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 arguments must be ordered so that n > m:
9 * n is the number of (more, shorter) input matches of length m
10 * m is the number of (fewer, longer) output matches of length n
14 * matchsticks/main.c Copyright 2014 Ian Jackson
16 * This program is free software: you can redistribute it and/or modify
17 * it under the terms of the GNU General Public License as published by
18 * the Free Software Foundation, either version 3 of the License, or
19 * (at your option) any later version.
21 * This program is distributed in the hope that it will be useful,
22 * but WITHOUT ANY WARRANTY; without even the implied warranty of
23 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
24 * GNU General Public License for more details.
40 #include <sys/types.h>
43 #include <sys/fcntl.h>
48 #define VERSION "(unknown-version)"
54 * Each input match contributes, or does not contribute, to each
55 * output match; we do not need to consider multiple fragments
56 * relating to the same input/output pair this gives an n*m adjacency
57 * matrix (bitmap). Given such an adjacency matrix, the problem of
58 * finding the best sizes for the fragments can be expressed as a
59 * linear programming problem.
61 * We search all possible adjacency matrices, and for each one we run
62 * GLPK's simplex solver. We represent the adjacency matrix as an
63 * array of bitmaps: one word per input stick, with one bit per output
66 * However, there are a couple of wrinkles:
68 * To best represent the problem as a standard LP problem, we separate
69 * out the size of each fragment into a common minimum size variable,
70 * plus a fragment-specific extra size variable. This reduces the LP
71 * problem size at the cost of making the problem construction, and
72 * interpretation of the results, a bit fiddly.
74 * Many of the adjacency matrices are equivalent. In particular,
75 * permutations of the columns, or of the rows, do not change the
76 * meaning. It is only necessasry to consider any one permutation.
77 * We make use of this by considering only adjacency matrices whose
78 * bitmap array contains bitmap words whose numerical values are
79 * nondecreasing in array order.
81 * Once we have a solution, we also avoid considering any candidate
82 * which involves dividing one of the input sticks into so many
83 * fragment that the smallest fragment would necessarily be no bigger
84 * than our best solution. That is, we reject candidates where any of
85 * the hamming weights of the adjacency bitmap words are too large.
87 * We further winnow the set of possible adjacency matrices, by
88 * ensuring the same bit is not set in too many entries of adjmatrix
89 * (ie, as above, only considering output sticks); and by ensuring
90 * that it is not set in too few: each output stick must consist
91 * of at least two fragments since the output sticks are longer than
94 * And, we want to do the search in order of increasing maximum
95 * hamming weight. This is because in practice optimal solutions tend
96 * to have low hamming weight, and having found a reasonable solution
97 * early allows us to eliminate a lot of candidates without doing the
101 typedef uint32_t AdjWord;
102 #define PRADJ "08"PRIx32
104 #define FOR_BITS(j,m) for (j=0, j##bit=1; j < (m); j++, j##bit<<=1)
106 static int n, m, maxhamweight;
107 static AdjWord *adjmatrix;
108 static AdjWord adjall;
111 static glp_prob *best_prob;
112 static AdjWord *best_adjmatrix;
114 static int n_max_frags, m_max_frags;
117 static unsigned printcounter;
119 static void iterate(void);
120 static void iterate_recurse(int i, AdjWord min);
121 static bool preconsider_ok(int nwords, bool doprint);
122 static bool maxhamweight_ok(void);
123 static void optimise(bool doprint);
125 static void progress_eol(void) {
126 fprintf(stderr," \r");
130 static void set_best(double new_best) {
133 * When computing n_max_frags, we want to set a value that will skip
134 * anything that won't provide strictly better solutions. So we
138 * <=> frags < | n / best |
140 * <=> frags <= | n / best | - 1
142 n_max_frags = ceil(n / best) - 1;
143 m_max_frags = ceil(m / best) - 1;
146 /*----- multicore support -----*/
157 * - one pipe ("work") from generator to workers
158 * - ever-extending file ("bus") containing new "best" values
159 * - one file for each worker giving maxhamweight and adjmatrix for best
161 * generator runs iterate_recurse to a certain depth and writes the
162 * candidates to a pipe
164 * workers read candidates from the pipe and resume iterate_recurse
165 * halfway through the recursion
167 * whenever a worker does a doprint, it checks the bus for new best
168 * value; actual best values are appended
170 * master waits for generator and all workers to finish and then
171 * runs optimise() for each worker's best, then prints
174 static int ncpus = 0, multicore_iteration_boundary = INT_MAX;
176 static int mc_bus, mc_work[2];
177 static off_t mc_bus_read;
184 static Worker *mc_us;
185 static bool mc_am_generator;
187 static void multicore_check_for_new_best(void);
190 static AdjWord mc_iter_min;
192 static size_t mc_iovlen;
193 static struct iovec mc_iov[MAX_NIOVS];
195 #define IOV0 (mc_niovs = mc_iovlen = 0)
197 #define IOV(obj, count) ({ \
198 assert(mc_niovs < MAX_NIOVS); \
199 mc_iov[mc_niovs].iov_base = &(obj); \
200 mc_iov[mc_niovs].iov_len = sizeof(obj) * (count); \
201 mc_iovlen += mc_iov[mc_niovs].iov_len; \
205 static void mc_rwvsetup_outer(void) {
207 IOV(maxhamweight, 1);
209 IOV(*adjmatrix, multicore_iteration_boundary);
213 static void mc_rwvsetup_full(void) {
218 static void vlprintf(const char *fmt, va_list al) {
219 vfprintf(stderr,fmt,al);
223 static void LPRINTF(const char *fmt, ...) {
230 static void mc_awaitpid(int wnum, pid_t pid) {
231 LPRINTF("master awaiting %2d [%ld]",wnum,(long)pid);
233 pid_t got = waitpid(pid, &status, 0);
236 fprintf(stderr,"\nFAILED SUBPROC %2d [%ld] %d\n",
237 wnum, (long)pid, status);
242 static void multicore_outer_iteration(int i, AdjWord min) {
243 static unsigned check_counter;
245 assert(i == multicore_iteration_boundary);
248 ssize_t r = writev(mc_work[1], mc_iov, mc_niovs);
249 assert(r == mc_iovlen);
250 /* effectively, this writev arranges to transfers control
251 * to some worker's instance of iterate_recurse via mc_iterate_worker */
253 if (!(check_counter++ & 0xff))
254 multicore_check_for_new_best();
257 static void mc_iterate_worker(void) {
260 ssize_t r = readv(mc_work[0], mc_iov, mc_niovs);
262 assert(r == mc_iovlen);
264 bool ok = maxhamweight_ok();
267 ok = preconsider_ok(multicore_iteration_boundary, 1);
271 /* stop iterate_recurse from trying to run multicore_outer_iteration */
272 int mc_org_it_bound = multicore_iteration_boundary;
273 multicore_iteration_boundary = INT_MAX;
274 iterate_recurse(mc_org_it_bound, mc_iter_min);
275 multicore_iteration_boundary = mc_org_it_bound;
277 if (best_adjmatrix) {
278 LPRINTF("worker %2d reporting",mc_us->w);
279 adjmatrix = best_adjmatrix;
281 ssize_t r = writev(fileno(mc_us->results), mc_iov, mc_niovs);
282 assert(r == mc_iovlen);
284 LPRINTF("worker %2d ending",mc_us->w);
288 static void multicore(void) {
293 multicore_iteration_boundary = n / 2;
295 FILE *busf = tmpfile(); assert(busf);
296 mc_bus = fileno(busf);
297 int r = fcntl(mc_bus, F_GETFL); assert(r >= 0);
299 r = fcntl(mc_bus, F_SETFL, r); assert(r >= 0);
301 r = pipe(mc_work); assert(!r);
303 mc_workers = xmalloc(sizeof(*mc_workers) * ncpus);
304 for (w=0; w<ncpus; w++) {
306 mc_workers[w].results = tmpfile(); assert(mc_workers[w].results);
307 mc_workers[w].pid = fork(); assert(mc_workers[w].pid >= 0);
308 if (!mc_workers[w].pid) {
309 mc_us = &mc_workers[w];
311 LPRINTF("worker %2d running", w);
319 genpid = fork(); assert(genpid >= 0);
322 LPRINTF("generator running");
328 mc_awaitpid(-1, genpid);
329 for (w=0; w<ncpus; w++)
330 mc_awaitpid(w, mc_workers[w].pid);
332 for (w=0; w<ncpus; w++) {
334 LPRINTF("reading report from %2d",w);
335 ssize_t sr = preadv(fileno(mc_workers[w].results), mc_iov, mc_niovs, 0);
337 LPRINTF("got report from %2d",w);
343 static void multicore_check_for_new_best(void) {
344 if (!(mc_us || mc_am_generator))
349 ssize_t got = pread(mc_bus, &msg, sizeof(msg), mc_bus_read);
351 assert(got == sizeof(msg));
354 mc_bus_read += sizeof(msg);
358 static void multicore_found_new_best(void) {
362 if (mc_us /* might be master */) fprintf(stderr," w%-2d ",mc_us->w);
363 ssize_t wrote = write(mc_bus, &best, sizeof(best));
364 assert(wrote == sizeof(best));
367 /*----- end of multicore support -----*/
369 static AdjWord *xalloc_adjmatrix(void) {
370 return xmalloc(sizeof(*adjmatrix)*n);
373 static void prep(void) {
374 adjall = ~((~(AdjWord)0) << m);
375 adjmatrix = xalloc_adjmatrix();
376 glp_term_out(GLP_OFF);
378 weight = calloc(sizeof(*weight), m); assert(weight);
379 n_max_frags = INT_MAX;
380 m_max_frags = INT_MAX;
384 static AdjWord one_adj_bit(int bitnum) {
385 return (AdjWord)1 << bitnum;
389 static int count_set_adj_bits(AdjWord w) {
393 total += !!(w & jbit);
397 #define PRINTF(...) if (!doprint) ; else fprintf(stderr, __VA_ARGS__)
399 static int totalfrags;
401 static bool maxhamweight_ok(void) {
402 return maxhamweight <= m_max_frags;
405 static bool preconsider_ok(int nwords, bool doprint) {
408 PRINTF("%2d ", maxhamweight);
411 for (i=0, totalfrags=0; i<nwords; i++) {
412 int frags = count_set_adj_bits(adjmatrix[i]);
413 PRINTF("%"PRADJ" ", adjmatrix[i]);
414 if (frags > m_max_frags) {
418 had_max += (frags >= maxhamweight);
422 /* Skip this candidate as its max hamming weight is lower than
423 * we're currently looking for (which means we must have done it
424 * already). (The recursive iteration ensures that none of the
425 * words have more than the max hamming weight.) */
435 static void optimise(bool doprint) {
436 /* Consider the best answer (if any) for a given adjacency matrix */
442 * Up to a certain point, optimise() can be restarted. We use this
443 * to go back and print the debugging output if it turns out that we
444 * have an interesting case. The HAVE_PRINTED macro does this: its
445 * semantics are to go back in time and make sure that we have
446 * printed the description of the search case.
448 #define HAVE_PRINTED ({ \
449 if (!doprint) { doprint = 1; goto retry_with_print; } \
453 glp_delete_prob(prob);
457 bool ok = preconsider_ok(n, doprint);
462 * We formulate our problem as an LP problem as follows.
463 * In this file "n" and "m" are the matchstick numbers.
465 * Each set bit in the adjacency matrix corresponds to taking a
466 * fragment from old match i and making it part of new match j.
468 * The structural variables (columns) are:
469 * x_minimum minimum size of any fragment (bounded below by 0)
470 * x_morefrag_i_j the amount by which the size of the fragment
471 * i,j exceeds the minimum size (bounded below by 0)
473 * The auxiliary variables (rows) are:
474 * x_total_i total length for each input match (fixed variable)
475 * x_total_j total length for each output match (fixed variable)
477 * The objective function is simply
480 * We use X_ and Y_ to refer to GLPK's (1-based) column and row indices.
481 * ME_ refers to entries in the list of constraint matrix elements
482 * which we build up as we go.
485 prob = glp_create_prob();
487 int Y_totals_i = glp_add_rows(prob, n);
488 int Y_totals_j = glp_add_rows(prob, m);
489 int X_minimum = glp_add_cols(prob, 1);
492 int next_matrix_entry = 1; /* wtf GLPK! */
493 int matrix_entries_size = next_matrix_entry + n + m + totalfrags*2;
494 double matrix_entries[matrix_entries_size];
495 int matrix_entries_XY[2][matrix_entries_size];
497 #define ADD_MATRIX_ENTRY(Y,X) ({ \
498 assert(next_matrix_entry < matrix_entries_size); \
499 matrix_entries_XY[0][next_matrix_entry] = (X); \
500 matrix_entries_XY[1][next_matrix_entry] = (Y); \
501 matrix_entries[next_matrix_entry] = 0; \
502 next_matrix_entry++; \
505 int ME_totals_i__minimum = next_matrix_entry;
506 for (i=0; i<n; i++) ADD_MATRIX_ENTRY(Y_totals_i+i, X_minimum);
508 int ME_totals_j__minimum = next_matrix_entry;
509 for (j=0; j<m; j++) ADD_MATRIX_ENTRY(Y_totals_j+j, X_minimum);
511 /* \forall_i x_total_i = m */
512 /* \forall_i x_total_j = n */
513 for (i=0; i<n; i++) glp_set_row_bnds(prob, Y_totals_i+i, GLP_FX, m,m);
514 for (j=0; j<m; j++) glp_set_row_bnds(prob, Y_totals_j+j, GLP_FX, n,n);
517 glp_set_col_bnds(prob, X_minimum, GLP_LO, 0, 0);
518 glp_set_col_name(prob, X_minimum, "minimum");
520 /* objective is maximising x_minimum */
521 glp_set_obj_dir(prob, GLP_MAX);
522 glp_set_obj_coef(prob, X_minimum, 1);
524 for (i=0; i<n; i++) {
526 if (!(adjmatrix[i] & jbit))
528 /* x_total_i += x_minimum */
529 /* x_total_j += x_minimum */
530 matrix_entries[ ME_totals_i__minimum + i ] ++;
531 matrix_entries[ ME_totals_j__minimum + j ] ++;
533 /* x_morefrag_i_j >= 0 */
534 int X_morefrag_i_j = glp_add_cols(prob, 1);
535 glp_set_col_bnds(prob, X_morefrag_i_j, GLP_LO, 0, 0);
538 snprintf(buf,sizeof(buf),"mf %d,%d",i,j);
539 glp_set_col_name(prob, X_morefrag_i_j, buf);
542 /* x_total_i += x_morefrag_i_j */
543 /* x_total_j += x_morefrag_i_j */
544 int ME_totals_i__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_i+i, X_morefrag_i_j);
545 int ME_totals_j__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_j+j, X_morefrag_i_j);
546 matrix_entries[ME_totals_i__mf_i_j] = 1;
547 matrix_entries[ME_totals_j__mf_i_j] = 1;
551 assert(next_matrix_entry == matrix_entries_size);
553 glp_load_matrix(prob, matrix_entries_size-1,
554 matrix_entries_XY[1], matrix_entries_XY[0],
557 int r = glp_simplex(prob, NULL);
558 PRINTF(" glp=%d", r);
561 case e: PRINTF(" " #e ); goto out;
563 case e: HAVE_PRINTED; printf(" " #e " CRASHING\n"); exit(-1);
565 default: HAVE_PRINTED; printf(" ! CRASHING\n"); exit(-1);
585 r = glp_get_status(prob);
586 PRINTF(" status=%d", r);
598 double got = glp_get_obj_val(prob);
606 multicore_found_new_best();
608 if (best_prob) glp_delete_prob(best_prob);
611 free(best_adjmatrix);
612 best_adjmatrix = xalloc_adjmatrix();
613 memcpy(best_adjmatrix, adjmatrix, sizeof(*adjmatrix)*n);
621 glp_delete_prob(prob);
622 if (doprint) progress_eol();
623 if (doprint) multicore_check_for_new_best();
626 static void iterate_recurse(int i, AdjWord min) {
636 optimise(!(printcounter & 0xfff));
639 if (i >= multicore_iteration_boundary) {
640 multicore_outer_iteration(i, min);
643 for (adjmatrix[i] = min;
646 if (count_set_adj_bits(adjmatrix[i]) > maxhamweight)
648 if (i == 0 && (adjmatrix[i] & (1+adjmatrix[i])))
652 if (adjmatrix[i] & jbit)
654 for (int j = 0; j < m; j++)
655 if (weight[j] >= n_max_frags)
658 iterate_recurse(i+1, adjmatrix[i]);
662 if (adjmatrix[i] & jbit)
666 if (adjmatrix[i] == adjall)
671 static void iterate(void) {
672 for (maxhamweight=1; maxhamweight<=m; maxhamweight++) {
673 if (!maxhamweight_ok())
676 iterate_recurse(0, 1);
680 static void report(void) {
681 fprintf(stderr, "\n");
683 double min = glp_get_obj_val(best_prob);
686 for (i = 0; i < n; i++)
687 for (j = 0; j < m; j++)
689 cols = glp_get_num_cols(best_prob);
690 for (i = 1; i <= cols; i++) {
692 if (2 != sscanf(glp_get_col_name(best_prob, i), "mf %d,%d", &x, &y))
694 a[x][y] = min + glp_get_col_prim(best_prob, i);
696 printf("%d into %d: min fragment %g [%s]\n", n, m, min, VERSION);
697 for (i = 0; i < n; i++) {
698 for (j = 0; j < m; j++) {
700 printf(" %9.3f", a[i][j]);
707 if (ferror(stdout) || fclose(stdout)) { perror("stdout"); exit(-1); }
710 int main(int argc, char **argv) {
712 while ((opt = getopt(argc,argv,"j:")) >= 0) {
714 case 'j': ncpus = atoi(optarg); break;
715 case '+': assert(!"bad option");
728 if (ncpus) multicore();