+/*
+ * 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 "08"PRIx32
-static int n, m;
-static AdjWord *adjmatrix_counters;
-static AdjWord *adjmatrix_offsets;
+static int n, m, maxhamweight;
static AdjWord *adjmatrix;
static AdjWord adjall;
}
static void prep(void) {
- int i;
adjall = ~((~(AdjWord)0) << m);
adjmatrix = xalloc_adjmatrix();
- adjmatrix_counters = xalloc_adjmatrix();
- adjmatrix_offsets = xalloc_adjmatrix();
glp_term_out(GLP_OFF);
- for (i=0; i<n; i++)
- adjmatrix_offsets[i] = ((i+1)*(AdjWord)0x12345678) & adjall;
}
static AdjWord one_adj_bit(int bitnum) {
}
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:
-#define PRINTF if (!doprint) ; else printf /* bodgy */
+ 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;
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.
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_totals_i = m */
- /* \forall_i x_totals_j = n */
+ /* \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);
best_adjmatrix = xalloc_adjmatrix();
memcpy(best_adjmatrix, adjmatrix, sizeof(*adjmatrix)*n);
- printf(" BEST \n");
+ PRINTF(" BEST \n");
return;
}
out:
if (prob)
glp_delete_prob(prob);
- if (doprint) { printf(" \r"); fflush(stdout); }
+ if (doprint) { PRINTF(" \r"); fflush(stdout); }
}
static void iterate_recurse(int i, AdjWord min) {
if (i >= n) {
- int ii;
- for (ii=0; ii<n; ii++) {
- adjmatrix[ii] = adjmatrix_counters[ii] ^ adjmatrix_offsets[ii];
- if (!adjmatrix[ii]) return;
- }
printcounter++;
optimise(!(printcounter & 0xfff));
return;
}
- for (adjmatrix_counters[i] = min;
+ for (adjmatrix[i] = min;
;
- adjmatrix_counters[i]++) {
- iterate_recurse(i+1, adjmatrix_counters[i]);
- if (adjmatrix_counters[i] == adjall)
+ adjmatrix[i]++) {
+ if (count_set_adj_bits(adjmatrix[i]) > maxhamweight)
+ goto again;
+ if (i == 0 && (adjmatrix[i] & (1+adjmatrix[i])))
+ goto again;
+
+ iterate_recurse(i+1, adjmatrix[i]);
+
+ again:
+ if (adjmatrix[i] == adjall)
return;
}
}
static void iterate(void) {
- iterate_recurse(0, 0);
+ 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();
- printf("\n");
- if (best_prob)
- glp_print_sol(best_prob,"/dev/stdout");
+ 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