X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ianmdlvl/git?a=blobdiff_plain;f=main.c;h=da9e5fa089503f9c3dfdbb44507316824f113fbd;hb=f4a5716e75bb34407a452675ef662e8d8c95c67f;hp=6465f9be57ed5ae331363a28ecd78271248c1c7e;hpb=ba89f161b9bc185cb1d6ce236fdfdb71f42e6a86;p=matchsticks-search.git

diff --git a/main.c b/main.c
index 6465f9b..da9e5fa 100644
--- a/main.c
+++ b/main.c
@@ -1,42 +1,162 @@
+/*
+ * 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 <unistd.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 int n, m, maxhamweight;
 static AdjWord *adjmatrix;
 static AdjWord adjall;
 
 static double best;
+static glp_prob *best_prob;
+static AdjWord *best_adjmatrix;
+
+static unsigned printcounter;
+
+static int ncpus = 1;
+
+static AdjWord *xalloc_adjmatrix(void) {
+  return xmalloc(sizeof(*adjmatrix)*n);
+}
 
 static void prep(void) {
   adjall = ~((~(AdjWord)0) << m);
-  adjmatrix = xmalloc(sizeof(*adjmatrix)*n);
+  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) {
   int j, total;
   for (j=0, total=0; j<m; j++)
-    total += !!(w & ((AdjWord)1 << j));
+    total += !!(w & one_adj_bit(j));
   return total;
 }
 
-static void optimise(void) {
-  int i;
-  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;
+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.
@@ -46,43 +166,240 @@ static void optimise(void) {
    * fragment from old match i and making it part of new match j.
    *
    * The structural variables (columns) are:
-   *   x_fragsz_i_j    the size of that fragment
-   *   x_minimum       minimum size of any fragment
+   *   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, constraints) are:
+   * 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)
-   *   x_fragmin_i_j   amount by which fragment is > minimum (lower bound 0)
    *
-   * The objective function is simply to maximise x_minimum
+   * 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.
    */
 
-  printf("nyi\n");
+  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++) {
+    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;
+    }
+  }
+
+  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();
+    printcounter++;
+    optimise(!(printcounter & 0xfff));
     return;
   }
   for (adjmatrix[i] = min;
        ;
        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, 1);
+  for (maxhamweight=1; maxhamweight<=m; maxhamweight++) {
+    double maxminsize = (double)m / maxhamweight;
+    if (maxminsize <= best)
+      continue;
+
+    iterate_recurse(0, 1);
+  }
 }
 
+static void report(void) {
+  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); }
+}
+ 
 int main(int argc, char **argv) {
+  int opt;
+  while ((opt = getopt(argc,argv,"j:")) >= 0) {
+    switch (opt) {
+    case 'j': ncpus = atoi(optarg); break;
+    case '+': assert(!"bad option");
+    default: abort();
+    }
+  }
+  argc -= optind-1;
+  argv += optind-1;
+  assert(argc==3);
   n = atoi(argv[1]);
   m = atoi(argv[2]);
+
   prep();
   iterate();
-  if (ferror(stdout) || fclose(stdout)) { perror("stdout"); exit(-1); }
+  report();
   return 0;
 }