+/*
+ * 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.
+ */
+
+#define _GNU_SOURCE
+
+#include <publib.h>
#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 <math.h>
+#include <sys/types.h>
+#include <sys/wait.h>
+#include <sys/uio.h>
+#include <sys/fcntl.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;
static glp_prob *best_prob;
static AdjWord *best_adjmatrix;
+static int n_over_best;
+static int *weight;
+
static unsigned printcounter;
+static void iterate(void);
+static void iterate_recurse(int i, AdjWord min);
+static bool preconsider_ok(int nwords, bool doprint);
+static bool maxhamweight_ok(void);
+static void optimise(bool doprint);
+
+static void progress_eol(void) {
+ fprintf(stderr," \r");
+ fflush(stderr);
+}
+
+static void set_best(double new_best) {
+ best = new_best;
+ n_over_best = n / best;
+}
+
+/*----- multicore support -----*/
+
+/*
+ * Multicore protocol
+ *
+ * We fork into:
+ * - master (parent)
+ * - generator
+ * - ncpu workers
+ *
+ * ipc facilities:
+ * - one pipe ("work") from generator to workers
+ * - ever-extending file ("bus") containing new "best" values
+ * - one file for each worker giving maxhamweight and adjmatrix for best
+ *
+ * generator runs iterate_recurse to a certain depth and writes the
+ * candidates to a pipe
+ *
+ * workers read candidates from the pipe and resume iterate_recurse
+ * halfway through the recursion
+ *
+ * whenever a worker does a doprint, it checks the bus for new best
+ * value; actual best values are appended
+ *
+ * master waits for generator and all workers to finish and then
+ * runs optimise() for each worker's best, then prints
+ */
+
+static int ncpus = 0, multicore_iteration_boundary = INT_MAX;
+
+static int mc_bus, mc_work[2];
+static off_t mc_bus_read;
+
+typedef struct {
+ int w;
+ FILE *results;
+ pid_t pid;
+} Worker;
+static Worker *mc_us;
+
+static void multicore_check_for_new_best(void);
+
+#define MAX_NIOVS 3
+static AdjWord mc_iter_min;
+static int mc_niovs;
+static size_t mc_iovlen;
+static struct iovec mc_iov[MAX_NIOVS];
+
+#define IOV0 (mc_niovs = mc_iovlen = 0)
+
+#define IOV(obj, count) ({ \
+ assert(mc_niovs < MAX_NIOVS); \
+ mc_iov[mc_niovs].iov_base = &(obj); \
+ mc_iov[mc_niovs].iov_len = sizeof(obj) * (count); \
+ mc_iovlen += mc_iov[mc_niovs].iov_len; \
+ mc_niovs++; \
+ })
+
+static void mc_rwvsetup_outer(void) {
+ IOV0;
+ IOV(maxhamweight, 1);
+ IOV(mc_iter_min, 1);
+ IOV(*adjmatrix, multicore_iteration_boundary);
+}
+
+static void mc_rwvsetup_full(void) {
+ IOV0;
+ IOV(*adjmatrix, n);
+}
+
+static void vlprintf(const char *fmt, va_list al) {
+ vfprintf(stderr,fmt,al);
+ progress_eol();
+}
+
+static void LPRINTF(const char *fmt, ...) {
+ va_list al;
+ va_start(al,fmt);
+ vlprintf(fmt,al);
+ va_end(al);
+}
+
+static void mc_awaitpid(int wnum, pid_t pid) {
+ LPRINTF("master awaiting %2d [%ld]",wnum,(long)pid);
+ int status;
+ pid_t got = waitpid(pid, &status, 0);
+ assert(got == pid);
+ if (status) {
+ fprintf(stderr,"\nFAILED SUBPROC %2d [%ld] %d\n",
+ wnum, (long)pid, status);
+ exit(-1);
+ }
+}
+
+static void multicore_outer_iteration(int i, AdjWord min) {
+ static unsigned check_counter;
+
+ assert(i == multicore_iteration_boundary);
+ mc_iter_min = min;
+ mc_rwvsetup_outer();
+ ssize_t r = writev(mc_work[1], mc_iov, mc_niovs);
+ assert(r == mc_iovlen);
+ /* effectively, this writev arranges to transfers control
+ * to some worker's instance of iterate_recurse via mc_iterate_worker */
+
+ if (!(check_counter++ & 0xff))
+ multicore_check_for_new_best();
+}
+
+static void mc_iterate_worker(void) {
+ for (;;) {
+ mc_rwvsetup_outer();
+ ssize_t r = readv(mc_work[0], mc_iov, mc_niovs);
+ if (r == 0) break;
+ assert(r == mc_iovlen);
+
+ bool ok = maxhamweight_ok();
+ if (!ok) continue;
+
+ ok = preconsider_ok(multicore_iteration_boundary, 1);
+ progress_eol();
+ if (!ok) continue;
+
+ /* stop iterate_recurse from trying to run multicore_outer_iteration */
+ int mc_org_it_bound = multicore_iteration_boundary;
+ multicore_iteration_boundary = INT_MAX;
+ iterate_recurse(mc_org_it_bound, mc_iter_min);
+ multicore_iteration_boundary = mc_org_it_bound;
+ }
+ LPRINTF("worker %2d reporting",mc_us->w);
+ if (best_adjmatrix) {
+ adjmatrix = best_adjmatrix;
+ mc_rwvsetup_full();
+ ssize_t r = writev(fileno(mc_us->results), mc_iov, mc_niovs);
+ assert(r == mc_iovlen);
+ }
+ LPRINTF("worker %2d ending",mc_us->w);
+ exit(0);
+}
+
+static void multicore(void) {
+ Worker *mc_workers;
+ int w;
+ pid_t genpid;
+
+ multicore_iteration_boundary = n / 2;
+
+ FILE *busf = tmpfile(); assert(busf);
+ mc_bus = fileno(busf);
+ int r = fcntl(mc_bus, F_GETFL); assert(r >= 0);
+ r |= O_APPEND;
+ r = fcntl(mc_bus, F_SETFL, r); assert(r >= 0);
+
+ r = pipe(mc_work); assert(!r);
+
+ mc_workers = xmalloc(sizeof(*mc_workers) * ncpus);
+ for (w=0; w<ncpus; w++) {
+ mc_workers[w].w = w;
+ mc_workers[w].results = tmpfile(); assert(mc_workers[w].results);
+ mc_workers[w].pid = fork(); assert(mc_workers[w].pid >= 0);
+ if (!mc_workers[w].pid) {
+ mc_us = &mc_workers[w];
+ close(mc_work[1]);
+ LPRINTF("worker %2d running", w);
+ mc_iterate_worker();
+ exit(0);
+ }
+ }
+
+ close(mc_work[0]);
+
+ genpid = fork(); assert(genpid >= 0);
+ if (!genpid) {
+ LPRINTF("generator running");
+ iterate();
+ exit(0);
+ }
+
+ close(mc_work[1]);
+ mc_awaitpid(-1, genpid);
+ for (w=0; w<ncpus; w++)
+ mc_awaitpid(w, mc_workers[w].pid);
+
+ for (w=0; w<ncpus; w++) {
+ mc_rwvsetup_full();
+ LPRINTF("reading report from %2d",w);
+ ssize_t sr = preadv(fileno(mc_workers[w].results), mc_iov, mc_niovs, 0);
+ if (!sr) continue;
+ maxhamweight = 0;
+ optimise(1);
+ }
+}
+
+static void multicore_check_for_new_best(void) {
+ if (!ncpus) return;
+
+ for (;;) {
+ double msg;
+ ssize_t got = pread(mc_bus, &msg, sizeof(msg), mc_bus_read);
+ if (!got) break;
+ assert(got == sizeof(msg));
+ if (msg > best)
+ set_best(msg);
+ mc_bus_read += sizeof(msg);
+ }
+}
+
+static void multicore_found_new_best(void) {
+ if (!ncpus) return;
+
+ if (mc_us /* might be master */) fprintf(stderr," w%-2d ",mc_us->w);
+ ssize_t wrote = write(mc_bus, &best, sizeof(best));
+ assert(wrote == sizeof(best));
+}
+
+/*----- end of multicore support -----*/
+
static AdjWord *xalloc_adjmatrix(void) {
return xmalloc(sizeof(*adjmatrix)*n);
}
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;
+ setlinebuf(stderr);
+ weight = calloc(sizeof(*weight), m); assert(weight);
+ n_over_best = INT_MAX;
}
static AdjWord one_adj_bit(int bitnum) {
return total;
}
-static void optimise(int doprint) {
- glp_prob *prob = 0;
- int i, j, totalfrags;
+#define PRINTF(...) if (!doprint) ; else fprintf(stderr, __VA_ARGS__)
-#define HAVE_PRINTED ({ \
- if (!doprint) { doprint = 1; goto retry_with_print; } \
- })
- retry_with_print:
-#define PRINTF if (!doprint) ; else printf /* bodgy */
+static int totalfrags;
+
+static bool maxhamweight_ok(void) {
+ double maxminsize = (double)m / maxhamweight;
+ return maxminsize > best;
+}
+
+static bool preconsider_ok(int nwords, bool doprint) {
+ int i;
- for (i=0, totalfrags=0; i<n; i++) {
+ PRINTF("%2d ", maxhamweight);
+
+ bool had_max = 0;
+ for (i=0, totalfrags=0; i<nwords; 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;
+ }
+ return 1;
+
+ out:
+ return 0;
+}
+
+static void optimise(bool doprint) {
+ /* Consider the best answer (if any) for a given adjacency matrix */
+ glp_prob *prob = 0;
+ int i, j;
+
+ /*
+ * 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;
+ }
+
+ bool ok = preconsider_ok(n, doprint);
+ if (!ok)
+ 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);
HAVE_PRINTED;
- best = got;
+ set_best(got);
+ multicore_found_new_best();
if (best_prob) glp_delete_prob(best_prob);
best_prob = prob;
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) progress_eol();
+ if (doprint) multicore_check_for_new_best();
}
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;
+ if (i >= multicore_iteration_boundary) {
+ multicore_outer_iteration(i, min);
+ return;
+ }
+ 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;
+
+ for (int j = 0; j < m; j++)
+ if (adjmatrix[i] & one_adj_bit(j))
+ weight[j]++;
+ for (int j = 0; j < m; j++)
+ if (weight[j] >= n_over_best)
+ goto takeout;
+
+ iterate_recurse(i+1, adjmatrix[i]);
+
+ takeout:
+ for (int j = 0; j < m; j++)
+ if (adjmatrix[i] & one_adj_bit(j))
+ weight[j]--;
+
+ again:
+ if (adjmatrix[i] == adjall)
return;
}
}
static void iterate(void) {
- iterate_recurse(0, 0);
+ for (maxhamweight=1; maxhamweight<=m; maxhamweight++) {
+ if (!maxhamweight_ok())
+ 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();
- printf("\n");
- if (best_prob)
- glp_print_sol(best_prob,"/dev/stdout");
- if (ferror(stdout) || fclose(stdout)) { perror("stdout"); exit(-1); }
+
+ if (ncpus) multicore();
+ else iterate();
+
+ report();
return 0;
}
~ianmdlvl / matchsticks-search.git / blobdiff