@s item int
@s node int
@s mod and
\let\Xmod=\bmod % this is CWEB magic for using "mod" instead of "%"
\def\dts{\mathinner{\ldotp\ldotp}}

\datethis
@*Intro. This program is part of a series of ``exact cover solvers'' that
I'm putting together for my own education as I prepare to write Section
7.2.2.1 of {\sl The Art of Computer Programming}. My intent is to
have a variety of compatible programs on which I can run experiments,
in order to learn how different approaches work in practice.

The basic input format for all of these solvers is described at the beginning
of program {\mc DLX1}, and you should read that description now if you are
unfamiliar with it. Please read also the opening paragraphs of {\mc DLX2},
which adds ``color controls'' to nonprimary items.

{\mc DLX3} extends {\mc DLX2} by allowing the item totals to be
more flexible: Instead of insisting that each primary item occurs
exactly once in the chosen options, we prescribe an {\it interval\/} of
permissible values $[a_j\dts b_j]$ for each primary item~$j$, and we find all
solutions in which the sum $s_1s_2\ldots s_n$ of chosen options satisfies
$a_j\le s_j\le b_j$ for such~$j$. 
(In a sense this represents a generalization from sets to
{\it multisets\/}, although the options themselves are still sets.)

These bounds appear in the first ``item-naming'' line of input:
You can write `$a_j$\.:$b_j$\.{\char"7C}' just before the item name,
where $a_j$ and $b_j$ are decimal integers.
But $a_j$ and the colon can be omitted if $a_j=b_j$;
both can be omitted if $a_j=b_j=1$.

Here, for example, is a simple test case:
$$
\vcenter{\halign{\tt#\cr
\char"7C\ A simple example of color controls\cr
A B 2:3{\char"7C}C \char"7C\ X Y\cr
A B X:0 Y:0\cr
A C X:1 Y:1\cr
C X:0\cr
B X:1\cr
C Y:1\cr}}
$$
The unique solution consists of options \.{A C X:1 Y:1}, \.{B X:1}, \.{C Y:1}.

There's a subtle distinction between a primary item
with bounds $[0\dts1]$ and a secondary item with no bounds, because
every option is required to include at least one primary item.

If the input contains no item-bound specifications, the behavior of {\mc DLX3}
will almost exactly match that of~{\mc DLX2}, except for having a
slightly longer program and taking a bit longer to input the options.

[{\it Historical note:\/} My first program for multiset exact
covering was {\mc MDANCE}, written in August 2004 when I was thinking
about packing various sizes of bricks into boxes. That program allowed
users to specify arbitrary item sums, and it had the same structure
as this one, but it was less general than
{\mc DLX3} because it didn't allow lower bounds to be less than upper bounds.
Later I came gradually to
realize that the ideas have many, many other applications.]

@ 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'' and
``cleansings'' were made.
The running time in ``mems'' is also reported, together with the approximate
number of bytes needed for data storage.
(An ``update'' is the removal of an option from its item list.
A ``cleansing'' is the removal of a satisfied color constraint from its option.
One ``mem'' essentially means a memory access to a 64-bit word.
The reported 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 10000 /* at most this many items */
@d max_nodes 100000000 /* 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>
#include "gb_flip.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_node,best_itm,stage,score,best_s,best_l;
  @<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_basics) @<Give statistics about the run@>;
  @<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{$\bullet$}
`\.s$\langle\,$integer$\,\rangle$' causes the algorithm to make
random choices in key places (thus providing some variety, although
the solutions are by no means uniformly random), and it also
defines the seed for any random numbers that are used;
\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 */

@<Glob...@>=
int random_seed=0; /* seed for the random words of |gb_rand| */
int randomizing; /* has `\.s' been specified? */
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,cmems,tmems; /* mem counts */
ullng updates; /* update counts */
ullng cleansings; /* cleansing 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 */

@ 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 's': k|=(sscanf(argv[j]+1,""O"d",&random_seed)-1),randomizing=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);
}
if (randomizing) gb_init_rand(random_seed);

@ @<Give statistics about the run@>=
{
  fprintf(stderr,"Altogether "O"llu solution"O"s",
                              count,count==1?"":"s");
  fprintf(stderr,", "O"llu+"O"llu mems,",imems,mems);
  fprintf(stderr," "O"llu updates, "O"llu cleansings,",
                              updates,cleansings);
  bytes=last_itm*sizeof(item)+last_node*sizeof(node)+maxl*sizeof(int);
  fprintf(stderr," "O"llu bytes, "O"llu nodes,",
                              bytes,nodes);
  fprintf(stderr," ccost "O"lld%%.\n",
                  (200*cmems+mems)/(2*mems));
}

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

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

More precisely, the nodes of individual options appear sequentially,
with ``spacer'' nodes between them. The nodes are also
linked circularly within each item, in doubly linked lists.
The item lists each include a header node, but the option lists do not.
Item header nodes are aligned with an \&{item} struct, which
contains further info about the item.

Each node contains four important fields. Two are the pointers |up|
and |down| of doubly linked lists, already mentioned.
A~third points directly to the item containing the node.
And the last specifies a color, or zero if no color is specified.

A ``pointer'' is an array index, not a \CEE/ reference (because the latter
would occupy 64~bits and waste cache space). The |cl| array is for
\&{item} structs, and the |nd| array is for \&{node}s. I assume that both of
those arrays are small enough to be allocated statically. (Modifications
of this program could do dynamic allocation if needed.)
The header node corresponding to |cl[c]| is |nd[c]|.

Notice that each \&{node} occupies two octabytes.
We count one mem for a simultaneous access to the |up| and |down| fields,
or for a simultaneous access to the |itm| and |color| fields.

Although the item-list pointers are called |up| and |down|, they need not
correspond to actual positions of matrix entries. The elements of
each item list can appear in any order, so that one option
needn't be consistently ``above'' or ``below'' another. Indeed, when
|randomizing| is set, we intentionally scramble each item list.

This program doesn't change the |itm| fields after they've first been set up.
But the |up| and |down| fields will be changed frequently, although preserving
relative order.

Exception: In the node |nd[c]| that is the header for the list of
item~|c|, we use the |itm| field to hold the {\it length\/} of that
list (excluding the header node itself).
We also might use its |color| field for special purposes.
The alternative names |len| for |itm| and |aux| for |color|
are used in the code so that this nonstandard semantics will be more clear.

A {\it spacer\/} node has |itm<=0|. Its |up| field points to the start
of the preceding option; its |down| field points to the end of the following option.
Thus it's easy to traverse an option circularly, in either direction.

The |color| field of a node is set to |-1| when that node has been cleansed.
In such cases its original color appears in the item header.
(The program uses this fact only for diagnostic outputs.)

@d len itm /* item list length (used in header nodes only) */
@d aux color /* an auxiliary quantity (used in header nodes only) */

@<Type...@>=
typedef struct node_struct {
  int up,down; /* predecessor and successor in item */
  int itm; /* the item containing this node */
  int color; /* the color specified by this node, if any */
} node;

@ Each \&{item} struct contains five fields:
The |name| is the user-specified identifier;
|next| and |prev| point to adjacent items, when this
item is part of a doubly linked list;
|bound| is the maximum number of options from this item that can
be added to the current partial solution;
|slack| is the difference between this item's given upper and lower bounds.
As computation proceeds, |bound| might change but |slack| will not.

An item can be removed from the active list of ``unfinished items'' when its
|bound| field is reduced to zero. A removed item is said to be ``covered'';
all of its remaining options are then hidden from further participation.
Furthermore, we will remove an item when we find that it has no unhidden
options; that situation can arise if |bound<=slack|.

As backtracking proceeds, nodes
will be deleted from item lists when their option has been hidden by
other options in the partial solution.
But when backtracking is complete, the data structures will be
restored to their original state.

We count one mem for a simultaneous access to the |prev| and |next| fields,
or for a simultaneous access to |bound| and |slack|.

The |bound| and |slack| fields of secondary items are not used.

@<Type...@>=
typedef struct itm_struct {
  char name[8]; /* symbolic identification of the item, for printing */
  int prev,next; /* neighbors of this item */
  int bound,slack; /* residual capacity of this item */
} item;

@ @<Glob...@>=
node nd[max_nodes]; /* the master list of nodes */
int last_node; /* the first node in |nd| that's not yet used */
item cl[max_cols+2]; /* the master list of items */
int second=max_cols; /* boundary between primary and secondary items */
int last_itm; /* the first item in |cl| that's not yet used */

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

@d root 0 /* |cl[root]| is the gateway to the unsettled items */

@ 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, relative
to a given head location.

@<Sub...@>=
void print_option(int p,FILE *stream,int head,int score) {
  register int k,q;
  if ((p<last_itm && p==head) || (head>=last_itm && p==nd[head].itm))
    fprintf(stream," null "O".8s",cl[p].name);
  else {
    if (p<last_itm || p>=last_node || nd[p].itm<=0) {
      fprintf(stderr,"Illegal option "O"d!\n",p);
      return;
    }
    for (q=p;;) {
      fprintf(stream," "O".8s",cl[nd[q].itm].name);
      if (nd[q].color)
        fprintf(stream,":"O"c",nd[q].color>0? nd[q].color: nd[nd[q].itm].color);
      q++;
      if (nd[q].itm<=0) q=nd[q].up;
          /* |-nd[q].itm| is actually the option number */
      if (q==p) break;
    }
  }
  for (q=head,k=1;q!=p;k++) {
    if (p>=last_itm && q==nd[p].itm) {
      fprintf(stream," (?)\n");@+return; /* option not in its item list! */
    }@+else q=nd[q].down;
  }
  fprintf(stream," ("O"d of "O"d)\n",k,score);
}
@#
void prow(int p) {
  print_option(p,stderr,nd[nd[p].itm].down,nd[nd[p].itm].len);
}

@ 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<root || c>=last_itm) {
    fprintf(stderr,"Illegal item "O"d!\n",c);
    return;
  }
  fprintf(stderr,"Item "O".8s",cl[c].name);
  if (c<second) {
    if (cl[c].slack || cl[c].bound!=1)
       fprintf(stderr," ("O"d,"O"d)",cl[c].bound-cl[c].slack,cl[c].bound);
    fprintf(stderr,", length "O"d, neighbors "O".8s and "O".8s:\n",
        nd[c].len,cl[cl[c].prev].name,cl[cl[c].next].name);
  }@+else fprintf(stderr,", length "O"d:\n",nd[c].len);
  for (p=nd[c].down;p>=last_itm;p=nd[p].down) prow(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,p,q,pp,qq,t;
  for (q=root,p=cl[q].next;;q=p,p=cl[p].next) {
    if (cl[p].prev!=q) fprintf(stderr,"Bad prev field at itm "O".8s!\n",
                                                            cl[p].name);
    if (p==root) break;
    @<Check item |p|@>;
  }
}    

@ @<Check item |p|@>=
for (qq=p,pp=nd[qq].down,k=0;;qq=pp,pp=nd[pp].down,k++) {
  if (nd[pp].up!=qq) fprintf(stderr,"Bad up field at node "O"d!\n",pp);
  if (pp==p) break;
  if (nd[pp].itm!=p) fprintf(stderr,"Bad itm field at node "O"d!\n",pp);
}
if (nd[p].len!=k) fprintf(stderr,"Bad len field in item "O".8s!\n",
                                                       cl[p].name);

@*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.

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

@<Input the item names@>=
if (max_nodes<=2*max_cols) {
  fprintf(stderr,"Recompile me: max_nodes must exceed twice max_cols!\n");
  exit(-999);
} /* every item will want a header node and at least one other node */
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];) {
  @<Scan an item name, possibly prefixed by bounds@>;
  @<Initialize |last_itm| to a new item with an empty list@>;
  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;
o,cl[root].prev=second-1; /* |cl[second-1].next=root| since |root=0| */
last_node=last_itm; /* reserve all the header nodes and the first spacer */
o,nd[last_node].itm=0;

@ @<Scan an item name, possibly prefixed by bounds@>=
if (second==max_cols) stage=0;@+else stage=2;
start_name:@+for (j=0;j<8 && (o,!isspace(buf[p+j]));j++) {
    if (buf[p+j]==':') {
      if (stage) panic("Illegal `:' in item name");
      @<Convert the prefix to an integer, |q|@>;
      r=q,stage=1;
      goto start_name;
    }@+else if (buf[p+j]=='|') {
      if (stage>1) panic("Illegal `|' in item name");
      @<Convert the prefix...@>;
      if (q==0) panic("Upper bound is zero");
      if (stage==0) r=q;
      else if (r>q) panic("Lower bound exceeds upper bound");
      stage=2;
      goto start_name;
    }
    o,cl[last_itm].name[j]=buf[p+j];
  }
  switch (stage) {
case 1: panic("Lower bound without upper bound");
case 0: q=r=1;
case 2: break;
  }
  if (j==0) panic("Item name empty");
  if (j==8 && !isspace(buf[p+j])) panic("Item name too long");
  @<Check for duplicate item name@>;

@ @<Convert the prefix to an integer, |q|@>=
for (q=0,pp=p;pp<p+j;pp++) {
  if (buf[pp]<'0' || buf[pp]>'9') panic("Illegal digit in bound spec");
  q=10*q+buf[pp]-'0';
}
p=pp+1;
while (j) cl[last_itm].name[--j]=0;

@ @<Check for duplicate item name@>=
for (k=1;o,strncmp(cl[k].name,cl[last_itm].name,8);k++) ;
if (k<last_itm) panic("Duplicate item name");

@ @<Initialize |last_itm| to a new item with an empty list@>=
if (last_itm>max_cols) panic("Too many items");
if (second==max_cols)
 oo,cl[last_itm-1].next=last_itm,cl[last_itm].prev=last_itm-1,
 o,cl[last_itm].bound=q,cl[last_itm].slack=q-r; 
else o,cl[last_itm].next=cl[last_itm].prev=last_itm;
o,nd[last_itm].up=nd[last_itm].down=last_itm;
 /* |nd[last_itm].len=0| */
last_itm++;

@ I'm putting the option number into the spacer that follows it, as a
possible debugging aid. But the program doesn't currently use that information.

@<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];) {
    for (j=0;j<8 && (o,!isspace(buf[p+j])) && buf[p+j]!=':';j++)
      o,cl[last_itm].name[j]=buf[p+j];
    if (!j) panic("Empty item name");
    if (j==8 && !isspace(buf[p+j]) && buf[p+j]!=':')
          panic("Item name too long");
    if (j<8) o,cl[last_itm].name[j]='\0';
    @<Create a node for the item named in |buf[p]|@>;
    if (buf[p+j]!=':') o,nd[last_node].color=0;
    else if (k>=second) {
      if ((o,isspace(buf[p+j+1])) || (o,!isspace(buf[p+j+2])))
        panic("Color must be a single character");
      o,nd[last_node].color=(unsigned char)buf[p+j+1];
      p+=2;
    }@+else panic("Primary item must be uncolored");
    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@>;
      last_node--;
    }
  }@+else {
    o,nd[i].down=last_node;
    last_node++; /* create the next spacer */
    if (last_node==max_nodes) panic("Too many nodes");
    options++;
    o,nd[last_node].up=i+1;
    o,nd[last_node].itm=-options;
  }
}

@ @<Create a node for the item named in |buf[p]|@>=
for (k=0;o,strncmp(cl[k].name,cl[last_itm].name,8);k++) ;
if (k==last_itm) panic("Unknown item name");
if (o,nd[k].aux>=i) panic("Duplicate item name in this option");
last_node++;
if (last_node==max_nodes) panic("Too many nodes");
o,nd[last_node].itm=k;
if (k<second) pp=1;
o,t=nd[k].len+1;
@<Insert node |last_node| into the list for item |k|@>;

@ Insertion of a new node is simple, unless we're randomizing.
In the latter case, we want to put the node into a random position
of the list.

We store the position of the new node into |nd[k].aux|, so that
the test for duplicate items above will be correct.

As in other programs developed for TAOCP, I assume that four mems are
consumed when 31 random bits are being generated by any of the {\mc GB\_FLIP}
routines.

@<Insert node |last_node| into the list for item |k|@>=
o,nd[k].len=t; /* store the new length of the list */
nd[k].aux=last_node; /* no mem charge for |aux| after |len| */
if (!randomizing) {
  o,r=nd[k].up; /* the ``bottom'' node of the item list */
  ooo,nd[r].down=nd[k].up=last_node,nd[last_node].up=r,nd[last_node].down=k;
}@+else {  
  mems+=4,t=gb_unif_rand(t); /* choose a random number of nodes to skip past */
  for (o,r=k;t;o,r=nd[r].down,t--) ;
  ooo,q=nd[r].up,nd[q].down=nd[r].up=last_node;
  o,nd[last_node].up=q,nd[last_node].down=r;  
}

@ @<Remove |last_node| from its item@>=
o,k=nd[last_node].itm;
oo,nd[k].len--,nd[k].aux=i-1;
o,q=nd[last_node].up,r=nd[last_node].down;
oo,nd[q].down=r,nd[r].up=q;

@ @<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-1,last_itm-second,last_node-last_itm);

@ 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.

@<Report the item totals@>=
{
  fprintf(stderr,"Item totals:");
  for (k=1;k<last_itm;k++) {
    if (k==second) fprintf(stderr," |");
    fprintf(stderr," "O"d",nd[k].len);
  }
  fprintf(stderr,"\n");
}

@*The dancing.
Our strategy for generating all exact covers will be to repeatedly
choose an active primary item and to branch on the ways to reduce
the possibilities for covering that item.
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 an item.'' This means removing it
from the list of items needing to be covered, and ``hiding'' its
options: removing nodes from other lists whenever they belong to an option of
a node in this item's list. We cover the chosen item when it has
|bound=1|.

There's also an auxiliary operation called ``tweaking an item,'' used when
covering is inappropriate. In that case we simply hide the topmost option
in the item's list; we also remove that option temporarily from the list.
(The tweaking operation, whose beauties will be described below,
is a new dance step! It was introduced in the {\mc MDANCE} program of 2004.)

@<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, and let |score| be
  its branching degree@>;
if (score<=0) goto backdown; /* not enough options left in this item */
if (score==infty) @<Visit a solution and |goto backdown|@>;
scor[level]=score,first_tweak[level]=0;
  /* for diagnostics only, so no mems charged */
oo,cur_node=choice[level]=nd[best_itm].down;
o,cl[best_itm].bound--; /* one mem will be charged later */
if (cl[best_itm].bound==0 && cl[best_itm].slack==0) cover(best_itm,1);
else {
  o,first_tweak[level]=cur_node;
  if (cl[best_itm].bound==0) cover(best_itm,1);
}
advance:@+@<If |cur_node| is off limits, |goto backup|; also tweak if needed@>;
if ((vbose&show_choices) && level<show_choices_max) @<Report the current move@>;
if (cur_node>last_itm)
  @<Cover or partially cover all other items of |cur_node|'s option@>;
@<Increase |level| and |goto forward|@>;
backup:@+@<Restore the original state of |best_itm|@>;
backdown:@+if (level==0) goto done;
level--;
oo,cur_node=choice[level],best_itm=nd[cur_node].itm,score=scor[level];
if (cur_node<last_itm) @<Reactivate |best_itm| and |goto backup|@>;
@<Uncover or partially uncover all other items of |cur_node|'s option@>;
oo,cur_node=choice[level]=nd[cur_node].down;@+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 */
int first_tweak[max_level]; /* original top of item before tweaking */
int scor[max_level]; /* for reports of progress */

@ @<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;
}

@ @<Increase |level| and |goto forward|@>=
if (++level>maxl) {
  if (level>=max_level) {
    fprintf(stderr,"Too many levels!\n");
    exit(-4);
  }
  maxl=level;
}
goto forward;

@ @<Report the current move@>=
{
  fprintf(stderr,"L"O"d:", level);
  if (cl[best_itm].bound==0 && cl[best_itm].slack==0)
    print_option(cur_node,stderr,nd[best_itm].down,score);
  else print_option(cur_node,stderr,first_tweak[level],score);
}

@ @<Reactivate |best_itm| and |goto backup|@>=
{
  best_itm=cur_node;
  o,p=cl[best_itm].prev,q=cl[best_itm].next;
  oo,cl[p].next=cl[q].prev=best_itm; /* reactivate |best_itm| */
  goto backup;
}

@ In the normal cases treated by {\mc DLX1} and {\mc DLX2}, we want to
back up after trying all options in the item; this happens when |cur_node|
has advanced to |best_itm|, the item's header node.

In the other cases, we've been tweaking this item. Then
we back up when fewer than |bound+1-slack| options remain in the item's list.
(The current value of |bound| is one less than its original value
on entry to this level.)

Notice that we might reach a situation where the list is empty
(that is, |cur_node=best_itm|), yet we don't want to back up.
This can happen when |bound-slack<0|. In such cases the move at
this level is null: No option is added to the solution, and the
item becomes inactive.

@<If |cur_node| is off limits, |goto backup|...@>=
if ((o,cl[best_itm].bound==0) && (cl[best_itm].slack==0)) {
  if (cur_node==best_itm) goto backup;
}@+else if (oo,nd[best_itm].len<=cl[best_itm].bound-cl[best_itm].slack)
  goto backup;
else if (cur_node!=best_itm) tweak(cur_node,cl[best_itm].bound);
else if (cl[best_itm].bound!=0) {
  o,p=cl[best_itm].prev,q=cl[best_itm].next;
  oo,cl[p].next=q,cl[q].prev=p; /* deactivate |best_itm| */
}

@ @<Restore the original state of |best_itm|@>=
if ((o,cl[best_itm].bound==0) && (cl[best_itm].slack==0)) uncover(best_itm,1);
else o,untweak(best_itm,first_tweak[level],cl[best_itm].bound);
oo,cl[best_itm].bound++;

@ 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.

We can save time by not removing nodes from secondary items that have been
purified. (Such nodes have |color<0|. Note that |color| and |itm| are
stored in the same octabyte; hence we pay only one mem to look at
them both.)

@<Sub...@>=
void cover(int c,int deact) {
  register int cc,l,r,rr,nn,uu,dd,t;
  if (deact) {
    o,l=cl[c].prev,r=cl[c].next;
    oo,cl[l].next=r,cl[r].prev=l;
  }
  updates++;
  for (o,rr=nd[c].down;rr>=last_itm;o,rr=nd[rr].down)
    for (nn=rr+1;nn!=rr;) {
      if (o,nd[nn].color>=0) {
        o,uu=nd[nn].up,dd=nd[nn].down;
        cc=nd[nn].itm;
        if (cc<=0) {
          nn=uu;
          continue;
        }
        oo,nd[uu].down=dd,nd[dd].up=uu;
        updates++;
        o,t=nd[cc].len-1;
        o,nd[cc].len=t;
      }
      nn++;
    }
}

@ I used to think that it was important to uncover an item by
processing its options from bottom to top, since covering was done
from top to bottom. But while writing this
program I realized that, amazingly, no harm is done if the
options are processed again in the same order. So I'll go downward again,
just to prove the point. Whether we go up or down, the pointers
execute an exquisitely choreo\-graphed dance that returns them almost
magically to their former state.

@<Subroutines@>=
void uncover(int c,int react) {
  register int cc,l,r,rr,nn,uu,dd,t;
  for (o,rr=nd[c].down;rr>=last_itm;o,rr=nd[rr].down)
    for (nn=rr+1;nn!=rr;) {
      if (o,nd[nn].color>=0) {
        o,uu=nd[nn].up,dd=nd[nn].down;
        cc=nd[nn].itm;
        if (cc<=0) {
          nn=uu;
          continue;
        }
        oo,nd[uu].down=nd[dd].up=nn;
        o,t=nd[cc].len+1;
        o,nd[cc].len=t;
      }
      nn++;
    }
  if (react) {
    o,l=cl[c].prev,r=cl[c].next;
    oo,cl[l].next=cl[r].prev=c;
  }
}

@ @<Cover or partially cover all other items...@>=
for (pp=cur_node+1;pp!=cur_node;) {
  o,cc=nd[pp].itm;
  if (cc<=0) o,pp=nd[pp].up;
  else {
    if (cc<second) {
      oo,cl[cc].bound--;
      if (cl[cc].bound==0) cover(cc,1);
    }@+else {
      if (!nd[pp].color) cover(cc,1);
      else if (nd[pp].color>0) purify(pp);
    }
    pp++;
  }
}

@ We must go leftward as we uncover the items, because we went
rightward when covering them.

@<Uncover or partially uncover all other items...@>=
for (pp=cur_node-1;pp!=cur_node;) {
  o,cc=nd[pp].itm;
  if (cc<=0) o,pp=nd[pp].down;
  else {
    if (cc<second) {
      if (o,cl[cc].bound==0) uncover(cc,1);
      o,cl[cc].bound++;
    }@+else {
      if (!nd[pp].color) uncover(cc,1);
      else if (nd[pp].color>0) unpurify(pp);
    }
    pp--;
  }
}
      
@ When we choose an option that specifies colors in one or more items,
we ``purify'' those items by removing all incompatible options.
All options that want the chosen color in a purified item are temporarily
given the color code~|-1| so that they won't be purified again.

@<Sub...@>=
void purify(int p) {
  register int cc,rr,nn,uu,dd,t,x;
  o,cc=nd[p].itm,x=nd[p].color;
  nd[cc].color=x; /* no mem charged, because this is for |print_option| only */
  cleansings++;
  for (o,rr=nd[cc].down;rr>=last_itm;o,rr=nd[rr].down) {
    if (o,nd[rr].color!=x) {
      for (nn=rr+1;nn!=rr;) {
        o,uu=nd[nn].up,dd=nd[nn].down;
        o,cc=nd[nn].itm;
        if (cc<=0) {
          nn=uu;@+continue;
        }
        if (nd[nn].color>=0) {
          oo,nd[uu].down=dd,nd[dd].up=uu;
          updates++;
          o,t=nd[cc].len-1;
          o,nd[cc].len=t;
        }
        nn++;
      }
    }@+else if (rr!=p) cleansings++,o,nd[rr].color=-1;
  }
}

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

@<Sub...@>=
void unpurify(int p) {
  register int cc,rr,nn,uu,dd,t,x;
  o,cc=nd[p].itm,x=nd[p].color; /* there's no need to clear |nd[cc].color| */
  for (o,rr=nd[cc].up;rr>=last_itm;o,rr=nd[rr].up) {  
    if (o,nd[rr].color<0) o,nd[rr].color=x;
    else if (rr!=p) {
      for (nn=rr-1;nn!=rr;) {
        o,uu=nd[nn].up,dd=nd[nn].down;
        o,cc=nd[nn].itm;
        if (cc<=0) {
          nn=dd;@+continue;
        }
        if (nd[nn].color>=0) {
          oo,nd[uu].down=nd[dd].up=nn;
          o,t=nd[cc].len+1;
          o,nd[cc].len=t;
        }
        nn--;
      }
    }
  }
}

@ Now let's look at tweaking, which is deceptively simple. When this
subroutine is called, node |n| is the topmost for its item.
Tweaking is important because the item remains active and on a par
with all other active items.

In the special case that the item was chosen for branching with
|bound=1| and |slack>=1|, we've already covered the item;
hence we shouldn't block its options again.

@<Sub...@>=
void tweak(int n,int block) {
  register int cc,nn,uu,dd,t;
  for (nn=(block?n+1:n);;) {
    if (o,nd[nn].color>=0) {
      o,uu=nd[nn].up,dd=nd[nn].down;
      cc=nd[nn].itm;
      if (cc<=0) {
        nn=uu;
        continue;
      }
      oo,nd[uu].down=dd,nd[dd].up=uu;
      updates++;
      o,t=nd[cc].len-1;
      o,nd[cc].len=t;
    }
    if (nn==n) break;
    nn++;
  }
}

@ The punch line occurs when we consider untweaking. Consider, for
example, an item $c$ whose options from top to bottom are $x$, $y$,~$z$.
Then the |up| fields for $(c,x,y,z)$ are initially $(z,c,x,y)$, and the
|down| fields are $(x,y,z,c)$. After we've tweaked $x$, they've become
$(z,c,c,y)$ and $(y,y,z,c)$; after we've subsequently tweaked $y$, they've
become $(z,c,c,c)$ and $(z,y,z,c)$. Notice that $x$ still points to~$y$,
and $y$ still points to~$z$. So we can restore the original state
if we restore the |up| pointers in $y$ and $z$, as well as the |down|
pointer in~$c$. The value of~$x$ has been saved in the |first_tweak|
array for the current level; and that's sufficient to solve the puzzle.

We also have to resuscitate the options by reinstating them in their items.
That can be done top-down, as in |uncover|; in essence, a sequence of
tweaks is like a partial covering. 

@<Sub...@>=
void untweak(int c,int x,int unblock) {
  register int z,cc,nn,uu,dd,t,k,rr,qq;
  oo,z=nd[c].down,nd[c].down=x;
  for (rr=x,k=0,qq=c; rr!=z; o,qq=rr,rr=nd[rr].down) {
    o,nd[rr].up=qq,k++;
    if (unblock) for (nn=rr+1;nn!=rr;) {
      if (o,nd[nn].color>=0) {
        o,uu=nd[nn].up,dd=nd[nn].down;
        cc=nd[nn].itm;
        if (cc<=0) {
          nn=uu;
          continue;
        }
        oo,nd[uu].down=nd[dd].up=nn;
        o,t=nd[cc].len+1;
        o,nd[cc].len=t;
      }
      nn++;
    }
  }
  o,nd[rr].up=qq; /* |rr=z| */
  oo,nd[c].len+=k;
  if (!unblock) uncover(c,0);
}

@ The ``best item'' is considered to be an item that minimizes the
branching degree. If there are several candidates, we
choose the leftmost --- unless we're randomizing, in which case we
select one of them at random.

Consider an item that has four options $\{w,x,y,z\}$, and suppose its |bound|
is~3. If the |slack| is zero, we've got to choose either |w| or |x|,
so the branching degree is~2. But if |slack=1|, we have three choices,
|w| or |x| or |y|; if |slack=2|, there are four choices; and if |slack>=3|,
there are five, including the ``null'' choice.

In general, the branching degree turns out to be $l+s-b+1$, where
$l$~is the length of the item, $b$ is the current bound, and
$s$ is the minimum of $b$ and the slack. This formula gives degree
$\le0$ if and only if |l| is too small to satisfy the item
constraint; in such cases we will backtrack immediately.
(It would have been possible to detect this condition early,
before updating all the data structures and increasing |level|. But that would
make the downdating process much more difficult and error-prone. Therefore
I wait to discover such anomalies until item-choosing time.)

Let's assign the score |l+s-b+1| to each item. If two items have the
same score, I prefer the one with smaller |s|, because slack items
are less constrained. If two items with the same |s| have the same
score, I (counterintuitively)
prefer the one with larger~|b| (hence larger~|l|), because
that tends to reduce the size of the final search tree. 

Consider, for instance, the following example taken from {\mc MDANCE}:
If we want to choose 2 options from 4 in one item, and 3 options from 5 in another,
where all slacks are zero, and if the items are otherwise independent,
it turns out that the number of nodes per level if we choose the smaller
item first is $(1,3,6,6\cdot3,6\cdot6,6\cdot10)$. But if we choose
the larger item first it is $(1,3,6,10,10\cdot3,10\cdot6)$, which is
smaller in the middle levels.

@d infty max_nodes /* the ``score'' of a completely unconstrained item */

@<Set |best_itm| to the best item for branching...@>=
score=infty,tmems=mems;
if ((vbose&show_details) &&
    level<show_choices_max && level>=maxl-show_choices_gap)
  fprintf(stderr,"Level "O"d:",level);
for (o,k=cl[root].next;k!=root;o,k=cl[k].next) {
  o,s=cl[k].slack;@+if (s>cl[k].bound) s=cl[k].bound;
  if ((vbose&show_details) &&
      level<show_choices_max && level>=maxl-show_choices_gap) {
    if (cl[k].bound!=1 || s!=0) fprintf(stderr," "O".8s("O"d:"O"d,"O"d)",
                      cl[k].name,cl[k].bound-s,cl[k].bound,
                                 nd[k].len+s-cl[k].bound+1);
    else fprintf(stderr," "O".8s("O"d)",cl[k].name,nd[k].len);
  }
  t=nd[k].len+s-cl[k].bound+1;
  if (t<=score) {
    if (t<score || s<best_s || (s==best_s && nd[k].len>best_l))
      score=t,best_itm=k,best_s=s,best_l=nd[k].len,p=1;
    else if (s==best_s && nd[k].len==best_l) {
      p++; /* this many items achieve the min */
      if (randomizing && (mems+=4,!gb_unif_rand(p))) best_itm=k;
    }
  }
}
if ((vbose&show_details) &&
    level<show_choices_max && level>=maxl-show_choices_gap) {
  if (score<infty)
    fprintf(stderr," branching on "O".8s("O"d)\n",cl[best_itm].name,score);
  else fprintf(stderr," solution\n");
}
if (shape_file && score<infty) {
  fprintf(shape_file,""O"d "O".8s\n",score>=0?score:0,cl[best_itm].name);
  fflush(shape_file);
}
cmems+=mems-tmems;

@ @<Visit a solution and |goto backdown|@>=
{
  if (shape_file) {
    fprintf(shape_file,"sol\n");@+fflush(shape_file);
  }
  @<Record a solution and |goto backdown|@>;
}

@ @<Record a solution and |goto backdown|@>=
{
  count++;
  if (spacing && (count mod spacing==0)) {
    printf(""O"lld:\n",count);
    for (k=0;k<level;k++) {
      pp=choice[k];
      cc=pp<last_itm? pp: nd[pp].itm;
      if (!first_tweak[k]) print_option(pp,stdout,nd[cc].down,scor[k]);
      else print_option(pp,stdout,first_tweak[k],scor[k]);
    }
    fflush(stdout);
  }
  if (count>=maxcount) goto done;
  goto backdown;
}

@ @<Sub...@>=
void print_state(void) {
  register int l,p,c,q;
  fprintf(stderr,"Current state (level "O"d):\n",level);
  for (l=0;l<level;l++) {
    p=choice[l];
    c=(p<last_itm? p: nd[p].itm);
    if (!first_tweak[l]) print_option(p,stderr,nd[c].down,scor[l]);
    else print_option(p,stderr,first_tweak[l],scor[l]);
    if (l>=show_levels_max) {
      fprintf(stderr," ...\n");
      break;
    }
  }
  fprintf(stderr," "O"lld sols, "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
tends to grow 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++) {
    p=choice[l],d=scor[l];
    c=(p<last_itm? p: nd[p].itm);
    if (!first_tweak[l]) p=nd[c].down;
    else p=first_tweak[l];
    for (k=1;p!=choice[l];k++,p=nd[p].down) ;
    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.