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.
39 * Each input match contributes, or does not contribute, to each
40 * output match; we do not need to consider multiple fragments
41 * relating to the same input/output pair this gives an n*m adjacency
42 * matrix (bitmap). Given such an adjacency matrix, the problem of
43 * finding the best sizes for the fragments can be expressed as a
44 * linear programming problem.
46 * We search all possible adjacency matrices, and for each one we run
47 * GLPK's simplex solver. We represent the adjacency matrix as an
50 * However, there are a couple of wrinkles:
52 * To best represent the problem as a standard LP problem, we separate
53 * out the size of each fragment into a common minimum size variable,
54 * plus a fragment-specific extra size variable. This reduces the LP
55 * problem size at the cost of making the problem construction, and
56 * interpretation of the results, a bit fiddly.
58 * Many of the adjacency matrices are equivalent. In particular,
59 * permutations of the columns, or of the rows, do not change the
60 * meaning. It is only necessasry to consider any one permutation.
61 * We make use of this by considering only adjacency matrices whose
62 * bitmap array contains bitmap words whose numerical values are
63 * nondecreasing in array order.
65 * Once we have a solution, we also avoid considering any candidate
66 * which involves dividing one of the output sticks into so many
67 * fragment that the smallest fragment would necessarily be no bigger
68 * than our best solution. That is, we reject candidates where any of
69 * the hamming weights of the adjacency bitmap words are too large.
71 * And, we want to do the search in order of increasing maximum
72 * hamming weight. This is because in practice optimal solutions tend
73 * to have low hamming weight, and having found a reasonable solution
74 * early allows us to eliminate a lot of candidates without doing the
78 typedef uint32_t AdjWord;
79 #define PRADJ "08"PRIx32
81 static int n, m, maxhamweight;
82 static AdjWord *adjmatrix;
83 static AdjWord adjall;
86 static glp_prob *best_prob;
87 static AdjWord *best_adjmatrix;
89 static unsigned printcounter;
91 static AdjWord *xalloc_adjmatrix(void) {
92 return xmalloc(sizeof(*adjmatrix)*n);
95 static void prep(void) {
96 adjall = ~((~(AdjWord)0) << m);
97 adjmatrix = xalloc_adjmatrix();
98 glp_term_out(GLP_OFF);
101 static AdjWord one_adj_bit(int bitnum) {
102 return (AdjWord)1 << bitnum;
105 static int count_set_adj_bits(AdjWord w) {
107 for (j=0, total=0; j<m; j++)
108 total += !!(w & one_adj_bit(j));
112 static void optimise(int doprint) {
113 /* Consider the best answer (if any) for a given adjacency matrix */
115 int i, j, totalfrags;
118 * Up to a certain point, optimise() can be restarted. We use this
119 * to go back and print the debugging output if it turns out that we
120 * have an interesting case. The HAVE_PRINTED macro does this: its
121 * semantics are to go back in time and make sure that we have
122 * printed the description of the search case.
124 #define HAVE_PRINTED ({ \
125 if (!doprint) { doprint = 1; goto retry_with_print; } \
129 glp_delete_prob(prob);
133 #define PRINTF(...) if (!doprint) ; else fprintf(stderr, __VA_ARGS__) /* bodgy */
135 PRINTF("%2d ", maxhamweight);
138 for (i=0, totalfrags=0; i<n; i++) {
139 int frags = count_set_adj_bits(adjmatrix[i]);
140 had_max += (frags == maxhamweight);
142 PRINTF("%"PRADJ" ", adjmatrix[i]);
143 double maxminsize = (double)m / frags;
144 if (maxminsize <= best) {
150 /* Skip this candidate as its max hamming weight is lower than
151 * we're currently looking for (which means we must have done it
152 * already). (The recursive iteration ensures that none of the
153 * words have more than the max hamming weight.) */
159 * We formulate our problem as an LP problem as follows.
160 * In this file "n" and "m" are the matchstick numbers.
162 * Each set bit in the adjacency matrix corresponds to taking a
163 * fragment from old match i and making it part of new match j.
165 * The structural variables (columns) are:
166 * x_minimum minimum size of any fragment (bounded below by 0)
167 * x_morefrag_i_j the amount by which the size of the fragment
168 * i,j exceeds the minimum size (bounded below by 0)
170 * The auxiliary variables (rows) are:
171 * x_total_i total length for each input match (fixed variable)
172 * x_total_j total length for each output match (fixed variable)
174 * The objective function is simply
177 * We use X_ and Y_ to refer to GLPK's (1-based) column and row indices.
178 * ME_ refers to entries in the list of constraint matrix elements
179 * which we build up as we go.
182 prob = glp_create_prob();
184 int Y_totals_i = glp_add_rows(prob, n);
185 int Y_totals_j = glp_add_rows(prob, m);
186 int X_minimum = glp_add_cols(prob, 1);
189 int next_matrix_entry = 1; /* wtf GLPK! */
190 int matrix_entries_size = next_matrix_entry + n + m + totalfrags*2;
191 double matrix_entries[matrix_entries_size];
192 int matrix_entries_XY[2][matrix_entries_size];
194 #define ADD_MATRIX_ENTRY(Y,X) ({ \
195 assert(next_matrix_entry < matrix_entries_size); \
196 matrix_entries_XY[0][next_matrix_entry] = (X); \
197 matrix_entries_XY[1][next_matrix_entry] = (Y); \
198 matrix_entries[next_matrix_entry] = 0; \
199 next_matrix_entry++; \
202 int ME_totals_i__minimum = next_matrix_entry;
203 for (i=0; i<n; i++) ADD_MATRIX_ENTRY(Y_totals_i+i, X_minimum);
205 int ME_totals_j__minimum = next_matrix_entry;
206 for (j=0; j<m; j++) ADD_MATRIX_ENTRY(Y_totals_j+j, X_minimum);
208 /* \forall_i x_total_i = m */
209 /* \forall_i x_total_j = n */
210 for (i=0; i<n; i++) glp_set_row_bnds(prob, Y_totals_i+i, GLP_FX, m,m);
211 for (j=0; j<m; j++) glp_set_row_bnds(prob, Y_totals_j+j, GLP_FX, n,n);
214 glp_set_col_bnds(prob, X_minimum, GLP_LO, 0, 0);
215 glp_set_col_name(prob, X_minimum, "minimum");
217 /* objective is maximising x_minimum */
218 glp_set_obj_dir(prob, GLP_MAX);
219 glp_set_obj_coef(prob, X_minimum, 1);
221 for (i=0; i<n; i++) {
222 for (j=0; j<m; j++) {
223 if (!(adjmatrix[i] & one_adj_bit(j)))
225 /* x_total_i += x_minimum */
226 /* x_total_j += x_minimum */
227 matrix_entries[ ME_totals_i__minimum + i ] ++;
228 matrix_entries[ ME_totals_j__minimum + j ] ++;
230 /* x_morefrag_i_j >= 0 */
231 int X_morefrag_i_j = glp_add_cols(prob, 1);
232 glp_set_col_bnds(prob, X_morefrag_i_j, GLP_LO, 0, 0);
235 snprintf(buf,sizeof(buf),"mf %d,%d",i,j);
236 glp_set_col_name(prob, X_morefrag_i_j, buf);
239 /* x_total_i += x_morefrag_i_j */
240 /* x_total_j += x_morefrag_i_j */
241 int ME_totals_i__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_i+i, X_morefrag_i_j);
242 int ME_totals_j__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_j+j, X_morefrag_i_j);
243 matrix_entries[ME_totals_i__mf_i_j] = 1;
244 matrix_entries[ME_totals_j__mf_i_j] = 1;
248 assert(next_matrix_entry == matrix_entries_size);
250 glp_load_matrix(prob, matrix_entries_size-1,
251 matrix_entries_XY[1], matrix_entries_XY[0],
254 int r = glp_simplex(prob, NULL);
255 PRINTF(" glp=%d", r);
258 case e: PRINTF(" " #e ); goto out;
260 case e: HAVE_PRINTED; printf(" " #e " CRASHING\n"); exit(-1);
262 default: HAVE_PRINTED; printf(" ! CRASHING\n"); exit(-1);
282 r = glp_get_status(prob);
283 PRINTF(" status=%d", r);
295 double got = glp_get_obj_val(prob);
304 if (best_prob) glp_delete_prob(best_prob);
307 free(best_adjmatrix);
308 best_adjmatrix = xalloc_adjmatrix();
309 memcpy(best_adjmatrix, adjmatrix, sizeof(*adjmatrix)*n);
317 glp_delete_prob(prob);
318 if (doprint) { PRINTF(" \r"); fflush(stdout); }
321 static void iterate_recurse(int i, AdjWord min) {
324 optimise(!(printcounter & 0xfff));
327 for (adjmatrix[i] = min;
330 if (count_set_adj_bits(adjmatrix[i]) > maxhamweight)
332 if (i == 0 && (adjmatrix[i] & (1+adjmatrix[i])))
335 iterate_recurse(i+1, adjmatrix[i]);
338 if (adjmatrix[i] == adjall)
343 static void iterate(void) {
344 for (maxhamweight=1; maxhamweight<=m; maxhamweight++) {
345 double maxminsize = (double)m / maxhamweight;
346 if (maxminsize <= best)
349 iterate_recurse(0, 1);
353 int main(int argc, char **argv) {
359 fprintf(stderr, "\n");
361 double min = glp_get_obj_val(best_prob);
364 for (i = 0; i < n; i++)
365 for (j = 0; j < m; j++)
367 cols = glp_get_num_cols(best_prob);
368 for (i = 1; i <= cols; i++) {
370 if (2 != sscanf(glp_get_col_name(best_prob, i), "mf %d,%d", &x, &y))
372 a[x][y] = min + glp_get_col_prim(best_prob, i);
374 printf("%d into %d: min fragment %g\n", n, m, min);
375 for (i = 0; i < n; i++) {
376 for (j = 0; j < m; j++) {
378 printf(" %9.3f", a[i][j]);
385 if (ferror(stdout) || fclose(stdout)) { perror("stdout"); exit(-1); }