@*An example of backtracking. Given a list of $m$-letter words and another
list of $n$-letter words, we find all $m\times n$ matrices whose rows and
columns are all listed. This program improves on {\mc BACK-MXN-WORDS}
by using a more sophisticated data structure for the $m$-letter words,
significantly decreasing the number of candidates tested (I~hope).

I'm thinking $m=5$ and $n=6$ as an interesting case to try in {\sl TAOCP},
but of course the problem makes sense in general.

The word list files are named on the command line. You can also restrict
the list length to, say, at most 500 words, by appending `\.{:500}' to
the file name.

@d maxm 7 /* largest permissible value of $m$ */
@d maxn 10 /* largest permissible value of $n$ */
@d maxmwds 20000 /* largest permissible number of $m$-letter words */
@d maxtriesize 1000000 /* largest permissible number of $n$-letter prefixes */
@d o mems++
@d oo mems+=2
@d ooo mems+=3
@d bufsize maxm+maxn

@c
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
unsigned long long mems; /* memory references */
unsigned long long thresh=10000000000; /* reporting time */
int maxmm=maxmwds, maxnn=maxtriesize;
char mword[maxmwds][maxm+1];
int mlink[maxmwds+1][maxm];
int head[maxm][26], size[maxm][26];
int trie[maxtriesize][27];
int trieptr;
char buf[bufsize];
unsigned int count; /* this many solutions found */
FILE *mfile,*nfile;
int a[maxn+1][maxn+1];
int x[maxn+1],y[maxn+1],z[maxn+1];
long long profile[maxn+2],weight[maxn+2];
main(int argc,char*argv[]) {
  register int i,j,k,l,m,n,p,q,mm,nn,t,xl,yl,zl;
  register char *w;
  @<Process the command line@>;
  @<Input the $m$-words@>;
  @<Input the $n$-words and make the trie@>;
  fprintf(stderr,"(%llu mems to initialize the data structures)\n",mems);
  @<Backtrack thru all solutions@>;
  fprintf(stderr,"Altogether %u solutions (%llu mems).\n",count,mems);
  @<Print the profile@>;
}

@ @<Print the profile@>=
fprintf(stderr,"Profile:          1%9.1f\n",(double)weight[1]);
for (k=2;k<=n+1;k++) fprintf(stderr,"%19lld%9.1f\n",
      profile[k],profile[k]?weight[k]/(double)profile[k]:0.0);

@ @<Process the command line@>=
if (argc!=3) {
  fprintf(stderr,"Usage: %s mwords[:mm] nwords[:nn]\n",argv[0]);
  exit(-1);
}
w=strchr(argv[1],':');
if (w) { /* colon in filename */
  if (sscanf(w+1,"%d",&maxmm)!=1) {
    fprintf(stderr,"I can't parse the m-file spec `%s'!\n",argv[1]);
    exit(-20);
  }
  *w=0;
}
if (!(mfile=fopen(argv[1],"r"))) {
  fprintf(stderr,"I can't open file `%s' for reading m-words!\n",argv[1]);
  exit(-2);
}
w=strchr(argv[2],':');
if (w) { /* colon in filename */
  if (sscanf(w+1,"%d",&maxnn)!=1) {
    fprintf(stderr,"I can't parse the n-file spec `%s'!\n",argv[1]);
    exit(-22);
  }
  *w=0;
}
if (!(nfile=fopen(argv[2],"r"))) {
  fprintf(stderr,"I can't open file `%s' for reading n-words!\n",argv[2]);
  exit(-3);
}

@ @<Input the $m$-words@>=
m=mm=0;
while (1) {
  if (mm==maxmm) break;
  if (!fgets(buf,bufsize,mfile)) break;
  mm++;
  for (k=0;o,buf[k]>='a' && buf[k]<='z';k++) o,mword[mm][k]=buf[k];
  if (buf[k]!='\n') {
    fprintf(stderr,"Illegal m-word: %s",buf);
    exit(-10);
  }
  if (m==0) {
    m=k;
    if (m>maxm) {
      fprintf(stderr,"Sorry, m should be at most %d!\n",maxm);
      exit(-16);
    }
  }@+else if (k!=m) {
    fprintf(stderr,"The m-file has words of lengths %d and %d!\n",m,k);
    exit(-4);
  }
  @<Build sublists for each character position@>;
}
fprintf(stderr,"OK, I've successfully read %d words of length m=%d.\n",mm,m);

@ For $0\le k<m$ we make 26 lists, one for each word that has a given
letter~|j+'a'|
in the $(k+1)$st position. The first such word is |mword| number
|head[k][j]|; the next such word following word~|x| is number
|mlink[x][k]|; these links terminate with zero.

The least significant bits of the characters in |buf| could have been
packed into a register, so we don't charge any mems for ``fetching'' them here.

@<Build sublists for each character position@>=
for (k=0;k<m;k++) {
  j=trunc(buf[k])-1; /* no mem charged, see above */
  o,p=head[k][j]; /* get the head of the |j|-list for position |k| */
  o,head[k][j]=mm; /* insert word |mm| into this list */
  o,mlink[mm][k]=p;
  oo,size[k][j]++; /* and remember the current list length */
}

@ For simplicity, I make a sparse trie with 27 branches at every node.
An $n$-letter word $w_1\ldots w_n$ leads to entries
|trie|$[p_{k-1}][[w_k]=p_k$ for $1\le k\le n$, where $p_0=0$ and $p_k>0$.
Here $1\le w_k\le 26$.

Slot 0 of |trie[p]| contains a bit pattern that will be helpful later:
If the other slots $j_1+1$, \dots, $j_r+1$ have nonzero entries,
we put the ``signature'' $\sum_{i=1}r\bigl(2^{j_i}$ into |trie[p][0]|.

Mems of statically allocated arrays like |trie|
are counted as if |trie[x][y]| is |array[27*x+y]|. (I mean,
`|trie[x]|' is not a pointer that must be fetched, it's a pointer
that the program can compute without fetching.)

@d trunc(c) ((c)&0x1f) /* convert |'a'| to 1, \dots, |'z'| to 26 */

@<Input the $n$-words and make the trie@>=
n=nn=0, trieptr=1;
while (1) {
  if (nn==maxnn) break;
  if (!fgets(buf,bufsize,nfile)) break;
  for (k=p=0;o,buf[k]>='a' && buf[k]<='z';k++,p=q) {
    o,q=trie[p][trunc(buf[k])];
    if (q==0) break;
  }
  for (j=k;o,buf[j]>='a' && buf[j]<='z';j++) {
    if (trieptr==maxtriesize) {
      fprintf(stderr,"Overflow (maxtriesize=%d)!\n",maxtriesize);
      exit(-66);
    }
    i=trunc(buf[j]);
    oo,trie[p][0]+=(1<<(i-1));
    if (j<n-1 || n==0) {
      o,trie[p][i]=trieptr;
      p=trieptr++;
    }
  }
  if (buf[j]!='\n') {
    fprintf(stderr,"Illegal n-word: %s",buf);
    exit(-11);
  }
  @<Check the length of the new line@>;
  o,trie[p][trunc(buf[n-1])]=nn+1; /* remember index of the word */
  mems-=3;
     /* we knew |trie[p]| when |p=0| and when |q=0|; |buf[j]| when |j=k| */
  nn++;
}      
fprintf(stderr,"Plus %d words of length n=%d.\n",nn,n);
fprintf(stderr,"(The trie has %d nodes.)\n",trieptr);

@ @<Check the length of the new line@>=
if (n==0) {
  n=j;
  p--,trieptr--; /* we allocated an unnecessary node, since |n| wasn't known */
  if (n>maxn) {
    fprintf(stderr,"Sorry, n should be at most %d!\n",maxn);
    exit(-17);
  }
}@+else {
  if (n!=j) {
    fprintf(stderr,"The n-file has words of lengths %d and %d!\n",n,j);
    exit(-5);
  }
  if (k==n) {
    buf[j]=0;
    fprintf(stderr,"The n-file has the duplicate word `%s'!\n",buf);
    exit(-6);
  }
}

@ Here I follow Algorithm 7.2.2B.

@<Backtrack thru all solutions@>=
b1: l=1;
  for (k=1;k<=m;k++) o,a[0][k]=0;
b2: profile[l]++;
  @<Report the current state, if |mems>=thresh|@>;
  if (l>n) @<Print a solution and |goto b5|@>;
  @<Choose a good position |zl| and its relevant signature |yl|@>;
  i=0;
next_i:@+while (((1<<i)&yl)==0) i++;
  o,xl=head[zl][i];
  if (xl==0) goto new_i;
b3: o,w=mword[xl]; /* think of |w|'s chars all in a register now, memwise */
  for (k=1;k<=m;k++) {
    oo,q=trie[a[l-1][k]][trunc(w[k-1])];
    if (!q) goto b4;@+else o,a[l][k]=q;
  }
  ooo,x[l]=xl,y[l]=yl,z[l]=zl,l++;
  goto b2;
b4: o,xl=mlink[xl][zl];
  if (xl) goto b3; /* move to the next $m$-word on sublist $i$ */
new_i:@+if ((1<<(++i))<=yl) goto next_i; 
b5: l--;
  if (l) {
    ooo,xl=x[l],yl=y[l],zl=z[l];
    o,i=mword[xl][zl]-'a'; /* this is the subtlest part */
    goto b4;
  }    

@ The $k$th letter of the next $m$-word must belong to the subset~$s_k$ that
is specified in slot~0 of |trie[a[l-1][k]]|.
We set |zl| to a |k-1| that minimizes the corresponding sum of sublist sizes,
and let |yl| be the corresponding subset.

@<Choose a good position |zl| and its relevant signature |yl|@>=
for (k=1,p=maxmm+1;k<=m;k++) {
  for (oo,t=trie[a[l-1][k]][0],q=0,i=0;(1<<i)<=t;i++)
    if ((1<<i)&t) o,q+=size[k-1][i];
  if (q<p) p=q,zl=k-1,yl=t;
}
weight[l]+=p; /* record the size of subdomain (for statistics only) */

@ @<Print a solution and |goto b5|@>=
{
  count++;@+printf("%d:",count);
  for (k=1;k<=n;k++) printf(" %s",mword[x[k]]);
  for (p=0,k=1;k<=n;k++) if (x[k]>=p) p=x[k];
  for (q=0,j=1;j<=m;j++) if (a[n][j]>q) q=a[n][j];
  printf(" (%06d,%06d; sum %07d, prod %012d)\n",p,q,p+q,p*q);
  goto b5;
}

@ @<Report the current state, if |mems>=thresh|@>=
if (mems>=thresh) {
   thresh+=10000000000;
   fprintf(stderr,"After %lld mems:",mems);
   for (k=2;k<=l;k++) fprintf(stderr," %lld",profile[k]);
   fprintf(stderr,"\n");
}

@*Index.