+/*
+ * Searches for "good" ways to divide n matchsticks up and reassemble them
+ * into m matchsticks. "Good" means the smallest fragment is as big
+ * as possible.
+ *
+ * Invoke as ./main n m
+ *
+ * The algorithm is faster if the arguments are ordered so that n > m.
+ */
+
+/*
+ * matchsticks/main.c Copyright 2014 Ian Jackson
+ *
+ * This program is free software: you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation, either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * This program is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ */
+
+#include <stdio.h>
+#include <stdint.h>
+#include <stdlib.h>
+#include <string.h>
+#include <assert.h>
+#include <stdbool.h>
+#include <inttypes.h>
+
+#include <publib.h>
+#include <glpk.h>
+
+/*
+ * Algorithm.
+ *
+ * Each input match contributes, or does not contribute, to each
+ * output match; we do not need to consider multiple fragments
+ * relating to the same input/output pair this gives an n*m adjacency
+ * matrix (bitmap). Given such an adjacency matrix, the problem of
+ * finding the best sizes for the fragments can be expressed as a
+ * linear programming problem.
+ *
+ * We search all possible adjacency matrices, and for each one we run
+ * GLPK's simplex solver. We represent the adjacency matrix as an
+ * array of bitmaps.
+ *
+ * However, there are a couple of wrinkles:
+ *
+ * To best represent the problem as a standard LP problem, we separate
+ * out the size of each fragment into a common minimum size variable,
+ * plus a fragment-specific extra size variable. This reduces the LP
+ * problem size at the cost of making the problem construction, and
+ * interpretation of the results, a bit fiddly.
+ *
+ * Many of the adjacency matrices are equivalent. In particular,
+ * permutations of the columns, or of the rows, do not change the
+ * meaning. It is only necessasry to consider any one permutation.
+ * We make use of this by considering only adjacency matrices whose
+ * bitmap array contains bitmap words whose numerical values are
+ * nondecreasing in array order.
+ *
+ * Once we have a solution, we also avoid considering any candidate
+ * which involves dividing one of the output sticks into so many
+ * fragment that the smallest fragment would necessarily be no bigger
+ * than our best solution. That is, we reject candidates where any of
+ * the hamming weights of the adjacency bitmap words are too large.
+ *
+ * And, we want to do the search in order of increasing maximum
+ * hamming weight. This is because in practice optimal solutions tend
+ * to have low hamming weight, and having found a reasonable solution
+ * early allows us to eliminate a lot of candidates without doing the
+ * full LP.
+ */
typedef uint32_t AdjWord;
-#define PRADJ PRIx32
+#define PRADJ "08"PRIx32
-static int n, m;
-static AdjWord adjmatrix[n];
+static int n, m, maxhamweight;
+static AdjWord *adjmatrix;
static AdjWord adjall;
static double best;
-static AdjWord adjmatrix_best[n];
+static glp_prob *best_prob;
+static AdjWord *best_adjmatrix;
+
+static unsigned printcounter;
+
+static AdjWord *xalloc_adjmatrix(void) {
+ return xmalloc(sizeof(*adjmatrix)*n);
+}
static void prep(void) {
adjall = ~((~(AdjWord)0) << m);
+ adjmatrix = xalloc_adjmatrix();
+ glp_term_out(GLP_OFF);
+}
+
+static AdjWord one_adj_bit(int bitnum) {
+ return (AdjWord)1 << bitnum;
}
static int count_set_adj_bits(AdjWord w) {
- for (int j=0, total=0; j<m; j++)
- total += !!(w & ((AdjWord)1 << j));
+ int j, total;
+ for (j=0, total=0; j<m; j++)
+ total += !!(w & one_adj_bit(j));
return total;
}
-static void optimise(void) {
+static void optimise(int doprint) {
+ /* Consider the best answer (if any) for a given adjacency matrix */
+ glp_prob *prob = 0;
+ int i, j, totalfrags;
+
+ /*
+ * Up to a certain point, optimise() can be restarted. We use this
+ * to go back and print the debugging output if it turns out that we
+ * have an interesting case. The HAVE_PRINTED macro does this: its
+ * semantics are to go back in time and make sure that we have
+ * printed the description of the search case.
+ */
+#define HAVE_PRINTED ({ \
+ if (!doprint) { doprint = 1; goto retry_with_print; } \
+ })
+ retry_with_print:
+ if (prob) {
+ glp_delete_prob(prob);
+ prob = 0;
+ }
+
+#define PRINTF(...) if (!doprint) ; else fprintf(stderr, __VA_ARGS__) /* bodgy */
+
+ PRINTF("%2d ", maxhamweight);
+
+ bool had_max = 0;
+ for (i=0, totalfrags=0; i<n; i++) {
+ int frags = count_set_adj_bits(adjmatrix[i]);
+ had_max += (frags == maxhamweight);
+ totalfrags += frags;
+ PRINTF("%"PRADJ" ", adjmatrix[i]);
+ double maxminsize = (double)m / frags;
+ if (maxminsize <= best) {
+ PRINTF(" too fine");
+ goto out;
+ }
+ }
+ if (!had_max) {
+ /* Skip this candidate as its max hamming weight is lower than
+ * we're currently looking for (which means we must have done it
+ * already). (The recursive iteration ensures that none of the
+ * words have more than the max hamming weight.) */
+ PRINTF(" nomaxham");
+ goto out;
+ }
+
+ /*
+ * We formulate our problem as an LP problem as follows.
+ * In this file "n" and "m" are the matchstick numbers.
+ *
+ * Each set bit in the adjacency matrix corresponds to taking a
+ * fragment from old match i and making it part of new match j.
+ *
+ * The structural variables (columns) are:
+ * x_minimum minimum size of any fragment (bounded below by 0)
+ * x_morefrag_i_j the amount by which the size of the fragment
+ * i,j exceeds the minimum size (bounded below by 0)
+ *
+ * The auxiliary variables (rows) are:
+ * x_total_i total length for each input match (fixed variable)
+ * x_total_j total length for each output match (fixed variable)
+ *
+ * The objective function is simply
+ * maximise x_minimum
+ *
+ * We use X_ and Y_ to refer to GLPK's (1-based) column and row indices.
+ * ME_ refers to entries in the list of constraint matrix elements
+ * which we build up as we go.
+ */
+
+ prob = glp_create_prob();
+
+ int Y_totals_i = glp_add_rows(prob, n);
+ int Y_totals_j = glp_add_rows(prob, m);
+ int X_minimum = glp_add_cols(prob, 1);
+
+ {
+ int next_matrix_entry = 1; /* wtf GLPK! */
+ int matrix_entries_size = next_matrix_entry + n + m + totalfrags*2;
+ double matrix_entries[matrix_entries_size];
+ int matrix_entries_XY[2][matrix_entries_size];
+
+#define ADD_MATRIX_ENTRY(Y,X) ({ \
+ assert(next_matrix_entry < matrix_entries_size); \
+ matrix_entries_XY[0][next_matrix_entry] = (X); \
+ matrix_entries_XY[1][next_matrix_entry] = (Y); \
+ matrix_entries[next_matrix_entry] = 0; \
+ next_matrix_entry++; \
+ })
+
+ int ME_totals_i__minimum = next_matrix_entry;
+ for (i=0; i<n; i++) ADD_MATRIX_ENTRY(Y_totals_i+i, X_minimum);
+
+ int ME_totals_j__minimum = next_matrix_entry;
+ for (j=0; j<m; j++) ADD_MATRIX_ENTRY(Y_totals_j+j, X_minimum);
+
+ /* \forall_i x_total_i = m */
+ /* \forall_i x_total_j = n */
+ for (i=0; i<n; i++) glp_set_row_bnds(prob, Y_totals_i+i, GLP_FX, m,m);
+ for (j=0; j<m; j++) glp_set_row_bnds(prob, Y_totals_j+j, GLP_FX, n,n);
+
+ /* x_minimum >= 0 */
+ glp_set_col_bnds(prob, X_minimum, GLP_LO, 0, 0);
+ glp_set_col_name(prob, X_minimum, "minimum");
+
+ /* objective is maximising x_minimum */
+ glp_set_obj_dir(prob, GLP_MAX);
+ glp_set_obj_coef(prob, X_minimum, 1);
+
for (i=0; i<n; i++) {
- printf("%"PRADJ" ", adjmatrix[i]);
- double maxminsize = (double)m / count_set_adj_bits(adjmatrix[i]);
- if (maxminsize < best) {
- printf(" too fine\n");
- return;
+ for (j=0; j<m; j++) {
+ if (!(adjmatrix[i] & one_adj_bit(j)))
+ continue;
+ /* x_total_i += x_minimum */
+ /* x_total_j += x_minimum */
+ matrix_entries[ ME_totals_i__minimum + i ] ++;
+ matrix_entries[ ME_totals_j__minimum + j ] ++;
+
+ /* x_morefrag_i_j >= 0 */
+ int X_morefrag_i_j = glp_add_cols(prob, 1);
+ glp_set_col_bnds(prob, X_morefrag_i_j, GLP_LO, 0, 0);
+ if (doprint) {
+ char buf[255];
+ snprintf(buf,sizeof(buf),"mf %d,%d",i,j);
+ glp_set_col_name(prob, X_morefrag_i_j, buf);
+ }
+
+ /* x_total_i += x_morefrag_i_j */
+ /* x_total_j += x_morefrag_i_j */
+ int ME_totals_i__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_i+i, X_morefrag_i_j);
+ int ME_totals_j__mf_i_j = ADD_MATRIX_ENTRY(Y_totals_j+j, X_morefrag_i_j);
+ matrix_entries[ME_totals_i__mf_i_j] = 1;
+ matrix_entries[ME_totals_j__mf_i_j] = 1;
}
}
- printf("nyi\n");
+ assert(next_matrix_entry == matrix_entries_size);
+
+ glp_load_matrix(prob, matrix_entries_size-1,
+ matrix_entries_XY[1], matrix_entries_XY[0],
+ matrix_entries);
+
+ int r = glp_simplex(prob, NULL);
+ PRINTF(" glp=%d", r);
+
+#define OKERR(e) \
+ case e: PRINTF(" " #e ); goto out;
+#define BADERR(e) \
+ case e: HAVE_PRINTED; printf(" " #e " CRASHING\n"); exit(-1);
+#define DEFAULT \
+ default: HAVE_PRINTED; printf(" ! CRASHING\n"); exit(-1);
+
+ switch (r) {
+ OKERR(GLP_ESING);
+ OKERR(GLP_ECOND);
+ OKERR(GLP_EBOUND);
+ OKERR(GLP_EFAIL);
+ OKERR(GLP_ENOPFS);
+ OKERR(GLP_ENODFS);
+ BADERR(GLP_EBADB);
+ BADERR(GLP_EOBJLL);
+ BADERR(GLP_EOBJUL);
+ BADERR(GLP_EITLIM);
+ BADERR(GLP_ETMLIM);
+ BADERR(GLP_EINSTAB);
+ BADERR(GLP_ENOCVG);
+ case 0: break;
+ DEFAULT;
+ }
+
+ r = glp_get_status(prob);
+ PRINTF(" status=%d", r);
+
+ switch (r) {
+ OKERR(GLP_NOFEAS);
+ OKERR(GLP_UNDEF);
+ BADERR(GLP_FEAS);
+ BADERR(GLP_INFEAS);
+ BADERR(GLP_UNBND);
+ case GLP_OPT: break;
+ DEFAULT;
+ }
+
+ double got = glp_get_obj_val(prob);
+ PRINTF(" %g", got);
+ if (got <= best)
+ goto out;
+
+ HAVE_PRINTED;
+
+ best = got;
+
+ if (best_prob) glp_delete_prob(best_prob);
+ best_prob = prob;
+
+ free(best_adjmatrix);
+ best_adjmatrix = xalloc_adjmatrix();
+ memcpy(best_adjmatrix, adjmatrix, sizeof(*adjmatrix)*n);
+
+ PRINTF(" BEST \n");
+ return;
+
+ }
+ out:
+ if (prob)
+ glp_delete_prob(prob);
+ if (doprint) { PRINTF(" \r"); fflush(stdout); }
}
static void iterate_recurse(int i, AdjWord min) {
- if (i > n) {
- optimise();
+ if (i >= n) {
+ printcounter++;
+ optimise(!(printcounter & 0xfff));
return;
}
for (adjmatrix[i] = min;
;
adjmatrix[i]++) {
+ if (count_set_adj_bits(adjmatrix[i]) > maxhamweight)
+ goto again;
+
iterate_recurse(i+1, adjmatrix[i]);
+
+ again:
+ if (adjmatrix[i] == adjall)
+ return;
}
}
static void iterate(void) {
- iterate_recurse(0, 1);
+ for (maxhamweight=1; maxhamweight<=m; maxhamweight++) {
+ double maxminsize = (double)m / maxhamweight;
+ if (maxminsize <= best)
+ continue;
+
+ iterate_recurse(0, 1);
+ }
}
int main(int argc, char **argv) {
+ assert(argc==3);
n = atoi(argv[1]);
m = atoi(argv[2]);
prep();
iterate();
+ fprintf(stderr, "\n");
+ if (best_prob) {
+ double min = glp_get_obj_val(best_prob);
+ double a[n][m];
+ int i, j, cols;
+ for (i = 0; i < n; i++)
+ for (j = 0; j < m; j++)
+ a[i][j] = 0;
+ cols = glp_get_num_cols(best_prob);
+ for (i = 1; i <= cols; i++) {
+ int x, y;
+ if (2 != sscanf(glp_get_col_name(best_prob, i), "mf %d,%d", &x, &y))
+ continue;
+ a[x][y] = min + glp_get_col_prim(best_prob, i);
+ }
+ printf("%d into %d: min fragment %g\n", n, m, min);
+ for (i = 0; i < n; i++) {
+ for (j = 0; j < m; j++) {
+ if (a[i][j])
+ printf(" %9.3f", a[i][j]);
+ else
+ printf(" ");
+ }
+ printf("\n");
+ }
+ }
if (ferror(stdout) || fclose(stdout)) { perror("stdout"); exit(-1); }
return 0;
}
~ianmdlvl / matchsticks-search.git / blobdiff