The owner of the Automatic Cellular Manufacturing
corporation has just patented a new process for the mass
production of identical parts. Her approach uses a
twodimensional lattice of twostate cells, each of which is
either “empty” or “filled.” The exact details are, of course,
proprietary.
Initially, a set of cells in the lattice is filled with a
copy of the part that is to be reproduced. In a sequence of
discrete steps, each cell in the lattice simultaneously updates
its state by examining its own state and those of its eight
surrounding neighbors. If an odd number of these nine cells are
filled, the cell’s state in the next time step will be filled,
otherwise it will be empty. Figure 1 shows several steps
in the replication process for a simple pattern consisting of
three filled cells.
However, a bug has crept into the process. After each update
step, one cell in the lattice might spontaneously flip its
state. For instance, Figure 2 shows what might happen if a
cell flipped its state after the first time step and another
flipped its state after the third time step.
Unfortunately, the original patterns were lost, and only the
(possibly corrupted) results of the replication remain. Can you
write a program to determine a smallest possible nonempty
initial pattern that could have resulted in a given final
pattern?
Input
The first line of input contains two integers $w$ ($1
\le w \le 300$) and $h$ ($1
\le h \le 300$), where $w$ and $h$ are the width and height of the
bounding box of the final pattern. Following that are
$h$ lines, each containing
$w$ characters, giving the
final pattern. Each character is either ‘.’ (representing an empty cell) or
‘#’ (representing a filled cell).
There is at least one filled cell in the first row, in the last
row, in the first column, and in the last column.
Output
Display a minimumsize nonempty pattern that could have
resulted in the given pattern, assuming that at each stage of
the replication process at most one cell spontaneously changed
state. The size of a pattern is the area of its bounding box.
If there is more than one possible minimumsize nonempty
starting pattern, any one will be accepted. Use the character
‘.’ for empty cells and
‘#’ for filled cells. Use the minimum
number of rows and columns needed to display the pattern.
Sample Input 1 
Sample Output 1 
10 10
.#...#...#
##..##..##
##.#.##...
##.#.##...
.#...#####
...##..#.#
......###.
##.#.##...
#..#..#..#
##..##..##

.#
##

Sample Input 2 
Sample Output 2 
8 8
##..#.##
#.####.#
.#.#.#..
.##.#.##
.#.#.#..
.##.#.##
#..#.###
##.#.##.

####
#..#
#.##
###.

Sample Input 3 
Sample Output 3 
5 4
#....
..###
..###
..###

#
