minimum);
480 /* \forall_i x_total_i = m */
481 /* \forall_i x_total_j = n */
482 for (i=0; i<n; i++) glp_set_row_bnds(prob, Y_totals_i+i, GLP_FX, m,m);
483 for (j=0; j<m; j++) glp_set_row_bnds(prob, Y_totals_j+j, GLP_FX, n,n);
486 glp_set_col_bnds(prob, X_minimum, GLP_LO, 0, 0);
487 glp_set_col_name(prob, X_minimum, "minimum");
489 /* objective is maximising x_minimum */
490 glp_set_obj_dir(prob, GLP_MAX);
491 glp_set_obj_coef(prob, X_minimum, 1);
493 for (i=0; i<n; i++) {
494 for (j=0; j<m; j++) {
495 if (!(adjmatrix[i] & one_adj_bit(j)))
497 /* x_total_i += x_minimum */
498 /* x_total_j += x_minimum */
499 matrix_entries[ ME_totals_i__minimum + i ] ++;
500 matrix_entries[ ME_totals_j__minimum + j ] ++;
502 /* x_morefrag_i_j >= 0 */
503 int X_morefrag_i_j = glp_add_cols(prob, 1);
504 glp_set_col_bnds(prob, X_morefrag_i_j, GLP_LO, 0, 0);
507 snprintf(buf,sizeof(buf),"mf %d,%d",i,j);
508 glp_set_col_name(prob, X_morefrag_i_j, buf);
511 /* x_total_i += x_morefrag_i_j */
512 /* x_total_j += x_morefrag_i_j */
513 int ME_totals_i__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_i+i, X_morefrag_i_j);
514 int ME_totals_j__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_j+j, X_morefrag_i_j);
515 matrix_entries[ME_totals_i__mf_i_j] = 1;
516 matrix_entries[ME_totals_j__mf_i_j] = 1;
520 assert(next_matrix_entry == matrix_entries_size);
522 glp_load_matrix(prob, matrix_entries_size-1,
523 matrix_entries_XY[1], matrix_entries_XY[0],
526 int r = glp_simplex(prob, NULL);
527 PRINTF(" glp=%d", r);
530 case e: PRINTF(" " #e ); goto out;
532 case e: HAVE_PRINTED; printf(" " #e " CRASHING\n"); exit(-1);
534 default: HAVE_PRINTED; printf(" ! CRASHING\n"); exit(-1);
554 r = glp_get_status(prob);
555 PRINTF(" status=%d", r);
567 double got = glp_get_obj_val(prob);
575 multicore_found_new_best();
577 if (best_prob) glp_delete_prob(best_prob);
580 free(best_adjmatrix);
581 best_adjmatrix = xalloc_adjmatrix();
582 memcpy(best_adjmatrix, adjmatrix, sizeof(*adjmatrix)*n);
590 glp_delete_prob(prob);
591 if (doprint) progress_eol();
592 if (doprint) multicore_check_for_new_best();
595 static void iterate_recurse(int i, AdjWord min) {
598 optimise(!(printcounter & 0xfff));
601 if (i >= multicore_iteration_boundary) {
602 multicore_outer_iteration(i, min);
605 for (adjmatrix[i] = min;
608 if (count_set_adj_bits(adjmatrix[i]) > maxhamweight)
610 if (i == 0 && (adjmatrix[i] & (1+adjmatrix[i])))
613 for (int j = 0; j < m; j++)
614 if (adjmatrix[i] & one_adj_bit(j))
616 for (int j = 0; j < m; j++)
617 if (weight[j] >= n_over_best)
620 iterate_recurse(i+1, adjmatrix[i]);
623 for (int j = 0; j < m; j++)
624 if (adjmatrix[i] & one_adj_bit(j))
628 if (adjmatrix[i] == adjall)
633 static void iterate(void) {
634 for (maxhamweight=1; maxhamweight<=m; maxhamweight++) {
635 if (!maxhamweight_ok())
638 iterate_recurse(0, 1);
642 static void report(void) {
643 fprintf(stderr, "\n");
645 double min = glp_get_obj_val(best_prob);
648 for (i = 0; i < n; i++)
649 for (j = 0; j < m; j++)
651 cols = glp_get_num_cols(best_prob);
652 for (i = 1; i <= cols; i++) {
654 if (2 != sscanf(glp_get_col_name(best_prob, i), "mf %d,%d", &x, &y))
656 a[x][y] = min + glp_get_col_prim(best_prob, i);
658 printf("%d into %d: min fragment %g [%s]\n", n, m, min, VERSION);
659 for (i = 0; i < n; i++) {
660 for (j = 0; j < m; j++) {
662 printf(" %9.3f", a[i][j]);
669 if (ferror(stdout) || fclose(stdout)) { perror("stdout"); exit(-1); }
672 int main(int argc, char **argv) {
674 while ((opt = getopt(argc,argv,"j:")) >= 0) {
676 case 'j': ncpus = atoi(optarg); break;
677 case '+': assert(!"bad option");
689 if (ncpus) multicore();