P16840 [GKS 2021 #A] Checksum

Grace and Edsger are constructing a $N \times N$ boolean matrix $A$. The element in the $i$-th row and $j$-th column is represented by $A_{i,j}$. They decide to note down the checksum (defined as bitwise XOR of given list of elements) along each row and column. Checksum of the $i$-th row is represented as $R_i$. Checksum of the $j$-th column is represented as $C_j$. For example, if $N = 2$, $A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$, then $R = [1 \quad 0]$ and $C = [0 \quad 1]$. Once they finished the matrix, Edsger stores the matrix in his computer. However, due to a virus, some of the elements in matrix $A$ are replaced with $-1$ in Edsger's computer. Luckily, Edsger still remembers the checksum values. He would like to restore the matrix, and reaches out to Grace for help. After some investigation, it will take $B_{i,j}$ hours for Grace to recover the original value of $A_{i,j}$ from the disk. Given the final matrix $A$, cost matrix $B$, and checksums along each row ($R$) and column ($C$), can you help Grace decide on the minimum total number of hours needed in order to restore the original matrix $A$?

The first line of the input gives the number of test cases, $T$. $T$ test cases follow. The first line of each test case contains a single integer $N$. The next $N$ lines each contain $N$ integers representing the matrix $A$. The $j$-th element on the $i$-th line represents $A_{i,j}$. The next $N$ lines each contain $N$ integers representing the matrix $B$. The $j$-th element on the $i$-th line represents $B_{i,j}$. The next line contains $N$ integers representing the checksum of the rows. The $i$-th element represents $R_i$. The next line contains $N$ integers representing the checksum of the columns. The $j$-th element represents $C_j$.

For each test case, output one line containing `Case #x: y`, where $x$ is the test case number (starting from $1$) and $y$ is the minimum number of hours to restore matrix $A$.

In Sample Case #$1$, $A_{1,2}$ can be restored using the checksum of either $1$-st row or $2$-nd column. Hence, Grace can restore the matrix without spending any time to recover the data. In Sample Case #$2$, Grace spends $1$ hour to recover $A_{1,1}$. After that, she can use checksums of $1$-st row and $1$-st column to restore $A_{1,2}$ and $A_{2,1}$ respectively. And then she can use checksum of $2$-nd row to restore $A_{2,2}$. Hence, Grace can restore the matrix by spending $1$ hour. In Sample Case #$3$, Grace can spend $1$ hour to recover $A_{1,1}$ and another hour to recover $A_{2,2}$. After that, she can use checksum to restore the rest of the matrix. Hence, Grace can restore the matrix by spending $2$ hours in total. ### Limits $1 \le T \le 100$. $-1 \le A_{i,j} \le 1$, for all $i$, $j$. $1 \le B_{i,j} \le 1000$, for $i$, $j$ where $A_{i,j} = -1$, otherwise $B_{i,j} = 0$. $0 \le R_i \le 1$, for all $i$. $0 \le C_j \le 1$, for all $j$. It is guaranteed that there exist at least one way to replace $-1$ in $A$ with $0$ or $1$ such that $R$ and $C$ are satisfied. **Test Set $1$** $1 \le N \le 4$. **Test Set $2$** $1 \le N \le 40$. **Test Set $3$** $1 \le N \le 500$.