ZOJ Problem Set - 1080
Statement of the Problem
Some algorithms on image processing are more efficient when applied to small patterns, such as 3 x 3 matrices. One way of decomposing a given figure into small components is to apply the operation of direct subtraction, which is described in the following.
Given a 0/1 matrix Amxm and a 0/1 matrix B3x3 we define the matrix C(m-2)x(m-2) = A - B obtained in the following way:
We call matrix C a valid direct subtraction of B from A if for every aij = 1 in matrix A, there is a 1 in matrix C which results from a subtraction of B from a submatrix of A containing aij, and the element of B which is subtracted from aij is equal to 1.
Example: Given matrices A and B,
the direct subtraction C = A - B is valid and is given by
Now, given matrices A and B,
The direct subtraction C = A - B is not valid and is given by
The objective of this problem is to determine if a matrix A can be transformed
into a 3 x 3 matrix through a sequence of valid direct subtractions of, possibly
different, 3 x 3 matrices.
Several input instances are given. Each instance begins with the dimension
0 < n < 20 of the matrix to be decomposed. The following n lines describe
the rows of that matrix, as a sequence of n 0's and 1's, with no blank spaces
between them. The input ends with a line with a single 0.
For each input instance your program must identify it by printing Instance
i: (where i is the number of the instance) and, in the next line a message Yes
or No for the case, resp. that the matrix is (resp. is not) decomposable.
Source: South America 2001