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.
40 * Each input match contributes, or does not contribute, to each
41 * output match; we do not need to consider multiple fragments
42 * relating to the same input/output pair this gives an n*m adjacency
43 * matrix (bitmap). Given such an adjacency matrix, the problem of
44 * finding the best sizes for the fragments can be expressed as a
45 * linear programming problem.
47 * We search all possible adjacency matrices, and for each one we run
48 * GLPK's simplex solver. We represent the adjacency matrix as an
51 * However, there are a couple of wrinkles:
53 * To best represent the problem as a standard LP problem, we separate
54 * out the size of each fragment into a common minimum size variable,
55 * plus a fragment-specific extra size variable. This reduces the LP
56 * problem size at the cost of making the problem construction, and
57 * interpretation of the results, a bit fiddly.
59 * Many of the adjacency matrices are equivalent. In particular,
60 * permutations of the columns, or of the rows, do not change the
61 * meaning. It is only necessasry to consider any one permutation.
62 * We make use of this by considering only adjacency matrices whose
63 * bitmap array contains bitmap words whose numerical values are
64 * nondecreasing in array order.
66 * Once we have a solution, we also avoid considering any candidate
67 * which involves dividing one of the output sticks into so many
68 * fragment that the smallest fragment would necessarily be no bigger
69 * than our best solution. That is, we reject candidates where any of
70 * the hamming weights of the adjacency bitmap words are too large.
72 * And, we want to do the search in order of increasing maximum
73 * hamming weight. This is because in practice optimal solutions tend
74 * to have low hamming weight, and having found a reasonable solution
75 * early allows us to eliminate a lot of candidates without doing the
79 typedef uint32_t AdjWord;
80 #define PRADJ "08"PRIx32
82 static int n, m, maxhamweight;
83 static AdjWord *adjmatrix;
84 static AdjWord adjall;
87 static glp_prob *best_prob;
88 static AdjWord *best_adjmatrix;
90 static unsigned printcounter;
94 static AdjWord *xalloc_adjmatrix(void) {
95 return xmalloc(sizeof(*adjmatrix)*n);
98 static void prep(void) {
99 adjall = ~((~(AdjWord)0) << m);
100 adjmatrix = xalloc_adjmatrix();
101 glp_term_out(GLP_OFF);
104 static AdjWord one_adj_bit(int bitnum) {
105 return (AdjWord)1 << bitnum;
108 static int count_set_adj_bits(AdjWord w) {
110 for (j=0, total=0; j<m; j++)
111 total += !!(w & one_adj_bit(j));
115 static void optimise(int doprint) {
116 /* Consider the best answer (if any) for a given adjacency matrix */
118 int i, j, totalfrags;
121 * Up to a certain point, optimise() can be restarted. We use this
122 * to go back and print the debugging output if it turns out that we
123 * have an interesting case. The HAVE_PRINTED macro does this: its
124 * semantics are to go back in time and make sure that we have
125 * printed the description of the search case.
127 #define HAVE_PRINTED ({ \
128 if (!doprint) { doprint = 1; goto retry_with_print; } \
132 glp_delete_prob(prob);
136 #define PRINTF(...) if (!doprint) ; else fprintf(stderr, __VA_ARGS__)
138 PRINTF("%2d ", maxhamweight);
141 for (i=0, totalfrags=0; i<n; i++) {
142 int frags = count_set_adj_bits(adjmatrix[i]);
143 had_max += (frags == maxhamweight);
145 PRINTF("%"PRADJ" ", adjmatrix[i]);
146 double maxminsize = (double)m / frags;
147 if (maxminsize <= best) {
153 /* Skip this candidate as its max hamming weight is lower than
154 * we're currently looking for (which means we must have done it
155 * already). (The recursive iteration ensures that none of the
156 * words have more than the max hamming weight.) */
162 * We formulate our problem as an LP problem as follows.
163 * In this file "n" and "m" are the matchstick numbers.
165 * Each set bit in the adjacency matrix corresponds to taking a
166 * fragment from old match i and making it part of new match j.
168 * The structural variables (columns) are:
169 * x_minimum minimum size of any fragment (bounded below by 0)
170 * x_morefrag_i_j the amount by which the size of the fragment
171 * i,j exceeds the minimum size (bounded below by 0)
173 * The auxiliary variables (rows) are:
174 * x_total_i total length for each input match (fixed variable)
175 * x_total_j total length for each output match (fixed variable)
177 * The objective function is simply
180 * We use X_ and Y_ to refer to GLPK's (1-based) column and row indices.
181 * ME_ refers to entries in the list of constraint matrix elements
182 * which we build up as we go.
185 prob = glp_create_prob();
187 int Y_totals_i = glp_add_rows(prob, n);
188 int Y_totals_j = glp_add_rows(prob, m);
189 int X_minimum = glp_add_cols(prob, 1);
192 int next_matrix_entry = 1; /* wtf GLPK! */
193 int matrix_entries_size = next_matrix_entry + n + m + totalfrags*2;
194 double matrix_entries[matrix_entries_size];
195 int matrix_entries_XY[2][matrix_entries_size];
197 #define ADD_MATRIX_ENTRY(Y,X) ({ \
198 assert(next_matrix_entry < matrix_entries_size); \
199 matrix_entries_XY[0][next_matrix_entry] = (X); \
200 matrix_entries_XY[1][next_matrix_entry] = (Y); \
201 matrix_entries[next_matrix_entry] = 0; \
202 next_matrix_entry++; \
205 int ME_totals_i__minimum = next_matrix_entry;
206 for (i=0; i<n; i++) ADD_MATRIX_ENTRY(Y_totals_i+i, X_minimum);
208 int ME_totals_j__minimum = next_matrix_entry;
209 for (j=0; j<m; j++) ADD_MATRIX_ENTRY(Y_totals_j+j, X_minimum);
211 /* \forall_i x_total_i = m */
212 /* \forall_i x_total_j = n */
213 for (i=0; i<n; i++) glp_set_row_bnds(prob, Y_totals_i+i, GLP_FX, m,m);
214 for (j=0; j<m; j++) glp_set_row_bnds(prob, Y_totals_j+j, GLP_FX, n,n);
217 glp_set_col_bnds(prob, X_minimum, GLP_LO, 0, 0);
218 glp_set_col_name(prob, X_minimum, "minimum");
220 /* objective is maximising x_minimum */
221 glp_set_obj_dir(prob, GLP_MAX);
222 glp_set_obj_coef(prob, X_minimum, 1);
224 for (i=0; i<n; i++) {
225 for (j=0; j<m; j++) {
226 if (!(adjmatrix[i] & one_adj_bit(j)))
228 /* x_total_i += x_minimum */
229 /* x_total_j += x_minimum */
230 matrix_entries[ ME_totals_i__minimum + i ] ++;
231 matrix_entries[ ME_totals_j__minimum + j ] ++;
233 /* x_morefrag_i_j >= 0 */
234 int X_morefrag_i_j = glp_add_cols(prob, 1);
235 glp_set_col_bnds(prob, X_morefrag_i_j, GLP_LO, 0, 0);
238 snprintf(buf,sizeof(buf),"mf %d,%d",i,j);
239 glp_set_col_name(prob, X_morefrag_i_j, buf);
242 /* x_total_i += x_morefrag_i_j */
243 /* x_total_j += x_morefrag_i_j */
244 int ME_totals_i__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_i+i, X_morefrag_i_j);
245 int ME_totals_j__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_j+j, X_morefrag_i_j);
246 matrix_entries[ME_totals_i__mf_i_j] = 1;
247 matrix_entries[ME_totals_j__mf_i_j] = 1;
251 assert(next_matrix_entry == matrix_entries_size);
253 glp_load_matrix(prob, matrix_entries_size-1,
254 matrix_entries_XY[1], matrix_entries_XY[0],
257 int r = glp_simplex(prob, NULL);
258 PRINTF(" glp=%d", r);
261 case e: PRINTF(" " #e ); goto out;
263 case e: HAVE_PRINTED; printf(" " #e " CRASHING\n"); exit(-1);
265 default: HAVE_PRINTED; printf(" ! CRASHING\n"); exit(-1);
285 r = glp_get_status(prob);
286 PRINTF(" status=%d", r);
298 double got = glp_get_obj_val(prob);
307 if (best_prob) glp_delete_prob(best_prob);
310 free(best_adjmatrix);
311 best_adjmatrix = xalloc_adjmatrix();
312 memcpy(best_adjmatrix, adjmatrix, sizeof(*adjmatrix)*n);
320 glp_delete_prob(prob);
321 if (doprint) { PRINTF(" \r"); fflush(stdout); }
324 static void iterate_recurse(int i, AdjWord min) {
327 optimise(!(printcounter & 0xfff));
330 for (adjmatrix[i] = min;
333 if (count_set_adj_bits(adjmatrix[i]) > maxhamweight)
335 if (i == 0 && (adjmatrix[i] & (1+adjmatrix[i])))
338 iterate_recurse(i+1, adjmatrix[i]);
341 if (adjmatrix[i] == adjall)
346 static void iterate(void) {
347 for (maxhamweight=1; maxhamweight<=m; maxhamweight++) {
348 double maxminsize = (double)m / maxhamweight;
349 if (maxminsize <= best)
352 iterate_recurse(0, 1);
356 static void report(void) {
357 fprintf(stderr, "\n");
359 double min = glp_get_obj_val(best_prob);
362 for (i = 0; i < n; i++)
363 for (j = 0; j < m; j++)
365 cols = glp_get_num_cols(best_prob);
366 for (i = 1; i <= cols; i++) {
368 if (2 != sscanf(glp_get_col_name(best_prob, i), "mf %d,%d", &x, &y))
370 a[x][y] = min + glp_get_col_prim(best_prob, i);
372 printf("%d into %d: min fragment %g\n", n, m, min);
373 for (i = 0; i < n; i++) {
374 for (j = 0; j < m; j++) {
376 printf(" %9.3f", a[i][j]);
383 if (ferror(stdout) || fclose(stdout)) { perror("stdout"); exit(-1); }
386 int main(int argc, char **argv) {
388 while ((opt = getopt(argc,argv,"j:")) >= 0) {
390 case 'j': ncpus = atoi(optarg); break;
391 case '+': assert(!"bad option");