\datethis
@*Generalized exact cover.
This program implements an extension of
the algorithm discussed in my paper about ``dancing
links.'' I hacked it together from the {\mc XCOVER} program that I wrote in
1994; I apologize for not having time to apply spit and polish.

Given a matrix whose elements are 0 or 1, the problem in that paper was to
find all subsets of its rows whose sum is at most~1 in all columns and
{\it exactly\/}~1 in all ``primary'' columns. The matrix is specified
in the standard input file as follows: Each column has a symbolic name,
up to seven characters long. The first line of input contains
the names of all primary columns, followed by `\.{\char"7C}', followed by
the names of all other columns.
(If all columns are primary, the~`\.{\char"7C}' may be omitted.)
The remaining lines represent the rows, by listing the columns where 1 appears.

Here I extend the idea so that nonprimary columns can have a different sort
of restriction: If a row specifies a ``color'' in a nonprimary column,
it rules out rows of all other colors in that column,
but any number of rows with the
same color are allowed. (The previous situation was the special case
in which all rows had a different color.)
If \.{xx} is a column name, a specification like \.{xx:a} as part of a row
stands for color~\.a in column~\.{xx}. Each color is specified by a
single character.

The program prints the number of solutions and the total number of link
updates. It also prints every $n$th solution, if the integer command
line argument $n$ is given. A second command-line argument causes the
full search tree to be printed, and a third argument makes the output
even more verbose.

@d max_level 1000 /* at most this many rows in a solution */
@d max_degree 6000 /* at most this many branches per search tree node */
@d max_cols 6000 /* at most this many columns */
@d max_nodes 1000000 /* at most this many nonzero elements in the matrix */
@d verbose Verbose

@c
#include <stdio.h>
#include <stdlib.h>
#include <ctype.h>
#include <string.h>
@<Type definitions@>@;
@<Global variables@>@;
@<Subroutines@>;
@#
main(argc,argv)
  int argc;
  char *argv[];
{
  @<Local variables@>;
  verbose=argc-1;
  if (verbose) sscanf(argv[1],"%d",&spacing);
  @<Initialize the data structures@>;
  @<Backtrack through all solutions@>;
  printf("Altogether %lld solutions,",count);
  printf(" after %lld updates", updates);
  printf(" and %lld cleansings.\n", purifs);
  if (verbose) @<Print a profile of the search tree@>;
  exit(0);
}

@ @<Global...@>=
int verbose; /* $>0$ to show solutions, $>1$ to show partial ones too */
unsigned long long count; /* number of solutions found so far */
unsigned long long updates; /* number of times we deleted a list element */
unsigned long long purifs; /* number of times we purified a list element */
int spacing=1; /* if |verbose|, we output solutions when |count%spacing==0| */
unsigned long long profile[max_level][max_degree]; /* tree nodes of given level and degree */
unsigned long long upd_prof[max_level]; /* updates at a given level */
unsigned long long pur_prof[max_level]; /* purifications at a given level */
int maxb=0; /* maximum branching factor actually needed */
int maxl=0; /* maximum level actually reached */

@*Data structures.
Each column of the input matrix is represented by a \&{column} struct,
and each row is represented as a linked list of \&{node} structs. There's one
node for each nonzero entry in the matrix.

More precisely, the nodes are linked circularly within each row, in
both directions. The nodes are also linked circularly within each column;
the column lists each include a header node, but the row lists do not.
Column header nodes are part of a \&{column} struct, which
contains further info about the column.

Each node contains five fields. Four are the pointers of doubly linked lists,
already mentioned; the fifth points to the column containing the node.

@s col_struct int

@<Type...@>=
typedef struct node_struct {
  struct node_struct *left,*right; /* predecessor and successor in row */
  struct node_struct *up,*down; /* predecessor and successor in column */
  struct col_struct *col; /* the column containing this node */
  int color; /* color, if specified */
} node;

@ Each \&{column} struct contains five fields:
The |head| is a node that stands at the head of its list of nodes;
the |len| tells the length of that list of nodes, not counting the header;
the |name| is a one-, two-, or \dots\ or seven-letter identifier;
|next| and |prev| point to adjacent columns, when this
column is part of a doubly linked list.

As backtracking proceeds, nodes
will be deleted from column lists when their row has been blocked by
other rows in the partial solution.
But when backtracking is complete, the data structures will be
restored to their original state.

@<Type...@>=
typedef struct col_struct {
  node head; /* the list header */
  int len; /* the number of non-header items currently in this column's list */
  char name[8]; /* symbolic identification of the column, for printing */
  struct col_struct *prev,*next; /* neighbors of this column */
} column;

@ One |column| struct is called the root. It serves as the head of the
list of columns that need to be covered, and is identifiable by the fact
that its |name| is empty.

@d root col_array[0] /* gateway to the unsettled columns */

@ A row is identified not by name but by the names of the columns it contains.
Here is a routine that prints a row, given a pointer to any of its
nodes. It also prints the position of the given node in its column.

@<Sub...@>=
print_row(p)
  node *p;
{@+register node *q=p;
  register int k;
  do {
    printf(" %s",q->col->name);
    if (q->color) @<Print the color of node |q|@>;
    q=q->right;
  } while (q!=p);
  for (q=p->col->head.down,k=1;q!=p;k++)
    if (q==&(p->col->head)) {
      printf("\n");@+return 0; /* row not in its column! */
    }@+else q=q->down;
  printf(" (%d of %d)\n",k,p->col->len);
}

@*Inputting the matrix.
Brute force is the rule in this part of the program.

@<Initialize the data structures@>=
@<Read the column names@>;
@<Read the rows@>;

@ @d buf_size 8*max_cols+3 /* upper bound on input line length */

@<Glob...@>=
column col_array[max_cols+2]; /* place for column records */
node node_array[max_nodes]; /* place for nodes */
char buf[buf_size];
column *first_nonprim_col; /* the first nonprimary column, if any */
column *last_nonprim_col; /* the first unused column */

@ @d panic(m) {@+fprintf(stderr,"%s!\n%s",m,buf);@+exit(-1);@+}

@<Read the column names@>=
cur_col=col_array+1;
fgets(buf,buf_size,stdin);
if (buf[strlen(buf)-1]!='\n') panic("Input line too long");
for (p=buf,primary=1;*p;p++) {
  while (isspace(*p)) p++;
  if (!*p) break;
  if (*p=='|') {
    primary=0;
    if (cur_col==col_array+1) panic("No primary columns");
    (cur_col-1)->next=&root, root.prev=cur_col-1;
    first_nonprim_col=cur_col;
    continue;
  }
  for (q=p+1;!isspace(*q);q++);
  if (q>p+7) panic("Column name too long");
  if (cur_col>=&col_array[max_cols]) panic("Too many columns");
  for (q=cur_col->name;!isspace(*p);q++,p++) *q=*p;
  cur_col->head.up=cur_col->head.down=&cur_col->head;
  cur_col->len=0;
  if (primary) cur_col->prev=cur_col-1, (cur_col-1)->next=cur_col;
  else cur_col->prev=cur_col->next=cur_col;
  cur_col++;
}
if (primary) {
  if (cur_col==col_array+1) panic("No primary columns");
  (cur_col-1)->next=&root, root.prev=cur_col-1;
    first_nonprim_col=cur_col;
}
last_nonprim_col=cur_col;

@ @<Local variables@>=
register column *cur_col;
register char *p,*q;
register node *cur_node;
int primary;

@ @<Read the rows@>=
cur_node=node_array;
while (fgets(buf,buf_size,stdin)) {
  register column *ccol;
  register node *row_start, *x;
  if (buf[strlen(buf)-1]!='\n') panic("Input line too long");
  row_start=NULL;
  for (p=buf;*p;p++) {
    while (isspace(*p)) p++;
    if (!*p) break;
    for (q=p+1; !isspace(*q) && *q != ':'; q++);
    if (q>p+7) panic("Column name too long");
    for (q=cur_col->name;!isspace(*p) && *p!=':';q++,p++) *q=*p;
    *q='\0';
    for (ccol=col_array;strcmp(ccol->name,cur_col->name);ccol++);
    if (ccol==cur_col) panic("Unknown column name");
    if (cur_node==&node_array[max_nodes]) panic("Too many nodes");
    if (!row_start) row_start=cur_node;
    else cur_node->left=cur_node-1, (cur_node-1)->right=cur_node;
    for (x=row_start; x!=cur_node; x++)
      if (x->col==ccol) panic("A row can't use a column twice");
    cur_node->col=ccol;
    cur_node->up=ccol->head.up, ccol->head.up->down=cur_node;
    ccol->head.up=cur_node, cur_node->down=&ccol->head;
    ccol->len++;
    if (*p==':') @<Read a color restriction@>;
    cur_node++;
  }
  if (!row_start) panic("Empty row");
  row_start->left=cur_node-1, (cur_node-1)->right=row_start;
}

@*Backtracking.
Our strategy for generating all exact covers will be to repeatedly
choose always the column that appears to be hardest to cover, namely the
column with shortest list, from all columns that still need to be covered.
And we explore all possibilities via depth-first search.

The neat part of this algorithm is the way the lists are maintained.
Depth-first search means last-in-first-out maintenance of data structures;
and it turns out that we need no auxiliary tables to undelete elements from
lists when backing up. The nodes removed from doubly linked lists remember
their former neighbors, because we do no garbage collection.

The basic operation is ``covering a column.'' This means removing it
from the list of columns needing to be covered, and ``blocking'' its
rows: removing nodes from other lists whenever they belong to a row of
a node in this column's list.

@<Backtrack through all solutions@>=
level=0;
forward: @<Set |best_col| to the best column for branching@>;
cover(best_col);
cur_node=choice[level]=best_col->head.down;
advance:if (cur_node==&(best_col->head)) goto backup;
if (verbose>1) {
  printf("L%d:",level);
  print_row(cur_node);
}
@<Cover all other columns of |cur_node|@>;
if (root.next==&root) @<Record solution and |goto recover|@>;
level++;
goto forward;
backup: uncover(best_col);
if (level==0) goto done;
level--;
cur_node=choice[level];@+best_col=cur_node->col;
recover: @<Uncover all other columns of |cur_node|@>;
cur_node=choice[level]=cur_node->down;@+goto advance;
done:if (verbose>3)
  @<Print column lengths, to make sure everything has been restored@>;

@ @<Local...@>=
register column *best_col; /* column chosen for branching */
register node *pp; /* traverses a row */

@ @<Glob...@>=
int level; /* number of choices in current partial solution */
node *choice[max_level]; /* the row and column chosen on each level */

@ When a row is blocked, it leaves all lists except the list of the
column that is being covered. Thus a node is never removed from a list
twice.

@<Subroutines@>=
cover(c)
  column *c;
{@+register column *l,*r;
  register node *rr,*nn,*uu,*dd;
  register int k=1; /* updates */
  l=c->prev;@+r=c->next;
  l->next=r;@+r->prev=l;
  for (rr=c->head.down;rr!=&(c->head);rr=rr->down)
    for (nn=rr->right;nn!=rr;nn=nn->right) {
      uu=nn->up;@+dd=nn->down;
      uu->down=dd;@+dd->up=uu;
      k++;
      nn->col->len--;
    }
  updates+=k;
  upd_prof[level]+=k;
}

@ Uncovering is done in precisely the reverse order. The pointers thereby
execute an exquisitely choreo\-graphed dance which returns them almost
magically to their former state.

@<Subroutines@>=
uncover(c)
  column *c;
{@+register column *l,*r;
  register node *rr,*nn,*uu,*dd;
  for (rr=c->head.up;rr!=&(c->head);rr=rr->up)
    for (nn=rr->left;nn!=rr;nn=nn->left) {
      uu=nn->up;@+dd=nn->down;
      uu->down=dd->up=nn;
      nn->col->len++;
    }
  l=c->prev;@+r=c->next;
  l->next=r->prev=c;
}

@ @<Cover all other columns of |cur_node|@>=
for (pp=cur_node->right;pp!=cur_node;pp=pp->right)
  if (!pp->color) cover(pp->col);
  else if (pp->color>0) purify(pp);

@ We included |left| links, thereby making the rows doubly linked, so
that columns would be uncovered in the correct LIFO order in this
part of the program. (The |uncover| routine itself could have done its
job with |right| links only.) (Think about it.)

@<Uncover all other columns of |cur_node|@>=
for (pp=cur_node->left;pp!=cur_node;pp=pp->left)
  if (!pp->color)  uncover(pp->col);
  else if (pp->color>0) unpurify(pp);

@ @<Set |best_col| to  the best column for branching@>=
minlen=max_nodes;
if (verbose>2) printf("Level %d:",level);
for (cur_col=root.next;cur_col!=&root;cur_col=cur_col->next) {
  if (verbose>2) printf(" %s(%d)",cur_col->name,cur_col->len);
  if (cur_col->len<minlen) best_col=cur_col,minlen=cur_col->len;
}
if (verbose) {
  if (level>maxl) {
    if (level>=max_level) panic("Too many levels");
    maxl=level;
  }
  if (minlen>maxb) {
    if (minlen>=max_degree) panic("Too many branches");
    maxb=minlen;
  }
  profile[level][minlen]++;
  if (verbose>2) printf(" branching on %s(%d)\n",best_col->name,minlen);
}

@ @<Local...@>=
register int minlen;
register int j,k,x;
long long xx,tt;

@ @<Record solution and |goto recover|@>=
{
  count++;
  if (verbose) {
    profile[level+1][0]++;
    if (count%spacing==0) {
      printf("%lld:\n",count);
      for (k=0;k<=level;k++) print_row(choice[k]);
    }
  }
  goto recover;
}

@ @<Print column lengths, to make sure everything has been restored@>=
{
  printf("Final column lengths");
  for (cur_col=root.next;cur_col!=&root;cur_col=cur_col->next)
    printf(" %s(%d)",cur_col->name,cur_col->len);
  printf("\n");
}

@ @<Print a profile...@>=
{
  xx=1; /* the root node doesn't show up in the profile */
  for (level=1;level<=maxl+1;level++) {
    tt=0;
    for (k=0;k<=maxb;k++) {
      printf("%10lld",profile[level][k]);
      tt+=profile[level][k];
    }
    printf("%16lld nodes, %llu updates, %llu cleansings\n",
      tt,upd_prof[level-1],pur_prof[level-1]);
    xx+=tt;
  }
  printf("Total %lld nodes.\n",xx);
}

@* Color barriers. Finally, here's the new material related to coloring.

@<Read a color restriction@>=
{
  if (ccol<first_nonprim_col) panic("Color isn't allowed in a primary column");
  if (isspace(*(p+1)) || !isspace(*(p+2)))
    panic("Color should be a single character");
  cur_node->color=*(p+1);
  p+=2;
}

@ When we choose a row that specifies colors in one or more columns, we
``purify'' those columns by removing all incompatible rows. All rows that
want the same color in a purified column will now be given the color code~$-1$
so that we need not purify the column again.

@<Subroutines@>=
purify(p)
  node *p;
{@+register column *c=p->col;
  register int x=p->color;
  register node *rr,*nn,*uu,*dd;
  register int k=0,kk=1; /* updates */
  c->head.color=x; /* this is used only to help |print_row| */
  for (rr=c->head.down;rr!=&(c->head);rr=rr->down)
    if (rr->color!=x) {
      for (nn=rr->right;nn!=rr;nn=nn->right) {
        uu=nn->up;@+dd=nn->down;
        uu->down=dd;@+dd->up=uu;
        k++;
        nn->col->len--;
      }
    }@+else if (rr!=p) kk++, rr->color=-1;
  updates+=k, purifs+=kk;
  upd_prof[level]+=k, pur_prof[level]+=kk;
}

@ Just as |purify| is analogous to |cover|, the inverse process is analogous
to |uncover|.

@<Subroutines@>=
unpurify(p)
  node *p;
{@+register column *c=p->col;
  register int x=p->color;
  register node *rr,*nn,*uu,*dd;
  for (rr=c->head.up;rr!=&(c->head);rr=rr->up)
    if (rr->color<0) rr->color=x;
    else if (rr!=p) {
      for (nn=rr->left;nn!=rr;nn=nn->left) {
        uu=nn->up;@+dd=nn->down;
        uu->down=dd->up=nn;
        nn->col->len++;
      }
    }
  c->head.color=0;
}

@ @<Print the color of node |q|@>=
printf(":%c",q->color>0? q->color: q->col->head.color);

@*Help for debugging. Here's a subroutine for when I'm
doing a long run and want to check the current progress.

@<Sub...@>=
void show_state()
{
  register int k;
  printf("Current state (level %d):\n",level);
  for (k=0;k<level;k++) print_row(choice[k]);
  printf("Max level so far: %d\n",maxl);
  printf("Max branching so far: %d\n",maxb);
  printf("Solutions so far: %lld\n",count);
}

@*Index.