@s mod and
\let\Xmod=\bmod % this is CWEB magic for using "mod" instead of "%"

\datethis
@*Intro. This program is an ``exact cover solver'' that I'm writing
as an experiment in the use of so-called sparse-set data structures
instead of the dancing links structures I've played with for thirty years.
I plan to write it as if I live on a planet where the sparse-set
ideas are well known, but doubly linked links are almost unheard-of.
As I begin, I have no idea what the tradeoffs will be.

I shall accept the {\mc DLX} input format used in the previous solvers,
without change, so that a fair comparison can be made.
(See the program {\mc DLX1} for definitions. Much of the code from
that program is used to parse the input for this one.)

@ After this program finds all solutions, it normally prints their total
number on |stderr|, together with statistics about how many
nodes were in the search tree, and how many ``updates'' were made.
The running time in ``mems'' is also reported, together with the approximate
number of bytes needed for data storage. One ``mem'' essentially means a
memory access to a 64-bit word.
(These totals don't include the time or space needed to parse the
input or to format the output.)

Here is the overall structure:

@d o mems++ /* count one mem */
@d oo mems+=2 /* count two mems */
@d ooo mems+=3 /* count three mems */
@d O "%" /* used for percent signs in format strings */
@d mod % /* used for percent signs denoting remainder in \CEE/ */

@d max_level 500 /* at most this many options in a solution */
@d max_cols 100000 /* at most this many items */
@d max_nodes 25000000 /* at most this many nonzero elements in the matrix */
@d bufsize (9*max_cols+3) /* a buffer big enough to hold all item names */

@c
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <ctype.h>
typedef unsigned int uint; /* a convenient abbreviation */
typedef unsigned long long ullng; /* ditto */
@<Type definitions@>;
@<Global variables@>;
@<Subroutines@>;
main (int argc, char *argv[]) {
  register int cc,i,j,k,p,pp,q,r,s,t,cur_choice,cur_node,best_itm;
  @<Process the command line@>;
  @<Input the item names@>;
  @<Input the options@>;
  if (vbose&show_basics)
    @<Report the successful completion of the input phase@>;
  if (vbose&show_tots)
    @<Report the item totals@>;
  imems=mems, mems=0;
  @<Solve the problem@>;
done:@+if (vbose&show_tots)
    @<Report the item totals@>;
  if (vbose&show_profile) @<Print the profile@>;
  if (vbose&show_max_deg)
    fprintf(stderr,"The maximum branching degree was "O"d.\n",maxdeg);
  if (vbose&show_basics) {
    fprintf(stderr,"Altogether "O"llu solution"O"s, "O"llu+"O"llu mems,",
                                count,count==1?"":"s",imems,mems);
    bytes=(itemlength+setlength)*sizeof(int)+last_node*sizeof(node)
        +maxl*sizeof(int);
    fprintf(stderr," "O"llu updates, "O"llu bytes, "O"llu nodes.\n",
                                updates,bytes,nodes);
  }
  @<Close the files@>;
}

@ You can control the amount of output, as well as certain properties
of the algorithm, by specifying options on the command line:
\smallskip\item{$\bullet$}
`\.v$\langle\,$integer$\,\rangle$' enables or disables various kinds of verbose
 output on |stderr|, given by binary codes such as |show_choices|;
\item{$\bullet$}
`\.m$\langle\,$integer$\,\rangle$' causes every $m$th solution
to be output (the default is \.{m0}, which merely counts them);
\item{$\bullet$}
`\.d$\langle\,$integer$\,\rangle$' sets |delta|, which causes periodic
state reports on |stderr| after the algorithm has performed approximately
|delta| mems since the previous report (default 10000000000);
\item{$\bullet$}
`\.c$\langle\,$positive integer$\,\rangle$' limits the levels on which
choices are shown during verbose tracing;
\item{$\bullet$}
`\.C$\langle\,$positive integer$\,\rangle$' limits the levels on which
choices are shown in the periodic state reports;
\item{$\bullet$}
`\.l$\langle\,$nonnegative integer$\,\rangle$' gives a {\it lower\/} limit,
relative to the maximum level so far achieved, to the levels on which
choices are shown during verbose tracing;
\item{$\bullet$}
`\.t$\langle\,$positive integer$\,\rangle$' causes the program to
stop after this many solutions have been found;
\item{$\bullet$}
`\.T$\langle\,$integer$\,\rangle$' sets |timeout| (which causes abrupt
termination if |mems>timeout| at the beginning of a level);
\item{$\bullet$}
`\.S$\langle\,$filename$\,\rangle$' to output a ``shape file'' that encodes
the search tree.

@d show_basics 1 /* |vbose| code for basic stats; this is the default */
@d show_choices 2 /* |vbose| code for backtrack logging */
@d show_details 4 /* |vbose| code for further commentary */
@d show_profile 128 /* |vbose| code to show the search tree profile */
@d show_full_state 256 /* |vbose| code for complete state reports */
@d show_tots 512 /* |vbose| code for reporting item totals at start and end */
@d show_warnings 1024 /* |vbose| code for reporting options without primaries */
@d show_max_deg 2048 /* |vbose| code for reporting maximum branching degree */

@<Glob...@>=
int vbose=show_basics+show_warnings; /* level of verbosity */
int spacing; /* solution $k$ is output if $k$ is a multiple of |spacing| */
int show_choices_max=1000000; /* above this level, |show_choices| is ignored */
int show_choices_gap=1000000; /* below level |maxl-show_choices_gap|,
    |show_details| is ignored */
int show_levels_max=1000000; /* above this level, state reports stop */
int maxl=0; /* maximum level actually reached */
char buf[bufsize]; /* input buffer */
ullng count; /* solutions found so far */
ullng options; /* options seen so far */
ullng imems,mems; /* mem counts */
ullng updates; /* update counts */
ullng bytes; /* memory used by main data structures */
ullng nodes; /* total number of branch nodes initiated */
ullng thresh=10000000000; /* report when |mems| exceeds this, if |delta!=0| */
ullng delta=10000000000; /* report every |delta| or so mems */
ullng maxcount=0xffffffffffffffff; /* stop after finding this many solutions */
ullng timeout=0x1fffffffffffffff; /* give up after this many mems */
FILE *shape_file; /* file for optional output of search tree shape */
char *shape_name; /* its name */
int maxdeg; /* the largest branching degree seen so far */

@ If an option appears more than once on the command line, the first
appearance takes precedence.

@<Process the command line@>=
for (j=argc-1,k=0;j;j--) switch (argv[j][0]) {
case 'v': k|=(sscanf(argv[j]+1,""O"d",&vbose)-1);@+break;
case 'm': k|=(sscanf(argv[j]+1,""O"d",&spacing)-1);@+break;
case 'd': k|=(sscanf(argv[j]+1,""O"lld",&delta)-1),thresh=delta;@+break;
case 'c': k|=(sscanf(argv[j]+1,""O"d",&show_choices_max)-1);@+break;
case 'C': k|=(sscanf(argv[j]+1,""O"d",&show_levels_max)-1);@+break;
case 'l': k|=(sscanf(argv[j]+1,""O"d",&show_choices_gap)-1);@+break;
case 't': k|=(sscanf(argv[j]+1,""O"lld",&maxcount)-1);@+break;
case 'T': k|=(sscanf(argv[j]+1,""O"lld",&timeout)-1);@+break;
case 'S': shape_name=argv[j]+1, shape_file=fopen(shape_name,"w");
  if (!shape_file)
    fprintf(stderr,"Sorry, I can't open file `"O"s' for writing!\n",
      shape_name);
  break;
default: k=1; /* unrecognized command-line option */
}
if (k) {
  fprintf(stderr, "Usage: "O"s [v<n>] [m<n>] [s<n>] [d<n>]"
       " [c<n>] [C<n>] [l<n>] [t<n>] [T<n>] [S<bar>] < foo.dlx\n",
                            argv[0]);
  exit(-1);
}

@ @<Close the files@>=
if (shape_file) fclose(shape_file);

@*Data structures.
Sparse-set data structures were introduced by Preston Briggs
and Linda Torczon [{\sl ACM Letters on Programming Languages and Systems\/
\bf2} (1993), 59--69], who realized that exercise 2.12 in
Aho, Hopcroft, and Ullman's classic text {\sl The Design and Analysis
of Computer Algorithms\/} (Addison--Wesley, 1974) was much more than
just a slick trick to avoid initializing an array.
(Indeed, {\sl TAOCP\/} exercise 2.2.6--24 calls it the ``sparse array trick.'')

The basic idea is amazingly simple, when specialized to the situations
that we need to deal with: We can represent a subset~$S$ of the universe
$U=\{x_0,x_1,\ldots,x_{n-1}\}$ by maintaining two $n$-element arrays
$p$ and $q$, each of which is a permutation of~$\{0,1,\ldots,n-1\}$,
together with an integer $s$ in the range $0\le s\le n$. In fact, $p$ is
the {\it inverse\/} of~$q$; and $s$ is the number of elements of~$S$.
The current value of the set $S$ is then simply
$\{x_{p_0},\ldots,x_{p_{s-1}}\}$. (Notice that every $s$-element subset can be
represented in $s!\,(n-s)!$ ways.)

It's easy to test if $x_k\in S$, because that's true if and only if $q_k<s$.
It's easy to insert a new element $x_k$ into~$S$: Swap indices
so that $p_s=k$, $q_k=s$, then increase $s$ by~1.
It's easy to delete an element $x_k$ that belongs to~$S$: Decrease $s$
by~1, then swap indices so that $p_s=k$ and $q_k=s$.
And so on.

Briggs and Torczon were interested in applications where $s$ begins at
zero and tends to remain small. In such cases, $p$ and $q$ need not
be permutations: The values of $p_s$, $p_{s+1}$, \dots, $p_{n-1}$ can
be garbage, and the values of $q_k$ need be defined only when $x_k\in S$.
(Such situations correspond to Aho, Hopcroft, and Ullman, who started
with an array full of garbage and used a sparse-set structure to remember
the set of nongarbage cells.) Our applications are different: Each set
begins equal to its intended universe, and gradually shrinks. In such
cases, we might as well maintain inverse permutations.
The basic operations go faster when we know in advance that we
aren't inserting an element that's already present (nor deleting an element
that isn't).

Many variations are possible. For example, $p$ could be a permutation
of $\{x_0,x_1,\ldots,x_{n-1}\}$ instead of permutation of
$\{0,1,\ldots,n-1\}$. The arrays that play the role of $q$
in the following routines don't have indices that are consecutive;
they live inside of other structures.

@ This program has an
array called |item|, with one entry for each item. The value of |item[k]|
is an index~|x| into a much larger array called |set|. The set of all
options that involve the $k$th item appears in that array beginning
at |set[x]|; and it continues for $s$ consecutive entries, where |s=size(x)|
is an abbreviation for |set[x-1]|. If |item[k]=x|, we maintain the
relation |pos(x)=k|, where |pos(x)| is an abbreviation for
|set[x-2]|. Thus |item| plays the role of array~$p$, in a
sparse-set data structure for the set of all currently active items;
and |pos| plays the role of~$q$.

Suppose the |k|th item |x| currently appears in |s| options. Those options
are indices into |nd|, which is an array of ``nodes.'' Each node
has two fields, |itm| and |loc|. If |x<=q<x+s|, let |y=set[q]|.
This is essentially a pointer to a node, and we have
|nd[y].itm=x|, |nd[y].loc=q|. In other words,
the sequential list of |s| elements that begins at
|x=item[k]| in the |set| array is the sparse-set representation of the
currently active options that contain the |k|th item.
The |itm| fields remain constant,
once we've initialized everything, but the |loc| fields will change.

The given options are stored sequentially in the |nd| array, with one node
per item, separated by ``spacer'' nodes. If |y| is the spacer node
following an option with $t$ items, we have |nd[y].itm=-t|.
If |y| is the spacer node {\it preceding\/} an option with $t$ items,
we have |nd[y].loc=t|.

This probably sounds confusing, until you can see some code.
Meanwhile, let's take note of the invariant relations that hold
whenever |k|, |q|, |x|, and |y| have appropriate values:
$$\hbox{|pos(item[k])=k|;\quad
|nd[set[q]].loc=q|;\quad
|item[pos(x)]=x|;\quad
|set[nd[y].loc]=y|.}$$
(These are the analogs of the invariant relations |p[q[k]]=q[p[k]]=k| in
the simple sparse-set scheme that we started with.)

The |set| array contains also the item names.

We count one mem for a simultaneous access to the |itm| and |loc| fields
of a node.

@d size(x) set[(x)-1] /* number of active options of the |k|th item, |x| */
@d pos(x) set[(x)-2] /* where that item is found in the |item| array */
@d lname(x) set[(x)-4] /* the first four bytes of |x|'s name */
@d rname(x) set[(x)-3] /* the last four bytes of |x|'s name */

@<Type...@>=
typedef struct node_struct {
  int itm; /* the item |x| corresponding to this node */
  int loc; /* where this node resides in |x|'s active set */
} node;

@ @<Glob...@>=
node nd[max_nodes]; /* the master list of nodes */
int last_node; /* the first node in |nd| that's not yet used */
int item[max_cols]; /* the master list of items */
int second=max_cols; /* boundary between primary and secondary items */
int last_itm; /* items seen so far during input, plus 1 */
int set[max_nodes+4*max_cols]; /* the sets of active options for active items */
int itemlength; /* number of elements used in |item| */
int setlength; /* number of elements used in |set| */
int active; /* current number of active items */

@ We're going to store string data (an item name) in the midst of
the integer array |set|. So we've got to do some type coercion using
low-level \CEE/-ness.

@<Type def...@>=
typedef struct {
  int l,r;
} twoints;
typedef union {
  unsigned char str[8]; /* eight one-byte characters */
  twoints lr; /* two four-byte integers */
} stringbuf;
stringbuf namebuf;

@ @<Subroutines@>=
void print_item_name(int k,FILE *stream) {
  namebuf.lr.l=lname(k),namebuf.lr.r=rname(k);
  fprintf(stream," "O".8s",namebuf.str);
}

@ An option is identified not by name but by the names of the items it contains.
Here is a routine that prints an option, given a pointer to any of its
nodes. It also prints the position of the option in its item list.

@<Sub...@>=
void print_option(int p,FILE *stream) {
  register int k,q,x;
  x=nd[p].itm;
  if (p>=last_node || x<=0) {
    fprintf(stderr,"Illegal option "O"d!\n",p);
    return;
  }
  for (q=p;;) {
    print_item_name(x,stream);
    q++;
    x=nd[q].itm;
    if (x<0) q+=x,x=nd[q].itm;
    if (q==p) break;
  }
  k=nd[q].loc;
  fprintf(stream," ("O"d of "O"d)\n",k-x+1,size(x));
}
@#
void prow(int p) {
  print_option(p,stderr);
}

@ When I'm debugging, I might want to look at one of the current item lists.

@<Sub...@>=
void print_itm(int c) {
  register int p;
  if (c<4 || c>=setlength ||
         pos(c)<0 || pos(c)>=itemlength || item[pos(c)]!=c) {
    fprintf(stderr,"Illegal item "O"d!\n",c);
    return;
  }
  fprintf(stderr,"Item");
  print_item_name(c,stderr);
  if (pos(c)<second) fprintf(stderr," ("O"d of "O"d), length "O"d:\n",
         pos(c)+1,active,size(c));
  else fprintf(stderr," (secondary "O"d), length "O"d:\n",
         pos(c)+1,size(c));
  for (p=c;p<c+size(c);p++) prow(set[p]);
}

@ Speaking of debugging, here's a routine to check if redundant parts of our
data structure have gone awry.

@d sanity_checking 0 /* set this to 1 if you suspect a bug */

@<Sub...@>=
void sanity(void) {
  register int k,x,i,l,r;
  for (k=0;k<itemlength;k++) {
    x=item[k];
    if (pos(x)!=k) {
      fprintf(stderr,"Bad pos field of item");
      print_item_name(x,stderr);
      fprintf(stderr," ("O"d,"O"d)!\n",k,x);
    }
  }
  for (i=0;i<last_node;i++) {
    l=nd[i].itm,r=nd[i].loc;
    if (l<=0) {
      if (nd[i+r+1].itm!=-r)
        fprintf(stderr,"Bad spacer in nodes "O"d, "O"d!\n",i,i+r+1);
    }@+else {
      if (l>r) fprintf(stderr,"itm>loc in node "O"d!\n",i);
      else if (set[r]!=i) {
        fprintf(stderr,"Bad loc field for option "O"d of item",r-l+1);
        print_item_name(l,stderr);
        fprintf(stderr," in node "O"d!\n",i);
      }
    }
  }
}

@*Inputting the matrix. Brute force is the rule in this part of the code,
whose goal is to parse and store the input data and to check its validity.

We use only four entries of |set| per item while reading the item-name line.

@d panic(m) {@+fprintf(stderr,""O"s!\n"O"d: "O".99s\n",m,p,buf);@+exit(-666);@+}

@<Input the item names@>=
while (1) {
  if (!fgets(buf,bufsize,stdin)) break;
  if (o,buf[p=strlen(buf)-1]!='\n') panic("Input line way too long");
  for (p=0;o,isspace(buf[p]);p++) ;
  if (buf[p]=='|' || !buf[p]) continue; /* bypass comment or blank line */
  last_itm=1;
  break;
}
if (!last_itm) panic("No items");
for (;o,buf[p];) {
  o,namebuf.lr.l=namebuf.lr.r=0;
  for (j=0;j<8 && (o,!isspace(buf[p+j]));j++) {
    if (buf[p+j]==':' || buf[p+j]=='|')
              panic("Illegal character in item name");
    o,namebuf.str[j]=buf[p+j];
  }
  if (j==8 && !isspace(buf[p+j])) panic("Item name too long");
  oo,lname(last_itm<<2)=namebuf.lr.l,rname(last_itm<<2)=namebuf.lr.r;
  @<Check for duplicate item name@>;
  last_itm++;
  if (last_itm>max_cols) panic("Too many items");
  for (p+=j+1;o,isspace(buf[p]);p++) ;
  if (buf[p]=='|') {
    if (second!=max_cols) panic("Item name line contains | twice");
    second=last_itm;
    for (p++;o,isspace(buf[p]);p++) ;
  }
}
if (second==max_cols) second=last_itm;

@ @<Check for duplicate item name@>=
for (k=last_itm-1;k;k--) {
  if (o,lname(k<<2)!=namebuf.lr.l) continue;
  if (rname(k<<2)==namebuf.lr.r) break;
}
if (k) panic("Duplicate item name");

@ @<Input the options@>=
while (1) {
  if (!fgets(buf,bufsize,stdin)) break;
  if (o,buf[p=strlen(buf)-1]!='\n') panic("Option line too long");
  for (p=0;o,isspace(buf[p]);p++) ;
  if (buf[p]=='|' || !buf[p]) continue; /* bypass comment or blank line */
  i=last_node; /* remember the spacer at the left of this option */
  for (pp=0;buf[p];) {
    o,namebuf.lr.l=namebuf.lr.r=0;
    for (j=0;j<8 && (o,!isspace(buf[p+j]));j++)
      o,namebuf.str[j]=buf[p+j];
    if (j==8 && !isspace(buf[p+j])) panic("Item name too long");
    @<Create a node for the item named in |buf[p]|@>;
    for (p+=j+1;o,isspace(buf[p]);p++) ;
  }
  if (!pp) {
    if (vbose&show_warnings)
      fprintf(stderr,"Option ignored (no primary items): "O"s",buf);
    while (last_node>i) {
      @<Remove |last_node| from its item list@>;
      last_node--;
    }
  }@+else {
    o,nd[i].loc=last_node-i; /* complete the previous spacer */
    last_node++; /* create the next spacer */
    if (last_node==max_nodes) panic("Too many nodes");
    options++;
    o,nd[last_node].itm=i+1-last_node;
  }
}
@<Initialize |item|@>;
@<Expand |set|@>;
@<Adjust |nb|@>;

@ We temporarily use |pos| to recognize duplicate items in an option.

@<Create a node for the item named in |buf[p]|@>=
for (k=(last_itm-1)<<2;k;k-=4) {
  if (o,lname(k)!=namebuf.lr.l) continue;
  if (rname(k)==namebuf.lr.r) break;
} 
if (!k) panic("Unknown item name");
if (o,pos(k)>i) panic("Duplicate item name in this option");
last_node++;
if (last_node==max_nodes) panic("Too many nodes");
o,t=size(k); /* how many previous options have used this item? */
o,nd[last_node].itm=k>>2,nd[last_node].loc=t;
if ((k>>2)<second) pp=1;
o,size(k)=t+1, pos(k)=last_node;

@ @<Remove |last_node| from its item list@>=
o,k=nd[last_node].itm<<2;
oo,size(k)--,pos(k)=i-1;

@ @<Initialize |item|@>=
itemlength=last_itm-1;
for (k=0,j=4;k<itemlength;k++)
  oo,item[k]=j,j+=4+size((k+1)<<2);
setlength=j-4;
active=second=second-1;

@ Going from high to low, we now move the item names and sizes
to their final positions (leaving room for the pointers into |nb|).

@<Expand |set|@>=
for (;k;k--) {
  o,j=item[k-1];
  oo,size(j)=size(k<<2);
  o,pos(j)=k-1;
  oo,rname(j)=rname(k<<2),lname(j)=lname(k<<2);
}

@ @<Adjust |nb|@>=
for (k=1;k<last_node;k++) {
  if (o,nd[k].itm<0) continue; /* skip over a spacer */
  o,j=item[nd[k].itm-1];
  i=j+nd[k].loc; /* no mem charged because we just read |nd[k].itm| */
  o,nd[k].itm=j,nd[k].loc=i;
  o,set[i]=k;
}

@ The ``number of entries'' includes spacers (because {\mc DLX1}
includes spacers in its reports). If you want to know the
sum of the option lengths, just subtract the number of options.

@<Report the successful completion of the input phase@>=
fprintf(stderr,
  "("O"lld options, "O"d+"O"d items, "O"d entries successfully read)\n",
                       options,second,last_itm-second-1,last_node);

@ The item lengths after input should agree with the item lengths
after this program has finished. I print them (on request), in order to
provide some reassurance that the algorithm isn't badly screwed up.

[Caution: They will probably appear in a different order than before!]

@<Report the item totals@>=
{
  fprintf(stderr,"Item totals:");
  for (k=0;k<itemlength;k++) {
    if (k==second) fprintf(stderr," |");
    fprintf(stderr," "O"d",size(item[k]));
  }
  fprintf(stderr,"\n");
}

@*The dancing.
Our strategy for generating all exact covers will be to repeatedly
choose always an item that appears to be hardest to cover, namely the
item with smallest set, from all items 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 sparse-set representations remember
enough of what was done so that we can undo it later.

The basic operation is ``covering an item.'' This means removing it
from the set of items needing to be covered, and ``hiding'' its
options: removing them from the sets of the other items they contain.

@<Solve the problem@>=
level=0;
forward: nodes++;
if (vbose&show_profile) profile[level]++;
if (sanity_checking) sanity();
@<Do special things if enough |mems| have accumulated@>;
@<Set |best_itm| to the best item for branching@>;
if (t==0) goto donewithlevel;
cover(best_itm);
cur_choice=best_itm;
oo,cur_node=choice[level]=set[best_itm];
goto tryit;
advance:@+if (o,cur_choice>=best_itm+size(best_itm)) goto backup;
oo,cur_node=choice[level]=set[cur_choice];
tryit:@+if ((vbose&show_choices) && level<show_choices_max) {
  fprintf(stderr,"L"O"d:",level);
  print_option(cur_node,stderr);
}
@<Cover all other items of |cur_node|@>;
if (active==0) @<Visit a solution and |goto recover|@>;
if (++level>maxl) {
  if (level>=max_level) {
    fprintf(stderr,"Too many levels!\n");
    exit(-4);
  }
  maxl=level;
}
goto forward;
backup: uncover(best_itm);
donewithlevel:@+if (level==0) goto done;
level--;
oo,cur_node=choice[level],best_itm=nd[cur_node].itm,cur_choice=nd[cur_node].loc;
recover: @<Uncover all other items of |cur_node|@>;
cur_choice++;@+goto advance;

@ @<Glob...@>=
int level; /* number of choices in current partial solution */
int choice[max_level]; /* the node chosen on each level */
ullng profile[max_level]; /* number of search tree nodes on each level */

@ @<Do special things if enough |mems| have accumulated@>=
if (delta && (mems>=thresh)) {
  thresh+=delta;
  if (vbose&show_full_state) print_state();
  else print_progress();
}
if (mems>=timeout) {
  fprintf(stderr,"TIMEOUT!\n");@+goto done;
}

@ When an option is hidden, it leaves all lists except the list of the
item that is being covered. Thus a node is never removed from a list
twice.

@<Sub...@>=
void cover(int c) {
  register int k,a,cc,s,rr,ss,nn,tt,uu,vv,nnp;
  o,k=pos(c);
  if (k<second) { /* update the active list, if |c| is primary */
    a=active-1,active=a;
    o,cc=item[a];
    oo,item[a]=c,item[k]=cc;
    oo,pos(cc)=k,pos(c)=a;
    updates++;
  }
  for (o,rr=c,s=c+size(c);rr<s;rr++)
    @<Remove the option |set[rr]| from the other sets it's in@>;
}

@ @<Remove the option |set[rr]| from the other sets it's in@>=
{
  for (o,tt=set[rr],nn=tt+1;nn!=tt;) {
    o,uu=nd[nn].itm,vv=nd[nn].loc;
    if (uu<0) {@+nn+=uu;@+continue;@+}
    o,ss=size(uu)-1;
    o,nnp=set[uu+ss];
    o,size(uu)=ss;
    oo,set[uu+ss]=nn,set[vv]=nnp;
    oo,nd[nn].loc=uu+ss,nd[nnp].loc=vv;
    nn++;
    updates++;
  }
}

@ To undo the |cover| operation, we need only increase the set size,
because the previously deleted element is in position to be seamlessly
reinstated. (Inactive elements are never moved.)
We need not swap that element back to its former position.

@<Subroutines@>=
void uncover(int c) {
  register int k,cc,s,rr,ss,nn,tt,uu;
  for (o,rr=c,s=c+size(c);rr<s;rr++)
    @<Unremove the option |set[rr]| from the other sets it was in@>;
  o,k=pos(c);
  if (k<second) active++;
}

@ @<Unremove the option |set[rr]| from the other sets it was in@>=
{
  for (o,tt=set[rr],nn=tt+1;nn!=tt;) {
    o,uu=nd[nn].itm;
    if (uu<0) {@+nn+=uu;@+continue;@+}
    o,ss=size(uu)+1;
    o,size(uu)=ss;
    nn++;
  }
}

@ @<Cover all other items of |cur_node|@>=
for (pp=cur_node+1;pp!=cur_node;) {
  o,cc=nd[pp].itm;
  if (cc<0) pp+=cc;
  else cover(cc),pp++;
}

@ Covering and uncovering both traverse options to the right.
That's okay---although it takes a bit of thought to verify that all
sets are restored correctly. (An item that has lost $k$ options
from its set will regain those $k$ options, but not necessarily
in the same order.)

But we do need to go left here, {\it not\/} right.

@<Uncover all other items of |cur_node|@>=
for (pp=cur_node-1;pp!=cur_node;) {
  o,cc=nd[pp].itm;
  if (cc<=0) pp+=nd[pp].loc;
  else uncover(cc),pp--;
}

@ The ``best item'' is considered to be an item that minimizes the
number of remaining choices. If there are several candidates, we
choose the leftmost.

(This program explores the search space in a different order
from {\mc DLX1}, because the ordering of items in the active list
is no longer fixed. Thus ties are broken in a different way.)

@<Set |best_itm| to the best item for branching@>=
t=max_nodes;
if ((vbose&show_details) &&
    level<show_choices_max && level>=maxl-show_choices_gap)
  fprintf(stderr,"Level "O"d:",level);
for (k=0;t&&(k<active);k++) {
  oo,s=size(item[k]);
  if ((vbose&show_details) &&
      level<show_choices_max && level>=maxl-show_choices_gap) {
    print_item_name(item[k],stderr);
    fprintf(stderr,"("O"d)",s);
  }
  if (s<t) best_itm=item[k],t=s;
}
if ((vbose&show_details) &&
    level<show_choices_max && level>=maxl-show_choices_gap) {
  fprintf(stderr," branching on");
  print_item_name(best_itm,stderr);
  fprintf(stderr,"("O"d)\n",t);
}
if (t>maxdeg) maxdeg=t;
if (shape_file) {
  fprintf(shape_file,""O"d",t);
  print_item_name(best_itm,shape_file);
  fprintf(shape_file,"\n");
  fflush(shape_file);
}

@ @<Visit a solution and |goto recover|@>=
{
  nodes++; /* a solution is a special node, see 7.2.2--(4) */
  if (level+1>maxl) {
    if (level+1>=max_level) {
      fprintf(stderr,"Too many levels!\n");
      exit(-5);
    }
    maxl=level+1;
  }
  if (vbose&show_profile) profile[level+1]++;
  if (shape_file) {
    fprintf(shape_file,"sol\n");@+fflush(shape_file);
  }
  @<Record solution and |goto recover|@>;
}

@ @<Record solution and |goto recover|@>=
{
  count++;
  if (spacing && (count mod spacing==0)) {
    printf(""O"lld:\n",count);
    for (k=0;k<=level;k++) print_option(choice[k],stdout);
    fflush(stdout);
  }
  if (count>=maxcount) goto done;
  goto recover;
}

@ @<Sub...@>=
void print_state(void) {
  register int l;
  fprintf(stderr,"Current state (level "O"d):\n",level);
  for (l=0;l<level;l++) {
    print_option(choice[l],stderr);
    if (l>=show_levels_max) {
      fprintf(stderr," ...\n");
      break;
    }
  }
  fprintf(stderr," "O"lld solutions, "O"lld mems, and max level "O"d so far.\n",
                              count,mems,maxl);
}

@ During a long run, it's helpful to have some way to measure progress.
The following routine prints a string that indicates roughly where we
are in the search tree. The string consists of character pairs, separated
by blanks, where each character pair represents a branch of the search
tree. When a node has $d$ descendants and we are working on the $k$th,
the two characters respectively represent $k$ and~$d$ in a simple code;
namely, the values 0, 1, \dots, 61 are denoted by
$$\.0,\ \.1,\ \dots,\ \.9,\ \.a,\ \.b,\ \dots,\ \.z,\ \.A,\ \.B,\ \dots,\.Z.$$
All values greater than 61 are shown as `\.*'. Notice that as computation
proceeds, this string will increase lexicographically.

Following that string, a fractional estimate of total progress is computed,
based on the na{\"\i}ve assumption that the search tree has a uniform
branching structure. If the tree consists
of a single node, this estimate is~.5; otherwise, if the first choice
is `$k$ of~$d$', the estimate is $(k-1)/d$ plus $1/d$ times the
recursively evaluated estimate for the $k$th subtree. (This estimate
might obviously be very misleading, in some cases, but at least it
grows monotonically.)

@<Sub...@>=
void print_progress(void) {
  register int l,k,d,c,p;
  register double f,fd;
  fprintf(stderr," after "O"lld mems: "O"lld sols,",mems,count);
  for (f=0.0,fd=1.0,l=0;l<level;l++) {
    c=nd[choice[l]].itm,d=size(c),k=nd[choice[l]].loc-c+1;
    fd*=d,f+=(k-1)/fd; /* choice |l| is |k| of |d| */
    fprintf(stderr," "O"c"O"c",
      k<10? '0'+k: k<36? 'a'+k-10: k<62? 'A'+k-36: '*',
      d<10? '0'+d: d<36? 'a'+d-10: d<62? 'A'+d-36: '*');
    if (l>=show_levels_max) {
      fprintf(stderr,"...");
      break;
    }
  }
  fprintf(stderr," "O".5f\n",f+0.5/fd);
}

@ @<Print the profile@>=
{
  fprintf(stderr,"Profile:\n");
  for (level=0;level<=maxl;level++)
    fprintf(stderr,""O"3d: "O"lld\n",
                              level,profile[level]);
}

@*Index.