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 * We further winnow the set of possible adjacency matrices, by
86 * ensuring the same bit is not set in too many entries of adjmatrix
87 * (ie, as above, only considering output sticks).
89 * And, we want to do the search in order of increasing maximum
90 * hamming weight. This is because in practice optimal solutions tend
91 * to have low hamming weight, and having found a reasonable solution
92 * early allows us to eliminate a lot of candidates without doing the
96 typedef uint32_t AdjWord;
97 #define PRADJ "08"PRIx32
99 #define FOR_BITS(j,m) for (j=0, j##bit=1; j < (m); j++, j##bit<<=1)
101 static int n, m, maxhamweight;
102 static AdjWord *adjmatrix;
103 static AdjWord adjall;
106 static glp_prob *best_prob;
107 static AdjWord *best_adjmatrix;
109 static int n_max_frags, m_max_frags;
112 static unsigned printcounter;
114 static void iterate(void);
115 static void iterate_recurse(int i, AdjWord min);
116 static bool preconsider_ok(int nwords, bool doprint);
117 static bool maxhamweight_ok(void);
118 static void optimise(bool doprint);
120 static void progress_eol(void) {
121 fprintf(stderr," \r");
125 static void set_best(double new_best) {
128 * When computing n_max_frags, we want to set a value that will skip
129 * anything that won't provide strictly better solutions. So we
133 * <=> frags < | n / best |
135 * <=> frags <= | n / best | - 1
137 n_max_frags = ceil(n / best) - 1;
138 m_max_frags = ceil(m / best) - 1;
141 /*----- multicore support -----*/
152 * - one pipe ("work") from generator to workers
153 * - ever-extending file ("bus") containing new "best" values
154 * - one file for each worker giving maxhamweight and adjmatrix for best
156 * generator runs iterate_recurse to a certain depth and writes the
157 * candidates to a pipe
159 * workers read candidates from the pipe and resume iterate_recurse
160 * halfway through the recursion
162 * whenever a worker does a doprint, it checks the bus for new best
163 * value; actual best values are appended
165 * master waits for generator and all workers to finish and then
166 * runs optimise() for each worker's best, then prints
169 static int ncpus = 0, multicore_iteration_boundary = INT_MAX;
171 static int mc_bus, mc_work[2];
172 static off_t mc_bus_read;
179 static Worker *mc_us;
180 static bool mc_am_generator;
182 static void multicore_check_for_new_best(void);
185 static AdjWord mc_iter_min;
187 static size_t mc_iovlen;
188 static struct iovec mc_iov[MAX_NIOVS];
190 #define IOV0 (mc_niovs = mc_iovlen = 0)
192 #define IOV(obj, count) ({ \
193 assert(mc_niovs < MAX_NIOVS); \
194 mc_iov[mc_niovs].iov_base = &(obj); \
195 mc_iov[mc_niovs].iov_len = sizeof(obj) * (count); \
196 mc_iovlen += mc_iov[mc_niovs].iov_len; \
200 static void mc_rwvsetup_outer(void) {
202 IOV(maxhamweight, 1);
204 IOV(*adjmatrix, multicore_iteration_boundary);
208 static void mc_rwvsetup_full(void) {
213 static void vlprintf(const char *fmt, va_list al) {
214 vfprintf(stderr,fmt,al);
218 static void LPRINTF(const char *fmt, ...) {
225 static void mc_awaitpid(int wnum, pid_t pid) {
226 LPRINTF("master awaiting %2d [%ld]",wnum,(long)pid);
228 pid_t got = waitpid(pid, &status, 0);
231 fprintf(stderr,"\nFAILED SUBPROC %2d [%ld] %d\n",
232 wnum, (long)pid, status);
237 static void multicore_outer_iteration(int i, AdjWord min) {
238 static unsigned check_counter;
240 assert(i == multicore_iteration_boundary);
243 ssize_t r = writev(mc_work[1], mc_iov, mc_niovs);
244 assert(r == mc_iovlen);
245 /* effectively, this writev arranges to transfers control
246 * to some worker's instance of iterate_recurse via mc_iterate_worker */
248 if (!(check_counter++ & 0xff))
249 multicore_check_for_new_best();
252 static void mc_iterate_worker(void) {
255 ssize_t r = readv(mc_work[0], mc_iov, mc_niovs);
257 assert(r == mc_iovlen);
259 bool ok = maxhamweight_ok();
262 ok = preconsider_ok(multicore_iteration_boundary, 1);
266 /* stop iterate_recurse from trying to run multicore_outer_iteration */
267 int mc_org_it_bound = multicore_iteration_boundary;
268 multicore_iteration_boundary = INT_MAX;
269 iterate_recurse(mc_org_it_bound, mc_iter_min);
270 multicore_iteration_boundary = mc_org_it_bound;
272 if (best_adjmatrix) {
273 LPRINTF("worker %2d reporting",mc_us->w);
274 adjmatrix = best_adjmatrix;
276 ssize_t r = writev(fileno(mc_us->results), mc_iov, mc_niovs);
277 assert(r == mc_iovlen);
279 LPRINTF("worker %2d ending",mc_us->w);
283 static void multicore(void) {
288 multicore_iteration_boundary = n / 2;
290 FILE *busf = tmpfile(); assert(busf);
291 mc_bus = fileno(busf);
292 int r = fcntl(mc_bus, F_GETFL); assert(r >= 0);
294 r = fcntl(mc_bus, F_SETFL, r); assert(r >= 0);
296 r = pipe(mc_work); assert(!r);
298 mc_workers = xmalloc(sizeof(*mc_workers) * ncpus);
299 for (w=0; w<ncpus; w++) {
301 mc_workers[w].results = tmpfile(); assert(mc_workers[w].results);
302 mc_workers[w].pid = fork(); assert(mc_workers[w].pid >= 0);
303 if (!mc_workers[w].pid) {
304 mc_us = &mc_workers[w];
306 LPRINTF("worker %2d running", w);
314 genpid = fork(); assert(genpid >= 0);
317 LPRINTF("generator running");
323 mc_awaitpid(-1, genpid);
324 for (w=0; w<ncpus; w++)
325 mc_awaitpid(w, mc_workers[w].pid);
327 for (w=0; w<ncpus; w++) {
329 LPRINTF("reading report from %2d",w);
330 ssize_t sr = preadv(fileno(mc_workers[w].results), mc_iov, mc_niovs, 0);
332 LPRINTF("got report from %2d",w);
338 static void multicore_check_for_new_best(void) {
339 if (!(mc_us || mc_am_generator))
344 ssize_t got = pread(mc_bus, &msg, sizeof(msg), mc_bus_read);
346 assert(got == sizeof(msg));
349 mc_bus_read += sizeof(msg);
353 static void multicore_found_new_best(void) {
357 if (mc_us /* might be master */) fprintf(stderr," w%-2d ",mc_us->w);
358 ssize_t wrote = write(mc_bus, &best, sizeof(best));
359 assert(wrote == sizeof(best));
362 /*----- end of multicore support -----*/
364 static AdjWord *xalloc_adjmatrix(void) {
365 return xmalloc(sizeof(*adjmatrix)*n);
368 static void prep(void) {
369 adjall = ~((~(AdjWord)0) << m);
370 adjmatrix = xalloc_adjmatrix();
371 glp_term_out(GLP_OFF);
373 weight = calloc(sizeof(*weight), m); assert(weight);
374 n_max_frags = INT_MAX;
375 m_max_frags = INT_MAX;
379 static AdjWord one_adj_bit(int bitnum) {
380 return (AdjWord)1 << bitnum;
384 static int count_set_adj_bits(AdjWord w) {
388 total += !!(w & jbit);
392 #define PRINTF(...) if (!doprint) ; else fprintf(stderr, __VA_ARGS__)
394 static int totalfrags;
396 static bool maxhamweight_ok(void) {
397 return maxhamweight <= m_max_frags;
400 static bool preconsider_ok(int nwords, bool doprint) {
403 PRINTF("%2d ", maxhamweight);
406 for (i=0, totalfrags=0; i<nwords; i++) {
407 int frags = count_set_adj_bits(adjmatrix[i]);
408 PRINTF("%"PRADJ" ", adjmatrix[i]);
409 if (frags > m_max_frags) {
413 had_max += (frags >= maxhamweight);
417 /* Skip this candidate as its max hamming weight is lower than
418 * we're currently looking for (which means we must have done it
419 * already). (The recursive iteration ensures that none of the
420 * words have more than the max hamming weight.) */
430 static void optimise(bool doprint) {
431 /* Consider the best answer (if any) for a given adjacency matrix */
437 * Up to a certain point, optimise() can be restarted. We use this
438 * to go back and print the debugging output if it turns out that we
439 * have an interesting case. The HAVE_PRINTED macro does this: its
440 * semantics are to go back in time and make sure that we have
441 * printed the description of the search case.
443 #define HAVE_PRINTED ({ \
444 if (!doprint) { doprint = 1; goto retry_with_print; } \
448 glp_delete_prob(prob);
452 bool ok = preconsider_ok(n, doprint);
457 * We formulate our problem as an LP problem as follows.
458 * In this file "n" and "m" are the matchstick numbers.
460 * Each set bit in the adjacency matrix corresponds to taking a
461 * fragment from old match i and making it part of new match j.
463 * The structural variables (columns) are:
464 * x_minimum minimum size of any fragment (bounded below by 0)
465 * x_morefrag_i_j the amount by which the size of the fragment
466 * i,j exceeds the minimum size (bounded below by 0)
468 * The auxiliary variables (rows) are:
469 * x_total_i total length for each input match (fixed variable)
470 * x_total_j total length for each output match (fixed variable)
472 * The objective function is simply
475 * We use X_ and Y_ to refer to GLPK's (1-based) column and row indices.
476 * ME_ refers to entries in the list of constraint matrix elements
477 * which we build up as we go.
480 prob = glp_create_prob();
482 int Y_totals_i = glp_add_rows(prob, n);
483 int Y_totals_j = glp_add_rows(prob, m);
484 int X_minimum = glp_add_cols(prob, 1);
487 int next_matrix_entry = 1; /* wtf GLPK! */
488 int matrix_entries_size = next_matrix_entry + n + m + totalfrags*2;
489 double matrix_entries[matrix_entries_size];
490 int matrix_entries_XY[2][matrix_entries_size];
492 #define ADD_MATRIX_ENTRY(Y,X) ({ \
493 assert(next_matrix_entry < matrix_entries_size); \
494 matrix_entries_XY[0][next_matrix_entry] = (X); \
495 matrix_entries_XY[1][next_matrix_entry] = (Y); \
496 matrix_entries[next_matrix_entry] = 0; \
497 next_matrix_entry++; \
500 int ME_totals_i__minimum = next_matrix_entry;
501 for (i=0; i<n; i++) ADD_MATRIX_ENTRY(Y_totals_i+i, X_minimum);
503 int ME_totals_j__minimum = next_matrix_entry;
504 for (j=0; j<m; j++) ADD_MATRIX_ENTRY(Y_totals_j+j, X_minimum);
506 /* \forall_i x_total_i = m */
507 /* \forall_i x_total_j = n */
508 for (i=0; i<n; i++) glp_set_row_bnds(prob, Y_totals_i+i, GLP_FX, m,m);
509 for (j=0; j<m; j++) glp_set_row_bnds(prob, Y_totals_j+j, GLP_FX, n,n);
512 glp_set_col_bnds(prob, X_minimum, GLP_LO, 0, 0);
513 glp_set_col_name(prob, X_minimum, "minimum");
515 /* objective is maximising x_minimum */
516 glp_set_obj_dir(prob, GLP_MAX);
517 glp_set_obj_coef(prob, X_minimum, 1);
519 for (i=0; i<n; i++) {
521 if (!(adjmatrix[i] & jbit))
523 /* x_total_i += x_minimum */
524 /* x_total_j += x_minimum */
525 matrix_entries[ ME_totals_i__minimum + i ] ++;
526 matrix_entries[ ME_totals_j__minimum + j ] ++;
528 /* x_morefrag_i_j >= 0 */
529 int X_morefrag_i_j = glp_add_cols(prob, 1);
530 glp_set_col_bnds(prob, X_morefrag_i_j, GLP_LO, 0, 0);
533 snprintf(buf,sizeof(buf),"mf %d,%d",i,j);
534 glp_set_col_name(prob, X_morefrag_i_j, buf);
537 /* x_total_i += x_morefrag_i_j */
538 /* x_total_j += x_morefrag_i_j */
539 int ME_totals_i__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_i+i, X_morefrag_i_j);
540 int ME_totals_j__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_j+j, X_morefrag_i_j);
541 matrix_entries[ME_totals_i__mf_i_j] = 1;
542 matrix_entries[ME_totals_j__mf_i_j] = 1;
546 assert(next_matrix_entry == matrix_entries_size);
548 glp_load_matrix(prob, matrix_entries_size-1,
549 matrix_entries_XY[1], matrix_entries_XY[0],
552 int r = glp_simplex(prob, NULL);
553 PRINTF(" glp=%d", r);
556 case e: PRINTF(" " #e ); goto out;
558 case e: HAVE_PRINTED; printf(" " #e " CRASHING\n"); exit(-1);
560 default: HAVE_PRINTED; printf(" ! CRASHING\n"); exit(-1);
580 r = glp_get_status(prob);
581 PRINTF(" status=%d", r);
593 double got = glp_get_obj_val(prob);
601 multicore_found_new_best();
603 if (best_prob) glp_delete_prob(best_prob);
606 free(best_adjmatrix);
607 best_adjmatrix = xalloc_adjmatrix();
608 memcpy(best_adjmatrix, adjmatrix, sizeof(*adjmatrix)*n);
616 glp_delete_prob(prob);
617 if (doprint) progress_eol();
618 if (doprint) multicore_check_for_new_best();
621 static void iterate_recurse(int i, AdjWord min) {
627 optimise(!(printcounter & 0xfff));
630 if (i >= multicore_iteration_boundary) {
631 multicore_outer_iteration(i, min);
634 for (adjmatrix[i] = min;
637 if (count_set_adj_bits(adjmatrix[i]) > maxhamweight)
639 if (i == 0 && (adjmatrix[i] & (1+adjmatrix[i])))
643 if (adjmatrix[i] & jbit)
645 for (int j = 0; j < m; j++)
646 if (weight[j] >= n_max_frags)
649 iterate_recurse(i+1, adjmatrix[i]);
653 if (adjmatrix[i] & jbit)
657 if (adjmatrix[i] == adjall)
662 static void iterate(void) {
663 for (maxhamweight=1; maxhamweight<=m; maxhamweight++) {
664 if (!maxhamweight_ok())
667 iterate_recurse(0, 1);
671 static void report(void) {
672 fprintf(stderr, "\n");
674 double min = glp_get_obj_val(best_prob);
677 for (i = 0; i < n; i++)
678 for (j = 0; j < m; j++)
680 cols = glp_get_num_cols(best_prob);
681 for (i = 1; i <= cols; i++) {
683 if (2 != sscanf(glp_get_col_name(best_prob, i), "mf %d,%d", &x, &y))
685 a[x][y] = min + glp_get_col_prim(best_prob, i);
687 printf("%d into %d: min fragment %g [%s]\n", n, m, min, VERSION);
688 for (i = 0; i < n; i++) {
689 for (j = 0; j < m; j++) {
691 printf(" %9.3f", a[i][j]);
698 if (ferror(stdout) || fclose(stdout)) { perror("stdout"); exit(-1); }
701 int main(int argc, char **argv) {
703 while ((opt = getopt(argc,argv,"j:")) >= 0) {
705 case 'j': ncpus = atoi(optarg); break;
706 case '+': assert(!"bad option");
718 if (ncpus) multicore();