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 #define FOR_BITS(j,m) for (j=0, j##bit=1; j < (m); j++, j##bit<<=1)
96 static int n, m, maxhamweight;
97 static AdjWord *adjmatrix;
98 static AdjWord adjall;
101 static glp_prob *best_prob;
102 static AdjWord *best_adjmatrix;
104 static int n_over_best, m_over_best;
107 static unsigned printcounter;
109 static void iterate(void);
110 static void iterate_recurse(int i, AdjWord min);
111 static bool preconsider_ok(int nwords, bool doprint);
112 static bool maxhamweight_ok(void);
113 static void optimise(bool doprint);
115 static void progress_eol(void) {
116 fprintf(stderr," \r");
120 static void set_best(double new_best) {
122 n_over_best = floor(n / best);
123 m_over_best = floor(m / best);
126 /*----- multicore support -----*/
137 * - one pipe ("work") from generator to workers
138 * - ever-extending file ("bus") containing new "best" values
139 * - one file for each worker giving maxhamweight and adjmatrix for best
141 * generator runs iterate_recurse to a certain depth and writes the
142 * candidates to a pipe
144 * workers read candidates from the pipe and resume iterate_recurse
145 * halfway through the recursion
147 * whenever a worker does a doprint, it checks the bus for new best
148 * value; actual best values are appended
150 * master waits for generator and all workers to finish and then
151 * runs optimise() for each worker's best, then prints
154 static int ncpus = 0, multicore_iteration_boundary = INT_MAX;
156 static int mc_bus, mc_work[2];
157 static off_t mc_bus_read;
164 static Worker *mc_us;
165 static bool mc_am_generator;
167 static void multicore_check_for_new_best(void);
170 static AdjWord mc_iter_min;
172 static size_t mc_iovlen;
173 static struct iovec mc_iov[MAX_NIOVS];
175 #define IOV0 (mc_niovs = mc_iovlen = 0)
177 #define IOV(obj, count) ({ \
178 assert(mc_niovs < MAX_NIOVS); \
179 mc_iov[mc_niovs].iov_base = &(obj); \
180 mc_iov[mc_niovs].iov_len = sizeof(obj) * (count); \
181 mc_iovlen += mc_iov[mc_niovs].iov_len; \
185 static void mc_rwvsetup_outer(void) {
187 IOV(maxhamweight, 1);
189 IOV(*adjmatrix, multicore_iteration_boundary);
193 static void mc_rwvsetup_full(void) {
198 static void vlprintf(const char *fmt, va_list al) {
199 vfprintf(stderr,fmt,al);
203 static void LPRINTF(const char *fmt, ...) {
210 static void mc_awaitpid(int wnum, pid_t pid) {
211 LPRINTF("master awaiting %2d [%ld]",wnum,(long)pid);
213 pid_t got = waitpid(pid, &status, 0);
216 fprintf(stderr,"\nFAILED SUBPROC %2d [%ld] %d\n",
217 wnum, (long)pid, status);
222 static void multicore_outer_iteration(int i, AdjWord min) {
223 static unsigned check_counter;
225 assert(i == multicore_iteration_boundary);
228 ssize_t r = writev(mc_work[1], mc_iov, mc_niovs);
229 assert(r == mc_iovlen);
230 /* effectively, this writev arranges to transfers control
231 * to some worker's instance of iterate_recurse via mc_iterate_worker */
233 if (!(check_counter++ & 0xff))
234 multicore_check_for_new_best();
237 static void mc_iterate_worker(void) {
240 ssize_t r = readv(mc_work[0], mc_iov, mc_niovs);
242 assert(r == mc_iovlen);
244 bool ok = maxhamweight_ok();
247 ok = preconsider_ok(multicore_iteration_boundary, 1);
251 /* stop iterate_recurse from trying to run multicore_outer_iteration */
252 int mc_org_it_bound = multicore_iteration_boundary;
253 multicore_iteration_boundary = INT_MAX;
254 iterate_recurse(mc_org_it_bound, mc_iter_min);
255 multicore_iteration_boundary = mc_org_it_bound;
257 if (best_adjmatrix) {
258 LPRINTF("worker %2d reporting",mc_us->w);
259 adjmatrix = best_adjmatrix;
261 ssize_t r = writev(fileno(mc_us->results), mc_iov, mc_niovs);
262 assert(r == mc_iovlen);
264 LPRINTF("worker %2d ending",mc_us->w);
268 static void multicore(void) {
273 multicore_iteration_boundary = n / 2;
275 FILE *busf = tmpfile(); assert(busf);
276 mc_bus = fileno(busf);
277 int r = fcntl(mc_bus, F_GETFL); assert(r >= 0);
279 r = fcntl(mc_bus, F_SETFL, r); assert(r >= 0);
281 r = pipe(mc_work); assert(!r);
283 mc_workers = xmalloc(sizeof(*mc_workers) * ncpus);
284 for (w=0; w<ncpus; w++) {
286 mc_workers[w].results = tmpfile(); assert(mc_workers[w].results);
287 mc_workers[w].pid = fork(); assert(mc_workers[w].pid >= 0);
288 if (!mc_workers[w].pid) {
289 mc_us = &mc_workers[w];
291 LPRINTF("worker %2d running", w);
299 genpid = fork(); assert(genpid >= 0);
302 LPRINTF("generator running");
308 mc_awaitpid(-1, genpid);
309 for (w=0; w<ncpus; w++)
310 mc_awaitpid(w, mc_workers[w].pid);
312 for (w=0; w<ncpus; w++) {
314 LPRINTF("reading report from %2d",w);
315 ssize_t sr = preadv(fileno(mc_workers[w].results), mc_iov, mc_niovs, 0);
317 LPRINTF("got report from %2d",w);
323 static void multicore_check_for_new_best(void) {
324 if (!(mc_us || mc_am_generator))
329 ssize_t got = pread(mc_bus, &msg, sizeof(msg), mc_bus_read);
331 assert(got == sizeof(msg));
334 mc_bus_read += sizeof(msg);
338 static void multicore_found_new_best(void) {
342 if (mc_us /* might be master */) fprintf(stderr," w%-2d ",mc_us->w);
343 ssize_t wrote = write(mc_bus, &best, sizeof(best));
344 assert(wrote == sizeof(best));
347 /*----- end of multicore support -----*/
349 static AdjWord *xalloc_adjmatrix(void) {
350 return xmalloc(sizeof(*adjmatrix)*n);
353 static void prep(void) {
354 adjall = ~((~(AdjWord)0) << m);
355 adjmatrix = xalloc_adjmatrix();
356 glp_term_out(GLP_OFF);
358 weight = calloc(sizeof(*weight), m); assert(weight);
359 n_over_best = INT_MAX;
360 m_over_best = INT_MAX;
364 static AdjWord one_adj_bit(int bitnum) {
365 return (AdjWord)1 << bitnum;
369 static int count_set_adj_bits(AdjWord w) {
373 total += !!(w & jbit);
377 #define PRINTF(...) if (!doprint) ; else fprintf(stderr, __VA_ARGS__)
379 static int totalfrags;
381 static bool maxhamweight_ok(void) {
382 return maxhamweight <= m_over_best;
385 static bool preconsider_ok(int nwords, bool doprint) {
388 PRINTF("%2d ", maxhamweight);
391 for (i=0, totalfrags=0; i<nwords; i++) {
392 int frags = count_set_adj_bits(adjmatrix[i]);
393 PRINTF("%"PRADJ" ", adjmatrix[i]);
394 if (frags > m_over_best) {
398 had_max += (frags >= maxhamweight);
402 /* Skip this candidate as its max hamming weight is lower than
403 * we're currently looking for (which means we must have done it
404 * already). (The recursive iteration ensures that none of the
405 * words have more than the max hamming weight.) */
415 static void optimise(bool doprint) {
416 /* Consider the best answer (if any) for a given adjacency matrix */
422 * Up to a certain point, optimise() can be restarted. We use this
423 * to go back and print the debugging output if it turns out that we
424 * have an interesting case. The HAVE_PRINTED macro does this: its
425 * semantics are to go back in time and make sure that we have
426 * printed the description of the search case.
428 #define HAVE_PRINTED ({ \
429 if (!doprint) { doprint = 1; goto retry_with_print; } \
433 glp_delete_prob(prob);
437 bool ok = preconsider_ok(n, doprint);
442 * We formulate our problem as an LP problem as follows.
443 * In this file "n" and "m" are the matchstick numbers.
445 * Each set bit in the adjacency matrix corresponds to taking a
446 * fragment from old match i and making it part of new match j.
448 * The structural variables (columns) are:
449 * x_minimum minimum size of any fragment (bounded below by 0)
450 * x_morefrag_i_j the amount by which the size of the fragment
451 * i,j exceeds the minimum size (bounded below by 0)
453 * The auxiliary variables (rows) are:
454 * x_total_i total length for each input match (fixed variable)
455 * x_total_j total length for each output match (fixed variable)
457 * The objective function is simply
460 * We use X_ and Y_ to refer to GLPK's (1-based) column and row indices.
461 * ME_ refers to entries in the list of constraint matrix elements
462 * which we build up as we go.
465 prob = glp_create_prob();
467 int Y_totals_i = glp_add_rows(prob, n);
468 int Y_totals_j = glp_add_rows(prob, m);
469 int X_minimum = glp_add_cols(prob, 1);
472 int next_matrix_entry = 1; /* wtf GLPK! */
473 int matrix_entries_size = next_matrix_entry + n + m + totalfrags*2;
474 double matrix_entries[matrix_entries_size];
475 int matrix_entries_XY[2][matrix_entries_size];
477 #define ADD_MATRIX_ENTRY(Y,X) ({ \
478 assert(next_matrix_entry < matrix_entries_size); \
479 matrix_entries_XY[0][next_matrix_entry] = (X); \
480 matrix_entries_XY[1][next_matrix_entry] = (Y); \
481 matrix_entries[next_matrix_entry] = 0; \
482 next_matrix_entry++; \
485 int ME_totals_i__minimum = next_matrix_entry;
486 for (i=0; i<n; i++) ADD_MATRIX_ENTRY(Y_totals_i+i, X_minimum);
488 int ME_totals_j__minimum = next_matrix_entry;
489 for (j=0; j<m; j++) ADD_MATRIX_ENTRY(Y_totals_j+j, X_minimum);
491 /* \forall_i x_total_i = m */
492 /* \forall_i x_total_j = n */
493 for (i=0; i<n; i++) glp_set_row_bnds(prob, Y_totals_i+i, GLP_FX, m,m);
494 for (j=0; j<m; j++) glp_set_row_bnds(prob, Y_totals_j+j, GLP_FX, n,n);
497 glp_set_col_bnds(prob, X_minimum, GLP_LO, 0, 0);
498 glp_set_col_name(prob, X_minimum, "minimum");
500 /* objective is maximising x_minimum */
501 glp_set_obj_dir(prob, GLP_MAX);
502 glp_set_obj_coef(prob, X_minimum, 1);
504 for (i=0; i<n; i++) {
505 for (j=0, jbit=1; j<m; j++, jbit<<=1) {
506 if (!(adjmatrix[i] & jbit))
508 /* x_total_i += x_minimum */
509 /* x_total_j += x_minimum */
510 matrix_entries[ ME_totals_i__minimum + i ] ++;
511 matrix_entries[ ME_totals_j__minimum + j ] ++;
513 /* x_morefrag_i_j >= 0 */
514 int X_morefrag_i_j = glp_add_cols(prob, 1);
515 glp_set_col_bnds(prob, X_morefrag_i_j, GLP_LO, 0, 0);
518 snprintf(buf,sizeof(buf),"mf %d,%d",i,j);
519 glp_set_col_name(prob, X_morefrag_i_j, buf);
522 /* x_total_i += x_morefrag_i_j */
523 /* x_total_j += x_morefrag_i_j */
524 int ME_totals_i__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_i+i, X_morefrag_i_j);
525 int ME_totals_j__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_j+j, X_morefrag_i_j);
526 matrix_entries[ME_totals_i__mf_i_j] = 1;
527 matrix_entries[ME_totals_j__mf_i_j] = 1;
531 assert(next_matrix_entry == matrix_entries_size);
533 glp_load_matrix(prob, matrix_entries_size-1,
534 matrix_entries_XY[1], matrix_entries_XY[0],
537 int r = glp_simplex(prob, NULL);
538 PRINTF(" glp=%d", r);
541 case e: PRINTF(" " #e ); goto out;
543 case e: HAVE_PRINTED; printf(" " #e " CRASHING\n"); exit(-1);
545 default: HAVE_PRINTED; printf(" ! CRASHING\n"); exit(-1);
565 r = glp_get_status(prob);
566 PRINTF(" status=%d", r);
578 double got = glp_get_obj_val(prob);
586 multicore_found_new_best();
588 if (best_prob) glp_delete_prob(best_prob);
591 free(best_adjmatrix);
592 best_adjmatrix = xalloc_adjmatrix();
593 memcpy(best_adjmatrix, adjmatrix, sizeof(*adjmatrix)*n);
601 glp_delete_prob(prob);
602 if (doprint) progress_eol();
603 if (doprint) multicore_check_for_new_best();
606 static void iterate_recurse(int i, AdjWord min) {
612 optimise(!(printcounter & 0xfff));
615 if (i >= multicore_iteration_boundary) {
616 multicore_outer_iteration(i, min);
619 for (adjmatrix[i] = min;
622 if (count_set_adj_bits(adjmatrix[i]) > maxhamweight)
624 if (i == 0 && (adjmatrix[i] & (1+adjmatrix[i])))
628 if (adjmatrix[i] & jbit)
630 for (int j = 0; j < m; j++)
631 if (weight[j] >= n_over_best)
634 iterate_recurse(i+1, adjmatrix[i]);
638 if (adjmatrix[i] & jbit)
642 if (adjmatrix[i] == adjall)
647 static void iterate(void) {
648 for (maxhamweight=1; maxhamweight<=m; maxhamweight++) {
649 if (!maxhamweight_ok())
652 iterate_recurse(0, 1);
656 static void report(void) {
657 fprintf(stderr, "\n");
659 double min = glp_get_obj_val(best_prob);
662 for (i = 0; i < n; i++)
663 for (j = 0; j < m; j++)
665 cols = glp_get_num_cols(best_prob);
666 for (i = 1; i <= cols; i++) {
668 if (2 != sscanf(glp_get_col_name(best_prob, i), "mf %d,%d", &x, &y))
670 a[x][y] = min + glp_get_col_prim(best_prob, i);
672 printf("%d into %d: min fragment %g [%s]\n", n, m, min, VERSION);
673 for (i = 0; i < n; i++) {
674 for (j = 0; j < m; j++) {
676 printf(" %9.3f", a[i][j]);
683 if (ferror(stdout) || fclose(stdout)) { perror("stdout"); exit(-1); }
686 int main(int argc, char **argv) {
688 while ((opt = getopt(argc,argv,"j:")) >= 0) {
690 case 'j': ncpus = atoi(optarg); break;
691 case '+': assert(!"bad option");
703 if (ncpus) multicore();