X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ianmdlvl/git?a=blobdiff_plain;ds=sidebyside;f=main.c;h=83728fa7a4aee7cb0168bbdc96d2bd689483669d;hb=3a1fd673483e2e2f17853cc73be3b7f4ffea459c;hp=6ccfea7bb0384f472c23befb0ec89bde375b0251;hpb=2e084dc7235e257b62d034b663e8466a2c7b6af0;p=matchsticks-search.git

diff --git a/main.c b/main.c
index 6ccfea7..83728fa 100644
--- a/main.c
+++ b/main.c
@@ -1,20 +1,84 @@
+/*
+ * 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 "04"PRIx32
+#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;
 
@@ -29,14 +93,9 @@ static AdjWord *xalloc_adjmatrix(void) {
 }
 
 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) {
@@ -51,25 +110,50 @@ static int count_set_adj_bits(AdjWord w) {
 }
 
 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"/%"PRADJ" ", adjmatrix_counters[i], adjmatrix[i]);
+    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.
@@ -121,8 +205,8 @@ static void optimise(int doprint) {
   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);
 
@@ -224,48 +308,78 @@ static void optimise(int doprint) {
   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;
+
+    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;
 }