D. 505time limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputA binary matrix is called good if every even length square sub-matrix has an odd number of ones. Given a binary matrix a consisting of n rows and m columns, determine the minimum number of cells you need to change to make it good, or report that there is no way to make it good at all. All the terms above have their usual meanings — refer to the Notes section for their formal definitions. InputThe first line of input contains two integers n and m (1 \leq n \leq m \leq 10^6 and n\cdot m \leq 10^6)  — the number of rows and columns in a, respectively. The following n lines each contain m characters, each of which is one of 0 and 1. If the j-th character on the i-th line is 1, then a_{i,j} = 1. Similarly, if the j-th character on the i-th line is 0, then a_{i,j} = 0.OutputOutput the minimum number of cells you need to change to make a good, or output -1 if it's not possible at all.ExamplesInput
3 3
101
001
110
Output
2Input
7 15
000100001010010
100111010110001
101101111100100
010000111111010
111010010100001
000011001111101
111111011010011
Output
-1NoteIn the first case, changing a_{1,1} to 0 and a_{2,2} to 1 is enough. You can verify that there is no way to make the matrix in the second case good. Some definitions —   A binary matrix is one in which every element is either 1 or 0.  A sub-matrix is described by 4 parameters — r_1, r_2, c_1, and c_2; here, 1 \leq r_1 \leq r_2 \leq n and 1 \leq c_1 \leq c_2 \leq m.  This sub-matrix contains all elements a_{i,j} that satisfy both r_1 \leq i \leq r_2 and c_1 \leq j \leq c_2.  A sub-matrix is, further, called an even length square if r_2-r_1 = c_2-c_1 and r_2-r_1+1 is divisible by 2. 
3 3
101
001
110

2
['bitmasks', 'brute force', 'constructive algorithms', 'dp', 'greedy', 'implementation', '*2000']