[picture of a maze] The art of obfuscation


Note: this text appears in the following, highly recommended book. Do not reproduce without permission from the author. Note that connections formed by a partial maze correspond to a "Murasaki diagram".
"Obfuscated C and Other Mysteries" by Don Libes, John Wiley & Sons, 1993, pp 425 including disk with complete source code. All source is in the public domain. ISBN 0-471-57805-3.

I have a long history of writing maze-generating programs. The first I ever saw must have been one in a book containing listings of Sinclair ZX-Spectrum programs. As I remember, it didn't make a whole lot of sense.

That was before I knew about recursion. As I became familiar with the notion, I had little trouble writing my own maze generator, taking care of the stacky bits with a simple array. Later still I suffered quite a few headaches from converting the program to Z-80 assembly for speed's sake. Developments of the program were since directed by availability of output devices. On occasion I took a long train-ride to a friend with an ink-jet printer to produce mazes in all kinds of color.

I perceived two problems with that method of generating mazes. First of all, the mazes produced were hardly random, having very long corridors with very few junctions. This made solving the mazes a bit too easy, in my opinion. Second, the generator had to allocate memory for the entire maze. This limited the length of the mazes I could print. Being rather obsessed with mazes, I wasn't satisfied with a `mere' 4 meters worth of maze.

This got me thinking about generating mazes on the fly, line by line, without having to remember much information. That way, you could conceivably just start printing a maze, and tell it to somehow finish it nicely as you notice the printer running out of paper.

My submission is an obfuscated and nicely laid out version of a program I wrote to generate mazes in this way. Let me now describe this program.

Its central data-structure is a collection of circular doubly linked lists, the elements of which correspond to the maze-cells of one row of the maze, the one currently being generated. More precisely, each list contains all the cells on the current row that are connected by a path in the, already generated, part of the maze above the current row. Thus the lists form a partition of the cells in the current row; each cell is contained in exactly one of these lists. The doubly linked lists are stored in the two arrays L[] (for left) and R[] (for right). For a set of cells i_1,i_2,...,i_k (from left to right) that are all connected in the part of the maze above the current row, R[i_1] = i_2, R[i_2] = i_3, ..., and R[i_k] = i_1. Similarly for the left `pointers'.

How do we make use of this data-structure? Well, for each new row of the maze, we will visit all its cells from left to right, and for each one decide whether it should be connected to its right neighbour (the next cell visited) and whether it should be connected to its neighbour below (in the next row). What do we base these decision on? Well, we would like the decision to be as random as possible, but at the same time, we want the resulting maze to be what is formally known as a spanning tree. That means that there should be exactly one path between any two cells in the maze, understandably a highly desirable property of mazes. Since this property prohibits having a cycle in the maze, connections which would result in a cycle will have to be avoided. Since it also requires all cells to be connected, certain vital connections will have to be enforced. It turns out that the prohibition of connections is an issue only for right-connections, while the enforcement of connections is an issue only for down-connections.

Our data-structure is ideally suited to testing for either condition, and for the necessary updates. Here's what we do at cell i with right neighbour i'.

Note that:

To decide on a right connection:

Check if i equals L[i']; if so then a right-connection is prohibited as it would create a cycle. otherwise, flip a (slightly biased) coin. to MAKE the connection, we join the two lists by the following operations:
R[L[i']]=R[i] ; L[R[i]]=L[i'] ; /* link L[i'] to R[i] */
R[  i  ]=  i' ; L[  i']=  i   ; /* link   i   to   i' */
We will later print a '.' (right-connection) or a '|' (no right-connection), after we print the character for the presence/absence of a down connection.

To decide on a down connection:

Check if i equals L[i]; if so then this cell is alone in its list and a down-connection is enforced to keep it connected to the rest of the maze. otherwise, flip a (slightly biased) coin. to OMIT the connection, we take the cell out of its list by the following operations:
R[L[i]]=R[i] ; L[R[i]]=L[i] ; /* link L[i] to R[i] */
R[  i ]=  i  ; L[  i ]=  i  ; /* link   i   to  i  */
We now print a ' ' (down-connection) or a '_' (no down-connection), followed by the character for the presence/absence of a right connection.

A trick is used to make sure that the last cell j in a row does not try to connect to its non-existing right neighbour. We ensure that the test for equality of j and L[j'] always succeeds by creating a dummy cell for j' and setting L[j'] to j. (in the algorithm j=1 and j'=0.)

To finish the maze, a special last row is created in which some right connections are enforced to ensure the connectedness of the entire maze. The workings of the corresponding piece of code are not hard to understand given the above, and are left as an exercise for the reader.

Here is a minimally obfuscated impression of the original code:

char M[3];		/* holds the 2 characters printed for each cell */
int H,			/* height of the maze */
    C,			/* current cell */
    E,			/* temporary pointer used in the updating */
    L[40],R[40];        /* left and right pointers */

main()
{
  L[0] = scanf("%d",&H);		/* reads height and sets L[0] to 1 */
  for (E = 40; --E; L[E] = R[E] = E)
    printf("._");			/* close top of maze */
  printf("\n|");
  while (--H)                           /* more rows to do */
  { for (C = 40; --C; printf(M))	/* visit cells from left to right */
    { if (C != (E=L[C-1]) && 6<<27<rand())	/* make right-connection ? */
      { R[E] = R[C];			/* link E */
        L[R[C]] = E;			/* to R[C] */
        R[C] = C-1;			/* link C */
        L[C-1] = C;			/* to C-1 */
        M[1] = '.';			/* no wall to the right */
      }
      else M[1] = '|';			/* wall to the right */
      if (C != (E=L[C]) && 6<<27<rand()) 	/* omit down-connection ? */
      { R[E] = R[C];			/* link E */
        L[R[C]] = E;			/* to R[C] */
        L[C] = C;			/* link C */
        R[C] = C;			/* to C */
        M[0] = '_';			/* wall downward */
      }
      else M[0] = ' ';			/* no wall downward */
    }
    printf("\n|");
  }
  M[0] = '_';				/* close bottom of maze */
  for (C = 40; --C; printf(M))		/* bottom row */
  { if (C != (E=L[C-1]) && (C == R[C] || 6<<27<rand()))
    { L[R[E]=R[C]]=E;
      L[R[C]=C-1]=C;
      M[1] = '.';
    }
    else M[1] = '|';
    E = L[C];
    R[E] = R[C];
    L[R[C]] = E;
    L[C] = C;
    R[C] = C;
  }
  printf("\n");
}
Let's look at the obfuscation process step by step.

First, we of course get rid of all the comments. Then we cut on the white space, nest a few assignments, abbreviate zero array indices, replace if-then-elses by the more concise conditional expression, initialize M[] with the scanf format string, and generally move things about a bit:

char M[]="%d",C,E=40,L[40],R[40];
main(H)
{
  for (*L=scanf(M,&H); --E; L[E]=R[E]=E)
    printf("._");
  printf("\n|");
  while (--H)
  { for (C=40; --C; printf(M))
    { M[1] = C!=(E=L[C-1]) && 6<<27<rand() ?
      L[R[E]=R[C]]=E,L[R[C]=C-1]=C,'.' : '|';
      *M = C!=(E=L[C]) && 6<<27<rand() ?
        L[R[E]=R[C]]=E,L[C]=R[C]=C,'_' : ' ';
    }
    printf("\n|");
  }
  *M='_';
  for (C=40; --C; printf(M))
  { M[1] = C!=(E=L[C-1]) && (C==R[C] || 6<<27<rand()) ?
      L[R[E]=R[C]]=E,L[R[C]=C-1]=C,'.' : '|';
    E=L[C]; L[R[E]=R[C]]=E; L[C]=R[C]=C;
  }
  printf("\n");
}
To shorten the program further, we integrate the last row loop into the main one. The test for omitting the down connection should always succeed on the last row, so we simply extend this test to C!=(E=L[C]) && 6<<27<rand() || !H.

The main loop then becomes:

      M[1] = C!=(E=L[C-1]) && 6<<27<rand() ?
        L[R[E]=R[C]]=E,L[R[C]=C-1]=C,'.' : '|';
      *M   = C!=(E=L[C  ]) && 6<<27<rand() || !H ?
        L[R[E]=R[C]]=E,L[C]=R[C]  =C,'_' : ' ';
and we notice a lot of similarity between the right and down cases. Once we get the idea to loop on a variable, say Z, from 1 to 0, we quickly arrive at the following huge improvement (in obfuscation):
    for (Z=1; Z>=0; Z--)
    { M[Z] = C!=(E=L[C-Z]) && 6<<27<rand() || !H&!Z ?
        L[R[E]=R[C]]=E,L[R[C]=C-Z]=C,"_."[Z] : " |"[Z];
    }
Due to the enormous amount of functionality captured in these few lines, it is already amazingly hard to comprehend them without knowing their evolution. This code is shortened further by reversing the string constants and their index Z, thus allowing Z to be lifted from the entire assignment. Also, the inequality is replaced by a subtraction, a technique discovered and used by many OCCC authors:
    for (Z=1; Z>=0; Z--)
    { M[Z] = Z[C-(E=L[C-Z]) && 6<<27<rand() || !H&!Z ?
        L[R[E]=R[C]]=E,L[R[C]=C-Z]=C,"_." : " |"];
    }
(the L[C-Z] will later be replaced by the more obfuscated C[L-Z]. restricting the obfuscation to only this occurrence of L hopefully adds to the mystery.)

The program now has 3 nested loops, one over the rows of the maze, one over the cells within this row, and one over the two directions in which connections can be made:

  while (--H)
  { for (C=40; --C; printf(M))
    { for (Z=1; Z>=0; Z--)
      {
      }
    }
    printf("\n|");
  }
With yet another trick, we will fold them all into one big loop! From the point of view of the innermost loop, Z toggles between 1 and 0; when it goes from 0 to 1, then M must be printed and C must be decremented; when C becomes 0, then \n| must be printed, C must be reset to 39 and H must be decremented. This can be expressed by the following code:
  if (Z == 0)
    printf(M);
  if ((C-=Z=!Z) == 0)
  { printf("\n|");
    C = 39;
    H--;
  }
Other useful obfuscation techniques are This yields:
  Z || printf(M);
  (C-=Z=!Z) || (printf("\n|"), C = 39, H--);
and these two statements will be used as the update and test components of our single for loop. We also rename our variables to By making all variables of the type (pointer to) char (except the maze height which is made the argument of main()), we avoid the need for declaring ints, saving another 4 characters.

We thus arrive at the following, arguably minimal length, program for generating arbitrary length mazes:

char*M,A,Z,E=40,J[40],T[40];main(C){for(*J=A=scanf(M="%d",&C);--E;J[E]=T[E]=E)
printf("._");for(;(A-=Z=!Z)||(printf("\n|"),A=39,C--);Z||printf(M))M[Z]=Z[A-(E=A
[J-Z])&&!C&A==T[A]|6<<27<rand()||!C&!Z?J[T[E]=T[A]]=E,J[T[A]=A-Z]=A,"_.":" |"];}
The only issue that remains to be considered is aesthetics. While the above code nicely emphasizes the minimality of the code, it's not particularly pleasing to the eye. We could do with a better layout.

After some thinking, an ingenious idea struck me. The layout of the code should resemble a maze. Not just any maze, but one whose very structure contains yet another reference to mazes! I first started experimenting with layouts where the internal walls of the maze would spell the word MAZE, but I somehow failed to be satisfied. Then I tried the other possibility of having the corridors spell MAZE. Since this requires less internal wall-structure, it allows for a much better impression of the four letters spelling MAZE. Here is the program in its final layout:

char*M,A,Z,E=40,J[40],T[40];main(C){for(*J=A=scanf(M="%d",&C);
--            E;             J[              E]             =T
[E   ]=  E)   printf("._");  for(;(A-=Z=!Z)  ||  (printf("\n|"
)    ,   A    =              39              ,C             --
)    ;   Z    ||    printf   (M   ))M[Z]=Z[A-(E   =A[J-Z])&&!C
&    A   ==             T[                                  A]
|6<<27<rand()||!C&!Z?J[T[E]=T[A]]=E,J[T[A]=A-Z]=A,"_.":" |"];}
Note how the two decrement operators suggest an entry and exit to the maze!

Running the translate command

tr " -}" "O "
(or "O[ *99]" on systems with a braindead tr) on the above code exposes the hidden reference:
  OOOOOOOOOOOO  OOOOOOOOOOOOO  OOOOOOOOOOOOOO  OOOOOOOOOOOOO
  OOO  OO  OOO             OO              OO  OO
 OOOO OOO OOOO OOOOOOOOOOOOOO  OOOOOOOOOOOOOO  OOOOOOOOOOOOO
 OOOO OOO OOOO  OOOO      OOO  OOO             OOO
 OOOO OOO  OOOOOOOOOOOOO  OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
                                                        O
The following program, by Carl Shapiro, won the Grand Prize for most well-rounded in confusion of the 1985 International Obfuscated C Code Contest:
#define P(X)j=write(1,X,1)
#define C 39
int M[5000]={2},*u=M,N[5000],R=22,a[4],l[]={0,-1,C-1,-1},m[]={1,-C,-1,C},*b=N,
*d=N,c,e,f,g,i,j,k,s;main(){for(M[i=C*R-1]=24;f|d>=b;){c=M[g=i];i=e;for(s=f=0;
s<4;s++)if((k=m[s]+g)>=0&&k<C*R&&l[s]!=k%C&&(!M[k]||!j&&c>=16!=M[k]>=16))a[f++
]=s;if(f){f=M[e=m[s=a[rand()/(1+2147483647/f)]]+g];j=j<f?f:j;f+=c&-16*!j;M[g]=
c|1<<s;M[*d++=e]=f|1<<(s+2)%4;}else e=d>b++?b[-1]:e;}P(" ");for(s=C;--s;P("_")
)P(" ");for(;P("\n"),R--;P("|"))for(e=C;e--;P("_ "+(*u++/8)%2))P("| "+(*u/4)%2
);}
The contest rules said that the judges would not like submissions that are similar to previous winners (or losers:-) ...
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tromp@cwi.nl