How do I solve this tromino tiling problem?

Question

I am trying to solve this tromino tiling problem for n x n board. Given Missing square co-ordinate(MS) x and y in n x n board, we have to fill the rest of the board with 'L' shaped tile.

I have managed to get the output for 2 x 2 board. But, I am stuck with the output for board sizes 4 x 4, 8 x 8...and so on.

Following is the function I wrote that recursively calls itself to print the board:

/* This program tiles with right trominoes an nxn
   board with one square missing, assuming that n
   is a power of 2.                                */

#include <stdio.h>
#include <stdlib.h>

// #define LR 0
// #define LL 1
// #define UR 3
// #define UL 2
// #define MS -1

const int Max = 256;

int board[Max][Max];

void tromino /* function to do tiling */
     ( int x_board,      /* x coordinate of board */
       int y_board,      /* y coordinate of board */
       int x_missing,    /* x coordinate of missing square */
       int y_missing,    /* y coordinate of missing square */
       int board_size);   /* size of board */

void show_Tromino(int size, int x_miss, int y_miss);

int main()
{
     int board_size,
         x_missing,  /* x coordinate of missing square */
         y_missing;  /* y coordinate of missing square */
     do {
          printf( "\n-------------------------------------" );
          printf( "\nEnter size of board (0 to quit): " );
          scanf( "%d", &board_size );
          if ( board_size ) {
               printf( "\nEnter coordinates of missing square: " );
               scanf( "%d%d", &x_missing, &y_missing );
               printf( "\n\nTiling\n" );
               // if(board[x_board][0] == x_missing && board[0][y_board] == y_missing)
               //  printf("MS\t");
               //board[x_missing][y_missing] = -1;
               tromino( 0, 0, x_missing, y_missing, board_size);
               show_Tromino(board_size, x_missing, y_missing);
          }
     } while ( board_size );

     return EXIT_SUCCESS;
}

void tromino( int x_board,     /* x coordinate of board */
              int y_board,     /* y coordinate of board */
              int x_missing,   /* x coordinate of missing square */
              int y_missing,   /* y coordinate of missing square */
              int board_size) /* size of board */

{
     int half_size = board_size/2, /* size of subboard */
         x_center,  /* x coordinate of center of board */
         y_center,  /* y coordinate of center of board */
         x_upper_left,  /* x coordinate of missing square in upper
                                                     left subboard */
         y_upper_left,  /* y coordinate of missing square in upper
                                                     left subboard */
         x_upper_right, /* x coordinate of missing square in upper
                                                    right subboard */
         y_upper_right, /* y coordinate of missing square in upper
                                                    right subboard */
         x_lower_right, /* x coordinate of missing square in lower
                                                    right subboard */
         y_lower_right, /* y coordinate of missing square in lower
                                                    right subboard */
         x_lower_left,  /* x coordinate of missing square in lower
                                                     left subboard */
         y_lower_left;  /* y coordinate of missing square in lower
                                                     left subboard */

     if ( board_size == 2 ) /* 2x2 board */
     {
       for(int i=board_size-1; i>=0; i--)
       {
         for(int j=0;j<board_size;j++)
         {
           if(i==x_missing && j==y_missing)
            //printf("MS\t");
          {  board[i][j] = -1;}
           //printf( "%d %d ", x_board + 1, y_board + 1 );
           /* find and print orientation of tromino */
         if(x_missing == x_board)
          {
            if(y_missing == y_board)
              // printf("UR\t");
              {board[i][j] = 2;}
            else
              // printf("LR\t");
              {board[i][j] = 0;}
          }

          else
          {
            if(y_missing == y_board)
              // printf("UL\t");
              {board[i][j] = 3;}
            else
              // printf("LL\t");
              {board[i][j] = 1;}
          }
       }
       printf("\n");
     }
     return;
    }

   // if(board_size>2)
   // {   /* compute x and y coordinates of center of board */
     x_center = x_board + half_size;
     y_center = y_board + half_size;


         if((x_missing < x_center) && (y_missing < y_center))
         {
           //printf("UR\t");
           x_upper_left = x_center - 1;
           y_upper_left = y_center;
           x_upper_right = x_center;
           y_upper_right = y_center;
           x_lower_left = x_missing;
           y_lower_left = y_missing;
           x_lower_right = x_center;
           y_lower_right = y_center - 1;

         }
         if((x_missing >= x_center) && (y_missing < y_center))
         {
           //printf("UL\t");
           x_upper_left = x_center - 1;
           y_upper_left = y_center;
           x_upper_right = x_center;
           y_upper_right = y_center;
           x_lower_left = x_center - 1;
           y_lower_left = y_center - 1;
           x_lower_right = x_missing;
           y_lower_right = y_missing;
         }
         if((x_missing >= x_center) && (y_missing >= y_center))
         {
           //printf("LL\t");
           x_upper_left = x_center - 1;
           y_upper_left = y_center;
           x_upper_right = x_missing;
           y_upper_right = y_missing;
           x_lower_left = x_center - 1;
           y_lower_left = y_center - 1;
           x_lower_right = x_center;
           y_lower_right = y_center - 1;
         }
         if((x_missing < x_center) && (y_missing >= y_center))
         {
           //printf("LR\t");
           x_upper_left = x_missing;
           y_upper_left = y_missing;
           x_upper_right = x_center;
           y_upper_right = y_center;
           x_lower_left = x_center - 1;
           y_lower_left = y_center - 1;
           x_lower_right = x_center;
           y_lower_right = y_center - 1;
         }
     /* tile the four subboards */
     tromino( x_board, y_board + half_size,
          x_upper_left, y_upper_left, half_size);
     tromino( x_board + half_size, y_board + half_size,
          x_upper_right, y_upper_right, half_size);
     tromino( x_board + half_size, y_board,
          x_lower_right, y_lower_right, half_size);
     tromino( x_board, y_board,
                    x_lower_left, y_lower_left, half_size);

}

void show_Tromino(int size, int x_miss, int y_miss)
{
  int bsize[Max][Max];
  for(int i=size-1; i>=0;i--)
  {
    for(int j=0; j<size;j++)
    {
      if(!(bsize[i][j] == -1 || (j==x_miss && i==y_miss)))
      {
        if(bsize[i][j] == 1)
        {
          printf("LL\t");
        }

        else if(bsize[i][j] == 2)
        {
          printf("UL\t");
        }

        else if(bsize[i][j] == 3)
        {
          printf("UR\t");
        }

        else if(bsize[i][j] == 0)
        {
          printf("LR\t");
        }
      }
      else if (bsize[i][j] == -1 || (j==x_miss && i==y_miss))
      {
        printf("MS\t");
      }
      bsize[i][j] += 1;
    }

    printf("\n");
  }
}

output1 : when board size 2 x 2(which is correct)

Enter size of board (0 to quit): 2

Enter coordinates of missing square: 0 1


Tiling
------
MS  LR  
LR  LR

output2 : when board size 4 x 4(which is incorrect). Though, I am getting the correct position of Missing Square(MS).

Enter size of board (0 to quit): 4

Enter coordinates of missing square: 0 1


Tiling
------
LR  LR  LR  LR  
LR  LR  LR  LR  
MS  LR  LR  LR  
LR  LR  LR  LR

output2 : when board size 4 x 4(which is expected)

Enter size of board (0 to quit): 4

Enter coordinates of missing square: 0 1

Tiling
------

UL UL UR UR
UL UR UR UR
MS LR UR LR 
LR LR LR LR

Note:

• UL- Upper Left
• UR- Upper Right
• LL- Lower Left
• LR- Lower Right
• MS- Missing Square

Answer 1

The problem is due to the base case. When solving for 2x2 you forgot to take the offset into consideration. Only [0,0],[0,1],[1,0],[1,1] are updated each time. The rest of the values (except the missing tile) are holding garbage values.

I believe your compiler sets all array values to zero by default. So the garbage values result in value LR.

Edit: Added the corrected code. Just map the outputs to the letters (I've printed the numbers)

/* This program tiles with right trominoes an nxn
   board with one square missing, assuming that n
   is a power of 2.                                */

#include <stdio.h>
#include <stdlib.h>

// #define LR 4
// #define LL 1
// #define UR 3
// #define UL 2
// #define MS -1

# define Max 256
int board[Max][Max];

void tromino /* function to do tiling */
     ( int x_board,      /* x coordinate of board */
       int y_board,      /* y coordinate of board */
       int x_missing,    /* x coordinate of missing square */
       int y_missing,    /* y coordinate of missing square */
       int board_size);   /* size of board */

void show_Tromino(int size, int x_miss, int y_miss);

int main()
{
     int board_size,
         x_missing,  /* x coordinate of missing square */
         y_missing;  /* y coordinate of missing square */
     do {
          printf( "\n-------------------------------------" );
          printf( "\nEnter size of board (0 to quit): " );
          scanf( "%d", &board_size );
          if ( board_size ) {
               printf( "\nEnter coordinates of missing square: " );
               scanf( "%d%d", &x_missing, &y_missing );
               printf( "\n\nTiling\n" );
               // if(board[x_board][0] == x_missing && board[0][y_board] == y_missing)
               //  printf("MS\t");
               //board[x_missing][y_missing] = -1;
               tromino( 0, 0, x_missing, y_missing, board_size);
               show_Tromino(board_size, x_missing, y_missing);
          }
     } while ( board_size );

     return EXIT_SUCCESS;
}

void tromino( int x_board,     /* x coordinate of board */
              int y_board,     /* y coordinate of board */
              int x_missing,   /* x coordinate of missing square */
              int y_missing,   /* y coordinate of missing square */
              int board_size) /* size of board */

{
     int half_size = board_size/2, /* size of subboard */
         x_center,  /* x coordinate of center of board */
         y_center,  /* y coordinate of center of board */
         x_upper_left,  /* x coordinate of missing square in upper
                                                     left subboard */
         y_upper_left,  /* y coordinate of missing square in upper
                                                     left subboard */
         x_upper_right, /* x coordinate of missing square in upper
                                                    right subboard */
         y_upper_right, /* y coordinate of missing square in upper
                                                    right subboard */
         x_lower_right, /* x coordinate of missing square in lower
                                                    right subboard */
         y_lower_right, /* y coordinate of missing square in lower
                                                    right subboard */
         x_lower_left,  /* x coordinate of missing square in lower
                                                     left subboard */
         y_lower_left;  /* y coordinate of missing square in lower
                                                     left subboard */

     if ( board_size == 2 ) /* 2x2 board */
     {
       //Theres too much of nesting for only 4 cases. Got rid of it
       if(x_board==x_missing && y_board==y_missing)
         board[x_board+1][y_board] = board[x_board][y_board+1]=board[x_board+1][y_board+1]=2;

       if(x_board+1==x_missing && y_board==y_missing)
         board[x_board][y_board]= board[x_board][y_board+1]=board[x_board+1][y_board+1]=3;

       if(x_board==x_missing && y_board+1==y_missing)
         board[x_board][y_board]= board[x_board+1][y_board] = board[x_board+1][y_board+1]=1;

       if(x_board+1==x_missing && y_board+1==y_missing)
         board[x_board][y_board]= board[x_board+1][y_board] = board[x_board][y_board+1]=0;
       return;
     }

   // if(board_size>2)
   // {   /* compute x and y coordinates of center of board */
     x_center = x_board + half_size;
     y_center = y_board + half_size;


         if((x_missing < x_center) && (y_missing < y_center))
         {
           //printf("UR\t");
           x_upper_left = x_center - 1;
           y_upper_left = y_center;
           x_upper_right = x_center;
           y_upper_right = y_center;
           x_lower_left = x_missing;
           y_lower_left = y_missing;
           x_lower_right = x_center;
           y_lower_right = y_center - 1;
           board[x_upper_left][y_upper_left]=2;     //CHANGE MADE. You forgot to tile the board
           board[x_upper_right][y_upper_right]=2;
           board[x_lower_right][y_lower_right]=2;

         }
         if((x_missing >= x_center) && (y_missing < y_center))
         {
           //printf("UL\t");
           x_upper_left = x_center - 1;
           y_upper_left = y_center;
           x_upper_right = x_center;
           y_upper_right = y_center;
           x_lower_left = x_center - 1;
           y_lower_left = y_center - 1;
           x_lower_right = x_missing;
           y_lower_right = y_missing;
           board[x_upper_left][y_upper_left]=3;     //CHANGE MADE. You forgot to tile the board
           board[x_upper_right][y_upper_right]=3;
           board[x_lower_left][y_lower_left]=3;
         }
         if((x_missing >= x_center) && (y_missing >= y_center))
         {
           //printf("LL\t");
           x_upper_left = x_center - 1;
           y_upper_left = y_center;
           x_upper_right = x_missing;
           y_upper_right = y_missing;
           x_lower_left = x_center - 1;
           y_lower_left = y_center - 1;
           x_lower_right = x_center;
           y_lower_right = y_center - 1;
           board[x_upper_left][y_upper_left]=1;     //CHANGE MADE. You forgot to tile the board
           board[x_lower_left][y_lower_left]=1;
           board[x_lower_right][y_lower_right]=1;
         }
         if((x_missing < x_center) && (y_missing >= y_center))
         {
           //printf("LR\t");
           x_upper_left = x_missing;
           y_upper_left = y_missing;
           x_upper_right = x_center;
           y_upper_right = y_center;
           x_lower_left = x_center - 1;
           y_lower_left = y_center - 1;
           x_lower_right = x_center;
           y_lower_right = y_center - 1;
           board[x_lower_left][y_lower_left]=0;  //CHANGE MADE. You forgot to tile the board
           board[x_upper_right][y_upper_right]=0;
           board[x_lower_right][y_lower_right]=0;
         }
     /* tile the four subboards */
     tromino( x_board, y_board + half_size,
          x_upper_left, y_upper_left, half_size);
     tromino( x_board + half_size, y_board + half_size,
          x_upper_right, y_upper_right, half_size);
     tromino( x_board + half_size, y_board,
          x_lower_right, y_lower_right, half_size);
     tromino( x_board, y_board,
                    x_lower_left, y_lower_left, half_size);

}

void show_Tromino(int size, int x_miss, int y_miss)
{
  for(int i=size-1;i>=0;i--)
  {
     for(int j=0; j<size;j++)
     {
       if(j==x_miss && i==y_miss)
        printf("-1");
       else
        printf("%d ",board[j][i]);
     }
    printf("\n");
  }
  //int bsize[Max][Max]; //CHANGE: WTH is bsize? All changes were made to board. Edited and changed them all

}

Answer 2

There is a dynamic recursion solution, and a similar dynamic iteration solution, which help to avoid stack overflow error due to deep recursive calls when input is very large.

---

Dynamic - recursion

Analysis:

* for a 2*2 cell, could simply fill it,
* for a 4*4 cell,
    * divide it into 4 2*2 smaller squares,
    * first fill the 2*2 square that already has 1 cell filled,
    * now the 2*2 square at the center of the original square has 3 empty cells, just fill it with a tile,
    * now all the 3 remain 2*2 squares divided in previous step, has 1 cell filled,
    * then for each of the 3 remain 2*2 squares, fill with a tile,
    * 
* for a 8*8 cell,
    * divide it into 4 4*4 smaller squares,
    * then fill the 4*4 square that already has 1 cell filled, in similar way as a 4*4 input,
    * now the 2*2 square at the center of the original square has 3 empty cells, just fill it with a tile,
    * now all the 3 remain 4*4 squares divided in previous step, has 1 cell filled,
    * then for each of the 3 remain 4*4 squares, fill it in similar way as a 4*4 input,
    * 
* for a n*n cell,
    repeat divide & conquer,
    steps:
    * if n = 2,
        this is base, simple fill a tile,
    * else
        * divide it into 4 (n/2 * n/2) squares,
        * work on the one already with has a tile, by a recursive call,
        * then put a tile at center of input cell, to fill the (2*2) square in center,
        * then for each of the 3 (n/2 * n/2) squares,
            * work on it, by a recursive call,
            * 
        * 
    * 
* 

---

Dynamic - iteration

Similar as Dynamic - recursion above, but it use loop to avoid deep recursive calls.

Thus it's preferred.

Analysis:

* if it's base case (2*2 square),
    it's the same as `Dynamic - recursion`,
* if it's not base case,
    * divide it into 4 (n/2 * n/2) squares, mark the one already has a cell filled as A,
    * then fill (2*2) square at the center of input square first, leave the cell belong to A empty,
        now all the 4 (n/2 * n/2) squares, has a single cell filled,
    * 
    * then loop the 4 (n/2 * n/2) squares, for each:
        * work on it, by a recursive call,
        * 
    * 
* 

---

Representation of Shapes

• Cell

Need its x and y position.

Thus the structure may look like this, in c:

typedef struct {
    int x;
    int y;
} Cell;

• L tile

There could be 4 direction, mark as: 1, 2, 3, 4;
And need the coordinate of the middle cell of L tile to locate it,

Thus the structure may look like this, in c:

typedef struct {
    char direction; // 1, 2, 3, 4;
    Cell mid; // position of middle cell of tile,
} Tile;

• Square

Need left/top cell to decide its position.
And length of side to decide its size.

Thus the structure may look like this, in c:

typedef struct {
    Cell start; // top/left cell,
    int size; // length of side,
} Square;

---